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34
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 29 (6 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
Implicit function theorem over free groups and genus problem
 AMS/IP Studies in Advanced Mathematics
"... groups ..."
Ordering Constraints on Trees
, 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 par ..."
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Cited by 20 (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.
Tarski’ problem about the elementary theory of free nonabelian groups has a positive solution
 ERAAMS
, 1998
"... 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. ..."
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Cited by 14 (5 self)
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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.
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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Cited by 9 (5 self)
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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 encompa ..."
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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 ...
Word equations with length constraints: what’s decidable
 In HVC’12
"... Abstract. We prove several decidability and undecidability results for the satisfiability and validity problems for languages that can express solutions of word equations with length constraints. The atomic formulas over this language are equality over string terms (word equations), linear inequalit ..."
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Abstract. We prove several decidability and undecidability results for the satisfiability and validity problems for languages that can express solutions of word equations with length constraints. The atomic formulas over this language are equality over string terms (word equations), linear inequality over the length function (length constraints), and membership in regular sets. These questions are important in logic, program analysis, and formal verification. Variants of these questions have been studied for many decades and practical satisfiability procedures (aka SMT solvers) for these formulas have become increasingly important in the analysis of stringmanipulating programs such as web applications and scripts. We prove three main theorems. First, we give a new proof of undecidability for the validity problem for the set of sentences written as a ∀ ∃ quantifier alternation applied to positive word equations. A corollary of this undecidability result is that this set is undecidable even with sentences with at most two occurrences of a string variable. Second, we consider Boolean combinations of quantifierfree formulas constructed out of word equations and length constraints. We show that if word equations can be converted to a solved form, a form relevant in practice, then the satisfiability problem for Boolean combinations of word equations and length constraints is decidable. Third, we show that the satisfiability problem for quantifierfree formulas over word equations in regular solved form, length constraints, and the membership predicate over regular expressions is also decidable. 1
Decidability and Definability Results Related to the Elementary Theory of Ordinal Multiplication
"... The elementary theory of hff; \Thetai, where ff is an ordinal and \Theta denotes ordinal multiplication, is decidable if and only if ff ! ! . Moreover if j r and j l respectively denote the right and lefthand divisibility relation, we show that Thh! ; j l i are decidable for every ordinal . ..."
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The elementary theory of hff; \Thetai, where ff is an ordinal and \Theta denotes ordinal multiplication, is decidable if and only if ff ! ! . Moreover if j r and j l respectively denote the right and lefthand divisibility relation, we show that Thh! ; j l i are decidable for every ordinal . Further related definability results are also presented.