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Grid Structures and Undecidable Constraint Theories
 In Proceedings of 6th Colloquium on Trees in Algebra and Programming, volume 1214 of LNCS
, 1999
"... We prove three new undecidability results for computational mechanisms over finite trees: There is a linear, ultrashallow, noetherian and strongly confluent rewrite system R such that the 9 8 fragment of the firstorder theory of onesteprewriting by R is undecidable; the emptiness problem ..."
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Cited by 10 (3 self)
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We prove three new undecidability results for computational mechanisms over finite trees: There is a linear, ultrashallow, noetherian and strongly confluent rewrite system R such that the 9 8 fragment of the firstorder theory of onesteprewriting by R is undecidable; the emptiness problem for tree automata with equality tests between cousins is undecidable; and the 9 8  fragment of the firstorder theory of set constraints with the union operator is undecidable. The common feature of these three computational mechanisms is that they allow us to describe the set of firstorder terms that represent grids. We extend our representation of grids by terms to a representation of linear twodimensional patterns by linear terms, which allows us to transfer classical techniques on the grid to terms and thus to obtain our undecidability results. 1 Introduction The grid structure provides convenient means for encoding computation sequences of Turing machines. A classical encoding...
EUnification by Means of Tree Tuple Synchronized Grammars
, 1996
"... : The goal of this paper is both to give a Eunification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructorbased rewrite system, and that four additional restrictions are satisfied. We give a proce ..."
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Cited by 9 (3 self)
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: The goal of this paper is both to give a Eunification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructorbased rewrite system, and that four additional restrictions are satisfied. We give a procedure that represents the (possibly infinite) set of solutions thanks to a tree tuple synchronized grammar, and that can decide unifiability thanks to an emptiness test. Moreover we show that if only three of the four additional restrictions are satisfied then unifiability is undecidable. 1 Introduction First order Eunification [29] is a tool that plays an important role in automated deduction, in particular in functional logic programming and for solving symbolic constraints (see [4] for an extensive survey of the area). It consists in finding instances to variables that make two terms equal modulo an equational theory given by a set of equalities, i.e. it amounts to solve an equation (ca...
Unification in extensions of shallow equational theories
 REWRITING TECHNIQUES AND APPLICATIONS, 9TH INTERNATIONAL CONFERENCE, RTA98', VOL. 1379 OF LNCS
, 1998
"... We show that unification in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equa ..."
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Cited by 9 (1 self)
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We show that unification in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equational theories is introduced. This class is a natural extension of tree automata with equality constraints between brother subterms as well as shallow sort theories. We show that saturation under sorted superposition is effective on sorted shallow equational theories. So called semilinear equational theories can be e ectively transformed into equivalent sorted shallow equational theories and generalize the classes of shallow and standard equational theories.
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
Termination of Narrowing Revisited
"... This paper describes several classes of term rewriting systems (TRS’s) where narrowing has a finite search space and is still (strongly) complete as a mechanism for solving reachability goals. These classes do not assume confluence of the TRS. We also ascertain purely syntactic criteria that suffice ..."
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Cited by 4 (4 self)
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This paper describes several classes of term rewriting systems (TRS’s) where narrowing has a finite search space and is still (strongly) complete as a mechanism for solving reachability goals. These classes do not assume confluence of the TRS. We also ascertain purely syntactic criteria that suffice to ensure the termination of narrowing and include several subclasses of popular TRS’s such as rightlinear TRS’s, almost orthogonal TRS’s, topmost TRS’s, and leftflat TRS’s. Our results improve and/or generalize previous criteria in the literature regarding narrowing termination.
Basic Syntactic Mutation
"... We give a set of inference rules for Eunification, similar to the inference rules for Syntactic Mutation. If the E is finitely saturated by paramodulation, then we can block certain terms from further inferences. Therefore, ..."
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Cited by 1 (0 self)
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We give a set of inference rules for Eunification, similar to the inference rules for Syntactic Mutation. If the E is finitely saturated by paramodulation, then we can block certain terms from further inferences. Therefore,
Multiple Congruence Relations, FirstOrder Theories on Terms, and the Frames of the Applied PiCalculus
"... Abstract. We investigate the problem of deciding firstorder theories of finite trees with several distinguished congruence relations, each of them given by some equational axioms. We give an automatabased solution for the case where the different equational axiom systems are linear and variabledi ..."
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Abstract. We investigate the problem of deciding firstorder theories of finite trees with several distinguished congruence relations, each of them given by some equational axioms. We give an automatabased solution for the case where the different equational axiom systems are linear and variabledisjoint (this includes the case where all axioms are ground), and where the logic does not permit to express tree relations x = f(y, z). We show that the problem is undecidable when these restrictions are relaxed. As motivation and application, we show how to translate the modelchecking problem of AπL, a spatial equational logic for the applied picalculus, to the validity of firstorder formulas in term algebras with multiple congruence relations. 1
Uniqueness of Normal Forms is Decidable for Shallow Term Rewrite Systems ∗
"... Uniqueness of normal forms (UN = ) is an important property of term rewrite systems. UN = is decidable for ground (i.e., variablefree) systems and undecidable in general. Recently it was shown to be decidable for linear, shallow systems. We generalize this previous result and show that this propert ..."
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Uniqueness of normal forms (UN = ) is an important property of term rewrite systems. UN = is decidable for ground (i.e., variablefree) systems and undecidable in general. Recently it was shown to be decidable for linear, shallow systems. We generalize this previous result and show that this property is decidable for shallow rewrite systems, in contrast to confluence, reachability and other properties, which are all undecidable for flat systems. Our result is also optimal in some sense, since we prove that the UN = property is undecidable for two superclasses of flat systems: leftflat, leftlinear systems in which righthand sides are of depth at most two and rightflat, rightlinear systems in which lefthand sides are of depth at most two.