J. Lambek. Deductive Systems and Categories I. J. Math. Systems Theory, 2:278--318, 1968.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[42] which therefore includes some content in common with the present paper, which nevertheless presents a quite distinct perspective. Our starting point in this paper is the monoidal semantics of substructural logics, which was independently discovered by several researchers in the late 1960s [27, 11, 52, 48]. The version of the semantics we use is based on a preordered commutative monoid M = M; e; v) of possible worlds. The basic idea is to use the monoidal structure to define the semantics of the multiplicative, or substructural, connectives (I , in BI s notation) in the standard way, ....

J. Lambek. Deductive Systems and Categories I. J. Math. Systems Theory, 2:278--318, 1968.

....[43] which therefore includes some content in common with the present paper, which nevertheless presents a quite distinct perspective. Our starting point in this paper is the monoidal semantics of substructural logics, which was independently discovered by several researchers in the late 1960s [28, 10, 51, 48]. The version of the semantics we use is based on a preordered commutative monoid M = M; e; v) of possible worlds. The basic idea is to use the monoidal structure to define the semantics of the multiplicative, or substructural, connectives (I , in BI s notation) in the standard way, ....

J. Lambek. Deductive Systems and Categories I. J. Math. Systems Theory, 2:278--318, 1968.

....Methods. Throughout the paper we use proof theoretical language. This is based on the possibility to introduce a structure of free SMC category on the Multiplicative Intuitionistic Linear Logic. The connection between categories with structure and logical calculi was first described by J. Lambek [10]. Let us recall that, for this connection, it is useful to keep in mind the following correspondence between notions of category theory and proof theory: See examples 3.6, 5.1 and section 7 ffl Formulas are objects; ffl Equivalence classes of derivations of the sequent A B are morphisms from ....

J. Lambek. Deductive Systems and Categories. I. Math. Systems Theory, 2, 287-318, 1968.

....can be regarded as a closed monoidal category, by interpreting propositions as objects, proofs as morphisms and the multiplicative conjunction as the binary operation Omega . This was first pointed out by J. Lambek and applied in the proof of the coherence theorem for closed monoidal categories [16, 20]. We can also interpret the additive conjunction in the given category, if we add one more binary operation which is a Cartesian product in the sense defined in category theory. Therefore, we can say that the intuitionistic linear logic without exponentials is a monoidal closed category with a ....

J. Lambek. "Deductive systems and categories." I. J. Math. Systems Theory, 2, 1968, 278-318.

....Research Council of Canada. We thank Narciso Mart i Oliet and Robert Seely for helpful comments on a preliminary version of this work. 1 ffl Natural Transformations from category theory. ffl Parametricity from the foundations of polymorphism. Familiar work of Curry, Howard, Lambek and others [12, 15, 17] has shown how we may consider constructive proofs as programs. For example, Gentzen s intuitionistic sequents A 1 ; A k B may be interpreted as functional programs mapping k inputs of types A i , 1 i k, to outputs of type B. More precisely, proofs are interpreted as certain terms of ....

.... treatment of coherence in categories, discusses this very situation in detail for a general calculus of transformations, while special cases of the general problem were already resolved in Eilenberg Kelly [5] Cut elimination theorems were successfully applied to coherence questions by Lambek [15, 16, 17, 18] and Mints [22] Of course, for the simple types of this paper, normalization or cut elimination poses no problems. But even for these types, we shall obtain more: cut elimination implies that the internal language supports a compositional dinatural interpretation (between definable functors) In ....

J. Lambek. Deductive Systems and Categories I, J. Math. Systems Theory2 (1968), pp. 278-318.

....commutation congruence axioms of the autonomous type theory characterise autonomous categories exactly. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also con uent. As a immediate corollary, we solve a Coherence Problem a la Lambek [12]: the equality of maps in any autonomous category freely generated from a discrete graph is decidable. Contents 1 Introduction 2 2 Autonomous and Autonomous Type Theories 4 3 A Brief Survey of Related Work 15 4 Metatheoretic Results and Rewrite Systems 15 5 Interpretation in Autonomous and ....

....category) whose theory is exactly that. In addition, the rewrite systems that are associated with the type theories are all strongly normalising; modulo a simple notion of congruence, they are also con uent. As an immediate corollary, we obtain a solution to a coherence problem a la Lambek [12]: the equality of maps in any autonomous category freely generated from a discrete graph is decidable. In Section 2 we give an informal introduction of the various constructs of the autonomous type theories and explain the motivations behind the equality and congruence axioms. A brief ....

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J. Lambek. Deductive systems and categories I. J. Math. Systems Theory, 2:178-318, 1968.

....categories. More precisely we prove that that there is a canonical interpretation of the type theories in autonomous categories which is complete i.e. for any type theory, there is a autonomous category whose theory is exactly that. As a corollary, we solve a Coherence Problem a la Lambek [15]: the equality of maps in any free autonomous category (generated from a discrete graph) is decidable. As another application of the internal language, we prove a MacLane style 1 Coherence Theorem for autonomous categories: in any free such category, if A ( B, regarded as a type, is linearly ....

....for autonomous categories in the sense that there is a canonical interpretation of the type theories in autonomous categories that is complete i.e. for any type theory, there is a autonomous category whose theory is exactly that. As a corollary, we solve a Coherence Problem a la Lambek [15]: Theorem There is a decision procedure for equality of maps of a autonomous category freely generated from a discrete graph. As another application of the internal language, we prove a MacLane style 4 Coherence Theorem for autonomous categories in Section 5. First some definitions. We say ....

[Article contains additional citation context not shown here]

J. Lambek. Deductive systems and categories I. J. Math. Systems Theory, 2:178--318, 1968.

....and three commutation congruence axioms of the autonomous type theory characterise autonomous categories exactly. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek [12]: the equality of maps in any autonomous category freely generated from a discrete graph is decidable. 1 Introduction In this paper we introduce a family of type theories which can be regarded as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the ....

....category) whose theory is exactly that. In addition, the rewrite systems that are associated with the type theories are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a simple corollary, we obtain a solution to a coherence problem a la Lambek [12]: the equality of maps in any autonomous category freely generated from a discrete graph is decidable. To our knowledge, the problem was first solved by Blute et al. in [6] In Section 2 we give an informal introduction of the various constructs of the autonomous type theories and explain the ....

[Article contains additional citation context not shown here]

J. Lambek. Deductive systems and categories I. J. Math. Systems Theory, 2:178--318, 1968.

....is complete i.e. for any type theory, there is a model (i.e. autonomous category) whose theory is exactly that. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek [11]: the equality of maps in any autonomous category freely generated from a discrete graph is decidable. 1 Introduction A symmetric monoidal category h C ; Omega ; ff; oe; l; r i is a category C , a bifunctor Omega : C Theta C C , an object (tensor unit) 2 C and four isomorphisms ff A;B;C ....

....and we are done. In the context of monoidal category theory, by a MacLane style coherence theorem [13] we mean informally a statement of the form every diagram of a certain class (e.g. those that are composed of canonical isos) commutes , as opposed to a Lambek style coherence theorem [11] which we take to mean the existence of a decision procedure for deciding the equality of maps e.g. Theorem 6.1. We say that a type is linearly balanced if every non unit atomic type that occurs in it does so exactly twice, one occurrence is positive, the other is negative. We say that a type is ....

J. Lambek. Deductive systems and categories I. J. Math. Systems Theory, 2:178--318, 1968.

....combination of [Dorre e.a. 94] where feature formulas play the role of global constraints over type logical derivations. The layered architecture of [Dorre Manandhar 95] realizes the interface between type logic and feature logic in terms of a subtyping discipline on the atomic types A. In [Lambek 68] such an extension of the original calculus is shown to be decidable, cf. Buszkowski 88] for discussion. The interpretation mapping respects the subtyping pre order, i.e. v(p) v(q) whenever p q for p; q 2 A. On the proof theoretic level, one refines the Axiom schema to take into account ....

Lambek, J. (1968), `Deductive systems and categories. I', J. Math. Systems Theory 2, 278--318.

....A Delta; Sigma Gamma A (restrictedweakening) Delta Gamma A A; Sigma Gamma B Delta; Sigma Gamma B (cut) permutation at the left side of the arrow no contraction rule Here Sigma , Delta are lists of formulas, A,B,C formulas. This calculus was introduced by Lambek (see, e.g. [5,6]) Its description may also be found in the book by Szabo [7] a natural deduction system, which is equivalent to it with respect to categories) was described by Mints [8] and investigated by Babaev [9,10] Babaev s works have no English translation) The structure of a free closed category on our ....

....by anothers, according to some rules. These parts must not be necessarily the terms, but only quasiterms. If we replace Omega by , then quasiterms here will be just the ordinary lambda terms for the ordinary intuitionistic propositional calculus, i.e. for the Cartesian Closed case cf. 2] [5]. In the first stage there are the following conversions. 1) s A)B x A (s A)B ; x A ) where s is a quasiterm,s is not of the form y:s 0 and the occur rences of s, replaced by x: s; x) are not occurrences of the form (f: fs; u 1 g; u 2 g; un g; vg with u 1 ; un of type I, ....

J.Lambek. Deductive Systems and Categories.II. Lect.Notes in Math.,86:76-122, 1969.

....Research Council of Canada. We thank Narciso Mart i Oliet and Robert Seely for helpful comments on a preliminary version of this work. ffl Natural Transformations from category theory. ffl Parametricity from the foundations of polymorphism. Familiar work of Curry, Howard, Lambek and others [12, 15, 17] has shown how we may consider constructive proofs as programs. For example, Gentzen s intuitionistic sequents A 1 ; A k B may be interpreted as functional programs mapping k inputs of types A i , 1 i k, to outputs of type B. More precisely, proofs are interpreted as certain terms of ....

.... treatment of coherence in categories, discusses this very situation in detail for a general calculus of transformations, while special cases of the general problem were already resolved in Eilenberg Kelly [5] Cut elimination theorems were successfully applied to coherence questions by Lambek [15, 16, 17, 18] and Mints [22] Of course, for the simple types of this paper, normalization or cut elimination poses no problems. But even for these types, we shall obtain more: cut elimination implies that the internal language supports a compositional dinatural interpretation (between definable functors) In ....

J. Lambek. Deductive Systems and Categories I, J. Math. Systems Theory2 (1968), pp. 278-318.

....properties of sets and binary relations (under the usual operation of composition of relations) have also been influential. The book by Freyd and Scedrov [ 1990 ] provides a wealth of material on categorical logic from this perspective. Categorical proof theory Both Lawvere [ 1969 ] and Lambek [ 1968 ] put forward the idea that proofs of logical entailment between propositions, OE , may be modelled by morphisms, OE] Gamma [ in categories. If one is only interested in the existence of such proofs, then one might as well only consider categories with at most one morphism between ....

J. Lambek. Deductive systems and categories i. Math. Systems Theory, 2:287--318, 1968.

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Lambek, J. (1968), `Deductive systems and categories. I', J. Math. Systems Theory 2, 278--318.

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J. Lambek, "Deductive systems and categories I", J. Math. Systems Theory 2 (1968), 278 -- 318.

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J. Lambek, "Deductive systems and categories I", J. Math. Systems Theory 2 (1968), 278 -- 318.

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J. Lambek. Deductive Systems and Categories. I. Math. Systems Theory, 2, 287-318, 1968.

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