Hereditarily finite sets and hypersets are characterized both as an algorithmic data structure and by means of a first-order axiomatization which, although rather weak, suffices to make the following two problems decidable: (1) Establishing whether a conjunction r of formulae of the form 8 y 1 \Delta \Delta \Delta 8 y m ((y 1 2 w 1 & \Delta \Delta \Delta & y m 2 wm ) ! q), with q quantifier-free and involving only the relators =; 2 and propositional connectives, and each y i distinct from all w j 's, is satisfiable. (2) Establishing whether a formula of the form 8 y q, q quantifier-free, is satisfiable. Concerning (1), an explicit decision algorithm is provided; moreover, significantly broad sub-problems of (1) are singled out in which a classification ---named the `syllogistic decomposition' of r--- of all possible ways of satisfying the input conjunction r can be obtained automatically. For one of these sub-problems, carrying out the decomposition results in producing a fi...

We investigate the relation between the set-theoretical description of coinduction based on Tarski Fixpoint Theorem, and the categorical description of coinduction based on coalgebras. In particular, we examine set-theoretic generalizations of the coinduction proof principle, in the spirit of Milner's bisimulation "up-to", and we discuss categorical counterparts for these. Moreover, we investigate the connection between these and the equivalences induced by T -coiterative functions. These are morphisms into final coalgebras, satisfying the T -coiteration scheme, which is a generalization of both the coiteration and the corecursion scheme. We generalize Rutten's transformation from coalgebraic bisimulations to set-theoretic bisimulations, in order to cover also the case of bisimulations "up-to". A list of examples of set-theoretic coinductive specifications which appear not to be easily expressible in coalgebraic terms are discussed. Introduction Coinductive definitions and ...

In this paper we show how to extend a set unification algorithm -- i.e., an extended unification algorithm incorporating the axioms of a simple theory of sets -- to hyperset unification, that is to sets in which, roughly speaking, membership can form cycles. This is obtained by enlarging the domain from that of terms (hence, trees) to that of graphs involving free as well as interpreted function symbols (namely, the set element insertion and the empty set), which can be regarded as a convenient denotation of hypersets. We present a hyperset unification algorithm which (non-deterministically) computes, for each given unification problem, a finite collection of systems of equations in solvable form whose solutions represent a complete set of solutions for the given unification problem. The crucial issue of termination of the algorithm is addressed and solved by the addition of simple non-membership constraints. Finally, the hyperset unification problem dealt with is proved to be NP-comp...

The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial F-algebras of locally continuous functors F on the category of cpo's with strict and continuous maps. Here a dual of that theorem is presented, giving a number of equivalent characterizations of final coalgebras of such functors. In particular, final coalgebras are order strongly-extensional (sometimes called internal full abstractness): the order is the union of all (ordered) F-bisimulations. (Since the initial fixed point for locally continuous functors is also final, both theorems apply.) Further a similar co-induction theorem is given for a category of complete metric spaces and locally contracting functors.

. We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. non-wellfounded sets. In particular we discuss how to solve recursive equations involving set-theoretic operators within hyperuniverses with atoms. Hyperuniverses are transitive sets which carry a uniform topological structure and include as a clopen subset their exponential space (i.e. the set of their closed subsets) with the exponential uniformity. This approach allows to solve many recursive domain equations of processes which cannot be even expressed in standard Zermelo-Fraenkel Set Theory, e.g. when the functors involved have negative occurrences of the argument. Such equations arise in the semantics of concurrrent programs in connection with function spaces and higher order assignment. Finally, we briefly compare our results to those which make use of complete metric spaces, due to de Bakker, America and Rutten. Introduction In the Semantics of ...

[41], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary set-theoretical model of the λβ-calculus. Functions are modelled in a similar way to that normally employed in set theory, by their graphs; difficulties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the λβη-calculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those non-trivial models of the λβη-calculus which are continuous complete lattices. In the second part we begin with a brief discussion of models of the λ-calculus in set theories with anti-foundation axioms. Next we review the model of the λβ-calculus of Part 1 and also the closely related—but different!—models of Scott [51, 52] and of Engeler [19, 20]. Then we discuss general frameworks in which elementary constructions of models can be given. Following Longo [36], one can employ certain Scott-Engeler algebras.

. The paper investigates the strength of the antifoundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-wellfounded sets, or hypersets (cf. [17], [5]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [7]). Instead of the Foundation Axiom these set theories adopt the so-called Anti-Foundation Axiom, AFA, which gives rise to a rich universe of sets. AFA provides an elegant tool for modeling all sorts of circular phenomena. The application areas range from knowledge representation and theoretical economics to the semantics of natural language and pr...

and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of

C'era una volta un re seduto in canap`e, che disse alla regina raccontami una storia. La regina cominci`o: "C'era una volta un re seduto in canap`e

We discuss new ways of characterizing, as maximal fixed points of monotone operators, observational congruences on -terms and, more in general, equivalences on applicative structures. These characterizations naturally induce new forms of coinduction principles, for reasoning on program equivalences, which are not based on Abramsky's applicative bisimulation. We discuss in particular, what we call, the cartesian coinduction principle, which arises when we exploit the elementary observation that functional behaviours can be expressed as cartesian graphs. Using the paradigm of final semantics, the soundness of this principle over an applicative structure can be expressed easily by saying that the applicative structure can be construed as a strongly extensional coalgebra for the functor (P( \Theta )) \Phi (P( \Theta )). In this paper, we present two general methods for showing the soundenss of this principle. The first applies to approximable applicative structures. Many c.p.o. -models in...