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Degrees of undecidability in term rewriting
 Proceedings of Computer 30 Logic (CSL09), volume 5771 of Lecture Notes in Computer Science
, 2009
"... Abstract. Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy cl ..."
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Abstract. Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas and continuing into the analytic hierarchy, where also quantification over function variables is allowed. In this paper we consider properties of first order term rewriting systems and classify them in this hierarchy. Most of the standard properties are Π 0 2complete, that is, of the same level as uniform halting of Turing machines. In this paper we show two exceptions. Weak confluence is Σ 0 1complete, and therefore essentially easier than ground weak confluence which is Π 0 2complete. The most surprising result is on dependency pair problems: we prove this to be Π 1 1complete, which means that this property exceeds the arithmetical hierarchy and is essentially analytic. A minor variant, dependency pair problems with minimality flag, turns out be Π 0 2complete again, just like the original termination problem for which dependency pair analysis was developed. 1
Infinitary rewriting: Foundations revisited
 Proceedings of the 21st International Conference on Rewriting Techniques and Applications, Leibniz International Proceedings in Informatics (LIPIcs
, 2010
"... Abstract. Infinitary Term Rewriting allows to express infinitary terms and infinitary reductions that converge to them. As their notion of transfinite reduction in general, and as binary relations in particular two concepts have been studied in the past: strongly and weakly convergent reductions, an ..."
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Abstract. Infinitary Term Rewriting allows to express infinitary terms and infinitary reductions that converge to them. As their notion of transfinite reduction in general, and as binary relations in particular two concepts have been studied in the past: strongly and weakly convergent reductions, and in the last decade research has mostly focused around the former. Finitary rewriting has a strong connection to the equational theory of its rule set: if the rewrite system is confluent this (implies consistency of the theory and) gives rise to a semidecision procedure for the theory, and if the rewrite system is in addition terminating this becomes a decision procedure. This connection is the original reason for the study of these properties in rewriting. For infinitary rewriting there is barely an established notion of an equational theory. The reason this issue is not trivial is that such a theory would need to include some form of “getting to limits”, and there are different options one can pursue. These options are being looked at here, as well as several alternatives for the notion of reduction relation and their relationships to these equational theories. 1.
Levels of Undecidability in Rewriting
, 2011
"... Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarc ..."
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Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2complete. The particular problem of local confluence turns out to be Π 0 2complete for ground terms, but only Σ 0 1complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.
UNIQUE NORMAL FORMS IN INFINITARY WEAKLY ORTHOGONAL TERM REWRITING
, 2010
"... We present some contributions to the theory of infinitary rewriting for weakly orthogonal term rewrite systems, in which critical pairs may occur provided they are trivial. We show that the infinitary unique normal form property (UN ∞ ) fails by a simple example of a weakly orthogonal TRS with two ..."
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We present some contributions to the theory of infinitary rewriting for weakly orthogonal term rewrite systems, in which critical pairs may occur provided they are trivial. We show that the infinitary unique normal form property (UN ∞ ) fails by a simple example of a weakly orthogonal TRS with two collapsing rules. By translating this example, we show that UN ∞ also fails for the infinitary λβηcalculus. As positive results we obtain the following: Infinitary confluence, and hence UN ∞ , holds for weakly orthogonal TRSs that do not contain collapsing rules. To this end we refine the compression lemma. Furthermore, we consider the triangle and diamond properties for infinitary multisteps (complete developments) in weakly orthogonal TRSs, by refining an earlier clusteranalysis for the finite case.
Modularity of convergence and strong convergence in infinitary rewriting
 Log. Methods Comput Sci
, 2010
"... Abstract. Properties of Term Rewriting Systems are called modular iff they are preserved under (and reflected by) disjoint union, i.e. when combining two Term Rewriting Systems with disjoint signatures. Convergence is the property of Infinitary Term Rewriting Systems that all reduction sequences con ..."
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Abstract. Properties of Term Rewriting Systems are called modular iff they are preserved under (and reflected by) disjoint union, i.e. when combining two Term Rewriting Systems with disjoint signatures. Convergence is the property of Infinitary Term Rewriting Systems that all reduction sequences converge to a limit. Strong Convergence requires in addition that redex positions in a reduction sequence move arbitrarily deep. In this paper it is shown that both Convergence and Strong Convergence are modular properties of noncollapsing Infinitary Term Rewriting Systems, provided (for convergence) that the term metrics are granular. This generalises known modularity results beyond metric d∞. 1.
Degrees of Undecidability in Rewriting
, 902
"... Abstract. Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy cl ..."
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Abstract. Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas and continuing into the analytic hierarchy, where also quantification over function variables is allowed. In this paper we consider properties of first order term rewriting systems and classify them in this hierarchy. Weak and strong normalization for single terms turn out to be Σ 0 1complete, while their uniform versions as well as dependency pair problems with minimality flag are Π 0 2complete. We find that confluence is Π 0 2complete both for single terms and uniform. Unexpectedly weak confluence for ground terms turns out to be harder than weak confluence for open terms. The former property is Π 0 2complete while the latter is Σ 0 1complete (and thereby recursively enumerable). The most surprising result is on dependency pair problems without minimality flag: we prove this to be Π 1 1complete, which means that this property exceeds the arithmetical hierarchy and is essentially analytic. 1
Productivity of NonOrthogonal Term Rewrite Systems
"... Productivity is the property that finite prefixes of an infinite constructor term can be computed using a given term rewrite system. Hitherto, productivity has only been considered for orthogonal systems, where nondeterminism is not allowed. This paper presents techniques to also prove productivity ..."
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Productivity is the property that finite prefixes of an infinite constructor term can be computed using a given term rewrite system. Hitherto, productivity has only been considered for orthogonal systems, where nondeterminism is not allowed. This paper presents techniques to also prove productivity of nonorthogonal term rewrite systems. For such systems, it is desired that one does not have to guess the reduction steps to perform, instead any outermostfair reduction should compute an infinite constructor term in the limit. As a main result, it is shown that for possibly nonorthogonal term rewrite systems this kind of productivity can be concluded from contextsensitive termination. This result can be applied to prove stabilization of digital circuits, as will be illustrated by means of an example. 1
2.1 Different Views of a Cell........................ 7
"... ter verkrijging van de graad van doctor aan de ..."