Abstract
A clause C is syntactically relevant in some clause set N, if it occurs in every refutation of N. A clause C is syntactically semirelevant, if it occurs in some refutation of N. While syntactic relevance coincides with satisfiability (if C is syntactically relevant then \(N\setminus \{C\}\) is satisfiable), the semantic counterpart for syntactic semirelevance was not known so far. Using the new notion of a conflict literal we show that for independent clause sets N a clause C is syntactically semirelevant in the clause set N if and only if it adds to the number of conflict literals in N. A clause set is independent, if no clause out of the clause set is the consequence of different clauses from the clause set.
Furthermore, we relate the notion of relevance to that of a minimally unsatisfiable subset (MUS) of some independent clause set N. In propositional logic, a clause C is relevant if it occurs in all MUSes of some clause set N and semirelevant if it occurs in some MUS. For firstorder logic the characterization needs to be refined with respect to ground instances of N and C.
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1 Introduction
In our previous work [11], we introduced a notion of syntactic relevance based on refutations while at the same time generalized the completeness result for resolution by the setofsupport strategy (SOS) [28, 33] as its test. Our notion of syntactic relevance is useful for explaining why a set of clauses is unsatisfiable. In this paper, we introduce a semantic counterpart of syntactic relevance that sheds further light on the relationship between a clause out of a clause set and the potential refutations of this clause set. In the following Sect. 1.1, we first recall syntactic relevance along with an example and then proceeds to explain it in terms of our new semantic relevance in the later Sect. 1.2.
1.1 Syntactic Relevance
Given an unsatisfiable set of clauses N, \(C\in N\) is syntactically relevant if it occurs in all refutations, it is syntactically semirelevant if it occurs in some refutation, otherwise it is called syntactically irrelevant. The clausebased notion of relevance is useful in relating the contribution of a clause to refutation (goal conjecture). This has in particular been shown in the context of product scenarios built out of construction kits as they are used in the car industry [8, 32].
For an illustration of our privous notions and results we now consider the following unsatisfiable firstorder clause set N where Fig. 1 presents a refutation of N.
In essence, inferences in an SOS refutation always involve at least one clause in the SOS and put the resulting clause back in it. So, this refutation is not an SOS refutation from the syntactically semirelevant clause \((3) \lnot B(c{,}a)\vee B(b{,}f(x_6))\), because only the shaded part represents an SOS refutation starting with this clause. More specifically, there are two inferences ended in \((8) \lnot B(b{,}f(a))\) which violates the condition for an SOS refutaiton. Nevertheless, it can be transformed into an SOS refutation where the clause \((3) \lnot B(c{,}a)\vee B(b{,}f(x_6))\) is in the SOS [11], Fig. 2. Please note that \(N\setminus \{(3) \lnot B(c,a) \vee B(b,f(x_6))\}\) is still unsatisfiable and classical SOS completeness [33] is not sufficient to guarantee the existence of a refutation with SOS \(\{(3) \lnot B(c{,}a)\vee B(b{,}f(x_6))\}\) [11].
In addition, \(N\setminus \{(3) \lnot B(c,a) \vee B(b,f(x_6))\}\) is also a minimally unsatisfiable subset (MUS), where Fig. 3 presents a respective refutation. A MUS is an unsatisfiable clause set such that removing a clause from this set would render it satisfiable. Consequently, a MUSbased defined notion of semirelevance on the level of the original firstorder clauses is not sufficient here. The clause \((3) \lnot B(c,a) \vee B(b,f(x_6))\) should not be disregarded, because it leads to a different grounding of the clauses. For example, in the refutation of Fig. 2 clause \((5) \lnot C(x_5)\) is necessarily instantiated with \(\{x_5\mapsto c\}\) where in the refutation of Fig. 3 it is necessarily instantiated with \(\{x_5\mapsto b\}\). Therefore, the two refutations are different and clause \((3) \lnot B(c,a) \vee B(b,f(x_6))\) should be considered semirelevant. Nevertheless, in propositional logic it is sufficient to consider MUSes to explain unsatisfiability on the original clause level, Lemma 18.
1.2 Semantic Relevance
We now illustrate how our new notion of relevance works on the previous example. First, different from the other works, we propose a way of characterizing semantic relevance by using our novel concept of a conflict literal. A ground literal L is a conflict literal in a clause set N if there are some satisfiable sets of instances \(N_1\) and \(N_2\) from N s.t. \(N_1\models L\) and \(N_2\models {\text {comp}}(L)\). On the one hand, explaining an unsatisfiable clause set as the absence of a model (as it is usually defined) is not that helpful since an absence means there is nothing to discuss in the first place. On the other hand, the contribution of a clause to unsatisfiability of a clause set can only partially be explained using the concept of a MUS which we have discussed before. A conflict literal provides a middle ground to explain the contribution of a clause to unsatisfiability between the absence of a model and MUSes. It also better reflects our intuition that there is a contradiction (in the form of two implied simple facts that cannot be both true at the same time) in an unsatisfiable set of clauses.
From Fig. 1, we can already see that C(c) and its complement \(\lnot C(c)\) are conflict literals because
Also, in addition to that \(\{\lnot C(x)\}\) is trivially satisfiable, \(N\setminus \{\lnot C(x)\}\) is also satisfiable. Based on the refutation in Fig. 3, \(\lnot C(x)\) is syntactically relevant due to \(N\setminus \{(3) \lnot B(c,a) \vee B(b,f(x_6))\}\) being a MUS. We will also show that for a ground MUS any ground literal occurring in it is a conflict literal, Lemma 20. For our ongoing example it is still possible to identify the conflict literals by means of ground MUSes by looking into the refutations from Fig. 1 and Fig. 3. This leads to the following conflict literals for N, see Definition 10:
These conflict literals can be identified by pushing the substitutions in the refutations from Fig. 1 and Fig. 3 towards the input clauses. They correspond to two firstorder MUSes \(M_1\) and \(M_2\). All the ground literals are conflict literals and all other ground conflict literals can be obtained by grounding the remaining variables.
One can see that, despite \((3) \lnot B(c,a) \vee B(b,f(x_6))\) is outside of the only MUS on the firstorder level, an instance of it does occur in some ground MUS, take \(M_1\) and an arbitrary grounding of \(x_3\) and \(x_7\) to the identical term t, and the conflict literal \((\lnot )B(c,a)\) depends on clause (3). Nevertheless, determining conflict literals is not so obvious in the general case since we do not necessarily know beforehand which ground terms should substitute the variables in the clauses. Moreover, there can be an infinite number of such ground MUSes of possibly unbounded size.
Based on conflict literals, here we introduce a notion of relevance that is semantic in nature, Definition 16. This will also serve as an alternative characterization to our previous refutationbased syntactic relevance. As redundant clauses, e.g., tautologies, can also be syntactically semirelevant, we require independent clause sets for the definition of semantic relevance. A clause set is independent, if it does not contain clauses with instances implied by satisfiable sets of instances of different clauses out of the set. Given an unsatisfiable independent set of clauses N, a clause C is relevant in N if N without C has no conflict literals, it is semirelevant if C is necessary to some conflict literals, and it is irrelevant otherwise.
Similar to our previous work, relevant clauses are the obvious ones because removing them would make our set satisfiable. On the other hand, irrelevant clauses can be freely identified once we know the semirelevant ones. For our running example, in fact \((3) \lnot B(c,a) \vee B(b,f(x_6))\) is semirelevant because it is necessary for the conflict literals \((\lnot )C(c)\) and \((\lnot ) B(c,a)\). More specifically, the set of conflicts for \(N\setminus \{\lnot B(c,a)\vee B(b,f(x_6))\}\) does not include \((\lnot )C(c)\) and \((\lnot ) B(c,a)\):
These are conflict literals identifiable from \(M_2\): Assume that the variables \(x_3\) and \(x_7\) in \(M_2\) are both grounded by an identical term t. Take some ground literal, for example, \(A(f(a))\in {\text {conflict}}(N\setminus \{\lnot B(c,a)\vee B(b,f(x_6)))\), and define
\(N_\emptyset \cup N_A(f(a))\) and \(N_\emptyset \cup N_{\lnot A(f(a))}\) are satisfiable because of the Herbrand model \(\{B(b,f(a)),A(f(a)) \}\) and \(\{B(b,f(a))\}\) respectively. In addition,
because A(f(a)) can be acquired using resolution between (1) and (2) for \(N_\emptyset \cup N_{A(f(a))}\) and \(\lnot A(f(a))\) can be acquired using resolution between (4) and (6) for \(N_\emptyset \cup N_{\lnot A(f(a))}\). In a similar manner, we can show that the other ground literals are also conflict literals.
Related Work: Other works which aim to explain unsatisfiability mostly rely on the notion of MUSes, mainly in propositional logic [14,15,16, 21, 26]. The complexity of determining whether a clause set is a MUS is \(D^p\)complete for a propositional clause set with at most three literals per clause and at most three occurrences of each propositional variable [25]. In [14], syntactically semirelevant clauses for propositional logic are called a plain clause set. Using the terminology in [16], a clause \(C\in N\) is necessary if it occurs in all MUSes, it is potentially necessary if it occurs in some MUS, otherwise, it is never necessary. In addition, a clause is defined to be usable if it occurs in some refutation. This is thus similar to our syntactic notion of semirelevance [11]: Given a clause \(C\in N\), C is usable ifandonlyif C is syntactically semirelevant. It is also argued that a usable clause that is not potentially necessary is semantically superfluous. A different but related notion has also been applied for propositional abduction [7]. The notion of a MUS has also been used for explaining unsatisfiability in firstorder logic [20]. There, it has been defined in a more general setting: If a set of clauses N is divided into \(N=N'\uplus N''\) with a nonrelaxable clause set \(N'\) and relaxable clause set \(N''\) (which must be satisfiable), a MUS is a subset M of \(N''\) s.t. \(N'\uplus M\) is unsatisfiable but removing a clause from M would render it satisfiable. There are also some works in satisfiability modulo theory (SMT) [5, 6, 9, 35]. A deletionbased approach wellknown in propositional logic has also been used for MUS extraction in SMT [9]. In [5, 6], a MUS is extracted by combining an SMT solver with an arbitrary external propositional core extractor. Another approach is to construct some graph representing the subformulas of the problem instance, recursively remove clauses in a depthfirstsearch manner and additionally use some heuristics to further improve the runtime[35]. For the functionfree and equalityfree firstorder fragment, there is a "decomposemerge" approach to compute all MUSes [19, 34]. In description logic, a notion that is related to MUS is called minimal axiom set (MinA) usually identified by the problem of axiom pinpointing [1, 4, 13, 30]. Its computation is usually divided into two categories: blackbox and whitebox. A blackbox approach picks some inputs, executes it using some sound and complete reasoner, and then interprets the output [13]. On the other hand, whitebox approach takes some reasoner and performs an internal modification for it. In this case, Tableau is mostly used [1, 30]. In addition, the concept of a lean kernel has also been used to approximate the union of such MinA’s [27]. The way relevance is defined is similar in spirit but usually used for an entailment problem instead of unsatisfiability. The notion of syntactic semirelevance has also been applied to description logics via a translation scheme to firstorder logic [10].
The paper is organized as follows. Section 2 fixes the notations, definitions and existing results in particular from [11]. Section 3 is reserved for our new notion of semantic relevance. Finally, we conclude our work in Sect. 4 with a discussion of our results.
2 Preliminaries
We assume a standard firstorder language without equality over a signature \(\varSigma =(\varOmega ,\varPi )\) where \(\varOmega \) is a nonempty set of functions symbols, \(\varPi \) a nonempty set of predicate symbols both coming with their respective fixed arities denoted by the function \({\text {arity}}\). The set of terms over an infinite set of variables \(\mathcal {X}\) is denoted by \(T(\varSigma , \mathcal {X})\). Atoms, literals, clauses, and clause sets are defined as usual, e.g., see [24]. We identify a clause with its multiset of literals. Variables in clauses are universally quantified. Then N denotes a clause set; C, D denote clauses; L, K denote literals; A, B denote atoms; P, Q, R, T denote predicates; t, s terms; f, g, h functions; a, b, c, d constants; and x, y, z variables, all possibly indexed. The complement of a literal is denoted by the function \({\text {comp}}\). Atoms, literals, clauses, and clause sets are ground if they do not contain any variable.
An interpretation \(\mathcal {I}\) with a nonempty domain (or universe) \(\mathcal {U}\) assigns (i) a total function \(f^\mathcal {I}:{\mathcal {U}}^n\mapsto \mathcal {U}\) for each \(f\in \varOmega \) with \({\text {arity}}(f)=n\) and (ii) a relation \(P\subseteq \mathcal {U}^m\) to every predicate symbol \(P^\mathcal {I}\in \varPi \) with \({\text {arity}}(P)=m\). A valuation \(\beta \) is a function \(\mathcal {X}\mapsto \mathcal {U}\) where the assignment of some variable x can be modified to \(e\in \mathcal {U}\) by \(\beta [x\mapsto e]\). It is extended to terms as \(\mathcal {I}(\beta ):T (\varSigma , \mathcal {X}) \mapsto \mathcal {U}\). Semantic entailment \(\models \) considers variables in clauses to be universally quantified. The extension to atoms, literals, disjunctions, clauses and sets of clauses is as follows: \(\mathcal {I}(\beta )(P (t_1 , \ldots , t_n ))=1\) if \((\mathcal {I}(\beta )(t_1 ), \ldots , \mathcal {I}(\beta )(t_n )) \in P^\mathcal {I}\) and 0 otherwise; \(\mathcal {I}(\beta )(\lnot \phi )=1\mathcal {I}(\beta )(\phi )\); for a disjunction \(L_1\vee \ldots \vee L_k\), \(\mathcal {I}(\beta )(L_1\vee \ldots \vee L_k)=\max (\mathcal {I}(\beta )(L_1),\ldots ,\mathcal {I}(\beta )(L_k))\); for a clause C, \(\mathcal {I}(\beta )(C) = 1\) if for all valuations \(\beta = \{x_1\mapsto e_1,\ldots , x_n\mapsto e_n\}\) where the \(x_i\) are the free variables in C there is a literal \(L\in C\) such that \(\mathcal {I}(\beta )(L)=1\); for a set of clauses \(N=\{C_1,\ldots ,C_k\}\), \(\mathcal {I}(\beta )(\{C_1,\ldots , C_k\})=\min (\mathcal {I}(\beta )(C_1),\ldots ,\mathcal {I}(\beta )(C_k))\). A set of clauses N is satisfiable if there is an \(\mathcal {I}\) of N such that \(\mathcal {I}(\beta )(N)=1\), \(\beta \) arbitrary, (in this case \(\mathcal {I}\) is called a model of N: \(\mathcal {I}\models N\)) otherwise N is called unsatisfiable.
Substitutions \(\sigma , \tau \) are total mappings from variables to terms, where \({\text {dom}}(\sigma ) := \{x \mid x\sigma \ne x\}\) is finite and \({\text {codom}}(\sigma ) := \{ t\mid x\sigma = t, x\in {\text {dom}}(\sigma )\}\). A renaming \(\sigma \) is a bijective substitution. The application of substitutions is extended to literals, clauses, and sets/sequences of such objects in the usual way. If \(C'=C\sigma \) for some substitution \(\sigma \), then \(C'\) is an instance of C. A unifier \(\sigma \) for a set of terms \(t_1,\dots ,t_k\) satisfies \(t_i\sigma =t_j\sigma \) for all \(1\le i,j\le k\) and it is called a most general unifier if for any unifier \(\sigma '\) of \(t_1,\dots ,t_k\) there is a substitution \(\tau \) s.t. \(\sigma '=\sigma \tau \). The function \({\text {mgu}}\) denotes the most general unifier of two terms, atoms, literals if it exists. We assume that any \({\text {mgu}}\) of two terms or literals does not introduce any fresh variables and is idempotent.
The resolution calculus consists of two inference rules: Resolution and Factoring [28, 29]. The rules operate on a state (N, S) where the initial state for a classical resolution refutation from a clause set N is \((\emptyset ,N)\) and for an SOS (Set Of Support) refutation with clause set N and initial SOS clause set S the initial state is (N, S). We describe the rules in the form of abstract rewrite rules operating on states (N, S). As usual we assume for the resolution rule that the involved clauses are variable disjoint. This can always be achieved by applying renamings into fresh variables.
Resolution \((N,S\uplus \{C\vee K\}) \Rightarrow _{\text {RES}} (N,S\cup \{C\vee K, (D\vee C)\sigma \})\)
provided \((D\vee L)\in (N\cup S)\) and \(\sigma ={\text {mgu}}(L,{\text {comp}}(K))\)
Factoring \((N,S\uplus \{C\vee L\vee K\})\) \(\Rightarrow _{\text {RES}}\) \((N,S\cup \{C\vee L\vee K\}\cup \{(C\vee L)\sigma \})\)
provided \(\sigma ={\text {mgu}}(L,K)\)
The clause \((D\vee C)\sigma \) is the result of a Resolution inference between its parents and called a resolvent. The clause \((C\vee L)\sigma \) is the result of a Factoring inference of its parent and called a factor. A sequence of rule applications \((N,S)\Rightarrow _{RES}^*(N,S')\) is called a resolution derivation. It is called an SOS resolution derivation if \(N\ne \emptyset \). In case \(\bot \in S'\) it is a called a (SOS) resolution refutation. If for two clauses C, D there exists a substitution \(\sigma \) such that \(C\sigma \subseteq D\), then we say that C subsumes D. In this case \(C\models D\).
Theorem 1
(Soundness and Refutational Completeness of (SOS) Resolution [11, 28, 33]). Resolution is sound and refutationally complete [28]. If for some clause set N and initial SOS S, N is satisfiable and \(N\cup S\) is unsatisfiable, then there is a (SOS) resolution derivation of \(\bot \) from (N, S) [33]. If for some clause set N and clause \(C\in N\) there exists a resolution refutation from N using C, then there is an SOS derivation of \(\bot \) from \((N\setminus \{C\},\{C\})\) [11].
Please note that the recent SOS completeness result of [11] generalizes the classical SOS completeness result by [33].
Theorem 2
(Deductive Completeness of Resolution [17, 22]). Given a set of clauses N and a clause D, if \(N\models D\), then there is a resolution derivation of some clause C from \((\emptyset ,N)\) such that C subsumes D.
For deductions we require every clause to be used exactly once, so deductions always have a tree form.
Definition 3
(Deduction [11]). A deduction \(\pi _{N}=[C_1,\ldots ,C_n]\) of a clause \(C_n\) from some clause set N is a finite sequence of clauses such that for each \(C_i\) the following holds:

1.1
\(C_i\) is a renamed, variablefresh version of a clause in N, or

1.2
there is a clause \(C_j\in \pi _{N}\), \(j<i\) s.t. \(C_i\) is the result of a Factoring inference from \(C_j\), or

1.3
there are clauses \(C_j,C_k\in \pi _{N}\), \(j<k<i\) s.t. \(C_i\) is the result of a Resolution inference from \(C_j\) and \(C_k\),
and for each \(C_i\in \pi _N\), \(i<n\):

2.1
there exists exactly one factor \(C_j\) of \(C_i\) with \(j>i\), or

2.2
there exists exactly one \(C_j\) and \(C_k\) such that \(C_k\) is a resolvent of \(C_i\) and \(C_j\) and \(i,j<k\).
We omit the subscript N in \(\pi _N\) if the context is clear.
A deduction \(\pi '\) of some clause \(C\in \pi \), where \(\pi \), \(\pi '\) are deductions from N is a subdeduction of \(\pi \) if \(\pi '\subseteq \pi \), where the subset relation is overloaded for sequences. A deduction \(\pi _{N}=[C_1,\ldots ,C_{n1},\bot ]\) is called a refutation. While the conditions 3.1.1, 3.1.2, and 3.1.3 are sufficient to represent a resolution derivation, the conditions 3.2.1 and 3.2.2 force deductions to be minimal with respect to \(C_n\).
Note that variable renamings are only applied to clauses from N such that all clauses from N that are introduced in the deduction are variable disjoint. Also recall that our notion of a deduction implies a tree structure. Both assumptions together admit the existence of overall grounding substitutions for a deduction.
Definition 4
(Overall Substitution of a Deduction [11]). Given a deduction \(\pi \) of a clause \(C_n\) the overall substitution \(\tau _{\pi ,i}\) of \(C_i\in \pi \) is recursively defined by

1
if \(C_i\) is a factor of \(C_j\) with \(j<i\) and mgu \(\sigma \), then \(\tau _{\pi ,i}=\tau _{\pi ,j}\circ \sigma \),

2
if \(C_i\) is a resolvent of \(C_j\) and \(C_k\) with \(j<k<i\) and mgu \(\sigma \), then \(\tau _{\pi ,i}=(\tau _{\pi ,j}\circ \tau _{\pi ,k})\circ \sigma \),

3
if \(C_i\) is an initial clause, then \(\tau _{\pi ,i}=\emptyset \),
and the overall substitution of the deduction is \(\tau _\pi = \tau _{\pi ,n}\). We omit the subscript \(\pi \) if the context is clear.
Overall substitutions are welldefined because clauses introduced from N into the deduction are variable disjoint and each clause is used exactly once in the deduction. A grounding of an overall substitution \(\tau \) of some deduction \(\pi \) is a substitution \(\tau \delta \) such that \({\text {codom}}(\tau \delta )\) only contains ground terms and \({\text {dom}}(\delta )\) is exactly the variables from \({\text {codom}}(\tau )\).
Definition 5
(SOS Deduction [11]). A deduction \(\pi _{N\cup S}=[C_1,\ldots , C_n]\) is called an SOS deduction with SOS S, if the derivation \((N,S_0) \Rightarrow ^*_{\text {RES}} (N,S_m)\) is an SOS derivation where \(C'_1,\ldots ,C'_m\) is the subsequence from \([C_1,\ldots , C_n]\) with input clauses removed, \(S_0 = S\), and \(S_{i+1} = S_i \cup C'_{i+1}\).
Oftentimes, it is of particular interest to identify the set of clauses that is minimally unsatisfiable, i.e., removing a clause would make it satisfiable. The earliest mention of such a notion is in [26] where it is introduced via a decision problem. Minimally unsatisfiable sets (MUS) have also gained a lot of attention in practice.
Definition 6
(Minimal Unsatisfiable Subset (MUS) [20]). Given an unsatisfiable set of clauses N, the subset \(N'\subseteq N\) is a minimally unsatisfiable subset (MUS) of N if any strict subset of \(N'\) is satisfiable.
In our previous work, we defined a notion of relevance based on how clauses may contribute to unsatisfiability by means of refutations.
Definition 7
(Syntactic Relevance [11]). Given an unsatisfiable set of clauses N, a clause \(C\in N\) is syntactically relevant if for all refutations \(\pi \) of N it holds that \(C\in \pi \). A clause \(C\in N\) is syntactically semirelevant if there exists a refutation \(\pi \) of N in which \(C\in \pi \). A clause \(C\in N\) is syntactically irrelevant if there is no refutation \(\pi \) of N in which \(C\in \pi \).
Syntactic relevance can be identified by using the resolution calculus. A clause \(C\in N\) is syntactically semirelevant if and only if there exists an SOS refutation from SOS \(\{C\}\) and \(N\setminus \{C\}\).
Theorem 8
(Syntactic Relevance [11]). Given an unsatisfiable set of clauses N, the clause \(C\in N\) is

1.
syntactically relevant if and only if \(N\setminus \{C\}\) is satisfiable,

2.
syntactically semirelevant if and only if \((N\setminus \{C\},\{C\})\Rightarrow ^*_{\text {RES}} (N\setminus \{C\},S\cup \{\bot \})\).
An open problem from [11] is the question of a semantic counterpart to syntactic semirelevance. Without any further properties of the clause set N, the notion of semirelevance can lead to unintuitive results. For example, a tautology could be semirelevant. Given a refutation showing semirelevance of some clause C, where, in the refutation, some unary predicate P occurs, the refutation can be immediately extended using the tautology \(P(x) \vee \lnot P(x)\). We may additionally stumble upon a problem in the case where our set of clauses contains a subsumed clause. For example, if both clauses Q(a) and Q(x) exist in a clause set, they may be both semirelevant, although from an intuition point of view one may only want to consider Q(x) to be semirelevant, or even relevant. On the other hand, in some cases, redundant clauses are welcome as semirelevant clauses.
Example 9
(Redundant Clauses). Given a set of clauses
all clauses are syntactically semirelevant while \(\lnot Q(a)\vee P(b)\) and \(\lnot P(b)\) are syntactically relevant. However, if we disregard the redundant clauses Q(a) and \(P(x) \vee \lnot P(x)\), then the clause Q(x) becomes a relevant clause. Therefore, for our semantic notion of relevance we will only consider clause sets without clauses implied by other, different clauses from the clause set.
3 Semantic Relevance
Except for the trivially false clause \(\bot \), the simplest form of a contradiction is two unit clauses K and L such that K and \({\text {comp}}(L)\) are unifiable. They will be called conflict literals, below. Then the idea for our semantic definition of semirelevance is to consider clauses that contribute to the number of conflict literals of a clause set. Furthermore, we will show that in any MUS every literal is a conflict literal.
While conflict literals could straightforwardly be defined in propositional logic having the above idea in mind, in firstorder logic we have always to relate properties of literals, clauses to their respective ground instances. This is simply due to the fact that unsatisfiability of a firstorder clause set is given by unsatisfiability of a finite set of ground instances from this set. Eventually, we will show that for independent clause sets a clause is semirelevant, if it contributes to the number of conflict literals.
Definition 10
(Conflict Literal). Given a set of clauses N over some signature \(\varSigma \), a ground literal L is a conflict literal in a clause set N if there are two satisfiable clause sets \(N_1, N_2\) such that

1.
the clauses in \(N_1, N_2\) are instances of clauses from N and

2.
\(N_1 \models L\) and \(N_2 \models {\text {comp}}(L)\).
\({\text {conflict}}(N)\) denotes the set of conflict literals in N.
Our notion of a conflict literal generalizes the respective notion in [12] defined for propositional logic.
Example 11
(Conflict Literal). Given an unsatisfiable set of clauses over the signature \(\varSigma =(\{a, b, c,d,f\},\{P\})\):
Consider the following satisfiable sets of instances from N
P(f(a, b)) is a conflict literal because \(N_1\models P(f(a,b))\) and \(N_2\models \lnot P(f(a,b))\).
We can show that \(N_1\models P(f(a,b))\) because the resolution calculus is sound. Resolving both literals of \(\lnot P(f(a,d)) \vee \lnot P(f(c,y))\) with the first literal of the clause \(P(f(x,d)) \vee P(f(a,b))\) results in the clause \(P(f(a,b)) \vee P(f(a,b))\) which can be factorized to P(f(a, b)). Moreover, \(N_1\) is satisfiable: An interpretation \(\mathcal {I}\) with \(\mathcal {I}(P(f(a,b))) = 1\) and \(\mathcal {I}(P(t)) = 0\) for all terms \(t \ne f(a,b)\) satisfies \(N_1\) and P(f(a, b)). \(N_2\models \lnot P(f(a,b))\) can also be shown in the same manner.
Example 12
(Conflict Literal). Given
its conflict literals are
In addition to a refutation, the existence of a conflict literal is another way to characterize unsatisfiability of a clause set. Obviously, conflict literals always come in pairs.
Lemma 13
(Minimal Unsatisfiable Ground Clause Sets and Conflict Literals). If N is a minimally unsatisfiable set of ground clauses (MUS) then any literal occurring in N is a conflict literal.
Proof
Take any ground atom A such that A occurs in N. N can be split into three disjoint clause sets:
Since N is minimal, \(N_A\) and \(N_{\lnot A}\) are nonempty, because otherwise A is a pure literal and its corresponding clauses can be removed from N preserving unsatisfiability. Obviously \(N_\emptyset \cup N_A\) must be satisfiable, for otherwise the initial choice of N was not minimal. However, \(N_\emptyset \cup N'_A\), where \(N'_A\) results from all \(N_A\) by deleting all A literals from the clauses of \(N_A\), must be unsatisfiable, for otherwise we can construct a satisfying interpretation for N. Thus, every model of \(N_\emptyset \cup N_A\) must also be a model of A: \(N_\emptyset \cup N_A\models A\). Using the same argument, \(N_\emptyset \cup N_{\lnot A}\) is satisfiable and \(N_\emptyset \cup N_{\lnot A} \models \lnot A\). Therefore, A is a conflict literal. \(\square \)
Lemma 14
(Conflict Literals and Unsatisfiability). Given a set of clauses N, \({\text {conflict}}(N)\ne \emptyset \) if and only if N is unsatisfiable.
Proof
“\(\Rightarrow \)" Let \(L\in {\text {conflict}}(N)\). By definition, there are two satisfiable subsets of instances \(N_1,N_2\) from N such that \(N_1\models L\) and \(N_2\models {\text {comp}}(L)\). Towards contradiction, suppose N is satisfiable. Then, there exists an interpretation \(\mathcal {I}\) with \(\mathcal {I}\models N\) and therefore it holds that \(\mathcal {I}\models N_1\) and \(\mathcal {I}\models N_2\). Furthermore, by definition of a conflict literal, \(\mathcal {I}\models L\) and \(\mathcal {I}\models {\text {comp}}(L)\), a contradiction.
“\(\Leftarrow \)" Given an unsatisfiable clause set N, we show that there is a conflict literal in N. Since N is unsatisfiable, by compactness of firstorder logic there is a minimal set of ground instances \(N'\) from N that is also unsatisfiable. The rest follows from Lemma 13. \(\square \)
Intuitively, a clause that is implied by other clauses is redundant and can be removed from the set of clauses. However, then applying a calculus generating new clauses, this intuitive notion of redundancy may destroy completeness [2, 23]. Still, the detection and elimination of redundant clauses, compatible or incompatible with completeness, is an important concept to the efficiency of automatic reasoning, e.g., in propositional logic [3, 18]. It is also apparently important when we try to define a semantic notion of relevance. For example, a syntactically relevant clause would step down to be syntactically semirelevant if it is duplicated. So, in order to have a semantically robust notion of relevance in firstorder logic, we need to use a strong notion of (in)dependency.
Definition 15
(Dependency). A clause C is dependent in N if there exists a satisfiable set of instances \(N'\) from \(N\setminus \{C\}\) such that \(N'\models C\sigma \) for some \(\sigma \). If C is not dependent in N it is independent in N. A clause set N is independent if it does not contain any dependent clauses.
A subsumed clause is obviously a dependent clause. However, there could also be nonsubsumed clauses that are dependent. For example, in the set of clauses
P(x, b) is dependent because P(a, b) is an instance of P(x, b) and it is entailed by P(a, y). Now, we are ready to define the semantic notion of relevance based on conflict literals and dependency.
In some way, our notion of independence of clause sets is a strong assumption because there might be nonredundant clauses that are considered dependent. While this holds by design in some scenarios (e.g. the mentioned car scenario) in others it is violated by design. In addition, one question that may arise is how to acquire an independent clause set out of a dependent one. For example, in a scenario where some theory is developed out of some independent axioms. Then of course proven lemmas, theorems are dependent with respect to the axioms. In this case one could trace out of the proofs the dependency relations between the intermediate lemmas, theorems and the axioms and this way calculate independent clause sets with respect to some proven conjecture. This would then lead again to independent (sub) clause sets with respect to the proven conjecture where our results are applicable.
Definition 16
(Semantic Relevance). Given an unsatisfiable set of independent clauses N, a clause \(C\in N\) is

1.
relevant, if \({\text {conflict}}(N\setminus \{C\})=\emptyset \)

2.
semirelevant, if \({\text {conflict}}(N\setminus \{C\})\subsetneq {\text {conflict}}(N)\)

3.
irrelevant, if \({\text {conflict}}(N\setminus \{C\})= {\text {conflict}}(N)\)
Example 17
(Dependent Clauses in Propositional Logic).
The existence of dependent clauses \(\lnot P\vee Q\) and \(\lnot R\vee P\) causes an independent clause \(\lnot Q\vee R\) to be a semirelevant clause. However, \(\lnot Q\vee R\) is not inside the only MUS \(\{P,\lnot P\}\).
Very often, concepts from propositional logic can be generalized to firstorder logic. However, in the context of relevance this is not the case. Our notion of (semi)relevance can also be characterized by MUSes in propositional logic, but not in firstorder logic without considering instances of clauses.
Lemma 18
(Propositional Clause Sets and Relevance). Given an independent unsatisfiable set of propositional clauses N, the relevant clauses coincide with the intersection of all MUSes and the semirelevant clauses coincide with the union of all MUSes.
Proof
For the case of relevance: Given \(C\in N\), C is relevant if and only if \({\text {conflict}}(N\setminus \{C\})=\emptyset \) if and only if \(N\setminus \{C\}\) is satisfiable by Lemma 14 if and only if C is contained in all MUSes \(N'\) of N.
For the case of semirelevance: Given \(C\in N\), we show C is semirelevant if and only if C is in some MUS \(N'\subseteq N\).
“\(\Rightarrow \)”: Towards contradiction, suppose there is a semirelevant clause C that is not in any MUS. By definition of semirelevant clauses, there are satisfiable sets \(N_1\) and \(N_2\) and a propositional variable P such that \(N_1\models P\), \(N_2\models \lnot P\) but the MUS M out of \(N_1\cup N_2\) does not contain C. By Theorem 2 there exist deductions \(\pi _1\) and \(\pi _2\) of P and \(\lnot P\) from \(N_1\) and \(N_2\), respectively. Since a deduction is connected, some clauses in M and \((N_1\cup N_2)\setminus M\) must have some complementary propositional literals Q and \(\lnot Q\), respectively to be eventually resolved upon in either \(\pi _1\) or \(\pi _2\). At least one of these deductions must contain this resolution step between a clause from M and one from \((N_1\cup N_2)\setminus M\). Now by Lemma 13 the literals Q and \(\lnot Q\) are conflict literals in M. Thus, there are satisfiable subsets from M which entail Q and \(\lnot Q\), respectively. Therefore, the clause containing Q or \(\lnot Q\) in \((N_1\cup N_2)\setminus M\) is dependent contradicting the assumption that N does not contain dependent clauses.
“\(\Leftarrow \)": If C is in some MUS \(N'\subseteq N\), then, \(N'\setminus \{C\}\) is satisfiable. So invoking Lemma 13 any literal \(L\in C \) is a conflict literal in \(N'\). In addition, L is not a conflict literal in \(N\setminus \{C\}\) for otherwise C is dependent: Suppose L is a conflict literal in \(N\setminus \{C\}\) then, by definition, there is satisfiable subset from \(N\setminus \{C\}\) which entails L. However, since \(L\models C\), it means C is dependent. \(\square \)
The next example demonstrates that the notion of a MUS cannot be carried over straightforwardly to the level of clauses with variables to characterize semirelevant clauses in firstorder logic.
Example 19
(FirstOrder Relevant Clauses). Given a set of clauses
over \(\varSigma =(\{a, b, c,d,e\},\{P,Q\})\). The conflict literals are
The clause P(a, y) is relevant. The literals entailed by some satisfiable instances \(N'\) from N such that \(P(a,y)\not \in N'\) are \(\{\lnot Q(b,d)\}\uplus \{\lnot P(t,c),\lnot Q(t,e) \mid t\in \{a,b,c,d,e\}\}\) and no two of them are complementary. Thus, \({\text {conflict}}(N\setminus \{P(a,y)\})=\emptyset \). The clause \(\lnot P(a,d) \vee Q(b,d)\) is semirelevant: \(Q(b,d)\not \in {\text {conflict}}(N\setminus \{\lnot P(a,d) \vee Q(b,d)\})\). The clause Q(z, e) is irrelevant.
With respect to a MUS, the clause \(\lnot P(a,d)\vee Q(b,d)\) from Example 19 is irrelevant. The only MUS from N is \(\{P(a,y),\lnot P(x,c)\}\) with grounding substitution \(\{x\mapsto a, y\mapsto c\}\). However, in firstorder logic we should not ignore the clauses \(\lnot P(a,d)\vee Q(b,d)\), \(\lnot Q(b,d) \vee P(d,c)\), because together with the clauses \(P(a,y),\lnot P(x,c)\) they result in a different grounding \(\{x\mapsto d, y\mapsto d\}\). So, we argue that MUSbased (semi)relevance on the original clause set is not sufficient to characterize the way clauses are used to derive a contradiction for full firstorder logic. However, it does so if ground instances are considered.
Lemma 20
(Relevance and MUSes on FirstOrder Clauses). Given an unsatisfiable set of independent firstorder clauses N. Then a clause C is relevant in N, if all MUSes of unsatisfiable sets of ground instances from N contain a ground instance of C. The clause C is semirelevant in N, if there exists a MUS of an unsatisfiable set of ground instances from N that contains a ground instance of C.
Proof
(Relevance) Since all ground instances from N contain a ground instance of C, then, if \(N\setminus \{C\}\) contains a ground MUS from N it means that some ground instance of C is entailed by \(N\setminus \{C\}\). This violates our assumption that N contains no dependent clauses. Thus, \(N\setminus \{C\}\) contains no ground MUSes. This further means that \(N\setminus \{C\}\) is satisfiable by the compactness theorem of firstorder logic. By Lemma 14 it therefore has no conflict literals and C is relevant.
(SemiRelevance) Take some ground MUS M containing some ground instance \(C'\) of C. Due to Lemma 13, any literal \(P\in C'\) is a conflict literal in M and consequently also in N. In addition, P is not a conflict literal in \(N\setminus \{C\}\) for otherwise C is dependent: Suppose P is a conflict literal in \(N\setminus \{C\}\). Then, by definition, there is some satisfiable instances from \(N\setminus \{C\}\) which entails P. However, since \(P\models C'\), it means C is dependent. In conclusion, \(P\in {\text {conflict}}(N)\setminus {\text {conflict}}(N\setminus \{C\})\) and thus C is semirelevant. \(\square \)
In Example 19, we could identify two ground MUSes:
and
Our notion of relevance is thus alternatively explainable using Lemma 20: P(a, y) is relevant because every MUS contains an instance of it (P(a, c) and P(a, d)). The clause \(\lnot P(a,d) \vee Q(b,d)\) is semirelevant as it is immediately contained in the second MUS. The clause Q(z, e) is irrelevant since no MUS contains any instance of Q(z, e). On the other hand, we may still encounter the case where a dependent clause is actually categorized as syntactically semirelevant. Therefore, by using the dependency notion while at the same time not restricting a refutation to only use MUS as the input set, we can show that (semi)relevance actually coincides with the syntactic (semi)relevance. So, the semidecidability result also follows.
Theorem 21
(Semantic versus Syntactic Relevance). Given an independent, unsatisfiable set of clauses N in firstorder logic, then (semi)relevant clauses coincide with syntactically (semi)relevant clauses.
Proof
We show the following: if N contains no dependent clause, C is (semi)relevant if and only if C is syntactically (semi)relevant. The case for relevant clauses is a consequence of Lemma 14. Now, we show it for semirelevant clauses.
“\(\Rightarrow \)" Let L be a ground literal with \(L\in {\text {conflict}}(N)\setminus {\text {conflict}}(N\setminus \{C\})\). We can construct a refutation using C. There are two satisfiable subsets of instances \(N_1,N_2\) from N such that \(N_1\models L\) and \(N_2\models {\text {comp}}(L)\) where \(N_1\cup N_2\) contains at least one instance of C, for otherwise \(L\not \in {\text {conflict}}(N)\setminus {\text {conflict}}(N\setminus \{C\})\). By the deductive completeness, Theorem 2, and the fact that L and \({\text {comp}}(L)\) are ground literals, there are two variable disjoint deductions \(\pi _1\) and \(\pi _2\) of some literals \(K_1\) and \(K_2\) such that \(K_1\sigma = L\) and \(K_2\sigma = {\text {comp}}(L)\) for some grounding \(\sigma \). Obviously, the two variable disjoint deductions can be combined to a refutation \(\pi _1.\pi _2.\bot \) containing C. Thus, C is syntactically semirelevant in N.
“\(\Leftarrow \)" Given an SOS refutation \(\pi \) using C, i.e., an SOS refutation \(\pi \) from \(N\setminus \{C\}\) with SOS \(\{C\}\) and overall grounding substitution \(\sigma \), we show that C is semantically semirelevant. Let \(N'\) be the variable renamed versions of clauses from \(N\setminus \{C\}\) used in the refutation and \(S'\) be the renamed copies of C used in the refutation. First, we show that \(N'\sigma \) is satisfiable. Towards contradiction, suppose \(N'\sigma \) is unsatisfiable and let \(M\sigma \subseteq N'\sigma \) be its MUS. Since \(\pi \) is connected, some clauses in \(M\sigma \) and \(S'\sigma \cup (N'\sigma \setminus M\sigma )\) contains literals L and \({\text {comp}}(L)\) respectively. By Lemma 13, L and \({\text {comp}}(L)\) are also conflict literals in \(M\sigma \). So, by Definition 15, the clause containing \({\text {comp}}(L)\) in \(S'\sigma \cup (N'\sigma \setminus M\sigma )\) is dependent violating our initial assumption.
Now, since \(N'\sigma \) is satisfiable, there is a ground MUS from \((N'\cup S')\sigma \) containing some \(C'\sigma \in S\sigma \). Due to Lemma 13, any \(L\in C'\sigma \) is a conflict literal in \(N'\) (and consequently also in N). In addition, L is not a conflict literal in \(N\setminus \{C\}\) for otherwise C is dependent: Suppose L is a conflict literal in \(N\setminus \{C\}\). Then, by definition, there is some satisfiable instances from \(N\setminus \{C\}\) which entails L. However, since \(L\models C'\sigma \), it means C is dependent. In conclusion, \(L\in {\text {conflict}}(N)\setminus {\text {conflict}}(N\setminus \{C\})\) and thus C is semirelevant. \(\square \)
When we have a ground MUS, identifification of conflict literals is obvious because all of the literals in it are. However, testing if a literal L is a conflict literal is not trivial, in general. One can try enumerating all MUSes and check if L is contained in some. This definitely works for propositional logic despite being computationally expensive. In firstorder logic, this is problematic because there could potentially be an infinite number of MUSes and determining a MUS is not even semidecidable, in general. The following lemma provides a semidecidable test using the SOS strategy.
Lemma 22
Given a ground literal L and an unsatisfiable set of clauses N with no dependent clauses, L is a conflict literal if and only if there is an SOS refutation from \((N,\{L\vee {\text {comp}}(L)\})\).
Proof
“\(\Rightarrow \)" By the deductive completeness, Theorem 2, and the fact that L and \({\text {comp}}(L)\) are ground literals, there are two variable disjoint deductions \(\pi _1\) and \(\pi _2\) of some literals \(K_1\) and \(K_2\) such that \(K_1\sigma = L\) and \(K_2\sigma = {\text {comp}}(L)\) for some grounding \(\sigma \). Obviously, the two variable disjoint deductions can be combined to a refutation \(\pi _1.\pi _2.\bot \). We can then construct a refutation \(\pi _1.\pi _2.(L\vee \lnot L).({\text {comp}}(L)).\bot \) where \(K_2\) is resolved with \(L\vee {\text {comp}}(L)\) to get \({\text {comp}}(L)\) which will be resolved with \(K_1\) from \(\pi _1\) to get \(\bot \). By Theorem 7, it means there is an SOS refutation from \((N,\{L\vee \lnot L\})\)
“\(\Leftarrow \)" Given an SOS refutation \(\pi \) using \(\{L\vee {\text {comp}}(L)\}\), i.e., an SOS refutation \(\pi \) from \(N\setminus \{\{L\vee {\text {comp}}(L)\}\}\) with SOS \(\{\{L\vee {\text {comp}}(L)\}\}\), Let \(N'\) be the variable renamed versions of clauses from N and overall grounding substitution \(\sigma \). \(N'\sigma \) is a MUS for otherwise there is a dependent clause: Suppose \(N'\sigma \setminus M\) is an MUS where M is nonempty. Since \(\pi \) is connected, some clause \(D'\) in M must be resolved with some \(D\in N'\sigma \) upon some literal K. Thus, by Lemma 13, K and \({\text {comp}}(K)\) are also conflict literals in \(N'\sigma \setminus M\). So, by Definition 15, the clause subsuming \(D'\) in N is dependent violating our initial assumption. Finally, because L occurs in \(N'\sigma \) and \(N'\sigma \) is an MUS, by Lemma 13, L is a conflict literal. \(\square \)
4 Conclusion
The main results of this paper are: (i) a semantic notion of relevance based on the existence of conflict literals, Definition 10, and Definition 16, (ii) its relationship to syntactic relevance, namely, both notions coincide for independent clause sets, Theorem 21, and (iii) the relationship of semantic relevance to minimal unsatisfiable sets, MUSes, both for propositional logic, Lemma 18, and firstorder logic, Lemma 20.
The semantic relevance notion sheds some further light on the way clauses may contribute to a refutation beyond what can be offered by the notion of MUSes. While the syntactic notion of semirelevance also considers redundant clauses such as tautologies to be semirelevant, the semantic notion rules out redundant clauses. Here, the notions only coincide for independent clause sets. Still, the syntactic notion is “easier” to test and there are applications where clause sets do not contain implied clauses by construction. Hence, the syntacticrelevance coincides with semantic relevance. For example, firstorder toolbox formalizations have this property because every tool is formalized by its own distinct predicate. Still a goal, refutation, can be reached by the use of different tools. The classic example is the toolbox for car/truck/tractor building [8, 31].
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Acknowledgments
This work was partly funded by DFG grant 389792660 as part of TRR 248. We thank Christopher Lynch and David Plaisted for a number of discussions on semantic relevance. We thank the anonymous reviewers for their constructive and detailed comments.
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Haifani, F., Weidenbach, C. (2022). Semantic Relevance. In: Blanchette, J., Kovács, L., Pattinson, D. (eds) Automated Reasoning. IJCAR 2022. Lecture Notes in Computer Science(), vol 13385. Springer, Cham. https://doi.org/10.1007/9783031107696_13
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