Giorgio Levi

Declarative modeling of the operational behavior of logic languages

Abstract The paper defines a new declarative semantics for logic programs, which is based on interpretations containing (possibly) non-ground atoms. Two different interpretations are introduced and the corresponding models are defined and compared. The classical results on the Herbrand model semantics of logic programs are shown to hold in the new models too (i.e. existence of a minimal model, fixpoint characterization, etc.). With the new models, we have a stronger soundness and completeness result for SLD-resolution. In particular, one of the two models allows the set of computed answer substitutions to be characterized precisely.

The s-semantics approach: theory and applications.

Abstract This paper is a general overview of an approach to the semantics of logic programs whose aim is to find notions of models which really capture the operational semantics, and are, therefore, useful for defining program equivalences and for semantics-based program analysis. The approach leads to the introduction of extended interpretations which are more expressive than Herbrand interpretations. The semantics in terms of extended interpretations can be obtained as a result of both an operational (top-down) and a fixpoint (bottom-up) construction. It can also be characterized from the model-theoretic viewpoint, by defining a set of extended models which contains standard Herbrand models. We discuss the original construction modeling computed answer substitutions, its compositional version, and various semantics modeling more concrete observables. We then show how the approach can be applied to several extensions of positive logic programs. We finally consider some applications, mainly in the area of semantics-based program transformation and analysis.

Kernel-LEAF: a logic plus functional language

Abstract Kernel-LEAF is a logic plus functional language based on the flattening technique. It differs from other similar languages because it is able to cope with partial (undefined or non-terminating) functions. This is achieved by introducing the distinction between data structures and (functional) term structures, and by using two kinds of equality. The language has a clean model-theoretic semantics, where the domains of the interpretations are the algebraic CPOs. In these domains the difference between the two equalities corresponds to a different behaviour with respect to continuity. The operational semantics (based on SLD-resolution) is proved sound and complete with respect to the model-theoretic one. Finally, an outermost strategy, more efficient than unrestricted SLD-resolution, but still complete, is presented.

The relation between logic and functional languages: A survey

Abstract The paper considers different methods of integrating the functional and logic programming paradigms, starting with the identification of their semantic differences. The main methods to extend functional programs with logic features (i.e. unification) are then considered. These include narrowing, completion, SLD-resolution of equational formulas, and set abstraction. The different techniques are analyzed from several viewpoints, including the ability to support both paradigms, lazy evaluation, and concurrency.

A general framework for semantics-based bottom-up abstract interpretation of logic programs

The theory of abstract interpretation provides a formal framework to develop advanced dataflow analysis tools. The idea is to define a nonstandard semantics which is able to compute, in finite time, an approximated model of the program. In this paper, we define an abstract interpretation framework based on a fixpoint approach to the semantics. This leads to the definition, by means of a suitable set of operators, of an abstract fixpoint characterization of a model associated with the program. Thus, we obtain a specializable abstract framework for bottom-up abstract interpretations of definite logic programs. The specialization of the framework is shown on two examples, namely, gound-dependence analysis and depth-k analysis.

A model-theoretic reconstruction of the operational semantics of logic programs

Abstract In this paper we define a new notion or truth on Herbrand interpretations extended with variables which allows us to capture, by means of suitable models, various observable properties, such as the ground success set, the set of atomic consequences, and the computed answer substitutions. The notion of truth extends the classical one to account for non-ground formulas in the interpretations. The various operational semantics are all models. An ordering on interpretations is defined to overcome the problem that the intersection of a set of models is not necessarily a model. The set of interpretations with this partial order is shown to be a complete lattice, and the greatest lower bound of any set of models is shown to be a model. Thus there exists a least model, which is the least Herbrand model and therefore the ground success set semantics. Richer operational semantics are non-least models, which can, however, be effectively defined by fixpoint constructions. The model corresponding to the computed answer substitutions operational semantics is the most primitive one (the others can easily be obtained from it).

A grey-weighted skeleton

Skeletons have been largely used as descriptors of shape in the field of image processing. Only binary pictures, however, have been considered so far. In this paper a grey-weighted skeleton is defined for grey-valued continuous and quantized images. In order to extend the invertibility property of the skeleton to the grey case, a transformation is defined, which is a generalization of both direct and inverse binary skeleton transformations. By taking advantage of the properties of this transformation, a procedure for the inversion of a grey-weighted skeleton (i.e., for obtaining a binary reconstructed image from a grey-weighted skeleton) is finally proposed.

A compositional semantics for logic programs

Bossi, A., M. Gabbrielli, G. Levi and M.C. Meo, A compositional semantics for logic programs, Theoretical Computer Science 122 (1994) 3-47. This paper considers open logic programs originally introduced as a tool to build an OR-compositional semantics of logic programs. We extend the original semantic definitions in the framework of the general approach to the semantics,of logic programs described by Gabbrielli and Levi (1991). We first define an operational semantic

Generalized semantics and abstract interpretation for constraint logic programs

Lno6P) which models computed answer substitutions and which is compositional w.r.t. the union 6f programs. Next, we consider the semantic domain of Q-denotations, which are sets of clauses with a suitable equivalence relation. The fixpoint semantics F;(P) given by Bossi and Menegus (1991) is proved equivalent to the operational semantics. From the model-theoretic viewpoint, an Q-denotation is mapped onto a set of Herbrand interpretations, thus leading to a definition of an R-model based on the classical notion of truth. Moreower, we consider a particular kind of Q-models (compositional modelsL and we show that I”,(P) is a (nonminimal) compositional Q-model. A suitable abstraction oq0,( P) is compositional and fully

Generalized and/or graphs

Abstract We present simple and powerful generalized algebraic semantics for constraint logic programs that are parameterized with respect to the underlying constraint system. The idea is to abstract away from standard semantic objects by focusing on the general properties of any—possibly nonstandard—semantic definition. In constraint logic programming, this corresponds to a suitable definition of the constraint system supporting the semantic definition. An algebraic structure is introduced to formalize the notion of a constraint system, thus making classical mathematical results applicable. Both top-down and bottom-up semantics are considered. Nonstandard semantics for constraint logic programs can then be formally specified using the same techniques used to define standard semantics. Different nonstandard semantics for constraint logic languages can be specified in this framework. In particular, abstract interpretation of constraint logic programs can be viewed as an instance of the constraint logic programming framework itself.