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Complexity and Expressive Power of Logic Programming
, 1997
"... This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results ..."
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Cited by 341 (58 self)
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This paper surveys various complexity results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming. The complexity of the unification problem is also addressed.
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 126 (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.
A Survey on Complexity Results for Nonmonotonic Logics
 Journal of Logic Programming
, 1993
"... This paper surveys the main results appeared in the literature on the computational complexity of nonmonotonic inference tasks. We not only give results about the tractability/intractability of the individual problems but we also analyze sources of complexity and explain intuitively the nature of e ..."
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Cited by 90 (6 self)
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This paper surveys the main results appeared in the literature on the computational complexity of nonmonotonic inference tasks. We not only give results about the tractability/intractability of the individual problems but we also analyze sources of complexity and explain intuitively the nature of easy/hard cases. We focus mainly on nonmonotonic formalisms, like default logic, autoepistemic logic, circumscription, closedworld reasoning and abduction, whose relations with logic programming are clear and well studied. Complexity as well as recursiontheoretic results are surveyed. Work partially supported by the ESPRIT Basic Research Action COMPULOG and the Progetto Finalizzato Informatica of the CNR (Italian Research Council). The first author is supported by a CNR scholarship 1 Introduction Nonmonotonic logics and negation as failure in logic programming have been defined with the goal of providing formal tools for the representation of default information. One of the ideas und...
The Complexity of Local Stratification
 Fundamenta Informaticae
, 1994
"... The class of locally stratified logic programs is shown to be Π 1 1complete by the construction of a reducibility of the class of infinitely branching nondeterministic finite register machines. 1 ..."
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Cited by 17 (0 self)
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The class of locally stratified logic programs is shown to be Π 1 1complete by the construction of a reducibility of the class of infinitely branching nondeterministic finite register machines. 1
The expressiveness of locally stratified programs
 Annals of Mathematics and Artificial Intelligence
, 1995
"... This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be defined by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is a ..."
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Cited by 15 (2 self)
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This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be defined by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is an optimal result, since the perfect model of a locally stratified program is itself an implicitly definable hyperarithmetic set (under a recursive coding of the Herbrand base); hence to
The Finite Stages of Inductive Definitions
 Logical Foundations of Mathematics, Computer Science and Physics — Kurt Gödel’s Legacy
, 1996
"... . In general, the least fixed point of a positive elementary inductive definition over the Herbrand universe is # 1 1 and has no computational meaning. The finite stages, however, are computable, since validity of equality formulas in the Herbrand universe is decidable. We set up a formal system BI ..."
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Cited by 1 (1 self)
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. In general, the least fixed point of a positive elementary inductive definition over the Herbrand universe is # 1 1 and has no computational meaning. The finite stages, however, are computable, since validity of equality formulas in the Herbrand universe is decidable. We set up a formal system BID for the finite stages of positive elementary inductive definitions over the Herbrand universe and show that the provably total functions of the system are exactly that of Peano arithmetic. The formal system BID contains the socalled inductive extension of a logic program as a special case. This firstorder theory can be used to prove termination and correctness properties of pure Prolog programs, since notions like negationasfailure and lefttermination can be turned into positive inductive definitions. 1 Why inductive definitions over the Herbrand universe? In traditional logic programming, the semantics of a program is always given by the least fixed point of a monotonic operator over the Herbrand universe. The first example is the wellknown van EmdenKowalski operator for definite Horn clause programs in [25]. This operator is defined by a purely existential formula and is therefore continuous. The least fixed point of the operator is recursively enumerable. Moreover, the finite stages of the inductive definition are exactly what is computed by SLDresolution. In [11], Fitting has generalized the van EmdenKowalski operator using threevalued logic to programs which may also contain negation in the bodies of the clauses. Although Fitting's operator is still monotonic it is no longer continuous. It follows from Blair [2] and Kunen [14] that the least fixed point of this operator can be # 1 1 complete and that the closure ordinal can be # CK 1 even for definite Horn...
Logic programming: From underspecification to undefinedness
 In Theory of Computing 2012
, 2012
"... The semantics of logic programs was originally described in terms of twovalued logic. Soon, however, it was realised that threevalued logic had some natural advantages, as it provides distinct values not only for truth and falsehood, but also for “undefined”. The threevalued semantics proposed b ..."
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The semantics of logic programs was originally described in terms of twovalued logic. Soon, however, it was realised that threevalued logic had some natural advantages, as it provides distinct values not only for truth and falsehood, but also for “undefined”. The threevalued semantics proposed by Fitting and by Kunen are closely related to what is computed by a logic program, the third truth value being associated with nontermination. A different threevalued semantics, proposed by Naish, shared much with those of Fitting and Kunen but incorporated allowances for programmer intent, the third truth value being associated with underspecification. Naish used an (apparently) novel “arrow ” operator to relate the intended meaning of left and right sides of predicate definitions. In this paper we suggest that the additional truth values of Fitting/Kunen and Naish are best viewed as duals. We use Fitting’s later fourvalued approach to unify the two threevalued approaches. The additional truth value has very little affect on the Fitting threevalued semantics, though it can be useful when finding approximations to this semantics for program analysis. For the Naish semantics, the extra truth value allows intended interpretations to be more expressive, allowing us to verify and debug a larger class of programs. We also explain that the “arrow ” operator of Naish (and our fourvalued extension) is essentially the information ordering. This sheds new light on the relationships between specifications and programs, and successive executions states of a program. 1
www.elsevier.com/locate/tcs Fixpoint semantics for logic programming a survey
"... 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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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 0xpoint 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 0xpoint treatments developed for other programming methodologies. c © 2002 Elsevier Science B.V. All rights reserved. 1.