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16
Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given ..."
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Cited by 70 (4 self)
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We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of CobhamSemenov, the original proof being published in Russian. 1
Decidable theories of the ordering of natural numbers with unary predicates
 Proceedings of Computer Science Logic (CSL ’06)
, 2006
"... Expansions of the natural number ordering by unary predicates are studied, using logics which in expressive power are located between firstorder and monadic secondorder logic. Building on the modeltheoretic composition method of Shelah, we give two characterizations of the decidable theories of t ..."
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Cited by 12 (9 self)
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Expansions of the natural number ordering by unary predicates are studied, using logics which in expressive power are located between firstorder and monadic secondorder 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 firstorder 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.
On decidability of monadic logic of order over the naturals extended by monadic predicates
, 2007
"... ..."
Church Synthesis Problem with Parameters
"... Abstract. The following problem is known as the Church Synthesis problem: Input: an MLO formula ψ(X, Y). Task: Check whether there is an operator Y = F (X) such that ..."
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Cited by 6 (1 self)
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Abstract. The following problem is known as the Church Synthesis problem: Input: an MLO formula ψ(X, Y). Task: Check whether there is an operator Y = F (X) such that
Extensions of Büchi’s problem : Questions of decidability for addition and kth powers
"... The authors thank the referee for his comments. The second author aknowledges the hospitality of the University of CreteHeraklion, where the main part of this work was done. Abstract. We generalize a question of Büchi: Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algo ..."
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Cited by 4 (3 self)
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The authors thank the referee for his comments. The second author aknowledges the hospitality of the University of CreteHeraklion, where the main part of this work was done. Abstract. We generalize a question of Büchi: Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algorithm to decide the solvability in R of any given system of polynomial equations, each of which is linear in the k−th powers of the unknowns, with coefficients in C? We state a numbertheoretical problem, depending on k, a positive answer to which would imply a negative answer to the question for R = C = Z. We reduce a negative answer for k = 2 and for R = F (t), a field of rational functions of zero characteristic, to the undecidability of the ring theory of F (t). We address the similar question, where we allow, along with the equations, also conditions of the form ‘x is a constant ’ and ‘x takes the value 0 at t = 0’, for k = 3 and for function fields R = F (t) of zero characteristic, with C = Z[t]. We prove that a negative answer to this question would follow from a negative answer for a ring between Z and the extension of Z by a primitive cube root of 1.
A Survey of Arithmetical Definability
, 2002
"... We survey definability and decidability issues related to firstorder fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions. ..."
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Cited by 2 (0 self)
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We survey definability and decidability issues related to firstorder fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions.
THE ANALOGUE OF BÜCHI’S PROBLEM FOR RATIONAL FUNCTIONS
, 2006
"... Büchi’s problem asked whether there exists an integer M such that the surface defined by a system of equations of the form x 2 n + x2n−2 =2x2n−1 +2, n =2,...,M − 1, has no integer points other than those that satisfy ±xn = ±x0 + n (the ± signs are independent). If answered positively, it would imply ..."
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Cited by 2 (1 self)
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Büchi’s problem asked whether there exists an integer M such that the surface defined by a system of equations of the form x 2 n + x2n−2 =2x2n−1 +2, n =2,...,M − 1, has no integer points other than those that satisfy ±xn = ±x0 + n (the ± signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system Q =(q1,...,qr) of integral quadratic forms and an arbitrary rtuple B =(b1,...,br) of integers, whether Q represents B (see T. Pheidas and X. Vidaux, Fund. Math. 185 (2005) 171–194). Thus it would imply the following strengthening of the negative answer to Hilbert’s tenth problem: the positiveexistential theory of the rational integers in the language of addition and a predicate for the property ‘x is a square ’ would be undecidable. Despite some progress, including a conditional positive answer (depending on conjectures of Lang), Büchi’s problem remains open. In this paper we prove the following: (A) an analogue of Büchi’s problem in rings of polynomials of characteristic either 0 or p � 17 and for fields of rational functions of characteristic 0; and (B) an analogue of Büchi’s problem in fields of rational functions of characteristic p � 19, but only for sequences that satisfy a certain additional hypothesis. As a consequence we prove the following result in logic. Let F be a field of characteristic either 0 or at least 17 and let t be a variable. Let Lt be the first order language which contains symbols for 0 and 1, a symbol for addition, a symbol for the property ‘x is a square ’ and symbols for multiplication by each element of the image of Z[t] in F [t]. Let R beasubringofF (t), containing the natural image of Z[t] inF (t). Assume that one of the following is true: (i) R ⊂ F [t]; (ii) the characteristic of F is either 0 or p � 19. Then multiplication is positiveexistentially definable over the ring R, in the language Lt. Hence the positiveexistential theory of R in Lt is decidable if and only if the positiveexistential ringtheory of R in the language of rings, augmented by a constantsymbol for t, is decidable. 1.
Decidable expansions of labelled linear orderings
, 2010
"... Let M =(A, <, P)where(A, <) is a linear ordering and P denotes a finite sequence of monadic predicates on A. We show that if A contains an interval of order type ω or −ω, and the monadic secondorder theory of M is decidable, then there exists a nontrivial expansion M ′ of M by a monadic pred ..."
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Let M =(A, <, P)where(A, <) is a linear ordering and P denotes a finite sequence of monadic predicates on A. We show that if A contains an interval of order type ω or −ω, and the monadic secondorder theory of M is decidable, then there exists a nontrivial expansion M ′ of M by a monadic predicate such that the monadic secondorder theory of M ′ is still decidable.
Almost Periodicity, Finite Automata Mappings and Related Effectiveness Issues
 Proceedings of WoWA’06, St
"... ..."
Theories of arithmetics in finite models
"... 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 Σ2– ..."
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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 Σ2–theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ1–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. 1