Semantic Relevance

. A clause C is syntactically relevant in some clause set N , if it occurs in every refutation of N . A clause C is syntactically semi-relevant, if it occurs in some refutation of N . While syntactic relevance coincides with satisﬁability (if C is syntactically relevant then N \ { C } is satisﬁable), the semantic counterpart for syntactic semi-relevance 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 semi-relevant 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 semi-relevant if it occurs in some MUS. For ﬁrst-order logic the characterization needs to be reﬁned with respect to ground instances of N and C .


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 set-of-support 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.

Syntactic Relevance
Given an unsatisfiable set of clauses N , C ∈ N is syntactically relevant if it occurs in all refutations, it is syntactically semi-relevant if it occurs in some refutation, otherwise it is called syntactically irrelevant. The clause-based 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 first-order 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 semi-relevant clause (3)¬B(c,a) ∨ 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)¬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)¬B(c,a) ∨ B(b,f (x 6 )) is in the SOS [11], Fig. 2. Please note that N \ {(3)¬B(c, a) ∨ 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)¬B(c,a) ∨ B(b,f (x 6 ))} [11].
In addition, N \ {(3)¬B(c, a) ∨ 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 MUS-based defined notion of semi-relevance on the level of the original first-order clauses is not sufficient here. The clause (2) ¬D(x7)

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 |= L and N 2 |= 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 ¬C(c) are conflict literals because )} 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 first-order 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.
¬A(f (a)) ∨ ¬B(b, f (a))} One can see that, despite (3)¬B(c, a) ∨ B(b, f (x 6 )) is outside of the only MUS on the first-order 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 (¬)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 refutation-based syntactic relevance. As redundant clauses, e.g., tautologies, can also be syntactically semi-relevant, 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 semi-relevant 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 semi-relevant ones. 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)) ∈ conflict(N \ {¬B(c, a) ∨ B(b, f (x 6 ))), and define because A(f (a)) can be acquired using resolution between (1) and (2) for N ∅ ∪ N A(f (a)) and ¬A(f (a)) can be acquired using resolution between (4) and (6) for . 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 semi-relevant clauses for propositional logic are called a plain clause set. Using the terminology in [16], a clause C ∈ 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 semi-relevance [11]: Given a clause C ∈ N , C is usable if-and-only-if C is syntactically semi-relevant. 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 N with a non-relaxable clause set N and relaxable clause set N (which must be satisfiable), a MUS is a subset M of N s.t. N 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 deletion-based approach well-known 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 depth-first-search manner and additionally use some heuristics to further improve the runtime [35]. For the function-free and equality-free first-order 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: black-box and white-box. A black-box approach picks some inputs, executes it using some sound and complete reasoner, and then interprets the output [13]. On the other hand, white-box 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 semi-relevance has also been applied to description logics via a translation scheme to first-order 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.

Preliminaries
We assume a standard first-order language without equality over a signature Σ = (Ω, Π) where Ω is a non-empty set of functions symbols, Π a non-empty set of predicate symbols both coming with their respective fixed arities denoted by the function arity. The set of terms over an infinite set of variables X is denoted by T (Σ, 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 comp. Atoms, literals, clauses, and clause sets are ground if they do not contain any variable.
An interpretation I with a nonempty domain (or universe) U assigns (i) a total function f I : U n → U for each f ∈ Ω with arity(f ) = n and (ii) a relation P ⊆ U m to every predicate symbol P I ∈ Π with arity(P ) = m. A valuation β is a function X → U where the assignment of some variable x can be modified to e ∈ U by β[x → e]. It is extended to terms as I(β) : T (Σ, X ) → U. Semantic entailment |= considers variables in clauses to be universally quantified. The extension to atoms, literals, disjunctions, clauses and sets of clauses is as follows: I(β)(P (t 1 , . . . , t n )) = 1 if (I(β)(t 1 ), . . . , I(β)(t n )) ∈ P I and 0 otherwise; Substitutions σ, τ are total mappings from variables to terms, where dom(σ) : A renaming σ 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 = Cσ for some substitution σ, then C is an instance of C. A unifier σ for a set of terms t 1 , . . . , t k satisfies t i σ = t j σ for all 1 ≤ i, j ≤ k and it is called a most general unifier if for any unifier σ of t 1 , . . . , t k there is a substitution τ s.t. σ = στ . The function mgu denotes the most general unifier of two terms, atoms, literals if it exists. We assume that any 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 (∅, 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.
The clause (D∨C)σ is the result of a Resolution inference between its parents and called a resolvent. The clause (C ∨ L)σ is the result of a Factoring inference of its parent and called a factor. A sequence of rule applications (N, In case ⊥ ∈ S it is a called a (SOS) resolution refutation. If for two clauses C, D there exists a substitution σ such that Cσ ⊆ D, then we say that C subsumes D. In this case C |= D. [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 ∪ S is unsatisfiable, then there is a (SOS) resolution derivation of ⊥ from (N, S) [33]. If for some clause set N and clause C ∈ N there exists a resolution refutation from N using C, then there is an SOS derivation of ⊥ from (N \ {C}, {C}) [11].

Theorem 1 (Soundness and Refutational Completeness of (SOS) Resolution
Please note that the recent SOS completeness result of [11] generalizes the classical SOS completeness result by [33]. [17,22]). Given a set of clauses N and a clause D, if N |= D, then there is a resolution derivation of some clause C from (∅, N) such that C subsumes D.

Theorem 2 (Deductive Completeness of Resolution
For deductions we require every clause to be used exactly once, so deductions always have a tree form. [11]). A deduction π N = [C 1 , . . . , 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:

Definition 3 (Deduction
1.1 C i is a renamed, variable-fresh version of a clause in N , or 1.2 there is a clause C j ∈ π 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 ∈ π 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 ∈ π 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 π N if the context is clear.
A deduction π of some clause C ∈ π, where π, π are deductions from N is a subdeduction of π if π ⊆ π, where the subset relation is overloaded for sequences. A deduction π N = [C 1 , . . . , C n−1 , ⊥] 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. [11]). Given a deduction π of a clause C n the overall substitution τ π,i of C i ∈ π is recursively defined by 1 if C i is a factor of C j with j < i and mgu σ, then τ π,i = τ π,j • σ, 2 if C i is a resolvent of C j and C k with j < k < i and mgu σ, then τ π,i = (τ π,j • τ π,k ) • σ, 3 if C i is an initial clause, then τ π,i = ∅, and the overall substitution of the deduction is τ π = τ π,n . We omit the subscript π if the context is clear.

Definition 4 (Overall Substitution of a Deduction
Overall substitutions are well-defined 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 τ of some deduction π is a substitution τ δ such that codom(τ δ) only contains ground terms and dom(δ) is exactly the variables from codom(τ ). [11]

Definition 5 (SOS Deduction
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. [20]). Given an unsatisfiable set of clauses N , the subset N ⊆ N is a minimally unsatisfiable subset (MUS) of N if any strict subset of N is satisfiable.

Definition 6 (Minimal Unsatisfiable Subset (MUS)
In our previous work, we defined a notion of relevance based on how clauses may contribute to unsatisfiability by means of refutations. [11]). Given an unsatisfiable set of clauses N , a clause C ∈ N is syntactically relevant if for all refutations π of N it holds that C ∈ π. A clause C ∈ N is syntactically semi-relevant if there exists a refutation π of N in which C ∈ π. A clause C ∈ N is syntactically irrelevant if there is no refutation π of N in which C ∈ π.

Definition 7 (Syntactic Relevance
Syntactic relevance can be identified by using the resolution calculus. A clause C ∈ N is syntactically semi-relevant if and only if there exists an SOS refutation from SOS {C} and N \ {C}. [11]). Given an unsatisfiable set of clauses N , the clause C ∈ N is

syntactically relevant if and only if N \ {C} is satisfiable, 2. syntactically semi-relevant if and only if (N \ {C}, {C}) ⇒ * RES (N \ {C}, S ∪ {⊥}).
An open problem from [11] is the question of a semantic counterpart to syntactic semi-relevance. Without any further properties of the clause set N , the notion of semi-relevance can lead to unintuitive results. For example, a tautology could be semi-relevant. Given a refutation showing semi-relevance of some clause C, where, in the refutation, some unary predicate P occurs, the refutation can be immediately extended using the tautology P (x) ∨ ¬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 semi-relevant, although from an intuition point of view one may only want to consider Q(x) to be semi-relevant, or even relevant. On the other hand, in some cases, redundant clauses are welcome as semi-relevant clauses.

Example 9 (Redundant Clauses). Given a set of clauses
all clauses are syntactically semi-relevant while ¬Q(a) ∨ P (b) and ¬P (b) are syntactically relevant. However, if we disregard the redundant clauses Q(a) and P (x)∨¬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.

Semantic Relevance
Except for the trivially false clause ⊥, the simplest form of a contradiction is two unit clauses K and L such that K and comp(L) are unifiable. They will be called conflict literals, below. Then the idea for our semantic definition of semi-relevance 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 first-order 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 first-order 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 semi-relevant, if it contributes to the number of conflict literals.

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. f (a, x)) ∨ ¬P (f (c, y)), P (f (x, d)) ∨ P (f (y, b))} Consider the following satisfiable sets of instances from N f (a, b)) is a conflict literal because N 1 |= P (f (a, b)) and N 2 |= ¬P (f (a, b)). N 1 |= P (f (a, b)) because the resolution calculus is sound. Resolving both literals of ¬P (f (a, d)) ∨ ¬P (f (c, y)) with the first literal of the clause P (f (x, d)) ∨ P (f (a, b)) results in the clause P (f (a, b)) ∨ P (f (a, b)) which can be factorized to P (f (a, b)). Moreover, N 1 is satisfiable: An interpretation I with I (P (f (a, b))) = 1 and I(P (t)) = 0 for all terms t = f (a, b) satisfies N 1 and P (f (a, b)). N 2 |= ¬P (f (a, b)) can also be shown in the same manner.  {P (a, b), ¬P (a, b),

R(c), ¬R(c), Q(a), ¬Q(a)}
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 ¬A are nonempty, because otherwise A is a pure literal and its corresponding clauses can be removed from N preserving unsatisfiability. Obviously N ∅ ∪ N A must be satisfiable, for otherwise the initial choice of N was not minimal. However, N ∅ ∪ 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 ∅ ∪ N A must also be a model of A: N ∅ ∪ N A |= A. Using the same argument, N ∅ ∪ N ¬A is satisfiable and N ∅ ∪ N ¬A |= ¬A. Therefore, A is a conflict literal.

Lemma 14 (Conflict Literals and Unsatisfiability). Given a set of clauses N , conflict(N ) = ∅ if and only if N is unsatisfiable.
Proof "⇒" Let L ∈ conflict(N ). By definition, there are two satisfiable subsets of instances N 1 , N 2 from N such that N 1 |= L and N 2 |= comp(L). Towards contradiction, suppose N is satisfiable. Then, there exists an interpretation I with I |= N and therefore it holds that I |= N 1 and I |= N 2 . Furthermore, by definition of a conflict literal, I |= L and I |= comp(L), a contradiction. "⇐" Given an unsatisfiable clause set N , we show that there is a conflict literal in N . Since N is unsatisfiable, by compactness of first-order logic there is a minimal set of ground instances N from N that is also unsatisfiable. The rest follows from Lemma 13.
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 semi-relevant if it is duplicated. So, in order to have a semantically robust notion of relevance in first-order logic, we need to use a strong notion of (in)dependency.

it does not contain any dependent clauses.
A subsumed clause is obviously a dependent clause. However, there could also be non-subsumed clauses that are dependent. For example, in the set of clauses ) 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 non-redundant 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.

Example 17 (Dependent Clauses in Propositional Logic).
The existence of dependent clauses ¬P ∨ Q and ¬R ∨ P causes an independent clause ¬Q ∨ R to be a semi-relevant clause. However, ¬Q ∨ R is not inside the only MUS {P, ¬P }.
Very often, concepts from propositional logic can be generalized to first-order 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 first-order logic without considering instances of clauses. For the case of semi-relevance: Given C ∈ N , we show C is semi-relevant if and only if C is in some MUS N ⊆ N . "⇒": Towards contradiction, suppose there is a semi-relevant clause C that is not in any MUS. By definition of semi-relevant clauses, there are satisfiable sets N 1 and N 2 and a propositional variable P such that N 1 |= P , N 2 |= ¬P but the MUS M out of N 1 ∪ N 2 does not contain C. By Theorem 2 there exist deductions π 1 and π 2 of P and ¬P from N 1 and N 2 , respectively. Since a deduction is connected, some clauses in M and (N 1 ∪ N 2 ) \ M must have some complementary propositional literals Q and ¬Q, respectively to be eventually resolved upon in either π 1 or π 2 . At least one of these deductions must contain this resolution step between a clause from M and one from (N 1 ∪ N 2 ) \ M . Now by Lemma 13 the literals Q and ¬Q are conflict literals in M . Thus, there are satisfiable subsets from M which entail Q and ¬Q, respectively. Therefore, the clause containing Q or ¬Q in (N 1 ∪ N 2 ) \ M is dependent contradicting the assumption that N does not contain dependent clauses. "⇐": If C is in some MUS N ⊆ N , then, N \ {C} is satisfiable. So invoking Lemma 13 any literal L ∈ C is a conflict literal in N . In addition, L is not a conflict literal in N \ {C} for otherwise C is dependent: Suppose L is a conflict literal in N \ {C} then, by definition, there is satisfiable subset from N \ {C} which entails L. However, since L |= C, it means C is dependent.
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 first-order logic.   N is {P (a, y), ¬P (x, c)} with grounding substitution {x → a, y → c}. However, in first-order logic we should not ignore the clauses ¬P (a, d) ∨ Q(b, d), ¬Q(b, d) ∨ P (d, c), because together with the clauses P (a, y), ¬P (x, c) they result in a different grounding {x → d, y → d}. So, we argue that MUS-based (semi-)relevance on the original clause set is not sufficient to characterize the way clauses are used to derive a contradiction for full first-order logic. However, it does so if ground instances are considered.

Lemma 20 (Relevance and MUSes on First-Order Clauses). Given an unsatisfiable set of independent first-order 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 semi-relevant 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 \ {C} contains a ground MUS from N it means that some ground instance of C is entailed by N \ {C}. This violates our assumption that N contains no dependent clauses. Thus, N \{C} contains no ground MUSes. This further means that N \ {C} is satisfiable by the compactness theorem of firstorder logic. By Lemma 14 it therefore has no conflict literals and C is relevant. (Semi-Relevance) Take some ground MUS M containing some ground instance C of C. Due to Lemma 13, any literal P ∈ C is a conflict literal in M and consequently also in N . In addition, P is not a conflict literal in N \ {C} for otherwise C is dependent: Suppose P is a conflict literal in N \ {C}. Then, by definition, there is some satisfiable instances from N \ {C} which entails P . However, since P |= C , it means C is dependent. In conclusion, P ∈ conflict(N )\ conflict(N \ {C}) and thus C is semi-relevant.
In Example 19, we could identify two ground MUSes: 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)). "⇒" Let L be a ground literal with L ∈ conflict(N ) \ conflict (N \ {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 |= L and N 2 |= comp(L) where N 1 ∪ N 2 contains at least one instance of C, for otherwise L ∈ conflict(N ) \ conflict(N \ {C}). By the deductive completeness, Theorem 2, and the fact that L and comp(L) are ground literals, there are two variable disjoint deductions π 1 and π 2 of some literals K 1 and K 2 such that K 1 σ = L and K 2 σ = comp(L) for some grounding σ.
Obviously, the two variable disjoint deductions can be combined to a refutation π 1 .π 2 .⊥ containing C. Thus, C is syntactically semi-relevant in N .
"⇐" Given an SOS refutation π using C, i.e., an SOS refutation π from N \ {C} with SOS {C} and overall grounding substitution σ, we show that C is semantically semi-relevant. Let N be the variable renamed versions of clauses from N \ {C} used in the refutation and S be the renamed copies of C used in the refutation. First, we show that N σ is satisfiable. Towards contradiction, suppose N σ is unsatisfiable and let Mσ ⊆ N σ be its MUS. Since π is connected, some clauses in Mσ and S σ ∪ (N σ \ Mσ) contains literals L and comp(L) respectively. By Lemma 13, L and comp(L) are also conflict literals in Mσ. So, by Definition 15, the clause containing comp(L) in S σ∪(N σ\Mσ) is dependent violating our initial assumption. Now, since N σ is satisfiable, there is a ground MUS from (N ∪ S )σ containing some C σ ∈ Sσ. Due to Lemma 13, any L ∈ C σ is a conflict literal in N (and consequently also in N ). In addition, L is not a conflict literal in N \ {C} for otherwise C is dependent: Suppose L is a conflict literal in N \ {C}. Then, by definition, there is some satisfiable instances from N \ {C} which entails L. However, since L |= C σ, it means C is dependent. In conclusion, L ∈ conflict(N ) \ conflict(N \ {C}) and thus C is semi-relevant.
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 first-order logic, this is problematic because there could potentially be an infinite number of MUSes and determining a MUS is not even semi-decidable, in general. The following lemma provides a semi-decidable 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 ∨ comp(L)}).
Proof "⇒" By the deductive completeness, Theorem 2, and the fact that L and comp(L) are ground literals, there are two variable disjoint deductions π 1 and π 2 of some literals K 1 and K 2 such that K 1 σ = L and K 2 σ = comp(L) for some grounding σ. Obviously, the two variable disjoint deductions can be combined to a refutation π 1 .π 2 .⊥. We can then construct a refutation π 1 .π 2 .(L ∨ ¬L).(comp(L)).⊥ where K 2 is resolved with L ∨ comp(L) to get comp(L) which will be resolved with K 1 from π 1 to get ⊥. By Theorem 7, it means there is an SOS refutation from (N, {L ∨ ¬L}) "⇐" Given an SOS refutation π using {L ∨ comp(L)}, i.e., an SOS refutation π from N \ {{L ∨ comp(L)}} with SOS {{L ∨ comp(L)}}, Let N be the variable renamed versions of clauses from N and overall grounding substitution σ. N σ is a MUS for otherwise there is a dependent clause: Suppose N σ \ M is an MUS where M is non-empty. Since π is connected, some clause D in M must be resolved with some D ∈ N σ upon some literal K. Thus, by Lemma 13, K and comp(K) are also conflict literals in N σ \ M . So, by Definition 15, the clause subsuming D in N is dependent violating our initial assumption. Finally, because L occurs in N σ and N σ is an MUS, by Lemma 13, L is a conflict literal.

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 semi-relevance also considers redundant clauses such as tautologies to be semi-relevant, 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, first-order 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].