Agustín Valverde

Strongly equivalent logic programs

A logic program <inline-equation><f><g>P</g><subscrpt>1</subscrpt></f> </inline-equation> is said to be equivalent to a logic program <inline-equation><f><g>P</g><subscrpt>2</subscrpt></f></inline-equation> in the sense of the answer set semantics if <inline-equation><f><g>P</g><subscrpt>1</subscrpt></f></inline-equation> and <inline-equation><f><g>P</g><subscrpt>2</subscrpt></f></inline-equation> have the same answer sets. We are interested in the following stronger condition: for every logic program, <inline-equation><f><g>P</g>, <g>P</g><subscrpt>1</subscrpt>, ∪ <g>P</g></f></inline-equation> has the same answer sets as <inline-equation><f><g>P</g><subscrpt>2</subscrpt> ∪ <g>P</g></f></inline-equation>. The study of strong equivalence is important, because we learn from it how one can simplify a part of a logic program without looking at the rest of it. The main theorem shows that the verification of strong equivalence can be accomplished by cheching the equivalence of formulas in a monotonic logic, called the logic of here-and-there, which is intermediate between classical logic and intuitionistic logic.

A characterization of strong equivalence for logic programs with variables

Two sets of rules are said to be strongly equivalent to each other if replacing one by the other within any logic program preserves the programs stable models. The familiar characterization of strong equivalence of grounded programs in terms of the propositional logic of here-and-there is extended in this paper to a large class of logic programs with variables. This class includes, in particular, programs with conditional literals and cardinality constraints. The first-order version of the logic of here-and-there required for this purpose involves two additional non-intuitionistic axiom schemas.

Quantified equilibrium logic and hybrid rules

In the ongoing discussion about combining rules and Ontologies on the Semantic Web a recurring issue is how to combine first-order classical logic with nonmonotonic rule languages. Whereas several modular approaches to define a combined semantics for such hybrid knowledge bases focus mainly on decidability issues, we tackle the matter from a more general point of view. In this paper we show how Quantified Equilibrium Logic (QEL) can function as a unified framework which embraces classical logic as well as disjunctive logic programs under the (open) answer set semantics. In the proposed variant of QEL we relax the unique names assumption, which was present in earlier versions of QEL. Moreover, we show that this framework elegantly captures the existing modular approaches for hybrid knowledge bases in a unified way.

Towards a First Order Equilibrium Logic for Nonmonotonic Reasoning

Equilibrium logic, introduced in [20], is a conservative extension of answer set semantics for logic programs to the full language of propositional logic. In this paper we initiate the study of first-order variants of equilibrium logic. In particular, we focus on a quantified version QN 5 of the propositional many-valued logic N 5 of here-and-there with strong negation, and define the condition of equilibrium via a minimal model construction. We verify Skolem forms and Herbrand theorems for QN 5 and show that, like its propositional counterpart, the quantified version of equilibrium logic also conservatively extends answer set semantics.

Uniform Equivalence for Equilibrium Logic and Logic Programs

For a given semantics, two logic programs Π1 and Π2 can be said to be equivalent if they have the same intended models and strongly equivalent if for any program X, Π1 ∪ X and Π2 ∪ X are equivalent. Eiter and Fink have recently studied and characterised under answer set semantics a further, related property of uniform equivalence, where the extension X is required to be a set of atoms. We extend their main results to propositional theories in equilibrium logic and describe a tableaux proof system for checking the property of uniform equivalence. We also show that no new forms of equivalence are obtained by varying the logical form of expressions in the extension X. Finally, some examples are studied including special cases of nested and generalized rules.

Quantified Equilibrium Logic and Foundations for Answer Set Programs

QHT is a first-order super-intuitionistic logic that provides afoundation for answer set programming (ASP) and a useful tool foranalysing and transforming non-ground programs. We recall someproperties of QHT and its nonmonotonic extension, quantifiedequilibrium logic (QEL). We show how the proof theory of QHT can beused to extend to non-ground programs previous results on thecompleteness of θ-subsumption. We also establish a reductionof QHT to classical logic and show how this can be used to obtainand extend classical encodings for concepts such as the strongequivalence of programs and theories. We pay special attention to aclass of general (disjunctive) logic programs that capture alluniversal theories in QEL.

A First Order Nonmonotonic Extension of Constructive Logic

Certain extensions of Nelsons constructive logic N with strong negation have recently become important in arti.cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelsons .rst-order logic as a basis for de.ning nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for .rst-order here-and-there logics, and their minimal strong negation extensions, for both constant and varying domains. We choose the constant domain version, which we denote by QNc5, as a basis for de.ning a .rst-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of QNc5, including Skolem forms and Herbrand theorems and Interpolation, and show that the .rst-oder version of equilibrium logic can be used as a foundation for answer set inference.

Towards Biresiduated Multi-adjoint Logic Programming

Multi-adjoint logic programs were recently proposed as a generalization of monotonic and residuated logic programs, in that simultaneous use of several implications in the rules and rather general connectives in the bodies are allowed. In this work, the need of biresiduated pairs is justified through the study of a very intuitive family of operators, which turn out to be not necessarily commutative and associative and, thus, might have two different residuated implications; finally, we introduce the framework of biresiduated multi-adjoint logic programming and sketch some considerations on its fixpoint semantics.

A Tableau Calculus for Equilibrium Entailment

We apply tableau methods to the problem of computing entailment in the nonmonotonic system of equilibrium logic, a generalisation of the inference relation associated with the stable model and answer set semantics for logic programs. We describe tableau calculi for the nonclassical logics underlying equilibrium entailment, namely here-and-there with strong negation and its strengthening classical logic with strong negation. A further tableau calculus is then presented for computing equilibrium entailment. This makes use of a new method for reducing the complexity of the tableau expansion rules, which we call signing-up.

Reducing propositional theories in equilibrium logic to logic programs

The paper studies reductions of propositional theories in equilibrium logic to logic programs under answer set semantics. Specifically we are concerned with the question of how to transform an arbitrary set of propositional formulas into an equivalent logic program and what are the complexity constraints on this process. We want the transformed program to be equivalent in a strong sense so that theory parts can be transformed independent of the wider context in which they might be embedded. It was only recently established [1] that propositional theories are indeed equivalent (in a strong sense) to logic programs. Here this result is extended with the following contributions. (i) We show how to effectively obtain an equivalent program starting from an arbitrary theory. (ii) We show that in general there is no polynomial time transformation if we require the resulting program to share precisely the vocabulary or signature of the initial theory. (iii) Extending previous work we show how polynomial transformations can be achieved if one allows the resulting program to contain new atoms. The program obtained is still in a strong sense equivalent to the original theory, and the answer sets of the theory can be retrieved from it.