Chapter 20. Software Verification

This chapter covers an application of propositional satisfiability to program analysis. We focus on the discovery of programming flaws in low-level programs, such as embedded software. The loops in the program are unwound together with a property to form a formula, which is then converted into CNF. The method supports low-level programming constructs such as bit-wise operators or pointer arithmetic.

Chapter 27. Non-Clausal SAT and ATPG

When studying the propositional satisfiability problem (SAT), that is, the problem of deciding whether a propositional formula is satisfiable, it is typically assumed that the formula is given in the conjunctive normal form (CNF). Also most software tools for deciding satisfiability of a formula (SAT solvers) assume that their input is in CNF. An important reason for this is that it is simpler to develop efficient data structures and algorithms for CNF than for arbitrary formulas. On the other hand, using CNF makes efficient modeling of an application cumbersome. Therefore one often employs a more general formula representation in modeling and then transforms the formula into CNF for SAT solvers. Transforming a propositional formula in CNF either increases the formula size exponentially or requires the use of auxiliary variables, which can have an negative effect on the performance of a SAT solver in the worst-case. Moreover, by translating to CNF one often loses information about the structure of the original problem. In this chapter we survey methods for solving propositional satisfiability problems when the input formula is not given in CNF but as a general formula or even more compactly as a Boolean circuit. We show how the techniques applied in CNF level Davis-Putnam-Loveland-Logemann algorithm generalize to Boolean circuits and how the problem structure available in the circuit form can be exploited. Then we consider a closely related area of automatic test pattern generation (ATPG) for digital circuits and review classical ATPG algorithms, formulation of ATPG as a SAT problem, and advanced techniques for SAT-based ATPG.

A Formal Approach for Cautious Reasoning in Answer Set Programming (Extended Abstract)

The issue of describing in a formal way solving algorithms in various fields such as Propositional Satisfiability (SAT), Quantified SAT, Satisfiability Modulo Theories, Answer Set Programming (ASP), and Constraint ASP, has been relatively recently solved employing abstract solvers. In this paper we deal with cautious reasoning tasks in ASP, and design, implement and test novel abstract solutions, borrowed from backbone computation in SAT. By employing abstract solvers, we also formally show that the algorithms for solving cautious reasoning tasks in ASP are strongly related to those for computing backbones of Boolean formulas. Some of the new solutions have been implemented in the ASP solver WASP, and tested.

Treewidth-aware Reductions of Normal ASP to SAT - Is Normal ASP Harder than SAT after All?

Answer Set Programming (ASP) is a paradigm and problem modeling/solving toolkit for KR that is often invoked. There are plenty of results dedicated to studying the hardness of (fragments of) ASP. So far, these studies resulted in characterizations in terms of computational complexity as well as in fine-grained insights presented in form of dichotomy-style results, lower bounds when translating to other formalisms like propositional satisfiability (SAT), and even detailed parameterized complexity landscapes. A quite generic and prominent parameter in parameterized complexity originating from graph theory is the so-called treewidth, which in a sense captures structural density of a program. Recently, there was an increase in the number of treewidth-based solvers related to SAT. While there exist several translations from (normal) ASP to SAT, yet there is no reduction preserving treewidth or at least being aware of the treewidth increase. This paper deals with a novel reduction from normal ASP to SAT that is aware of the treewidth, and guarantees that a slight increase of treewidth is indeed sufficient. Then, we also present a new result establishing that when considering treewidth, already the fragment of normal ASP is slightly harder than SAT (under reasonable assumptions in computational complexity). This also confirms that our reduction probably cannot be significantly improved and that the slight increase of treewidth is unavoidable.

Abstract Solvers for Computing Cautious Consequences of ASP programs

Abstract solvers are a method to formally analyze algorithms that have been profitably used for describing, comparing and composing solving techniques in various fields such as Propositional Satisfiability (SAT), Quantified SAT, Satisfiability Modulo Theories, Answer Set Programming (ASP), and Constraint ASP.In this paper, we design, implement and test novel abstract solutions for cautious reasoning tasks in ASP. We show how to improve the current abstract solvers for cautious reasoning in ASP with new techniques borrowed from backbone computation in SAT, in order to design new solving algorithms. By doing so, we also formally show that the algorithms for solving cautious reasoning tasks in ASP are strongly related to those for computing backbones of Boolean formulas. We implement some of the new solutions in the ASP solver wasp and show that their performance are comparable to state-of-the-art solutions on the benchmark problems from the past ASP Competitions.

Sharpness of the Satisfiability Threshold for Non-Uniform Random k-SAT

We study a more general model to generate random instances of Propositional Satisfiability (SAT) with n Boolean variables, m clauses, and exactly k variables per clause. Additionally, our model is given an arbitrary probability distribution (p_1, ..., p_n) on the variable occurrences. Therefore, we call it non-uniform random k-SAT. The number m of randomly drawn clauses at which random formulas go from asymptotically almost surely (a.a.s.) satisfiable to a.a.s. unsatisfiable is called the satisfiability threshold. Such a threshold is called sharp if it approaches a step function as n increases. We identify conditions on the variable probability distribution (p_1, ..., p_n) under which the satisfiability threshold is sharp if its position is already known asymptotically. This result generalizes Friedgut’s sharpness result from uniform to non-uniform random k -SAT and implies sharpness for thresholds of a wide range of random k -SAT models with heterogeneous probability distributions, for example such models where the variable probabilities follow a power-law.

Evolving Solutions to Community-Structured Satisfiability Formulas

We study the ability of a simple mutation-only evolutionary algorithm to solve propositional satisfiability formulas with inherent community structure. We show that the community structure translates to good fitness-distance correlation properties, which implies that the objective function provides a strong signal in the search space for evolutionary algorithms to locate a satisfying assignment efficiently. We prove that when the formula clusters into communities of size s ∈ ω(logn) ∩O(nε/(2ε+2)) for some constant 0

Identifying Bottlenecks in Practical SAT-Based Model Finding for First-Order Logic Ontologies with Datasets

Satisfiability of first-order logic (FOL) ontologies is typically verified by translation to propositional satisfiability (SAT) problems, which is then tackled by a SAT solver. Unfortunately, SAT solvers often experience scalability issues when reasoning with FOL ontologies and even moderately sized datasets. While SAT solvers have been found to capably handle complex axiomatizations, finding models of datasets gets considerably more complex and time-intensive as the number of clause exponentially increases with increase in individuals and axiomatic complexity. We identify FOL definitions as a specific bottleneck and demonstrate via experiments that the presence of many defined terms of the highest arity significantly slows down model finding. We also show that removing optional definitions and substituting these terms by their definiens leads to a reduction in the number of clauses, which makes SAT-based model finding practical for over 100 individuals in a FOL theory.