Logical Theories of One-Place Functions on the Set of Natural Numbers

Semenov, A. L.

doi:10.1070/im1984v022n03abeh001456

Cited by 50 publications

(54 citation statements)

References 13 publications

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“…By the same way, using the result of Semenov in [Sem83], we can deduce that for an arbitrary natural number n theories T h(FM ((N, +, n x ))) and sl(FM ((N, +))) are decidable. Also results contained in [Büc60], [Sem83], [Kor95] and [Bés97] provide a large set of examples of theories of arithmetics decidable in finite models.…”

Section: ) If the Standard Ordering Is Definable In A And T H(a) Admimentioning

confidence: 74%

Theories of arithmetics in finite models

Krynicki

Zdanowski

2005

J. symb. log.

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We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ 2 -theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ 1 -theory of multiplication and order is decidable in finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation.We consider also the spectrum problem. We show that the spectrum of arithmetic with multiplication and arithmetic with exponentiation is strictly contained in the spectrum of arithmetic with addition and multiplication.

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Section: ) If the Standard Ordering Is Definable In A And T H(a) Admimentioning

confidence: 74%

Theories of arithmetics in finite models

Krynicki

Zdanowski

2005

J. symb. log.

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“…The modular nature of our solution makes our contributions orthogonal with respect to the state of the art: we can enrich P with various definable or even not definable symbols [24] and get from our Theorems 1,2 decidable classes which are far from the scope of existing results. Still, it is interesting to notice that also the special cases of the decidable classes covered by Corollary 1 and Theorem 3 are orthogonal to the results from the literature.…”

Section: Conclusion and Related Workmentioning

confidence: 99%

Decision Procedures for Flat Array Properties

Alberti

Ghilardi

Sharygina

2014

Tools and Algorithms for the Construction and Analysis of Systems

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“…A larger class of such predicates was presented in [3,4]; another comprehensive study is [11]. Contrary to the case of functions, no "natural" recursive predicate P is known such that the monadic theory of (N, <, P ) is undecidable.…”

Section: Introductionmentioning

confidence: 99%

Decidable Theories of the Ordering of Natural Numbers with Unary Predicates

Rabinovich

Thomas

2006

Lecture Notes in Computer Science

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Abstract.Expansions of the natural number ordering by unary predicates are studied, using logics which in expressive power are located between first-order and monadic second-order logic. Building on the modeltheoretic composition method of Shelah, we give two characterizations of the decidable theories of this form, in terms of effectiveness conditions on two types of "homogeneous sets". We discuss the significance of these characterizations, show that the first-order theory of successor with extra predicates is not covered by this approach, and indicate how analogous results are obtained in the semigroup theoretic and the automata theoretic framework.

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Logical Theories of One-Place Functions on the Set of Natural Numbers

Cited by 50 publications

References 13 publications

Theories of arithmetics in finite models

Theories of arithmetics in finite models

Decision Procedures for Flat Array Properties

Decidable Theories of the Ordering of Natural Numbers with Unary Predicates

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