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On The Algebraic Models Of Lambda Calculus
 Theoretical Computer Science
, 1997
"... . The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory ..."
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Cited by 20 (11 self)
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. The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory logic in this regard since it is a firstorder algebraic description of lambda calculus, which allows to keep the lambda notation and hence all the functional intuitions. In this paper we show that the lattice of the subvarieties of lambda abstraction algebras is isomorphic to the lattice of lambda theories of the lambda calculus; for every variety of lambda abstraction algebras there exists exactly one lambda theory whose term algebra generates the variety. For example, the variety generated by the term algebra of the minimal lambda theory is the variety of all lambda abstraction algebras. This result is applied to obtain a generalization of the genericity lemma of finitary lambda calculus...
A rewriting calculus for cyclic higherorder term graphs
 in "2nd International Workshop on Term Graph Rewriting  TERMGRAPHâ€™2004
, 2004
"... graphs ..."
NonStandard Semantics for Program Slicing
 Special issue on Partial Evalution and SemanticsBased Program Manipulation
, 2003
"... In this paper we generalize the notion of compositional semantics to cope with trans nite reductions of a transition system. Standard denotational and predicate transformer semantics, even though compositional, provide inadequate models for some known program manipulation techniques. We are interes ..."
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Cited by 10 (1 self)
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In this paper we generalize the notion of compositional semantics to cope with trans nite reductions of a transition system. Standard denotational and predicate transformer semantics, even though compositional, provide inadequate models for some known program manipulation techniques. We are interested in the systematic design of extended compositional semantics, observing possible trans  nite computations, i.e. computations that may occur after a given number of in nite loops. This generalization is necessary to deal with program manipulation techniques modifying the termination status of programs, such as program slicing. We include the trans nite generalization of semantics in the hierarchy developed in 1997 by P. Cousot, where semantics at dierent levels of abstraction are related with each other by abstract interpretation. We prove that a specular hierarchy of nonstandard semantics modeling trans nite computations of programs can be speci ed in such a way that the standard hierarchy can be derived by abstract interpretation. We prove that nonstandard trans nite denotational and predicate transformer semantics can be both systematically derived as solutions of simple abstract domain equations involving the basic operation of reduced power of abstract domains. This allows us to prove the optimality of these semantics, i.e. they are the most abstract semantics in the hierarchy which are compositional and observe respectively the terminating and initial states of trans nite computations, providing an adequate mathematical model for program manipulation.
Innocent Game Models of Untyped λCalculus
 Theoretical Computer Science
, 2000
"... We present a new denotational model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [9] and Nickau [17], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and ..."
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Cited by 3 (1 self)
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We present a new denotational model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [9] and Nickau [17], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and DREC as global sections of a reexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these are sensible algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a strong connexion between the economical form of the denotation of a term in the game models and a variablefree form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call eectively almosteverywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a model D...