Abstract
We propose a new calculus SCL(EQ) for firstorder logic with equality that only learns nonredundant clauses. Following the idea of CDCL (Conflict Driven Clause Learning) and SCL (Clause Learning from Simple Models) a ground literal model assumption is used to guide inferences that are then guaranteed to be nonredundant. Redundancy is defined with respect to a dynamically changing ordering derived from the ground literal model assumption. We prove SCL(EQ) sound and complete and provide examples where our calculus improves on superposition.
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1 Introduction
There has been extensive research on sound and complete calculi for firstorder logic with equality. The current prime calculus is superposition [2], where ordering restrictions guide paramodulation inferences and an abstract redundancy notion enables a number of clause simplification and deletion mechanisms, such as rewriting or subsumption. Still this “syntactic” form of superposition infers many redundant clauses. The completeness proof of superposition provides a “semantic” way of generating only nonredundant clauses, however, the underlying ground model assumption cannot be effectively computed in general [35]. It requires an ordered enumeration of infinitely many ground instances of the given clause set, in general. Our calculus overcomes this issue by providing an effective way of generating ground model assumptions that then guarantee nonredundant inferences on the original clauses with variables.
The underlying ordering is based on the order of ground literals in the model assumption, hence changes during a run of the calculus. It incorporates a standard rewrite ordering. For practical redundancy criteria this means that both rewriting and redundancy notions that are based on literal subset relations are permitted to dynamically simplify or eliminate clauses. Newly generated clauses are nonredundant, so redundancy tests are only needed backwards. Furthermore, the ordering is automatically generated by the structure of the clause set. Instead of a fixed ordering as done in the superposition case, the calculus finds and changes an ordering according to the currently easiest way to make progress, analogous to CDCL (Conflict Driven Clause Learning) [11, 12, 28, 33, 39].
Typical for CDCL and SCL (Clause Learning from Simple Models) [1, 15, 21] approaches to reasoning, the development of a model assumption is done by decisions and propagations. A decision guesses a ground literal to be true whereas a propagation concludes the truth of a ground literal through an otherwise false clause. While propagations in CDCL and propositional logic are restricted to the finite number of propositional variables, in firstorder logic there can already be infinite propagation sequences [21], as there might exist an infinite set of ground instances. In order to overcome this issue, model assumptions in SCL(EQ) are at any point in time restricted to a finite number of ground literals, hence to a finite number of ground instances of the clause set at hand. Therefore, without increasing the number of considered ground literals, the calculus either finds a refutation or runs into a stuck state where the current model assumption satisfies the finite number of ground instances. In this case one can check whether the model assumption can be generalized to a model assumption of the overall clause set or the information of the stuck state can be used to appropriately increase the number of considered ground literals and continue search for a refutation. SCL(EQ) does not require exhaustive propagation, in general, it just forbids the decision of the complement of a literal that could otherwise be propagated.
For an example of SCL(EQ) inferring clauses, consider the three firstorder clauses
with a Knuth–Bendix Ordering (KBO), unique weight 1, and precedence \(d\prec c\prec b\prec a \prec g \prec h \prec f\). A Superposition Left [2] inference between \(C_2\) and \(C_3\) results in
For SCL(EQ) we start by building a partial model assumption, called a trail, with two decisions
where \(\sigma {:}{=} \{x\mapsto a\}\). Decisions and propagations are always ground instances of literals from the firstorder clauses, and are annotated with a level and a justification clause, in case of a decision a tautology. Now with respect to \(\Gamma \) clause \(C_3\) is false with grounding \(\sigma \), and rule Conflict is applicable; see Sect. 3.1 for details on the inference rules. In general, clauses and justifications are considered variable disjoint, but for simplicity of the presentation of this example, we repeat variable names here as long as the same ground substitution is shared. The maximal literal in \(C_3\sigma \) is \((f(x) \not \approx h(x))\sigma \) and a rewrite refutation using the ground equations from the trail results in the justification clause
where for the refutation justification clauses and all otherwise inferred clauses we use the grounding \(\sigma \) for guidance, but operate on the clauses with variables. The respective ground clause is smaller than \((f(x) \not \approx h(x))\sigma \), false with respect to \(\Gamma \) and becomes our new conflict clause by an application of our inference rule ExploreRefutation. It is simplified by our inference rules EqualityResolution and Factorize, resulting in the finally learned clause
which is then used to apply rule Backtrack to the trail. Further details on this example are available from the Appendix, Example 6. Observe that \(C_4\) is strictly stronger than \(C'_4\) the clause inferred by superposition and that \(C_4\) cannot be inferred by superposition. Thus SCL(EQ) can infer stronger clauses than superposition for this example.
1.1 Related Work
SCL(EQ) is based on ideas of SCL [1, 15, 17, 21] but for the first time includes a native treatment of firstorder equality reasoning. Similar to [15] propagations need not to be exhaustively applied, the trail is built out of decisions and propagations of ground literals annotated by firstorder clauses, SCL(EQ) only learns nonredundant clauses, but for the first time conflicts resulting out of a decision have to be considered, due to the nature of the equality relation.
There have been suggested several approaches to lift the idea of an inference guiding model assumption from propositional to full firstorder logic [8, 13, 14, 21]. They do not provide a native treatment of equality, e.g., via paramodulation or rewriting.
Baumgartner et al. describe multiple calculi that handle equality by using unit superposition style inference rules and are based on either hyper tableaux [5] or DPLL [18, 19]. Hyper tableaux fix a major problem of the wellknown free variable tableaux, namely the fact that free variables within the tableau are rigid, i.e., substitutions have to be applied to all occurrences of a free variable within the entire tableau. Hyper tableaux with equality [9] in turn integrates unit superposition style inference rules into the hyper tableau calculus.
Another approach that is related to ours is the model evolution calculus with equality (\(\mathcal {ME_E}\)) by Baumgartner et al. [6, 10] which lifts the DPLL calculus to firstorder logic with equality. Similar to our approach, \(\mathcal {ME_E}\) creates a candidate model until a clause instance contradicts this model or all instances are satisfied by the model. The candidate model results from a socalled context, which consists of a finite set of nonground rewrite literals. Roughly speaking, a context literal specifies the truth value of all its ground instances unless a more specific literal specifies the complement. Initially the model satisfies the identity relation over the set of all ground terms. Literals within a context may be universal or parametric, where universal literals guarantee all its ground instances to be true. If a clause contradicts the current model, it is repaired by a nondeterministic split which adds a parametric literal to the current model. If the added literal does not share any variables in the contradictory clause it is added as a universal literal.
Another approach by Baumgartner and Waldmann [7] combined the superposition calculus with the Model Evolution calculus with equality. In this calculus the atoms of the clauses are labeled as “split atoms” or “superposition atoms”. The superposition part of the calculus then generates a model for the superposition atoms while the model evolution part generates a model for the split atoms. Conversely, this means that if all atoms are labeled as “split atom”, the calculus behaves similar to the model evolution calculus. If all atoms are labeled as “superposition atom”, it behaves like the superposition calculus.
Both the hyper tableaux calculus with equality and the model evolution calculus with equality allow only unit superposition applications, while SCL(EQ) inferences are guided paramodulation inferences on clauses of arbitrary length. The model evolution calculus with equality was revised and implemented in 2011 [10] and compares its performance with that of hyper tableaux. Model evolution performed significantly better, with more problems solved in all relevant TPTP [34] categories, than the implementation of the hyper tableaux calculus.
Plaisted et al. [31] present the Ordered Semantic HyperLinking (OSHL) calculus. OSHL is an instantiation based approach that repeatedly chooses ground instances of a nonground input clause set such that the current model does not satisfy the current ground clause set. A further step repairs the current model such that it satisfies the ground clause set again. The algorithm terminates if the set of ground clauses contains the empty clause. OSHL supports rewriting and narrowing, but only with unit clauses. In order to handle nonunit clauses it makes use of other mechanisms such as Brand’s Transformation [4].
InstGen [25] is an instantiation based calculus, that creates ground instances of the input firstorder formulas which are forwarded to a SAT solver. If a ground instance is unsatisfiable, then the firstorder set is as well. If not then the calculus creates more instances. The InstGenEQ calculus [26] creates instances by extracting instantiations of unit superposition refutations of selected literals of the firstorder clause set. The ground abstraction is then extended by the extracted clauses and an SMT solver then checks the satisfiability of the resulting set of equational and nonequational ground literals.
On ground equational clauses, the behavior of SCL(EQ) is similar to SMT (Satisfiability Modulo Theories) [30]. SCL(EQ) rigorously searches for implied equalities and does not explicitely consider the propositional abstraction that drives SMT. Therefore, SCL(EQ) only learns nonredundant clauses that is not guaranteed by standard SMT reasoning. On the other hand the level of laziness in reasoning that is offered by SMT is currently not supported by SCL(EQ). On equational clauses with variables, SCL(EQ) learns only nonredundant clauses with variables whereas SMT solely operates on ground instances.
This article is an extended version of our proceedings paper [27]. It includes more examples and an extended background and discussion section. In favor of a better structure we have moved all proofs to an Appendix. The rest of the paper is organized as follows. Section 2 provides basic formalisms underlying SCL(EQ). The rules of the calculus are presented in Sect. 3. Soundness and completeness results are provided in Sect. 4. We end with a discussion of obtained results and future work, Sect. 5. The main contribution of this paper is the SCL(EQ) calculus that only learns nonredundant clauses, permits subset based redundancy elimination and rewriting, and its soundness and completeness.
2 Preliminaries
We assume a standard firstorder language with equality and signature \(\Sigma = (\Omega , \emptyset )\) where the only predicate symbol is equality \(\approx \). N denotes a set of clauses, C, D denote clauses, L, K, H denote equational literals, A, B denote equational atoms, t, s terms from \(T(\Omega ,\mathcal {X})\) for an infinite set of variables \(\mathcal {X}\), f, g, h function symbols from \(\Omega \), a, b, c constants from \(\Omega \) and x, y, z variables from \(\mathcal {X}\). The function comp denotes the complement of a literal. We write \(s\not \approx t\) as a shortcut for \(\lnot (s \approx t)\). The literal \(s\,\#\, t\) may denote both \(s\approx t\) and \(s\not \approx t\). The semantics of firstorder logic and semantic entailment \(\models \) is defined as usual.
By \(\sigma ,\tau ,\delta \) we denote substitutions, which are total mappings from variables to terms. Let \(\sigma \) be a substitution, then its finite domain is defined as \(dom(\sigma ) {:}{=} \{x \mid x\sigma \not = x\}\) and its codomain is defined as \(codom(\sigma ) = \{t \mid x\sigma = t, x \in dom (\sigma )\}\). We extend their application to literals, clauses and sets of such objects in the usual way. A term, literal, clause or sets of these objects is ground if it does not contain any variable. A substitution \(\sigma \) is ground if \(codom(\sigma )\) is ground. A substitution \(\sigma \) is grounding for a term t, literal L, clause C if \(t\sigma \), \(L\sigma \), \(C\sigma \) is ground, respectively. By \(C\cdot \sigma \), \(L\cdot \sigma \) we denote a closure consisting of a clause C, literal L and a grounding substitution \(\sigma \), respectively. The function gnd computes the set of all ground instances of a literal, clause, or clause set. The function \({\text {mgu}}\) denotes the most general unifier of terms, atoms, literals, respectively. We assume that mgus do not introduce fresh variables and that they are idempotent.
The set of positions \({{\text {pos}}}(L)\) of a literal (term \({{\text {pos}}}(t)\)) is inductively defined as usual. The notion \(L\vert _p\) denotes the subterm of a literal L (\(t\vert _p\) for term t) at position \(p\in {{\text {pos}}}(L)\) (\(p\in {{\text {pos}}}(t)\)). The replacement of a subterm of a literal L (term t) at position \(p\in {{\text {pos}}}(L)\) (\(p\in {{\text {pos}}}(t)\)) by a term s is denoted by \(L[s]_p\) (\(t[s]_p\)). For example, the term f(a, g(x)) has the positions \(\{\epsilon , 1, 2, 21\}\), \(f(a,g(x))\vert _{21} = x\) and \(f(a,g(x))[b]_2\) denotes the term f(a, b).
Let R be a set of rewrite rules \(l \rightarrow r\), called a term rewrite system (TRS). The rewrite relation \(\rightarrow _R \subseteq T(\Omega , \mathcal {X})\times T(\Omega , \mathcal {X})\) is defined as usual by \(s\rightarrow _R t\) if there exists \((l\rightarrow r)\in R\), \(p\in {{\text {pos}}}(s)\), and a matcher \(\sigma \), such that \(s\vert _p=l\sigma \) and \(t = s[r\sigma ]_p.\) We write \(s=t{\downarrow _R}\) if s is the normal form of t in the rewrite relation \(\rightarrow _R\). We write \(s\,\#\, t =(s'\,\#\, t'){\downarrow _R}\) if s is the normal form of \(s'\) and t is the normal form of \(t'\). A rewrite relation is terminating if there is no infinite descending chain \(t_0\rightarrow t_1\rightarrow ...\) and confluent if \(t ^{*}\hspace{1.5ex}\leftarrow s \rightarrow ^* t'\) implies \(t \leftrightarrow ^* t'\). A rewrite relation is convergent if it is terminating and confluent. A rewrite order is a irreflexive and transitive rewrite relation. A TRS R is terminating, confluent, convergent, if the rewrite relation \(\rightarrow _R\) is terminating, confluent, convergent, respectively. A term t is called irreducible by a TRS R if no rule from R rewrites t. Otherwise it is called reducible. A literal, clause is irreducible if all of its terms are irreducible, and reducible otherwise. A substitution \(\sigma \) is called irreducible if any \(t\in codom(\sigma )\) is irreducible, and reducible otherwise.
Let \(\prec _T\) denote a wellfounded rewrite ordering on terms which is total on ground terms and for all ground terms t there exist only finitely many ground terms \(s \prec _T t\). We call \(\prec _T\) a desired term ordering. We extend \(\prec _T\) to equations by assigning the multiset \(\{s,t\}\) to positive equations \(s\approx t\) and \(\{s,s,t,t\}\) to inequations \(s\not \approx t\). Furthermore, we identify \(\prec _T\) with its multiset extension comparing multisets of literals. For a (multi)set of terms \(\{t_1,\ldots ,t_n\}\) and a term t, we define \(\{t_1,\ldots ,t_n\} \prec _T t\) if \(\{t_1,\ldots ,t_n\} \prec _T \{t\}\). For a (multi)set of Literals \(\{L_1,\ldots ,L_n\}\) and a term t, we define \(\{L_1,\ldots ,L_n\} \prec _T t\) if \(\{L_1,\ldots ,L_n\} \prec _T \{\{t\}\}\). Given a ground term \(\beta \) then \({\text {gnd}}_{\prec _T\beta }\) computes the set of all ground instances of a literal, clause, or clause set where the groundings are smaller than \(\beta \) according to the ordering \(\prec _T\). Given a set (sequence) of ground literals \(\Gamma \) let \({\text {conv}}(\Gamma )\) be a convergent rewrite system from the positive equations in \(\Gamma \) using \(\prec _T\).
Let \(\prec \) be a wellfounded, total, strict ordering on ground literals, which is lifted to clauses and clause sets by its respective multiset extension. We overload \(\prec \) for literals, clauses, clause sets if the meaning is clear from the context. The ordering is lifted to the nonground case via instantiation: we define \(C \prec D\) if for all grounding substitutions \(\sigma \) it holds \(C\sigma \prec D\sigma \). Then we define \(\preceq \) as the reflexive closure of \(\prec \) and \(N^{\preceq C} {:}{=} \{D \mid D\in N \;\text {and}\; D\preceq C\}\) and use the standard superposition style notion of redundancy [2].
Definition 1
(Clause Redundancy) A ground clause C is redundant with respect to a set N of ground clauses and an ordering \(\prec \) if \(N^{\preceq C} \models C\). A clause C is redundant with respect to a clause set N and an ordering \(\prec \) if for all \(C'\in gnd(C)\), \(C'\) is redundant with respect to gnd(N).
The Superposition calculus [2] is defined by the following rules. Given a reduction order \(\succ \) on terms which is extended to an ordering on literals and clauses in the usual way, the basic Superposition rule is as follows:
Inferences are only allowed if the left premise is not greater than or equal to the right one, the last literal of each premise is greater than the remaining literals of the respective clause and the lefthand side of these literals is not smaller or equal to the righthand side. Two more rules are needed for completeness:
3 The SCL(EQ) Calculus
We start the introduction of the calculus by defining the ingredients of an SCL(EQ) state.
Definition 2
(Trail) A trail \(\Gamma {:}{=}[L_1^{i_1:C_1\cdot \sigma _1},...,L_n^{i_n:C_n\cdot \sigma _n}]\) is a consistent sequence of ground equations and inequations where \(L_j\) is annotated by a level \(i_j\) with \(i_{j1}\le i_j\), and a closure \(C_j\cdot \sigma _j\). We omit the annotations if they are not needed in a certain context. A ground literal L is true in \(\Gamma \) if \(\Gamma \models L\). A ground literal L is false in \(\Gamma \) if \(\Gamma \models comp(L)\). A ground literal L is undefined in \(\Gamma \) if \(\Gamma \not \models L\) and \(\Gamma \not \models comp(L)\). Otherwise it is defined. For each literal \(L_j\) in \(\Gamma \) it holds that \(L_j\) is undefined in \([L_1,...,L_{j1}]\) and irreducible by \({\text {conv}}(\{L_1,...,L_{j1}\})\).
The above definition of truth and undefinedness is extended to clauses in the obvious way. The notions of true, false, undefined can be parameterized by a ground term \(\beta \) by saying that L is \(\beta \)undefined in a trail \(\Gamma \) if \(\beta \prec _T L\) or L is undefined. The notions of a \(\beta \)true, \(\beta \)false term are restrictions of the above notions to literals smaller \(\beta \), respectively. All SCL(EQ) reasoning is layered with respect to a ground term \(\beta \).
Definition 3
Let \(\Gamma \) be a trail and L a ground literal such that L is defined in \(\Gamma \). By \({\text {core}}(\Gamma ;L)\) we denote a minimal subsequence \(\Gamma ' \subseteq \Gamma \) such that L is defined in \(\Gamma '\). By \({\text {cores}}(\Gamma ;L)\) we denote the set of all cores.
Note that \(core(\Gamma ;L)\) is not necessarily unique. There can be multiple cores for a given trail \(\Gamma \) and ground literal L.
Definition 4
(Trail Ordering) Let \(\Gamma {:}{=}[L_1,...,L_n]\) be a trail. The (partial) trail ordering \(\prec _\Gamma \) is the sequence ordering given by \(\Gamma \), i.e., (\(L_i\prec _\Gamma L_j\) if \(i < j\)) for all \(1\le i,j\le n\).
Definition 5
(Defining Core and Defining Literal) For a trail \(\Gamma \) and a sequence of literals \(\Delta \subseteq \Gamma \) we write \(max_{\prec _\Gamma }(\Delta )\) for the largest literal in \(\Delta \) according to the trail ordering \(\prec _\Gamma \). Let \(\Gamma \) be a trail and L a ground literal such that L is defined in \(\Gamma \). Let \(\Delta \in cores(\Gamma ;L)\) be a sequence of literals where \(max_{\prec _\Gamma }(\Delta ) \preceq _\Gamma max_{\prec _\Gamma }(\Lambda )\) for all \(\Lambda \in cores(\Gamma ;L)\), then \(\max _\Gamma (L) {:}{=} \max _{\prec _\Gamma }(\Delta )\) is called the defining literal and \(\Delta \) is called a defining core for L in \(\Gamma \). If \(cores(\Gamma ;L)\) contains only the empty core, then L has no defining literal and no defining core.
As an example, consider the trail \(\Gamma {:}{=}[L_1,L_2,L_3]\) and literal L, such that \([L_1,L_2]\models L\) and \([L_2,L_3]\models L\). Then both \([L_1,L_2]\) and \([L_2,L_3]\) are cores, but only \([L_1,L_2]\) is a defining core and only \(L_2\) is the defining literal. Note that there can be multiple defining cores but only one defining literal for any defined literal L. For example, consider a trail \(\Gamma {:}{=}[f(a)\approx f(b)^{1:C_1\cdot \sigma _1},a\approx b^{2:C_2\cdot \sigma _2},b\approx c^{3:C_3\cdot \sigma _3}]\) with an ordering \(\prec _T\) that orders the terms of the equations from left to right, and a literal \(g(f(a))\approx g(f(c))\). Then the defining cores are \(\Delta _1 {:}{=}[a\approx b,b\approx c]\) and \(\Delta _2 {:}{=}[f(a)\approx f(b),b\approx c]\). The defining literal, however, is in both cases \(b\approx c\). Defined literals that have no defining core and therefore no defining literal are literals that are trivially false or true. Consider, for example, \(g(f(a))\approx g(f(a))\). This literal is trivially true in \(\Gamma \). Thus an empty subset of \(\Gamma \) is sufficient to show that \(g(f(a))\approx g(f(a))\) is defined in \(\Gamma \).
Definition 6
(Literal Level) Let \(\Gamma \) be a trail. A ground literal \(L \in \Gamma \) is of level i if L is annotated with i in \(\Gamma \). A defined ground literal \(L\not \in \Gamma \) is of level i if the defining literal of L is of level i. If L has no defining literal, then L is of level 0. A ground clause D is of level i if i is the maximum level of a literal in D.
The restriction to minimal subsequences for the defining literal and definition of a level eventually guarantee that learned clauses are smaller in the trail ordering. This enables completeness in combination with learning nonredundant clauses as shown later.
Lemma 1
Let \(\Gamma _1\) be a trail and K a defined literal that is of level i in \(\Gamma _1\). Then K is of level i in a trail \(\Gamma {:}{=}\Gamma _1,\Gamma _2\).
Definition 7
Let \(\Gamma \) be a trail and \(L\in \Gamma \) a literal. L is called a decision literal if \(\Gamma = \Gamma _0,K^{i:C\cdot \tau }, L^{i+1:C'\cdot \tau '}, \Gamma _1\). Otherwise L is called a propagated literal.
In other words: L is a decision literal if the level of L is one greater than the level of the preceeding literal K. In our above example \(g(f(a))\approx g(f(c))\) is of level 3 since the defining literal \(b\approx c\) is annotated with 3. \(a\not \approx b\) on the other hand is of level 2.
We define a wellfounded total strict ordering which is induced by the trail and with which nonredundancy is proven in Sect. 4. Unlike SCL [15, 21] we use this ordering for the inference rules as well. In previous SCL calculi, conflict resolution automatically chooses the greatest literal and resolves with this literal. In SCL(EQ) this is generalized. Coming back to our running example above, suppose we have a conflict clause \(f(b)\not \approx f(c)\vee b\not \approx c\). The defining literal for both inequations is \(b\approx c\). So we could do paramodulation inferences with both literals. The following ordering makes this nondeterministic choice deterministic.
Definition 8
(Trail Induced Ordering) Let \(\Gamma {:}{=}[L_1^{i_1:C_1\cdot \sigma _1},...,L_n^{i_n:C_n\cdot \sigma _n}]\) be a trail, \(\beta \) a ground term such that \(\{L_1,...,L_n\}\prec _T \beta \) and \(M_{i,j}\) all \(\beta \)defined ground literals not contained in \(\Gamma \,\cup \, comp(\Gamma )\): for a defining literal \(max_\Gamma (M_{i,j}) = L_i\) and for two literals \(M_{i,j}\), \(M_{i,k}\) we have \(j<k\) if \(M_{i,j} \prec _T M_{i,k}\).
The trail induces a total wellfounded strict order \(\prec _{\Gamma ^*}\) on \(\beta \text {} defined \) ground literals \(M_{k,l}, M_{m,n}\), \(L_i\), \(L_j\) of level greater than zero, where

1.
\(M_{i,j}\prec _{\Gamma ^*} M_{k,l}\) if \(i<k\) or (\(i=k\) and \(j<l\));

2.
\(L_i\prec _{\Gamma ^*} L_j\) if \(L_i\prec _\Gamma L_j\);

3.
\(comp(L_i)\prec _{\Gamma ^*} L_j\) if \(L_i\prec _\Gamma L_j\);

4.
\(L_i\prec _{\Gamma ^*} comp(L_j)\) if \(L_i\prec _\Gamma L_j\) or \(i = j\);

5.
\(comp(L_i)\prec _{\Gamma ^*} comp(L_j)\) if \(L_i\prec _\Gamma L_j\);

6.
\(L_i\prec _{\Gamma ^*} M_{k,l}\), \({\text {comp}}(L_i)\prec _{\Gamma ^*} M_{k,l}\) if \(i\le k\);

7.
\(M_{k,l}\prec _{\Gamma ^*} L_i\), \(M_{k,l}\prec _{\Gamma ^*} {\text {comp}}(L_i)\) if \(k<i\);
and for all \(\beta \text {} defined \) literals L of level zero:

8.
\(\prec _{\Gamma ^*} {:}{=} \prec _T\);

9.
\(L\prec _{\Gamma ^*} K\) if K is of level greater than zero and K is \(\beta {\text {}} defined \);
and can eventually be extended to \(\beta {\text {}} undefined \) ground literals K, H by

10.
\(K \prec _{\Gamma ^*} H\) if \(K\prec _T H\);

11.
\(L \prec _{\Gamma ^*} H\) if L is \(\beta {\text {}} defined \).
The literal ordering \(\prec _{\Gamma ^*}\) is extended to ground clauses by multiset extension and identified with \(\prec _{\Gamma ^*}\) as well.
Note, that in the above definition for a given i we can always put the \(M_{i,j}\) in an order corresponding to the integers. This is due to the fact, that the total number of ground literals is countable and since there are only finitely many \(\beta \)defined literals.
Lemma 2
(Properties of \(\prec _{\Gamma ^*}\))

1
\(\prec _{\Gamma ^*}\) is welldefined.

2
\(\prec _{\Gamma ^*}\) is a total strict order, i.e. \(\prec _{\Gamma ^*}\) is irreflexive, transitive and total.

3
\(\prec _{\Gamma ^*}\) is a wellfounded ordering.
Example 1
Assume a trail \(\Gamma {:}{=} [a\approx b^{1:C_0\cdot \sigma _0}, c\approx d^{1:C_1\cdot \sigma _1}, f(a')\not \approx f(b')^{1:C_2\cdot \sigma _2}]\), select KBO as the term ordering \(\prec _T\) where all symbols have weight one and \(a\prec a'\prec b\prec b'\prec c\prec d\prec f\) and a ground term \(\beta {:}{=} f(f(a))\). According to the trail induced ordering we have that \(a\approx b\prec _{\Gamma ^*} c\approx d\prec _{\Gamma ^*} f(a')\not \approx f(b')\) by 8.2. Furthermore we have that
by 8.3 and 8.4. Now for any literal L that is \(\beta {\text {}} defined \) in \(\Gamma \) and the defining literal is \(a\approx b\) it holds that \(a\not \approx b\prec _{\Gamma ^*} L\prec _{\Gamma ^*} c\approx d\) by 8.6 and 8.7. This holds analogously for all literals that are \(\beta \text {} defined \) in \(\Gamma \) and the defining literal is \(c\approx d\) or \(f(a')\not \approx f(b')\). Thus we get:
where \(K_i\) are the \(\beta {\text {}} undefined \) literals and \(L_j\) are the trivially defined literals.
Table 1 summarizes the various orders presented so far.
Definition 9
(Rewrite Step) A rewrite step is a fivetuple \((s\# t\cdot \sigma , s\# t\vee C\cdot \sigma , R, S, p)\) and inductively defined as follows. The tuple \((s\# t\cdot \sigma , s\# t\vee C\cdot \sigma , \epsilon , \epsilon , \epsilon )\) is a rewrite step. Given rewrite steps R, S and a position p then \((s\# t\cdot \sigma , s\# t\vee C\cdot \sigma , R, S, p)\) is a rewrite step. The literal \(s\# t\) is called the rewrite literal. In case R, S are not \(\epsilon \), the rewrite literal of R is an equation.
Note that R and S in the above definition describe the “history” of a rewrite step, i.e. they contain all preceeding rewrite steps.
Rewriting is one of the core features of our calculus. The following definition describes a rewrite inference between two clauses. Note that unlike the superposition calculus we allow rewriting below variable level.
Definition 10
(Rewrite Inference) Let \(I_1 {:}{=} (l_1\approx r_1\cdot \sigma _1,l_1\approx r_1\vee C_1\cdot \sigma _1, R_1,L_1, p_1)\) and \(I_2 {:}{=} (l_2\# r_2\cdot \sigma _2,l_2\# r_2\vee C_2\cdot \sigma _2, R_2, L_2, p_2)\) be two variable disjoint rewrite steps where \(r_1\sigma _1\prec _T l_1\sigma _1\), \((l_2\# r_2)\sigma _2\vert _p = l_1\sigma _1\) for some position p. We distinguish two cases:

1.
if \(p\in {{\text {pos}}}(l_2\# r_2)\) and \(\mu {:}{=} {\text {mgu}}((l_2\# r_2)\vert _p,l_1)\) then \((((l_2\# r_2)[r_1]_{p})\mu \cdot \sigma _1\sigma _2,\)
\(((l_2\# r_2)[r_1]_{p})\mu \vee C_1\mu \vee C_2\mu \cdot \sigma _1\sigma _2,I_1,I_2,p) \) is the result of a rewrite inference.

2.
if \(p\not \in {{\text {pos}}}(l_2\# r_2)\) then let \((l_2\# r_2)\delta \) be the most general instance of \(l_2\# r_2\) such that \(p\in {{\text {pos}}}((l_2\# r_2)\delta )\), \(\delta \) introduces only fresh variables and \((l_2\# r_2)\delta \sigma _2\rho = (l_2\# r_2)\sigma _2\) for some minimal \(\rho \). Let \(\mu {:}{=} {\text {mgu}}((l_2\# r_2)\delta \vert _p,l_1)\). Then
\(((l_2\# r_2)\delta [r_1]_{p}\mu \cdot \sigma _1\sigma _2\rho , (l_2\# r_2)\delta [r_1]_{p}\mu \vee C_1\mu \vee C_2\delta \mu \cdot \sigma _1\sigma _2\rho , I_1, I_2,p) \) is the result of a rewrite inference.
Note that case 1 describes rewriting above or at a variable and case 2 describes rewriting inside a variable.
Lemma 3
Let \(I_1 {:}{=} (l_1\approx r_1\cdot \sigma _1,l_1\approx r_1\vee C_1\cdot \sigma _1, R_1, L_1, p_1)\) and \(I_2 {:}{=} (l_2\# r_2\cdot \sigma _2,l_2\# r_2\vee C_2\cdot \sigma _2, R_2, L_2, p_2)\) be two variable disjoint rewrite steps where \(r_1\sigma _1\prec _T l_1\sigma _1\), \((l_2\# r_2)\sigma _2\vert _p = l_1\sigma _1\) for some position p. Let \(I_3 {:}{=} (l_3\# r_3\cdot \sigma _3,l_3\# r_3\vee C_3\cdot \sigma _3, I_1, I_2, p)\) be the result of a rewrite inference. Then:

1.
\(C_3\sigma _3 = (C_1\vee C_2)\sigma _1\sigma _2\) and \(l_3\# r_3\sigma _3 = (l_2\# r_2)\sigma _2[r_1\sigma _1]_p\).

2.
\((l_3\# r_3)\sigma _3\prec _T (l_2\# r_2)\sigma _2\)

3.
If \(N\models ( l_1\approx r_1\vee C_1) \wedge (l_2\# r_2\vee C_2)\) for some set of clauses N, then \(N\models l_3\# r_3\vee C_3\)
Now that we have defined rewrite inferences we can use them to define a reduction chain application and a refutation, which are sequences of rewrite steps. Intuitively speaking, a reduction chain application reduces a literal in a clause with literals in \(conv(\Gamma )\) until it is irreducible. A refutation for a literal L that is \(\beta \text {} false \) in \(\Gamma \) for a given \(\beta \), is a sequence of rewrite steps with literals in \(\Gamma ,L\) such that \(\bot \) is inferred. Refutations for the literals of the conflict clause will be examined during conflict resolution by the rule ExploreRefutation.
Definition 11
(Reduction Chain) Let \(\Gamma \) be a trail. A reduction chain \(\mathcal {P}\) from \(\Gamma \) is a sequence of rewrite steps \([I_1,...,I_m]\) such that for each \(I_i = (s_i\# t_i\cdot \sigma _i, s_i\# t_i\vee C_i\cdot \sigma _i, I_j, I_k, p_i)\) either

1.
\(s_i\# t_i^{n_i:s_i\# t_i\vee C_i\cdot \sigma }\) is contained in \(\Gamma \) and \(I_j = I_k = p_i = \epsilon \) or

2.
\(I_i\) is the result of a rewriting inference from rewrite steps \(I_j, I_k\) out of \([I_1,...,I_m]\) where \(j, k < i\).
Let \((l\,\#\, r)\delta ^{o:l\,\#\, r \vee C \cdot \delta }\) be an annotated ground literal. A reduction chain application from \(\Gamma \) to \(l\,\#\, r\) is a reduction chain \([I_1,...,I_m]\) from \(\Gamma ,(l\,\#\, r)\delta ^{o:l\,\#\, r \vee C \cdot \delta }\) such that \(l\delta {\downarrow _{{\text {conv}}(\Gamma )}} = s_m\sigma _m\) and \(r\delta {\downarrow _{{\text {conv}}(\Gamma )}} = t_m\sigma _m\). We assume reduction chain applications to be minimal, i.e., if any rewrite step is removed from the sequence it is no longer a reduction chain application.
Definition 12
(Refutation) Let \(\Gamma \) be a trail and \((l\,\#\, r)\delta ^{o:l\,\#\, r \vee C \cdot \delta }\) an annotated ground literal that is \(\beta \text {} false \) in \(\Gamma \) for a given \(\beta \). A refutation \(\mathcal {P}\) from \(\Gamma \) and \(l\,\#\, r \) is a reduction chain \([I_1,...,I_m]\) from \(\Gamma ,(l\,\#\, r)\delta ^{o:l\,\#\, r \vee C \cdot \delta }\) such that \((s_m\# t_m)\sigma _m = s \not \approx s\) for some s. We assume refutations to be minimal, i.e., if any rewrite step \(I_k\), \(k<m\) is removed from the refutation, it is no longer a refutation.
3.1 The SCL(EQ) Inference Rules
We can now define the rules of our calculus based on the previous definitions. A state is a sixtuple \((\Gamma ; N; U; \beta ; k; D)\) similar to the SCL calculus, where \(\Gamma \) a sequence of annotated ground literals, N and U the sets of initial and learned clauses, \(\beta \) is a ground term such that for all \(L\in \Gamma \) it holds \(L\prec _T \beta \), k is the decision level, and D a status that is \(\top \), \(\bot \) or a closure \(C\,\cdot \sigma \). Before we propagate or decide any literal, we make sure that it is irreducible in the current trail. Together with the design of \(\prec _{\Gamma ^*}\) this eventually enables rewriting as a simplification rule.
The rule Propagate finds a ground instance of a clause which is propagable, i.e. it contains (possibly multiple occurences of) a literal that is \( undefined \) and all other literals are \( false \) in the trail. The multiple occurences of the undefined literal are factored. Then it adds the normal form of this literal to the trail. The propagating clause is reduced by the corresponding paramodulation steps. Note that the definition of Propagate also includes the case where \(L\sigma \) is irreducible by \(\Gamma \). In this case \(L = s_m\# t_m\) and \(m=1\). The rule Decide below, is similar to Propagate, except for the subclause \(C_0\) which must be \(\beta \text {} undefined \) or \(\beta \text {} true \) in \(\Gamma \), i.e., Propagate cannot be applied and the decision literal is annotated by a tautology.
For the nonequational case, when a conflict clause is found by an SCL calculus [15, 21], the complements of its firstorder ground literals are contained in the trail. For equational literals this is not the case, in general. The proof showing D to be \(\beta \text {} false \) with respect to \(\Gamma \) is a rewrite proof with respect to \({\text {conv}}(\Gamma )\). This proof needs to be analyzed to eventually perform paramodulation steps on D or to replace D by a \(\prec _{\Gamma ^*}\) smaller \(\beta \text {} false \) clause showing up in the proof.
The ExploreRefutation rule is the FOL with Equality counterpart to the resolve rule in CDCL or SCL. While in CDCL or SCL complementary literals of the conflict clause are present on the trail and can directly be used for resolution steps, this needs a generalization for FOL with Equality. Here, in general, we need to look at (rewriting) refutations of the conflict clause and pick an appropriate clause from the refutation as the next conflict clause.
For backtracking we have to make sure, that the learned clause is not \( false \) in the resulting trail. It is not sufficient to backtrack to the point where the clause with the current substitution is no longer false, but where it is no longer false with all possible substitutions.
In addition to soundness and completeness of the SCL(EQ) rules their tractability in practice is an important property for a successful implementation. In particular, finding propagating literals or detecting a false clause under some grounding. It turns out that these operations are NPcomplete, similar to firstorder subsumption which has been shown to be tractable in practice.
Lemma 4
Assume that all ground terms t with \(t\prec _T \beta \) for any \(\beta \) are polynomial in the size of \(\beta \). Then testing Propagate (Conflict) is NPComplete, i.e., the problem of checking for a given clause C whether there exists a grounding substitution \(\sigma \) such that \(C\sigma \) propagates (is false) is NPComplete.
Example 2
(SCL(EQ) vs. Superposition: Saturation) Consider the following clauses:
where again we assume a KBO with all symbols having weight one, precedence \(d\prec c\prec b\prec a\prec g\prec f\) and \(\beta {:}{=} f(f(g(a)))\). Suppose that we first decide \(c\approx d\) and then propagate \(a\approx b\): \(\Gamma = [c\approx d^{1:c\approx d\vee c\not \approx d}, a\approx b^{1:C_2}]\). Now we have a conflict with \(C_3\). ExploreRefutation applied to the conflict clause \(C_3\) results in a paramodulation inference between \(C_3\) and \(C_2\). Another application of EqualityResolution gives us the new conflict clause \(C_4 {:}{=} c\not \approx d\vee g(c)\not \approx g(d)\). Now we can Skip the last literal on the trail, which gives us \(\Gamma = [c\approx d^{1:c\approx d\vee c\not \approx d}]\). Another application of the ExploreRefutation rule to \(C_4\) using the decision justification clause followed by EqualityResolution and Factorize gives us \(C_5 {:}{=} c\not \approx d\). Thus with SCL(EQ) the following clauses remain:
where we derived \(C'_1\) out of \(C_1\) by subsumption resolution [37] using \(C_5\). Actually, subsumption resolution is compatible with the general redundancy notion of SCL(EQ), see Lemma 6. Now we consider the same example with superposition and the very same ordering (\(N_i\) is the clause set of the previous step and \(N_0\) the initial clause set N).
Thus superposition ends up with the following clauses:
The superposition calculus generates more and larger clauses.
Example 3
(SCL(EQ) vs. Superposition: Refutation) Suppose the following set of clauses: \(N {:}{=} \{C_1 {:}{=} f(x)\not \approx a\vee f(x)\approx b, C_2 {:}{=} f(f(y))\approx y, C_3 {:}{=} a\not \approx b\}\) where again we assume a KBO with all symbols having weight one, precedence \(b\prec a\prec f\) and \(\beta {:}{=} f(f(f(a)))\). A long refutation by the superposition calculus results in the following (\(N_i\) is the clause set of the previous step and \(N_0\) the initial clause set N):
The shortest refutation by the superposition calculus is as follows:
In SCL(EQ) on the other hand we would always first propagate \(a\not \approx b,f(f(a))\approx a\) and \(f(f(b))\approx b\). As soon as \(a\not \approx b\) and \(f(f(a))\approx a\) are propagated we have a conflict with \(C_1\{x\rightarrow f(a)\}\). So suppose in the worst case we propagate:
Now we have a conflict with \(C_1\{x\rightarrow f(a)\}\). Since there is no decision literal on the trail, \( Conflict \) rule immediately returns \(\bot \) and we are done.
4 Soundness and Completeness
In this section we show soundness and refutational completeness of SCL(EQ) under the assumption of a regular run. We provide the definition of a regular run and show that for a regular run all learned clauses are nonredundant according to our trail induced ordering. We start with the definition of a sound state.
Definition 13
A state \((\Gamma ; N; U; \beta ; k; D)\) is sound if the following conditions hold:

1
\(\Gamma \) is a consistent sequence of annotated literals,

2
for each decomposition \(\Gamma = \Gamma _1, L\sigma ^{i:(C \vee L)\cdot \sigma }, \Gamma _2\) where \(L\sigma \) is a propagated literal, we have that \(C\sigma \) is \(\beta \text {} false \) in \(\Gamma _1\), \(L\sigma \) is \(\beta \text {} undefined \) in \(\Gamma _1\) and irreducible by \(conv(\Gamma _1)\), \(N\cup U \models (C \vee L)\) and \((C \vee L) \sigma \prec _T \beta \),

3
for each decomposition \(\Gamma = \Gamma _1, L\sigma ^{i:(L\vee comp(L))\cdot \sigma }, \Gamma _2\) where \(L\sigma \) is a decision literal, we have that \(L\sigma \) is \(\beta \text {} undefined \) in \(\Gamma _1\) and irreducible by \(conv(\Gamma _1)\), \(N\cup U \models (L\vee comp(L))\) and \((L \vee comp(L))\sigma \prec _T \beta \),

4
\(N \models U\),

5
if \(D = C\cdot \sigma \), then \(C\sigma \) is \(\beta \text {} false \) in \(\Gamma \), \(N \cup U\models C\)
Lemma 5
The initial state \((\epsilon ; N; \emptyset ; \beta ; 0; \top )\) is sound.
Definition 14
A run is a sequence of applications of SCL(EQ) rules starting from the initial state.
Theorem 1
Assume a state \((\Gamma ;N;U;\beta ;k;D)\) resulting from a run. Then \((\Gamma ;N;U;\beta ;k;D)\) is sound.
Next, we give the definition of a regular run. Intuitively speaking, in a regular run we are always allowed to do decisions except if

1.
a literal can be propagated before the first decision and

2.
the negation of a literal can be propagated.
To ensure nonredundant learning we enforce at least one application of Skip during conflict resolution except for the special case of a conflict after a decision.
Definition 15
(Regular Run) A run is called regular if

1.
the rules \( Conflict \) and \( Factorize \) have precedence over all other rules,

2.
If \(k=0\) in a state \((\Gamma ; N;U;\beta ;k;D)\), then \( Propagate \) has precedence over \( Decide \),

3.
If an annotated literal \(L^{k:C\cdot \sigma }\) could be added by an application of Propagate on \(\Gamma \) in a state \((\Gamma ; N;U;\beta ;k;D)\) and \(C\in N\cup U\), then the annotated literal \(comp(L)^{k+1:C'\cdot \sigma '}\) is not added by Decide on \(\Gamma \),

4.
during conflict resolution \( Skip \) is applied at least once, except if \( Conflict \) is applied immediately after an application of \( Decide \).

5.
if \( Conflict \) is applied immediately after an application of \( Decide \), then Backtrack is only applied in a state \((\Gamma ,L'; N;U;\beta ;k;D\cdot \sigma )\) if \(L\sigma = comp(L')\) for some \(L\in D\).
Now we show that any learned clause in a regular run is nonredundant according to our trail induced ordering.
Lemma 6
(NonRedundant Clause Learning) Let N be a clause set. The clauses learned during a regular run in SCL(EQ) are not redundant with respect to \(\prec _{\Gamma ^*}\) and \(N \cup U\). For the trail only nonredundant clauses need to be considered.
The proof of Lemma 6 is based on the fact that conflict resolution eventually produces a clause smaller then the original conflict clause with respect to \(\prec _{\Gamma ^*}\). All simplifications, e.g., contextual rewriting, as defined in [2, 24, 37, 38, 40, 41], are therefore compatible with Lemma 6 and may be applied to the newly learned clause as long as they respect the induced trail ordering. In detail, let \(\Gamma \) be the trail before the application of rule Backtrack. The newly learned clause can be simplified according to the induced trail ordering \(\prec _{\Gamma ^*}\) as long as the simplified clause is smaller with respect to \(\prec _{\Gamma ^*}\).
Another important consequence of Lemma 6 is that newly learned clauses need not to be considered for redundancy. Furthermore, the SCL(EQ) calculus always terminates, Lemma 11, because there only finitely many nonredundant clauses with respect to a fixed \(\beta \).
For dynamic redundancy, we have to consider the fact that the induced trail ordering changes. At this level, only redundancy criteria and simplifications that are compatible with all induced trail orderings may be applied. Due to the construction of the induced trail ordering, it is compatible with \(\prec _T\) for unit clauses.
Lemma 7
(Unit Rewriting) Assume a state \((\Gamma ;N;U;\beta ;k;D)\) resulting from a regular run where the current level \(k>0\) and a unit clause \(l\approx r\in N\). Now assume a clause \(C\vee L[l']_p \in N\) such that \(l' = l\mu \) for some matcher \(\mu \). Now assume some arbitrary grounding substitutions \(\sigma '\) for \(C\vee L[l']_p\), \(\sigma \) for \(l\approx r\) such that \(l\sigma =l'\sigma '\) and \(r\sigma \prec _T l\sigma \). Then \( (C\vee L[r\mu \sigma \sigma ']_p)\sigma '\prec _{\Gamma ^*} (C\vee L[l']_p)\sigma '\).
In addition, any notion that is based on a literal subset relationship is also compatible with ordering changes. The standard example is subsumption.
Lemma 8
Let C, D be two clauses. If there exists a substitution \(\sigma \) such that \(C\sigma \subset D\), then D is redundant with respect to C and any \(\prec _{\Gamma ^*}\).
The notion of redundancy, Definition 1, only supports a strict subset relation for Lemma 8, similar to the superposition calculus. However, the newly generated clauses of SCL(EQ) are the result of paramodulation inferences [32]. In a recent contribution to dynamic, abstract redundancy [36] it is shown that also the nonstrict subset relation in Lemma 8, i.e., \(C\sigma \subseteq D\), preserves completeness.
If all stuck states, see below Definition 16, with respect to a fixed \(\beta \) are visited before increasing \(\beta \) then this provides a simple dynamic fairness strategy.
When unit reduction or any other form of supported rewriting is applied to clauses smaller than the current \(\beta \), it can be applied independently from the current trail. If, however, unit reduction is applied to clauses larger than the current \(\beta \) then the calculus must do a restart to its initial state, in particular the trail must be emptied, as for otherwise rewriting may result generating a conflict that did not exist with respect to the current trail before the rewriting. This is analogous to a restart in CDCL once a propositional unit clause is derived and used for simplification. More formally, we add the following new Restart rule to the calculus to reset the trail to its initial state after a unit reduction.
Next we show refutation completeness of SCL(EQ). To achieve this we first give a definition of a stuck state. Then we show that stuck states only occur if all ground literals \(L\prec _T\beta \) are \(\beta \text {} defined \) in \(\Gamma \) and not during conflict resolution. Finally we show that conflict resolution will always result in an application of Backtrack. This allows us to show termination (without application of Grow) and refutational completeness.
Definition 16
(Stuck State) A state \((\Gamma ;N;U;\beta ;k;D)\) is called stuck if \(D\ne \bot \) and none of the rules of the calculus, except for Grow, is applicable.
Lemma 9
(Form of Stuck States) If a regular run (without rule Grow) ends in a stuck state \((\Gamma ;N;U;\beta ;k;D)\), then \(D=\top \) and all ground literals \(L\sigma \prec _T \beta \), where \(L\vee C\in (N\cup U)\) are \(\beta \text {} defined \) in \(\Gamma \).
Lemma 10
Suppose there is a sound state \((\Gamma ; N;U;\beta ;k; D)\) resulting from a regular run where \(D\not \in \{\top ,\bot \}\). If Backtrack is not applicable then any set of applications of \( Explore\text {}Refutation \), Skip, \( Factorize \), \( Equality\text {}Resolution \) will finally result in a sound state \((\Gamma '; N;U;\beta ;k;D')\), where \(D' \prec _{\Gamma ^*} D\). Then Backtrack will be finally applicable.
Corollary 1
(Satisfiable Clause Sets) Let N be a satisfiable clause set. Then any regular run without rule Grow will end in a stuck state, for any \(\beta \).
Thus a stuck state can be seen as an indication for a satisfiable clause set. Of course, it remains to be investigated whether the clause set is actually satisfiable. Superposition is one of the strongest approaches to detect satisfiability and constitutes a decision procedure for many decidable firstorder fragments [3, 23]. Now given a stuck state and some specific ordering such as KBO, LPO, or some polynomial ordering [20], it is decidable whether the ordering can be instantiated from a stuck state such that \(\Gamma \) coincides with the superposition model operator on the ground terms smaller than \(\beta \). In this case it can be effectively checked whether the clauses derived so far are actually saturated by the superposition calculus with respect to this specific ordering. In this sense, SCL(EQ) has the same power to decide satisfiability of firstorder clause sets as superposition.
Definition 17
A regular run terminates in a state \((\Gamma ; N;U;\beta ;k; D)\) if \(D=\top \) and no rule is applicable, or \(D = \bot \).
Lemma 11
Let N be a set of clauses and \(\beta \) be a ground term. Then any regular run that never uses Grow terminates.
Lemma 12
If a regular run reaches the state \((\Gamma ;N;U;\beta ;k;\bot )\) then N is unsatisfiable.
Theorem 2
(Refutational Completeness) Let N be an unsatisfiable clause set, and \(\prec _T\) a desired term ordering. For any ground term \(\beta \) where \(gnd_{\prec _T\beta }(N)\) is unsatisfiable, any regular \({\text {SCL(EQ)}}\) run without rule Grow will terminate by deriving \(\bot \).
5 Discussion
We presented SCL(EQ), a new sound and complete calculus for reasoning in firstorder logic with equality. We will now discuss some of its aspects and present ideas for future work beyond the scope of this paper.
The SCL(EQ) calculus can be viewed as a generalization of the firstorder without equality SCL calculus [17], where syntactic equality with respect to trail literals is replaced with equality modulo the presented equational theory. If standard firstorder literals like R(x, y) are represented by equations like \(f_R(x,y)\approx \texttt {true}\) then performing SCL(EQ) on the latter simplifies to classical SCL reasoning on the firstorder literals with a slightly different strategy.
The trail induced ordering, Definition 8, is the result of letting the calculus follow the logical structure of the clause set on the literal level and at the same time supporting rewriting at the term level. It can already be seen by examples on ground clauses over (in)equations over constants that this combination requires a layered approach as suggested by Definition 8, see Example 4.
Example 4
(Propagate Smaller Equation) Assume a term ordering \(\prec _{kbo}\), unique weight 1 and with precedence \(d\prec c\prec b\prec a\). Further assume \(\beta \) to be large enough. Assume the ground clause set N solely built out of constants
and the trail \(\Gamma {:}{=} [c\approx d^{0:C_1}, a\approx b^{0:C_2}, b\approx d^{0:C_3}]\). Now, although the first two steps propagated equations that are strictly maximal in the ordering in their respective clauses, the finally propagated equation \(b\approx d\) is smaller in the term ordering \(\prec _{kbo}\) than \(a\approx b\). Thus the structure of the clause set forces propagation of a smaller equation in the term ordering. So the more complicated trail ordering is a result of following the structure of the clause set rather than employing an a priori fixed ordering.
In case the calculus runs into a stuck state, i.e., the current trail is a model for the set of considered ground instances, then the trail information can be effectively used for a guided continuation. For example, in order to use the trail to certify a model, the trail literals can be used to guide the design of a lifted ordering for the clauses with variables such that propagated trail literals are maximal in respective clauses. Then it could be checked by superposition, if the current clause is saturated by such an ordering. If this is not the case, then there must be a superposition inference larger than the current \(\beta \), thus giving a hint on how to extend \(\beta \). Another possibility is to try to extend the finite set of ground terms considered in a stuck state to the infinite set of all ground terms by building extended equivalence classes following patterns that ensure decidability of clause testing, similar to the ideas in [15]. If this fails, then again this information can be used to find an appropriate extension term \(\beta \) for rule Grow.
In contrast to superposition, SCL(EQ) does also inferences below variable level. In general, single superposition inferences below variables are redundant [2]. Inferences in SCL(EQ) are guided by a false clause with respect to a partial model assumption represented by the trail. They are typically not single superposition steps, but a sequence of superposition inferences eventually resulting in a nonredundant clause changing the partial model assumption. Therefore, compared to the syntactic style of superpositionbased theorem proving, in SCL(EQ) reasoning below variables does not result in an explosion in the number of possibly inferred clauses but also rather in the derivation of more general clauses, see Example 5.
Example 5
(Rewriting below variable level) Assume a term ordering \(\prec _{kbo}\), unique weight 1 and with precedence \(d\prec c\prec b\prec a \prec g \prec h \prec f\). Further assume \(\beta \) to be large enough. Assume the clause set N:
Let \(\sigma =\{x\rightarrow g(a)\}\) be a substitution. \(C_1\sigma \) must be propagated: \(\Gamma = [f(g(a))\approx h(b)^{0:C_1\sigma }]\). Now suppose that we decide \(f(g(b))\not \approx h(b)\). Then \(\Gamma = [f(g(a))\approx h(b)^{0:C_1\sigma },f(g(b))\not \approx h(b)^{1:f(g(b))\not \approx h(b)\vee f(g(b))\approx h(b)}]\) and \(C_3\) is a conflict clause. ExploreRefutation now creates the following refutation for \(a\approx b\):
Multiple applications of EqualityResolution and Factorize result in the final conflict clause \(C_4 {:}{=} f(g(b)) \approx h(b)\) with which we can backtrack. The clause set resulting from this new clause is:
where \(C'_2\) is the result of a unit reduction between \(C_4\) and \(C_2\). Note that the refutation required rewriting below variable level in step \(I_4\). Superposition would create the following clauses (EqualityResolution and Factorization steps are implicitly done):
For superposition the resulting clause set is thus:
Currently, the reasoning with solely positive equations is done on and with respect to the trail. It is wellknown that also inferences from this type of reasoning can be used to speed up the overall reasoning process. The SCL(EQ) calculus already provides all information for such a type of reasoning, because it computes the justification clauses for trail reasoning via rewriting inferences. By an assessment of the quality of these clauses, e.g., their reduction potential with respect to trail literals, they could also be added, independently from resolving a conflict.
The trail reasoning is currently defined with respect to rewriting. It could also be performed by congruence closure [29]. However, we still need a ground term rewrite system to propagate literals. A possible solution to this is an algorithm by Gallier et al. [22] which creates a ground rewrite system out of the congruence classes in polynomial time.
Bromberger et al. [16] showed how to lift the twowatched literal scheme to SCL. We could make use of this in an implementation as well. In general, an implementation of SCL(EQ) requires both the infrastructure of a superpositonbased prover and an CDCL/SMT solver. The aspect of how to find interesting ground decision or propagation literals for the trail including the respective grounding substituion \(\sigma \) can be treated similar to CDCL [11, 12, 28, 33]. A simple heuristic may be used from the start, like counting the number of instance relationships of some ground literal with respect to the clause set, but later on a bonus system can focus the search towards the structure of the clause sets. Ground literals involved in a conflict or the process of learning a new clause get a bonus or preference. However, since the number of ground literals is not fixed from the beginning with growing \(\beta \), all these operations need to be done via hashing or indexing operations modulo matching/unification in contrast to simple lookups in the CDCL case. The regular strategy requires the propagation of all ground unit clauses smaller than \(\beta \). For an implementation a propagation of the (explicit and implicit) unit clauses with variables to the trail will be a better choice. This complicates the implementation of refutation proofs and rewriting (congruence closure), but because every reasoning is layered by a ground term \(\beta \) this can still be efficiently done.
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Acknowledgements
This work was partly funded by DFG Grant 389792660 as part of TRR 248, see https://perspicuouscomputing.science. The authors thank the anonymous reviewers and Martin Desharnais for their thorough reading, detailed comments, and corrections.
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Appendices
Appendix: A Proofs and Auxiliary Lemmas
Proof of Lemma 1
Let \(\Gamma _1\) be a trail and K a defined literal that is of level i in \(\Gamma _1\). Then K is of level i in a trail \(\Gamma {:}{=}\Gamma _1,\Gamma _2\).
Proof
Assume a trail \(\Gamma _1\) and a literal K that is of level i in \(\Gamma _1\). Let \(\Gamma {:}{=}\Gamma _1,\Gamma _2\) be a trail. Then we have two cases:

1.
K has no defining literal in \(\Gamma _1\). Then \(cores(\Gamma _1;K) = \{[]\}\) contains only the empty core and K is of level 0 in \(\Gamma _1\). Then \(cores(\Gamma ;K) = \{[]\}\) as well and thus K is of level 0 in \(\Gamma \).

2.
K has a defining literal \(L{:}{=}max_{\Gamma _1}(K)\) and L is of level i. Then there exists a core \(\Delta \in cores(\Gamma _1;K)\) such that L is the maximum literal in \(\Delta \) according to \(\prec _{\Gamma }\) and for all \(\Lambda \in cores(\Gamma _1;K)\) it holds \(max_{\prec _{\Gamma }}(\Delta )\preceq _{\Gamma }max_{\prec _{\Gamma }}(\Lambda )\). Thus \(\Delta \) is a defining core. Now any \(\Lambda \in (cores(\Gamma ;K){\setminus } cores(\Gamma _1;K))\) has a higher maximum literal according to \(\prec _\Gamma \). Thus \(\Delta \) is also a defining core in \(\Gamma \) and L is the defining literal of K in \(\Gamma \) and thus K is of level i in \(\Gamma \).
\(\square \)
1.1 Auxiliary Lemmas for the Proofs of Lemma 2
Lemma 13
Let \(\Gamma \) be a trail. Then any literal in \(\Gamma \) occurs exactly once.
Proof
Let \(\Gamma {:}{=}[L_1^{i_1:C_1\cdot \sigma _1},...,L_n^{i_n:C_n\cdot \sigma _n}]\). Now suppose there exist \(L_i, L_j\) with \(i<j\) and \(1\le i,j\le n\) such that \(L_i=L_j\). By definition of \(\Gamma \), \(L_j\) is undefined in \([L_1,...,L_i,...,L_{j1}]\). But obviously \(L_j\) is defined in \(\Gamma \). Contradiction. \(\square \)
Lemma 14
Let \(\Gamma \) be a trail. If a literal L is of level i, then it is not of level \(j\not = i\).
Proof
Let \(\Gamma \) be a trail. By Lemma 13 any literal in \(\Gamma \) is unique. Suppose there exists a literal L such that L is of level i and of level j. If the core is empty for L then L is of level 0 by definition. Otherwise there must exist cores \(\Delta ,\Lambda \in cores(\Gamma ;L)\) such that \(max_{\prec _\Gamma }(\Delta )\preceq _{\Gamma } max_{\prec _\Gamma }(\Lambda ')\) and \(max_{\prec _\Gamma }(\Lambda )\preceq _{\Gamma } max_{\prec _\Gamma }(\Lambda ')\) for all \(\Lambda '\in cores(\Gamma ;L)\). But then \(max_{\prec _\Gamma }(\Lambda )=max_{\prec _\Gamma }(\Delta )\). Contradiction. \(\square \)
Lemma 15
Let L be a ground literal and \(\Gamma \) a trail. If L is defined in \(\Gamma \) then L has a level.
Proof
Let \(\Gamma \) be a trail. Suppose that L is defined in \(\Gamma \). Then it either has a defining literal or it has no defining literal. If it has a defining literal K, then the level of K is the level of L. Since \(K\in \Gamma \) it is annotated by a level. Thus L has a level. If L does not have a defining literal, then L is of level 0 by definition of a literal level. \(\square \)
Proof of Lemma 21
\(\prec _{\Gamma ^*}\) is welldefined.
Proof
Suppose a trail \(\Gamma {:}{=}[L_1^{i_1:C_1\cdot \sigma _1},...,L_n^{i_n:C_n\cdot \sigma _n}]\) and a term \(\beta \) such that \(\{L_1,...,L_n\}\prec _T\beta \). We have to show that the rules 8.1–8.11 are pairwise disjunct. Consider the rules 8.1–8.7. These rules are pairwise disjunct, if the sets \(\{L_1,...,L_n\}\), \(\{comp(L_1),...,comp(L_n)\}\) and \(\{M_{i,j}~\vert ~i\le n\}\) are pairwise disjunct. Obviously, \(\{L_1,...,L_n\} \cap \{comp(L_1),...,comp(L_n)\} = \emptyset \). Furthermore \((\{L_1,...,L_n\}\cup \) \(\{comp(L_1),...,comp(L_n)\})\cap \{M_{i,j}~\vert ~i\le n\} = \emptyset \) follows directly from the definition of a trail induced ordering. 8.8 and 8.9 are disjunct since \(\{L~\vert ~L~is~of~level~0\}\) and \(\{L~\vert ~L~is~of~level~greater~0\}\) are disjunct by Lemma 14. It follows that 8.1–8.9 are pairwise disjunct, since all relations in 8.1–8.7 contain only \(\beta \text {} defined \) literals of level 1 or higher and all relations in 8.8, 8.9 contain at least one \(\beta \text {} defined \) literal of level 0. 8.10 and 8.11 are disjunct since a literal cannot be both \(\beta \text {} defined \) and \(\beta \text {} undefined \). It follows that 8.1–8.11 are pairwise disjunct, since all relations in 8.1–8.9 contain only \(\beta \text {} defined \) literals and all relations in 8.10, 8.11 contain at least one \(\beta \text {} undefined \) literal. \(\square \)
Proof of Lemma 22
\(\prec _{\Gamma ^*}\) is a total strict order, i.e. \(\prec _{\Gamma ^*}\) is irreflexive, transitive and total.
Proof
Suppose a trail \(\Gamma {:}{=}[L_1^{i_1:C_1\cdot \sigma _1},...,L_n^{i_n:C_n\cdot \sigma _n}]\) and a term \(\beta \) such that \(\{L_1,...,L_n\}\prec _T\beta \).
Irreflexivity We have to show that there is no ground literal L such that \(L\prec _{\Gamma ^*} L\). Suppose two literals L and K such that \(L\prec _{\Gamma ^*} K\) and \(L=K\). Now we have several cases:

1.
Suppose that L, K are \(\beta \text {} defined \) and of level 1 or higher. Then we have several cases:

(a)
\(L=M_{i,j}\) and \(K=M_{k,l}\). Then by 8.1 \(M_{i,j}\prec _{\Gamma ^*} M_{k,l}\) if \(i<k\) or \((i=k\) and \(j<l)\). Thus \(i\not =k\) or \(j\not =l\). We show that for \(M_{i,j}\), \(M_{k,l}\) with \(i\not =k\) or \(j\not =l\) it holds \(M_{i,j}\not =M_{k,l}\). Assume that \(M_{i,j} = M_{k,l}\) and \(k\not =i\) or \(j\not = l\). Assume that \(k=i\). Then, by Definition 8\(M_{i,j} \prec _T M_{k,l}\) or \(M_{k,l} \prec _T M_{i,j}\). Thus \(M_{i,j} \not = M_{k,l}\) since \(\prec _T\) is a rewrite ordering. Now assume that \(k\not =i\). Since \(M_{i,j} = M_{k,l}\) it holds \(max_\Gamma (M_{i,j}) = max_\Gamma (M_{k,l})\), since both have the same level by Lemma 14. But then \(k=i\). Thus \(M_{i,j} \not = M_{k,l}\) for \(k\not =i\) or \(j\not = l\). Thus if by 8.1 \(M_{i,j}\prec _{\Gamma ^*} M_{k,l}\) if \(i<k\) or (\(i=k\) and \(j<l\)), then \(M_{i,j}\not = M_{k,l}\).

(b)
\(L=L_i\) and \(K=L_j\). Then by 8.2 \(L_i\prec _{\Gamma ^*} L_j\) if \(L_i\prec _\Gamma L_j\). Then by Lemma 13\(L_i\not =L_j\).

(c)
\(L=comp(L_i)\) and \(K=L_j\). Then by 8.3 \(comp(L_i)\prec _{\Gamma ^*} L_j\) if \(L_i\prec _\Gamma L_j\). \(L_i\not =L_j\) by Lemma 13. \(L\not = K\) has to hold since \(\Gamma \) is consistent.

(d)
\(L=L_i\) and \(K=comp(L_j)\). Then by 8.4 \(L_i\prec _{\Gamma ^*} comp(L_j)\) if \(L_i\prec _\Gamma L_j\) or \(i=j\). If \(i\not =j\) then we can proceed analogous to the previous step. If \(i=j\) then obviously \(L_i\not =comp(L_i)\).

(e)
\(L=comp(L_i)\) and \(K=comp(L_j)\). Then by 8.5 \(comp(L_i)\prec _{\Gamma ^*} comp(L_j)\) if \(L_i\prec _\Gamma L_j\). By Lemma 13\(L_i\not = L_j\). Thus \(comp(L_i)\not = comp(L_j)\).

(f)
\(L=L_i\) and \(K=M_{k,l}\). Then by 8.6 \(L_i\prec _{\Gamma ^*} M_{k,l}\), \({\text {comp}}(L_i)\prec _{\Gamma ^*} M_{k,l}\) if \(i\le k\). \(M_{k,l}\not =L_i\) and \(M_{k,l}\not =comp(L_i)\) follows directly from the Definition 8. Thus if \(L_i\prec _{\Gamma ^*} M_{k,l}\) or \({\text {comp}}(L_i)\prec _{\Gamma ^*} M_{k,l}\) if \(i\le k\) by 8.6, then \(L_i\not = M_{k,l}\) and \({\text {comp}}(L_i)\not = M_{k,l}\).

(g)
\(L=M_{k,l}\) and \(K=L_i\). Then we can proceed analogous to the previous step for 8.7.

(a)

2.
Suppose that L and K are \(\beta \text {} defined \) and of level zero. Since \(\prec _T\) is irreflexive, \(L\not \prec _T K\) has to hold. Since \(\prec _{\Gamma ^*} = \prec _T\) for literals of level zero \(L\not \prec _{\Gamma ^*} K\) has to hold too.

3.
Suppose that L, K are \(\beta \text {} defined \) and L is of level zero and K is of level greater than zero. But then \(L\not =K\) has to hold by Lemma 14. Thus \(L\not \prec _{\Gamma ^*} K\) for 8.9.

4.
Suppose that L and K are \(\beta \text {} undefined \). Then by 8.10 \(K\prec _{\Gamma ^*} H\) if \(K\prec _T H\). Since \(\prec _T\) is a rewrite ordering \(K\prec _T H\) iff \(K\not = H\).

5.
Suppose that L is \(\beta \text {} defined \) and K is \(\beta \text {} undefined \). Then by 8.11 \(L\prec _{\Gamma ^*} K\). Then \(L\not =K\) has to hold since otherwise L, K would be both \(\beta \text {} defined \) and \(\beta \text {} undefined \), contradicting consistency of \(\Gamma \).
Transitivity Suppose there exist literals L, K, H such that \(H\prec _{\Gamma ^*} K\) and \(K \prec _{\Gamma ^*} L\) but not \(H\prec _{\Gamma ^*} L\). We have several cases:

1.
Suppose all literals are \(\beta \text {} undefined \). Then \(K \prec _T L\) and \(H\prec _T K\). Otherwise \(K \prec _{\Gamma ^*} L\) and \(H\prec _{\Gamma ^*} K\) would not hold. Thus also \(H\prec _T L\) by transitivity of \(\prec _T\). Thus \(H\prec _{\Gamma ^*} L\) by 8.10.

2.
Suppose two literals are \(\beta \text {} undefined \). If K would be \(\beta \text {} defined \), then \(K\prec _{\Gamma ^*} H\) by 8.11 contradicting assumption. If L would be \(\beta \text {} defined \), then \(L \prec _{\Gamma ^*} K\) by 8.11 again contradicting assumption. Thus H has to be \(\beta \text {} defined \). Then \(H \prec _{\Gamma ^*} L\) by definition 8.11.

3.
Suppose one literal is \(\beta \text {} undefined \). If K would be \(\beta \text {} undefined \), then \(L \prec _{\Gamma ^*} K\) by Definition 8.11 contradicting assumption. If H would be \(\beta \text {} undefined \), then \(K \prec _{\Gamma ^*} H\) by Definition 8.11 again contradicting assumption. Thus L has to be \(\beta \text {} undefined \). Then \(H \prec _{\Gamma ^*} L\) by Definition 8.11.

4.
Suppose all literals are \(\beta \text {} defined \). Then we have multiple subcases:

(a)
Suppose all literals have the same defining literal \(L_i\) and \(L_i\) is of level 1 or higher. By 8.6 \(L_i\prec _{\Gamma ^*} M_{i,j}\) and \(comp(L_i)\prec _{\Gamma ^*} M_{i,j}\) for all j. By 8.4 \(L_i\prec _{\Gamma ^*} comp(L_i)\). Thus \(L_i\prec _{\Gamma ^*} comp(L_i)\prec _{\Gamma ^*} M_{i,j}\) for all j. Since \(K \prec _{\Gamma ^*} L\) either \(K=L_i\) and \(L\not =L_i\) or \(L=M_{i,j}\) and \(K=comp(L_i)\) or \(L=M_{i,j}\) and \(K=M_{i,k}\) with \(k<j\).

(i)
Assume \(K=L_i\) and \(L\not =L_i\). Since K is the smallest literal with defining literal \(L_i\), \(K = H\) has to hold. But then \(K \prec _{\Gamma ^*} K\) contradicting irreflexivity.

(ii)
Assume \(L=M_{i,j}\) and \(K=comp(L_i)\). Since \(H \prec _{\Gamma ^*} K\) and all literals have the same defining literal, \(H = L_i\) has to hold by 8.6 and 8.4. Then, again by 8.6, \(H \prec _{\Gamma ^*} L\).

(iii)
\(L=M_{i,j}\) and \(K=M_{i,k}\) with \(k<j\). Since \(H \prec _{\Gamma ^*} K\) and all have the same defining literal either \(H = M_{i,l}\) with \(l<k\) by 8.1, or \(H=L_i\) or \(H=comp(L_i)\) by 8.6. In both cases \(H \prec _{\Gamma ^*} L\) holds by 8.1 and 8.6.

(i)

(b)
Suppose H, K, L have at least one different defining literal and \(max_\Gamma (L) = L_i\) with \(L_i\) of level 1 or higher. First, we have to show that if \(L_j = max_\Gamma (K') \prec _\Gamma max_\Gamma (L') = L_i\) and \(L_i\) is of level 1 or higher, then \(K' \prec _{\Gamma ^*} L'\). Suppose that \(L_j\) is of level 0. Then \(K' \prec _{\Gamma ^*} L'\) by 8.9. Suppose that \(L'= M_{i,k}\) and \(K'= M_{j,l}\). Then \(K' \prec _{\Gamma ^*} L'\) by 8.1. Suppose that \(L'= M_{i,k}\) and \(K'=L_j\) or \(K'=comp(L_j)\). Then \(K' \prec _{\Gamma ^*} L'\) by 8.6. Suppose that \(L'= L_i\) or \(L'=comp(L_i)\) and \(K'=L_j\) or \(K'=comp(L_j)\). Then \(K' \prec _{\Gamma ^*} L'\) by 8.2–8.5. Suppose that \(L'= L_i\) or \(L'=comp(L_i)\) and \(K'= M_{j,l}\). Then \(K' \prec _{\Gamma ^*} L'\) by 8.7.
Now by assumption \(H \prec _{\Gamma ^*} K\) and \(K \prec _{\Gamma ^*} L\). If \(max_\Gamma (K) \prec _\Gamma max_\Gamma (H)\) then \(K \prec _{\Gamma ^*} H\) contradicting assumption. The same holds for L and K. Thus either \(max_\Gamma (H) \prec _\Gamma max_\Gamma (L)\) or \(max_\Gamma (K) \prec _\Gamma max_\Gamma (L)\). In the first case \(H \prec _{\Gamma ^*} L\) follows from above. In the second case \(max_\Gamma (H) \preceq _\Gamma max_\Gamma (K)\) has to hold. Thus \(H \prec _{\Gamma ^*} L\) follows again.

(c)
Suppose that \(max_\Gamma (L) = L_i\) where \(L_i\) is of level 0. Since \(K \prec _{\Gamma ^*} L\), \(max_\Gamma (K) = L_j\) with \(L_j\) of level 0 has to hold by 8.9 and 8.11. Now assume that \(L\prec _T K\). Then \(L\prec _{\Gamma ^*} K\) by 8.8 contradicting assumption. Thus \(K\prec _T L\) has to hold since \(K\not = L\). Since \(H \prec _{\Gamma ^*} K\), \(max_\Gamma (H) = L_k\) with \(L_k\) of level 0 has to hold by 8.9 and 8.11. Now assume that \(K\prec _T H\). Then \(K\prec _{\Gamma ^*} H\) by 8.8 contradicting assumption. Thus \(H\prec _T K\) has to hold since \(H\not = K\). By transitivity of \(\prec _T\), \(H\prec _T L\) and thus \(H \prec _{\Gamma ^*} L\) has to hold.

(a)
Totality First we show that any ground literal is either \(\beta {\text {}} defined \) and has a level or \(\beta \text {} undefined \). Since \(\Gamma \) is consistent, a literal is either \(\beta \text {} defined \) or \(\beta \text {} undefined \). We just need to show that if a literal is \(\beta \text {} defined \), it has a level. By Lemma 15 all defined literals have a level. \(\beta \text {} definedness \) implies definedness. Thus all \(\beta \text {} defined \) literals have a level. Now assume some arbitrary ground literals \(L\ne K\). We have several cases:

1.
L, K are \(\beta \text {} undefined \). Since \(L\not =K\) we have \(L\prec _T K\) or \(K\prec _T L\) by totality of \(\prec _T\) on ground literals. Thus by 8.10 \(L\prec _{\Gamma ^*} K\) or \(K\prec _{\Gamma ^*} L\).

2.
One is \(\beta \text {} defined \). Then either \(L\prec _{\Gamma ^*} K\) or \(K\prec _{\Gamma ^*} L\) by 8.11.

3.
Both are \(\beta \text {} defined \). Then we have several subcases:

(a)
L is of level zero and K is of level greater than zero or vice versa. Then by 8.9 \(L\prec _{\Gamma ^*} K\) or \(K\prec _{\Gamma ^*} L\) has to hold.

(b)
\(max_\Gamma (L)= L_i\) and \(max_\Gamma (K)= L_j\) and both are of level 1 or higher.

(i)
\(L=M_{i,k}\) and \(K=M_{j,l}\). Then either \(M_{i,k}\prec _{\Gamma ^*} M_{j,l}\) or \(M_{j,l}\prec _{\Gamma ^*} M_{i,k}\) by 8.1.

(ii)
\(L=M_{i,k}\) and \(K=L_j\) or \(K=comp(L_j)\). If \(i\ge j\) then by 8.6 \(L_j\prec _{\Gamma ^*} M_{i,k}\) or \( comp(L_j)\prec _{\Gamma ^*} M_{i,k}\). If \(i<j\) then by 8.7 \(M_{i,k}\prec _{\Gamma ^*} L_j\) or \(M_{i,k}\prec _{\Gamma ^*} comp(L_j)\).

(iii)
\(K=M_{i,k}\) and \(L=L_j\) or \(L=comp(L_j)\). Analogous to previous step.

(iv)
\(L=L_i\) and \(K=L_j\). Then if \(i<j\) by 8.2 \(L_i\prec _{\Gamma ^*} L_j\) and \(L_j\prec _{\Gamma ^*} L_i\) otherwise.

(v)
\(L=comp(L_i)\) and \(K=comp(L_j)\) analogous to previous step for 8.5.

(vi)
\(L=L_i\) and \(K=comp(L_j)\). Then if \(i\le j\) by 8.4 \(L_i\prec _{\Gamma ^*} comp(L_j)\). If \(j<i\) by 8.3 \(comp(L_j)\prec _{\Gamma ^*} L_i\).

(vii)
\(L=comp(L_i)\) and \(K=L_j\). Then if \(i< j\) by 8.3 \(comp(L_i)\prec _{\Gamma ^*} L_j\). If \(j\le i\) by 8.4 \(L_j\prec _{\Gamma ^*} comp(L_i)\).

(i)

(c)
\(max_\Gamma (L)= L_i\) and \(max_\Gamma (K)= L_j\) and both are of level 0. Now either \(L\prec _T K\) or \(K\prec _T L\). Thus by 8.8 \(L\prec _{\Gamma ^*} K\) or \(K\prec _{\Gamma ^*} L\).

(a)
\(\square \)
Proof of Lemma 23
\(\prec _{\Gamma ^*}\) is a wellfounded ordering.
Proof
Suppose some arbitrary subset M of all ground literals, a trail \(\Gamma {:}{=}[L_1^{i_1:C_1\cdot \sigma _1},...,L_n^{i_n:C_n\cdot \sigma _n}]\) and a term \(\beta \) such that \(\{L_1,...,L_n\}\prec _T \beta \). We have to show that M has a minimal element. We have several cases:

1.
L is \(\beta \text {} undefined \) in \(\Gamma \) for all literals \(L\in M\). Then \(\prec _{\Gamma ^*} = \prec _T\). Since \(\prec _T\) is wellfounded there exists a minimal element in M. Thus there exists a minimal element in M with regard to \(\prec _{\Gamma ^*}\).

2.
there exists at least one literal in M that is \(\beta \text {} defined \). Then we have two cases:

(a)
there exists a literal in M that is of level zero. Then let \(L\in M\) be the literal of level zero, where \(L\prec _T K\) for all \(K\in M\) with K of level zero. We show that L is the minimal element. Suppose there exists a literal \(L'\in M\) that is smaller than L. Since L is of level zero, \(L\prec _{\Gamma ^*} K\) for all literals K of level greater than zero by 8.9 and \(L\prec _{\Gamma ^*} H\) for all \(\beta \text {} undefined \) literals H by 8.11. Thus \(L'\) must be of level zero. But then \(L'\prec _T L\) has to hold, contradicting assumption.

(b)
There exists no literal in M that is of level zero. Let \(L\in M\) be the literal where \(max_\Gamma (L)\preceq _\Gamma max_\Gamma (K)\) for all \(K\in M\) and

(i)
\(L= max_\Gamma (L)\) or;

(ii)
\(L= comp(max_\Gamma (L))\) and \(max_\Gamma (L)\not \in M\) or;

(iii)
\(L\prec _T H\) for all \(H\in M\) such that \(max_\Gamma (L)= max_\Gamma (H)\) and \(max_\Gamma (L)\not \in M\) and \(comp(max_\Gamma (L))\not \in M\).
We show that L is the minimal element. Suppose there exists a literal \(L'\in M\) that is smaller than L. We have three cases:

(i)
\(max_\Gamma (L) = L = L_i\). Since \(L_i\prec _T \beta \) we have either \(L' = L_j\) with \(j < i\) by 8.2 or \(L'= comp(L_j)\) with \(j< i\) by 8.3 or \(L' = M_{k,l}\) with \(k < i\) by 8.7. In all three cases we have \(max_\Gamma (L')\prec _\Gamma max_\Gamma (L)\) contradicting assumption that the defining literal of L is minimal in M.

(ii)
\(L=comp(L_i) = comp(max_\Gamma (L))\) and \(max_\Gamma (L)\not \in M\). Since \(L_i\prec _T\beta \) either \(L' = L_j\) with \(j < i\) by 8.4 or \(L'= comp(L_j)\) with \(j< i\) by 8.5 or \(L' = M_{k,l}\) with \(k < i\) by 8.7. In all three cases we have \(max_\Gamma (L')\prec _\Gamma max_\Gamma (L)\) contradicting assumption that the defining literal of L is minimal in M.

(iii)
\(L = M_{k,l}\) and \(max_\Gamma (L)\not \in M\) and \(comp(max_\Gamma (L))\not \in M\). Then either \(L'=M_{i,j}\) with \(i<k\) or \((i=k\) and \(j<l)\) by 8.1 or \(L'=L_i\) or \(L'=comp(L_i)\) with \(i\le k\). Suppose that \(L'=M_{i,j}\) and \(i<k\). Then \(max_\Gamma (L')\prec _\Gamma max_\Gamma (L)\) contradicting assumption. Suppose that \(L'=M_{i,j}\) and \(i=k\) and \(j<l\). Then \(L'\prec _T L\) and \(max_\Gamma (L)= max_\Gamma (L')\). For L it holds \(L\prec _T H\) for all \(H\in M\) such that \(max_\Gamma (L)= max_\Gamma (H)\). Contradiction. Suppose that \(L'=L_i\) or \(L'=comp(L_i)\) with \(i = k\). Then \(max_\Gamma (L) = L_i\). By assumption \(max_\Gamma (L)\not \in M\) and \(comp(max_\Gamma (L))\not \in M\). Contradiction. Suppose that \(L'=L_i\) or \(L'=comp(L_i)\) with \(i < k\). Then we have \(max_\Gamma (L')\prec _\Gamma max_\Gamma (L)\) contradicting assumption that the defining literal of L is minimal in M. \(\square \)

(i)

(a)
Proof of Lemma 4
Assume that all ground terms t with \(t\prec _T \beta \) for any \(\beta \) are polynomial in the size of \(\beta \). Then testing Propagate (Conflict) is NPComplete, i.e., the problem of checking for a given clause C whether there exists a grounding substitution \(\sigma \) such that \(C\sigma \) propagates (is false) is NPComplete.
Proof
Let \(C\sigma \) be propagable (false). The problem is in NP because \(\beta \) is constant and for all \(t\in codom(\sigma )\) it holds that t is polynomial in the size of \(\beta \). Checking if \(C\sigma \) is propagable (false) can be done in polynomial time with Congruence Closure [29] since \(\sigma \) has polynomial size.
We reduce 3SAT to testing rule Conflict. Consider a 3place predicate R, a unary function g, and a mapping from propositional variables P to firstorder variables \(x_P\). Assume a 3SAT clause set \(N =\{ \{L_0,L_1,L_2\},...,\{L_{n2},L_{n1},L_{n}\}\}\), where \(L_i\) may denote both \(P_i\) and \(\lnot P_i\). Now we create the clause
where \(t_{i} {:}{=} x_{P_{i}}\) if \(L_{i} = P_{i}\) and \(t_{i} {:}{=} g(x_{P_{i}})\) otherwise. Now let \(\Gamma {:}{=}\{R(x_0,x_1,x_2)\mid x_i\in \{0,1,g(0),g(1)\}\) such that \((x_0\vee x_1 \vee x_2)\downarrow _{\{g(x)\mapsto (\lnot x)\}}\) is true \(\}\) be the set of all Ratoms that evaluate to true if considered as a three literal propositional clause. Now N is satisfiable if and only if Conflict is applicable to the new clause. The reduction is analogous for Propagate. \(\square \)
Proof of Theorem 1
Assume a state \((\Gamma ;N;U;\beta ;k;D)\) resulting from a run. Then \((\Gamma ;N;U;\beta ;k;D)\) is sound.
Proof
Proof by structural induction on \((\Gamma ;N;U;\beta ;k;D)\). Let \((\Gamma ;N;U;\beta ;k; D) = (\epsilon , N, \emptyset , \beta , 0, \top )\), the initial state. Then it is sound according to Lemma 5. Now assume that \((\Gamma ;N;U;\beta ;k;D)\) is sound. We need to show that any application of a rule results in a sound state.
Propagate Assume \( Propagate \) is applicable. Then there exists \(C\in N\cup U\) such that \(C=C_0\vee C_1\vee L\), \(L\sigma \) is \(\beta \text {} undefined \) in \(\Gamma \), \(C_1\sigma = L\sigma \vee ...\vee L\sigma \), \(C_1=L_1\vee ...\vee L_n\),\(\mu =mgu(L_1,...,L_n,L)\) and \(C_0\sigma \) is \(\beta \text {} false \) in \(\Gamma \). Then a reduction chain application \([I_1,...,I_m]\) from \(\Gamma \) to \(L\sigma ^{k:(C_0\vee L)\mu \cdot \sigma }\) is created with \(I_m {:}{=} (s_m\# t_m\cdot \sigma _m, s_m\# t_m\vee C_m\cdot \sigma _m, I_j, I_k, p_m)\). Finally \(s_m\# t_m\sigma _m^{k:(s_m\# t_m\vee C_m)\cdot \sigma _m}\) is added to \(\Gamma \).
By definition of a reduction chain application \((s_m\# t_m)\sigma _m = L\sigma {\downarrow _{conv(\Gamma )}}\). Thus, \((s_m\# t_m)\sigma _m\) must be \(\beta \text {} undefined \) in \(\Gamma \) and irreducible by \(conv(\Gamma )\), since \((C_0\vee L)\mu \sigma \prec _T \beta \) by definition of Propagate.

13.1: Since \((s_m\# t_m)\sigma _m\) is \(\beta \text {} undefined \) in \(\Gamma \), adding \((s_m\# t_m)\sigma _m\) does not make \(\Gamma \) inconsistent. Thus \(\Gamma ,(s_m\# t_m)\sigma _m\) remains consistent.

13.2: \((s_m\# t_m)\sigma _m\) is \(\beta \text {} undefined \) in \(\Gamma \) and irreducible by \(conv(\Gamma )\). It remains to show that \(C_m\sigma _m\) is \(\beta \text {} false \) in \(\Gamma \), \(N \cup U \models s_m\# t_m\vee C_m\) and \((s_m\# t_m\vee C_m)\sigma _m\prec _T \beta \). By i.h. for all \(L'\sigma '^{l:(L'\vee C')\cdot \sigma '}\in \Gamma \) it holds that \(C'\sigma '\) is \(\beta \text {} false \) in \(\Gamma \), \((L'\vee C')\sigma '\prec _T \beta \) and \(N\cup U \models (L'\vee C')\). By definition of Propagate \(C_0\sigma \) is \(\beta \text {} false \) in \(\Gamma \) and \(C\sigma \prec _T \beta \) and \(N\cup U \models C\). \((C_0\vee C_1\vee L)\mu \) is an instance of C. Thus \(C \models (C_0\vee C_1\vee L)\mu \). \(C_0\mu = L\mu \vee ...\vee L\mu \) by definition of Propagate. Thus \(C \models (C_1\vee L)\mu \) and by this \(N\cup U \models (C_1\vee L)\mu \). By definition of a reduction chain application \(I_j\) either contains a clause annotation from \(\Gamma ,L\sigma ^{k:(C_0\vee L)\cdot \sigma }\) or it is a rewriting inference from smaller rewrite steps for all \(1\le j\le m\). Thus, by Lemma 3 it follows by induction that for any rewriting inference \(I_j {:}{=} (s_j\# t_j\cdot \sigma _j, s_j\# t_j\vee C_j\cdot \sigma _j, I_i, I_k, p_j)\) it holds \(C_j\sigma _j\) is \(\beta \text {} false \) in \(\Gamma \), \(N \cup U \models s_j\# t_j\vee C_j\) and \((s_j\# t_j\vee C_j)\sigma _j\prec _T \beta \).

13.5: trivially holds since \(D = \top \).
Decide Assume Decide is applicable. Then there exists \(C\in N\cup U\) such that \(C=C_0\vee L\), \(L\sigma \) is ground and \(\beta \text {} undefined \) in \(\Gamma \) and \(C_0\sigma \) is ground and \(\beta \text {} undefined \) or \(\beta \text {} true \) in \(\Gamma \). Then a reduction chain application \([I_1,...,I_m]\) from \(\Gamma \) to \(L\sigma ^{k+1:(C_0\vee L)\cdot \sigma }\) is created with \(I_m {:}{=} (s_m\# t_m\cdot \sigma _m, s_m\# t_m\vee C_m\cdot \sigma _m, I_j, I_k, p_m)\). Finally \(s_m\# t_m\sigma _m^{k+1:(s_m\# t_m\vee comp(s_m\# t_m))\cdot \sigma _m}\) is added to \(\Gamma \).
By definition of a reduction chain application \((s_m\# t_m)\sigma _m = L\sigma {\downarrow _{conv(\Gamma )}}\). Thus, \((s_m\# t_m)\sigma _m\) must be \(\beta \text {} undefined \) in \(\Gamma \) and irreducible by \(conv(\Gamma )\), since \((C_0\vee L)\sigma \prec _T \beta \) by definition of Decide.

13.1: Since \((s_m\,\#\, t_m)\sigma _m\) is \(\beta \text {} undefined \) in \(\Gamma \) adding \((s_m\# t_m)\sigma _m\) does not make \(\Gamma \) inconsistent. Thus \(\Gamma ,(s_m\# t_m)\sigma _m\) remains consistent.

13.3: \((s_m\# t_m)\sigma _m\) is \(\beta \text {} undefined \) in \(\Gamma \) and irreducible by \(conv(\Gamma )\). \(N\cup U\models (s_m\# t_m)\vee comp(s_m\# t_m)\) obviously holds. \((s_m\# t_m)\sigma _m\prec _T \beta \) holds inductively by Lemma 3 and since \(L\sigma \prec _T \beta \).

13.5: trivially holds since \(D = \top \).
Conflict Assume \( Conflict \) is applicable. Then there exists a \(D'\sigma \) such that \(D'\sigma \) is \(\beta \text {} false \) in \(\Gamma \). Then:

13.5: \(D'\sigma \) is \(\beta \text {} false \) in \(\Gamma \) by definition of \( Conflict \). Now we have two cases:

1.
\(D'\sigma \) is of level greater than zero. Then \(N \cup U \models D'\) since \(D'\in N\cup U\) by definition of \( Conflict \).

2.
\(D'\sigma \) is of level zero. Then we have to show that \(N \cup U \models \bot \). For any literal \(L_0^{0:(L_0\vee D_0)\cdot \sigma }\in \Gamma \) it holds \(N\models L_0\), since any literal of level 0 is a propagated literal. By definition of a level, for any \(K\in D'\sigma \) there exists a core \(core(\Gamma ;K)\) that contains only literals of level 0. Thus \(N\cup U\models core(\Gamma ;K)\) and \(core(\Gamma ;K)\models \lnot K\) for any such K. Then \(N\cup U\models \lnot D'\sigma \) and \(N\cup U\models D'\sigma \) and therefore \(N\cup U\models \bot \).

1.
Skip Assume \( Skip \) is applicable. Then \(\Gamma = \Gamma ', L\) and \(D = D'\cdot \sigma \) and \(D'\sigma \) is \(\beta \text {} false \) in \(\Gamma '\).

13.1: By i.h. \(\Gamma \) is consistent. Thus \(\Gamma '\) is consistent as well.

13.2–13.4: trivially hold by induction hypothesis and since \(\Gamma '\) is a prefix of \(\Gamma \).

13.5: By i.h. \(D'\sigma \) is \(\beta \text {} false \) in \(\Gamma \) and \(N\cup U\models D'\). By definition of \( Skip \) \(D'\sigma \) is \(\beta \text {} false \) in \(\Gamma '\).
ExploreRefutation Assume \( Explore\text {}Refutation \) is applicable. Then \(D=(D'\vee s\,\#\,t)\cdot \sigma \), \((s\,\#\,t)\sigma \) is strictly \(\prec _{\Gamma ^*}\) maximal in \((D'\vee s\,\#\,t)\sigma \), \([I_1,...,I_m]\) is a refutation from \(\Gamma \) and \((s\,\#\,t)\sigma \), \(I_j = (s_j\# t_j\cdot \sigma _j, (s_j\# t_j\vee C_j)\cdot \sigma _j, I_l, I_k, p_j)\), \(1\le j\le m\), \( (s_j\,\#\,t_j\vee C_j)\sigma _j \prec _{\Gamma ^*} (D'\vee s\,\#\,t)\sigma \), \((s_j\# t_j\vee C_j)\sigma _j\) is \(\beta \text {} false \) in \(\Gamma \).

13.5: By definition \((C_j\vee s_j\,\#\,t_j)\sigma _j\) is \(\beta \text {} false \) in \(\Gamma \). By i.h. for all \(L'\sigma '^{l:(L'\vee C')\cdot \sigma '}\) \(\in \Gamma \) it holds that \(N\cup U \models (L'\vee C')\). By i.h. \(N\cup U \models D'\vee s\,\#\,t\). By definition of a refutation \(I_j {:}{=} (s_j\# t_j\cdot \sigma _j, s_j\# t_j\vee C_j\cdot \sigma _j, I_i, I_k, p_j)\) either contains a clause annotation from \(\Gamma ,(s\,\#\,t)\sigma ^{k:(D'\vee s\,\#\,t)\cdot \sigma }\) or it is a rewriting inference from smaller rewrite steps for all \(1\le j\le m\). Thus it follows inductively by Lemma 3 that \(N\cup U \models (s_j\# t_j\vee C_j)\).
Factorize Assume \( Factorize \) is applicable. Then \(D=D'\cdot \sigma \).

13.5: By i.h. \(D'\sigma \) is \(\beta \text {} false \) in \(\Gamma \) and \(N\cup U\models D'\). By the definition of \( Factorize \) \(D' = D_0 \vee L \vee L'\) such that \(L\sigma =L'\sigma \) and \(\mu =mgu(L,L')\). \((D_0\vee L\vee L')\mu \) is an instance of \(D'\). Thus \(N \cup U\models (D_0 \vee L\vee L')\mu \). Since \(L\mu = L'\mu \), \((D_0\vee L\vee L')\mu \models (D_0\vee L)\mu \). Thus \(N \cup U\models (D_0 \vee L)\mu \) and \((D_0 \vee L)\mu \sigma \) is \(\beta \text {} false \) since \((D_0\vee L)\mu \sigma = (D_0\vee L)\sigma \) by definition of an mgu.
EqualityResolution Assume \( Equality\text {}Resolution \) is applicable. Then \(D= (D'\vee s \not \approx s')\sigma \) and \(s\sigma =s'\sigma \), \(\mu =mgu(s,s')\). Then

13.5: By i.h. \((D'\vee s\not \approx s')\sigma \) is \(\beta \text {} false \) in \(\Gamma \) and \(N\cup U\models (D'\vee s \not \approx s')\). \(D'\mu \) is an instance of \((D'\vee s \not \approx s')\). Thus \((D'\vee s \not \approx s')\models D'\mu \). Thus \(N\cup U\models D'\mu \). \(D'\mu \sigma \) is \(\beta \text {} false \) since \((D'\vee s\not \approx s')\sigma \) is \(\beta \text {} false \) and \(D'\mu \sigma = D'\sigma \) by definition of a mgu.
Backtrack Assume Backtrack is applicable. Then \(\Gamma = \Gamma ', K, \Gamma ''\) and \(D=(D'\vee L)\sigma \), where \(L\sigma \) is of level k, and \(D'\sigma \) is of level i.

13.1: By i.h. \(\Gamma \) is consistent. Thus \(\Gamma ' \subseteq \Gamma \) is consistent.

13.2–13.3: Since \(\Gamma '\) is a prefix of \(\Gamma \) by i.h. this holds.

13.4: By i.h. \(N\cup U \models D'\vee L\) and \(N\models U\). Thus \(N\models U \cup \{D'\vee L\}\).

13.5: trivially holds since \(D = \top \) after backtracking.
\(\square \)
Proof of Lemma 7
Assume a state \((\Gamma ;N;U;\beta ;k;D)\) resulting from a regular run where the current level \(k>0\) and a unit clause \(l\approx r\in N\). Now assume a clause \(C\vee L[l']_p \in N\) such that \(l' = l\mu \) for some matcher \(\mu \). Now assume some arbitrary grounding substitutions \(\sigma '\) for \(C\vee L[l']_p\), \(\sigma \) for \(l\approx r\) such that \(l\sigma =l'\sigma '\) and \(r\sigma \prec _T l\sigma \). Then \( (C\vee L[r\mu \sigma \sigma ']_p)\sigma '\prec _{\Gamma ^*} (C\vee L[l']_p)\sigma '\).
Proof
Let \((\Gamma ;N;U;\beta ;k;D)\) be a state resulting from a regular run where \(k> 0\) and \(\Gamma = [L_1,...,L_n]\). Now we have two cases:

1.
\(\beta \prec _T (l\approx r)\sigma \). Since \((l\approx r)\sigma \) rewrites \(L[l']_p\sigma '\), \(\beta \prec _T L[l']_p\sigma '\) has to hold as well. Thus \((l\approx r)\sigma \) is \(\beta \text {} undefined \) in \(\Gamma \) and \(L[l']_p\sigma '\) is \(\beta \text {} undefined \) in \(\Gamma \). By definition of a trail induced ordering \(\prec _{\Gamma ^*} {:}{=} \prec _T\) for \(\beta \text {} undefined \) literals. Thus, in case that \(L[r\mu ]_p)\sigma \sigma '\) is still undefined, \((L[r\mu ]_p)\sigma \sigma ' \prec _{\Gamma ^*} (L[l']_p)\sigma '\) has to hold since \((L[r\mu ]_p)\sigma \sigma ' \prec _T (L[l']_p)\sigma '\). Thus, according to the definition of multiset orderings, \( (C\vee L[r\mu ]_p)\sigma \sigma ' \prec _{\Gamma ^*} (C\vee L[l']_p)\sigma '\). In the case that \((L[r\mu ]_p)\sigma \sigma '\) is defined, \((L[r\mu ]_p)\sigma \sigma ' \prec _{\Gamma ^*} (L[l']_p)\sigma '\) has to hold as well by Definition 8.11. Thus, according to the definition of multiset orderings, \( (C\vee L[r\mu ]_p)\sigma \sigma ' \prec _{\Gamma ^*} (C\vee L[l']_p)\sigma '\).

2.
\((l\approx r)\sigma \prec _T \beta \). Since propagation is exhaustive for literals of level 0 (cf. 15.2) \((l\approx r)\sigma \) is on the trail or defined and of level 0. Now we have two cases:

(a)
\((L[l']_p)\sigma '\) is of level 1 or higher. Since \((L[l']_p)\sigma '\) is reducible by \((l\approx r)\sigma \), \((L[l']_p)\sigma ' \ne {L_i}\) and \((L[l']_p)\sigma ' \ne {comp(L_i)}\) for all \(L_i\in \Gamma \). Since \((L[l']_p)\sigma '\) is of level 1 or higher, rewriting with \((l\approx r)\sigma \) does not change the defining literal of \((L[l']_p)\sigma '\). Thus \((L[r\mu ]_p)\sigma \sigma ' \prec _{\Gamma ^*} (L[l']_p)\sigma '\) has to hold since \((L[r\mu ]_p)\sigma \sigma ' \prec _T (L[l']_p)\sigma '\). Thus, according to the definition of multiset orderings, \((C\vee L[r\mu ]_p)\sigma \sigma ' \prec _{\Gamma ^*} (C\vee L[l']_p)\sigma '\).

(b)
\((L[l']_p)\sigma '\) is of level 0. First we show that \((L[r\mu ]_p)\sigma \sigma '\) is still of level 0. Suppose that \((L[l']_p)\sigma ' = s\,\#\, s\). Then rewriting either the left or right side of the equation results in \((L[r\mu ]_p)\sigma \sigma '\). Then \(core(\Gamma ; (l\approx r)\sigma )\) is also a core for \((L[r\mu ]_p)\sigma \sigma '\) and thus \((L[r\mu ]_p)\sigma \sigma '\) must be of level 0. Now suppose that \((L[r\mu ]_p)\sigma \sigma ' = s\,\#\, s\). Then it is of level 0 by definition of a level. Finally suppose that \((L[r\mu ]_p)\sigma \sigma ' \not = s\,\#\, s\) and \((L[l']_p)\sigma ' \not = s\,\#\, s\). Then \(core(\Gamma ;(L[l']_p)\sigma ')\cup core(\Gamma ; (l\approx r)\sigma )\) is a core for \((L[r\mu ]_p)\sigma \sigma '\). Thus \((L[r\mu ]_p)\sigma \sigma '\) is of level 0. Since \((L[r\mu ]_p)\sigma \sigma ' \prec _T (L[l']_p)\sigma '\), \((L[r\mu ]_p)\sigma \sigma ' \prec _{\Gamma ^*} (L[l']_p)\sigma '\) according to the definition of \(\prec _{\Gamma ^*}\). Thus, according to the definition of multiset orderings, \((C\vee L[r\mu ]_p)\sigma \sigma ' \prec _{\Gamma ^*} (C\vee L[l']_p)\sigma '\).

(a)
\(\square \)
Proof of Lemma 8
Let C, D be two clauses. If there exists a substitution \(\sigma \) such that \(C\sigma \subset D\), then D is redundant with respect to C and any \(\prec _{\Gamma ^*}\).
Proof
Let \(\tau \) be a grounding substitution for D. Since \(C\sigma \subset D\), \(C\sigma \tau \subset D\tau \). Thus, for any \(L\in C\sigma \tau \) it holds \(L\in D\tau \) and \(C\sigma \tau \not = D\tau \). Thus, \(C\sigma \tau \prec _{\Gamma ^*}D\tau \) by definition of a multiset extension and \(C\sigma \tau \) makes \(D\tau \) redundant by Definition 1. \(\square \)
1.2 Auxiliary Lemmas for the Proof of Lemma 6
Lemma 16
During a regular run, if \((\Gamma ;N;U;\beta ;k;\top )\) is the immediate result of an application of Backtrack, then there exists no clause \(C\in N\cup U\) and no substitution \(\sigma \) such that \(C\sigma \) is \(\beta \text {} false \) in \(\Gamma \).
Proof
We prove this by induction. For the induction start assume the state \((\Gamma ';N;U\cup \{D\};\beta ;i;\top )\) after the first application of Backtrack in a regular run, where D is the learned clause. Since Backtrack was not applied before, the previous (first) application of \( Conflict \) in a state \((\Gamma ,K;N;U;\beta ;k;\top )\) was immediately preceded by an application of Propagate or Decide. By the definition of a regular run there is no clause \(C\in N\) with substitution \(\sigma \) such that \(C\sigma \) is \(\beta \text {} false \) in \(\Gamma \). Otherwise \( Conflict \) would have been applied earlier. By the definition of Backtrack, there exists no substition \(\tau \) such that \(D\tau \) is \(\beta \text {} false \) in \(\Gamma '\). Since there existed such a substitution before the application of Backtrack, \(\Gamma '\) has to be a prefix of \(\Gamma \) and \(\Gamma \not =\Gamma '\). Thus there exists no clause \(C\in N\cup U\cup \{D\}\) and a grounding substitution \(\delta \) such that \(C\delta \) is \(\beta \text {} false \) in \(\Gamma '\).
For the induction step assume the state \((\Gamma ';N;U\cup \{D\};\beta ;i;\top )\) after nth application of Backtrack. By i.h. the previous application of Backtrack did not produce any \(\beta \text {} false \) clause. It follows that the the previous application of \( Conflict \) in a state \((\Gamma ,K;N;U;\beta ;k;\top )\) was immediately preceded by an application of Propagate or Decide. By the definition of a regular run there is no clause \(C\in N\cup U\) with substitution \(\sigma \) such that \(C\sigma \) is \(\beta \text {} false \) in \(\Gamma \). Otherwise \( Conflict \) would have been applied earlier. By the definition of Backtrack, there exists no substition \(\tau \) such that \(D\tau \) is \(\beta \text {} false \) in \(\Gamma '\). Since there existed such a substitution before the application of Backtrack, \(\Gamma '\) has to be a prefix of \(\Gamma \) and \(\Gamma \not =\Gamma '\). Thus there exists no clause \(C\in N\cup U\cup \{D\}\) and a grounding substitution \(\delta \) such that \(C\delta \) is \(\beta \text {} false \) in \(\Gamma '\). \(\square \)
Corollary 2
If \( Conflict \) is applied in a regular run, then it is immediately preceded by an application of Propagate or Decide, except if it is applied to the initial state.
Lemma 17
Assume a state \((\Gamma ;N;U;\beta ;k;D)\) resulting from a regular run. Then there exists no clause \((C\vee L)\in N\cup U\) and no grounding substitution \(\sigma \) such that \((C\vee L)\sigma \) is \(\beta \text {} false \) in \(\Gamma \), \(comp(L\sigma )\) is a decision literal of level i in \(\Gamma \) and \(C\sigma \) is of level \(j<i\).
Proof
Proof is by induction. Assume the initial state \((\epsilon ;N;\emptyset ;\beta ;0;\top )\). Then any clause \(C\in N\) is undefined in \(\Gamma \). Then this trivially holds.
Now for the induction step assume a state \((\Gamma ;N;U;\beta ;k;D)\). Only Propagate, Decide, Backtrack and Skip change the trail and only Backtrack adds a new literal to U. By i.h. there exists no clause with the above properties in \(N\cup U\).
Now assume that Propagate is applied. Then a literal L is added to the trail. Let \(C_1\vee L_1,...,C_n\vee L_n\) be the ground clause instances that get \(\beta \text {} false \) in \(\Gamma \) by the application such that L is the defining literal of \(L_1,...,L_n\). Then \(L_i\) is of level k for \(1\le i\le n\). Thus \(L_i\not =comp(K)\) for the decision literal \(K\in \Gamma \) of level k. Thus \(C_1\vee L_1,...,C_n\vee L_n\) do not have the above properties.
Now assume that Decide is applied. Then a literal L of level \(k+1\) is added to the trail. Let \(C_1\vee L_1,...,C_n\vee L_n\) be the (ground) clause instances that get \(\beta \text {} false \) in \(\Gamma \) by the application such that L is the defining literal of \(L_1,...,L_n\). By the definition of a regular run for all \(L_i\) with \(1\le i\le n\) it holds that \(L_i\not =comp(L)\) or there exists another literal \(K_i\in C_i\) such that \(K_i\) is of level \(k+1\) and \(L_i\not =K_i\), since otherwise Propagate must be applied. Thus \(C_1\vee L_1,...,C_n\vee L_n\) do not have the above properties.
Now assume that Skip is applied. Then there are no new clauses that get \(\beta \text {} false \) in \(\Gamma \). Thus this trivially holds.
Now assume that Backtrack is applied. Then a new clause \(D\vee L\) is added to U and \(\Gamma =\Gamma ',K,\Gamma ''\) such that there is a grounding substitution \(\tau \) with \((D\vee L)\tau \) \(\beta \text {} false \) in \(\Gamma ',K\), there is no grounding substitution \(\delta \) with \((D\vee L)\delta \) \(\beta \text {} false \) in \(\Gamma '\). \(\Gamma '\) is the trail resulting from the application of Backtrack. By Lemma 16, after application of Backtrack there exists no clause \(C\in N\cup U\) and a substitution \(\sigma \) such that \(C\sigma \) is \(\beta \text {} false \) in \(\Gamma '\). Thus there exists no clause with the above properties. \(\square \)
Proof of Lemma 6
Let N be a clause set. The clauses learned during a regular run in SCL(EQ) are not redundant with respect to \(\prec _{\Gamma ^*}\) and \(N \cup U\). For the trail only nonredundant clauses need to be considered.
We first prove that learned clauses are nonredundant and then that only nonredundant clauses need to be considered, Lemma 21, below.
Proof
Consider the following fragment of a derivation learning a clause:
Assume there are clauses in \(N' \subseteq (gnd(N\cup U)^{\preceq _{\Gamma ^*} C\sigma })\) such that \(N' \models C\sigma \). Since \(N'\preceq _{\Gamma ^*} C\sigma \) and \(C\sigma \) is \(\beta \text {} defined \) in \(\Gamma \), there is no \(\beta \text {} undefined \) literal in \(N'\), as all \(\beta \text {} undefined \) literals are greater than all \(\beta \text {} defined \) literals. If \(\Gamma \models N'\) then \(\Gamma \models C\sigma \), a contradiction. Thus there is a \(C'\in N'\) with \(C'\preceq _{\Gamma ^*} C\sigma \) such that \(C'\) is \(\beta \text {} false \) in \(\Gamma \). Now we have two cases:

1.
\(\Gamma ' \not = \Gamma \). Then \(\Gamma = \Gamma ',\Delta \). Thus at least one Skip was applied, so \(C\sigma \) does not contain a literal that is \(\beta \text {} undefined \) without the rightmost literal of \(\Gamma \), therefore \(C\sigma \not =D\sigma \). Suppose that this is not the case, so \(C\sigma =D\sigma \). Then \(D\sigma \) is \(\beta \text {} false \) in \(\Gamma '\). But since Backtrack does not produce any \(\beta \text {} false \) clauses by Lemma 16, \( Conflict \) could have been applied earlier on \(D\sigma \) contradicting a regular run. Since \(C' \preceq _{\Gamma ^*} C\sigma \) we have that \(C'\not =D\sigma \) as well. Thus, again since Backtrack does not produce any \(\beta \text {} false \) clauses by Lemma 16, at a previous point in the derivation there must have been a state such that \(C'\) was \(\beta \text {} false \) under the current trail and \( Conflict \) was applicable but not applied, a contradiction to the definition of a regular run.

2.
\(\Gamma ' = \Gamma \), then conflict was applied immediately after an application of \( Decide \) by Corollary 2 and the definition of a regular run. Thus \(\Gamma = \Delta ,K^{(k1):D'\cdot \delta },L^{k:D''\cdot \tau }\). \(C'\) does not have any \(\beta \text {} undefined \) literals. Suppose that \(C'\) has no literals of level k. Then all literals in \(C'\) are of level \(i<k\). Since \(C'\) is \(\beta \text {} false \) in \(\Gamma \), \(C'\) is \(\beta \text {} false \) in \(\Delta ,K\) as well, since it does not have any literals of level k. Thus, again since Backtrack does not produce any \(\beta \text {} false \) clauses by Lemma 16, at a previous point in the derivation there must have been a state such that \(C'\) was \(\beta \text {} false \) under the current trail and \( Conflict \) was applicable but not applied, a contradiction to the definition of a regular run.
Since \(C'\preceq _{\Gamma ^*} C\sigma \), it may have at most one literal of level k, namely comp(L), since \(comp(L) \in C\sigma \) by definition of a regular run, since Skip was not applied, and there exists only L such that \(L \prec _{\Gamma ^*} comp(L)\) and L is of level k. But L is \(\beta \text {} true \) in \(\Gamma \). Thus \(L\not \in C'\) has to hold.
Now suppose that \(C'\) has one literal of level k. Thus \(C' = C''\vee comp(L)\), where \(C''\) is \(\beta \text {} false \) in \(\Delta ,K\). But by Lemma 17 there does not exist such a clause. Contradiction.
\(\square \)
1.3 Auxiliary Lemma for the Proof of Lemma 19
Lemma 18
Assume a clause \(L_1\vee ...\vee L_m\), a trail \(\Gamma \) resulting from a regular run starting from the initial state, and a reducible (by \(conv(\Gamma )\)) grounding substitution \(\sigma \), such that \(L_i\sigma \) is \(\beta \text {} false \) (\(\beta \text {} true \) or \(\beta \text {} undefined \)) in \(\Gamma \) and \(L_i\sigma \prec _T \beta \) for \(1\le i \le m\). Then there exists a substitution \(\sigma '\) that is irreducible by \(conv(\Gamma )\) such that \(L_i\sigma '\) is \(\beta \text {} false \) (\(\beta \text {} true \) or \(\beta \text {} undefined \)) in \(\Gamma \), \(L_i\sigma '\prec _T \beta \) and \(L_i\sigma {\downarrow _{conv(\Gamma )}} = L_i\sigma '{\downarrow _{conv(\Gamma )}}\).
Proof
Let \(L_1\vee ...\vee L_m\) be a clause, \(\Gamma \) a trail resulting from a regular run. Let \(\sigma {:}{=} \{x_1\rightarrow t_1,..., x_n\rightarrow t_n\}\). Now set \(\sigma ' {:}{=} \{x_1\rightarrow ({t_1}{\downarrow _{conv(\Gamma )}}),..., x_n\rightarrow ({t_n}{\downarrow _{conv(\Gamma )}})\}\). Obviously \(\sigma '\) is irreducible by \(conv(\Gamma )\) and \(L_i\sigma '\prec _T \beta \) for all \(1\le i \le m\). By definition, \(conv(\Gamma )\) is a confluent and terminating rewrite system. Since \(\Gamma \) is consistent, \({t_j}{\downarrow _{conv(\Gamma )}}\approx t_j\) is \(\beta \text {} true \) in \(\Gamma \) for \(1\le j \le n\). Thus there exists a chain such that \(L_i\sigma \rightarrow _{conv(\Gamma )} \cdots \rightarrow _{conv(\Gamma )}L_i\sigma '\) and \(L_i\sigma '\) is \(\beta \text {} false \) (\(\beta \text {} true \) or \(\beta \text {} undefined \)) in \(\Gamma \). Now there also exists a chain \(L_i\sigma \rightarrow _{conv(\Gamma )} \cdots \rightarrow _{conv(\Gamma )}L_i\sigma {\downarrow _{conv(\Gamma )}}\). By definition of convergence there must exist a chain \(L_i\sigma '\rightarrow _{conv(\Gamma )} \cdots \rightarrow _{conv(\Gamma )}L_i\sigma {\downarrow _{conv(\Gamma )}}\). Thus \(L_i\sigma {\downarrow _{conv(\Gamma )}} = L_i\sigma '{\downarrow _{conv(\Gamma )}}\). \(\square \)
1.4 Auxiliary Lemma for the Proof of Lemma 9 and Theorem 2
Lemma 19
Suppose a sound state \((\Gamma ; N;U;\beta ;k; \top )\) resulting from a regular run. If there exists a \(C\in N\cup U\) and a grounding substitution \(\sigma \) such that \(C\sigma \) is \(\beta \text {} false \) in \(\Gamma \), then \( Conflict \) is applicable. Otherwise, if there exists a \(C\in N\cup U\) and a grounding substitution \(\sigma \) such that \(C\sigma \prec _T \beta \) and there exists at least one \(L\in C\) such that \(L\sigma \) is \(\beta \text {} undefined \), then one of the rules Propagate or Decide is applicable and a \(\beta \text {} undefined \) literal \(K\in D\), where \(D\in gnd_{\prec _T}\beta (N\cup U)\) is \(\beta \text {} defined \) after application.
Proof
Let \((\Gamma ; N;U;\beta ;k; \top )\) be a state resulting from a regular run. Suppose there exists a \(C\in N\cup U\) and a grounding \(\sigma \) such that \(C\sigma \) is \(\beta \text {} false \) in \(\Gamma \), then by Lemma 18 there exists an irreducible substitution \(\sigma '\) such that \(C\sigma '\) is \(\beta \text {} false \). Thus \( Conflict \) is applicable. Now suppose there exists a \(C\in N\cup U\) and a grounding substitution \(\sigma \) such that \(C\sigma \prec _T \beta \) and there exists at least one \(L\in C\) such that \(L\sigma \) is \(\beta \text {} undefined \). By Lemma 18 there exists a irreducible substitution \(\sigma '\) such that \(L\sigma '\) is \(\beta \text {} undefined \). Now assume that \(C = C_0\vee C_1\vee L\) such that \(C_1\sigma ' = L\sigma '\vee ...\vee L\sigma '\) and \(C_0\sigma '\) is \(\beta \text {} false \) in \(\Gamma \). Then \( Propagate \) is applicable. Let \(C_1 = L_1,...,L_n\) and \(\mu = mgu(L_1,...,L_n,L)\). Now let \([I_1,...,I_m]\) be the reduction chain application from \(\Gamma \) to \(L\sigma '^{k:(L\vee C_0)\mu \cdot \sigma '}\). Let \(I_m = (s_m\# t_m\cdot \sigma _m, (s_m\# t_m\vee C_m)\cdot \sigma _m, I_j, I_k, p_m)\). Then \(L\sigma '{\downarrow _{conv(\Gamma )}} = s_m\# t_m\sigma _m\) by definition of a reduction chain application. Thus \(L\sigma '\) is \(\beta \text {} true \) in \(\Gamma ,s_m\# t_m\sigma _m\). Since \(L\sigma {\downarrow _{conv(\Gamma )}} = L\sigma '{\downarrow _{conv(\Gamma )}}\) by Lemma 18, \(L\sigma \) is \(\beta \text {} true \) in \(\Gamma ,s_m\# t_m\sigma _m\) as well. If \(C_0\sigma \) is \(\beta \text {} undefined \) or \(\beta \text {} true \) in \(\Gamma \) then Propagate is not applicable to \(C\sigma '\). If Decide is not applicable by definition of a regular run, then there exists a clause \(C'\in (N\cup U)\) and a substitution \(\delta \) such that Propagate is applicable. Then we can apply Propagate by definition of a regular run and a previously undefined literal gets defined after application as seen above and we are done. Now suppose that there exists no such clause. Then let \([I'_1,...,I'_l]\) be the reduction chain application from \(\Gamma \) to \(L\sigma '^{k+1:C\cdot \sigma '}\) and \(I'_l = (s_l\# t_l\cdot \sigma _l, (s_l\# t_l\vee C_l)\cdot \sigma _l, I'_j, I'_k, p_l)\). Then \(L\sigma '{\downarrow _{conv(\Gamma )}} = (s_l\# t_l)\sigma _l\) by definition of a reduction chain application. Thus \(L\sigma '\) is \(\beta \text {} true \) in \(\Gamma ,(s_l\# t_l)\sigma _l\). Since \(L\sigma {\downarrow _{conv(\Gamma )}} = L\sigma '{\downarrow _{conv(\Gamma )}}\) by lemma 18, \(L\sigma \) is \(\beta \text {} true \) in \(\Gamma ,(s_l\# t_l)\sigma _l\) as well. \((s_l\# t_l)\sigma _l^{k+1:(s_l\# t_l\vee comp(s_l\# t_l))\cdot \sigma _l}\) can be added to \(\Gamma \) by definition of a regular run and also by definition of Decide since \(C\in N\cup U\), \(\sigma '\) is grounding for C and irreducible in \(conv(\Gamma )\), \(L\sigma '\) is \(\beta \text {} undefined \) in \(\Gamma \) and \(C\sigma '\prec _T \beta \). \(\square \)
1.5 Auxiliary Lemma for the Proof of Lemma 9
Lemma 20
Suppose a sound state \((\Gamma ; N;U;\beta ;k; D\cdot \sigma )\) resulting from a regular run. Then \(D\sigma \) is of level 1 or higher.
Proof
Let \((\Gamma ; N;U;\beta ;k; D\cdot \sigma )\) be a state resulting from a regular run. Suppose that \(D\sigma \) is not of level 1 or higher, thus \(D\sigma \) is of level 0. Then \( Conflict \) was applied earlier to a clause that was of level 1 or higher. Thus there must have been an application of \( Explore\text {}Refutation \) on a state \((\Gamma ,\Gamma ',L; N;U;\beta ; l; D'\cdot \sigma ')\) between the state after the application of \( Conflict \) and the current state resulting in a state \((\Gamma ,\Gamma ',L^{l:(L\vee C)\cdot \delta }; N;U;\beta ; l; D''\cdot \sigma '')\) such that \(D'\sigma '\) is of level l and \(D''\sigma ''\) is of level 0, since no other rule can reduce the level of \(D'\sigma '\). Then there exists a \(K\in D'\sigma '\) such that L is the defining literal of K. Let \([I_1,...,I_m]\) be the refutation of K and \(I_j = (s_j\# t_j\cdot \sigma _j, (s_j\# t_j\vee C_j)\cdot \sigma _j, I_i, I_k, p_j)\) be the step that was chosen by \( Explore\text {}Refutation \). Then \(D''\sigma '' =(s_j\# t_j\vee C_j)\sigma _j\). \(C\delta \subset C_j\sigma _j\) has to hold since L is the defining literal of K. Then \(C\delta \) must be of level 0 or empty. Note that \(C\delta \) is of level l if L is a decision literal. But then, by the definition of a regular run, \(L^{l:(L\vee C)\cdot \delta }\) must have been propagated before the first decision, since propagation is exhaustive at level 0. Contradiction. \(\square \)
Proof of Lemma 9
If a regular run (without rule Grow) ends in a stuck state \((\Gamma ;N;U;\beta ;k;D)\), then \(D=\top \) and all ground literals \(L\sigma \prec _T \beta \), where \(L\vee C\in N\cup U\) are \(\beta \text {} defined \) in \(\Gamma \).
Proof
First we prove that stuck states never appear during conflict resolution. Assume a sound state \((\Gamma ;N;U;\beta ;k;D\cdot \sigma )\) resulting from a regular run. Now we show that we can always apply a rule. Suppose that \(D\sigma = (D'\vee L\vee L')\sigma \) such that \(L\sigma = L'\sigma \). Then we must apply \( Factorize \) by the definition of a regular run. Now suppose that \( Factorize \) is not applicable and \(\Gamma {:} {=} \Gamma ', L\) and \(D\sigma \) is false in \(\Gamma '\). If \(D\sigma = (D'\vee s\not \approx s')\sigma \) such that \(s\sigma = s'\sigma \), we can apply \( Equality\text {}Resolution \). So suppose that \( Equality\text {}Resolution \) is not applicable. Then we can apply Skip. Now suppose that \(\Gamma {:} {=} \Gamma ', L^{k:(L\vee C)\delta }\) and L is the defining literal of at least one literal in \(D\sigma \), so Skip is not applicable. If \(D\sigma = (D'\vee L')\sigma \) where \(D'\sigma \) is of level \(i<k\) and \(L'\sigma \) is of level k and Skip was applied at least once during this conflict resolution, then Backtrack is applicable. If Skip was not applied and \(L=comp(L'\sigma )\) and L is a decision literal, then Backtrack is also applicable. Otherwise, let \((s\,\#\, t)\sigma \in D\sigma \) such that \(K \prec _{\Gamma ^*} (s\,\#\, t)\sigma \) for all \(K\in D\sigma \). \((s\,\#\, t)\sigma \) exists since Factorize is not applicable. By Lemma 20, \((s\,\#\, t)\sigma \) must be of level 1 or higher. By the definition of \(\prec _{\Gamma ^*}\), L must be the defining literal of \((s\,\#\, t)\sigma \) since L is of level 1 or higher and any literal in \(D\sigma \) that has another defining literal is smaller than \((s\,\#\, t)\sigma \). Now suppose that L is a decision literal and \((s\,\#\, t)\sigma = comp(L)\). Then \((s\,\#\, t)\sigma \) is of level k and all other literals \(K\in D\sigma \) are of level \(i<k\), since \((s\,\#\, t)\sigma \) is the smallest \(\beta \text {} false \) literal of level k and Factorize is not applicable. In this case ExploreRefutation is not applicable since a paramodulation step with the decision literal does not make the conflict clause smaller. But Backtrack is applicable in this case even if Skip was not applied earlier by the definition of a regular run. Thus \((s\,\#\, t)\sigma \not =comp(L)\) or L is a propagated literal has to hold. We show that in this case ExploreRefutation is applicable. Let \([I_1,...,I_m]\) be a refutation of \((s\,\#\,t)\sigma \) from \(\Gamma \), \(I_m = (s_m\# t_m\cdot \sigma _m, (s_m\# t_m\vee C_m)\cdot \sigma _m, I_j, I_k, p_m)\). Since \([I_1,...,I_m]\) is a refutation \(s_m\# t_m\sigma _m = s'\not \approx s'\). Furthermore any \(I_i\) either contains a clause annotation from \(\Gamma ,(s\,\#\, t)\sigma ^{k:D\cdot \sigma }\) or it is a rewrite inference from \(I_{j'},I_{k'}\) with \(j',k'<i\). Thus by Lemma 3 it inductively follows that \(C_m\sigma _m = D'\sigma _m\vee ...\vee D'\sigma _m \vee C'_1\sigma _m\vee ... \vee C'_n\sigma _m\), where \(C'_1\sigma _m,...,C'_n\sigma _m\) are clauses from \(\Gamma \) without the leading trail literal and \(D\sigma =D'\sigma _m\vee (s\# t)\sigma \). Since L is the defining literal of \((s\,\#\, t)\sigma \) there must exist at least one \(C'_i\) such that \(C'_i\sigma _m=C\delta \). If L is a propagated literal, then any literal in \(C'_i\sigma _m\) is smaller than \((s\,\#\, t)\sigma \), since they are already \( false \) in \(\Gamma '\). If L is a decision literal, then \(C'_i\sigma _m =comp(L)\). Then comp(L) is smaller, since \((s\,\#\, t)\sigma \not =comp(L)\) and \((s\,\#\, t)\sigma \not = L\). Thus \(comp(L)\prec _{\Gamma ^*}(s\,\#\, t)\sigma \). Any other literal in \(C_1\sigma _m,...,C'_n\sigma _m\) is smaller in \(\prec _{\Gamma ^*}\), since they are already defined in \(\Gamma '\). Since Factorize is not applicable \((s\,\#\, t)\sigma \) is also strictly maximal in \(D'\sigma _m\). Thus \((s_m\# t_m\vee C_m)\sigma _m\prec _{\Gamma ^*} D\sigma \) which makes \( Explore\text {}Refutation \) applicable.
Now by Lemma 19 it holds that if there exists an \(\beta \text {} undefined \) literal in \(gnd_{\prec _T \beta }(N\cup U)\), we can always apply at least one of the rules \( Propagate \) or Decide which makes a previously \(\beta \text {} undefined \) literal in \(gnd_{\prec _T \beta }(N\cup U)\) \(\beta \text {} defined \). \(\square \)
Proof of Lemma 10
Suppose a sound state \((\Gamma ; N;U;\beta ;k; D)\) resulting from a regular run where \(D\not \in \{\top ,\bot \}\). If Backtrack is not applicable then any set of applications of \( Explore\text {}Refutation \), Skip, \( Factorize \), \( Equality\text {}Resolution \) will finally result in a sound state \((\Gamma '; N;U;\beta ;k;D')\), where \(D' \prec _{\Gamma ^*} D\). Then Backtrack will be finally applicable.
Proof
Assume a sound state \((\Gamma ;N;U;\beta ;k;D\cdot \sigma )\) resulting from a regular run. Let \((s\,\#\, t)\sigma \in D\sigma \) such that \(L \preceq _{\Gamma ^*} (s\,\#\, t)\sigma \) for all \(L\in D\sigma \). If \((s\,\#\, t)\sigma \) occurs twice in \(D\sigma \), then Factorize is applicable. Suppose that it is applied. Then \(D\sigma = (D' \vee (s\,\#\, t) \vee L)\sigma \), where \(L\sigma = (s\,\#\,t)\sigma \). Then \(\mu = mgu(s\,\#\,t, L)\) and the new conflict clause is \((D' \vee s\,\#\, t)\mu \sigma \prec _{\Gamma ^*} D\sigma \). Thus in this case we are done. If Factorize is not applicable, then the only remaining applicable rules are Skip, \( Explore\text {}Refutation \) and \( Equality\text {}Resolution \). If \(\Gamma =\Gamma ',L,\Gamma ''\) where L is the defining literal of \((s\,\#\, t)\sigma \), then Skip is applicable \(\vert \Gamma ''\vert \) times, since otherwise \((s\,\#\, t)\sigma \) would not be maximal in \(D\sigma \). So at some point it is no longer applicable. Since \(D\sigma \) is finite, \( Equality\text {}Resolution \) can be applied only finitely often. Thus we finally have to apply \( Explore\text {}Refutation \). Then \([I_1,...,I_m]\) is a refutation of \((s\,\#\,t)\sigma \) from \(\Gamma \), and there exists an \(1\le j\le m\), such that \(I_j = (s_j\# t_j\cdot \sigma _j, (s_j\# t_j\vee C_j)\cdot \sigma _j, I_l, I_k, p_j)\), \( (C_j\vee s_j\,\#\,t_j)\sigma _j \prec _{\Gamma ^*} (D'\vee s\,\#\,t)\sigma \). Otherwise \( Explore\text {}Refutation \) would not be applicable, contradicting Lemma 9. Thus in this case we are done.
Now we show that Backtrack is finally applicable. Since \(\prec _{\Gamma ^*}\) is wellfounded and \(\Gamma \) is finite there must be a state where \( Explore\text {}Refutation \), Skip, \( Factorize \), \( Equality\text {}Resolution \) are no longer applicable. By Lemma 20 the conflict clause in this state must be of level 1 or higher, thus \(\bot \) cannot be inferred. Suppose that it is always of level \(i\ge l\) for some l. The smallest literal of level l that is \( false \) in \(\Gamma \) is comp(L), where L is the decision literal of level l. Since we can always reduce if \( Backtrack \) is not applicable and since we can always apply a rule by Lemma 9, we must finally reach a conflict clause \(comp(L)\vee C\), where C is of level \(j<l\). Thus \( Backtrack \) is applicable. \(\square \)
Proof of Lemma 11
Let N be a set of clauses and \(\beta \) be a ground term. Then any regular run that never uses Grow terminates.
Proof
Assume a new ground clause \(D\sigma \) is learned. By Lemma 6 all learned clauses are nonredundant. Thus \(D\sigma \) is nonredundant. By the definition of a regular run \( Factorize \) has precedence over all other rules. Thus \(D\sigma \) does not contain any duplicate literals. By Theorem 1, \(D\sigma \prec _T\beta \) has to hold. There are only finitely many clauses \(C\sigma \prec _T\beta \), where \(C\sigma \) is neither a tautology nor does it contain any duplicate literals. Thus there are only finitely many clauses \(D\sigma \) that can be learned. Thus there are only finitely many literals that can be decided or propagated. \(\square \)
Proof of Lemma 12
If a regular run reaches the state \((\Gamma ;N;U;\beta ;k;\bot )\) then N is unsatisfiable.
Proof
By definition of soundness, all learned clauses are consequences of \(N\cup U\), Definition 13.5, and \(\Gamma \) is satisfiable, Definition 13.1. \(\square \)
Proof of Theorem 2
Let N be an unsatisfiable clause set, and \(\prec _T\) a desired term ordering. For any ground term \(\beta \) where \(gnd_{\prec _T\beta }(N)\) is unsatisfiable, any regular \({\text {SCL(EQ)}}\) run without rule Grow will terminate by deriving \(\bot \).
Proof
Since regular runs of \({\text {SCL(EQ)}}\) terminate we just need to prove that it terminates in a failure state. Assume by contradiction that we terminate in a state \((\Gamma ;N;U;\beta ;k;\top )\). If no rule can be applied in \(\Gamma \) then for all \(s\,\#\, t\in C\) for some arbitrary \(C\in gnd_{\prec _T \beta }(N)\) it holds that \(s \,\#\, t\) is \(\beta \text {} defined \) in \(\Gamma \) (otherwise Propagate or Decide woud be applicable, see Lemma 19) and there aren’t any clauses in \(gnd_{\prec _T \beta }(N)\) \(\beta \text {} false \) under \(\Gamma \) (otherwise \( Conflict \) would be applicable, see again Lemma 19). Thus, for each \(C\in gnd_{\prec _T \beta }(N)\) it holds that C is \(\beta \text {} true \) in \(\Gamma \). So we have \(\Gamma \models gnd_{\prec _T \beta }(N)\), but by hypothesis there is a superposition refutation of N that only uses ground literals from \(gnd_{\prec _T \beta }(N)\), so also \(gnd_{\prec _T \beta }(N)\) is unsatisfiable, a contradiction. \(\square \)
Lemma 21
(Only NonRedundant Clauses Building the Trail) Let \(\Gamma =[L_1^{i_1:C_1\cdot \sigma _1},...,L_n^{i_n:C_n\cdot \sigma _n}]\) be a trail. If \(L_j^{i_j:C_j\cdot \sigma _j}\) is a propagated literal and there exist clauses \(\{D_1\vee K_1,...,D_m\vee K_m\}\) with grounding substitutions \(\delta _1,...,\delta _m\) such that \(N {:}{=} \{(D_1\vee K_1)\delta _1,...,(D_m\vee K_m)\delta _m\} \prec _{\Gamma ^*} C_j\sigma _j\) and \(\{(D_1\vee K_1)\delta _1,...,(D_m\vee K_m)\delta _m\}\models C_j\sigma _j\), then there exists a \((D_k\vee K_k)\delta _k\in N\) such that
is a trail.
Proof
Let \(N = \{(D_1\vee K_1)\delta _1,...,(D_m\vee K_m)\delta _m\}\) and \(L_j^{i_j:C_j\cdot \sigma _j}\) be as above. Let \(\Gamma ' = [L_1^{i_1:C_1\cdot \sigma _1},...,L_{j1}^{i_{j1}:C_{j1}\cdot \sigma _{j1}}]\). Now suppose that for every literal \(L\in N\) it holds \(L \prec _{\Gamma ^*} L_j\). Then every literal in N is defined in \(\Gamma '\) and \(\Gamma ' \models N\), otherwise \( Conflict \) would have been applied to a clause in N. Thus \(\Gamma '\models C_j\sigma _j\) would have to hold as well. But by definition of a trail \(L_j\) is undefined in \(\Gamma '\). Thus there must be at least one clause \((D_k\vee K_k)\delta _k\in N\) with \(K_k = L_j\) and \(D_k\delta _k \prec _{\Gamma ^*} L_j\) (otherwise \((D_k\vee K_k)\delta _k \not \prec _{\Gamma ^*} C_j\sigma _j\)), such that \(\Gamma ' \not \models D_k\). Suppose that \(\Gamma ' \models D_k\). Then \(N\not \models C_j\sigma _j\), since there exists an allocation, namely \(\Gamma ',\lnot L_k\) such that \(\Gamma ',\lnot L_k \models N\) but \(\Gamma ',\lnot L_k \not \models C_j\sigma _j\). Thus we can replace \(L_j^{i_j:C_j\cdot \sigma _j}\) by \(K_k^{i_j:(D_k\vee K_k)\cdot \delta _k}\) in \(\Gamma \). \(\square \)
Further Examples
For the following examples we assume a term ordering \(\prec _{kbo}\), unique weight 1 and with precedence \(d\prec c\prec b\prec a \prec a_1 \prec \cdots \prec a_n \prec g \prec h \prec f\). Further assume \(\beta \) to be large enough.
Example 6
(Implicit Conflict after Decision) Consider the following clause set N
Suppose we apply the rule Decide first to \(C_1\) and then to \(C_2\) with substitution \(\sigma = \{x\rightarrow a\}\). Then we yield a conflict with \(C_3\sigma \), resulting in the following state:
According to \(\prec _{\Gamma ^*}\), \(f(a) \not \approx h(a)\) is the greatest literal in \(C_3\sigma \). Since \(f(a) \approx g(a)\) is the defining literal of \(f(a)\not \approx h(a)\) we can not apply Skip. Factorize is also not applicable, since \(f(a)\not \approx h(a)\) and \(f(a) \not \approx g(a)\) are not equal. Thus we must apply ExploreRefutation to the greatest literal \(f(a) \not \approx h(a)\). The rule first creates a refutation \([I_1,...,I_5]\), where:
ExploreRefutation can now choose either \(I_4\) or \(I_5\). Both, \((h(x)\not \approx g(x))\sigma \) and \((g(x)\not \approx g(x))\sigma \) are smaller than \((f(x)\not \approx h(x))\sigma \) according to \(\prec _{\Gamma ^*}\) and false in \(\Gamma \). Suppose we choose \(I_5\). Now our new conflict state is:
Now we apply \( Equality\text {}Resolution \) and Factorize to get the new state
Now we can backtrack. Note, that this clause is nonredundant according to our ordering, although conflict was applied immediately after decision.
Example 7
(Undefined conflict clause) Consider the following ground clause set N:
Suppose that we decide \(f(a,a)\not \approx f(b,b)\). Then \(C_2\) is false in \(\Gamma \). Conflict state is as follows: \(([f(a,a)\not \approx f(b,b)^{1:f(a,a)\not \approx f(b,b)\vee f(a,a)\approx f(b,b)}]; N; \{\}; 2; C_2)\). ExploreRefutation creates the following ground refutation for \(a\approx b\), since it is greatest literal in the conflict clause:
As one can see, the intermediate result \(f(b,a)\not \approx f(b,b)\) is not false in \(\Gamma \). Thus it is no candidate for the new conflict clause. We have to choose \(I_4\). The new state is thus:
Now we can apply EqualityResolution and two times Factorize to get the final clause \(f(a,a)\approx f(b,b)\) with which we can backtrack.
Example 8
(SCL(EQ) vs. Superposition: Clause learning) Assume clauses:
The completeness proof of superposition requires that adding a new literal to an interpretation does not make any smaller literal true. In this example, however, after adding \(b\approx c\) to the interpretation, we cannot any further literal, since it breaks this invariant. So in superposition we would have to add the following clauses with the help of the EqualityFactoring rule:
In SCL(EQ) on the other hand we can just decide a literal in each clause to get a model for this clause set. As we support undefined literals we do not have to bother with this problem at all. For example if we add \(b\approx c\) to our model, both literals \(a_1\approx b\) and \(a_1\approx c\) are undefined in our model. Thus we need to decide one of these literals to add it to our model.
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Leidinger, H., Weidenbach, C. SCL(EQ): SCL for FirstOrder Logic with Equality. J Autom Reasoning 67, 22 (2023). https://doi.org/10.1007/s10817023096733
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DOI: https://doi.org/10.1007/s10817023096733