David G. Mitchell

Generating hard satisfiability problems

Abstract We report results from large-scale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to generate random formulas that are hard, that is, for which satisfiability testing is quite difficult. Our results provide a benchmark for the evaluation of satisfiability testing procedures.

SOME PITFALLS FOR EXPERIMENTERS WITH RANDOM SAT

Abstract We consider the use of random CNF formulas in evaluating the performance of SAT testing algorithms, and in particular the role that the phase transition phenomenon plays in this use. Examples from the literature illustrate the importance of understanding the properties of formula distributions prior to designing an experiment. We expect this to be of increasing importance in the field.

Resolution and constraint satisfaction

We study two resolution-like refutation systems for finite-domain constraint satisfaction problems, and the efficiency of these and of common CSP algorithms. By comparing the relative strength of these systems, we show that for instances with domain size d, backtracking with 2-way branching is super-polynomially more powerful than backtracking with d-way branching. We compare these systems with propositional resolution, and show that every family of CNF formulas which are hard for propositional resolution induces families of CSP instances that are hard for most of the standard CSP algorithms in the literature.

The resolution complexity of random graph k -colorability

We consider the resolution proof complexity of propositional formulas which encode random instances of graph k-colorability. We obtain a tradeoff between the graph density and the resolution proof complexity. For random graphs with linearly many edges we obtain linear-exponential lower bounds on the size of resolution refutations. For random graphs with n vertices and any @e>0, we obtain a lower-bound tradeoff between graph density and refutation size that implies subexponential lower bounds of the form 2^n^^^@d for some @d>0 for non-k-colorability proofs of graphs with n vertices and O(n^3^/^2^-^1^/^k^-^@e) edges. We obtain sharper lower bounds for Davis-Putnam-DPLL proofs and for proofs in a system considered by McDiarmid. These proof complexity bounds imply that many natural algorithms for k-coloring or k-colorability have essentially the same exponential tradeoff lower bounds on their running times. We also show that very simple algorithms for k-colorability have upper bounds on their running times that are qualitatively similar to the lower bounds as a function of the graph density.

Enfragmo: a system for modelling and solving search problems with logic

In this paper, we present the Enfragmo system for specifying and solving combinatorial search problems. It supports natural specification of problems by providing users with a rich language, based on an extension of first order logic. Enfragmo takes as input a problem specification and a problem instance and produces a propositional CNF formula representing solutions to the instance, which is sent to a SAT solver. Because the specification language is high level, Enfragmo provides combinatorial problem solving capability to users without expertise in use of SAT solvers or algorithms for solving combinatorial problems. Here, we describe the specification language and implementation of Enfragmo, and give experimental evidence that its performance is comparable to that of related systems.

Resolution Complexity of Random Constraints

Random instances are widely used as benchmarks in evaluating algorithms for finite-domain constraint satisfaction problems (CSPs). We present an analysis that shows why deciding satisfiability of instances from some distributions is challenging for current complete methods. For a typical random CSP model, we show that when constraints are not too tight almost all unsatisfiable instances have a structural property which guarantees that unsatisfiability proofs in a certain resolution-like system must be of exponential size. This proof system can efficiently simulate the reasoning of a large class of CSP algorithms which will thus have exponential running time on these instances.

Expressive power and abstraction in Essence

Development of languages for specifying or modelling problems is an important direction in constraint modelling. To provide greater abstraction and modelling convenience, these languages are becoming more syntactically rich, leading to a variety of questions about their expressive power. In this paper, we consider the expressiveness of Essence, a specification language with a rich variety of syntactic features. We identify natural fragments of Essence that capture the complexity classes P, NP, all levels

On the complexity of model expansion

\Sigma_i^p

Minimum witnesses for unsatisfiable 2CNFs

of the polynomial-time hierarchy, and all levels k-NEXP of the nondeterministic exponential-time hierarchy. The union of these classes is the very large complexity class ELEMENTARY. One goal is to begin to understand which features play a role in the high expressive power of the language and which are purely features of convenience. We also discuss the formalization of arithmetic in Essence and related languages, a notion of capturing NP-search which is slightly different than that of capturing NP, and a conjectured limit to the expressive power of Essence. Our study is an application of descriptive complexity theory, and illustrates the value of taking a logic-based view of modelling and specification languages.

Grounding formulas with complex terms

We study the complexity of model expansion (MX), which is the problem of expanding a given finite structure with additional relations to produce a finite model of a given formula. This is the logical task underlying many practical constraint languages and systems for representing and solving search problems, and our work is motivated by the need to provide theoretical foundations for these. We present results on both data and combined complexity of MX for several fragments and extensions of FO that are relevant for this purpose, in particular the guarded fragment GFk of FO and extensions of FO and GFk with inductive definitions. We present these in the context of the two closely related, but more studied, problems of model checking and finite satisfiability. To obtain results on FO(ID), the extension of FO with inductive definitions, we provide translations between FO(ID) with FO(LFP), which are of independent interest.