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32
Normalisation control in deep inference via atomic flows
, 2008
"... Abstract. We introduce ‘atomic flows’: they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a n ..."
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Cited by 23 (11 self)
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Abstract. We introduce ‘atomic flows’: they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a new and very general normalisation theorem, which contains cut elimination as a special case. We operate in deep inference, which is more general than other syntactic paradigms, and where normalisation is more difficult to control. We argue that atomic flows are a significant technical advance for normalisation theory, because 1) the technique they support is largely independent of syntax; 2) indeed, it is largely independent of logical inference rules; 3) they constitute a powerful geometric formalism, which is more intuitive than syntax. 1.
Termination orders for 3dimensional rewriting
 J. of
, 2004
"... Abstract: This paper studies 3polygraphs as a framework for rewriting on twodimensional words. A translation of term rewriting systems into 3polygraphs with explicit resource management is given, and the respective computational properties of each system are studied. Finally, a convergent 3polyg ..."
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Cited by 12 (4 self)
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Abstract: This paper studies 3polygraphs as a framework for rewriting on twodimensional words. A translation of term rewriting systems into 3polygraphs with explicit resource management is given, and the respective computational properties of each system are studied. Finally, a convergent 3polygraph for the (commutative) theory of Z/2Zvector spaces is given. In order to prove these results, it is explained how to craft a class of termination orders for 3polygraphs. This paper starts with the introductory section 1 on equational theories and term rewriting systems. It gives notations and graphical representations that are used in the sequel. Then, it focuses on one major restriction of term rewriting, namely the fact that it cannot provide convergent presentations for commutative equational theories: equational theories that contain a commutative binary operator. Section 2 studies the resource management operations of permutation, erasure and duplication: they are implicit and global in term rewriting and it is sketched there how to make them explicit. However, the framework for rewriting in algebraic structures needs to be extended to include this change; section 3 proposes 3polygraphs to fulfill this role. Here, these objects, introduced in [Burroni 1993], are used as equational presentations of a special case of 2categories: MacLane’s product categories, called PROs,
Polarized unification grammars
 In: Proceedings of the 21st International Conference on Computational Linguistics and 44th Annual Meeting of the Association for Computational Linguistics
, 2006
"... This paper proposes a generic mathematical formalism for the combination of various structures: strings, trees, dags, graphs and products of them. The polarization of the objects of the elementary structures controls the saturation of the final structure. This formalism is both elementary and powerf ..."
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Cited by 11 (0 self)
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This paper proposes a generic mathematical formalism for the combination of various structures: strings, trees, dags, graphs and products of them. The polarization of the objects of the elementary structures controls the saturation of the final structure. This formalism is both elementary and powerful enough to strongly simulate many grammar formalisms, such as rewriting systems, dependency grammars, TAG, HPSG and LFG. 1
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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Cited by 7 (0 self)
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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
INTENSIONAL PROPERTIES OF POLYGRAPHS
"... Abstract – We present Albert Burroni’s polygraphs as a computational model, showing how these objects can be seen as functional programs. First, we prove that the model is Turing complete. Then, we use a notion of termination proof introduced by the second author to characterize polygraphs that comp ..."
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Cited by 5 (0 self)
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Abstract – We present Albert Burroni’s polygraphs as a computational model, showing how these objects can be seen as functional programs. First, we prove that the model is Turing complete. Then, we use a notion of termination proof introduced by the second author to characterize polygraphs that compute in polynomial time and, going further, polynomial time functions. 1
Homomorphisms of higher categories
 U.U.D.M. REPORT 2008:47
, 2008
"... We describe a construction that to each algebraically specified notion of higherdimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associativ ..."
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Cited by 5 (0 self)
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We describe a construction that to each algebraically specified notion of higherdimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associative and unital composition. We give two applications of this construction. The first is to tricategories; and here we do not obtain the trihomomorphisms defined by Gordon, Power and Street, but rather something which is equivalent in a suitable sense. The second application is to Batanin’s weak ωcategories.
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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Cited by 5 (2 self)
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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
Resolutions By Polygraphs
, 2003
"... A notion of resolution for higherdimensional categories is defined, by using polygraphs, and basic invariance theorems are proved. ..."
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A notion of resolution for higherdimensional categories is defined, by using polygraphs, and basic invariance theorems are proved.