Maarten Mariën

SAT(ID): satisfiability of propositional logic extended with inductive definitions

We investigate the satisfiability problem, SAT(ID), of an extension of propositional logic with inductive definitions. We demonstrate how to extend existing SAT solvers to become SAT(ID) solvers, and provide an implementation on top of MiniSat. We also report on a performance study, in which our implementation exhibits the expected benefits: full use of the underlying SAT solvers potential.

Grounding FO and FO(ID) with bounds

Grounding is the task of reducing a first-order theory and finite domain to an equivalent propositional theory. It is used as preprocessing phase in many logic-based reasoning systems. Such systems provide a rich first-order input language to a user and can rely on efficient propositional solvers to perform the actual reasoning. Besides a first-order theory and finite domain, the input for grounders contains in many applications also additional data. By exploiting this data, the size of the grounders output can often be reduced significantly. A common practice to improve the efficiency of a grounder in this context is by manually adding semantically redundant information to the input theory, indicating where and when the grounder should exploit the data. In this paper we present a method to compute and add such redundant information automatically. Our method therefore simplifies the task of writing input theories that can be grounded efficiently by current systems. We first present our method for classical first-order logic (FO) theories. Then we extend it to FO(ID), the extension of FO with inductive definitions, which allows for more concise and comprehensive input theories. We discuss implementation issues and experimentally validate the practical applicability of our method.

On the Relation Between ID-Logic and Answer Set Programming

This paper is an analysis of two knowledge representation extensions of logic programming, namely Answer Set Programming and ID-Logic. Our aim is to compare both logics on the level of declarative reading, practical methodology and formal semantics. At the level of methodology, we put forward the thesis that in many (but not all) existing applications of ASP, an ASP program is used to encode definitions and assertions, similar as in ID-Logic. We illustrate this thesis with an example and present a formal result that supports it, namely an equivalence preserving translation from a class of ID-Logic theories into ASP. This translation can be exploited also to use the current efficient ASP solvers to reason on ID-Logic theories and it has been used to implement a model generator for ID-Logic.

Satisfiability Checking for PC(ID)

The logic FO(ID) extends classical first order logic with inductive definitions. This paper studies the satisifiability problem for PC(ID), its propositional fragment. We develop a framework for model generation in this logic, present an algorithm and prove its correctness. As FO(ID) is an integration of classical logic and logic programming, our algorithm integrates techniques from SAT and ASP. We report on a prototype system, called MidL, experimentally validating our approach.

Predicate introduction under stable and well-founded semantics

This paper studies the transformation of “predicate introduction”: replacing a complex formula in an existing logic program by a newly defined predicate. From a knowledge representation perspective, such transformations can be used to eliminate redundancy or to simplify a theory. From a more practical point of view, they can also be used to transform a theory into a normal form imposed by certain inference programs or theorems, e.g., through the elimination of universal quantifiers. In this paper, we study when predicate introduction is equivalence preserving under the stable and well-founded semantics. We do this in the algebraic framework of “approximation theory”; this is a fixpoint theory for non-monotone operators that generalizes all main semantics of various non-monotone logics, including Logic Programming, Default Logic and Autoepistemic Logic. We prove an abstract, algebraic equivalence result and then instantiate this abstract theorem to Logic Programming under the stable and well-founded semantics.

Integrating inductive definitions in SAT

We investigate techniques for supporting inductive definitions (IDs) in SAT, and report on an implementation, called MidL, of the resulting solver. This solver was first introduced in [11], as a part of a declarative problem solving framework. We go about our investigation by proposing a new formulation of the semantics of IDs as presented in [2]. This new formulation suggests a way to perform the computational task involved, resulting in an algorithm supporting IDs. We show in detail how to integrate our algorithm with traditional SAT solving techniques. We also point out the similarities with another algorithm that was recently developed for ASP [1]. Indeed, our formulation reveals a very tight relation with stable model semantics. We conclude by an experimental validation of our approach using MidL.