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Partial Order Infinitary Term Rewriting and Böhm Trees
, 2010
"... We study an alternative model of in nitary term rewriting. Instead of a metric on terms, a partial order on partial terms is employed to formalise convergence of reductions. We consider both a weak and a strong notion of convergence and show that the metric model of convergence coincides with the p ..."
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We study an alternative model of in nitary term rewriting. Instead of a metric on terms, a partial order on partial terms is employed to formalise convergence of reductions. We consider both a weak and a strong notion of convergence and show that the metric model of convergence coincides with the partial model restricted to total terms. Hence, partial order convergence constitutes a conservative extension of metric convergence that additionally offers a finegrained distinction between di erent levels of divergence. In the second part, we focus our investigation on strong convergence of orthogonal systems. The main result is that the gap between the metric model and the partial order model can be bridged by simply extending the term rewriting system by additional rules. These extensions are the wellknown Böhm extensions. Based on this result, we are able to establish that contrary to the metric setting orthogonal systems are both infinitarily confluent and infinitarily normalising in the partial order setting. The unique infinitary normal forms that the partial order model admits are Böhm trees.
What to Do Next? Analysing and Optimising System Behaviour in Time ACADEMISCH PROEFSCHRIFT
"... any form. No Zebra Finches were harmed during the development of this thesis. For the cover image: © ≫Unfold II≪, 100 x 100 cm, diasec, 2006 by sascha weidner The research in this thesis has been carried out at the Centre for Mathematics ..."
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Cited by 3 (2 self)
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any form. No Zebra Finches were harmed during the development of this thesis. For the cover image: © ≫Unfold II≪, 100 x 100 cm, diasec, 2006 by sascha weidner The research in this thesis has been carried out at the Centre for Mathematics
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.
OF PROBABILISTIC MODELS
, 1980
"... PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus prof. dr. H. Brinksma, volgens besluit van het College voor Promoties, in het openbaar te verdedigen op vrijdag 25 september 2009 om 13:15 uur door ..."
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PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus prof. dr. H. Brinksma, volgens besluit van het College voor Promoties, in het openbaar te verdedigen op vrijdag 25 september 2009 om 13:15 uur door
Printed by Wöhrmann Print Service.
"... has been carried out within the context of the Center for Telematics and Information Technology (CTIT) and under the auspices of the research school IPA (Institute for Programming research and Algorithmics). Keywords: wireless sensor networks, service discovery protocols, clustering algorithms, cont ..."
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has been carried out within the context of the Center for Telematics and Information Technology (CTIT) and under the auspices of the research school IPA (Institute for Programming research and Algorithmics). Keywords: wireless sensor networks, service discovery protocols, clustering algorithms, context awareness, movement detection
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"... This research is supported by the SEDAN project, ..."
Strict Ideal Completions of the Lambda Calculus
"... We present a family of infinitary lambda calculi with varying strictness. This family of calculi corresponds to the infinitary lambda calculi of Kennaway et al. but instead of metric completion our calculi are based on ideal completion. We show that each of our calculi conservatively extends the cor ..."
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We present a family of infinitary lambda calculi with varying strictness. This family of calculi corresponds to the infinitary lambda calculi of Kennaway et al. but instead of metric completion our calculi are based on ideal completion. We show that each of our calculi conservatively extends the corresponding metricbased calculus. The extension that our calculi provide is characterised in the form of ’⊥rules’: βreduction in the metric calculi extended by these ⊥rules, also know as Böhm reduction, coincides with βreduction in the corresponding ideal completion calculi. Thus, our calculi allow reasoning over partially converging infinite reductions without the need for additional rules, while their total fragments still coincide with the metric calculi.
door
, 2007
"... About the cover: picture of the SISYPHUS II machine, constructed by Bruce Shapiro. ..."
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About the cover: picture of the SISYPHUS II machine, constructed by Bruce Shapiro.
Noname manuscript No. (will be inserted by the editor)
"... the date of receipt and acceptance should be inserted later Abstract When infinitary rewriting was introduced by Kaplan et. al. [7] at the beginning of the 1990s, its term universe was explained as the metric completion of a metric on finite terms. The motivation for this connection to topology was ..."
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the date of receipt and acceptance should be inserted later Abstract When infinitary rewriting was introduced by Kaplan et. al. [7] at the beginning of the 1990s, its term universe was explained as the metric completion of a metric on finite terms. The motivation for this connection to topology was that it allowed to import other wellstudied notions from metric spaces, in particular the notion of convergence as a replacement for normalisation. This paper generalises the approach by parameterising it with a term metric, and applying the process of metric completion not only to terms but also to operations on and relations between terms. The resulting metatheory is studied, leading to a revised notion of infinitary rewrite system. For these systems a method is devised to prove their convergence. 1