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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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A Theory of Modules for Logic Programming
 In Symp. Logic Programming
, 1986
"... Abstract: We present a logical language which extends the syntax of positive Horn clauses by permitting implications in goals and in the bodies of clauses. The operational meaning of a goal which is an implication is given by the deduction theorem. That is, a goal D ⊃ G is satisfied by a program P i ..."
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Abstract: We present a logical language which extends the syntax of positive Horn clauses by permitting implications in goals and in the bodies of clauses. The operational meaning of a goal which is an implication is given by the deduction theorem. That is, a goal D ⊃ G is satisfied by a program P if the goal G is satisfied by the larger program P ∪ {D}. If the formula D is the conjunction of a collection of universally quantified clauses, we interpret the goal D ⊃ G as a request to load the code in D prior to attempting G, and then unload that code after G succeeds or fails. This extended use of implication provides a logical explanation of parametric modules, some uses of Prolog’s assert predicate, and certain kinds of abstract datatypes. Both a modeltheory and prooftheory are presented for this logical language. We show how to build a possibleworlds (Kripke) model for programs by a fixed point construction and show that the operational meaning of implication mentioned above is sound and complete for intuitionistic, but not classical, logic. 1. Implications as Goals Let A be a syntactic variable which ranges over atomic formulas of firstorder logic. Let G range over a class of formulas, called goal formulas, to be specified shortly. We shall assume, however, that this class always contains ⊤ (true) and all atomic formulas. The formulas represented by A and G may contain free variables. Given these two classes, we define definite clauses, denoted by the syntactic variable D, as follows: D: = G ⊃ A  ∀x D  D1 ∧ D2 A program is defined to be a finite set of closed definite clauses. P will be a syntactic variable for programs. A clause of the form ⊤ ⊃ A will often be written as simply A. Let P be a program. Define [P] to be the smallest set of formulas satisfying the following recursive definitions. (i) P ⊆ [P].
A Kripkelike Model for Negation as Failure
 In Proceedings of the North American Conference on Logic Programming (NACLP
, 1989
"... We extend the Kripkelike model theory given in [10] for a fragment of firstorder hereditary Harrop formulae to include negated atoms in goals. This gives us a formal framework in which to study the role of Negation as Failure rule. The class of predicates for which Negation As Failure is applicable ..."
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Cited by 7 (1 self)
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We extend the Kripkelike model theory given in [10] for a fragment of firstorder hereditary Harrop formulae to include negated atoms in goals. This gives us a formal framework in which to study the role of Negation as Failure rule. The class of predicates for which Negation As Failure is applicable is discussed, as well as the predicates for which some other form of negation will need to be used. We show how the former class may be incorporated into the model theory, giving a generalisation of the usual T ! construction. No restriction on the class of programs is needed for this approach; the construction may be used for programs which are not locally stratified[14]. This is accomplished by the use of a success level and a failure level of a goal, either or both of which may be infinite. The resulting T operator is not monotonic, which necessitates a slight departure from the standard procedure, but the important properties of the construction still hold. 1 Introduction A classic ...
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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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...
Submodels of Kripke Models
 Arch. Math. Logic
, 1998
"... A Kripke model K is a submodel of another Kripke modelMif K is obtained by restricting the set of nodes of M. In this paper we showthat the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas. T ..."
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A Kripke model K is a submodel of another Kripke modelMif K is obtained by restricting the set of nodes of M. In this paper we showthat the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas. This result is an analogue of the LosTarski theorem for the Classical Predicate Calculus. In appendix A we provethat for theories with decidable identity we can take as the embeddings between domains in Kripke models of the theory, the identical embeddings. This is a well known fact, but we know of no correct proof in the literature. In appendix B we answer, negatively, a question posed by Sam Buss: whether there is a classical theory T, such that HT is HA. HereHT is the theory of all Kripke modelsMsuch that the structures assigned to the nodes of M all satisfy T in the sense of
THE MODAL LOGIC OF FORCING
, 2007
"... Abstract. A set theoretical assertion ψ is forceable or possible, written ♦ ψ, if ψ holds in some forcing extension, and necessary, written � ψ, ifψ holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFCprovable principle ..."
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Abstract. A set theoretical assertion ψ is forceable or possible, written ♦ ψ, if ψ holds in some forcing extension, and necessary, written � ψ, ifψ holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFCprovable principles of forcing are exactly those in the modal theory S4.2. 1.
Decidability Extracted: Synthesizing ``CorrectbyConstruction'' Decision Procedures from Constructive Proofs
, 1998
"... The topic of this thesis is the extraction of efficient and readable programs from formal constructive proofs of decidability. The proof methods employed to generate the efficient code are new and result in clean and readable Nuprl extracts for two nontrivial programs. They are based on the use of ..."
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The topic of this thesis is the extraction of efficient and readable programs from formal constructive proofs of decidability. The proof methods employed to generate the efficient code are new and result in clean and readable Nuprl extracts for two nontrivial programs. They are based on the use of Nuprl's set type and techniques for extracting efficient programs from induction principles. The constructive formal theories required to express the decidability theorems are of independent interest. They formally circumscribe the mathematical knowledge needed to understand the derived algorithms. The formal theories express concepts that are taught at the senior college level. The decidability proofs themselves, depending on this material, are of interest and are presented in some detail. The proof of decidability of classical propositional logic is relative to a semantics based on Kleene's strong threevalued logic. The constructive proof of intuitionistic decidability presented here is the first machine formalization of this proof. The exposition reveals aspects of the Nuprl tactic collection relevant to the creation of readable proofs; clear extracts and efficient code are illustrated in the discussion of the proofs.
Independence Results for Weak systems of Intuitionistic Arithmetic
 Math. Logic Quart
, 2003
"... This paper proves some independence results for weak fragments of Heyting arithmetic by using Kripke models. We present a necessary condition for linear Kripke models of arithmetical theories which are closed under the negative translation and use it to show that the union of the worlds in any linea ..."
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This paper proves some independence results for weak fragments of Heyting arithmetic by using Kripke models. We present a necessary condition for linear Kripke models of arithmetical theories which are closed under the negative translation and use it to show that the union of the worlds in any linear Kripke model of HA satisfies P A. We construct a twonode P Anormal Kripke structure which does not force iΣ2. We prove i∀1 � i∃1, i∃1 � i∀1, iΠ2 � iΣ2 and iΣ2 � iΠ2. We use Smorynski’s operation Σ ′ to show HA � lΠ1.
A Modal Analysis of some Principles of the Provability Logic of Heyting Arithmetic
 In Proceedings of AiML’98
, 1998
"... In this paper four principles and one scheme of the Provability Logic of (intuitionistic) Heyting Arithmetic HA are studied from a modal logical point of view. These are, besides the wellknown principles K;K4 and L of the Provability Logic of Peano Arihtmetic, the principle 2(A B) ! 2(A 2B) and ..."
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In this paper four principles and one scheme of the Provability Logic of (intuitionistic) Heyting Arithmetic HA are studied from a modal logical point of view. These are, besides the wellknown principles K;K4 and L of the Provability Logic of Peano Arihtmetic, the principle 2(A B) ! 2(A 2B) and the scheme 2::(2A ! W 2B i ) ! 2(2A ! W 2B i ). Intuitionistic modal logic is introduced. And for all principles and for the scheme considered (seperately and together) frame completeness is proved, and the finite model property is established. 1 Introduction In contrast to the situation in classical Peano Arithmetic PA, the provability logic, to be defined in the next section, of intuitionistic Heyting Arithmetic HA appears to be a rich collection of principles. The full provability logic of HA is not known (yet?), but some principles have already been found [Leivant 75, Visser 94], among which are the following. K 2(A ! B) ! (2A ! 2B) K4 2A ! 22A L 2(2A ! A) ! 2A (Lob's princip...
Propositional Logics of Closed and Open Substitutions over Heyting’s Arithmetic
, 2005
"... In this note we compare propositional logics for closed substitutions and propositional logics for open substitutions in constructive arithmetical theories. We provide a strong example where these logics diverge in an essential way. We prove that for Markov’s Arithmetic, i.e. Heyting’s Arithmetic pl ..."
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In this note we compare propositional logics for closed substitutions and propositional logics for open substitutions in constructive arithmetical theories. We provide a strong example where these logics diverge in an essential way. We prove that for Markov’s Arithmetic, i.e. Heyting’s Arithmetic plus Markov’s principle plus Extended Church’s Thesis, the logic of closed and the logic of open substitutions are the same. 1