While syntactic inference restrictions don't play an important role for SAT, they are an essential reasoning technique for more expressive logics, such as first-order logic, or fragments thereof. In particular, they can result in short proofs or model representations. On the other hand, semantically guided inference systems enjoy important properties, such as the generation of solely non-redundant clauses. I discuss to what extend the two paradigms may be unifiable

Weidenbach, C.

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M. Suda, S. Winkler (Eds.): Second International Workshopon Automated Reasoning: Challenges, Applications,Directions, Exemplary Achievements (ARCADE 2019)EPTCS 311, 2019, pp. 5–10, doi:10.4204/EPTCS.311.1The Challenge of Unifying Semantic and Syntactic InferenceRestrictionsChristoph WeidenbachMax Planck Institute for Informatics, Saarland Informatics Campus E1 466123 Saarbru¨cken, Germanyweidenb@mpi-inf.mpg.deWhile syntactic inference restrictions don’t play an important role for SAT, they are an essentialreasoning technique for more expressive logics, such as first-order logic, or fragments thereof. Inparticular, they can result in short proofs or model representations. On the other hand, semanticallyguided inference systems enjoy important properties, such as the generation of solely non-redundantclauses. I discuss to what extend the two paradigms may be unifiable.1 IntroductionIn [29] I discussed the differences between simple-syntactic portfolio solvers and sophisticated-sematicportfolio solvers. In this paper I present the challenges in combining a solver based on syntactic inferencerestrictions with a solver based on the semantic guidance of inferences. More concretely, I investigatethe differences between CDCL-style solvers building an explicit partial model assumption and solversbased on ordering and selection restrictions on inferences.2 NP-Complete ProblemsThe prime calculus for SAT is CDCL (Conflict Driven Clause Learning) [26, 19, 6, 7]. The calculuscan be viewed as a resolution variant where resolution inferences are selected via explicit partial modelassumptions. The CDCL calculus operates on a five tuple (M,N,U,k,C) where M is a sequence ofliterals representing the current model assumption, called the trail, N the clause set under consideration,U a set of learned clauses, i.e., clauses derived via resolution, k the number guessed/decided literals inM, called the decision level, and C a clause that is > if no conflict has occurred yet, some non-emptyclause representing a found conflict, or ⊥ in case of an overall contradiction. Consider the clause setN = {P∨Q∨R, ¬R∨S, ¬S∨P∨Q, . . .}where a CDCL run starting from the empty trail ε may result in(ε;N; /0;0;>)⇒∗CDCL ([¬P1¬Q2 RP∨Q∨R S¬R∨S];N; /0;2;¬S∨P∨Q)where the literals ¬P, ¬Q were decided (guessed) at decision levels 1, 2, respectively; R is a prop-agated literal via the clause P∨Q∨R; S is propagated via ¬R∨ S and finally in the partial assignment[¬P1¬Q2 RP∨Q∨R S¬R∨S] the clause ¬S∨P∨Q is false. The conflict is solved by resolving the falseclause with clauses propagating literals from the trail. First with the clause ¬R∨S resulting in the clause6 Unifying Semantic and Syntactic Inference Restricions¬R∨P∨Q and then with the clause P∨Q∨R resulting in the clause P∨Q. Now this clause is learnedyielding the new CDCL state⇒∗CDCL ([¬P1 QP∨Q];N;{P∨Q};1;>).Most importantly, the new clause P∨Q is non-redundant, i.e., it is not implied by smaller clausesfrom N, where the ordering is a lifting of the literal ordering generated by the trail [1, 14]. Propositionalnon-redundancy is itself an NP-complete property. Hence, the CDCL polynomial time model generationprocedure either finds a model or eventually leads to learned clause that enjoys an NP-complete property.This has several immediate consequences: (i) termination (ii) the approach of forgetting clauses works.Note that since only non-redundant clauses are learned by CDCL, a forgotten clause that has becomeredundant will not be generated a second time. Hence, learning plus forgetting can be seen as an efficientway to getting rid of redundant clauses.Please recall, the classical first-order notion of redundancy means “not needed to find a proof ormodel”. Therefore, a redundant clause can be deleted. In the superposition or ordered resolution contextthis abstract definition is instantiated with “not implied by smaller clauses” [3, 20]. However, in thecontext of SAT, redundancy is often defined as a synonym for “satisfiability preserving” [16]. So in thecontext of many SAT related papers redundant clauses must not be removed, in general, or completenessis lost. These two notions of redundancy must not be confused and I stick here to the classical first-ordernotion.The prerequisites for learning non-redundant clauses are twofold:1. Propagation needs to be exhaustive.2. Conflict detection needs to be eager.The number of literals of a SAT problem is typically small compared to the respective clause setsize. Therefore, exhaustive propagation in SAT is not an efficiency issue, as well as the detection ofconflicts, i.e., false clauses. Surprisingly, what holds for SAT, does not hold for NP-complete problems,in general. It seems that the language of propositional logic is a nice compromise between expressivity,succinctness, and the efficiency of propagation.Consider the NP-complete problem of testing satisfiability of a system of linear arithmetic inequa-tions over the integers [17, 21]. The language of linear integer arithmetic is more succinct than propo-sitional logic, i.e., the encoding of on integer variable x requires linearly many propositional variables,because the a priori bounds for solvability are simply exponential. Now consider a CDCL style proce-dure testing solvability of a LIA (Linear Integer Arithmetic) system of inequations [11]. Consider theexample systemN = {1− x− y≤ 0, x− y≤ 0, . . .}and a CDCL-style run via CATSAT+ [11], where the partial model assumption is represented bysimple bounds, i.e., inequations of the form x#c, c ∈ Z, # ∈ {<,>,≤,≥}.(ε;N; /0;0;>)⇒∗CATSAT+ ([x≥ 51 y≥ 61−x−y≤0 x≥ 6x−y≤0 . . .];N; /0;1;>)Obviously, the propagation does not terminate, except for a priori simply exponential bounds: if m isthe number of inequations in N, n the number of different variables in N and a the maximal coefficientin N, then the problem has a solution iff −n · (m · a)2m+1 ≤ x ≤ n · (m · a)2m+1 for every variable x inWeidenbach 7N [21]. Hence, there is currently no CDCL-style procedure known that does exhaustive propagation.Therefore, the learned clauses [11] are not non-redundant. The current state-of-the-art procedures don’tuse a CDCL-style approach but apply a relaxation of LIA to LRA (linear rational arithmetic) and useLRA solutions for a branch-and-bound approach. This general idea is complemented via simplificationsand fast, sufficient tests for the existence of a solution [9, 18, 12, 10].3 NEXPTIME-Complete ProblemsThe same phenomenon that occurs at LIA, see Section 2, also shows up if the complexity is “slightly” in-creased from SAT to the satisfiability of the Bernays-Schoenfinkel (BS) fragment of first-order logic [5],which is NEXPTIME-Complete [25]. For example, consider the following clause set [22]N =1 : P(0,0,0,0),2 : ¬P(x1,x2,x3,0)∨P(x1,x2,x3,1),3 : ¬P(x1,x2,0,1)∨P(x1,x2,1,0),4 : ¬P(x1,0,1,1)∨P(x1,1,0,0),5 : ¬P(0,1,1,1)∨P(1,0,0,0),6 : ¬P(1,1,1,1)where a CDCL-style run, using ground literals for the partial model assumptions [14], will propagateall values of the 4-bit counter represented by P:(ε;N; /0;0;>)⇒∗SCL ([P(0,0,0,0)C1 P(0,0,0,1)C2 P(0,0,1,0)C3 . . .P(1,1,1,1)C2],N, /0,0,¬P(1,1,1,1))before detecting the conflict, where for the propagation justifications the notation C < clause number >was used. Thus, there are exponentially many propagations, in general. Still, the clause set can be refutedin linearly many steps by starting with resolution steps between the clauses 2−4:2.2 Res 3.1 7 : ¬P(x1,x2,0,0)∨P(x1,x2,1,0)7.2 Res 2.1 8 : ¬P(x1,x2,0,0)∨P(x1,x2,1,1)8.2 Res 4.1 9 : ¬P(x1,0,0,0)∨P(x1,1,0,0)9.2 Res 8.1 10 : ¬P(x1,0,0,0)∨P(x1,1,1,1)10.2 Res 5.1 11 : ¬P(0,0,0,0)∨P(1,0,0,0)11.2 Res 10.1 12 : ¬P(0,0,0,0)∨P(1,1,1,1)12.1 Res 6.1 13 :⊥.where this proof can be implemented by a standard ordering restriction on the P atoms, e.g., via aKBO (Knuth Bendix Ordering), plus a respective selection strategy [2]. For example, for the first stepthe second literal out of clause 2 is maximal with respect to a KBO instance where 1 >KBO 0 and the firstliteral of clause 3 is selected.The above clause set N without clause 6 is obviously satisfiable. Still, a CDCL-style calculus usingground literals generates exponentially many ground atoms before it detects satisfiability. An orderedresolution calculus using a standard KBO instance where 1 >KBO 0 does not generate any clause at all,because exactly the positive literals are maximal. Extending the model representation language fromground atoms [14] to more a more expressive language including variables [4, 13, 24, 1, 8] solves the8 Unifying Semantic and Syntactic Inference Restricionsissue with the above example by starting with a model assumption [P(x1,x2,x3,x4)1]. However, themore expressive model language requires more complex operations in order to guarantee consistency ofthe model assumption and to detect propagating literals and false clauses. The discrepancy between acompact model representation and complex calculations cannot be resolved in general, because NP 6=NEXPTIME.In addition, for “practically relevant” instances of the BS fragment the situation is the same. We haveshown [28] that the YAGO [27] fragment of BS can be effectively saturated by a variant of ordered res-olution with chaining, whereas all model-guided approaches fail. Again for the reason that propagationwith respect to millions of constants cannot be efficiently done, yet.4 UnificationThe two frameworks, syntactic inference restrictions and model-guided inferences cannot be easily com-bined. Obviously, since ordered resolution may generate redundant clauses, but model-guided inferenceswith exhaustive propagation and eager conflict detection do not, model-guided inferences cannot sim-ulate resolution inferences. The resolution inferences in Section 3 refuting the clause set encoding acounter are not redundant. Still, they cannot be simulated by a CDCL-style calculus [14] because it willimmediately run into the exponentially many propagation steps and find a refutation of exponential sizethis way. It seems to be non-trivial and non-obvious how the two paradigms can be unified. If eagerpropagation is dropped, then model-guided inferences can simulate resolution [23, 14], however, in thiscase non-redundancy of learned clauses is lost, in general.One way out of this dilemma could be to limit the amount of propagations by limiting the number ofliterals that may be derived by propagation. For example, the InstGen calculus [15] limits ground instan-tiation and the generation of new ground literals to using exactly one constant. Then the potential sizeof a model assumption remains linear in the size of the investigated clause set. However, a terminatingmodel assumption search does not result in an overall model for the clause set anymore. Thus the modelgeneration process itself needs to turn into a “learning” procedure.References[1] Ga´bor Alagi & Christoph Weidenbach (2015): NRCL - A Model Building Approach to the Bernays-Scho¨nfinkel Fragment. 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The Challenge of Unifying Semantic and Syntactic Inference Restrictions

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