Mauricio Osorio

Preferred extensions as stable models

Given an argumentation framework AF, we introduce a mapping function that constructs a disjunctive logic program P, such that the preferred extensions of AF correspond to the stable models of P, after intersecting each stable model with the relevant atoms. The given mapping function is of polynomial size w.r.t. AF. In particular, we identify that there is a direct relationship between the minimal models of a propositional formula and the preferred extensions of an argumentation framework by working on representing the defeated arguments. Then we show how to infer the preferred extensions of an argumentation framework by using UNSAT algorithms and disjunctive stable model solvers. The relevance of this result is that we define a direct relationship between one of the most satisfactory argumentation semantics and one of the most successful approach of nonmonotonic reasoning i.e., logic programming with the stable model semantics.

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.

A general theory of confluent rewriting systems for logic programming and its applications

Abstract Recently, Brass and Dix showed (J. Automat. Reason. 20(1) (1998) 143–165) that the well founded semantics WFS can be defined as a confluent calculus of transformation rules. This led not only to a simple extension to disjunctive programs (J. Logic Programming 38(3) (1999) 167–213), but also to a new computation of the well-founded semantics which is linear for a broad class of programs. We take this approach as a starting point and generalize it considerably by developing a general theory of Confluent LP-systems CS . Such a system CS is a rewriting system on the set of all logic programs over a fixed signature L and it induces in a natural way a canonical semantics. Moreover, we show four important applications of this theory: (1) most of the well-known semantics are induced by confluent LP-systems, (2) there are many more transformation rules that lead to confluent LP-systems, (3) semantics induced by such systems can be used to model aggregation, (4) the new systems can be used to construct interesting counterexamples to some conjectures about the space of well-behaved semantics.

Updates in answer set programming: An approach based on basic structural properties

We have studied the update operator ⊕1 defined for update sequences by Eiter et al. without tautologies and we have observed that it satisfies an interesting property. This property, which we call Weak Independence of Syntax (WIS), is similar to one of the postulates proposed by Alchourron, Gardenfors, and Makinson (AGM); only that in this case it applies to nonmonotonic logic. In addition, we consider other five additional basic properties about update programs and we show that ⊕1 satisfies them. This work continues the analysis of the AGM postulates with respect to the ⊕1 operator under a refined view that considers N2 as a monotonic logic which allows us to expand our understanding of answer sets. Moreover, N2 helped us to derive an alternative definition of ⊕1 avoiding the use of unnecessary extra atoms.

Inferring acceptable arguments with answer set programming

Following the argumentation framework and semantics proposed by Dung, we are interested in the problem of deciding which set of acceptable arguments support the decision making in an agent-based platform called CARREL. It is an agent-agency which mediates organ transplants. We present two possible ways to infer the stable and preferred extensions of an argumentation framework, one in a declarative way using answer set programming (ASP) and the other one in a procedure way.

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.

Semantics for possibilistic disjunctive programs

In this paper by considering answer set programming approach and some basic ideas from possibilistic logic, we introduce a possibilistic disjunctive logic programming approach able to deal with reasoning under uncertain and incomplete information. Our approach permits to use explicitly labels like possible, probable, plausible, etc., for capturing the incomplete state of a belief in a disjunctive logic program.

Theory of partial-order programming

Abstract This paper shows the use of partial-order program clauses and lattice domains for declarative programming. This paradigm is particularly useful for expressing concise solutions to problems from graph theory, program analysis, and database querying. These applications are characterized by a need to solve circular constraints and perform aggregate operations, a capability that is very clearly and efficiently provided by partial-order clauses. We present a novel approach to their declarative and operational semantics, as well as the correctness of the operational semantics. The declarative semantics is model-theoretic in nature, but the least model for any function is not the classical intersection of all models, but the greatest lower bound/least upper bound of the respective terms defined for this function in the different models. The operational semantics combines top-down goal reduction with memo-tables . In the partial-order programming framework, however, memoization is primarily needed in order to detect circular circular function calls. In general we need more than simple memoization when functions are defined circularly in terms of one another through monotonic functions. In such cases, we accumulate a set of functional-constraints and solve them by general fixed-point-finding procedure. In order to prove the correctness of memoization, a straightforward induction on the length of the derivation will not suffice because of the presence of the memo-table. However, since the entries in the table grow monotonically, we identify a suitable table invariant that captures the correctness of the derivation. The partial-order programming paradigm has been implemented and all examples shown in this paper have been tested using this implementation.

Pstable semantics for possibilistic logic programs

Uncertain information is present in many real applications e.g., medical domain, weather forecast, etc. The most common approaches for leading with this information are based on probability however some times; it is difficult to find suitable probabilities about some events. In this paper, we present a possibilistic logic programming approach which is based on possibilistic logic and PStable semantics. Possibilistic logic is a logic of uncertainty tailored for reasoning under incomplete evidence and Pstable Semantics is a solid semantics which emerges from the fusion of non-monotonic reasoning and logic programming; moreover it is able to express answer set semantics, and has strong connections with paraconsistent logics.