Meaningless terms in rewriting

Kennaway, Richard; Oostrom, Vincent van; Vries, Fred De

doi:10.1007/3-540-61735-3_17

Cited by 22 publications

(74 citation statements)

References 9 publications

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“…The three infinitary lambda calculi mentioned in the first three rows of Figure 1 capture the well-known cases of Böhm, Lévy-Longo and Berarducci trees [4,9,10]. In the fourth row, there is an uncountable class of infinitary lambda calculi with a ⊥-rule parametrised by a set U of meaningless terms [11,12]. By changing the parameter set U of the ⊥-rule, we obtain different infinitary lambda calculi.…”

Section: Introductionmentioning

confidence: 95%

“…This set contains the three sets of Böhm, Lévy-Longo and Berarducci trees. In [10][11][12], an alternative definition of the set Λ ∞ ⊥ is given using a metric. The coinductive and metric definitions are equivalent [3].…”

Section: Infinite Lambda Calculusmentioning

confidence: 99%

“…The coinductive and metric definitions are equivalent [3]. In this paper we consider only one set of λ-terms, namely Λ ∞ ⊥ , in contrast to the formulations in [10,11] where several sets (which are all subsets of Λ ∞ ⊥ ) are considered. The paper [12] shows that the infinitary lambda calculi can be formulated using a common set Λ ∞ ⊥ , confluence and normalisation still hold since the extra terms added by the superset Λ ∞ ⊥ are meaningless and equated to ⊥.…”

Section: Infinite Lambda Calculusmentioning

confidence: 99%

“…The η!-rule does not appear in the finite lambda calculus. The ⊥-rule is parametric on a set U ⊂ Λ ∞ of meaningless terms [11,12] where Λ ∞ is the set of terms in Λ ∞ ⊥ that do not contain ⊥. The notions of head normal form, weak head normal form and top normal form are defined as follows:…”

Section: Infinite Lambda Calculusmentioning

confidence: 99%

“…Let U be a set of meaningless terms satisfying the axioms of [11,12]. If the theory Eq(T U ) is consistent then T ⊆ U ⊆ H.…”

Section: Note That T ⊃ W ⊃ H and T ⊂ W ⊂ Hmentioning

confidence: 99%

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Continuity and Discontinuity in Lambda Calculus

Severi

Vries

2005

Lecture Notes in Computer Science

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Abstract. This paper studies continuity of the normal form and the context operators as functions in the infinitary lambda calculus. We consider the Scott topology on the cpo of the finite and infinite terms with the prefix relation. We prove that the only continuous parametric trees are Böhm and Lévy-Longo trees. We also prove a general statement: if the normal form function is continuous then so is the model induced by the normal form; as well as the converse for parametric trees. This allows us to deduce that the only continuous models induced by the parametric trees are the ones of Böhm and Lévy-Longo trees. As a first application, we prove that there is an injective embedding from the infinitary lambda calculus of the ∞η-Böhm trees in D∞. As a second application, we study the relation between the Scott topology on the prefix relation and the tree topologies. This allows us to prove that the only parametric tree topologies in which all context operators are continuous and the approximation property holds are the ones of Böhm and Lévy-Longo. As a third application, we give an explicit characterisation of the open sets of the Böhm and Lévy-Longo tree topologies.

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