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22
Open Problems in Rewriting
 Proceeding of the Fifth International Conference on Rewriting Techniques and Application (Montreal, Canada), LNCS 690
, 1991
"... Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27 ..."
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Cited by 19 (2 self)
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Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27, 28, 44, 56,57,76, 82, 81#; one edited collection of papers #1#; four monographs #3, 12,55,65#; and seven books #four of them still in progress# #8,9, 35, 54, 60,75, 84#. To encourage and stimulate continued progress in this area, wehave collected #with the help of colleagues# a number of problems that appear to us to be of interest and regarding whichwe do not know the answer. Questions on rewriting and other equational paradigms have been included; manyhave not aged su#ciently to be accorded the appellation #open problem". Wehave limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementation
Algorithms and Reductions for Rewriting Problems
 IN PROCEEDINGS OF THE 9TH INTERNATIONAL CONFERENCE ON REWRITING TECHNIQUES AND APPLICATIONS
, 1997
"... In this paper we initiate a systematic study of polynomialtime reductions for some basic decision problems of rewrite systems. We then give a polynomialtime algorithm for Uniquenormalform property of ground systems for the first time. Next we prove undecidability of these problems for string r ..."
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Cited by 10 (4 self)
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In this paper we initiate a systematic study of polynomialtime reductions for some basic decision problems of rewrite systems. We then give a polynomialtime algorithm for Uniquenormalform property of ground systems for the first time. Next we prove undecidability of these problems for string rewriting using our reductions. Finally, we prove partial decidability results for Confluence of commutative semithue systems. The Confluence and Uniquenormalform property are also shown Expspacehard for commutative semithue systems.
The Confluence of Ground Term Rewrite Systems is Decidable in Polynomial Time
 IN 42ND ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS). IEEE COMPUTER SOCIETY PRESS, LAS VEGAS
, 2001
"... The confluence property of ground (i.e., variablefree) term rewrite systems (GTRS) is wellknown to be decidable. This was proved independently in [4, 3] and in [13] using tree automata techniques and ground tree transducer techniques (originated from this problem), yielding EXPTIME decision proced ..."
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Cited by 10 (1 self)
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The confluence property of ground (i.e., variablefree) term rewrite systems (GTRS) is wellknown to be decidable. This was proved independently in [4, 3] and in [13] using tree automata techniques and ground tree transducer techniques (originated from this problem), yielding EXPTIME decision procedures (PSPACE for strings). Since then, it has been a wellknown longstanding open question whether this bound is optimal (see, e.g., [15]). Here we give
The Confluence Problem for Flat TRSs
 in &quot;Proceedings of the 8th International Conference on Artificial Intelligence and Symbolic Computation (AISC’06
, 2006
"... Abstract. We prove that the properties of reachability, joinability and confluence are undecidable for flat TRSs. Here, a TRS is flat if the heights of the left and righthand sides of each rewrite rule are at most one. Key words: Term rewriting system, Decision problem, Confluence, Flat. 1 ..."
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Cited by 7 (2 self)
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Abstract. We prove that the properties of reachability, joinability and confluence are undecidable for flat TRSs. Here, a TRS is flat if the heights of the left and righthand sides of each rewrite rule are at most one. Key words: Term rewriting system, Decision problem, Confluence, Flat. 1
Reachability Problems on Regular Ground Tree Rewriting Graphs
 THEORY OF COMPUTING SYSTEMS
, 2004
"... We consider the transition graphs of regular ground tree (or term) rewriting systems. The vertex set of such a graph is a (possibly infinite) set of trees. Thus, with a finite tree automaton one can represent a regular set of vertices. It is known that the backward closure of sets of vertices unde ..."
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Cited by 7 (0 self)
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We consider the transition graphs of regular ground tree (or term) rewriting systems. The vertex set of such a graph is a (possibly infinite) set of trees. Thus, with a finite tree automaton one can represent a regular set of vertices. It is known that the backward closure of sets of vertices under the rewriting relation preserves regularity, i.e., for a regular set T of vertices the set of vertices from which one can reach T can be accepted by a tree automaton. The main contribution of this paper is to lift this result to the recurrence problem, i.e., we show that the set of vertices from which one can reach infinitely often a regular set T is regular, too. Since this result is effective, it implies that the problem whether, given a tree t and a regular set T, there is a path starting in t that infinitely often reaches T, is decidable. Furthermore, it is shown that the problems whether all paths starting in t eventually (respectively, infinitely often) reach T, are undecidable. Based on the decidability result we define a fragment of temporal logic with a decidable modelchecking problem for the class of regular ground tree rewriting graphs.
Deciding Confluence of Certain Term Rewriting Systems in Polynomial Time
, 2002
"... We present a polynomial time algorithm for deciding confluence of ground term rewrite systems. We generalize the decision procedure to get a polynomial time algorithm, assuming that the maximum arity of a symbol in the signature is a constant, for deciding confluence of rewrite systems where each ru ..."
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Cited by 5 (2 self)
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We present a polynomial time algorithm for deciding confluence of ground term rewrite systems. We generalize the decision procedure to get a polynomial time algorithm, assuming that the maximum arity of a symbol in the signature is a constant, for deciding confluence of rewrite systems where each rule contains a shallow linear term on one side and a ground term on the other. The existence of a polynomial time algorithm for deciding confluence of ground rewrite systems was open for a long time and was independently solved only recently [4]. Our decision procedure is based on the concepts of abstract congruence closure [2] and abstract rewrite closure [12].
Classes of Term Rewrite Systems with Polynomial Confluence Problems
 ACM Transactions on Computational Logic
, 2002
"... this article, we have assumed that all terms and the TRS R are built over a fixed signature F , which is not part of the input of the confluence problem. Indeed, if the arities of the input symbols are not bounded, we have the following result ..."
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Cited by 3 (1 self)
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this article, we have assumed that all terms and the TRS R are built over a fixed signature F , which is not part of the input of the confluence problem. Indeed, if the arities of the input symbols are not bounded, we have the following result
The Complexity of Verifying Ground Tree Rewrite Systems
"... Abstract—Ground tree rewrite systems (GTRS) are an extension of pushdown systems with the ability to spawn new subthreads that are hierarchically structured. In this paper, we study the following problems over GTRS: (1) model checking EFlogic, (2) weak bisimilarity checking against finite systems, ..."
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Cited by 2 (2 self)
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Abstract—Ground tree rewrite systems (GTRS) are an extension of pushdown systems with the ability to spawn new subthreads that are hierarchically structured. In this paper, we study the following problems over GTRS: (1) model checking EFlogic, (2) weak bisimilarity checking against finite systems, and (3) strong bisimilarity checking against finite systems. While they are all known to be decidable, we show that problems (1) and (2) have nonelementary complexity, whereas problem (3) is shown to be in coNEXP by finding a syntactic fragment of EF whose model checking complexity is complete for P NEXP. The same problems are studied over a more general but decidable extension of GTRS called regular GTRS (RGTRS), where regular rewriting is allowed. Over RGTRS we show that all three problems have nonelementary complexity. We also apply our techniques to problems over PAprocesses, a wellknown class of infinite systems in Mayr’s PRS (Process Rewrite Systems) hierarchy. For example, strong bisimilarity checking of PAprocesses against finite systems is shown to be in coNEXP, yielding a first elementary upper bound for this problem. I.
Decidable CallbyNeed Computations in Term Rewriting
, 2004
"... The theorem of Huet and Lévy stating that for orthogonal rewrite systems (i) every reducible term contains a needed redex and (ii) repeated contraction of needed redexes results in a normal form if the term under consideration has a normal form, forms the basis of all results on optimal normalizing ..."
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Cited by 2 (2 self)
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The theorem of Huet and Lévy stating that for orthogonal rewrite systems (i) every reducible term contains a needed redex and (ii) repeated contraction of needed redexes results in a normal form if the term under consideration has a normal form, forms the basis of all results on optimal normalizing strategies for orthogonal rewrite systems. However, needed redexes are not computable in general. In the paper we show how the use of approximations and elementary tree automata techniques allows one to obtain decidable conditions in a simple and elegant way. Surprisingly, by avoiding complicated concepts like index and sequentiality we are able to cover much larger classes of rewrite systems. We also study modularity aspects of the classes in our hierarchy. It turns out that none of the classes is preserved under signature extension. By imposing various conditions we recover the preservation under signature extension. By imposing some more conditions we are able to strengthen the signature extension results to modularity for disjoint and constructorsharing combinations.