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Higherdimensional categories with finite derivation type
"... We study convergent (terminating and confluent) presentations of ncategories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for ncategories, generalising the one introduced by Squier for word rewriting systems. We characterise this pr ..."
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We study convergent (terminating and confluent) presentations of ncategories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for ncategories, generalising the one introduced by Squier for word rewriting systems. We characterise this property by using the notion of critical branching. In particular, we define sufficient conditions for an ncategory to have finite derivation type. Through examples, we present several techniques based on derivations of 2categories to study convergent presentations by 3polygraphs.
The three dimensions of proofs
 Annals of Pure and Applied Logic
, 2006
"... Abstract: In this document, we study a 3polygraphic translation for the proofs of SKS, a formal system for classical propositional logic. We prove that the free 3category generated by this 3polygraph describes the proofs of classical propositional logic modulo structural bureaucracy. We give a 3 ..."
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Abstract: In this document, we study a 3polygraphic translation for the proofs of SKS, a formal system for classical propositional logic. We prove that the free 3category generated by this 3polygraph describes the proofs of classical propositional logic modulo structural bureaucracy. We give a 3dimensional generalization of Penrose diagrams and use it to provide several pictures of a proof. We sketch how local transformations of proofs yield a non contrived example of 4dimensional rewriting. Outline In the first section of this paper, we give a 2dimensional translation of the formulas of system SKS, a formal system for propositional classical logic [Brünnler 2004] expressed in the style of the calculus of structures [Guglielmi 2004]. The idea consists in the replacement of formulas by circuitlike objects organized in a 2polygraph [Burroni 1993]. This construction is formalized in theorem 1.4.16. We proceed to section 2, whose purpose is to translate the proofs of SKS into 3dimensional objects that form a 3polygraph. There we note that every inference rule can be interpreted as a directed 3cell between two circuits. We prove theorem 2.4.3 stating that the 3polygraph we have built can be equipped with a proof theory which is the same as the SKS one. Section 3 is where the 3dimensional nature of
Computing Critical Pairs in 2Dimensional Rewriting Systems
, 2010
"... Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for ..."
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Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for the elements of the presented monoid. Polygraphs are a higherdimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of ncategories. Here, we are interested in proving confluence for polygraphs presenting 2categories, which can be seen as a generalization of termrewriting systems. For this purpose, we propose an adaptation of the usual algorithm for computing critical pairs. Interestingly, this framework is much richer than term rewriting systems and requires the elaboration of a new theoretical framework for representing critical pairs, based on contexts in compact 2categories. Term rewriting systems have proven very useful to reason about terms modulo equations. In some cases, the equations can be oriented and completed in a way giving rise to a normalizing (i.e. confluent and terminating) rewriting system, thus providing a notion of canonical representative of equivalence classes of terms. Usually, terms are freely generated by a signature (Σn)n∈N, which consists of a family of sets Σn of generators of arity n, and one considers equational theories on such a signature, which are formalized by sets of pairs of terms called equations. For example, the equational theory of monoids contains two generators m and e, whose arities are respectively 2 and 0, and three equations
TWO POLYGRAPHIC PRESENTATIONS OF PETRI NETS
, 2005
"... Abstract: This document gives an algebraic and two polygraphic translations of Petri nets, all three providing an easier way to describe reductions and to identify some of them. The first one sees places as generators of a commutative monoid and transitions as rewriting rules on it: this setting is ..."
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Abstract: This document gives an algebraic and two polygraphic translations of Petri nets, all three providing an easier way to describe reductions and to identify some of them. The first one sees places as generators of a commutative monoid and transitions as rewriting rules on it: this setting is totally equivalent to Petri nets, but lacks any graphical intuition. The second one considers places as 1dimensional cells and transitions as 2dimensional ones: this translation recovers a graphical meaning but raises many difficulties since it uses explicit permutations. Finally, the third translation sees places as degenerated 2dimensional cells and transitions as 3dimensional ones: this is a setting equivalent to Petri nets, equipped with a graphical interpretation. Outline In this document, we study Petri nets in order to give two possible polygraphic presentations for them. This work follows Albert Burroni’s intuitions: many computer science and proof theory objects have natural translations into polygraphs. These are topologyflavoured objects consisting of collections of directed cells of various dimensions, equipped with a rich algebraic structure. In section 1, we recall some basic facts about Petri nets, describe their representations and associate them reduction graphs, equipped with a relation that identifies paths that intuitively represent the same
Operads, clones, and distributive laws
, 2008
"... Abstract We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a monad on profunctors thanks to a distributivelaw. The presentation builds upon recent work by Fiore, Gambino, Hyland, and Winskel on a theory ..."
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Abstract We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a monad on profunctors thanks to a distributivelaw. The presentation builds upon recent work by Fiore, Gambino, Hyland, and Winskel on a theory of generalized species of structures,but, for the multicategory case, the general idea goes back to Burroni's Tcategories (1971). We show how other previous categorical analysesof operad (via Day's tensor products, or via analytical functor) fit with the profunctor approach.
The Structure of FirstOrder Causality
"... Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in firstorder propositional logic. One of the main difficulties that has to be fac ..."
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Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in firstorder propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterize definable strategies, that is strategies which actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task, which requires to combine advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model, by the means of generators and relations: those strategies can be generated from a finite set of atomic strategies and the equality between strategies admits a finite axiomatization, this equational structure corresponding to a polarized variation of the notion of bialgebra. This work thus bridges algebra and denotational semantics in order to reveal the structure of dependencies induced by firstorder quantifiers, and lays the foundations for a mechanized analysis of causality in programming languages. Denotational semantics were introduced to provide useful abstract invariants of proofs and programs modulo cutelimination or reduction. In particular, game semantics, introduced in the nineties, have been very successful in capturing precisely the interactive behaviour of programs. In these semantics, every type is interpreted as a game (that is as a set of moves that can be played during the game) together with the rules of the game (formalized by a partial order on the moves of the game indicating the dependencies between them). Every move is to be played by one of the two players, called Proponent and Opponent, who should be thought respectively as the program and its environment. A program is characterized by the sequences of moves that it can exchange with its environment during an
Open Graphs and Computational Reasoning
"... We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of halfedges (edges which are drawn with an unconnected end) and enjo ..."
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We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of halfedges (edges which are drawn with an unconnected end) and enjoy rich compositional principles by connecting graphs along these halfedges. In particular, this allows equations and rewrite rules to be specified between graphs. Particular computational models can then be encoded as an axiomatic set of such rules. Further rules can be derived graphically and rewriting can be used to simulate the dynamics of a computational system, e.g. evaluating a program on an input. Examples of models which can be formalised in this way include traditional electronic circuits as well as recent categorical accounts of quantum information. 1
Diagram rewriting and operads
, 2009
"... We introduce an explicit diagrammatic syntax for PROs and PROPs, which are used in the theory of operads. By means of diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. This diagrammatic syntax is useful for prac ..."
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We introduce an explicit diagrammatic syntax for PROs and PROPs, which are used in the theory of operads. By means of diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. This diagrammatic syntax is useful for practical computations, but also for theoretical results. Moreover, rewriting is strongly related to homotopy theory. For instance, it can be used to compute homological invariants of algebraic structures, or to prove coherence results.
Computing Critical Pairs in Polygraphs
 In Workshop on Computer Algebra Methods and Commutativity of Algebraic Diagrams (CAMCAD
, 2009
"... Polygraphs generalize to 2categories the usual notion of equational theory, by describing them as quotients, modulo equations, of freely generated 2categories on a given set of generators. In order to work with morphisms modulo the equations, it is often convenient to orient the equations into a c ..."
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Polygraphs generalize to 2categories the usual notion of equational theory, by describing them as quotients, modulo equations, of freely generated 2categories on a given set of generators. In order to work with morphisms modulo the equations, it is often convenient to orient the equations into a confluent rewriting system. In the case of a terminating system, confluence can be checked by showing that critical pairs are joinable. However, the computation of the critical pairs is more complicated for polygraphs than for term rewriting systems: in particular, two left members of a rule don’t necessarily have a finite number of unifiers. We advocate here that a more general notion of rewriting system should be considered instead, and introduce an operad of compact contexts in a 2category, in which two rules have a finite number of unifiers. A concrete representation of contexts is proposed, as well as an unification algorithm for these.