José Arrazola

Equivalence in Answer Set Programming

We study the notion of strong equivalence between two Answer Set programs and we show how some particular cases of testing strong equivalence between programs can be reduced to verify if a formula is a theorem in intuitionistic or classical logic. We present some program transformations for disjunctive programs, which can be used to simplify the structure of programs and reduce their size. These transformations are shown to be of interest for both computational and theoretical reasons. Then we propose how to generalize such transformations to deal with free programs (which allow the use of default negation in the head of clauses). We also present a linear time transformation that can reduce an augmented logic program (which allows nested expressions in both the head and body of clauses) to a program consisting only of standard disjunctive clauses and constraints.

Applications of intuitionistic logic in Answer Set Programming

We present some applications of intermediate logics in the field of Answer Set Programming (ASP). A brief, but comprehensive introduction to the answer set semantics, intuitionistic and other intermediate logics is given. Some equivalence notions and their applications are discussed. Some results on intermediate logics are shown, and applied later to prove properties of answer sets. A characterization of answer sets for logic programs with nested expressions is provided in terms of intuitionistic provability, generalizing a recent result given by Pearce. It is known that the answer set semantics for logic programs with nested expressions may select non-minimal models. Minimal models can be very important in some applications, therefore we studied them; in particular we obtain a characterization, in terms of intuitionistic logic, of answer sets which are also minimal models. We show that the logic G3 characterizes the notion of strong equivalence between programs under the semantic induced by these models. Finally we discuss possible applications and consequences of our results. They clearly state interesting links between ASP and intermediate logics, which might bring research in these two areas together.

Safe beliefs for propositional theories

We propose an extension of answer sets, that we call safe beliefs, that can be used to study several properties and notions of answer sets and logic programming from a more general point of view. Our definiti on, based on intuitionistic logic and following ideas from D. Pearce [Stable inference as intuitionistic validity, Logic Programming 38 (1999) 79–91], also provides a general approach to define several semantics based on different logics or inference systems. We prove that, in particular, intuitionistic logic can be replaced with any other proper intermediate logic without modifying the resulting semantics. We also show that the answer set semantics satisfies an important property, the “extension by definition”, that can be used to construct program translations. As a result we are able to provide a polynomial translation from propositional theories into the class of disjunctive programs.

Equivalence for the G 3 ′ -stable models semantics

Abstract We study the notion of strong equivalence between two disjunctive logic programs under the G 3 ′ -stable model semantics, also called the p-stable semantics, and we show how some particular cases of testing strong equivalence between programs can be reduced to checking whether a formula is a theorem in some paraconsistent logic, or in some cases in classical logic. We also present some program transformations for disjunctive programs which can be used to reduce the size of a program.

Closing the Gap between the Stable Semantics and Extensions of WFS

In order to really understand all aspects of logic-based program development of different semantics, it would be useful to have a common solid logical foundation. The stable semantics has one already based on intuitionistic logic I and using the notion of completions. Since S4 expresses I then the stable semantics can be fully represented in S4. We propose the same approach to define extensions of the WFS semantics. We distinguish a particular semantics that we call AS-WFS wich is defined over general propositional theories, can be defined via completions using S4. Interesting AS-WFS seems to satisfy most of the principles of a well behaved semantics. Our general goal is to propose S4 and completions to study the formal behavior of different semantics.

Three valued logic of Łukasiewicz for modeling semantics of logic programs

In order to really understand all aspects of logic-based program development of different semantics, it would be useful to have a common solid logical foundation.

A logical approach to A-Prolog

It has been recently provided a characterization of Answer Sets by intuitionistic logic as follows: a literal is entailed by a program in the stable model semantics if and only if it belongs to every intuitionistically complete and consistent extension of the program formed by adding only negated literals. We show that if we replace intuitionistic logic by any si-logic the result still holds.

Debugging in A-Prolog: A Logical Approach

A-Prolog, Answer Set Programming or Stable Model Programming, is an important outcome of the theoretical work on Nonmonotonic Reasoning and AI applications of Logic Programming in the last 15 years. In the full version of this paper we study interesting applications of logic in the field of answer sets.