A Free Logic for Stable Models with Partial Intensional Functions (original) (raw)

Quantified equilibrium logic and the first order logic of here-and-there

2006

Abstract This report continues the study of quantified equilibrium logic, QEL, introduced in [25, 26], and its monotonic base logic, here-and-there. We present a slightly modified version of QEL where the so-called unique name assumption or UNA is not assumed from the outset but may be added as a special requirement for specific applications. We also consider here an alternative axiom set for first-order here-and-there.

A Denotational Semantics for Equilibrium Logic

Theory and Practice of Logic Programming

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 ∧ γ are also equilibrium models of β ∧ 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.

A New Logical Characterisation of Stable Models and Answer Sets

1996

This paper relates inference in extended logic programming with nonclassical, nonmonotonic logics. We define a nonmonotonic logic, called equilibrium logic, based on the least constructive extension, N2, of the intermediate logic of “here-and-there”. We show that on logic programs equilibrium logic coincides with the inference operation associated with the stable model and answer set semantics of Gelfond and Lifschitz. We thereby obtain a very simple characterisation of answer set semantics as a form of minimal model reasoning in N2, while equilibrium logic itself provides a natural generalisation of this semantics to arbitrary theories. We discuss briefly some consequences and applications of this result.

Interpretability and Equivalence in Quantified Equilibrium Logic

2007

Interpretability and Equivalence in Quantified Equilibrium Logic David Pearce1* and Agustın Valverde2** 1 Computing Science and Artificial Intelligence, Univ. Rey Juan Carlos,(Móstoles, Madrid), Spain. davidandrew. pearce@ urjc. es 2 Dept. of Applied Mathematics, Univ. of Málaga, Spain. a valverde@ ctima. uma. es Abstract. The study of synonymy among propositional theories in equilibrium logic, begun in [36], is extended to the first-order case.

Revised Stable Models - A Semantics for Logic Programs

2005

This paper introduces an original 2-valued semantics for Normal Logic Programs (NLP), which conservatively extends the Stable Model semantics (SM) to all normal programs. The distinction consists in the revision of one feature of SM, namely its treatment of odd loops, and of infinitely long support chains, over default negation. This single revised aspect, addressed by means of a Reductio ad Absurdum approach, affords a number of fruitful consequences, namely regarding existence, relevance and top-down querying, cumulativity, and implementation. The paper motivates and defines the Revised Stable Models semantics (rSM), justifying and exemplifying it. Properties of rSM are given and contrasted with those of SM. Furthermore, these results apply to SM whenever odd loops and infinitely long chains over negation are absent, thereby establishing significant, not previously known, properties of SM. Conclusions, further work, terminate the paper.

Revised Stable Models-a new semantics for logic programs

2004

Abstract This paper introduces an original 2-valued semantics for Normal Logic Programs (NLP), important on its own. Nevertheless, its name draws attention to that it is inspired by and generalizes Stable Model semantics (SM). The definitional distinction consists in the revision of one feature of SM, namely its treatment of odd loops over default negation.

On the logic and computation of partial equilibrium models

2006

The nonmonotonic formalism of partial equilibrium logic (PEL) has recently been proposed as a logical foundation for the partial stable and well-founded semantics of logic programs [1, 2]. We study certain logical properties of PEL and some techniques to compute partial equilibrium models.

On the Logic and Computation of Partial Equilibrium Models (extended version)

Abstract. The nonmonotonic formalism of partial equilibrium logic (PEL) was introduced and studied in [1, 2] and proposed as a logical foundation for the partial stable and well-founded semantics of logic programs. Here we study further logical properties of PEL and some techniques to compute partial equilibrium models.

Epistemic Foundation of Stable Model Semantics

Eprint Arxiv Cs 0403002, 2004

Stable model semantics has become a very popular approach for the management of negation in logic programming. This approach relies mainly on the closed world assumption to complete the available knowledge and its formulation has its basis in the so-called Gelfond-Lifschitz transformation. The primary goal of this work is to present an alternative and epistemic-based characterization of stable model semantics, to the Gelfond-Lifschitz transformation. In particular, we show that stable model semantics can be defined entirely as an extension of the Kripke-Kleene semantics. Indeed, we show that the closed world assumption can be seen as an additional source of `falsehood' to be added cumulatively to the Kripke-Kleene semantics. Our approach is purely algebraic and can abstract from the particular formalism of choice as it is based on monotone operators (under the knowledge order) over bilattices only.