Unsatisfiable Formulas Research Articles

Classes of propositional UMU formulas and their extensions to minimal unsatisfiable formulas

The class UMU consists of propositional formulas which are a union of minimal unsatisfiable formulas (MU formulas). The structural complexity of UMU formulas depends essentially on the type and degree of intertwining of the MU subformulas. Starting from class of formulas that consist only of clause (or variable) disjoint minimal unsatisfiable subformulas, we study various UMU subclasses given by weakening and these conditions.Generalizing an idea from [6], we investigate a characterization of UMU formulas by whether they allow transformation into a MU formula by adding literals and/or clauses. It can be shown that simplicity in constructing of such extensions correlates with the degree of intertwining of the MU subformulas. For UMU formulas, however, we can give only extensions that have an exponential size in the worst case. The question of the existence of short MU combinations of MU-formulas by adding literals and clauses remains open.

Preprocessing of Propagation Redundant Clauses

The propagation redundant (PR) proof system generalizes the resolution and resolution asymmetric tautology proof systems used by conflict-driven clause learning (CDCL) solvers. PR allows short proofs of unsatisfiability for some problems that are difficult for CDCL solvers. Previous attempts to automate PR clause learning used hand-crafted heuristics that work well on some highly-structured problems. For example, the solver SaDiCaL incorporates PR clause learning into the CDCL loop, but it cannot compete with modern CDCL solvers due to its fragile heuristics. We present PReLearn, a preprocessing technique that learns short PR clauses. Adding these clauses to a formula reduces the search space that the solver must explore. By performing PR clause learning as a preprocessing stage, PR clauses can be found efficiently without sacrificing the robustness of modern CDCL solvers. On a large portion of SAT competition benchmarks we found that preprocessing with PReLearn improves solver performance. In addition, there were several satisfiable and unsatisfiable formulas that could only be solved after preprocessing with PReLearn. PReLearn supports proof logging, giving a high level of confidence in the results. Lastly, we tested the robustness of PReLearn by applying other forms of preprocessing as well as by randomly permuting variable names in the formula before running PReLearn, and we found PReLearn performed similarly with and without the changes to the formula.

Classes of Hard Formulas for QBF Resolution

To date, we know only a few handcrafted quantified Boolean formulas (QBFs) that are hard for central QBF resolution systems such as Q-Res and QU-Res, and only one specific QBF family to separate Q-Res and QU-Res. Here we provide a general method to construct hard formulas for Q-Res and QU-Res. The construction uses simple propositional formulas (e.g. minimally unsatisfiable formulas) in combination with easy QBF gadgets (Σb2 formulas without constant winning strategies). This leads to a host of new hard formulas, including new classes of hard random QBFs. We further present generic constructions for formulas separating Q-Res and QU-Res, and for separating Q-Res and LD-Q-Res.

Computing minimal unsatisfiable core for LTL over finite traces

Abstract In this paper, we consider the minimal unsatisfiable core (MUC) problem for linear temporal logic over finite traces (LTL$_{f}$), which nowadays is a popular formal-specification language for AI-related systems. Efficient algorithms to compute such MUCs can help locate the inconsistency rapidly in the written LTL$_{f}$ specification and are very useful for the system designers to amend the flawed requirement. As far as we know, there are no available tools off-the-shelf so far that provide MUC computation for LTL$_{f}$. We present here two generic approaches NaiveMUC and BinaryMUC to compute an MUC for LTL$_{f}$. Moreover, we introduce heuristics that are based on the Boolean unsatisfiable core (UC) technique to accelerate the two approaches, which are named NaiveMUC+UC and BinaryMUC+UC, respectively. In particular, for global LTL$_{f}$ formulas, we show that the MUC computation can be reduced to the pure Boolean MUC computation, which therefore conducts the GlobalMUC approach. Our experiments show that GlobalMUC performs the best to compute an MUC for global formulas, and BinaryMUC+UC is the best for an arbitrary unsatisfiable formula.

On Tseitin Formulas, Read-Once Branching Programs and Treewidth

We show that any nondeterministic read-once branching program that decides a satisfiable Tseitin formula based on an n × n grid graph has size at least 2Ω(n). Then using the Excluded Grid Theorem by Robertson and Seymour we show that for an arbitrary graph G(V, E) any nondeterministic read-once branching program that computes a satisfiable Tseitin formula based on G has size at least $2^{\Omega (\text {tw}(G)^{\delta })}$ for all δ < 1/36, where tw(G) is the treewidth of G (for planar graphs and some other classes of graphs the statement holds for δ = 1). We apply the mentioned results to the analysis of the complexity of derivations in the proof system OBDD(∧,reordering) and show that any OBDD(∧,reordering)-refutation of an unsatisfiable Tseitin formula based on a graph G has size at least $2^{\Omega (\text {tw}(G)^{\delta })}$ . We also show an upper bound O(|E|2pw(G)) on the size of OBDD representations of a satisfiable Tseitin formula based on G and an upper bound $O(|E||V| 2^{\text {pw}(G)}+|\text {TS}_{G,c}|^{2})$ on the size of OBDD(∧)-refutation of an unsatisifable Tseitin formula TSG, c, where pw(G) is the pathwidth of G.

SAT-based explicit [formula omitted] satisfiability checking

Linear Temporal Logic over finite traces (LTLf) was proposed in 2013 and has attracted increasing interest around the AI community. Though the theoretic basis for LTLf has been thoroughly explored since that time, there are still few algorithmic tools that are able to provide an efficient reasoning strategy for LTLf. In this paper, we present a SAT-based framework for LTLf satisfiability checking, which is the foundation of LTLf reasoning. We use propositional SAT-solving techniques to construct a transition system, which is an automata-style structure, for an input LTLf formula; satisfiability checking is then reduced to a path-search problem over this transition system. Based on this framework, we further present CDLSC (Conflict-Driven LTLf Satisfiability Checking), a novel algorithm (heuristic) that leverages information produced by propositional SAT solvers, utilizing both satisfiability and unsatisfiability results. More specifically, the satisfiable results of the SAT solver are used to create new states of the transition system and the unsatisfiable results to accelerate the path search over the system. We evaluate all 5 off-the-shelf LTLf satisfiability algorithms against our new approach CDLSC. Based on a comprehensive evaluation over 4 different LTLf benchmark suits with a total amount of 9317 formulas, our time-cost analysis shows that 1) CDLSC performs best on checking unsatisfiable formulas by achieving approximately a 4X time speedup, compared to the second-best solution (K-LIVE [1]); 2) Although no approaches dominate checking satisfiable formulas, CDLSC performs best on 2 of the total 4 tested satisfiable benchmark suits; and 3) CDLSC gains the best overall performance when considering both satisfiable and unsatisfiable instances.

NAE-resolution: A new resolution refutation technique to prove not-all-equal unsatisfiability

AbstractIn this paper, we analyze Boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation schemes. A read-once (resolution) refutation is one in which each clause is used at most once. Derived clauses can be used as many times as they are deduced. However, clauses in the original formula can only be used as part of one derivation. It is well known that ROR is not complete; that is, there exist unsatisfiable formulas for which no ROR exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called not-all-equal satisfiability (NAE-satisfiability). A CNF formula is NAE-satisfiable if it has a satisfying assignment in which at least one literal in each clause is set to false. It is well known that the problem of checking NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. The focus of this paper is on a variant of NAE-resolution called read-once NAE-resolution in which each clause (input or derived) can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for 2CNF formulas. Furthermore, we provide an algorithm to determine the smallest such NAE-resolution in polynomial time. This is in stark contrast to the corresponding problem concerning 2CNF formulas and ROR refutations. We also show that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.

An Improved Generator for 3-CNF Formulas

We consider the satisfiability problem (SAT) for Boolean formulas given in conjunctive normal form with the restriction that each clause contains three literals (3-CNF). Generation of random formulas with a fixed clause length is widely used in empirical studies. An interesting phenomenon of this method is the repeatedly confirmed linear dependence of the number of clauses in the formula on the number of Boolean variables at the point of the “phase transition” from satisfiable to unsatisfiable formulas (when the fraction of unsatisfiable formulas becomes prevailing). We propose and study a method for generating random formulas that has a smaller coefficient (3.49) of proportionality between the number of clauses and the number of variables at the point of the “phase transition” (for the previously known generation method this coefficient is 4.23).

Hyperresolution for Gödel logic with truth constants

In the paper, we generalise the well-known hyperresolution principle to the general first-order Gödel logic with explicit partial truth. We propose a hyperresolution calculus suitable for automated deduction in a useful expansion of Gödel logic by intermediate truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators. We solve the deduction problem of a formula from a countable theory in this expansion. We expand Gödel logic by a countable set of intermediate truth constants of the form c¯, c∈(0,1). Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory, consisting of order clauses. An order clause is a finite set of order literals of the form ε1⋄ε2 where εi is an atom or a truth constant or a quantified atom, and ⋄ is a connective ≖, equality, or ≺, strict order. ≖ and ≺ are interpreted by the equality and standard strict linear order on [0,1], respectively. We shall investigate the canonical standard completeness, where the semantics of Gödel logic is given by the standard G-algebra, and truth constants are interpreted by ‘themselves’. The hyperresolution calculus is refutation sound and complete for a countable order clausal theory under a certain condition for the set of truth constants occurring in the theory. As an interesting consequence, we get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in Gödel logic with truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators.

The Impact of Entropy and Solution Density on Selected SAT Heuristics.

We present a new characterization of propositional formulas called entropy, which approximates the freedom we have in assigning the variables. Like several other such measures (e.g., back-door and back-door-key variables), it is computationally expensive to compute. Nevertheless, for small and medium-size satisfiable formulas, it enables us to study the effect of this freedom on the impact of various SAT heuristics, following up on a recent study by C. Oh (Oh, SAT’15, LNCS 9340, 307–323). Oh’s findings were that the expected success of various heuristics depends on whether the input formula is satisfiable or not. With entropy, and also with the measure of solution density, we are able to refine these findings for the case of satisfiable formulas. Specifically, we found empirically that satisfiable formulas with small entropy “behave” similarly to unsatisfiable formulas.

Automated Deduction in Gödel Logic

This article addresses the deduction problem of a formula from a countable theory in the first-order Gödel logic. We generalise the well-known hyperresolution principle for deduction in Gödel logic. Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory, consisting of order clauses. We introduce a notion of quantified atom: a formula a is a quantified atom if a = Qx , p ( t 0 , …, t n ), where Q is a quantifier (∀, ∃), p ( t 0 , …, t n ) is an atom, and x is a variable occurring in p ( t 0 , …, t n ); for all i ≤ n , either t i = x or x does not occur in t i . Then an order clause is a finite set of order literals of the form ϵ 1 ⋄ ϵ 2 , where ϵ i is either an atom, or a truth constant ( 0 , 1 ), or a quantified atom, and ⋄ is either a connective ≖, equality, or ≺ strict order. ≖ and ≺ are interpreted by the equality and standard strict linear order on [ 0 , 1 ], respectively. On the basis of the hyperresolution principle, a calculus operating over order clausal theories is devised. The calculus is proved to be refutation sound and complete for the countable case. As an interesting consequence, we get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in Gödel logic.

The Local Lemma Is Asymptotically Tight for SAT

The Local Lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science, yet there are hardly any natural problems known where it provides an asymptotically tight answer. The main theme of our article is to identify several of these problems, among them a couple of widely studied extremal functions related to certain restricted versions of the k -SAT problem, where the Local Lemma does give essentially optimal answers. As our main contribution, we construct unsatisfiable k -CNF formulas where every clause has k distinct literals and every variable appears in at most 2/e + o (1)) 2 k / k clauses. The Lopsided Local Lemma, applied with an assignment of random values according to counterintuitive probabilities, shows that this is asymptotically best possible. The determination of this extremal function is particularly important, as it represents the value where the corresponding k -SAT problem exhibits a complexity hardness jump: From having every instance being a YES-instance it becomes NP-hard just by allowing each variable to occur in one more clause. The construction of our unsatisfiable CNF formulas is based on the binary tree approach of Gebauer [2012], and thus the constructed formulas are in the class MU(1) of minimal unsatisfiable formulas having one more clause than variables. The main novelty of our approach here comes in setting up an appropriate continuous approximation of the problem. This leads us to a differential equation, the solution of which we are able to estimate. The asymptotically optimal binary trees are then obtained through a discretization of this solution. The importance of the binary trees constructed is also underlined by their appearance in many other scenarios. In particular, they give asymptotically precise answers for seemingly unrelated problems like the European Tenure Game introduced by Doerr [2004] and a search problem allowing a limited number of consecutive lies. As yet another consequence, we slightly improve the best-known bounds on the maximum degree and maximum edge-degree of a k -uniform Maker’s win hypergraph in the Neighborhood Conjecture of Beck.

Strong ETH and Resolution via Games and the Multiplicity of Strategies

We consider a proof system intermediate between regular Resolution, in which no variable can be resolved more than once along any refutation path, and general Resolution. We call \(\delta \)-regular Resolution such system, in which at most a fraction \(\delta \) of the variables can be resolved more than once along each refutation path (however, the re-resolved variables along different paths do not need to be the same). We show that when for \(\delta \) not too large, \(\delta \)-regular Resolution is consistent with the Strong Exponential Time Hypothesis (SETH). More precisely, for large n and k, we show that there are unsatisfiable k-CNF formulas which require \(\delta \)-regular Resolution refutations of size \(2^{(1 - \epsilon _k)n}\), where n is the number of variables and \(\epsilon _k=\widetilde{O}(k^{-1/4})\) and \(\delta =\widetilde{O}(k^{-1/4})\) is the number of variables that can be resolved multiple times.

Enhancing unsatisfiable cores for LTL with information on temporal relevance

LTL is frequently used to express specifications in many domains such as embedded systems or business processes. Witnesses can help to understand why an LTL specification is satisfiable, and a number of approaches exist to make understanding a witness easier. In the case of unsatisfiable specifications unsatisfiable cores (UCs), i.e., parts of an unsatisfiable formula that are themselves unsatisfiable, are a well established means for debugging. However, little work has been done to help understanding a UC of an unsatisfiable LTL formula. In this paper we suggest to enhance a UC of an unsatisfiable LTL formula with information about the time points at which the subformulas of the UC are relevant for unsatisfiability. In previous work we showed how to obtain a UC in LTL by translating the LTL formula into a clausal normal form, applying temporal resolution, extracting a clausal UC from the resolution proof, and mapping the clausal UC back to a UC in LTL. In this paper we extend that method by extracting information at which time points the clauses of a clausal UC are relevant for unsatisfiability from a resolution proof and by transferring that information to a UC in LTL. We implement our method in TRP++, and we experimentally evaluate it.

CONSISTENCY OF CIRCUIT EVALUATION, EXTENDED RESOLUTION AND TOTAL NP SEARCH PROBLEMS

We consider sets ${\it\Gamma}(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps. We show that every CNF with a short refutation in extended R, ER, can be easily reduced to an instance of ${\it\Gamma}(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that ${\it\Gamma}(0,s,k)$ when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory $V_{1}^{1}$. We use the ideas of implicit proofs from Krajíček [J. Symbolic Logic, 69 (2) (2004), 387–397; J. Symbolic Logic, 70 (2) (2005), 619–630] to define from ${\it\Gamma}(0,s,k)$ a nonrelativized NP search problem $i{\it\Gamma}$ and we show that it is complete among all such problems provably total in bounded arithmetic theory $V_{2}^{1}$. The reductions are definable in theory $S_{2}^{1}$. We indicate how similar results can be proved for some other propositional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and we formulate two open problems about them.

Random model sampling: Making craig interpolation work when it should not

One of the most serious problems when doing program analyses is dealing with function calls. While function inlining is the traditional approach to this problem, it nonetheless suffers from the increase in analysis complexity due to the state space explosion. Craig interpolation has been successfully used in recent years in the context of bounded model checking to do function summarization which allows one to replace the complete function body with its succinct summary and, therefore, reduce the complexity, but unfortunately this technique can be applied only to a pair of unsatisfiable formulae. In this work-in-progress paper we present an approach to function summarization based on Craig interpolation that overcomes its limitation by using random model sampling. It captures interesting input/output relations, strengthening satisfiable formulae into unsatisfiable ones and thus allowing the use of Craig interpolation. Evaluation results show the applicability of this approach; in our future work we plan to do a more extensive evaluation on real-world examples.

Weakening Cardinality Constraints Creates Harder Satisfiability Benchmarks

For some time, the satisfiability formulae that have been the most difficult to solve for their size have been crafted to be unsatisfiable by the use of cardinality constraints. Recent solvers have introduced explicit checking of such constraints, rendering previously difficult formulae trivial to solve. A family of unsatisfiable formulae is described that is derived from the sgen4 family but cannot be solved using cardinality constraints detection and reasoning alone. These formulae were found to be the most difficult during the SAT2014 competition by a significant margin and include the shortest unsolved benchmark in the competition, sgen6-1200-5-1.cnf.

Non-Restarting SAT Solvers with Simple Preprocessing Can Efficiently Simulate Resolution

Propositional satisfiability (SAT) solvers based on conflict directed clause learning (CDCL) implicitly produce resolution refutations of unsatisfiable formulas. The precise class of formulas for which they can produce polynomial size refutations has been the subject of several studies, with special focus on the clause learning aspect of these solvers. The results, however, assume the use of non-standard and non-asserting learning schemes, or rely on polynomially many restarts for simulating individual steps of a resolution refutation, or work with a theoretical model that significantly deviates from certain key aspects of all modern CDCL solvers such as learning only one asserting clause from each conflict and other techniques such as conflict guided backjumping and phase saving. We study non-restarting CDCL solvers that learn only one asserting clause per conflict and show that, with simple preprocessing that depends only on the number of variables of the input formula, such solvers can polynomially simulate resolution. We show, moreover, that this preprocessing allows one to convert any CDCL solver to one that is non-restarting.

Relaxation Search: A Simple Way of Managing Optional Clauses

A number of problems involve managing a set of optional clauses. For example, the soft clauses in a MAXSAT formula are optional—they can be falsified for a cost. Similarly, when computing a Minimum Correction Set for an unsatisfiable formula, all clauses are optional—some can be falsified in order to satisfy the remaining. In both of these cases the task is to find a subset of the optional clauses that achieves some criteria, and whose removal leaves a satisfiable formula. Relaxation search is a simple method of using a standard SAT solver to solve this task. Relaxation search is easy to implement, sometimes requiring only a simple modification of the variable selection heuristic in the SAT solver; it offers considerable flexibility and control over the order in which subsets of optional clauses are examined; and it automatically exploits clause learning to exchange information between the two phases of finding a suitable subset of optional clauses and checking if their removal yields satisfiability. We demonstrate how relaxation search can be used to solve MAXSAT and to compute Minimum Correction Sets. In both cases relaxation search is able to achieve state-of-the-art performance and solve some instances other solvers are not able to solve.

Faithful and Full Translations Between Logics

PDF HTML阅读 XML下载 导出引用 引用提醒 逻辑之间的语义忠实语义满翻译 DOI: 10.3724/SP.J.1001.2013.04285 作者: 作者单位: 作者简介: 通讯作者: 中图分类号: 基金项目: 国家自然科学基金(60496326, 60573063, 60573064, 60773059, 61103169); 国家高技术研究发展计划(863)(2007AA01Z325) Faithful and Full Translations Between Logics Author: Affiliation: Fund Project: 摘要 | 图/表 | 访问统计 | 参考文献 | 相似文献 | 引证文献 | 资源附件 | 文章评论 摘要:翻译在计算机科学中的一个重要应用是实现一个逻辑与另一个逻辑在表达能力上的比较,以及利用目标逻辑的推理机实现源逻辑的推理.现有逻辑之间的翻译理论和性质没有深入研究逻辑的语义翻译,以及翻译是否保持不可满足性等问题.该文研究了一类同时保持公式的可满足性和不可满足性的翻译——语义忠实语义满翻译,给出了语义忠实语义满翻译的定义,比较了语义忠实语义满翻译与已有文献中翻译定义的区别和联系,讨论了逻辑的可靠性、完备性、可判定性、紧致性、公式的逻辑等价性,以及模型的初等等价性在语义忠实语义满翻译下被保持的问题.运用语义忠实语义满翻译的定义给出了逻辑之间的同义性定义,并证明了同义关系是逻辑之间的一个等价关系. Abstract:In computer science, an important application of translations includes comparing the expressive power among logics to achieve reasoning tasks of a logic in another defined logic. General properties of translations found in the literature do not give a comprehensive study on semantically translations and the preservation of the unsatisfiability. To preserve the satisfiability and the unsatisfiability of formulas, the definition of faithful and full translation is given, in this paper, and connections between the faithful and full translation, and other definitions of translations in the literature are discussed. Some properties of logics, such as the soundness, the completeness, the decidability, the compactness, the logical equivalence of formulas and the elementary equivalence of models, which are characterized by the existence of faithful and full translations between logics are also studied. By definition of faithful and full translation, the concept of synonymous logics is introduced and the proof for the equivalence of the synonymous relation is also given. 参考文献 相似文献 引证文献