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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.
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.
Simultaneous Rigid EUnification and Related Algorithmic Problems
 in ``Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS'96
, 1996
"... The notion of simultaneous rigid Eunification was introduced in 1987 in the area of automated theorem proving with equality in sequentbased methods, for example the connection method or the tableau method. Recently, simultaneous rigid Eunification was shown undecidable. Despite the importance of ..."
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The notion of simultaneous rigid Eunification was introduced in 1987 in the area of automated theorem proving with equality in sequentbased methods, for example the connection method or the tableau method. Recently, simultaneous rigid Eunification was shown undecidable. Despite the importance of this notion, for example in theorem proving in intuitionistic logic, very little is known of its decidable fragments. We prove decidability results for fragments of monadic simultaneous rigid Eunification and show the connections between this notion and some algorithmic problems of logic and computer science. 1 Introduction Simultaneous rigid Eunification plays a crucial role in automatic proof methods for firstorder logic with equality based on sequent calculi, such as semantic tableaux [13], the connection method [6] (also known as the mating method [1]), model elimination [26] and a dozen other procedures. All these methods are based on the Herbrand theorem and express the idea that ...
The Logic in Computer Science Column
 In Bulletin of the European Association for Theoretical Computer Science
, 1996
"... The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of ..."
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The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the socalled simultaneous rigid Eunification problem. In the second part, we survey recent result on the simultaneous rigid Eunification problem.
The Logic in Computer Science Column
 In Bulletin of the European Association for Theoretical Computer Science
, 1988
"... The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of ..."
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The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the socalled simultaneous rigid Eunification problem. In the second part, we survey recent result on the simultaneous rigid Eunification problem. 1 Problems This simultaneous rigid Eunification : : : It has too procedural a definition. Vaughan Pratt, private communication. We start with some decision problems having nonprocedural definitions. Most of these problems naturally stem from the famous Herbrand theorem. 1.1 Herbrand's theorem and six decision problems The Herbrand theorem [28] asserts that an existential firstorder formula 9¯x'(¯x) is provable (in classical firstorder logic) if and only if a particular disjunction '( ¯ t 1 ) : : : '( ¯ t n ) is provab...
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 ...
The Logic in Computer Science Column
 In Bulletin of the European Association for Theoretical Computer Science
, 1988
"... The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One ..."
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The famous Herbrand's theorem of mathematical logic plays an important role in automated theorem proving. In the first part of this article, we recall the theorem and formulate a number of natural decision problems related to it. Somewhat surprisingly, these problems happen to be equivalent. One of these problems is the socalled simultaneous rigid Eunification problem. In the second part, we survey recent result on the simultaneous rigid problem.
Complexity of undecidable formulæ in the rationals and inertial Zsigmondy theorems for elliptic curves
, 2004
"... ..."
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