S. Vorobyov. The First-order Theory of One-step Rewriting in Linear Noetherian Systems is Undecidable. In Proceedings 8th Conference RTA, Sitges (Spain), volume 1232 of LNCS, pages 241-253. Springer-Verlag, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that uses the only predicate symbol , where x y means x rewrites into y by one step. It has been shown undecidable in [16] Sharper undecidability results have been obtained for some subclasses of rewrite systems, about the 9 8 fragment [15, 12] and the 9 8 9 fragment [17]. It has been shown decidable in the case of unary signatures [6] in the case of linear rewrite systems whose left and right members do not share any variables [2] 7 , for the positive existential fragment [13] for the whole existential fragment in the case of quasi shallow 8 rewrite systems ....

S. Vorobyov. The First-order Theory of One-step Rewriting in Linear Noetherian Systems is Undecidable. In Proceedings 8th Conference RTA, Sitges (Spain), volume 1232 of LNCS, pages 241-253. Springer-Verlag, 1997.

....is also true, which shows that stratified context unification and one step rewriting constraints are interreducible. It was also noticed in [18] that the first order theory of context unification is undecidable, using the fact that the first order theory of one step rewriting is undecidable [27, 28, 16]. This result was improved by S. Vorobyov [29] who showed that the 89 8 equational theory of context unification is co recursively enumerable hard. The proof of Theorem 1.1 uses a series of four non deterministic translation steps. First we restrict considerations to single context equations. ....

S. Vorobyov. The first-order theory of one step rewriting in linear noetherian systems is undecidable. In H. Comon, editor, International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science. Springer-Verlag, June 1997.

.... been refined to the 9 8 fragment for the class of linear rewrite systems in [31] to the 9 8 fragment for the class of right ground and for the class of linear Noetherian rewrite systems in [20] and to the 9 8 9 fragment for the class of linear Noetherian rewrite systems in [32, 33]. In this paper we restrict the class of rewrite systems for which the theory of onestep rewriting is undecidable to the class of linear, ultra shallow and convergent term rewriting system, a rule being ultra shallow if variables occurs at depth one. This undecidability result is surprising in ....

....crossing the border between decidability and undecidability just consists in adding universal quantifiers, leading to a formula with one quantifier alternation. A part of the construction we use in this paper for checking some properties of terms is very similar to the one used by Vorobyov in [32]. Tree automata with equality tests Tree automata with equality tests have been introduced by Dauchet and Mongy to tackle non linearity problems in various fields such as rewriting, program approximation, and partial evaluation [21] On the one hand, the class of languages recognized by tree ....

Vorobyov, S. The first-order theory of one-step rewriting in linear noetherian systems is undecidable. In Proceedings. Eighth International Conference on Rewriting Techniques and Applications (Sitges, Spain, 1997), H. Comon, Ed., vol. 1232 of Lecture Notes in Computer Science, pp. 254--268.

.... terms that uses the only predicate symbol , where x y means x rewrites into y by one step. It has been shown undecidable in [11] Sharper undecidability results have been obtained for some subclasses of rewrite systems, about the 9 8 fragment [10, 8] and the 9 8 9 fragment [12]. It has been shown decidable for the positive existential fragment [9] in the case of unary signatures [3] in the case of linear rewrite systems whose left and right members do not share any variables [2] 3 , and for the whole existential fragment in the case of shallow rewrite systems ....

S. Vorobyov. The First-order Theory of One-step Rewriting in Linear Noetherian Systems is Undecidable. In Proceedings 8th Conference RTA, Sitges (Spain), volume 1232 of LNCS, pages 241--253. Springer-Verlag, 1997. This article was processed using the L a T E X macro package with LLNCS style

....application we give is to term rewriting. In 1995 Ralf Treinen [56] proved that the first order theory of one step rewriting was undecidable, thus solving a problem stated in [9] and listed in [14, 15] This result was independently and roughly in the same time, obtained by Sergei Vorobyov [57]. The paper of Treinen has attracted considerable attention, because of the widespread belief in the decidability of the theory. In Section 3 we first show that the theorem of Treinen can be obtained as an easy consequence of our main technical result (Theorem 2.15) and then we use Theorem 2.15 to ....

....9 8 ) part of the theory of one step rewriting is undecidable. This is important, since many interesting examples, as for example strong confluence or ground reducibility (see[56] are expressible in this fragment of first order theory. The result of Treinen was improved by Vorobyov [57], who gave undecidability of the 9 8 9 part of the theory of linear Noetherian rewrite systems. Another improvement was obtained by F. Seynhaeve and M. Tommasi and R. Treinen [50] who proved undecidability of the 9 8 part of the theory for linear shallow systems. Some decidability ....

S. Vorobyov. The the first-order theory of one step rewriting in linear noetherian systems is undecidable. In Proceedings of Rewriting Techniques and Applications, volume 1232 of Lecture Notes in Computer Science, pages 254--268. Springer Verlag, 1997.

....system R. This result has been refined to the 9 8 fragment for the class of linear rewriting systems in [22] to the 9 8 fragment for the class of right ground rewriting systems in [16] and to the 9 8 9 fragment for the class of linear noetherian rewriting systems in [24]. Recently, decidability of the positive existential fragment has been shown in [12] In this paper we restrict the class of rewriting systems for which the theory of one step rewriting is undecidable to the class of linear and shallow term rewriting systems. This undecidability result is ....

S. Vorobyov. The first-order theory of one step rewriting in linear noetherian systems is undecidable. To appear in RTA'97, 1997.

....of one step rewriting. Recall in this connection that the confluence is undecidable, in general, but becomes decidable for finite finitely terminating systems. The similar decidability problem was put forward for the subclass of linear systems. The decidability conjecture was first dispelled in (Vorobyov 1997), where a fixed finite, simultaneously finitely terminating and linear system with undecidable theory of one step rewriting was constructed. The proof again was given by reduction from the theory of binary concatenation (finitely generated free semigroups) well known to be undecidable (Quine ....

.... We also construct a fixed finite linear canonical system with undecidable 98 theory of one step rewriting (strong undecidability) Recall that the weak undecidability results of (Treinen 1996, Marcinkowski 1997) do not imply existence of such systems (Section 5) whereas (Vorobyov 1995, Vorobyov 1997) used much more complicated quantifier prefixes and non confluent systems. As a methodological advantage of the proof presented here let us mention the use of reduction from the well known undecidable halting problem for the 1 i.e. containing repeated variable occurrences on the left (or right) ....

[Article contains additional citation context not shown here]

Vorobyov, S. (1997), The first-order theory of one step rewriting in linear noetherian systems is undecidable, in `Rewriting Techniques and Applications '97', Vol. 1232 of Lect. Notes Comput. Sci., Springer-Verlag, pp. 254--268. Available from http://www.mpi-sb.mpg.de/sv.

....in general finite linear finitely terminating systems have undecidable 989 theories of one step rewriting. Up until now it was open problem whether such systems exist. All systems with undecidable theories of one step rewriting known so far were diverging and nonlinear [2, 3] In a recent paper [4] we demonstrated that the full first order theory of one step rewriting in one particular finite linear finitely terminating systems is undecidable. We used the reduction from the first order theory of binary concatenation (free semigroups) known to be undecidable [1] In the current note, by ....

S. Vorobyov. The first-order theory of one-step rewriting in linear noetherian systems is undecidable. Submitted, November 1996.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC