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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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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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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.
Simultaneous rigid Eunification and related algorithmic problems
 in ‘Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS’96)’, IEEE Computer
, 1996
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Complexity of undecidable formulæ in the rationals and inertial Zsigmondy theorems for elliptic curves
, 2004
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FraenkelMostowski Sets with NonHomogeneous Atoms
"... FraenkelMostowski sets (FM sets) are a variant of set theory, where sets can contain atoms. The existence of atoms is postulated as an axiom. The key role in the theory of FM sets is played by permutations of atoms. For instance, if a, b, c, d are atoms, then the sets {a, {a, b, c}, {a, c}} {b, {b, ..."
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FraenkelMostowski sets (FM sets) are a variant of set theory, where sets can contain atoms. The existence of atoms is postulated as an axiom. The key role in the theory of FM sets is played by permutations of atoms. For instance, if a, b, c, d are atoms, then the sets {a, {a, b, c}, {a, c}} {b, {b, c, d}, {b, d}} are equal up to permutation of atoms. In a more general setting, the atoms have some structure, and instead of permutations one talks about automorphisms of the atoms. Suppose for instance that the atoms are real numbers, equipped with the successor relation x = y + 1 and linear order x < y. Then the sets {−1, 0, 0.3} {5.2, 6.2, 6.12} are equal up to automorphism of the atoms, but the sets {0, 2} {5.3, 8.3} are not. (In the second example, the two sets can be mapped to each other by a partial automorphism, but not by one that extends to a automorphism of the
On TwoWay Nondeterministic Finite Automata with One ReversalBounded Counter Abstract
"... We show that the emptiness problem for twoway nondeterministic finite automata augmented with one reversalbounded counter (i.e., the counter alternates between nondecreasing and nonincreasing modes a fixed number of times) operating on bounded languages (i.e., subsets ¢¤£¥§¦¨¦¨¦©¢� £ � of for som ..."
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We show that the emptiness problem for twoway nondeterministic finite automata augmented with one reversalbounded counter (i.e., the counter alternates between nondecreasing and nonincreasing modes a fixed number of times) operating on bounded languages (i.e., subsets ¢¤£¥§¦¨¦¨¦©¢� £ � of for some nonnull ¢ ¥� � ¦¨¦¨ ¦ � ¢ � words) is decidable, resolving a problem left open in [4,7]. The proof is a rather involved reduction to the solution of a special class of Diophantine systems of degree 2 via a class of programs called twophase programs.
Communication Gap for Finite Memory Devices ⋆
"... Abstract. So far, not much is known on communication issues for computations on distributed systems, where the components are weak and simultaneously the communication bandwidth is severely limited. We consider synchronous systems consisting of finite automata which communicate by sending messages w ..."
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Abstract. So far, not much is known on communication issues for computations on distributed systems, where the components are weak and simultaneously the communication bandwidth is severely limited. We consider synchronous systems consisting of finite automata which communicate by sending messages while working on a shared readonly data. We consider the number of messages necessary to recognize a language as a its complexity measure. We present an interesting phenomenon that the systems described require either a constant number of messages or at least Ω((logloglogn) c) of them (in the worst case) for input data of length n and some constant c. Thus, in the hierarchy of message complexity classes there is a gap between the languages requiring only O(1) messages and those that need a nonconstant number of messages. We show a similar result for systems of oneway automata. In this case, there is no language that requires ω(1) and o(logn) messages (in the worst case). These results hold for any fixed number of automata in the system. The lower bound arguments presented in this paper depend very much on results concerning solving systems of diophantine equations and inequalities.
Simultaneous Rigid EUnification is not so Simple
, 1995
"... Simultaneous rigid Eunification has been introduced in the area of theorem proving with equality. It is used in extension procedures, like the tableau method or the connection method. Many articles in this area tacitly assume the existence of an algorithm for simultaneous rigid Eunification. There ..."
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Simultaneous rigid Eunification has been introduced in the area of theorem proving with equality. It is used in extension procedures, like the tableau method or the connection method. Many articles in this area tacitly assume the existence of an algorithm for simultaneous rigid Eunification. There were several faulty proofs of the decidability of this problem. In this article we prove several results about the simultaneous rigid Eunification. Two results are reductions of known problems to simultaneous rigid Eunification. Both these problems are very hard. The word equation solving (unification under associativity) is reduced to the monadic case of simultaneous rigid Eunification. The variablebounded semiunification problem is reduced to the general simultaneous rigid Eunification. The word equation problem used in the first reduction is known to be decidable, but the decidability result is extremely nontrivial. As for the variablebounded semiunification, its decidability is ...