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BöhmLike Trees for Rewriting
"... The work in this thesis has been carried out under the auspices of the research school IPA (Institute for Programming research and Algorithmics).vrije universiteit BöhmLike Trees for Rewriting academisch proefschrift ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag ..."
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The work in this thesis has been carried out under the auspices of the research school IPA (Institute for Programming research and Algorithmics).vrije universiteit BöhmLike Trees for Rewriting academisch proefschrift ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus prof.dr. T. Sminia, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der Exacte Wetenschappen op maandag 20 maart 2006 om 15.45 uur in de aula van de universiteit, De Boelelaan 1105 door
Infinitary Normalization
 We Will Show Them: Essays in Honour of Dov Gabbay
, 2005
"... abstract. In infinitary orthogonal firstorder term rewriting the properties confluence (CR), Uniqueness of Normal forms (UN), Parallel Moves Lemma (PML) have been generalized to their infinitary versions CR ∞ , UN ∞ , PML ∞ , and so on. Several relations between these properties have been establish ..."
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abstract. In infinitary orthogonal firstorder term rewriting the properties confluence (CR), Uniqueness of Normal forms (UN), Parallel Moves Lemma (PML) have been generalized to their infinitary versions CR ∞ , UN ∞ , PML ∞ , and so on. Several relations between these properties have been established in the literature. Generalization of the termination properties, Strong Normalization (SN) and Weak Normalization (WN) to SN ∞ and WN ∞ is less straightforward. We present and explain the definitions of these infinitary normalization notions, and establish that as a global property of orthogonal TRSs they coincide, so at that level there is just one notion of infinitary normalization. Locally, at the level of individual terms, the notions are still different. In the setting of orthogonal term rewriting we also provide an elementary proof of UN ∞ , the infinitary Unique Normal form property. 12
Infinite rewriting: from syntax to semantics
 In Processes, Terms and Cycles: Steps on the Road to Infinity: Essays Dedicated to Jan Willem Klop on the Occasion of His 60th Birthday
, 2005
"... Rewriting is the repeated transformation of a structured object according to a set of rules. This simple concept has turned out to have a rich variety of elaborations, giving rise to many different theoretical frameworks for reasoning about computation. Aside from its theoretical importance, rewriti ..."
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Rewriting is the repeated transformation of a structured object according to a set of rules. This simple concept has turned out to have a rich variety of elaborations, giving rise to many different theoretical frameworks for reasoning about computation. Aside from its theoretical importance, rewriting has also
Weakening the Axiom of Overlap in Infinitary Lambda Calculus
"... In this paper we present a set of necessary and sufficient conditions on a set of lambda terms to serve as the set of meaningless terms in an infinitary bottom extension of lambda calculus. So far only a set of sufficient conditions was known for choosing a suitable set of meaningless terms to make ..."
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In this paper we present a set of necessary and sufficient conditions on a set of lambda terms to serve as the set of meaningless terms in an infinitary bottom extension of lambda calculus. So far only a set of sufficient conditions was known for choosing a suitable set of meaningless terms to make this construction produce confluent extensions. The conditions covered the three main known examples of sets of meaningless terms. However, the much later construction of many more examples of sets of meaningless terms satisfying the sufficient conditions renewed the interest in the necessity question and led us to reconsider the old conditions. The key idea in this paper is an alternative solution for solving the overlap between beta reduction and bottom reduction. This allows us to reformulate the Axiom of Overlap, which now determines together with the other conditions a larger class of sets of meaningless terms. We show that the reformulated conditions are not only sufficient but also necessary for obtaining a confluent and normalizing infinitary lambda beta bottom calculus. As an interesting consequence of the necessity proof we obtain for infinitary lambda calculus with beta and bot reduction that confluence implies normalization.
Skew and ωSkew Confluence and Abstract Böhm Semantics
"... Abstract. Skew confluence was introduced as a characterization of nonconfluent term rewriting systems that had unique infinite normal forms or Böhm like trees. This notion however is not expressive enough to deal with all possible sources of nonconfluence in the context of infinite terms or terms e ..."
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Abstract. Skew confluence was introduced as a characterization of nonconfluent term rewriting systems that had unique infinite normal forms or Böhm like trees. This notion however is not expressive enough to deal with all possible sources of nonconfluence in the context of infinite terms or terms extended with letrec. We present a new notion called ωskew confluence which constitutes a sufficient and necessary condition for uniqueness. We also present a theory that can lift uniqueness results from term rewriting systems to rewriting systems on terms with letrec. We present our results in the setting of Abstract Böhm Semantics, which is a generalization of Böhm like trees to abstract reduction systems. 1
On Modularity in Infinitary Term Rewriting
"... We study modular properties in strongly convergent infinitary term rewriting. In particular, we show that: • Confluence is not preserved across direct sum of a finite number of systems, even when these are noncollapsing. • Confluence modulo equality of hypercollapsing subterms is not preserved acro ..."
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We study modular properties in strongly convergent infinitary term rewriting. In particular, we show that: • Confluence is not preserved across direct sum of a finite number of systems, even when these are noncollapsing. • Confluence modulo equality of hypercollapsing subterms is not preserved across direct sum of a finite number of systems. • Normalization is not preserved across direct sum of an infinite number of leftlinear systems. • Unique normalization with respect to reduction is not preserved across direct sum of a finite number of leftlinear systems. Together, these facts constitute a radical departure from the situation in finitary term rewriting. Positive results are: • Confluence is preserved under the direct sum of an infinite number of leftlinear systems iff at most one system contains a collapsing rule. • Confluence is preserved under the direct sum of a finite number of noncollapsing systems if only terms of finite rank are considered. • Toptermination is preserved under the direct sum of a finite number of leftlinear systems. • Normalization is preserved under the direct sum of a finite number of leftlinear systems. All of the negative results above hold in the setting of weakly convergent rewriting as well, as do the positive results concerning modularity of toptermination and normalization for leftlinear systems.
A lambda calculus for D∞
, 2002
"... We define an extension of lambda calculus which is fully abstract for Scott's D_infinitymodels. We do so by constructing an infinitary lambda calculus which not only has the confluence property, but also is normalising: every term has its infinity etaBöhm tree as unique normal form. The extension ..."
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We define an extension of lambda calculus which is fully abstract for Scott's D_infinitymodels. We do so by constructing an infinitary lambda calculus which not only has the confluence property, but also is normalising: every term has its infinity etaBöhm tree as unique normal form. The extension incorporates...
UNIQUE NORMAL FORMS IN INFINITARY WEAKLY ORTHOGONAL TERM REWRITING
"... Abstract. 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 ..."
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Abstract. 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. 1.
Substitution in nonwellfounded . . .
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 82 NO. 1 (2003)
, 2003
"... Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable bin ..."
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Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution. This is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category. In the nonwellfounded case, the fundamental operation of substitution is more beneficially defined in terms of primitive corecursion than coiteration.
Decomposing the Lattice of Meaningless Sets in the Infinitary Lambda Calculus
"... Abstract. The notion of a meaningless set has been defined for infinitary lambda calculus axiomatically. Standard examples of meaningless sets are sets of terms that have no head normal form, the set of terms without weak head normal form and the set of rootactive terms. The collection of meaningles ..."
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Abstract. The notion of a meaningless set has been defined for infinitary lambda calculus axiomatically. Standard examples of meaningless sets are sets of terms that have no head normal form, the set of terms without weak head normal form and the set of rootactive terms. The collection of meaningless sets is a lattice. In this paper, we study the way this lattices decompose as union of more elementary key intervals. We also analyse the distribution of the sets of meaningless terms in the lattice by selecting some sets as key vertices and study the cardinality in the intervals between key vertices. As an application, we prove that the lattice of meaningless sets is neither distributive nor modular. Interestingly, the example translates into a counterexample that the lattice of lambda theories is not modular. 1