Temperley-Lieb Algebra: From Knot Theory to Logic and Computation via Quantum Mechanics Export

@article{citeulike:5968633, abstract = {Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of apparently disparate topics: Knot Theory, Categorical Quantum Mechanics, and Logic and Computation. We shall focus in particular on the following two topics: - The Temperley-Lieb algebra has always hitherto been presented as a quotient of some sort: either algebraically by generators and relations as in Jones' original presentation, or as a diagram algebra modulo planar isotopy as in Kauffman's presentation. We shall use tools from Geometry of Interaction, a dynamical interpretation of proofs under Cut Elimination developed as an off-shoot of Linear Logic, to give a direct description of the Temperley-Lieb category -- a "fully abstract presentation", in Computer Science terminology. This also brings something new to the Geometry of Interaction, since we are led to develop a planar version of it, and to verify that the interpretation of Cut-Elimination (the "Execution Formula", or "composition by feedback") preserves planarity. - We shall also show how the Temperley-Lieb algebra provides a natural setting in which computation can be performed diagrammatically as geometric simplification -- "yanking lines straight". We shall introduce a "planar lambda-calculus" for this purpose, and show how it can be interpreted in the Temperley-Lieb category.}, archivePrefix = {arXiv}, author = {Abramsky, Samson}, citeulike-article-id = {5968633}, citeulike-linkout-0 = {http://arxiv.org/abs/0910.2737}, citeulike-linkout-1 = {http://arxiv.org/pdf/0910.2737}, day = {14}, eprint = {0910.2737}, keywords = {algebra, category-theory, computation, geometry, knots, logic, mathematics, quantum}, month = {Oct}, posted-at = {2009-10-18 23:55:44}, priority = {2}, title = {Temperley-Lieb Algebra: From Knot Theory to Logic and Computation via Quantum Mechanics}, url = {http://arxiv.org/abs/0910.2737}, year = {2009} }

TY - JOUR ID - citeulike:5968633 L3 - citeulike-article-id:5968633 N2 - Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of apparently disparate topics: Knot Theory, Categorical Quantum Mechanics, and Logic and Computation. We shall focus in particular on the following two topics: - The Temperley-Lieb algebra has always hitherto been presented as a quotient of some sort: either algebraically by generators and relations as in Jones' original presentation, or as a diagram algebra modulo planar isotopy as in Kauffman's presentation. We shall use tools from Geometry of Interaction, a dynamical interpretation of proofs under Cut Elimination developed as an off-shoot of Linear Logic, to give a direct description of the Temperley-Lieb category -- a "fully abstract presentation", in Computer Science terminology. This also brings something new to the Geometry of Interaction, since we are led to develop a planar version of it, and to verify that the interpretation of Cut-Elimination (the "Execution Formula", or "composition by feedback") preserves planarity. - We shall also show how the Temperley-Lieb algebra provides a natural setting in which computation can be performed diagrammatically as geometric simplification -- "yanking lines straight". We shall introduce a "planar lambda-calculus" for this purpose, and show how it can be interpreted in the Temperley-Lieb category. TI - Temperley-Lieb Algebra: From Knot Theory to Logic and Computation via Quantum Mechanics KW - algebra KW - category-theory KW - computation KW - geometry KW - knots KW - logic KW - mathematics KW - quantum AU - Abramsky, Samson PY - 2009/Oct/14/ UR - http://arxiv.org/abs/0910.2737 ER -