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Higherorder logic programming
 HANDBOOK OF LOGIC IN AI AND LOGIC PROGRAMMING, VOLUME 5: LOGIC PROGRAMMING. OXFORD (1998
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Lexical scoping as universal quantification
 In Sixth International Logic Programming Conference
, 1989
"... Abstract: A universally quantified goal can be interpreted intensionally, that is, the goal ∀x.G(x) succeeds if for some new constant c, the goal G(c) succeeds. The constant c is, in a sense, given a scope: it is introduced to solve this goal and is “discharged ” after the goal succeeds or fails. Th ..."
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Cited by 66 (19 self)
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Abstract: A universally quantified goal can be interpreted intensionally, that is, the goal ∀x.G(x) succeeds if for some new constant c, the goal G(c) succeeds. The constant c is, in a sense, given a scope: it is introduced to solve this goal and is “discharged ” after the goal succeeds or fails. This interpretation is similar to the interpretation of implicational goals: the goal D ⊃ G should succeed if when D is assumed, the goal G succeeds. The assumption D is discharged after G succeeds or fails. An interpreter for a logic programming language containing both universal quantifiers and implications in goals and the body of clauses is described. In its nondeterministic form, this interpreter is sound and complete for intuitionistic logic. Universal quantification can provide lexical scoping of individual, function, and predicate constants. Several examples are presented to show how such scoping can be used to provide a Prologlike language with facilities data types, and encapsulation of state.
A ProofTheoretic Analysis of GoalDirected Provability
 Journal of Logic and Computation
, 1992
"... One of the distinguishing features of logic programming seems to be the notion of goaldirected provability, i.e. that the structure of the goal is used to determine the next step in the proof search process. It is known that by restricting the class of formulae it is possible to guarantee that a ..."
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Cited by 14 (7 self)
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One of the distinguishing features of logic programming seems to be the notion of goaldirected provability, i.e. that the structure of the goal is used to determine the next step in the proof search process. It is known that by restricting the class of formulae it is possible to guarantee that a certain class of proofs, known as uniform proofs, are complete with respect to provability in intuitionistic logic. In this paper we explore the relationship between uniform proofs and classes of formulae more deeply. Firstly we show that uniform proofs arise naturally as a normal form for proofs in firstorder intuitionistic sequent calculus. Next we show that the class of formulae known as hereditary Harrop formulae are intimately related to uniform proofs, and that we may extract such formulae from uniform proofs in two different ways. We also give results which may be interpreted as showing that hereditary Harrop formulae are the largest class of formulae for which uniform proo...
On Normal Forms and Equivalence for Logic Programs
 Proceedings of the Joint International Conference and Symposium on Logic Programming
, 1992
"... It is known that larger classes of formulae than Horn clauses may be used as logic programming languages. One such class of formulae is hereditary Harrop formulae, for which an operational notion of provability has been studied, and it is known that operational provability corresponds to provability ..."
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Cited by 8 (7 self)
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It is known that larger classes of formulae than Horn clauses may be used as logic programming languages. One such class of formulae is hereditary Harrop formulae, for which an operational notion of provability has been studied, and it is known that operational provability corresponds to provability in intuitionistic logic. In this paper we discuss the notion of a normal form for this class of formulae, and show how this may be given by removing disjunctions and existential quantifications from programs. Whilst the normal form of the program preserves operational provability, there are operationally equivalent programs which are not intuitionistically equivalent. As it is known that classical logic is too strong to precisely capture operational provability for larger classes of programs than Horn clauses, the appropriate logic in which to study questions of equivalence is an intermediate logic. We explore the nature of the required logic, and show that this may be obtained by the addit...
Success and Failure for Hereditary Harrop Formulae
 Journal of Logic Programming
, 1993
"... We introduce the foundational issues involved in incorporating the Negation as Failure (NAF) rule into the framework of firstorder hereditary Harrop formulae of Miller et al. This is a larger class of formulae than Horn clauses, and so the technicalities are more intricate than in the Horn claus ..."
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Cited by 5 (0 self)
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We introduce the foundational issues involved in incorporating the Negation as Failure (NAF) rule into the framework of firstorder hereditary Harrop formulae of Miller et al. This is a larger class of formulae than Horn clauses, and so the technicalities are more intricate than in the Horn clause case. As programs may grow during execution in this framework, the role of NAF and the Closed World Assumption (CWA) need some modification, and for this reason we introduce the notion of a completely defined predicate, which may be thought of as a localisation of the CWA. We also show how this notion may be used to define a notion of NAF for a more general class of goals than literals alone. We also show how an extensional notion of universal quantification may be incorporated. This makes our framework somewhat different from that of Miller et al., but not essentially so. We also show how to construct a Kripkelike model for the extended class of programs. This is essentially a de...
On GoalDirected Provability in Classical Logic
, 1994
"... this paper we explore the possibilities for a notion of goaldirected proof in classical logic. The technical point to consider is how to deal with the multipleconclusioned nature of classical sequents, i.e. that classical succedents may contain more than one formula. This means that there may be mo ..."
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Cited by 5 (1 self)
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this paper we explore the possibilities for a notion of goaldirected proof in classical logic. The technical point to consider is how to deal with the multipleconclusioned nature of classical sequents, i.e. that classical succedents may contain more than one formula. This means that there may be more than one "candidate" right rule, as there may be several nonatomic formulae in the succedent, and so a choice has to be made as to which formula is to be reduced in the next step. The question of whether this choice may be free or restricted is one of the key decisions to be made. The free choice, i.e. that the order in which the formulae are reduced does not matter, will clearly constrain the logic programming language more than the restricted, one, and is arguably more declarative; on the other hand, the weaker notion is arguably more goaldirected, and there is no obvious reason to insist on the stronger version. We will refer to the free choice as rightreductive proofs, and to the restricted one as rightdirected proofs. Thus there seems to be more than one notion of goaldirected proof in classical logic, and clearly the corresponding logic programming languages may differ according to which class of proofs is used. However, as we shall see, there are do not seem to be any "interesting" languages for which the weaker notion is complete but the stronger one is not, and so it appears the stronger version (which requires that all right rules permute over each other) is the more useful notion.
Herbrand Methods in Sequent Calculi: Unification in LL
 Proc. of the Joint International Conference and Symposium on Logic Programming
, 1992
"... We propose a reformulation of quantifiers rules in sequent calculi which allows to replace blind existential instantiation with unification, thereby reducing nondeterminism and complexity in proofsearch. Our method, based on some ideas underlying the proof of Herbrand theorem for classical logic, m ..."
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Cited by 5 (2 self)
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We propose a reformulation of quantifiers rules in sequent calculi which allows to replace blind existential instantiation with unification, thereby reducing nondeterminism and complexity in proofsearch. Our method, based on some ideas underlying the proof of Herbrand theorem for classical logic, may be applied to any "reasonable" nonclassical sequent calculus, but here we focus on sequent calculus for linear logic, in view of an application to linear logic programming. We prove that the new linear proofsystem which we propose, the so called system LLH, is equivalent to standard linear sequent calculus LL. 1 Introduction A result in classical logic which has been widely exploited in logic programming is Herbrand theorem. Several versions of this result are present in the literature; we recall here one of them (see [13]). Herbrand Theorem Let F be a prenex formula of the form 9w8x9y8zA[w; x; y; z] with A quantifierfree. F is provable in predicate calculus if and only if a disjun...
Dialgebraic Specification and Modeling
"... corecursive functions COALGEBRA state model constructors destructors data model recursive functions reachable hidden abstraction observable hidden restriction congruences invariants visible abstraction ALGEBRA visible restriction!e Swinging Cube ..."
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Cited by 4 (4 self)
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corecursive functions COALGEBRA state model constructors destructors data model recursive functions reachable hidden abstraction observable hidden restriction congruences invariants visible abstraction ALGEBRA visible restriction!e Swinging Cube
A strong logic programming view for static embedded implication
 In Proc. of the 2 nd Int. Conf. of Foundation of Software Science and Computation Structures (FOSSACS'99
, 1999
"... Abstract. A strong (L) logic programming language ([14, 15]) is given by two subclasses of formulas (programs and goals) of the underlying logic L, provided that: firstly, any program P (viewed as a Ltheory) has a canonical model MP which is initial in the category of all its Lmodels; secondly, th ..."
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Cited by 2 (0 self)
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Abstract. A strong (L) logic programming language ([14, 15]) is given by two subclasses of formulas (programs and goals) of the underlying logic L, provided that: firstly, any program P (viewed as a Ltheory) has a canonical model MP which is initial in the category of all its Lmodels; secondly, the Lsatisfaction of a goal G in MP is equivalent to the Lderivability of G from P, and finally, there exists an effective (computable) proofsubcalculus of the Lcalculus which works out for derivation of goals from programs. In this sense, Horn clauses constitute a strong (firstorder) logic programming language. Following the methodology suggested in [15] for designing logic programming languages, an extension of Horn clauses should be made by extending its underlying firstorder logic to a richer logic which supports a strong axiomatization of the extended logic programming language. A wellknown approach for extending Horn clauses with embedded implications is the static scope programming language presented in [8]. In this paper we show that such language can be seen as a strong FO ⊃ logic programming language, where FO ⊃ is a very natural extension of firstorder logic with intuitionistic implication. That is, we present a new characterization of the language in [8] which shows that Horn clauses extended with embedded implications, viewed as FO ⊃theories, preserves all the attractive mathematical and computational properties that Horn clauses satisfy as firstordertheories. 1