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POVM and Naimark's theorem without sums
"... We introduce an abstract notion of POVM within the categorical quantum mechanical semantics in terms of ..."
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We introduce an abstract notion of POVM within the categorical quantum mechanical semantics in terms of
A sequent calculus for compact closed categories
"... In this paper, we introduce the system CMLL of sequent calculus and establish its correspondence with compact closed categories. CMLL is equivalent in provability to the system MLL of classical linear logic with the tensor and par connectives identified. We show that the system allows a fairly simpl ..."
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In this paper, we introduce the system CMLL of sequent calculus and establish its correspondence with compact closed categories. CMLL is equivalent in provability to the system MLL of classical linear logic with the tensor and par connectives identified. We show that the system allows a fairly simple cutelimination, and the proofs in the system have a natural interpretation in compact closed categories. However, the soundness of the cutelimination procedure in terms of the categorical interpretation is by no means evident. We answer to this question affirmatively and establish the soundness by using the coherence result on compact closed categories by Kelly and Laplaza. 1
A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES
"... Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “l ..."
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Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras ” or “Kleisli monoids ” relative to a “monad ” on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous
Coherence for categorified operadic theories
"... It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a definition of weak Pcategory for any strongly regular (operadic) ..."
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It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a definition of weak Pcategory for any strongly regular (operadic) theory P, and show that every weak Pcategory is equivalent via Pfunctors and Ptransformations to a strict Pcategory. This strictification functor is then shown to have an interesting universal property. 1
Towards Hilbert's 24th Problem: Combinatorial Proof Invariants
, 2006
"... Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for ..."
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Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination. Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand’s theorem.
Reasoning about meaning in natural language with compact closed categories and frobenius algebras
 Logic and Algebraic Structures in Quantum Computing and Information, Association for Symbolic Logic Lecture Notes in Logic
, 2013
"... Compact closed categories have found applications in modeling quantum information protocols by AbramskyCoecke. They also provide semantics for Lambek’s pregroup algebras, applied to formalizing the grammatical structure of natural language, and are implicit in a distributional model of word meaning ..."
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Compact closed categories have found applications in modeling quantum information protocols by AbramskyCoecke. They also provide semantics for Lambek’s pregroup algebras, applied to formalizing the grammatical structure of natural language, and are implicit in a distributional model of word meaning based on vector spaces. Specifically, in previous work CoeckeClarkSadrzadeh used the product category of pregroups with vector spaces and provided a distributional model of meaning for sentences. We recast this theory in terms of strongly monoidal functors and advance it via Frobenius algebras over vector spaces. The former are used to formalize topological quantum field theories by Atiyah and BaezDolan, and the latter are used to model classical data in quantum protocols by CoeckePavlovicVicary. The Frobenius algebras enable us to work in a single space in which meanings of words, phrases, and sentences of any structure live. Hence we can compare meanings of different language constructs and enhance the applicability of the theory. We report on experimental results on a number of language tasks and verify the theoretical predictions. 1
CATEGORICAL ANALYSIS
, 2001
"... Abstract. We propose the categorification of algebraic analysis in terms of a specific 2category, called here the Leibniz 2category given by generators and relations which include the Leibnizlike relation (strict 3cell) among extended 2cells. The Leibniz 2category offers the ‘most general ’ no ..."
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Abstract. We propose the categorification of algebraic analysis in terms of a specific 2category, called here the Leibniz 2category given by generators and relations which include the Leibnizlike relation (strict 3cell) among extended 2cells. The Leibniz 2category offers the ‘most general ’ notion of a ‘(co)derivation’, as a strict 3cell, for a general (al co)gebra, not necessarily (co)associative, not necessarily (co)unital, nor necessarily (co)commutative. We outline a program in which every 2cell related to a partial (co)derivation (called a Leibniz strict 3cell) is translated into an appropriate 2cell related to a Cartan’slike (co)derivation (called a Cartan strict 3cell), and vice versa. We found also that a nonLeibniz component (2cell related to a Leibniz 3cell), responsible
Gdinaturality
, 2008
"... An extension of the notion of dinatural transformation is introduced in order to give a criterion for preservation of dinaturality under composition. An example of an application is given by proving that all bicartesian closed canonical transformations are dinatural. An alternative sequent system fo ..."
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An extension of the notion of dinatural transformation is introduced in order to give a criterion for preservation of dinaturality under composition. An example of an application is given by proving that all bicartesian closed canonical transformations are dinatural. An alternative sequent system for intuitionistic propositional logic is introduced as a device, and a cut elimination procedure is established for this system. 1
Generalised ProofNets for Compact Categories with Biproducts
, 2009
"... Just as conventional functional programs may be understood as proofs in an intuitionistic logic, so quantum processes can also be viewed as proofs in a suitable logic. We describe such a logic, the logic of compact closed categories and biproducts, presented both as a sequent calculus and as a syste ..."
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Just as conventional functional programs may be understood as proofs in an intuitionistic logic, so quantum processes can also be viewed as proofs in a suitable logic. We describe such a logic, the logic of compact closed categories and biproducts, presented both as a sequent calculus and as a system of proofnets. This logic captures much of the necessary structure needed to represent quantum processes under classical control, while remaining agnostic to the fine details. We demonstrate how to represent quantum processes as proofnets, and show that the dynamic behaviour of a quantum process is captured by the cutelimination procedure for the logic. We show that the cut elimination procedure is strongly normalising: that is, that every legal way of simplifying a proofnet leads to the same, unique, normal form. Finally, taking some initial set of operations