Horn Logic Programs Research Articles

Well limit behaviors of term rewriting systems

The limit behaviors of computations have not been fully explored. It is necessary to consider such limit behaviors when we consider the properties of infinite objects in computer science, such as infinite logic programs, the symbolic solutions of infinite polynomial equations. Usually, we can use finite objects to approximate infinite objects, and we should know what kinds of infinite objects are approximable and how to approximate them effectively. A sequence {R k : k e ω} of term rewriting systems has the well limit behavior if under the condition that the sequence has the set-theoretic limit or the distance-based limit, the sequence {Th(R k ): k e ω} of corresponding theoretic closures of R k has the set-theoretic or distance-based limit, and lim k→∞ Th(R k ) is equal to the theoretic closure of the limit of {R k : k e ω}. Two kinds of limits of term rewriting systems are considered: one is based on the set-theoretic limit, the other is on the distance-based limit. It is proved that given a sequence {R k : κ e ω} of term rewriting systems R k , if there is a well-founded ordering ≺ on terms such that every R k is ≺-well-founded, and the set-theoretic limit of {R k : κ e ω} exists, then {R k : κ e ω} has the well limit behavior; and if (1) there is a well-founded ordering ≺ on terms such that every R k is ≺-well-founded, (2) there is a distance d on terms which is closed under substitutions and contexts and (3) {R k : k e ω} is Cauchy under d then {R κ: κ e ω} has the well limit behavior. The results are used to approximate the least Herbrand models of infinite Horn logic programs and real Horn logic programs, and the solutions and Grobner bases of (infinite) sets of real polynomials by sequences of (finite) sets of rational polynomials.

Induction from answer sets in nonmonotonic logic programs

Inductive logic programming (ILP) realizes inductive machine learning in computational logic. However, the present ILP mostly handles classical clausal programs, especially Horn logic programs, and has limited applications to learning nonmonotonic logic programs . This article studies a method for realizing induction in nonmonotonic logic programs. We consider an extended logic program as a background theory, and introduce techniques for inducing new rules using answer sets of the program. The produced new rules explain positive/negative examples in the context of inductive logic programming. The proposed methods extend the present ILP techniques to a syntactically and semantically richer framework, and contribute to a theory of nonmonotonic ILP.

Ontologies, Frames and Logical Theories in NKI

NKI (The National Knowledge Infrastructure) is a large-scale knowledge base, which uses frames to represent concepts in ontologies, and uses Horn logic programs as the automated reasoning. The formalizations of ontologies, frames and logical theories in NKI, and the transformations between the formalizations are given, and proved to be functors between ontologies, frames and logical theories if they are taken as categories in the theory of category. The result proved in this paper guarantees that in NKI, the inference based on the Horn logic programs is correct with respect to the knowledge base represented by ontologies and frames.

Recurrence Domains: Their Unification and Application to Logic Programming

In this paper we present a formalism for finitely representing infinite sets of terms. This formalism, called ρ- terms , enables us to reason finitely about certain recursive types. We present an extension of Horn logic programs, called ρ-Prolog, which allows a finite schematization of infinitely many clauses via predicates with ρ-terms as arguments. We show that for every ρ-Prolog program there is an equivalent Horn logic program. That is, incorporating ρ-terms into first order logic programming does not change its denotational semantics. Computationally, however, ρ-Prolog has the advantages of (1) representing infinitely many answers finitely, (2) avoiding repetition in computation and thus achieving better efficiency, (3) allowing infinite queries, and (4) avoiding certain kinds of non-termination of Prolog programs. The ρ-terms play a similar role to regular-trees and sort-expressions in explicitly defining abstract data types. It differs from the others in that it allows us to define certain non-regular-tree sets such as {( a n , b n , c n ): n ϵ N }. We present a finite and complete algorithm for unification between ρ-terms, with which we can also compute the intersections of the sets defined by ρ-terms.

On the relationship between annotated logic programs and nonmonotonic formalisms

Abstract In the past we developed a semantics for a restricted annotated logic language for inheritance reasoning. Here we generalize it to annotated Horn logic programs. We first provide a formal account of the language, describe its semantics, and provide an interpreter written in Prolog for it. We then investigate its relationship to Belnap's 4-valued logic, Gelfond and Lifschitz's semantics for logic programs with negation, Brewka's prioritized default logics and other annotated logics due to Kifer et al.

MODULARITY AND CORRECTNESS FOR LOGIC PROGRAMS AND KNOWLEDGE BASES

A modularity concept for structuring and developing large logic programs and logical knowledge bases is presented. The concept is motivated from work in the field of algebraic specification, and enforces an extreme modularity discipline that goes beyond the one found in imperative or logic programming languages. As concrete programming languages (respectively knowledge representation formalisms), we consider Horn logic and equational logic programming. We give formal semantics for single modules and discuss correctness and verification issues. Large systems are constructed as interconnections of single modules. We introduce the so-called module operations of composition, actualization, and union, and give results concerning compositionality of semantics and correctness preservation.

A domain-theoretic approach to functional and logic programming

AbstractThe integration of functional and logic programming languages has been a topic of great interest in the last decade. Many proposals have been made, yet none is completely satisfactory especially in the context of higher order functions and lazy evaluation. This paper addresses these shortcomings via a new approach: domain theory as a common basis for functional and logic programming. Our integrated language remains essentially within the functional paradigm. The logic programming capability is provided by set abstraction (via Zermelo-Frankel set notation), using the Herbrand universe as a set abstraction generator, but for efficiency reasons our proposed evaluation procedure treats this generator's enumeration parameter as a logical variable. The language is defined in terms of (computable) domain-theoretic constructions and primitives, using the lower (or angelic) powerdomain to model the set abstraction facility. The result is a simple, elegant and purely declarative language that successfully combines the most important features of both pure functional programming and pure Horn logic programming. Referential transparency with respect to the underlying mathematical model is maintained throughout. An implicitly correct operational semantics is obtained by direct execution of the denotational semantic definition, modified suitably to permit logical variables whenever the Herbrand universe is being generated within a set abstraction. Completeness of the operational semantics requires a form of parallel evaluation, rather than the more familiar left-most rule.

Semantics of Horn and disjunctive logic programs

Van Emden and Kowalski proposed a fixpoint semantics based on model-theory and an operational semantics based on proof-theory for Horn logic programs. They prove the equivalence of these semantics using fixpoint techniques. The main goal of this paper is to present a unified theory for the semantics of Horn and disjunctive logic programs. For this, we extend the fixpoint semantics and the operational or procedural semantics to the class of disjunctive logic programs and prove their equivalence using techniques similar to the ones used for Horn programs.

SYGRAF: implementing logic programs in a database style

It is shown how Horn logic programs can be implemented using database techniques, namely, mostly bottom-up in combination with certain top-down elements (as opposed to the top-down implementations of logic programs prevailing so far). The proposed method is sound and complete. It easily lends itself to a parallel implementation and is free of nonlogical features like backtracking. As an extension to the common approach to deductive databases, function symbols are allowed to appear in programs, and it is shown that much of database query optimization can be applied to optimize logic programs. An important advantage of present approach is its ability to evaluate successfully many programs that terminate under neither pure top-down nor bottom-up evaluation strategies.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Semantics of EqL

The formal semantics of a novel language, called EqL, are presented for first-order functional and Horn logic programming. An EqL program is a set of conditional pattern-directed rules, where the conditions are expressed as a conjunction of equations. The programming paradigm provided by this language may be called equational programming. The declarative semantics of equations is given in terms of their complete set of solutions, and the operational semantics for solving equations is an extension of reduction, called object refinement. The correctness of the operational semantics is established through the soundness and completeness theorems. Examples are given to illustrate the language and its semantics.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>