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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 ..."
Abstract

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...
Forcing for IZF in sheaf toposes
 Georgian Mathematical Journal
"... To Mamuka at the Occasion of his 50 th Birthday In [Sco] D. Scott has shown how the interpretation of intuitionistic set theory IZF in presheaf toposes can be reformulated in a more concrete fashion à la forcing as known to set theorists. In this note we show how this can be adapted to the more gene ..."
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To Mamuka at the Occasion of his 50 th Birthday In [Sco] D. Scott has shown how the interpretation of intuitionistic set theory IZF in presheaf toposes can be reformulated in a more concrete fashion à la forcing as known to set theorists. In this note we show how this can be adapted to the more general case of Grothendieck toposes dealt with abstractly in [Fou, Hay]. 1