MathSAT: Tight Integration of SAT and Mathematical Decision Procedures

Recent improvements in propositional satisfiability techniques (SAT) made it possible to tackle successfully some hard real-world problems (e.g., model-checking, circuit testing, propositional planning) by encoding into SAT. However, a purely Boolean representation is not expressive enough for many other real-world applications, including the verification of timed and hybrid systems, of proof obligations in software, and of circuit design at RTL level. These problems can be naturally modeled as satisfiability in linear arithmetic logic (LAL), that is, the Boolean combination of propositional variables and linear constraints over numerical variables. In this paper we present MathSAT, a new, SAT-based decision procedure for LAL, based on the (known approach) of integrating a state-of-the-art SAT solver with a dedicated mathematical solver for LAL. We improve MathSAT in two different directions. First, the top‐level line procedure is enhanced and now features a tighter integration between the Boolean search and the mathematical solver. In particular, we allow for theory-driven backjumping and learning, and theory-driven deduction; we use static learning in order to reduce the number of Boolean models that are mathematically inconsistent; we exploit problem clustering in order to partition mathematical reasoning; and we define a stack-based interface that allows us to implement mathematical reasoning in an incremental and backtrackable way. Second, the mathematical solver is based on layering; that is, the consistency of (partial) assignments is checked in theories of increasing strength (equality and uninterpreted functions, linear arithmetic over the reals, linear arithmetic over the integers). For each of these layers, a dedicated (sub)solver is used. Cheaper solvers are called first, and detection of inconsistency makes call of the subsequent solvers superfluous. We provide a through experimental evaluation of our approach, by taking into account a large set of previously proposed benchmarks. We first investigate the relative benefits and drawbacks of each proposed technique by comparison with respect to a reference option setting. We then demonstrate the global effectiveness of our approach by a comparison with several state-of-the-art decision procedures. We show that the behavior of MathSAT is often superior to its competitors, both on LAL and in the subclass of difference logic. This work has been partly supported by ISAAC, a European-sponsored project, contract no. AST3-CT-2003-501848; by ORCHID, a project sponsored by Provincia Autonoma di Trento; and by a grant from Intel Corporation. The work of T. Junttila has also been supported by the Academy of Finland, project 53695. S. Schulz has also been supported by a grant of the Italian Ministero dell'Istruzione, dell'Università e della Ricerca and the University of Verona.     

Solving parity games by a reduction to SAT

This paper presents a reduction from the problem of solving parity games to the satisfiability problem in propositional logic (SAT). The reduction is done in two stages, first into difference logic, i.e. SAT combined with the theory of integer differences, an instance of the SAT modulo theories (SMT) framework. In the second stage the integer variables and constraints of the difference logic encoding are replaced with a set of Boolean variables and constraints on them, giving rise to a pure SAT encoding of the problem. The reduction uses Jurdziński?s characterisation of winning strategies via progress measures. The reduction is motivated by the success of SAT solvers in symbolic verification, bounded model checking in particular. The paper reports on prototype implementations of the reductions and presents some experimental results.     

Complexity results for answer set programming with bounded predicate arities and implications

Answer set programming is a declarative programming paradigm rooted in logic programming and non-monotonic reasoning. This formalism has become a host for expressing knowledge representation problems, which reinforces the interest in efficient methods for computing answer sets of a logic program. The complexity of various reasoning tasks for general answer set programming has been amply studied and is understood quite well. In this paper, we present a language fragment in which the arities of predicates are bounded by a constant. Subsequently, we analyze the complexity of various reasoning tasks and computational problems for this fragment, comprising answer set existence, brave and cautious reasoning, and strong equivalence. Generally speaking, it turns out that the complexity drops significantly with respect to the full non-ground language, but is still harder than for the respective ground or propositional languages. These results have several implications, most importantly for solver implementations: Virtually all currently available solvers have exponential (in the size of the input) space requirements even for programs with bounded predicate arities, while our results indicate that for those programs polynomial space should be sufficient. This can be seen as a manifestation of the “grounding bottleneck” (meaning that programs are first instantiated and then solved) from which answer set programming solvers currently suffer. As a final contribution, we provide a sketch of a method that can avoid the exponential space requirement for programs with bounded predicate arities. Some results in this paper have been presented in preliminary form at KR 2004 [15].     

Bounded model checking of software using SMT solvers instead of SAT solvers

C bounded model checking (cbmc) has proved to be a successful approach to automatic software analysis. The key idea is to (i) build a propositional formula whose models correspond to program traces (of bounded length) that violate some given property and (ii) use state-of-the-art SAT solvers to check the resulting formulae for satisfiability. In this paper, we propose a generalisation of the cbmc approach on the basis of an encoding into richer (but still decidable) theories than propositional logic. We show that our approach may lead to considerably more compact formulae than those obtained with cbmc. We have built a prototype implementation of our technique that uses a satisfiability modulo theories (SMT) solver to solve the resulting formulae. Computer experiments indicate that our approach compares favourably with—and on some significant problems outperforms—cbmc.     

General default logic

In this paper, we propose a new nonmonotonic logic called general default logic. On the one hand, it generalizes Reiter’s default logic by adding to it rule-like operators used in logic programming. On the other hand, it extends logic programming by allowing arbitrary propositional formulas. We show that with this new logic, one can formalize naturally rule constraints, generalized closed world assumptions, and conditional defaults. We show that under a notion of strong equivalence, sentences of this new logic can be converted to a normal form. We also investigate the computational complexity of various reasoning tasks in the logic, and relate it to some other nonmonotonic formalisms such as Lin and Shoham’s logic of GK and Moore’s autoepistemic logic.     

Special section on advances in reachability analysis and decision procedures: contributions to abstraction-based system verification

Reachability analysis asks whether a system can evolve from legitimate initial states to unsafe states. It is thus a fundamental tool in the validation of computational systems—be they software, hardware, or a combination thereof. We recall a standard approach for reachability analysis, which captures the system in a transition system, forms another transition system as an over-approximation, and performs an incremental fixed-point computation on that over-approximation to determine whether unsafe states can be reached. We show this method to be sound for proving the absence of errors, and discuss its limitations for proving the presence of errors, as well as some means of addressing this limitation. We then sketch how program annotations for data integrity constraints and interface specifications—as in Bertrand Meyer’s paradigm of Design by Contract—can facilitate the validation of modular programs, e.g., by obtaining more precise verification conditions for software verification supported by automated theorem proving. Then we recap how the decision problem of satisfiability for formulae of logics with theories—e.g., bit-vector arithmetic—can be used to construct an over-approximating transition system for a program. Programs with data types comprised of bit-vectors of finite width require bespoke decision procedures for satisfiability. Finite-width data types challenge the reduction of that decision problem to one that off-the-shelf tools can solve effectively, e.g., SAT solvers for propositional logic. In that context, we recall the Tseitin encoding which converts formulae from that logic into conjunctive normal form—the standard format for most SAT solvers—with only linear blow-up in the size of the formula, but linear increase in the number of variables. Finally, we discuss the contributions that the three papers in this special section make in the areas that we sketched above.     

The satisfiability problem in regular CNF-formulas

 In this paper we deal with the propositional satisfiability (SAT) problem for a kind of multiple-valued clausal forms known as regular CNF-formulas and extend some known results from classical logic to this kind of formulas. We present a Davis–Putnam-style satisfiability checking procedure for regular CNF-formulas equipped with suitable data structures and prove its completeness. Then, we describe a series of experiments for regular random 3-SAT instances. We observe that, for the regular 3-SAT problem with this procedure, there exists a threshold of the ratio of clauses to variables such that (i) the most computationally difficult instances tend to be found near the threshold, (ii) there is a sharp transition from satisfiable to unsatisfiable instances at the threshold and (iii) the value of the threshold increases as the number of truth values considered increases. Instances in the hard part provide benchmarks for the evaluation of regular satisfiability solvers.     

Comparing parameter schemes for propositional reasoning: An empirical study

Complete propositional reasoning is impractical as a tool in artificial intelligence, because it is computationally intractable. Most current approaches to limited propositional reasoning cannot easily be adjusted to use more (or less) time to prove more (fewer) theorems when the task requires it. This difficulty can be solved byparameterizing the reasoner: designing in a ‘power dial’ giving the user fine control over cost and performance. System designers face the significant problem of choosing the best parameter scheme to use. This paper proposes an empirical methodology for comparing parameter schemes and illustrates its use in comparing eight such schemes for a given complete, resolution-based propositional reasoner. From the data, a clear choice emerges as the most preferable of the eight.     

An introduction to inductive programming

The research field of inductive programming is concerned with the design of algorithms for learning computer programs with complex flow of control (typically recursive calls) from incomplete specifications such as examples. We introduce a basic algorithmic approach for inductive programming and illustrate it with three systems: dialogs learns logic programs by combining inductive and abductive reasoning; the classical thesys system and its extension igor1 learn functional programs based on a recurrence detection mechanism in traces; igor2 learns functional programs over algebraic data-types making use of constructor-term rewriting systems. Furthermore, we give a short history of inductive programming, discuss related approaches, and give hints about current applications and possible future directions of research. A short, non-technical version of this paper appears in C. Sammut, editor, Encyclopedia of Machine Learning, Springer–Verlag, forthcoming. The paper was written while the first author was on sabbatical in 2006/2007 at Sabancı University in İstanbul, Turkey.     

Limitations of restricted branching in clause learning

The techniques for making decisions, that is, branching, play a central role in complete methods for solving structured instances of constraint satisfaction problems (CSPs). In this work we consider branching heuristics in the context of propositional satisfiability (SAT), where CSPs are expressed as propositional formulas. In practice, there are cases when SAT solvers based on the Davis-Putnam-Logemann-Loveland procedure (DPLL) benefit from limiting the set of variables the solver is allowed to branch on to so called input variables which provide a strong unit propagation backdoor set to any SAT instance. Theoretically, however, restricting branching to input variables implies a super-polynomial increase in the length of the optimal proofs for DPLL (without clause learning), and thus input-restricted DPLL cannot polynomially simulate DPLL. In this paper we settle the case of DPLL with clause learning. Surprisingly, even with unlimited restarts, input-restricted clause learning DPLL cannot simulate DPLL (even without clause learning). The opposite also holds, and hence DPLL and input-restricted clause learning DPLL are polynomially incomparable. Additionally, we analyze the effect of input-restricted branching on clause learning solvers in practice with various structured real-world benchmarks. This is an extended version of a paper [27] presented at the 13th International Conference on Principles and Practice of Constraint Programming (CP 2007) in Providence, RI, USA. The first author gratefully acknowledges financial support from Helsinki Graduate School in Computer Science and Engineering, Academy of Finland (grants #211025 and #122399), Emil Aaltonen Foundation, Jenny and Antti Wihuri Foundation, Finnish Foundation for Technology Promotion TES, and Nokia Foundation. The second author gratefully acknowledges the financial support from Academy of Finland (grant #112016).     

Don’t care in SMT: building flexible yet efficient abstraction/refinement solvers

This paper describes a method for combining “off-the-shelf” SAT and constraint solvers for building an efficient Satisfiability Modulo Theories (SMT) solver for a wide range of theories. Our method follows the abstraction/refinement approach to simplify the implementation of custom SMT solvers. The expected performance penalty by not using an interweaved combination of SAT and theory solvers is reduced by generalising a Boolean solution of an SMT problem first via assigning don’t care to as many variables as possible. We then use the generalised solution to determine a thereby smaller constraint set to be handed over to the constraint solver for a background theory. We show that for many benchmarks and real-world problems, this optimisation results in considerably smaller and less complex constraint problems. The presented approach is particularly useful for assembling a practically viable SMT solver quickly, when neither a suitable SMT solver nor a corresponding incremental theory solver is available. We have implemented our approach in the ABsolver framework and applied the resulting solver successfully to an industrial case-study: the verification problems arising in verifying an electronic car steering control system impose non-linear arithmetic constraints, which do not fall into the domain of any other available solver.     

Function in Device Representation

We explore the meanings of the terms ‘structure’, ‘behaviour’, and, especially, ‘function’ in engineering practice. Computers provide great assistance in calculation tasks in engineering practice, but they also have great potential for helping with reasoning tasks. However, realising this vision requires precision in representing engineering knowledge, in which the terms mentioned above play a central role. We start with a simple ontology for representing objects and causal interactions between objects. Using this ontology, we investigate a range of meanings for the terms of interest. Specifically, we distinguish between function as effect on the environment, and a device-centred view of device function. In the former view, function is seen as an intended or desired role that an artifact plays in its environment. We identify an important concept called mode of deployment that is often left implicit, but whose explicit representation is necessary for correct and complete reasoning. We discuss the task of design and design verification in this framework. We end with a discussion that relates needs in the world to functions of artifacts created to satisfy the needs.     

Performance Rating via the Feast Indices

Based on algorithmic and computational studies, we present the Feast Indices which are new indicators for the realistic performance of many recent processors, as typically used in ‘low cost’ PC's/workstations up to (parallel) supercomputers. Since these tests are very specifically designed for the evaluation of modern numerical simulation techniques, with a special emphasis on `large scale' FEM computations and iterative solvers of Krylov-space/multigrid type, they examine new aspects in comparison to the standard benchmark tests (LINPACK, SPEC FP95, NAS, STREAM) and allow new qualitative and particularly quantitative ratings of the various hardware platforms. We explore the computational efficiency of certain Linear Algebra components, hereby applying the typical sparse approaches and additionally the sparse banded style from Feast which enables us to exploit a significant percentage of the available Peak performance. The tests include ‘simple’ matrix-vector applications as well as complete multigrid solvers with very robust smoothers which are necessary for the efficient treatment of highly adapted meshes. Received: February 10, 1999; revised: June 14, 1999     

Some Computational Aspects of distance-sat

In many AI fields, one must face the problem of finding a solution that is as close as possible to a given configuration. This paper addresses this problem in a propositional framework. We introduce the decision problem distance-sat, which consists in determining whether a propositional formula admits a model that disagrees with a given partial interpretation on at most d variables. The complexity of distance-sat and of several restrictions of it are identified. Two algorithms based on the well-known Davis/Logemann/Loveland search procedure for the satisfiability problem sat are presented so as to solve distance-sat for CNF formulas. Their computational behaviors are compared with the ones offered by sat solvers on sat encodings of distance-sat instances. The empirical evaluation allows us to draw firm conclusions about the respective performances of the algorithms and to relate the difficulty of distance-sat with the difficulty of sat from the practical side. A preliminary version of this paper appeared with the title “distance-sat: Complexity and Algorithms” in the proceedings of the 16 ^th National Conference on Artificial Intelligence (AAAI’99), pages 642–647, 1999.     

Implementing the Davis–Putnam Method

The method proposed by Davis, Putnam, Logemann, and Loveland for propositional reasoning, often referred to as the Davis–Putnam method, is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the Davis–Putnam method efficiently using the trie data structure for propositional clauses. A new technique of indexing only the first and last literals of clauses yields a unit propagation procedure whose complexity is sublinear to the number of occurrences of the variable in the input. We also show that the Davis–Putnam method can work better when unit subsumption is not used. We illustrate the performance of our programs on some quasigroup problems. The efficiency of our programs has enabled us to solve some open quasigroup problems.     

Solving satisfiability problems with preferences

Propositional satisfiability (SAT) is a success story in Computer Science and Artificial Intelligence: SAT solvers are currently used to solve problems in many different application domains, including planning and formal verification. The main reason for this success is that modern SAT solvers can successfully deal with problems having millions of variables. All these solvers are based on the Davis–Logemann–Loveland procedure (dll). In its original version, dll is a decision procedure, but it can be very easily modified in order to return one or all assignments satisfying the input set of clauses, assuming at least one exists. However, in many cases it is not enough to compute assignments satisfying all the input clauses: Indeed, the returned assignments have also to be “optimal” in some sense, e.g., they have to satisfy as many other constraints—expressed as preferences—as possible. In this paper we start with qualitative preferences on literals, defined as a partially ordered set (poset) of literals. Such a poset induces a poset on total assignments and leads to the definition of optimal model for a formula ψ as a minimal element of the poset on the models of ψ. We show (i) how dll can be extended in order to return one or all optimal models of ψ (once converted in clauses and assuming ψ is satisfiable), and (ii) how the same procedures can be used to compute optimal models wrt a qualitative preference on formulas and/or wrt a quantitative preference on literals or formulas. We implemented our ideas and we tested the resulting system on a variety of very challenging structured benchmarks. The results indicate that our implementation has comparable performances with other state-of-the-art systems, tailored for the specific problems we consider.     

Heuristic-Based Backtracking Relaxation for Propositional Satisfiability

In recent years backtrack search algorithms for propositional satisfiability (SAT) have been the subject of dramatic improvements. These improvements allowed SAT solvers to successfully solve instances with thousands or tens of thousands of variables. However, many new challenging problem instances are still too hard for current SAT solvers. As a result, further improvements to SAT technology are expected to have key consequences in solving hard real-world instances. This paper introduces a new idea: choosing the backtrack variable using a heuristic approach with the goal of diversifying the regions of the space that are explored during the search. The proposed heuristics are inspired by the heuristics proposed in recent years for the decision branching step of SAT solvers, namely, VSIDS and its improvements. Completeness conditions are established, which guarantee completeness for the new algorithm, as well as for any other incomplete backtracking algorithm. Experimental results on hundreds of instances derived from real-world problems show that the new technique is able to speed SAT solvers, while aborting fewer instances. These results clearly motivate the integration of heuristic backtracking in SAT solvers.     

Reducing the size of resolution proofs in linear time

DPLL-based SAT solvers progress by implicitly applying binary resolution. The resolution proofs that they generate are used, after the SAT solver’s run has terminated, for various purposes. Most notable uses in formal verification are: extracting an unsatisfiable core, extracting an interpolant, and detecting clauses that can be reused in an incremental satisfiability setting (the latter uses the proof only implicitly, during the run of the SAT solver). Making the resolution proof smaller can benefit all of these goals: it can lead to smaller cores, smaller interpolants, and smaller clauses that are propagated to the next SAT instance in an incremental setting. We suggest two methods that are linear in the size of the proof for doing so. Our first technique, called Recycle-Units uses each learned constant (unit clause) (x) for simplifying resolution steps in which x was the pivot, prior to when it was learned. Our second technique, called   simplifies proofs in which there are several nodes in the resolution graph, one of which dominates the others, that correspond to the same pivot. Our experiments with industrial instances show that these simplifications reduce the core by ≈5% and the proof by ≈13%. It reduces the core less than competing methods such as run- till- fix, but whereas our algorithms are linear in the size of the proof, the latter and other competing techniques are all exponential as they are based on SAT runs. If we consider the size of the proof (the resolution graph) as being polynomial in the number of variables (it is not necessarily the case in general), this gives our method an exponential time reduction comparing to existing tools for small core extraction. Our experiments show that this result is evident in practice more so for the second method: rarely it takes more than a few seconds, even when competing tools time out, and hence it can be used as a cheap proof post-processing procedure.