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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 such a ..."
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Cited by 23 (0 self)
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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.
LoopFree Construction of CounterModels for Intuitionistic Propositional Logic
, 1995
"... . We present a nonlooping method to construct Kripke trees refuting the nontheorems of intuitionistic propositional logic, using a contractionfree sequent calculus. 1991 Mathematics Subject Classification: 03B20, 03B35, 03C25, 03F03, 68T15 1. Introduction It is well known that IPL (Intuitionistic ..."
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. We present a nonlooping method to construct Kripke trees refuting the nontheorems of intuitionistic propositional logic, using a contractionfree sequent calculus. 1991 Mathematics Subject Classification: 03B20, 03B35, 03C25, 03F03, 68T15 1. Introduction It is well known that IPL (Intuitionistic Propositional Logic) has the finite model property; in fact, any nontheorem of IPL can be invalidated by means of a finite Kripke tree [2]. The standard method (see [9] for a formal treatment) for constructing such countermodels requires a loopchecker. Here, we present a method for constructing countermodels not requiring a loopchecker, based on the contractionfree sequent calculi LJT and LJT* [1]. LJT provides a very simple but reasonably effective decision procedure for IPL. The ideas were first presented in the work of Vorob'ev [10] and more recently also in [3] and [5]. LJT differs from traditional formulations of Gentzen sequent calculi for IPL only in the rule oe) for introduct...
Linear Logic and Computation: A Survey
 Proof and Computation, Proceedings Marktoberdorf Summer School
, 1993
"... . This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth ..."
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Cited by 16 (6 self)
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. This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth (emphasized in classical logic) and proof construction (emphasized in intuitionistic logic) remain important in linear logic, it might be said that the emphasis in linear logic is on state. Linear logic is sometimes described as being resource sensitive because it provides an intrinsic and natural accounting of process states, events, and resources. Linear logic also sheds new light on classical logic and its relationship to intuitionistic logic, see Girard [15, 16] and Danos et al. [11]. An evocative semantic paradigm for linear logic by means of games is proposed by Blass [7] and by Abramsky and Jagadeesan [2]. As an intuitive motivation, let us consider reading logical deductions so tha...
Linearizing Intuitionistic Implication
 In Proc. 6th Annual IEEE Symposium on Logic in Computer Science
, 1993
"... An embedding of the implicational propositional intuitionistic logic (iil) into the nonmodal fragment of intuitionistic linear logic (imall) is given. The embedding preserves cutfree proofs in a proof system that is a variant of iil. The embedding is efficient and provides an alternative proof of t ..."
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Cited by 16 (5 self)
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An embedding of the implicational propositional intuitionistic logic (iil) into the nonmodal fragment of intuitionistic linear logic (imall) is given. The embedding preserves cutfree proofs in a proof system that is a variant of iil. The embedding is efficient and provides an alternative proof of the pspacehardness of imall. It exploits several prooftheoretic properties of intuitionistic implication that analyze the use of resources in iil proofs. Linear logic is a refinement of classical and intuitionistic logic that provides an intrinsic and natural accounting of resources. In Girard's words [12], "linear logic is a logic behind logic." A convenient way to present linear logic is by modifying the traditional Gentzenstyle sequent calculus axiomatization of classical logic (see, e.g., [15, 22]). The modification may be briefly described in three steps. The first step is to remove two structural rules, contraction and weakening, which manipulate the use of hypotheses and conclusi...
Lambda Terms for Natural Deduction, Sequent Calculus and Cut Elimination
"... It is wellknown that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequ ..."
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Cited by 13 (3 self)
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It is wellknown that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequent calculus derivations and natural deduction derivations is, however, not a oneone map, which causes some syntactic technicalities. The correspondence is best explained by two extensionally equivalent type assignment systems for untyped lambda terms, one corresponding to natural deduction (N) and the other to sequent calculus (L). These two systems constitute different grammars for generating the same (type assignment relation for untyped) lambda terms. The second grammar is ambiguous, but the first one is not. This fact explains the manyone correspondence mentioned above. Moreover, the second type assignment system has a `cutfree' fragment (L cf ). This fragment generates exactly the typeable lambda terms in normal form. The cut elimination theorem becomes a simple consequence of the fact that typed lambda terms posses a normal form.
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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Cited by 11 (4 self)
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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.
Sufficient conditions for cut elimination with complexity analysis
 Annals of Pure and Applied Logic
, 2007
"... Sufficient conditions for first order based sequent calculi to admit cut elimination by a SchütteTait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related with the calculus. The con ..."
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Cited by 9 (4 self)
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Sufficient conditions for first order based sequent calculi to admit cut elimination by a SchütteTait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related with the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and intuitionistic modal logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus.
Program development by proof transformation
"... We begin by reviewing the natural deduction rules for the!^8fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. ..."
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Cited by 7 (3 self)
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We begin by reviewing the natural deduction rules for the!^8fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. We also consider two alternative methods to deal with the strong or constructive existential quantifier 9\Lambda. Finally we discuss the wellknown notion of an extracted program of a derivation involving 9\Lambda, in order to set up a relation between the two alternatives. Section 2 deals with the computational content of classical proofs. As is wellknown a proof of a 89theorem with a quantifierfree kernel where 9 is viewed as defined by:8: can be used as a program. We describe a "direct method " to use such a proof as a program, and compare it with Harvey Friedman's Atranslation [3] followed by program extraction from the resulting constructive proof. It is shown that both algorithms coincide. In section 3 Goad's method of pruning of proof trees is introduced. It is shown how a proof can be simplified after addition of some further assumptions. In a first step some subproofs are replaced by different ones using the additional assumptions. In a second step parts of the proof tree are pruned, i.e. cut out. Note that the first step involves searching for new proofs using the new assumptions of formulas in the proof tree. Hence we also have to discuss proof search in minimal logic. Finally section 4 treats an example already considered by Goad in his thesis [5], the binpacking problem. The main difference to Goad's work is that he used a logic with the strong existential quantifier, whereas we work within the!8fragment. This example is particularly wellsuited to demonstrate that the pruning method can be applied to adapt programs to particular situations, and moreover that pruning can change the functions computed by programs. In this sense this method is essentially different from program development by program transformation. We would like to thank Michael Bopp and KarlHeinz Niggl for their help in preparing these notes.
A local system for intuitionistic logic
 of Lecture Notes in Artificial Intelligence
, 2006
"... Abstract. This paper presents systems for firstorder intuitionistic logic and several of its extensions in which all the propositional rules are local, in the sense that, in applying the rules of the system, one needs only a fixed amount of information about the logical expressions involved. The ma ..."
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Abstract. This paper presents systems for firstorder intuitionistic logic and several of its extensions in which all the propositional rules are local, in the sense that, in applying the rules of the system, one needs only a fixed amount of information about the logical expressions involved. The main source of nonlocality is the contraction rules. We show that the contraction rules can be restricted to the atomic ones, provided we employ deepinference, i.e., to allow rules to apply anywhere inside logical expressions. We further show that the use of deep inference allows for modular extensions of intuitionistic logic to Dummett’s intermediate logic LC, Gödel logic and classical logic. We present the systems in the calculus of structures, a proof theoretic formalism which supports deepinference. Cut elimination for these systems are proved indirectly by simulating the cutfree sequent systems, or the hypersequent systems in the cases of Dummett’s LC and Gödel logic, in the cut free systems in the calculus of structures.