16 research outputs found

    A Comparative Study of Coq and HOL

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    This paper illustrates the differences between the style of theory mechanisation of Coq and of HOL. This comparative study is based on the mechanisation of fragments of the theory of computation in these systems. Examples from these implementations are given to support some of the arguments discussed in this paper. The mechanisms for specifying definitions and for theorem proving are discussed separately, building in parallel two pictures of the different approaches of mechanisation given by these systems

    Provable and unprovable cases of transfinite induction in a theory obtained by adding to HAω so-called "term-forms" of the kind introduced by M. Yasugi

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    I begin by discussing several of the existing ways of proving the validity of transfinite induction up to ε₀ and argue that it is at least conceivable that there is room for a new proof that is more constructive than any of them. An attempt which I pay particular attention to is that made by Mariko Yasugi (1982). The centrepiece of her theory is the so-called "construction principle", a principle for defining computable functionals. I argue that, in principle, it ought to be possible to set up a theory whose terms denote or range over functionals of a sort constructed by a similar principle, in which the accessibility (a term to be defined below) of ε₀ is provable, yet which dispenses with quantifiers as well as with some strong axioms which she uses in order to achieve the same result. My theory, described in chapter 2, is called TF (for "term-forms"). In chapters 3, 4 and 5, a proof of the accessibility of ε₀ in TF is presented. This thesis ends (chapter 6) with a proof of the computability of the functionals that can be represented in TF

    On the Correspondence Between Proofs and Lambda-Terms

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    The correspondence between natural deduction proofs and λ-terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) λ-terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed λ-calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors →, ×, +, ∀,∃ and ⊥ (falsity) (with or without η-like rules)

    Bounded Linear Logic

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    A typed, modular paradigm for polynomial time computation is proposed

    The language theory of Automath

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    Acta Cybernetica : Volume 18. Number 1.

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    Proceedings of the Workshop on Linear Logic and Logic Programming

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    Declarative programming languages often fail to effectively address many aspects of control and resource management. Linear logic provides a framework for increasing the strength of declarative programming languages to embrace these aspects. Linear logic has been used to provide new analyses of Prolog\u27s operational semantics, including left-to-right/depth-first search and negation-as-failure. It has also been used to design new logic programming languages for handling concurrency and for viewing program clauses as (possibly) limited resources. Such logic programming languages have proved useful in areas such as databases, object-oriented programming, theorem proving, and natural language parsing. This workshop is intended to bring together researchers involved in all aspects of relating linear logic and logic programming. The proceedings includes two high-level overviews of linear logic, and six contributed papers. Workshop organizers: Jean-Yves Girard (CNRS and University of Paris VII), Dale Miller (chair, University of Pennsylvania, Philadelphia), and Remo Pareschi, (ECRC, Munich)

    Representing logics in type theory

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