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13
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 68 (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
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.
Extensions of Büchi’s problem: questions of decidability for addition and kth
, 2005
"... Abstract. We generalize a question of Büchi: Let R be an integral domain and k ≥ 2 an integer. Is there an algorithm to solve in R any given system of polynomial equations, each of which is linear in the k−th powers of the unknowns? We examine variances of this problem for k = 2, 3 and for R a field ..."
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Abstract. We generalize a question of Büchi: Let R be an integral domain and k ≥ 2 an integer. Is there an algorithm to solve in R any given system of polynomial equations, each of which is linear in the k−th powers of the unknowns? We examine variances of this problem for k = 2, 3 and for R a field of rational functions of characteristic zero. We obtain negative answers, provided that the analogous problem over Z has a negative answer. In particular we prove that the generalization of Büchi’s question for fields of rational functions over a realclosed field F, for k = 2, has a negative answer if the analogous question over Z has a negative answer. 1
Almost Periodicity, Finite Automata Mappings and Related Effectiveness Issues
 Proceedings of WoWA’06, St
"... The paper studies different variants of almost periodicity notion. We introduce the class of eventually strongly almost periodic sequences where some suffix is strongly almost periodic (=uniformly recurrent). The class of almost periodic sequences includes the class of eventually strongly almost per ..."
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The paper studies different variants of almost periodicity notion. We introduce the class of eventually strongly almost periodic sequences where some suffix is strongly almost periodic (=uniformly recurrent). The class of almost periodic sequences includes the class of eventually strongly almost periodic sequences, and we prove this inclusion to be strict. We prove that the class of eventually strongly almost periodic sequences is closed under finite automata mappings and finite transducers. Moreover, an effective form of this result is presented. Finally we consider some algorithmic questions concerning almost periodicity. 1
Decidable expansions of labelled linear orderings
"... Dedicated to Yuri Gurevich on the occasion of his seventieth birthday Abstract. 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 decid ..."
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Dedicated to Yuri Gurevich on the occasion of his seventieth birthday Abstract. 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.
doi:10.1112/S0024610706023283 THE ANALOGUE OF BÜCHI’S PROBLEM FOR RATIONAL FUNCTIONS
"... 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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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 Extensions of Church’s Problem
"... Abstract. For a twovariable formula B(X,Y) of Monadic Logic of Order (MLO) the Church Synthesis Problem concerns the existence and construction of a finitestate operator Y=F(X) such that B(X,F(X)) is universally valid over Nat. Büchi and Landweber (1969) proved that the Church synthesis problem is ..."
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Abstract. For a twovariable formula B(X,Y) of Monadic Logic of Order (MLO) the Church Synthesis Problem concerns the existence and construction of a finitestate operator Y=F(X) such that B(X,F(X)) is universally valid over Nat. Büchi and Landweber (1969) proved that the Church synthesis problem is decidable. We investigate a parameterized version of the Church synthesis problem. In this extended version a formula B and a finitestate operator F might contain as a parameter a unary predicate P. A large class of predicates P is exhibited such that the Church problem with the parameter P is decidable. Our proofs use Composition Method and game theoretical techniques. 1