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24
A New Method for Undecidability Proofs of First Order Theories
 Journal of Symbolic Computation
, 1992
"... this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction ..."
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Cited by 30 (7 self)
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this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction
Ordering Constraints on Trees
 Colloquium on Trees in Algebra and Programming
, 1994
"... . We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, wellfounded and compatible with the tree constructors. The paper includes some new results, in p ..."
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Cited by 21 (1 self)
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. We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, wellfounded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a nonunary signature. 1 Symbolic Constraints Constraints on trees are becoming popular in automated theorem proving, logic programming and in other fields thanks to their potential to represent large or even infinite sets of formulae in a nice and compact way. More precisely, a symbolic constraint system, also called a constraint system on trees, consists of a fragment of firstorder logic over a set of predicate symbols P and a set of function symbols F , together with a fixed interpretation of the predicate symbols in the algebra of finite trees T (F) (or sometimes the algebra of infinite trees I(F)) ov...
Implicit function theorem over free groups and genus problem
 AMS/IP Studies in Advanced Mathematics
"... groups ..."
Tarski’ problem about the elementary theory of free nonabelian groups has a positive solution
 ERAAMS
, 1998
"... Abstract. We prove that the elementary theories of all nonabelian free groups coincide and that the elementary theory of a free group is decidable. These results answer two old questions that were raised by A. Tarski around 1945. The object of this announcement is to sketch proofs of the following t ..."
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Cited by 13 (6 self)
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Abstract. We prove that the elementary theories of all nonabelian free groups coincide and that the elementary theory of a free group is decidable. These results answer two old questions that were raised by A. Tarski around 1945. The object of this announcement is to sketch proofs of the following two theorems. Theorem 1. The elementary theories of all nonabelian free groups coincide. Theorem 2. The elementary theory of a free group is decidable. These theorems answer two old questions that were raised by A. Tarski around 1945. We recall that the elementary theory Th(G) of a group G is the set of all first order sentences in the language of group theory which are true in G. Notice that in the language of group theory every sentence is equivalent to a sentence of the following type: r ∨ s∧ (1) Φ=∀X1∃Y1...∀Xk∃Yk ( upi(X1,Y1,...,Xk,Yk)=1 p=1 i=1 t∧ vpj(X1,Y1,...,Xk,Yk)=1). j=1 A discussion of this problem can be found in several textbooks on model theory (see, for example, C. Chang and H. Keisler [5] or Yu. Ershov and E. Palutin [8]) as well as in several textbooks on group theory (see, for example, R. Lyndon and P. Schupp [22]). Our solution of Tarski’s problem takes on the strongest possible, positive form, namely: thefreegroupF(a1,...,an) freely generated by a1,...,an is an elementary subgroup of F (a1,...,an,...,an+p) for every n ≥ 2 and p ≥ 0. Moreover, we prove also that: the elementary theory Th(F) of a free group F even with constants from F in the language is decidable. Observe, by comparison, that it is relatively easy to prove that free abelian groups of finite rank are elementarily equivalent if and only if their ranks coincide. The same is true for free nilpotent groups of finite rank and for free semigroups
Word equations over graph products
 In Proceedings of the 23rd Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2003), Mumbai (India), number 2914 in Lecture Notes in Computer Science
, 2003
"... For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those fac ..."
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For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and the existential theories of the remaining factors. Both results also include suitable constraints for the variables. Larger classes of constraints lead in many cases to undecidability results.
How to make SQL stand for String Query Language
 IN PROCEEDINGS OF DBPL'99, SPRINGER LNCS
, 1999
"... A string database is simply a collection of tables, the columns of which contain strings over some given alphabet. We address in this paper the issue of designing a simple, user friendly query language for string databases. We focus on the language FO(ffl), which is classical first order logic exten ..."
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Cited by 5 (0 self)
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A string database is simply a collection of tables, the columns of which contain strings over some given alphabet. We address in this paper the issue of designing a simple, user friendly query language for string databases. We focus on the language FO(ffl), which is classical first order logic extended with a concatenation operator, and where quantifiers range over the set of all strings. We wish to capture all string queries, i.e., welltyped and computable mappings involving a notion of string genericity. Unfortunately, unrestricted quantification may allow some queries to have infinite output. This leads us to study the "safety" problem for FO(ffl), that is, how to build syntactic and/or semantic restrictions so as to obtain a language expressing only queries with finite output, hopefully all string queries. We introduce a family of such restrictions and study their expressivness and complexity. We prove that none of these languages express all string queries. We prov...
The FirstOrder Theory of One Step Rewriting in Linear Noetherian Systems is Undecidable
 In 8th Int. Conference on Rewriting Techniques and Applications, volume 1232 of LNCS
, 1997
"... . We construct a finite linear finitely terminating rewrite rule system with undecidable theory of one step rewriting. 1 Preliminaries Given a functional signature \Sigma with constants and a finite rewrite rule system R, consider the model M = hT (\Sigma ); Ri, where T (\Sigma ) is the Herbrand ..."
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Cited by 5 (1 self)
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. We construct a finite linear finitely terminating rewrite rule system with undecidable theory of one step rewriting. 1 Preliminaries Given a functional signature \Sigma with constants and a finite rewrite rule system R, consider the model M = hT (\Sigma ); Ri, where T (\Sigma ) is the Herbrand universe over \Sigma and R = fhs; ti j s; t 2 T (\Sigma ) s !R tg ` T (\Sigma ) \Theta T (\Sigma ) is the one step rewrite relation on T (\Sigma ) generated by R. The presence of constants in \Sigma is necessary to guarantee that T (\Sigma ) is not empty. Let L be the firstorder language without equality containing the only binary relation symbol R. The firstorder theory of one step rewriting in R is the set of all sentences (closed formulas) of L true in M . Notice that the only nonlogical symbol used in formulas of the theory is R, and the symbols of \Sigma are not allowed in formulas. The aim of this note is to give an example of a finite linear finitely terminating system with unde...
Elementary Theory of OneStep Rewriting is Undecidable
, 1995
"... . We settle in the negative the following Problem 51 (ComonDauchet, RTA'93 [3], RTA'95 [4]). Given an arbitrary finite term rewriting system R, is the firstorder theory of onestep rewriting (!R ) decidable? Decidability would imply the decidability of the firstorder theory of encompassment (th ..."
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. We settle in the negative the following Problem 51 (ComonDauchet, RTA'93 [3], RTA'95 [4]). Given an arbitrary finite term rewriting system R, is the firstorder theory of onestep rewriting (!R ) decidable? Decidability would imply the decidability of the firstorder theory of encompassment (that is, being an instance of a subterm) [1], as well as several known decidability results in rewriting. (It is well known that the theory of ! R is in general undecidable. ) ut The positive solution to the problem would have given a powerful unified decision method for many problems in rewriting. The decidability of the problem was widely believed and conjectured. We present a finite term rewriting system R such that the dyadic word concatenation 1 is expressible via the onestep rewriting relation in R. Consequently (Quine [5], Smullyan [6]), the elementary theory of onestep rewriting is undecidable, in general. 1 Dyadic Concatenation Let f0; 1g be the set of finite words over a ...
Constraints in Term Algebras (Short Survey)
 Proc. Conf. on Algebraic Methodology and Software Technology, Univ. of Twente
, 1993
"... this paper. References ..."
Some notes on subword quantification and induction thereof. Typeset Manuscript
 Lecture Notes in Pure and Applied Mathematics 180
, 1996
"... The first section of this paper consists of a defense of the binary string notation for the formulation of weak theories of arithmetic which have computational significance. We defend that a stringlanguage is the most natural framework and that the usual arithmetic setting suffers from some troubles ..."
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The first section of this paper consists of a defense of the binary string notation for the formulation of weak theories of arithmetic which have computational significance. We defend that a stringlanguage is the most natural framework and that the usual arithmetic setting suffers from some troubles when dealing with very low complexity classes. Having introduced in the first section the theory Th − FO associated with a rather robust uniform version of the class of problems that can be decided by constant depth,polynomial size circuit families (the socalled AC 0class) we prove in the second section that the deletion of a crucial axiom from Th − FO results in a theory which is unsuitable from the computational point of view.