Domenico Cantone

Decision procedures for elementary sublanguages of set theory. XV. multilevel syllogistic extended by the predicate finite and the operators Singleton and Pred

AbstractIn this paper we prove the decidability of the class of unquantified formulae of set theory involving the operators ϕ, ∪, ∩, , {·}, pred< and the predicates =, ∈, n

Decision procedures for elementary sublanguages of set theory. V. Multilevel syllogistic extended by the general union operator

subseteq

Propositional- and Predicate-Calculus Preliminaries

n, Finite, where pred<(s) denotes the collection of all sets having rank strictly less than the rank of s.This work generalizes and combines earlier results published in the same series of papers.

A Closer Examination of the Sequence of Definitions and Theorems Presented in this Book

In the first paper of this series it was shown that any unquantified formula p in the collection MLSSF (multilevel syllogistic extended with the singleton operator and the predicate Finite) can be decomposed as a disjunction of set-theoretic formulae called syllogistic schemes. The syllogistic schemes are satisfiable and no two of them have a model in common, therefore the previous result already implied the decidability of the class MLSSF by simply checking if the set of syllogistic schemes associated with the given formula is empty.In the first section of this paper a new and improved searching algorithm for syllogistic schemes is introduced, based on a proof of existence of a ‘minimum effort’ scheme for any given satisfiable formula in MLSF. The algorithm addressed above can be piloted quite effectively even though it involves backtracking.In the second part of the paper, complexity issues are studied by showing that the class of (∀)o1-simple prenex formulae (an extension of MLS) has a decision problem which is NP-complete. The decision algorithm that proves the membership of this decision problem to NP can be seen as a different decision algorithm for MLS.

Set theory for computing : from decision procedures to declarative programming with sets

Etude des procedures de decision pour differents sous-langages restreints quantifies et non quantifies de la theorie des ensembles

Computational Logic and Set Theory: Applying Formalized Logic to Analysis

This chapter provides an extended survey of inference mechanisms which are candidates for inclusion in the initial endowment of a proof-verifier based on set theory, and points up some efficiency considerations which limit the complexity of the sets of statements to which each inference mechanism can be applied.

Decision procedures for elementary sublanguages of set theory. VI. Multi-level syllogistic extended by the powerset operator

This chapter prepares for the extensive account of a proof-verification system based on set theory which will be given later. Two of the system’s basic ingredients are described and analyzed: n n nthe propositional calculus, from which all necessary properties of the logical operations &, ∨, ¬, →, and ↔ are taken, and n n nthe (first order) predicate calculus, which adds compound functional and predicate constructions and the two quantifiers ∀ and ∃ to the propositional mechanisms. n n n nA gradual account of the proof of Godel’s completeness theorem for predicate calculus is provided. Notions which are relevant for computational logic, such as reduction of sentences to prenex normal form, conservative extensions of a first-order theory, etc., are also introduced.

Decision procedures for elementary sublanguages of set theory VIII. A semidecision procedure for finite satisfiability of unqualified set-theoretic formulae

In parallel with the design and implementation (at least as a prototype) of a proof-verification system based on set theory, the authors undertook the development of a large-scale proof scenario. Ideally, to demonstrate that the verifier can certify the correctness of a substantial body of mathematical analysis, this proof scenario should have culminated in the proof of the celebrated Cauchy integral theorem on analytic functions (whose statement is recalled at the end of this chapter). This presupposed proofs of the basic properties of the real and complex number systems defined in set-theoretic terms, the fundamental properties of limits, continuity and the differential and integral calculus.