arXiv subjects

Pedro Cabalar

Publications and source records attributed to Pedro Cabalar.

45 records · Page 3Linked to original sources

arXiv2016

Enablers and Inhibitors in Causal Justifications of Logic Programs

To appear in Theory and Practice of Logic Programming (TPLP). In this paper we propose an extension of logic programming (LP) where each default literal derived from the well-founded model is associated to a justification represented as an algebraic expression. This expression contains both causal explanations (in the form of proof graphs built with rule labels) and terms under the scope of negation that stand for conditions that enable or disable the application of causal rules. Using some examples, we discuss how these new conditions, we respectively call "enablers" and "inhibitors", are intimately related to default negation and have an essentially different nature from regular cause-effect relations. The most important result is a formal comparison to the recent algebraic approaches for justifications in LP: "Why-not Provenance" (WnP) and "Causal Graphs" (CG). We show that the current approach extends both WnP and CG justifications under the Well-Founded Semantics and, as a byproduct, we also establish a formal relation between these two approaches.

cs.LO↗

arXiv2015

A Denotational Semantics for Equilibrium Logic

In this paper we provide an alternative semantics for Equilibrium Logic and its monotonic basis, the logic of Here-and-There (also known as Gödel's G3 logic) that relies on the idea of "denotation" of a formula, that is, a function that collects the set of models of that formula. Using the three-valued logic G3 as a starting point and an ordering relation (for which equilibrium/stable models are minimal elements) we provide several elementary operations for sets of interpretations. By analysing structural properties of the denotation of formulas, we show some expressiveness results for G3 such as, for instance, that conjunction is not expressible in terms of the other connectives. Moreover, the denotational semantics allows us to capture the set of equilibrium models of a formula with a simple and compact set expression. We also use this semantics to provide several formal definitions for entailment relations that are usual in the literature, and further introduce a new one called "strong entailment". We say that $α$ strongly entails $β$ when the equilibrium models of $α\wedge γ$ are also equilibrium models of $β\wedge γ$ for any context $γ$. We also provide a characterisation of strong entailment in terms of the denotational semantics, and give an example of a sufficient condition that can be applied in some cases.

cs.LO↗

arXiv2014

Causal Graph Justifications of Logic Programs

In this work we propose a multi-valued extension of logic programs under the stable models semantics where each true atom in a model is associated with a set of justifications. These justifications are expressed in terms of causal graphs formed by rule labels and edges that represent their application ordering. For positive programs, we show that the causal justifications obtained for a given atom have a direct correspon- dence to (relevant) syntactic proofs of that atom using the program rules involved in the graphs. The most interesting contribution is that this causal information is obtained in a purely semantic way, by algebraic op- erations (product, sum and application) on a lattice of causal values whose ordering relation expresses when a justification is stronger than another. Finally, for programs with negation, we define the concept of causal stable model by introducing an analogous transformation to Gelfond and Lifschitz's program reduct. As a result, default negation behaves as "absence of proof" and no justification is derived from negative liter

cs.AI↗

arXiv2013

An Algebra of Causal Chains

In this work we propose a multi-valued extension of logic programs under the stable models semantics where each true atom in a model is associated with a set of justifications, in a similar spirit than a set of proof trees. The main contribution of this paper is that we capture justifications into an algebra of truth values with three internal operations: an addition '+' representing alternative justifications for a formula, a commutative product '*' representing joint interaction of causes and a non-commutative product '.' acting as a concatenation or proof constructor. Using this multi-valued semantics, we obtain a one-to-one correspondence between the syntactic proof tree of a standard (non-causal) logic program and the interpretation of each true atom in a model. Furthermore, thanks to this algebraic characterization we can detect semantic properties like redundancy and relevance of the obtained justifications. We also identify a lattice-based characterization of this algebra, defining a direct consequences operator, proving its continuity and that its least fix point can be computed after a finite number of iterations. Finally, we define the concept of causal stable model by introducing an analogous transformation to Gelfond and Lifschitz's program reduct.

cs.AI↗

arXiv2010

A Logical Charaterisation of Ordered Disjunction

In this paper we consider a logical treatment for the ordered disjunction operator 'x' introduced by Brewka, Niemelä and Syrjänen in their Logic Programs with Ordered Disjunctions (LPOD). LPODs are used to represent preferences in logic programming under the answer set semantics. Their semantics is defined by first translating the LPOD into a set of normal programs (called split programs) and then imposing a preference relation among the answer sets of these split programs. We concentrate on the first step and show how a suitable translation of the ordered disjunction as a derived operator into the logic of Here-and-There allows capturing the answer sets of the split programs in a direct way. We use this characterisation not only for providing an alternative implementation for LPODs, but also for checking several properties (under strongly equivalent transformations) of the 'x' operator, like for instance, its distributivity with respect to conjunction or regular disjunction. We also make a comparison to an extension proposed by Kärger, Lopes, Olmedilla and Polleres, that combines 'x' with regular disjunction.

cs.LO↗

arXiv2010

Functional Answer Set Programming

In this paper we propose an extension of Answer Set Programming (ASP), and in particular, of its most general logical counterpart, Quantified Equilibrium Logic (QEL), to deal with partial functions. Although the treatment of equality in QEL can be established in different ways, we first analyse the choice of decidable equality with complete functions and Herbrand models, recently proposed in the literature. We argue that this choice yields some counterintuitive effects from a logic programming and knowledge representation point of view. We then propose a variant called QELF where the set of functions is partitioned into partial and Herbrand functions (we also call constructors). In the rest of the paper, we show a direct connection to Scott's Logic of Existence and present a practical application, proposing an extension of normal logic programs to deal with partial functions and equality, so that they can be translated into function-free normal programs, being possible in this way to compute their answer sets with any standard ASP solver.

cs.LO↗

arXiv2007

Propositional theories are strongly equivalent to logic programs

This paper presents a property of propositional theories under the answer sets semantics (called Equilibrium Logic for this general syntax): any theory can always be reexpressed as a strongly equivalent disjunctive logic program, possibly with negation in the head. We provide two different proofs for this result: one involving a syntactic transformation, and one that constructs a program starting from the countermodels of the theory in the intermediate logic of here-and-there.

cs.AI↗

arXiv2002

Alternative Characterizations for Strong Equivalence of Logic Programs

In this work we present additional results related to the property of strong equivalence of logic programs. This property asserts that two programs share the same set of stable models, even under the addition of new rules. As shown in a recent work by Lifschitz, Pearce and Valverde, strong equivalence can be simply reduced to equivalence in the logic of Here-and-There (HT). In this paper we provide two alternatives respectively based on classical logic and 3-valued logic. The former is applicable to general rules, but not for nested expressions, whereas the latter is applicable for nested expressions but, when moving to an unrestricted syntax, it generally yields different results from HT.

cs.AI↗

arXiv2000

PAL: Pertinence Action Language

The current document contains a brief description of a system for Reasoning about Actions and Change called PAL (Pertinence Action Language) which makes use of several reasoning properties extracted from a Temporal Expert Systems tool called Medtool.

cs.AI↗