Satisfiability Checking for PC(ID)

Artificial Intelligence

Ternovska

2007

Self Cite

Temporal reasoning has always been a major test case for knowledge representation formalisms. In this paper, we develop an inductive variant of the situation calculus in ID-logic, classical logic extended with Inductive Definitions. This logic has been proposed recently and is an extension of classical logic. It allows for a uniform representation of various forms of definitions, including monotone inductive definitions and non-monotone forms of inductive definitions such as iterated induction and induction over well-founded posets. We show that the role of such complex forms of definitions is not limited to mathematics but extends to commonsense knowledge representation. In the ID-logic axiomatization of the situation calculus, fluents and causality predicates are defined by simultaneous induction on the well-founded poset of situations. The inductive approach allows us to solve the ramification problem for the situation calculus in a uniform and modular way. Our solution is among the most general solutions for the ramification problem in the situation calculus. Using previously developed modularity techniques, we show that the basic variant of the inductive situation calculus without ramification rules is equivalent to Reiter-style situation calculus.

“…Finally, observe that in the latter theory, the predicate symbols I F i , C F i and C ¬F i occur only at the left hand side of the explicit definitions in (21).…”

Section: Theorem 5 a Basic Action Theory Of The Inductive Situation Cmentioning

confidence: 98%

“…Depending on the expressiveness allowed, the framework captures various complexity classes, including P and NP. Several ID-logic solvers have been developed [30,21].…”

Section: Introductionmentioning

confidence: 99%

Inductive situation calculus

Artificial Intelligence

Ternovska

2007

Self Cite

“…for the Hamiltonian circuit problem. 10 The propositional instances obtained this way are over 50 times the size of those using an ID encoding, for graphs with 150 nodes, and were simply too big for graphs with 200 nodes. In Table 1, the column for MiniSAT refers to these instances.…”

Section: Discussionmentioning

confidence: 99%

“…An extension of classical logic with inductive definitions, FO(ID), is given in [2]. A first model generator for the propositional fragment of this logic, SAT(ID), was presented in [10]. An improved version of this solver was reported on in [11], as a part of a declarative problem solving framework.…”

Section: Introductionmentioning

confidence: 99%

Integrating Inductive Definitions in SAT

Mariën

Wittocx

Logic for Programming, Artificial Intelligence, and Reasoning

Self Cite

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.

“…Another application of this work could be the development of tools to check the correctness of the outputs generated by PC(ID) model generators such as MidL [14]. Given a PC(ID) theory T as input, MidL outputs a model for T or concludes that T is unsatisfiable.…”

Section: Introductionmentioning

confidence: 99%

A Deductive System for PC(ID)

Hou

Wittocx

Logic Programming and Nonmonotonic Reasoning

Self Cite

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.