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Solvable set/hyperset contexts: I. Some decision procedures for the pure, finite case
 Comm. Pure App. Math
, 1995
"... Hereditarily finite sets and hypersets are characterized both as an algorithmic data structure and by means of a firstorder 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 \D ..."
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Hereditarily finite sets and hypersets are characterized both as an algorithmic data structure and by means of a firstorder 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 quantifierfree 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 quantifierfree, is satisfiable. Concerning (1), an explicit decision algorithm is provided; moreover, significantly broad subproblems 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 subproblems, carrying out the decomposition results in producing a fi...
From Set to Hyperset Unification
, 1999
"... 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 ..."
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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 (nondeterministically) 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 nonmembership constraints. Finally, the hyperset unification problem dealt with is proved to be NPcomp...
A Structural CoInduction Theorem
 PROC. MFPS '93, SPRINGER LNCS 802
, 1993
"... The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial Falgebras 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 c ..."
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Cited by 7 (1 self)
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The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial Falgebras 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 stronglyextensional (sometimes called internal full abstractness): the order is the union of all (ordered) Fbisimulations. (Since the initial fixed point for locally continuous functors is also final, both theorems apply.) Further a similar coinduction theorem is given for a category of complete metric spaces and locally contracting functors.
A settheoretical definition of application
 University of Edinburgh
, 1972
"... [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 settheoretical model of the λβcalculus. Functions are modelled in a similar way to that normally employed in ..."
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[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 settheoretical model of the λβcalculus. 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 λβηcalculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those nontrivial models of the λβηcalculus which are continuous complete lattices. In the second part we begin with a brief discussion of models of the λcalculus in set theories with antifoundation axioms. Next we review the model of the λβcalculus 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 ScottEngeler algebras.
STS: A Structural Theory of Sets
 Logic Journal of the IGPL
, 1999
"... and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of ..."
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and their applications. SMC is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a member of
Processes and Hyperuniverses
 Proceedings of the 19th Symposium on Mathematical Foundations of Computer Science 1994, volume 841 of LNCS
, 1994
"... . We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. nonwellfounded sets. In particular we discuss how to solve recursive equations involving settheoretic operators within hyperuniverses with atoms. Hyperuniverses ar ..."
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. We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. nonwellfounded sets. In particular we discuss how to solve recursive equations involving settheoretic 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 ZermeloFraenkel 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 ...
The AntiFoundation Axiom In Constructive Set Theories
 Stanford University Press
, 2003
"... . 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 inte ..."
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Cited by 6 (5 self)
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. 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 nonwellfounded 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 socalled AntiFoundation 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...
Themes in Final Semantics
 Dipartimento di Informatica, Università di
, 1998
"... 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 ..."
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Cited by 5 (2 self)
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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
Axiomatic Characterizations of Hyperuniverses and Applications
 University of Southern
, 1996
"... Hyperuniverses are topological structures exhibiting strong closure properties under formation of subsets. They have been used both in Computer Science, for giving denotational semantics `a la Scottde Bakker, and in Mathematical Logic, in order to show the consistency of set theories which do not a ..."
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Cited by 4 (2 self)
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Hyperuniverses are topological structures exhibiting strong closure properties under formation of subsets. They have been used both in Computer Science, for giving denotational semantics `a la Scottde Bakker, and in Mathematical Logic, in order to show the consistency of set theories which do not abide by the "limitation of size" principle. We present correspondences between settheoretic properties and topological properties of hyperuniverses. We give existence theorems and discuss applications and generalizations to the non compact case. Work partially supported by 40% and 60% MURST grants, CNR grants, and EEC Science MASK, and BRA Types 6453 contracts. y Member of GNSAGA of CNR. z The main results of this paper have been communicated by this author at the "11 th Summer Conference on General Topology and Applications" August 1995, Portland, Maine. Introduction Natural frameworks for dicussing Selfreference and other circular phenomena are extremely useful in areas such ...
A logical approach to security in the context of ambient calculus. to appear in Electronic Notes in Computer Science, Elsevier, available at http://dit.unitn.it/ mardare/publications.htm
 open n Q m[ ]  u[ ]  s[ ] • n[ ] out m • in m n[ ] • ′ open t  • ′′ out s  • open s P
, 2003
"... In this paper 2 we advocate the use of a CTL * logic, built upon Ambient Calculus to analyze security properties. Our logic is a more expressive alternative to Ambient Logic, based on a single modality, but still powerful enough to handle mobility and dynamic hierarchies of locations. Moreover, havi ..."
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In this paper 2 we advocate the use of a CTL * logic, built upon Ambient Calculus to analyze security properties. Our logic is a more expressive alternative to Ambient Logic, based on a single modality, but still powerful enough to handle mobility and dynamic hierarchies of locations. Moreover, having a temporal logic to express properties of computation, we can reuse the algorithms for model checking temporal logics in analyzing models for security problems. We resort to syntax trees of Ambient Calculus and enrich them with some labeling functions in order to obtain what we called labeled syntax trees. The labeled syntax trees will be used as possible worlds in a Kripke structure developed for a propositional branching temporal logic. The accessibility relation is generated by the reduction of Ambient Calculus considered as reduction between syntax trees. Providing the algorithms for calculating the accessibility relation between states, we open the perspective of model checking Ambient Calculus by using our algorithms together with the algorithms for model checking temporal logic. Key words: ambient calculus, temporal logic, set theory, model checking. 1