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Linear Logic, Autonomous Categories and Cofree Coalgebras
 In Categories in Computer Science and Logic
, 1989
"... . A brief outline of the categorical characterisation of Girard's linear logic is given, analagous to the relationship between cartesian closed categories and typed calculus. The linear structure amounts to a autonomous category: a closed symmetric monoidal category G with finite products and a c ..."
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Cited by 102 (7 self)
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. A brief outline of the categorical characterisation of Girard's linear logic is given, analagous to the relationship between cartesian closed categories and typed calculus. The linear structure amounts to a autonomous category: a closed symmetric monoidal category G with finite products and a closed involution. Girard's exponential operator, ! , is a cotriple on G which carries the canonical comonoid structure on A with respect to cartesian product to a comonoid structure on !A with respect to tensor product. This makes the Kleisli category for ! cartesian closed. 0. INTRODUCTION. In "Linear logic" [1987], JeanYves Girard introduced a logical system he described as "a logic behind logic". Linear logic was a consequence of his analysis of the structure of qualitative domains (Girard [1986]): he noticed that the interpretation of the usual conditional ")" could be decomposed into two more primitive notions, a linear conditional "\Gammaffi" and a unary operator "!" (called "of cours...
Natural Deduction and Coherence for Weakly Distributive Categories
 Journal of Pure and Applied Algebra
, 1991
"... This paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categorie ..."
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Cited by 67 (25 self)
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This paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categories which have the twotensor structure (times/par) of linear logic, but lack a negation operator. Representing morphisms in weakly distributive categories as such nets, we derive a coherence theorem for such categories. As part of this process, we develop a theory of expansionreduction systems with equalities and a term calculus for proof nets, each of which is of independent interest. In the symmetric case the expansion reduction system on the term calculus yields a decision procedure for the equality of maps for free weakly distributive categories. The main results of this paper are these. First we have proved coherence for the full theory of weakly distributive categories, extending simi...
Discrete Quantum Causal Dynamics
 International Journal of Theoretical Physics
, 2003
"... We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolut ..."
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Cited by 9 (4 self)
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We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolution, driven both by unitary operators and measurements, is coded in a precise mathematical structure in such a way that the crucial properties of causality, covariance and entanglement are faithfully represented. We show how our framework may be expressed using the language of (poly)categories and functors. Remarkably, important physical consequences  such as covariance  follow directly from the functoriality of our axioms. We establish strong links between the physical picture we propose and linear logic. Specifically we show that the rened logical connectives of linear logic can be used to describe the entanglements of subsystems in a precise way. Furthermore, we show that there is a precise correspondence between the evolution of a given system and deductions in a certain formal logical system based on the rules of linear logic. This framework generalizes and enriches both causal posets and the histories approach to quantum mechanics. 1
OPERADS AND PROPS
, 2006
"... We review definitions and basic properties of operads, PROPs and algebras over these structures. ..."
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Cited by 8 (0 self)
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We review definitions and basic properties of operads, PROPs and algebras over these structures.
A Logical Calculus for Polynomialtime Realizability
 Journal of Methods of Logic in Computer Science
, 1991
"... A logical calculus, not unlike Gentzen's sequent calculus for intuitionist logic, is described which is sound for polynomialtime realizability as defined by Crossley and Remmel. The sequent calculus admits cut elimination, thus giving a decision procedure for the propositional fragment. 0 Introduct ..."
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Cited by 3 (0 self)
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A logical calculus, not unlike Gentzen's sequent calculus for intuitionist logic, is described which is sound for polynomialtime realizability as defined by Crossley and Remmel. The sequent calculus admits cut elimination, thus giving a decision procedure for the propositional fragment. 0 Introduction In [4], a restricted notion of realizability is introduced, a special case of which is polynomialtime realizability: this is like Kleene's original realizability, save for three features. First, closed atomic formulae are realized by realizers that give a measure of the resources required to establish the formula, unlike Kleene's system which only reflects the fact that the formula is provable. Second, open formulae are treated as the corresponding closed formulae with all free variables universally quantified simultaneously. (There is a difference between the quantifiers 8h¸; ji and 8¸8j.) And third, the realizers code polynomialtime ("ptime") functions, rather than arbitrary recurs...
Graded Multicategories of Polynomialtime Realizers (Extended Abstract)
 Department of Mathematics Department of Mathematics and Computer Science McGill University John Abbott College Monash University 805 Sherbrooke St
, 1989
"... Preliminary Version Abstract We present a logical calculus which imposes a grading on a sequentstyle calculus to account for the runtime of the programmes represented by the sequents. This system is sound for a notion of polynomialtime realizability. An extension of the grading is also considered ..."
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Cited by 1 (1 self)
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Preliminary Version Abstract We present a logical calculus which imposes a grading on a sequentstyle calculus to account for the runtime of the programmes represented by the sequents. This system is sound for a notion of polynomialtime realizability. An extension of the grading is also considered, giving a notion of "dependant grades", which is also sound. Furthermore, we define a notion of closed graded multicategory, and show how the structure of polynomialtime realizers has that structure. 0 Introduction In [4], a restricted notion of realizability is defined, a special case of which is polynomialtime realizability: this is like Kleene's original realizability, save for three features. First, closed atomic formulae are realized only by realizers that express a reason for the "truth" (or provability) of the formula, unlike Kleene's system which only reflects the fact that the formula is provable. Second, open formulae are treated as the corresponding closed formulae with all fre...
Universal Properties of Impure Programming Languages
"... We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums ..."
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We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums and functions — can be characterized by universal properties in the setting of ‘premulticategories’, multicategories where the commutativity law may fail. This leads us to new, universal characterizations of two earlier equational theories of impure programming languages: the premonoidal categories of Power and Robinson, and the monadbased models of Moggi. Our analysis thus puts these earlier abstract ideas on a canonical foundation, bringing them to a new, syntactic level. F.3.2 [Semantics of Pro
Abstract
, 2008
"... The consistent histories approach to quantum mechanics is traditionally based on linearly ordered sequences of events. We extend the histories formalism to sets of events whose causal ordering is described by directed acyclic graphs. The need for a global time is eliminated and our construction refl ..."
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The consistent histories approach to quantum mechanics is traditionally based on linearly ordered sequences of events. We extend the histories formalism to sets of events whose causal ordering is described by directed acyclic graphs. The need for a global time is eliminated and our construction reflects the causal structure faithfully. 1