Propositional dynamic logic of context-free programs and fixpoint logic with chop

This paper compares propositional dynamic logic of non-regular programs and fixpoint logic with chop. It identifies a fragment of the latter which is equi-expressive to the former. This relationship transfers several decidability and complexity results between the two logics.     

The Least Fixpoint Transformation for Disjunctive Logic Programs

The paradigm of disjunctive logic programming(DLP)enhances greatly the expressive power of normal logic programming(NLP)and many(declarative)semantics have been defined for DLP to cope with various problems of knowledge representation in artificial intelligence.However,the expressive ability of the semantics and the soundness of program transformations for DLP have been rarely explored.This paper defines an immediate consequence operatro T^GP for each disjunctive program and shows that T^GP has the least and computable fixpoint Lft(P),Lft is,in fact,a program transformation for DLP,which transforms all disjunctive programs into negative programs.It is shown that Lft preserves many key semantics,including the disjunctive stable models,well-founded model,disjunctive argunent semantics DAS,three-valued models,ect.Thic means that every disjunctive program P has a unique canonical form Lft(P)with respect to these semantics.As a result,the work in this paper provides a unifying framework for studying the expressive ability of various semantics for DLP On the other hand,the computing of the above semantics for negative programs is ust a trivial task,therefore,Lft(P)is also an optimization method for DLP.Another application of Lft is to derive some interesting semantic results for DLP.     

Compiling bottom-up and mixed derivations into top-down executable logic programs

We present a technique for the compilation of bottom-up and mixed logic derivations into PROLOG-programs. It is obtained as an extension of a program transformation technique called Compiling Control. We illustrate its applications in three different domains: solving numerical problems, integrity checking in deductive databases and theorem proving. The aim is to obtain efficient PROLOG programs for problems in which a non-top-down control is most appropriate.Work partly supported by ESPRIT BRA COMPULOG (project 3012).Supported by the Belgian I.W.O.N.L.-I.R.S.I.A. under contract number 5203. Author for correspondence.Supported by the Belgian National Fund for Scientific Research.     

SLDNF resolution with non-safe rule and fixpoint semantics for general logic programs

The SLDNF resolution (SLD resolution with negation as failure) is often restricted to yield a safe rule, that is, negation as failure rule is adopted only in the case that the selected negative literal in each goal should be in ground. In this paper we investigate extensions of goals in SLDNF resolutions with the case of selecting non-ground negative literals. Since Shepherdson's proposal is thought of as most general [16, 18] we formally show how the success and failure sets by Sherpherdson's SLDNFS resolution are related with a fixpoint semantics, which is generalized to be concerned with atom sets involving the variables.     

Finite satisfiability for guarded fixpoint logic

The finite satisfiability problem for guarded fixpoint logic is decidable and complete for 2ExpTime (resp. ExpTime for formulas of bounded width).     

A new fixpoint semantics for general logic programs compared with the well-founded and the stable model semantics

We study a new fixpoint semantics for logic programs with negation. Our construction is intermediate between Van Gelder’s well-founded model and Gelfond and Lifschitz’s stable model semantics. We show first that the stable models of a logic programP are exactly the well-supported models ofP, i.e. the supported models with loop-free finite justifications. Then we associate to any logic programP a non-monotonic operator over the semilattice of justified interpretations, and we define in an abstract form its ordinal powers. We show that the fixpoints of this operator are the stable models ofP, and that its ordinal powers after some ordinala are extensions of the well-founded partial model ofP. In particular ifP has a well-founded model then that canonical model is also an ordinal power and the unique fixpoint of our operator. We show with examples of logic programs which have a unique stable model but no well-founded model that the converse is false. We relate also our work to Doyle’s truth maintenance system and some implementations of rule-based expert systems.     

The alternation hierarchy in fixpoint logic with chop is strict too

Fixpoint Logic with Chop extends the modal μ-calculus with a sequential composition operator which results in an increase in expressive power. We develop a game-theoretic characterisation of its model checking problem and use these games to show that the alternation hierarchy in this logic is strict. The structure of this result follows the lines of Arnold’s proof showing that the alternation hierarchy in the modal μ-calculus is strict over the class of binary trees.     

A data-driven machine for OR-parallel evaluation of logic programs

Research in the area of parallel evaluation mechanisms for logic programs have led to the proposal of a number of schemes exploiting various forms of parallelism. Many of the early models have been based on the conventional approach of organising concurrent components of computations as communicating processes. More recently, however, models based on more novel computation organisations, in particular, data-driven organisations, have been proposed. This paper describes the development of one such model, its implementation and the design of a data-driven machine to support it. The model exploits a form of parallelism known as OR-parallelism and is particularly suited to database applications, although it would also support general applications. It is envisaged that the proposed machine may be refined into an efficient database engine, which can then be a component of a more powerful and integrated logic programming machine.     

Parallelism in logic programs

There is a tension between the objectives of avoiding irrelevant computation and extracting parallelism, in that a computational step used to restrict another must precede the latter. Our thesis, following [3], is that evaluation methods can be viewed as implementing a choice ofsideways information propagation graphs, or sips, which determines the set of goals and facts that must be evaluated. Two evaluation methods that implement the same sips can then be compared to see which obtains a greater degree of parallelism, and we provide a formal measure of parallelism to make this comparison.Using this measure, we prove that transforming a program using the Magic Templates algorithm and then evaluating the fixpoint bottom-up provides a most parallel implementation for a given choice of sips, without taking resource constraints into account. This result, taken in conjunction with earlier results from [3,27], which show that bottom-up evaluation performs no irrelevant computation and is sound and complete, suggests that a bottom-up approach to parallel evaluation of logic programs is very promising. A more careful analysis of the relative overheads in the top-down and bottom-up evaluation paradigms is needed, however, and we discuss some of the issues.The abstract model allows us to establish several results comparing other proposed parallel evaluation methods in the logic programming and deductive database literature, thereby showing some natural, and sometimes surprising, connections. We consider the limitations of the abstract model and of the proposed bottom-up evaluation method, including the inability of sips to describe certain evaluation methods, and the effect of resource constraints. Our results shed light on the limits of the sip paradigm of computation, which we extend in the process.     

OR-parallel evaluation of logic programs on a multi-ring dataflow machine

Logic programming languages have gained wide acceptance because of two reasons. First is their clear declarative semantics and the second is the wide scope for parallelism they provide which can be exploited by building suitable parallel architectures. In this paper, we propose a multi-ring dataflow machine to support theOR-parallelism and theArgument parallelism of logic programs. A new scheme is suggested for handling the deferred read mechanism of the dataflow architecture. The required data structures, the dataflow actors and the builtin dataflow procedures for OR-parallel execution are discussed. Multiple binding environments arising in the OR-parallel execution are handled by a new scheme called thetagged variable scheme. Schemes for constrained OR-parallel execution are also discussed.     

Canonical logic programs

Consider the class of programs P where the greatest fixpoint of T_P is equal to the complement of the finite failure set of P. Programs in this calss possess some important properties which others do not. The main result in this paper proves that this class is representative of all programs. Specifically, we call the programs in this class canonical and we prove that for any program P₁, there exists a semantically equivalent program P₂ which is canonical.     

Transformations of logic programs

This paper introduces a new concept of computation trees of logic programs that will be used in reasoning about programs. Three types of transformations improving the structure of logic programs are described. There are two natural measures of complexity suggested by computation trees, namely, the number of nodes called by recursion and the maximal number of AND/OR alternations on a branch. Both measures are shown to collapse, or more precisely, it is shown how every logic program can be transformed to a program computing the same function of which the computation tree has at most one called node and at most two alternations on every branch. Some conclusions related to this Normal Form Theorem are discussed. Theoretical results with the attempts to develop a practical methodology for the construction of logic programs are compared.     

A residualizing semantics for the partial evaluation of functional logic programs

Recent proposals for multi-paradigm declarative programming combine the most important features of functional, logic and concurrent programming into a single framework. The operational semantics of these languages is usually based on a combination of narrowing and residuation. In this paper, we introduce a non-standard, residualizing semantics for multi-paradigm declarative programs and prove its equivalence with a standard operational semantics. Our residualizing semantics is particularly relevant within the area of program transformation where it is useful, e.g., to perform computations during partial evaluation. Thus, the proof of equivalence is a crucial result to demonstrate the correctness of (existing) partial evaluation schemes.     

Equivalent logic programs and symmetric homogeneous forms of logic programs with equality

This article introduces the notion of CAS-equivalent logic programs: logic programs with identical correct answer substitutions. It is shown that the notions CAS-equivalence, refutational equivalence, and logical equivalence do not coincide in the case of definite clause logic programs. Least model criteria for refutational and CAS-equivalence are suggested and their correctness is proved. The least model approach is illustrated by two proofs of CAS-equivalence. It is shown that the symmetric extension of a logic program subsumes the symmetry axiom and the symmetric homogeneous form of a logic program with equality subsumes the symmetry, transitivity, and predicate substitutivity axioms of equality. These results contribute towards the goal of building equality into standard Prolog without introducing additional inference rules.     

Hypothetical reasoning in logic programs

In order to express incomplete knowledge, extended logic programs have been proposed as logic programs with classical negation along with negation as failure. This paper discusses ways to deal with a broad class of common sense knowledge by using extended logic programs. For this purpose, we present a uniform approach for dealing with both incomplete and contradictory programs, as a simple framework of hypothetical reasoning in which some rules are dealt with as candidate hypotheses that can be used to augment the background theory. This theory formation framework can be used for default reasoning, contradiction removals, the closed world assumption, and abduction. We also show a translation of the theory formation framework to an extended logic program whose answer sets correspond to the consistent belief sets of augmented theories.     

Nested expressions in logic programs

We extend the answer set semantics to a class of logic programs with nested expressions permitted in the bodies and heads of rules. These expressions are formed from literals using negation as failure, conjunction (,) and disjunction (;) that can be nested arbitrarily. Conditional expressions are introduced as abbreviations. The study of equivalent transformations of programs with nested expressions shows that any such program is equivalent to a set of disjunctive rules, possibly with negation as failure in the heads. The generalized answer set semantics is related to the Lloyd–Topor generalization of Clark’s completion and to the logic of minimal belief and negation as failure. This revised version was published online in June 2006 with corrections to the Cover Date.     

Detection of And-Parallelism in Logic Programs

In this paper,we present a detection technique of and-parallelism in logic programs.The detection consists of three phases:analysis of entry modes,derivation of exit modes and determination of execution graph expressions.Compared with other techniques^[2,4,5],our approach with the compile-time program-level data-dependence analysis of logic programs,can efficiently exploit and-parallelism in logic programs.Two precompilers,based on our technique and DeGroot‘s approach^[3] respectively,have been implemented in SES-PIM system^[12],Through compiling and running some typical benchmarks in SES-PIM,we conclude that our technique can,in most cases,exploit as much and-parallelism as the dynamic approach^[13]does under“produces-consumer”scheme,and needs less dynamic overhead while exploiting more and parallelism than DeGroot‘s approach does.     

Most specific logic programs

More specific versions of definite logic programs are introduced. These are versions of a program in which each clause is further instantiated or removed and which have an equivalent set of successful derivations to those of the original program, but a possibly increased set of finitely failed goals. They are better than the original program because failure in a non-successful derivation may be detected more quickly. Furthermore, information about allowed variable bindings which is hidden in the original program may be made explicit in a more specific version of it. This allows better static analysis of the program's properties and may reveal errors in the original program. A program may have several more specific versions but there is always a most specific version which is unique up to variable renaming. Methods to calculate more specific versions are given and it is characterized when they give the most specific version.     

On computing logic programs

In this paper we present and compare some classical problem-solving methods for computing the stable models of logic programs with negation. Using a graph theoretic representation of logic programs and their stable models, we discuss and compare linear programming, propositional satisfiability, constraint satisfaction, and graph methods.