Abstract
The paper focuses on extending existing decision procedures for set theory and related theories commonly used in mathematics to handle such notions as monotonicity, ordering, inverse functions, etc. After presenting two decision procedures for the basic multilevel syllogistic fragment of set theory and studying the computational complexity of its decision problem, we illustrate a technique based on a syntactic translation of formulae with the special function and predicate symbols above into multilevel syllogistic that, in most cases, yields nondeterministic polynomial-time decision procedures. Such results can be quite useful for tool developers who aim at providing assistance to common mathematical reasoning. A semantically oriented approach is illustrated in the second part of the paper, where nondeterministic exponential-time decision procedures, of theoretical interest only, are briefly sketched for two extensions of multilevel syllogistic, with the general union operator and with the powerset operator.
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Notes
- 1.
Thanks are due to Eugenio Omodeo and Andrea Formisano for their contribution to a very preliminary version of this paper which was presented at the Workshop on Pragmatics of Decision Procedures in Automated Reasoning held in Miami in 2003.
- 2.
From a practical point of view, it would be more convenient to eliminate equalities of the form x=y by replacing all occurrences of x and y in our set of statements by a selected representative of x,y, and then drop the atom x=y.
- 3.
NP-completeness of flat conjunctions of MLSS-literals was first proved in [10], using a different approach.
- 4.
Note, in passing, that a choice set for any family \(\mathcal{F}\) of disjoint non-null sets can be formed as \(\{\,\mathsf {arb}(x)\,:\ x\in\mathcal{F}\,\}\) ; hence, the assumed availability of arb, jointly with the replacement axiom of set theory, yields as a consequence the somewhat controversial postulate of choice.
- 5.
Note that these imply the single-valuedness conditions for all functions involved.
- 6.
To avoid intersections of empty families of sets, it is convenient to assume that C contains a clause x=f(∅), for each function symbol f present in C.
- 7.
In fact, introduction of ‘single-valuedness’ conditions (16) can easily be constrained so as only a linear number of such clauses need to be added.
- 8.
Thus, in this case, the function f is just the identity.
- 9.
For simplicity, we are assuming that the ∼-representative of the variable x in ¬Finite(x) is x itself.
- 10.
A partial negative result for the satisfiability problem of MLSS with the Cartesian product is contained in [5], when a cardinality operator is also admitted.
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Cantone, D. (2013). Decision Procedures for Elementary Sublanguages of Set Theory. XVII. Commonly Occurring Decidable Extensions of Multi-level Syllogistic. In: Davis, M., Schonberg, E. (eds) From Linear Operators to Computational Biology. Springer, London. https://doi.org/10.1007/978-1-4471-4282-9_5
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