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26
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.
Static Dependency Pair Method based on Strong Computability for HigherOrder Rewrite Systems
, 2008
"... Higherorder rewrite systems (HRSs) and simplytyped term rewriting systems (STRSs) are computational models of functional programs. We recently proposed an extremely powerful method, the static dependency pair method, which is based on the notion of strong computability, in order to prove terminati ..."
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Cited by 8 (0 self)
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Higherorder rewrite systems (HRSs) and simplytyped term rewriting systems (STRSs) are computational models of functional programs. We recently proposed an extremely powerful method, the static dependency pair method, which is based on the notion of strong computability, in order to prove termination in STRSs. In this paper, we extend the method to HRSs. Since HRSs include λabstraction but STRSs do not, we restructure the static dependency pair method to allow λabstraction, and show that the static dependency pair method also works well on HRSs without new restrictions.
Meadows and the equational specification of division
, 901
"... The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of equational specifications of data types cannot use field theory in ap ..."
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The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of equational specifications of data types cannot use field theory in applications to number systems based upon rational, real and complex numbers. We study a new axiomatic concept for number systems with division that uses only equations: a meadow is a commutative ring with a total inverse operator satisfying two equations which imply 0 −1 = 0. All fields and products of fields can be viewed as meadows. After reviewing alternate axioms for inverse, we start the development of a theory of meadows. We give a general representation theorem for meadows and find, as a corollary, that the conditional equational theory of meadows coincides with the conditional equational theory of zero totalized fields. We also prove representation results for meadows of finite characteristic.
A discrete Nash theorem with quadratic complexity and dynamic equilibria
, 2006
"... Nash’s Theorem guarantees the existence of Nash equilibria for strategicform games. The typical proof of the result uses Brouwer’s Fixed Point Theorem on probabilistic strategies. We show that Tarski’s Fixed Point Theorem can be used to establish a similar result for discrete equilibria in a large ..."
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Cited by 4 (2 self)
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Nash’s Theorem guarantees the existence of Nash equilibria for strategicform games. The typical proof of the result uses Brouwer’s Fixed Point Theorem on probabilistic strategies. We show that Tarski’s Fixed Point Theorem can be used to establish a similar result for discrete equilibria in a larger class of games that we call conversion/preference games. Our result rests on a graph characterisation of Nash equilibria that i) reifies the decision procedure for pure Nash equilibria, ii) allows us to compute the equilibria in quadratic time in the number of game situations, and iii) makes the equilibria explicitly dynamic in nature. We conclude by discussing the extended range of technical applications of noncooperative game theory that results from the new theorem, including for gene regulation and celllevel signal transduction. 1
Union of reducibility candidates for orthogonal constructor rewriting
 In CiE’08
, 2008
"... Abstract. We revisit Girard’s reducibility candidates by proposing a general of the notion of neutral terms. They are the terms which do not interact with some contexts called elimination contexts. We apply this framework to constructor rewriting, and show that for orthogonal constructor rewriting, ..."
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Abstract. We revisit Girard’s reducibility candidates by proposing a general of the notion of neutral terms. They are the terms which do not interact with some contexts called elimination contexts. We apply this framework to constructor rewriting, and show that for orthogonal constructor rewriting, Girard’s reducibility candidates are stable by union. 1
R.: Unfolding CSP
 In: Reflections on the Work of
, 2010
"... Appreciation from the second author Tony Hoare and I have exchanged ideas on concurrent processes for three decades. To a sufficiently distant observer it has seemed that we were doing the same thing, and that therefore we should not have made it look different. A closer view shows this to be false. ..."
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Appreciation from the second author Tony Hoare and I have exchanged ideas on concurrent processes for three decades. To a sufficiently distant observer it has seemed that we were doing the same thing, and that therefore we should not have made it look different. A closer view shows this to be false. Complementary things, yes; and both have been enriched by the crossfertilisation. A little history shows something of these different approaches. Around 1979 we first discovered our complementary interests in concurrency. Tony at first expressed his ideas through the medium of a programming language [6], and I through a prototypical algebraic theory [8]. The difference became plainer as time went on. Tony was keen to find a single formalism in which specifications of concurrent systems could be refined into programs. I, on the other hand, was keen to find a mathematical concept of process that could stand in analogy with the familiar notion of (single valued) function, and I was happy that specification should be done in an associated logic. We enjoyed discussing these things. I recall one discussion at a blackboard (or was it a whiteboard?) where Tony hinted to me his first ideas about failures semantics. The present short paper, in honour of Tony and continuing our long, friendly and sometimes rivalrous collaboration, is another step towards harmonising our approaches. 1
The lambdacontext calculus
 In: LFMTP’07: International Workshop on Logical Frameworks and MetaLanguages
"... We present a simple lambdacalculus whose syntax is populated by variables which behave like metavariables. It can express both captureavoiding and capturing substitution (instantiation). To do this requires several innovations, including a key insight in the confluence proof and a set of reductio ..."
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We present a simple lambdacalculus whose syntax is populated by variables which behave like metavariables. It can express both captureavoiding and capturing substitution (instantiation). To do this requires several innovations, including a key insight in the confluence proof and a set of reduction rules which manages the complexity of a calculus of contexts over the ‘vanilla ’ lambdacalculus in a very simple and modular way. This calculus remains extremely close in look and feel to a standard lambdacalculus with explicit substitutions, and good properties of the lambdacalculus are preserved. These include a HindleyMilner type system with principal typings and subject reduction, and an applicative characterisation of contextual equivalence. Key words: Lambdacalculus, calculi of contexts, functional programming,
Argument Filterings and Usable Rules in HigherOrder Rewrite Systems
, 2011
"... Abstract: The static dependency pair method is a method for proving the termination of higherorder rewrite systems à la Nipkow. It combines the dependency pair method introduced for firstorder rewrite systems with the notion of strong computability introducedfortyped λcalculi. Argument filterings ..."
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Abstract: The static dependency pair method is a method for proving the termination of higherorder rewrite systems à la Nipkow. It combines the dependency pair method introduced for firstorder rewrite systems with the notion of strong computability introducedfortyped λcalculi. Argument filteringsand usable rulesare twoimportant methods ofthe dependencypairframeworkusedbycurrentstateoftheart firstorder automated termination provers. In this paper, we extend the class of higherorder systems on which the static dependency pair method can be applied. Then, we extend argument filterings and usable rules to higherorder rewriting, hence providing the basis for a powerful automated termination prover for higherorder rewrite systems.
Formal Languages]: Mathematical Logic
"... Abstract. We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0−1 = 0, an interesting equation consistent with the ring axioms and many prop ..."
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Abstract. We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0−1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting system?