Formative processes with applications to the decision problem in set theory I: powerset and singleton operators

Authors:
Domenico Cantone;Pietro Ursino
Affiliations:
Univ. di Catania, Catania, Italy;Univ. di Catania, Catania, Italy
Venue:
Information and Computation
Year:
2002

Citing 11
Cited 2

Programming with sets; an introduction to SETL

Programming with sets; an introduction to SETL
Understanding Z: a specification language and its formal semantics

Understanding Z: a specification language and its formal semantics
Set-theoretic reduction of Hilbert's tenth problem

CSL '89 Proceedings of the third workshop on Computer science logic
Computable set theory

Computable set theory
Decision procedures for elementary sublanguages of set theory: X. Multilevel syllogistic extended by the singleton and powerset operators

Journal of Automated Reasoning - Special issue on new trends in automated reasoning
Decision procedures for elementary sublanguages of set theory: XIII. Model graphs, reflection and decidability

Journal of Automated Reasoning - Special issue on new trends in automated reasoning
The Go¨del programming language

The Go¨del programming language
T-theorem proving I

Journal of Symbolic Computation
On the representation and management of finite sets in CLP languages

JICSLP'98 Proceedings of the 1998 joint international conference and symposium on Logic programming
Set theory for computing: from decision procedures to declarative programming with sets

Set theory for computing: from decision procedures to declarative programming with sets
T-Resolution: Refinements and Model Elimination

Journal of Automated Reasoning

Decidability results for sets with atoms

ACM Transactions on Computational Logic (TOCL)
Theory-specific automated reasoning

A 25-year perspective on logic programming

Quantified Score

Hi-index	0.00

Visualization

Abstract

This paper introduces formative processes, composed by transitivepartitions. Given a family F of sets, a formative processending in the Venn partition &egr; of F is shown toexist. Sufficient criteria are also singled out for a transitivepartition to model (via a function from set variables to unions ofsets in the partition) all set-literals modeled by &egr;. Onthe basis of such criteria a procedure is designed that mimics agiven formative process by another where sets have finite rankbounded by C (|&egr;|), with C a specificcomputable function. As a by-product, one of the core results ondecidability in computable set theory is rediscovered, namely theone that regards the satisfiability of unquantified set-theoreticformulae involving Boolean operators, the singleton-former, and thepowerset operator. The method described (which is able to exhibit aset-solution when the answer is affirmative) can be extended tosolve the satisfiability problem for broader fragments of settheory.