H. Comon, R. Treinen. The first-order theory of lexicographic path orderings is undecidable. TCS 176 (1997), 67--87.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... is decidable and admits QE [53, 44] However, adding the prefix relation is not necessarily a trivial addition: for arbitrary term algebras with prefix (subterm) only the existential theory is decidable, but the full theory is undecidable [68] similar results hold for other orderings on terms [23]) The undecidability result of [68] requires at least one binary term constructor; our results indicate that in the simpler case of strings one recovers QE with the prefix relation. 3.2.1 A Normal Form for S We start with a result that gives a normal form for formulae of FO(S) For that, we ....

H. Comon, R. Treinen. The first-order theory of lexicographic path orderings is undecidable. TCS 176 (1997), 67--87.

....is often the case that they can be expressed with ordering constraints (although this might be inefficient) We prove that the first order theory of the recursive path ordering is decidable in the case of unary signatures with total precedence. This solves a problem that was mentioned as open in [6]. The result has to be contrasted with the undecidability results of the lexicographic path ordering [6] for the case of symbols with arity 2 and total precedence and for the case of unary signatures with partial precedence. We recall that lexicographic path ordering (lpo) and the recursive path ....

....We prove that the first order theory of the recursive path ordering is decidable in the case of unary signatures with total precedence. This solves a problem that was mentioned as open in [6] The result has to be contrasted with the undecidability results of the lexicographic path ordering [6] for the case of symbols with arity 2 and total precedence and for the case of unary signatures with partial precedence. We recall that lexicographic path ordering (lpo) and the recursive path ordering and many other orderings such as [13, 10] coincide in the unary case. Among the positive ....

H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Theoretical Computer Science 176, April 1997.

.... recursive path ordering, of which the existential fragment of the first order constraint system was then shown decidable in a collaboration between UPC and UPS [33] Finally, the first order theory of total recursive path ordering constraints was shown undecidable in a joint work of DFKI and UPS [34]. These algorithms are now implemented in the system SATURATE. The last problem is higher order unification, which was for a very long time considered to be of a different nature than first order (possibly equational) unification. By showing that higher order unification was indeed an instance of ....

....make as many simple constraints explicit as feasible. These extensions make the algorithm suitable for type checking of a full fledged programming language. 6. 2 Cooperations, visits, lectures, seminars within the working group Common work with UPS on symbolic ordering constraints is published in [34]. Ralf Treinen, from DFKI, is now working at UPS. We are also in close collaboration with the Max Planck Institut fur Informatik on constraint based programming paradigms [80, 82] 6.3 Implementations Oz is a concurrent constraint programming language designed for applications that require ....

Hubert Comon and Ralf Treinen. The first-order theory of lexicographic path orderings is undecidable. Theoretical Computer Science, 1995. To Appear.

....the satisfiability test for the existential fragment is in the total case. Though the first order theory of encompassment is decidable [ 19 ] the first order ( Sigma 2 ) theory of the recursive (lexicographic status) path ordering, assuming certain simple conditions on the precedence, is not [ 27 ] . Rephrased Problem 25 (R. Treinen [ 100 ] Consider a finite set of function symbols containing at least one AC (associative commutative) function symbol. Let T be the corresponding set of terms (modulo the AC properties) It is known from [ 101 ] that the first order theory ( Sigma 3 ....

H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Rapport de Recherche 867, Laboratoire de Recherche en Informatique, Universite de Paris-Sud, Orsay, France, November 1993.

....ordering over finite trees. The existential fragment is shown to be NP complete and to be undecidable [43] Muller et al. study the first order theory of feature trees and show it undecidable [25] Comon and Treinen show the first order theory of lexicographic path ordering is undecidable [11]. Automatatheoretic constructions are used to obtain decidability results for many theories. Buchi uses finite word automata to show the decidability of WS1S and S1S [7] Finite automata are also used to construct alternative proofs of decidability of Presburger arithmetic [6, 44] and Rabin s ....

H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Theoretical Computer Science, 176:67--87, April 1997.

....be considered, but arbitrary simplification by rewriting is not allowed any more. In this context, our problem is to decide joinability of the critical pairs by rewriting with a set of constrained equations. This is not easy, because simplification with an arbitrary constrained rule is undecidable [CT97]: given a term s and a rule l r j c, it is undecidable whether or not all instances of s can be reduced. Going further, we investigate the use of ordered rewriting in a classical application of rewrite systems: the proof by consistency approach to proving in 4 ductive theorems ....

H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Theoretical Computer Science, 176(1-2):67--87, April 1997.

....for fragments of the theory of a constraint system. In most of the cases it is not necessary to have a decision procedure for the complete theory. Decision procedures for complex formulae are however needed for deciding properties of constraint systems, see for instance the motivating example of [8]. When proving the decidability of the theory of a first order structure one often shows the completeness of some axiomatization of the theory. A complete axiomatization of a theory T is a decidable subset of T such that every sentence of T can be derived from it. In almost all of the cases, a ....

Hubert Comon and Ralf Treinen. The first order theory of lexicographic path orderings is undecidable. Theoretical Computer Science. To Appear.

....be considered, but arbitrary simplification by rewriting is not allowed any more. In this context, our problem is to decide joinability of the critical pairs by rewriting with a set of constrained equations. This is not easy, because simplification with an arbitrary constrained rule is undecidable [CT97]: given a term s and a rule l r j c, it is undecidable whether or not all instances of s can be reduced. Going further, we investigate the use of ordered rewriting in a classical application of rewrite systems: the proof by consistency approach to proving inductive theorems [KM87, JK86, Bac88] ....

H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Theoretical Computer Science, 176(1-2):67--87, April 1997.

....possible to show the following: Theorem 8 The Sigma 2 fragment of the theory of any (partial or total) lexicographic path ordering is undecidable, as soon as there is at least a binary function symbol. We give a sketch of the proof, the full (quite technical) proof of this result can be found in [8]. We reduce the Post Correspondence Problem (PCP) to the theory of a lexicographic path ordering following the line of [24] Let F be a finite set of function symbols, such that 0 is a minimal constant, f is a binary function symbol which is minimal in F Gamma f0g and g is a minimal unary symbol ....

....as can be seen from the definition of sub given below. The formula 9x OE 1 (x; y) does the job but introduces an existential quantifier at the wrong place, which would throw solv out of the Sigma 2 fragment. A working formula (y) using only universal quantifiers can be found in the full paper [8]. Now it can be shown that always j= OE 1 (x; cs(t 0 ; t n ) x = t n (4) which gives us access to the greatest pair of a list. Note that in our representation of lists, the greatest term stands at an innermost position; it is by no means obvious that we can access this term when the ....

H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Research Report RR-93-42, Deutsches Forschungszentrum fur Kunstliche Intelligenz, Stuhlsatzenhausweg 3, D-66123 Saarbrucken, Germany, Sept. 1993. Anonymous ftp from duck.dfki.uni-sb.de:/pub/ccl/dfki-saarbruecken.

....conditions under which the equation is derivable. The problem now is to decide whether the original set of equations may reduce (using only ordered rewriting) all instances of s = t, satisfying c, to a tautology. This is not so easy because simplification of constrained equations is undecidable [4]: given a term s and a constrained rule l r j c, whether or not all instances of s can be reduced by the rule is undecidable. Our confluence result hence heavily relies on the fact that c is not any constraint, but the constraint l r. On the other hand, we do not consider only a given term s, ....

...., we cannot simplify s = t using u v. We have to show that all instances of s = t j C 0 can be reduced by some instance of u v j C, and that all results of the simplification can be packed together in a single (or a finite number of) constrained equation(s) This is undecidable in general [4]. One possibility to overcome this difficulty would be to use the confluence trees: if all leaves of the confluence tree are trivial equations, then the root of the tree is a redundant equation and can be removed. More generally, we could replace the root of the confluence tree with its non ....

H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Theoretical Computer Science, 176, Apr. 1997.

....to show the following: Theorem 8. The Sigma 2 fragment of the theory of any (partial or total) lexicographic path ordering is undecidable, as soon as there is at least a binary function symbol. We give a sketch of the proof, the full (quite technical) proof of this result can be found in [8]. We reduce the Post Correspondence Problem (PCP) to the theory of a lexicographic path ordering following the line of [24] Let F be a finite set of function symbols, such that 0 is a minimal constant, f is a binary function symbol which is minimal in F Gamma f0g and g is a minimal unary symbol ....

....as can be seen from the definition of sub given below. The formula 9x OE 1 (x; y) does the job but introduces an existential quantifier at the wrong place, which would throw solv out of the Sigma 2 fragment. A working formula (y) using only universal quantifiers can be found in the full paper [8]. Now it can be shown that always j= OE 1 (x; cs(t 0 ; t n ) x = t n (4) which gives us access to the greatest pair of a list. Note that in our representation of lists, the greatest term stands at an innermost position; it is by no means obvious that we can access this term when the ....

H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Research Report RR-93-42, Deutsches Forschungszentrum fur Kunstliche Intelligenz, Stuhlsatzenhausweg 3, D-66123 Saarbrucken, Germany, Sept. 1993. Anonymous ftp from duck.dfki.uni-sb.de:/pub/ccl/dfki-saarbruecken.

....be considered, but arbitrary simplification by rewriting is not allowed any more. In this context, our problem is to decide joinability of the critical pairs by rewriting with a set of constrained equations. This is not easy, because simplification with an arbitrary constrained rule is undecidable [6]: given a term s and a rule l r j c, it is undecidable whether or not all instances of s can be reduced. Going further, we investigate the use of ordered rewriting in a classical application of rewrite systems: the proof by consistency approach to proving inductive theorems [14, 13, 1] Here ....

H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Theoretical Computer Science, 176, Apr. 1997.

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Hubert Comon and Ralf Treinen. The first-order theory of lexicographic path orderings is undecidable. Theoretical Computer Science, To appear.

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H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Theoretical Computer Science 176, Apr. 1997.

No context found.

Comon, Treinen Comon H., Treinen R. The first order theory of lexicographic path orderings is undecidable. Theoretical Computer Science 176. april 1997. To appear.

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