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Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
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Cited by 46 (18 self)
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
A SetTheoretic Translation Method for Polymodal Logics
, 1995
"... The paper presents a settheoretic translation method for polymodal logics that reduces the derivability problem of a large class of propositional polymodal logics to the derivability problem of a very weak firstorder set theory\Omega\Gamma Unlike most existing translation methods, the one we propos ..."
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Cited by 19 (12 self)
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The paper presents a settheoretic translation method for polymodal logics that reduces the derivability problem of a large class of propositional polymodal logics to the derivability problem of a very weak firstorder set theory\Omega\Gamma Unlike most existing translation methods, the one we proposed applies to any normal complete finitelyaxiomatizable polymodal logic, regardless if it is firstorder complete or if an explicit semantics is available for it. Moreover, the finite axiomatizability of\Omega makes it possible to implement mechanical proof search procedures via the deduction theorem or more specialized and efficient techniques. In the last part of the paper, we briefly discuss the application of set T resolution to support automated derivability in (a suitable extension of) \Omega\Gamma This work has been supported by funds MURST 40% and 60%. The second author was supported by a grant from the Italian Consiglio Nazionale delle Ricerche (CNR). 1 Introduction The paper...
Formative processes with applications to the decision problem in set theory: I. Powerset and singleton operators.
"... This paper introduces formative processes, composed by transitive partitions. Given a family F of sets, a formative process ending in the Venn partition of F is shown to exist. Sufficient criteria are also singled out for a transitive partition to model (via a function from set variables to unions o ..."
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Cited by 3 (3 self)
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This paper introduces formative processes, composed by transitive partitions. Given a family F of sets, a formative process ending in the Venn partition of F is shown to exist. Sufficient criteria are also singled out for a transitive partition to model (via a function from set variables to unions of sets in the partition) all setliterals modeled by . On the basis of such criteria a procedure is designed that mimics a given formative process by another where sets have finite rank bounded by C(j j), with C a specific computable function. As a byproduct, one of the core results on decidability in computable set theory is rediscovered, namely the one that regards the satis ability of unquanti ed settheoretic formulae involving Boolean operators, the singletonformer, and the powerset operator. The method described can be extended to decide the satisfiability problem for broader fragments of set theory.
A tableau calculus for quantifierfree set theoretic formulae
 In Proceedings, International Conference on Theorem Proving with Analytic Tableaux and Related Methods, Oisterwijk, The Netherlands, LNCS 1397
, 1998
"... Abstract. Set theory is the common language of mathematics. Therefore, set theory plays an important rôle in many important applications of automated deduction. In this paper, we present an improved tableau calculus for the decidable fragment of set theory called multilevel syllogistic with singlet ..."
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Cited by 2 (0 self)
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Abstract. Set theory is the common language of mathematics. Therefore, set theory plays an important rôle in many important applications of automated deduction. In this paper, we present an improved tableau calculus for the decidable fragment of set theory called multilevel syllogistic with singleton (MLSS). Furthermore, we describe an extension of our calculus for the bigger fragment consisting of MLSS enriched with free (uninterpreted) function symbols (MLSSF). 1
Transitive Venn diagrams with applications to the decision problem in set theory
 in set theory, Proc. APPIA GULP PRODE '99 Joint Conference on Declarative Programming
, 2000
"... This paper introduces the notions of transitive partition and of transitive Venn diagram of a collection of sets, and gives sufficient conditions in order that two transitive Venn diagrams satisfy a same family of settheoretic literals. As a byproduct, one of the core results on decidability in co ..."
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Cited by 1 (0 self)
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This paper introduces the notions of transitive partition and of transitive Venn diagram of a collection of sets, and gives sufficient conditions in order that two transitive Venn diagrams satisfy a same family of settheoretic literals. As a byproduct, one of the core results on decidability in computable set theory is rediscovered, namely the one that regards the satis ability of unquantified settheoretic formulae involving Boolean operators, the singletonformer, and the powerset operator. The method described is amenable to extensions able to deal with more general satisfiability decision problems for fragments of set theory.
of MultiLevel Syllogistic with the Cartesian
"... A decision procedure for a twosorted extension ..."