In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (Non-Well-Founded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of non-wellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken

. Algebraic structures which are generated by a collection of constructors--- like natural numbers (generated by a zero and a successor) or finite lists and trees--- are of well-established importance in computer science. Formally, they are initial algebras. Induction is used both as a definition principle, and as a proof principle for such structures. But there are also important dual "coalgebraic" structures, which do not come equipped with constructor operations but with what are sometimes called "destructor" operations (also called observers, accessors, transition maps, or mutators). Spaces of infinite data (including, for example, infinite lists, and non-well-founded sets) are generally of this kind. In general, dynamical systems with a hidden, black-box state space, to which a user only has limited access via specified (observer or mutator) operations, are coalgebras of various kinds. Such coalgebraic systems are common in computer science. And "coinduction" is the appropriate te...

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We consider foundational questions related to the definition of functions by corecursion. This method is especially suited to functions into the greatest fixed point of some monotone operator, and it is most applicable in the context of non-wellfounded sets. We review the work on the Special Final Coalgebra Theorem of Aczel [1] and the Corecursion Theorem of Barwise and Moss [4]. We offer a condition weaker than Aczel's condition of uniformity on maps, and then we prove a result relating the operators satisfying the new condition to the smooth operators of [4]. Keywords: corecursion, coalgebra, operator on sets 1 Introduction By a stream of natural numbers we mean a pair hn; si where n 2 N and s is again a stream of natural numbers. Let f : N ! N . Consider the following function which purports to define a function from N into the streams: iter f (n) = hn; iter f f(n)i (1.1) For each n, iter f (n) is a stream, so iter f itself is a function from numbers to streams. This is an examp...

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We study extensional models of the untyped lambda calculus in the setting of game semantics. In particular, we show that, somewhat unexpectedly and contrary to what happens in ordinary categories of domains, all reflexive objects in the category of games G, introduced by Abramsky, Jagadeesan and Malacaria, induce the same -theory. This is H , the maximal theory induced already by the classical CPO model D1 , introduced by Scott in 1969. This results indicates that the current notion of game carries a very specific bias towards head reduction.

In this paper we discuss final semantics for the -calculus, a process algebra which models systems that can dynamically change the topology of the channels. We show that the final semantics paradigm, originated by Aczel and Rutten for CCS-like languages, can be successfully applied also here. This is achieved by suitably generalizing the standard techniques so as to accommodate the mechanism of name creation and the behaviour of the binding operators peculiar to the -calculus. As a preliminary step, we give a higher order presentation of the -calculus using as metalanguage LF , a logical framework based on typed -calculus. Such a presentation highlights the nature of the binding operators and elucidates the role of free and bound channels. The final semantics is defined making use of this higher order presentation, within a category of hypersets.

. We study models of the untyped lambda calculus in the setting of game semantics. In particular, we show that, in the category of games G, introduced by Abramsky, Jagadeesan and Malacaria, all - models can be partitioned in three disjoint classes, and each model in a class induces the same theory (i.e. the set of equations between terms), that are the theory H , the theory which identies two terms i they have the same Bohm tree and the theory which identies all the terms which have the same Levy-Longo tree. Key Words: Games Semantics, Lambda Calculus, Bohm-trees Introduction In this paper we explore the methodology for giving denotational semantics based on games, recently introduced by Abramsky, Jagadeesan, Malacaria and Hyland, Ong (see [AJM96,HO00]). We use game semantics to build models of the untyped -calculus, focusing on which -theories can be modeled. -theories are congruences over -terms, which extend pure -conversion. Their interest lies in the fact that t...

A-calculus is defined, which is parametric with respect to a set V of input values and subsumes all the different-calculi given in the literature, in particular the classical one and the call-by-value-calculus of Plotkin. It is proved that it enjoy the confluence property, and a necessary and sufficient condition is given, under which it enjoys the standardization property. Its operational semantics is given through a reduction machine, parametric with respect to both V and a set Vo of output values. © 2003 Elsevier Inc. All rights reserved.

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