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Categorical Logic
 A CHAPTER IN THE FORTHCOMING VOLUME VI OF HANDBOOK OF LOGIC IN COMPUTER SCIENCE
, 1995
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Metalogical Frameworks
, 1992
"... In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the me ..."
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Cited by 57 (15 self)
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In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the metalogic of the object language being implemented. We also reason about the implementation itself, say to know it is correct; this is done in a programming logic. How do all these logics relate? This paper considers that question and more. We show that by taking the view that the metalogic is primary, these other parts are related in standard ways. The metalogic should be suitably rich so that the object logic can be presented as an abstract data type, and it must be suitably computational (or constructive) so that an instance of that type is an implementation. The data type abstractly encodes all that is relevant for metareasoning, i.e., not only the term constructing functions but also the...
Normal Forms and CutFree Proofs as Natural Transformations
 in : Logic From Computer Science, Mathematical Science Research Institute Publications 21
, 1992
"... What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax g ..."
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Cited by 12 (4 self)
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What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cutelimination and asymmetrical interpretations of cutfree proofs. This viewpoint is connected to Reynolds' relational interpretation of parametricity ([27], [2]), and to the KellyLambekMac LaneMints approach to coherence problems in category theory. 1 Introduction In the past several years, there has been renewed interest and research into the interconnections of proof theory, typed lambda calculus (as a functional programming paradigm) and category theory. Some of these connections can be surprisingly subtle. Here we a...
Coherence in Category Theory and the ChurchRosser Property
, 1993
"... Szabo's derivation systems on sequent calculi with exchange and product are not ChurchRosser. Thus his coherence results for categories having a symmetric product (either monoidal or cartesian) are false. 1 Introduction Gentzen's sequent calculi [9] have been applied extensively in category theory ..."
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Cited by 2 (0 self)
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Szabo's derivation systems on sequent calculi with exchange and product are not ChurchRosser. Thus his coherence results for categories having a symmetric product (either monoidal or cartesian) are false. 1 Introduction Gentzen's sequent calculi [9] have been applied extensively in category theory, e.g [2, 3, 4, 6, 7, 8]. Sequents correspond to morphisms of a category, and the rules of the calculus correspond to categorical structures (e.g. having an associative tensor product). Cutelimination was then used to put bounds on the complexity of these structures, e.g. to produce exhaustive lists (perhaps with duplications) of the canonical natural transformations between given functors. For symmetric, monoidal closed categories it was shown in [12] how to decide in principle whether two such transformations are equal, while an effective, lineartime decision procedure was given in [1]. Derivation systems (reduction rules) can be used to eliminate some duplicates in the list of cutfree ...
Uniqueness of Normal Proofs in Implicational Intuitionistic Logic
 Journal of Logic, Language and Information
, 1999
"... . A minimal theorem in a logic L is an Ltheorem which is not a nontrivial substitution instance of another Ltheorem. Komori (1987) raised the question whether every minimal implicational theorem in intuitionistic logic has a unique normal proof in the natural deduction system NJ. The answer has be ..."
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. A minimal theorem in a logic L is an Ltheorem which is not a nontrivial substitution instance of another Ltheorem. Komori (1987) raised the question whether every minimal implicational theorem in intuitionistic logic has a unique normal proof in the natural deduction system NJ. The answer has been known to be partially positive and generally negative. It is shown here that a minimal implicational theorem A in intuitionistic logic has a unique finormal proof in NJ whenever A is provable without nonprime contraction. The nonprime contraction rule in NJ is the implication introduction rule whose cancelled assumption differs from a propositional variable and appears more than once in the proof. Our result improves the known partial positive solutions to Komori's problem. Also, we present another simple example of a minimal implicational theorem in intuitionistic logic which does not have a unique fijnormal proof in NJ. Key words: natural deduction, uniqueness of normal proofs, coh...
Categorical Logic
, 2001
"... This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists. ..."
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This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists.
Abstract Obstructions to Coherence: Natural Noncoherent Associativity
, 2008
"... We study what happens when coherence fails. Categories with a tensor product and a natural associativity isomorphism that does not necessarily satisfy the pentagon coherence requirements (called associative categories) are considered. Categorical versions of associahedra where naturality squares com ..."
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We study what happens when coherence fails. Categories with a tensor product and a natural associativity isomorphism that does not necessarily satisfy the pentagon coherence requirements (called associative categories) are considered. Categorical versions of associahedra where naturality squares commute and pentagons do not, are constructed (called Catalan groupoids, An). These groupoids are used in the construction of the free associative category. They are also used in the construction of the theory of associative categories (given as a 2sketch). Generators and relations are given for the fundamental group, π(An), of the Catalan groupoids – thought of as a simplicial complex. These groups are shown to be more than just free groups. Each associative category, B, has related fundamental groups π(Bn) and homomorphisms π(Pn) : π(An) − → π(Bn). If the images of the π(Pn) are trivial, i.e. there is only one associativity path between any two objects, then the category is coherent. Otherwise the images of π(Pn) are obstructions to coherence. Some progress is made in classifying noncoherence of associative categories.