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Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λCalculi
, 1992
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On an interpretation of second order quantification in first order intuitionistic propositional logic
 JOURNAL OF SYMBOLIC LOGIC, 57: 33 { 52
, 1992
"... We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, , built up from propositional variables (p; q; r; : ::) and falsity (?) using conjunction (^), disjunction (_) and implication (!). Write ` to indicate that s ..."
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We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, , built up from propositional variables (p; q; r; : ::) and falsity (?) using conjunction (^), disjunction (_) and implication (!). Write ` to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula there exists a formula Ap (e ectively computable from), containing only variables not equal to p which occur in, and such that for all formulas not involving p, ` ! Ap if and only if ` !. Consequently quanti cation over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on rst order propositions. An immediate corollary is the strengthening of the usual Interpolation Theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC² can be constructed whose algebra of truthvalues is equal to any given Heyting algebra.
A Deterministic Terminating Sequent Calculus for GödelDummett logic
, 1999
"... We give a short prooftheoretic treatment of a terminating contractionfree calculus G4LC for the zeroorder GödelDummett logic LC. This calculus is a slight variant of a calculus given by Avellone et al, who show its completeness by modeltheoretic techniques. In our calculus, all the rules of G4 ..."
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We give a short prooftheoretic treatment of a terminating contractionfree calculus G4LC for the zeroorder GödelDummett logic LC. This calculus is a slight variant of a calculus given by Avellone et al, who show its completeness by modeltheoretic techniques. In our calculus, all the rules of G4LC are invertible, thus allowing a deterministic proofsearch procedure.
Admissibility of Structural Rules for ContractionFree Systems of Intuitionistic Logic
 Journal of Symbolic Logic
, 2000
"... We give a direct proof of admissibility of cut and contraction for the contractionfree sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multisuccedent calculus; this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs, i.e. ..."
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We give a direct proof of admissibility of cut and contraction for the contractionfree sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multisuccedent calculus; this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs, i.e. those which use induction on sequent weight or appeal to admissibility of rules in other calculi.
Decision Procedures for Intuitionistic Propositional Logic by Program Extraction
 In TABLEAUX 1998, LNCS 1397
, 1998
"... . We present two constructive proofs of the decidability of intuitionistic propositional logic by simultaneously constructing either a countermodel or a derivation. From these proofs, we extract two programs which have a sequent as input and return a derivation or a counter model. The search ..."
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. We present two constructive proofs of the decidability of intuitionistic propositional logic by simultaneously constructing either a countermodel or a derivation. From these proofs, we extract two programs which have a sequent as input and return a derivation or a counter model. The search tree of these algorithms is linearly bounded by the number of connectives of the input. Soundness of these programs follows from giving a correct construction of the derivations, similarly to Hudelmaier 's work [7]; completeness from giving a correct construction of the countermodels, inspired by Miglioli, Moscato, and Ornaghi [8]. 1 Introduction Intuitionistic proofs can be considered as programs together with their verification. Consequently intuitionistic logic is a method for developing correct programs. To demonstrate the advantage of this approach, we construct two theorem provers for the propositional part by extracting them from a decidability proof. Taking up Fitting's [2] ...
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. Theexposition 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.
Strong cutelimination systems for Hudelmaier’s depthbounded sequent calculus for implicational logic
 PROCEEDINGS OF THE 3RD INTERNATIONAL JOINT CONFERENCE ON AUTOMATED REASONING (IJCAR’06), VOLUME 4130 OF LNAI
, 2006
"... Inspired by the CurryHoward correspondence, we study normalisation procedures in the depthbounded intuitionistic sequent calculus of Hudelmaier (1988) for the implicational case, thus strengthening existing approaches to Cutadmissibility. We decorate proofs with proofterms and introduce various ..."
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Inspired by the CurryHoward correspondence, we study normalisation procedures in the depthbounded intuitionistic sequent calculus of Hudelmaier (1988) for the implicational case, thus strengthening existing approaches to Cutadmissibility. We decorate proofs with proofterms and introduce various termreduction systems representing proof transformations. In contrast to previous papers which gave different arguments for Cutadmissibility suggesting weakly normalising procedures for Cutelimination, our main reduction system and all its variations are strongly normalising, with the variations corresponding to different optimisations, some of them with good properties such as confluence.
Formalising a Proof of Contraction Admissibility for G4ip
"... We will, in this document, give a report of a formalisation, in Isabelle, of Contraction admissibility for the calculus G4ip. Specifically, we formalise sections of [3], which goes on to directly prove Cut admissibility as well. We ..."
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We will, in this document, give a report of a formalisation, in Isabelle, of Contraction admissibility for the calculus G4ip. Specifically, we formalise sections of [3], which goes on to directly prove Cut admissibility as well. We