Filip Marić

Instance-Based Selection of Policies for SAT Solvers

Execution of most of the modern DPLL-based SAT solvers is guided by a number of heuristics. Decisions made during the search process are usually driven by some fixed heuristic policies. Despite the outstanding progress in SAT solving in recent years, there is still an appealing lack of techniques for selecting policies appropriate for solving specific input formulae. In this paper we present a methodology for instance-based selection of solvers policies that uses a data-mining classification technique. The methodology also relies on analysis of relationships between formulae, their families, and their suitable solving strategies. The evaluation results are very good, demonstrate practical usability of the methodology, and encourage further efforts in this direction.

Formalization and Implementation of Modern SAT Solvers

Most, if not all, state-of-the-art complete SAT solvers are complex variations of the DPLL procedure described in the early 1960’s. Published descriptions of these modern algorithms and related data structures are given either as high-level state transition systems or, informally, as (pseudo) programming language code. The former, although often accompanied with (informal) correctness proofs, are usually very abstract and do not specify many details crucial for efficient implementation. The latter usually do not involve any correctness argument and the given code is often hard to understand and modify. This paper aims to bridge this gap by presenting SAT solving algorithms that are formally proved correct and also contain information required for efficient implementation. We use a tutorial, top-down, approach and develop a SAT solver, starting from a simple design that is subsequently extended, step-by-step, with a requisite series of features. The heuristic parts of the solver are abstracted away, since they usually do not affect solver correctness (although they are very important for efficiency). All algorithms are given in pseudo-code and are accompanied with correctness conditions, given in Hoare logic style. The correctness proofs are formalized within the Isabelle theorem proving system and are available in the extended version of this paper. The given pseudo-code served as a basis for our SAT solver argo-sat.

meSAT: multiple encodings of CSP to SAT

One approach for solving Constraint Satisfaction Problems (CSP) (and related Constraint Optimization Problems (COP)) involving integer and Boolean variables is reduction to propositional satisfiability problem (SAT). A number of encodings (e.g., direct, log, support, order) for this purpose exist as well as specific encodings for some constraints that are often encountered (e.g., cardinality constraints, global constraints). However, there is no single encoding that performs well on all classes of problems and there is a need for a system that supports multiple encodings. We present a system that translates specifications of finite linear CSP problems into SAT instances using several well-known encodings, and their combinations. We also present a methodology for selecting a suitable encoding based on simple syntactic features of the input CSP instance. Thorough evaluation has been performed on large publicly available corpora and our encoding selection method improves upon the efficiency of existing encodings and state-of-the-art tools used in comparison.

URBiVA: uniform reduction to bit-vector arithmetic

We describe a system URBiVA for specifying and solving a range of problems by uniformly reducing them to bit-vector arithmetic (bva). A problem description is given in a C-like specification language and this high-level specification is transformed to a bva formula by symbolic execution. The formula is passed to a bva solver and, if it is satisfiable, its models give solutions of the problem. The system can be used for efficient modelling (specifying and solving) of a wide class of problems. Several state-of-the-art solvers for bva are currently used (Boolector, MathSAT, Yices) and additional solvers can be easily included. Hence, the system can be used not only as a specification and solving tool, but also as a platform for evaluation and comparison between bva solvers.

argo-lib: A Generic Platform for Decision Procedures

argo-lib is a C++ library that provides support for using decision procedures and for schemes for combining and augmenting decision procedures. This platform follows the smt-lib initiative which aims at establishing a library of benchmarks for satisfiability modulo theories. The platform can be easily integrated into other systems. It also enables comparison and unifying of different approaches, evaluation of new techniques and, hopefully, can help in advancing the field.

Simple algorithm portfolio for SAT

The importance of algorithm portfolio techniques for SAT has long been noted, and a number of very successful systems have been devised, including the most successful one—SATzilla. However, all these systems are quite complex (to understand, reimplement, or modify). In this paper we present an algorithm portfolio for SAT that is extremely simple, but in the same time so efficient that it outperforms SATzilla. For a new SAT instance to be solved, our portfolio finds its k-nearest neighbors from the training set and invokes a solver that performs the best for those instances. The main distinguishing feature of our algorithm portfolio is the locality of the selection procedure—the selection of a SAT solver is based only on few instances similar to the input one. An open source tool that implements our approach is publicly available.

Formalization of Abstract State Transition Systems for SAT

We present a formalization of modern SAT solvers and their properties in a form of abstract state transition systems. SAT solving procedures are described as transition relations over states that represent the values of the solvers global variables. Several different SAT solvers are formalized, including both the classical DPLL procedure and its state-of-the-art successors. The formalization is made within the Isabelle/HOL system and the total correctness (soundness, termination, completeness) is shown for each presented system (with respect to a simple notion of satisfiability that can be manually checked). The systems are defined in a general way and cover procedures used in a wide range of modern SAT solvers. Our formalization builds up on the previous work on state transition systems for SAT, but it gives machine-verifiable proofs, somewhat more general specifications, and weaker assumptions that ensure the key correctness properties. The presented proofs of formal correctness of the transition systems can be used as a key building block in proving correctness of SAT solvers by using other verification approaches.

Formalization of Incremental Simplex Algorithm by Stepwise Refinement

We present an Isabelle/HOL formalization and total correctness proof for the incremental version of the Simplex algorithm which is used in most state-of-the-art SMT solvers. Formalization relies on stepwise program and data refinement, starting from a simple specification, going through a number of fine refinement steps, and ending up in a fully executable functional implementation. Symmetries present in the algorithm are handled with special care.

Formalizing complex plane geometry

Deep connections between complex numbers and geometry had been well known and carefully studied centuries ago. Fundamental objects that are investigated are the complex plane (usually extended by a single infinite point), its objects (points, lines and circles), and groups of transformations that act on them (e.g., inversions and Möbius transformations). In this paper, we treat the geometry of complex numbers formally and present a fully mechanically verified development within the theorem prover Isabelle/HOL. Apart from applications in formalizing mathematics and in education, this work serves as a ground for formally investigating various non-Euclidean geometries and their intimate connections. We discuss different approaches to formalization and discuss the major advantages of the more algebraically oriented approach.

Proving Correctness of a KRK Chess Endgame Strategy by Using Isabelle/HOL and Z3

We describe an executable specification and a total correctness proof of a King and Rook vs King (KRK) chess endgame strategy within the proof assistant Isabelle/HOL. This work builds upon a previous computer-assisted correctness analysis performed using the constraint solver URSA. The distinctive feature of the present machine verifiable formalization is that all central properties have been automatically proved by the SMT solver Z3 integrated into Isabelle/HOL, after being suitably expressed in linear integer arithmetic. This demonstrates that the synergy between the state-of-the-art automated and interactive theorem proving is mature enough so that very complex conjectures from various AI domains can be proved almost in a “push-button” manner, yet in a rich logical framework offered by the modern ITP systems.