Johan Wittocx

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. n nBesides 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. n nWe 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.

A prototype of a knowledge-based programming environment

This paper presents a proposal for a knowledge-based programming environment. Within this environment, declarative background knowledge, procedures, and concrete data are represented in suitable languages and combined in a flexible manner, which leads to a highly declarative programming style. We illustrate our approach with an example application and report on our prototype implementation.

Finite domain and symbolic inference methods for extensions of first-order logic

In this dissertation, we investigate various sorts of reasoning on finite structures and theories in the logic FO(·), a rich extension of classical logic with, amongst others, inductive definitions and aggregates. In particular, we study the tasks of constraint propagation, grounding, model revision, and debugging for FO(·).

Debugging for Model Expansion

Due to the development of efficient solvers, declarative problem solving frameworks based on model generation are becoming more and more applicable in practice. However, there are almost no tools to support debugging in these frameworks. For several reasons, current solvers are not suitable for debugging by tracing. In this paper, we propose a new solver algorithm for one of these frameworks, namely Model Expansion, that allows for debugging by tracing. We explain how to explore the trace of this solver in order to quickly locate a bug and we compare our debugging method with existing ones for Answer Set Programming and the Alloy system.

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.

Constraint Propagation for First-Order Logic and Inductive Definitions

In Constraint Programming, constraint propagation is a basic component of constraint satisfaction solvers. Here we study constraint propagation as a basic form of inference in the context of first-order logic (FO) and extensions with inductive definitions (FO(ID)) and aggregates (FO(AGG)). In a first, semantic approach, a theory of propagators and constraint propagation is developed for theories in the context of three-valued interpretations. We present an algorithm with polynomial-time data complexity. We show that constraint propagation in this manner can be represented by a datalog program. In a second, symbolic approach, the semantic algorithm is lifted to a constraint propagation algorithm in symbolic structures, symbolic representations of classes of structures. The third part of the article is an overview of existing and potential applications of constraint propagation for model generation, grounding, interactive search problems, approximate methods for ∃∀SO problems, and approximate query answering in incomplete databases.

Answer Set Programming’s Contributions to Classical Logic

Much research in logic programming and non-monotonic reasoning originates from dissatisfaction with classical logic as a knowledge representation language, and with classical deduction as a mode for automated reasoning. Discarding these classical roots has generated many interesting and fruitful ideas. However, to ensure the lasting impact of the results that have been achieved, it is important that they should not remain disconnected from their classical roots. Ultimately, a clear picture should emerge of what the achievements of answer set programming mean in the context of classical logic, so that they may be given their proper place in the canon of science. In this paper, a look at different aspects of ASP, in an effort to identify precisely the limitations of classical logic that they exposed and investigate how the ASP approaches can be transferred back to the classical setting. Among the issues we thus address are the closed world assumption, “classical” and default negation, default reasoning with exceptions, definitions, lp-functions and the interpolation technique and the strong introspection operator. We investigate the ASP-methodology to encode knowledge using these language constructs and come across a dichotomy in the ASP-methodology.

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.

A deductive system for PC(ID)

The logic FO(ID) uses ideas from the field of logic programming to extend first order logic with non-monotone inductive definitions. This paper studies a deductive inference method for PC(ID), its propositional fragment.We introduce a formal proof system based on the sequent calculus (Gentzen-style deductive system) for this logic. As PC(ID) is an integration of classical propositional logic and propositional inductive definitions, our deductive system integrates inference rules for propositional calculus and definitions.We prove the soundness and completeness of this deductive system for a slightly restricted fragment of PC(ID). We also give a counter-example to show that cut-elimination does not hold in this proof system.