Minimal Unsatisfiable Subsets Research Articles

ASP and subset minimality: Enumeration, cautious reasoning and MUSes

Answer Set Programming (ASP) is a well-known logic-based formalism that has been used to model and solve a variety of AI problems. For several years, ASP implementations primarily focused on the main computational task: the computation of one answer set of a (logic) program. Nonetheless, several AI problems, that can be conveniently modelled in ASP, require to enumerate solutions characterized by an optimality property that can be expressed in terms of subset-minimality with respect to some objective atoms. In this context, solutions are often either (i) answer sets that are subset-minimal w.r.t. the objective atoms or (ii) atoms that are contained in all subset-minimal answer sets, or (iii) sets of atoms that enforce the absence of answer sets on the ASP program at hand — such sets are referred to as minimal unsatisfiable subsets (MUSes). In all the above-mentioned cases, the corresponding computational task is currently not supported by plain state-of-the-art ASP solvers. In this paper, we study formally these tasks and fill the gap in current implementations by proposing several algorithms to enumerate MUSes and subset-minimal answer sets, as well as perform cautious reasoning on subset-minimal answer sets. We implement our algorithms on top of wasp and perform an experimental analysis on several hard benchmarks showing the good performance of our implementation.

Hashing-based approximate counting of minimal unsatisfiable subsets

Hashing-based approximate counting of minimal unsatisfiable subsets

Computation of minimal unsatisfiable subformulas for SAT-based digital circuit error diagnosis

The explanation of infeasibilities formed in Minimal Unsatisfiable Subformulas (MUSes) is a core task in the analysis of over-constrained Boolean formulas. A wide range of applications necessitate MUS detection including knowledge-based validation, software design, verification and error diagnosis in digital VLSI circuits. Consequently, various enhanced algorithms for determining MUS have been utilized for solving Maximum Satisfiability algorithms and Conjunction Normal Form (CNF) redundancies. Three enhancements are proposed in this paper. The first is a CPU-GPU algorithm for computation of Minimal Correction Subsets (MCSes) optimized for NVIDIA General Purpose Graphics Processing Unit paradigm. The proposed algorithm of generating all MCSes from the encoded CNF instance was developed using our parallel SSGPU solver and implemented using CUDA. The second enhancement is an algorithm for MUS computation based on auto-reduction of the enhanced MCSes for faster MUS detection. The third improved algorithm is for computing MUS directly without using MCSes. The two proposed algorithms outperform techniques as they could locate and explain design faults in digital VLSI circuits in earlier stages of IC design flow. The second proposed routine of MUS extraction was performed by avoiding a non-critical step in calling the SAT solver during MUS computation, leading to improving the performance of the MUS extraction algorithm. The third proposed mechanism for direct extraction of MUS was optimized by reducing the required SAT-solver calls using a classification of clauses in the input formula. Comparative analysis of the proposed algorithm against the Compute All Minimal Unsatisfiable Subsets (CAMUS) algorithm determined 1.54 × faster detection of MUS using ISCAS'85, ISCAS'89 and synthetic benchmarks. Also, the third algorithm for direct MUS computation delivers 17.05 × faster than shrinking algorithm used in MUST (minimal unsatisfiable subset) tool using ISCAS'85 and synthetic benchmarks. Moreover, it was observed that the CPU-GPU algorithm for MCSes computation based on the SSGPU solver delivered 1.92 × faster than its conventional counterpart, based on CUD@SAT equivalent on GPU using ISCAS'85, ISCAS'89 and synthetic Benchmarks.

Accelerating MUS enumeration by inconsistency graph partitioning

The problem of finding minimal unsatisfiable subsets (MUSes) has been studied frequently because of itstheoretical importance and wide range of applications in domains such as electronic design automation,software, and integrated circuit verification. In this paper, a method for accelerating theenumeration of MUSes based on inconsistency graph partitioning is proposed. First, an inconsistency graphof a set of clauses is constructed by extracting the inconsistencyrelations between literals of different clauses. In this paper, we show that by partitioning the inconsistency graphinto small connected components through a vertex cut, the enumeration of MUSes in different componentsbecomes independent and it is possible to compute them separately. Moreover, the MUSes of the original clause setcan be constructed by merging the unit clauses in the MUSes of these connected components back into the clauses inthe vertex cut. Experiments show that by integrating the acceleration method into the MARCO MUSes enumerator,there is a 2–3 times improvement in the average runtime of solved instances for randomly generatedbenchmarks. By integrating the acceleration method into itself as an MUS enumerator, there isanother 3–4 times improvement when compared with the accelerated MARCO.

Minimal sets on propositional formulae. Problems and reductions

Boolean Satisfiability (SAT) is arguably the archetypical NP-complete decision problem. Progress in SAT solving algorithms has motivated an ever increasing number of practical applications in recent years. However, many practical uses of SAT involve solving function as opposed to decision problems. Concrete examples include computing minimal unsatisfiable subsets, minimal correction subsets, prime implicates and implicants, minimal models, backbone literals, and autarkies, among several others. In most cases, solving a function problem requires a number of adaptive or non-adaptive calls to a SAT solver. Given the computational complexity of SAT, it is therefore important to develop algorithms that either require the smallest possible number of calls to the SAT solver, or that involve simpler instances. This paper addresses a number of representative function problems defined on Boolean formulae, and shows that all these function problems can be reduced to a generic problem of computing a minimal set subject to a monotone predicate. This problem is referred to as the Minimal Set over Monotone Predicate (MSMP) problem. This work provides new ways for solving well-known function problems, including prime implicates, minimal correction subsets, backbone literals, independent variables and autarkies, among several others. Moreover, this work also motivates the development of more efficient algorithms for the MSMP problem. Finally the paper outlines a number of areas of future research related with extensions of the MSMP problem.

On the query complexity of selecting minimal sets for monotone predicates

Propositional Satisfiability (SAT) solvers are routinely used for solving many function problems. A natural question that has seldom been addressed is: what is the number of calls to a SAT solver for solving some target function problem? This article improves upper bounds on the query complexity of solving several function problems defined on propositional formulas. These include computing the backbone of a formula and computing the set of independent variables of a formula. For the general case of monotone predicates, the article improves upper bounds on the query complexity of computing a minimal set when the number of minimal sets is constant. This applies for example to the computation of a minimal unsatisfiable subset (MUS) for CNF formulas, but also to the computation of prime implicants and implicates, with immediate application in a number of AI settings.

On computing minimal independent support and its applications to sampling and counting

Constrained sampling and counting are two fundamental problems arising in domains ranging from artificial intelligence and security, to hardware and software testing. Recent approaches to approximate solutions for these problems rely on employing SAT solvers and universal hash functions that are typically encoded as XOR constraints of length n/2 for an input formula with n variables. As the runtime performance of SAT solvers heavily depends on the length of XOR constraints, recent research effort has been focused on reduction of length of XOR constraints. Consequently, a notion of Independent Support was proposed, and it was shown that constructing XORs over independent support (if known) can lead to a significant reduction in the length of XOR constraints without losing the theoretical guarantees of sampling and counting algorithms. In this paper, we present the first algorithmic procedure (and a corresponding tool, called MIS) to determine minimal independent support for a given CNF formula by employing a reduction to group minimal unsatisfiable subsets (GMUS). By utilizing minimal independent supports computed by MIS, we provide new tighter bounds on the length of XOR constraints for constrained counting and sampling. Furthermore, the universal hash functions constructed from independent supports computed by MIS provide two to three orders of magnitude performance improvement in state-of-the-art constrained sampling and counting tools, while still retaining theoretical guarantees.

Introduction to the special issue on CSP technologies in artificial intelligence

Although constraint programming has evolved as an independent domain of research per se, it remains at the same time one of the most fertile and successful areas of artificial intelligence (A.I.). This special issue is intended to illustrate various current topics and research trends in constraint programming that pertain to A.I. It gathers a series of research contributions that range from the study of tractable classes of non-binary CSPs, to efficient algorithms for the extraction of minimal unsatisfiable subsets in constraint networks, and include the application of constraint programming for deciding the robustness of wireless sensor networks and interval methods for numerical constraint programming. The papers in this issue are all revised and expanded research contributions based on results that were presented at the SAT and CSP technologies track at IEEE ICTAI 2013. They were selected as the papers with the highest review scores amongst the 47 full papers submitted to the track, after all the papers submitted to the track had been thoroughly reviewed by at least three reviewers. The authors of the selected papers in the track were invited to submit substantially expanded and revised contributions for this issue. Each of these expanded contributions underwent two additional reviews. We gratefully acknowledge the reviewers for providing deep insights and useful suggestions that helped improve the papers. Finally, we would like to thank Gilles Pesant and Michela Milano, Editors in chief of Constraints, for inviting us to edit this special issue and for their expert help at the different stages of the process.

Fast, flexible MUS enumeration

The problem of enumerating minimal unsatisfiable subsets (MUSes) of an infeasible constraint system is challenging due first to the complexity of computing even a single MUS and second to the potentially intractable number of MUSes an instance may contain. In the face of the latter issue, when complete enumeration is not feasible, a partial enumeration of MUSes can be valuable, ideally with a time cost for each MUS output no greater than that needed to extract a single MUS. Recently, two papers independently presented a new MUS enumeration algorithm well suited to partial MUS enumeration (Liffiton and Malik, 2013, Previti and Marques-Silva, 2013). The algorithm exhibits good anytime performance, steadily producing MUSes throughout its execution; it is constraint agnostic, applying equally well to any type of constraint system; and its flexible structure allows it to incorporate advances in single MUS extraction algorithms and eases the creation of further improvements and modifications. This paper unifies and expands upon the earlier work, presenting a detailed explanation of the algorithm's operation in a framework that also enables clearer comparisons to previous approaches, and we present a new optimization of the algorithm as well. Expanded experimental results illustrate the algorithm's improvement over past approaches and newly explore some of its variants.

Partial MUS Enumeration

Minimal explanations of infeasibility find a wide range of uses. In the Boolean domain, these are referred to as Minimal Unsatisfiable Subsets (MUSes). In some settings, one needs to enumerate MUSes of a Boolean formula. Most often the goal is to enumerate all MUSes. In cases where this is computationally infeasible, an alternative is to enumerate some MUSes. This paper develops a novel approach for partial enumeration of MUSes, that complements existing alternatives. If the enumeration of all MUSes is viable, then existing alternatives represent the best option. However, for formulas where the enumeration of all MUSes is unrealistic, our approach provides a solution for enumerating some MUSes within a given time bound. The experimental results focus on formulas for which existing solutions are unable to enumerate MUSes, and shows that the new approach can in most cases enumerate a non-negligible number of MUSes within a given time bound.

A hybrid algorithm for finding minimal unsatisfiable subsets in over-constrained CSPs

Minimal Unsatisfiable Subsets (MUSes) are the subsets of constraints of an overconstrained constraint satisfaction problem (CSP) that cannot be satisfied simultaneously and therefore are responsible for the conflict in the CSP. In this paper, we present a hybrid algorithm for finding MUSes in overconstrained CSPs. The hybrid algorithm combines the direct and the indirect approaches to finding MUSes in overconstrained CSPs. Experimentation with random CSPs reveals that the hybrid approach is not only quite efficient but when operating under a time bound it finds a more representative set of MUSes. © 2011 Wiley Periodicals, Inc.

DIRECT ALGORITHMS FOR FINDING MINIMAL UNSATISFIABLE SUBSETS IN OVER-CONSTRAINED CSPs

In many situations, an explanation of the reasons behind inconsistency in an overconstrained CSP is required. This explanation can be given in terms of minimal unsatisfiable subsets (MUSes) of constraints. This paper presents algorithms for finding minimal unsatisfiable subsets (MUSes) of constraints in overconstrained CSPs with finite domains and binary constraints. The approach followed is to generate subsets in the subset space, test them for consistency and record the inconsistent subsets found. We present three algorithms as variations of this basic approach. Each algorithm generates subsets in the subset space in a different order and curtails search by employing various search pruning mechanisms. The proposed algorithms are anytime algorithms: a time limit can be set on an algorithm's search and the algorithm can be made to find a subset of MUSes. Experimental evaluation of the proposed algorithms demonstrates that they perform two to three orders of magnitude better than the existing indirect algorithms. Furthermore, the algorithms are able to find MUSes in large CSP benchmarks.

Incremental Satisfiability and Implication for UTVPI Constraints

Unit two-variable-per-inequality (UTVPI) constraints form one of the largest class of integer constraints that are polynomial time solvable (unless P = NP). There is considerable interest in their use for constraint solving, abstract interpretation, spatial database algorithms, and theorem proving. In this paper we develop new incremental algorithms for UTVPI constraint satisfaction and implication checking that require ℴ(m + n log n + p) time and ℴ(n + m + p) space to incrementally check satisfiability of m UTVPI constraints on n variables, and we check the implication of p UTVPI constraints. The algorithms can be straightforwardly extended to create nonincremental implication checking and generation of all (nonredundant) implied constraints, as well as generate minimal unsatisfiable subsets and minimal implicants.

Using heuristics to find minimal unsatisfiable subformulas in satisfiability problems

In this paper, we propose efficient algorithms to extract minimal unsatisfiable subsets of clauses or variables in unsatisfiable propositional formulas. Such subsets yield unsatisfiable propositional subformulas that become satisfiable when any of their clauses or variables is removed. These subformulas have numerous applications, including proving unsatisfiability and post-infeasibility analysis. The algorithms we propose are based on heuristics, and thus, can be applied to large instances. Furthermore, we show that, in some cases, the minimality of the subformulas can be proven with these algorithms. We also present an original algorithm to find minimum cardinality unsatisfiable subformulas in smaller instances. Finally, we report computational experiments on unsatisfiable instances from various sources, that demonstrate the effectiveness of our algorithms.

Algorithms for Computing Minimal Unsatisfiable Subsets of Constraints

Much research in the area of constraint processing has recently been focused on extracting small unsatisfiable "cores" from unsatisfiable constraint systems with the goal of finding minimal unsatis...

R-SATCHMO: Refinements on I-SATCHMO

In this paper, three refinements on I-SATCHMO are presented. I-SATCHMO is an application of the backjumping strategy for SATCHMO, which is a model generation theorem prover and is formalized as positive unit hyperresolution tableaux, to pruning away redundant branches. However, I-SATCHMO does not completely implement backjumping. Since marking an atom as useful whenever it contributes to a forward chaining reasoning, I-SATCHMO suffers from two problems. One is that the atoms that only contribute to irrelevant forward chaining reasoning are marked as 'useful'. The other is that the atoms that are only useful for the sibling branches of the branch under consideration are taken as 'useful' for the branch. In both cases, I-SATCHMO might take some irrelevant violated clauses as 'relevant' and therefore use them to expand the search space. Moreover, not utilizing the partial reasoning results derivable during reasoning, I-SATCHMO might make repeated reasoning. Addressing these problems, our extension, called R-SATCHMO, refines I-SATCHMO in three ways: (1) only making the atoms that contribute to relevant forward chaining reasoning as useful; (2) it distinguishes between the set of useful atoms in different branches of a proof tree; (3) it summarizes the derived refutations as nogoods and utilizes them to avoid repeated reasoning. Hence, R-SATCHMO always searches a subspace of that which I-SATCHMO does. R-SATCHMO is optimal in the sense that during reasoning no same complete subtree is constructed in a proof tree. Moreover, R-SATCHMO can be used to generate minimal unsatisfiable subsets of an unsatisfiable set of clauses, and the strategies proposed in R-SATCHMO are applicable to minimal model generation procedures. We describe our refinements, present our implementation, prove the correctness, provide examples to demonstrate the power of our refinements, and discuss the application to minimal model generation.

The satisfiability constraint gap

We describe an experimental investigation of the satisfiability phase transition for several different classes of randomly generated problems. We show that the “conventional” picture of easy-hard-easy problem difficulty is inadequate. In particular, there is a region of very variable problem difficulty where problems are typically underconstrained and satisfiable. Within this region, problems can be orders of magnitude harder than problems in the middle of the satisfiability phase transition. These extraordinarily hard problems appear to be associated with a “constraint gap”. That is, a region where search is a maximum as the amount of constraint propagation is a minimum. We show that the position and shape of this constraint gap change little with problem size. Unlike hard problems in the middle of the satisfiability phase transition, hard problems in the variable region are not critically constrained between satisfiability and unsatisfiability. Indeed, hard problems in the variable region often contain a small and unique minimal unsatisfiable subset or reduce at an early stage in search to a hard unsatisfiable subproblem with a small and unique minimal unsatisfiable subset. The difficulty in solving such problems is thus in identifying the minimal unsatisfiable subset from the many irrelevant clauses. The existence of a constraint gap greatly hinders our ability to find such minimal unsatisfiable subsets. However, it remains open whether these problems remain hard for more intelligent backtracking procedures. We conjecture that these results will generalize both to other SAT problem classes, and to the phase transitions of other NP-hard problems.