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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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Cited by 12 (3 self)
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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 7 (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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Cited by 6 (5 self)
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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.
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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Cited by 2 (0 self)
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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
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?
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.
Author manuscript, published in "Second international conference on Certified Programs and Proofs (2012)" Proof Pearl: Abella Formalization of λCalculus Cube Property
, 2013
"... Abstract. In 1994 Gerard Huet formalized in Coq the cube property of λcalculus residuals. His development is based on a clever idea, a beautiful inductive definition of residuals. However, in his formalization there is a lot of noise concerning the representation of terms with binders. We reinterp ..."
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Abstract. In 1994 Gerard Huet formalized in Coq the cube property of λcalculus residuals. His development is based on a clever idea, a beautiful inductive definition of residuals. However, in his formalization there is a lot of noise concerning the representation of terms with binders. We reinterpret his work in Abella, a recent proof assistant based on higherorder abstract syntax and provided with a nominal quantifier. By revisiting Huet’s approach and exploiting the features of Abella, we get a strikingly compact and natural development, which makes Huet’s idea really shine. 1
The Netherlands.
"... Abstract. From the range of techniques available for algebraic specifications we select a core set of features which we define to be the elementary algebraic specifications. These include equational specifications with hidden functions and sorts and initial algebra semantics. We give an elementary e ..."
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Abstract. From the range of techniques available for algebraic specifications we select a core set of features which we define to be the elementary algebraic specifications. These include equational specifications with hidden functions and sorts and initial algebra semantics. We give an elementary equational specification of the field operations and conjugation operator on the rational complex numbers Q(i) and discuss some open problems.
Philippe de Groote
, 2009
"... présentée en vue de l’obtention du grade de Docteur de l’Université de Savoie Spécialité: Mathématiques et Informatique par ..."
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présentée en vue de l’obtention du grade de Docteur de l’Université de Savoie Spécialité: Mathématiques et Informatique par