Discovering Classes of Strongly Equivalent Logic Programs

In this paper we apply computer-aided theorem discovery technique to discover theorems about strongly equivalent logic programs under the answer set semantics. Our discovered theorems capture new classes of strongly equivalent logic programs that can lead to new program simplification rules that preserve strong equivalence. Specifically, with the help of computers, we discovered exact conditions that capture the strong equivalence between a rule and the empty set, between two rules, between two rules and one of the two rules, between two rules and another rule, and between three rules and two of the three rules

Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms

In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (c-atoms). These approaches generalize the fixpoint-based and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of c-atoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpoint-based semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complement. These definitions, which differ from each other in the treatment of negation-as-failure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of c-atoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly well-supported models, are generalizations of the notion of well-supported models of normal logic programs to the case of programs with c-atoms. As for the case of fixpoint-based semantics, the difference between these two definitions is rooted in the treatment of naf atoms. We prove that answer sets by reduct (resp. by complement) are equivalent to weakly (resp. strongly) well-supported models of a program, thus generalizing the theorem on the correspondence between stable models and well-supported models of a normal logic program to the class of programs with c-atoms. We show that the newly defined semantics coincide with previously introduced semantics for logic programs with monotone c-atoms, and they extend the original answer set semantics of normal logic programs. We also study some properties of answer sets of programs with c-atoms, and relate our definitions to several semantics for logic programs with aggregates presented in the literature

Strong Equivalence of Logic Programs with Abstract Constraint Atoms

Abstract. Logic programs with abstract constraint atoms provide a unifying framework for studying logic programs with various kinds of constraints. Establishing strong equivalence between logic programs is a key property for program maintenance and optimization, and for guaranteeing the same behavior for a revised original program in any context. In this paper, we study strong equivalence of logic programs with abstract constraint atoms. We first give a general characterization of strong equivalence based on a new definition of program reduct for logic programs with abstract constraints. Then we consider a particular kind of program revision-constraint replacements addressing the question: under what conditions can a constraint in a program be replaced by other constraints, so that the resulting program is strongly equivalent to the original one

On Compiling Logic Programs Into Relational Algebra

The combination of logic programming methods and database systems technology will result in knowledge bases of increased size and improved efficiency: this topic has received a lot of attention [Zaniolo 1985, Reiter 1978, Chang 1986, Minker 1978, Henschen 1984, Parker 1986, Brodie 1986]. Our approach to integrating logic programming languages (e. g. PROLOG) and database systems is to compile logic programming languages into conventional relational algebra. There are many technical problems which must be addressed and solved when compiling logic programs into relational algebra. Mainly, we are interested in the following problems: the finiteness (i. e. safety) of a logic program's executions and the differences between logic programing languages and database systems in data representation and typing systems. Our approach to safety checking integrates the rule/goal graph of [Ullman 1985] with the magic basis of a variable [Zaniolo 1986]. This approach allows us, effectively, to check the safety of a logic program at compile time, for those programs which are strongly safe. Otherwise, the safety of the program with respect to a query must be checked at execution time. Relational database systems are well typed, whilst logic programming languages are not. We overcome this difference by adding types to PROLOG (i. e TPROLOG). TPROLOG allows the user to define enumerated types, sub-types, structured types, and variant types. Our approach to compiling typed logic programs into conventional relational algebra expressions is to translate the logic program containing complex clauses into an equivalent complex-free program, and then to translate it into a form suitable for storage and manipulation by conventional relational database systems. The normalization of logic programs is achieved by removing complex arguments from facts and rules and replacing them with simplified (i. e. normalized) facts and rules. The normalized facts are stored in a conventional relational database (i. e. extensional database), and the normalized rules are stored in a rule base (i. e. intentional database). The translation of a complex-free program into conventional relational algebra is based on [Reiter 1978, Chang 1986, Henschen 1984, Bancilhon 1986]

Temporal Answer Set Programming

Answer Set Programming (ASP) has become a popular way for representing different kinds of scenarios from knowledge representation in Artificial Intelligence. Frequently, these scenarios involve a temporal component which must be considered. In ASP, time is usually represented as a variable whose values are defined by an extensional predicate with a finite domain. Dealing with a finite temporal interval has some disadvantages. First, checking the existence of a plan is not possible and second, it also makes difficult to decide whether two programs are strongly equivalent. If we extend the syntax of Answer Set Programming by using temporal operators from temporal modal logics, then infinite time can be considered, so the aforementioned disadvantages can be overcome. This extension constitutes, in fact, a formalism called Temporal Equilibrium Logic, which is based on Equilibrium Logic (a logical characterisation of ASP). Although recent contributions have shown promising results, Temporal Equilibrium Logic is still a novel paradigm and there are many gaps to fill. Our goal is to keep developing this paradigm, filling those gaps and turning it into a suitable framework for temporal reasoning