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A Relevant Analysis of Natural Deduction
 Journal of Logic and Computation
, 1999
"... Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and ..."
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Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lLcalculus; the representation mechanism is judgementsastypes, developed for relevant logics. The lLcalculus type theory is a firstorder dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required prooftheoretic metatheory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I calculus and ML with references. We describe the CurryHowardde Bruijn correspondence of the lLcalculus with a s...
A Notion of Classical Pure Type System
 Proc. of 13th Ann. Conf. on Math. Found. of Programming Semantics, MFPS'97
, 1997
"... We present a notion of classical pure type system, which extends the formalism of pure type system with a double negation operator. 1 Introduction It is an old idea that proofs in formal logics are certain functions and objects. The BrowerHeytingKolmogorov (BHK) interpretation [15,51,40], in the ..."
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Cited by 10 (6 self)
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We present a notion of classical pure type system, which extends the formalism of pure type system with a double negation operator. 1 Introduction It is an old idea that proofs in formal logics are certain functions and objects. The BrowerHeytingKolmogorov (BHK) interpretation [15,51,40], in the form stated by Heyting [40], states that a proof of an implication P ! Q is a "construction " which transforms any proof of P into a proof of Q. This idea was formalized independently by Kleene's realizability interpretation [46,47] in which proofs of intuitionistic number theory are interpreted as numbers, by the CurryHoward (CH) isomorphism [21,43] in which proofs of intuitionistic implicational propositional logic are interpreted as simply typed terms, and by the LambekLawvere (LL) isomorphism [52,55] in which proofs of intuitionistic positive propositional logic are interpreted as morphisms in a cartesian closed category. In the latter cases, the interpretations have an inverse, in th...
Weak Normalization Implies Strong Normalization in Generalized NonDependent Pure Type Systems
 Comput. Sci
, 1997
"... The BarendregtGeuversKlop conjecture states that every weakly normalizing pure type system is also strongly normalizing. We show that this is true for a uniform class of systems which includes, e.g., the left hand side of Barendregt's cube as well as the system U . This seems to be the first ..."
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The BarendregtGeuversKlop conjecture states that every weakly normalizing pure type system is also strongly normalizing. We show that this is true for a uniform class of systems which includes, e.g., the left hand side of Barendregt's cube as well as the system U . This seems to be the first result giving a positive answer to the conjecture not merely for some concrete systems for which strong normalization is known to hold, but for a uniform class of systems in which not all systems are strongly normalizing. 1.
On the Conservativity of Leibniz Equality
, 1996
"... We embed a first order theory with equality in the Pure Type System L that is a subsystem of the wellknown type system PRED2. The embedding is based on the CurryHoward isomorphism, i.e. \Gamma\Gamma\Gamma? and 8 coincide with ! and \Pi. Formulas of the form t1 s = t2 are treated as Leibniz equal ..."
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We embed a first order theory with equality in the Pure Type System L that is a subsystem of the wellknown type system PRED2. The embedding is based on the CurryHoward isomorphism, i.e. \Gamma\Gamma\Gamma? and 8 coincide with ! and \Pi. Formulas of the form t1 s = t2 are treated as Leibniz equalities. That is, t1 s = t2 is identified with the second order formula 8P : P (t1 )\Gamma\Gamma\Gamma?P (t2 ), which contains only \Gamma\Gamma\Gamma?'s and 8's and can hence be embedded straightforwardly. We give a syntactic proof for the equivalence between derivability in the logic and inhabitance in L. The idea of the proof is to introduce extra reduction steps, that reduce those proofterms that do not correspond to derivations in the logic to ones that do correspond to derivations in the logic. Introduction Many logics can be interpreted in type systems. For instance Implicational Propositional Logic can be interpreted in ! . In [Ber89], Berardi designed a type system (PRED!) in w...