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16
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
Reachability and Confluence Are Undecidable for Flat Term Rewriting Systems
, 2003
"... Ground reachability, ground joinability and conuence are shown undecidable for at term rewriting systems, i.e. systems in which all left and right members of rule have depth at most one. ..."
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Cited by 12 (2 self)
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Ground reachability, ground joinability and conuence are shown undecidable for at term rewriting systems, i.e. systems in which all left and right members of rule have depth at most one.
Syntacticness, CycleSyntacticness and Shallow Theories
 INFORMATION AND COMPUTATION
, 1994
"... Solving equations in the free algebra T (F; X) (i.e. unification) uses the two rules: f(~s) = f( ~ t) ! ~s = ~ t (decomposition) and s[x] = x !? (occurcheck). These two rules are not correct in quotients of T (F; X) by a finitely generated congruence =E . Following C. Kirchner, we first define cl ..."
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Cited by 11 (0 self)
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Solving equations in the free algebra T (F; X) (i.e. unification) uses the two rules: f(~s) = f( ~ t) ! ~s = ~ t (decomposition) and s[x] = x !? (occurcheck). These two rules are not correct in quotients of T (F; X) by a finitely generated congruence =E . Following C. Kirchner, we first define classes of equational theories (called syntactic and cycle syntactic respectively) for which it is possible to derive some rules replacing the two above ones. Then, we show that these abstract classes are relevant: all shallow theories, i.e. theories which can be generated by equations in which variables occur at depth at most one, are both syntactic and cycle syntactic. Moreover, the new set of unification rules is terminating, which proves that unification is decidable and finitary in shallow theories. We give still further extensions. If the set of equivalence classes is infinite, a problem which turns out to be decidable in shallow theories, then shallow theories fulfill Colmerauer's indep...
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
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.
Reachability is Decidable for Ground AC Rewrite Systems
"... The reachability problem for ground associativecommutative (AC) rewrite systems is decidable. We show that ground AC rewrite systems are equivalent to Process Rewrite Systems (PRS) for which reachability is decidable [4]. However, the decidability proofs for PRS are cumbersome and thus we present a ..."
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Cited by 9 (0 self)
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The reachability problem for ground associativecommutative (AC) rewrite systems is decidable. We show that ground AC rewrite systems are equivalent to Process Rewrite Systems (PRS) for which reachability is decidable [4]. However, the decidability proofs for PRS are cumbersome and thus we present a simpler and more readable proof in the framework of ground AC rewrite systems. Moreover, we show decidability of reachability of states with certain properties and decidability of the boundedness problem.
The Confluence Problem for Flat TRSs
 in "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
A New Parallel Closed Condition for ChurchRosser of LeftLinear Term Rewriting Systems
 Proc. 8th RTA, Sitges, Spain), LNCS 1232
, 1997
"... . G.Huet (1980) showed that a leftlinear termrewriting system (TRS) is ChurchRosser (CR) if P ! j Q for every critical pair < P; Q > where P ! j Q is a parallel reduction from P to Q. But, it remains open whether it is CR when Q ! j P for every critical pair < P; Q >. In this paper, we give a par ..."
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Cited by 6 (0 self)
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. G.Huet (1980) showed that a leftlinear termrewriting system (TRS) is ChurchRosser (CR) if P ! j Q for every critical pair < P; Q > where P ! j Q is a parallel reduction from P to Q. But, it remains open whether it is CR when Q ! j P for every critical pair < P; Q >. In this paper, we give a partial solution to this problem, that is, a leftlinear TRS is CR if Q W ! j P for every critical pair < P; Q > where Q W ! j P is a parallel reduction with the set W of redex occurrences satisfying that if the critical pair is generated from two rules overlapping at an occurrence u, then the length jwj juj for every w 2 W . Furthermore, a leftlinear TRS is CR if Q W ! j P or P " ! Q for every critical pair < P; Q > where W satises the same condition as the above and P " ! Q is a reduction whose redex occurrence is " ( i.e., the root). As a corollary, any leftlinear TRS is CR if P = Q, P " ! Q or Q " ! P for every critical pair < P; Q >, so that we have a critical pair com...
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].