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27
Fixpoint semantics for logic programming  a survey
, 2000
"... The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and manyvalued logic, lattice theory, game theory, and topology. One source of this richness is the inherent nonmonotonicity of its negation, something that does not have close para ..."
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Cited by 106 (0 self)
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The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and manyvalued logic, lattice theory, game theory, and topology. One source of this richness is the inherent nonmonotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact between communities. In this paper we summarize one variety of approaches to the semantics of logic programs: that based on fixpoint theory. We do not attempt to cover much beyond this single area, which is already remarkably fruitful. We hope readers will see parallels with, and the divergences from the better known fixpoint treatments developed for other programming methodologies.
Hypothetical Datalog: Complexity and Expressibility
 Theoretical Computer Science
, 1988
"... We present an extension of Hornclause logic which can hypothetically add and delete tuples from a database. Such logics have been discussed in the literature, but their complexities and expressibilities have remained an open question. This paper examines two such logics in the functionfree, predic ..."
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Cited by 36 (15 self)
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We present an extension of Hornclause logic which can hypothetically add and delete tuples from a database. Such logics have been discussed in the literature, but their complexities and expressibilities have remained an open question. This paper examines two such logics in the functionfree, predicate case. It is shown, in particular, that augmenting Hornclause logic with hypothetical addition increases its datacomplexity from PTIME to PSPACE. When deletions are added as well, complexity increases again, to EXPTIME. We then augment the logic with negationasfailure and develop the notion of stratified hypothetical rulebases. It is shown that negation does not increase complexity. To establish expressibility, we view the logic as a query language for relational databases. It is shown that any typed generic query that is computable in PSPACE can be expressed as a stratified rulebase of hypothetical additions. Similarly, any typed generic query that is computable in EXPTIME can be exp...
A Proof Procedure for the Logic of Hereditary Harrop Formulas
 JOURNAL OF AUTOMATED REASONING
, 1993
"... A proof procedure is presented for a class of formulas in intuitionistic logic. These formulas are the socalled goal formulas in the theory of hereditary Harrop formulas. Proof search inintuitionistic logic is complicated by the nonexistence of a Herbrandlike theorem for this logic: formulas cann ..."
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Cited by 30 (12 self)
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A proof procedure is presented for a class of formulas in intuitionistic logic. These formulas are the socalled goal formulas in the theory of hereditary Harrop formulas. Proof search inintuitionistic logic is complicated by the nonexistence of a Herbrandlike theorem for this logic: formulas cannot in general be preprocessed into a form such as the clausal form and the construction of a proof is often sensitive to the order in which the connectives and quantifiers are analyzed. An interesting aspect of the formulas we consider here is that this analysis can be carried out in a relatively controlled manner in their context. In particular, the task of finding a proof can be reduced to one of demonstrating that a formula follows from a set of assumptions with the next step in this process being determined by the structure of the conclusion formula. An acceptable implementation of this observation must utilize unification. However, since our formulas may contain universal and existential quantifiers in mixed order, care must be exercised to ensure the correctness of unification. One way of realizing this requirement involves labelling constants and variables and then using these labels to constrain unification. This form of unification is presented and used in a proof procedure for goal formulas in a firstorder version of hereditary Harrop formulas. Modifications to this procedure for the relevant formulas in a higherorder logic are also described. The proof procedure that we present has a practical value in that it provides the basis for an implementation of the logic programming language lambdaProlog.
Representing Objects in a Logic Programming Language with Scoping Constructs
 International Conference in Logic Programming
, 1990
"... We present a logic programming language that uses implications and universal quantifiers in goals and the bodies of clauses to provide a simple scoping mechanism for program clauses and constants. Within this language it is possible to define a simple notion of parametric module and local constant. ..."
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Cited by 28 (8 self)
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We present a logic programming language that uses implications and universal quantifiers in goals and the bodies of clauses to provide a simple scoping mechanism for program clauses and constants. Within this language it is possible to define a simple notion of parametric module and local constant. Given this ability to structure programs, we explore how objectoriented programming, where objects are viewed as abstractions with behaviors, state, and inheritance, might be accommodated. To capture the notion of mutable state, we depart from the pure logic setting by adding a declaration that certain local predicates are deterministic (they succeed at most once). This declaration, along with a goalcontinuation passing style of programming is adequate to model the state of objects. We also examine a few aspects of how having objects embedded in logic programming can be used to enrich the notion of object: for examples, objects may be partial (that is, may contain free variables) and nond...
A Logic for Hypothetical Reasoning
 Department of Computer Science, Rutgers University
, 1988
"... This paper shows that classical logic is inappropriate for hypothetical reasoning and develops an alternative logic for this purpose. The paper focuses on a form of hypothetical reasoning which appears computationally tractable. Specifically, Hornclause logic is augmented with rules, called embedde ..."
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Cited by 15 (9 self)
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This paper shows that classical logic is inappropriate for hypothetical reasoning and develops an alternative logic for this purpose. The paper focuses on a form of hypothetical reasoning which appears computationally tractable. Specifically, Hornclause logic is augmented with rules, called embedded implications, which can hypothetically add atomic formulas to a rulebase. By introducing the notion of rulebase independence, we show that these rules can express hypothetical queries which classical logic cannot; and by adopting methods from modal logic, we show these rules to be intuitionistic. In particular, they form a subset of intuitionistic logic having semantic properties similar to those of Hornclause logic. This report is an expanded version of a paper published in the Proceedings of the Seventh National Conference on Artificial Intelligence, St. Paul, Minnesota, August 2126 1988, American Association for Artificial Intelligence (AAAI). 1 Introduction Several researchers...
Expressing Database Queries with Intuitionistic Logic
 Proceedings of the North American Conference on Logic Programming
, 1989
"... This paper develops a declarative language with intuitionistic semantics which expresses exactly the generic database queries. Syntactically, the language is an extension of Datalog (functionfree Horn logic) which allows rules themselves to appear in the bodies of other rules. Such rules are called ..."
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Cited by 12 (3 self)
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This paper develops a declarative language with intuitionistic semantics which expresses exactly the generic database queries. Syntactically, the language is an extension of Datalog (functionfree Horn logic) which allows rules themselves to appear in the bodies of other rules. Such rules are called embedded implications. Several researchers have studied restricted versions of these rules, but in their full incarnation, universal quantifiers may appear in the premises, as in the rule A / 8 x [B(x) / C(x)]. This paper focuses on these embedded universal quantifiers. It is shown, for instance, that such quantifiers give the logic the ability to create new constant symbols hypothetically during inference. This, in turn, allows the logic to simulate unbounded counters and arbitrary Turing machines. In addition, when the logic is augmented with negationasfailure, it becomes expressively complete, that is, it can express any database query which is typed and generic. Similar results exist...
Hypothetical Datalog: Negation and Linear Recursion
 In Proceedings of the ACM Symposium on the Principles of Database Systems (PODS
, 1989
"... This paper examines an extension of Horn logic in which rules can add entries to a database hypothetically. Several researchers have developed logical systems along these lines, but the complexity and expressibility of such logics is only now being explored. It has been shown, for instance, that the ..."
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Cited by 11 (7 self)
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This paper examines an extension of Horn logic in which rules can add entries to a database hypothetically. Several researchers have developed logical systems along these lines, but the complexity and expressibility of such logics is only now being explored. It has been shown, for instance, that the datacomplexity of these logics is PSPACEcomplete in the functionfree, predicate case. This paper extends this line of research by developing syntactic restrictions with lower complexity. These restrictions are based on two ideas from Hornclause logic: linear recursion and stratified negation. In particular, a notion of stratification is developed in which negationasfailure alternates with linear recursion. The complexity of such rulebases depends on the number of layers of stratification. The result is a hierarchy of syntactic classes which corresponds exactly to the polynomialtime hierarchy of complexity classes. In particular, rulebases with k strata are datacomplete for \Sigma P...
Adding NegationasFailure to Intuitionistic Logic Programming
 Proc. NACLP
, 1992
"... Intuitionistic logic programming is an extension of Hornclause logic programming in which implications may appear "embedded" on the righthand side of a rule. Thus, rules of the form A(x) / [B(x) / C(x)] are allowed. These rules are called embedded implications . In this paper, we develop a languag ..."
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Cited by 11 (4 self)
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Intuitionistic logic programming is an extension of Hornclause logic programming in which implications may appear "embedded" on the righthand side of a rule. Thus, rules of the form A(x) / [B(x) / C(x)] are allowed. These rules are called embedded implications . In this paper, we develop a language in which negationasfailure is combined with embedded implications in a principled way. Although this combination has been studied by other researchers, Gabbay has argued in [10] that the entire idea is logically incoherent since modus ponens would not be valid in such a system. We show how to solve this problem by drawing a distinction between rules and goals. To specify the semantics of rules and goals, we then develop an analogue of Przymusinski's perfect model semantics for stratified Hornclause logic [20]. Several modifications are necessary to adapt this idea from classical logic to intuitionistic logic, but we eventually show how to define a preferred model of a stratified intui...
Elimination of Negation in a Logical Framework
, 2000
"... Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negationasfailure does not fit well in a logical framework, especially one endowed with ..."
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Cited by 10 (3 self)
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Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negationasfailure does not fit well in a logical framework, especially one endowed with hypothetical and parametric judgements, we adapt the idea of elimination of negation introduced in [21] for Horn logic to a fragment of higherorder HHF. This entails finding a middle ground between the Closed World Assumption usually associated with negation and the Open World Assumption typical of logical frameworks; the main technical idea is to isolate a set of programs where static and dynamic clauses do not overlap.