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A NonStandard Semantics for Kahn Networks in Continuous Time
"... In a seminal article, Kahn has introduced the notion of process network and given a semantics for those using Scott domains whose elements are (possibly infinite) sequences of values. This model has since then become a standard tool for studying distributed asynchronous computations. From the beginn ..."
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In a seminal article, Kahn has introduced the notion of process network and given a semantics for those using Scott domains whose elements are (possibly infinite) sequences of values. This model has since then become a standard tool for studying distributed asynchronous computations. From the beginning, process networks have been drawn as particular graphs, but this syntax is never formalized. We take the opportunity to clarify it by giving a precise definition of these graphs,
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
HIGHERDIMENSIONAL NORMALISATION STRATEGIES FOR ACYCLICITY
"... Abstract – We introduce acyclic track polygraphs, a notion of complete categorical cellular models for small categories: they are polygraphs containing generators, with additional invertible cells for relations and higherdimensional globular syzygies. We give a rewriting method to realise such a mo ..."
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Abstract – We introduce acyclic track polygraphs, a notion of complete categorical cellular models for small categories: they are polygraphs containing generators, with additional invertible cells for relations and higherdimensional globular syzygies. We give a rewriting method to realise such a model by proving that a convergent presentation canonically extends to an acyclic track polygraph. For that, we introduce normalising strategies, defined as homotopically coherent ways to relate each cell of a track polygraph to its normal form, and we prove that acyclicity is equivalent to the existence of a normalisation strategy. Using track polygraphs, we extend to every dimension the homotopical finiteness condition of finite derivation type, introduced by Squier in string rewriting theory, and we prove that it implies a new homological finiteness condition that we introduce here. The proof is based on normalisation strategies and relates acyclic track polygraphs to free abelian resolutions of the small categories they present.
A Homotopical Completion Procedure with Applications to Coherence of Monoids
"... One of the most used algorithm in rewriting theory is the KnuthBendix completion procedure which starts from a terminating rewriting system and iteratively adds rules to it, trying to produce an equivalent convergent rewriting system. It is in particular used to study presentations of monoids, sinc ..."
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One of the most used algorithm in rewriting theory is the KnuthBendix completion procedure which starts from a terminating rewriting system and iteratively adds rules to it, trying to produce an equivalent convergent rewriting system. It is in particular used to study presentations of monoids, since normal forms of the rewriting system provide canonical representatives of words modulo the congruence generated by the rules. Here, we are interested in extending this procedure in order to retrieve information about the lowdimensional homotopy properties of a monoid. We therefore consider the notion of coherent presentation, which is a generalization of rewriting system that keeps track of the cells generated by confluence diagrams. We extend the KnuthBendix completion procedure to this setting, resulting in a homotopical completion procedure. It is based on a generalization of Tietze transformations, which are operations that can be iteratively applied to relate any two presentations of the same monoid. We also explain how these transformations can be used to remove useless generators, rules, or confluence diagrams in a coherent presentation, thus leading to a homotopical reduction procedure. Finally, we apply these techniques to the study of some examples coming from representation theory, to compute minimal coherent presentations for them: braid, plactic and Chinese monoids.
A POLYGRAPHIC SURVEY ON FINITENESS CONDITIONS FOR REWRITING SYSTEMS
"... Abstract – In 1987, Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition leftFP3. Using this result, he has constructed finitely presented decidable monoids that cannot be presented by finite co ..."
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Abstract – In 1987, Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition leftFP3. Using this result, he has constructed finitely presented decidable monoids that cannot be presented by finite convergent rewriting systems. In 1994, Squier introduced the condition of finite derivation type, which is a homotopical finiteness property on the derivation graph associated to a monoid presentation. He showed that this condition is an invariant of finite presentations and he gave a constructive way to prove this finiteness property based on the computation of the critical branchings: being of finite derivation type is a necessary condition for a finitely presented monoid to admit a finite convergent presentation. This selfcontained survey presents those results in the language of polygraphs.
Compressing Polarized Boxes
"... Abstract—The sequential nature of sequent calculus provides a simple definition of cutelimination rules that duplicate or erase subproofs. The parallel nature of proof nets, instead, requires the introduction of explicit boxes, which are global and synchronous constraints on the structure of graph ..."
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Abstract—The sequential nature of sequent calculus provides a simple definition of cutelimination rules that duplicate or erase subproofs. The parallel nature of proof nets, instead, requires the introduction of explicit boxes, which are global and synchronous constraints on the structure of graphs. We show that logical polarity can be exploited to obtain an implicit, compact, and natural representation of boxes: in an expressive polarized dialect of linear logic, boxes may be represented by simply recording some of the polarity changes occurring in the box at level 0. The content of the box can then be recovered locally and unambiguously. Moreover, implicit boxes are more parallel than explicit boxes, as they realize a larger quotient. We provide a correctness criterion and study the novel and subtle cutelimination dynamics induced by implicit boxes, proving confluence and strong normalization.
POLYGRAPHS OF FINITE DERIVATION TYPE
, 2014
"... In 1987, Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition leftFP3. Using this result, he has constructed finitely presented decidable monoids that cannot be presented by finite convergent ..."
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In 1987, Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition leftFP3. Using this result, he has constructed finitely presented decidable monoids that cannot be presented by finite convergent rewriting systems. In 1994, he introduced the condition of finite derivation type, which is a homotopical finiteness property on the presentation complex associated to a monoid presentation. He showed that this condition is an invariant of finite presentations and he gave a constructive way to prove this finiteness property based on the computation of the critical branchings: being of finite derivation type is a necessary condition for a finitely presented monoid to admit a finite convergent presentation. This selfcontained survey presents those results in the contemporary language of polygraphs and higherdimensional categories, providing new proofs and relations between them.