An Extensional Böhm Model

The infinitary lambda calculus of the infinite eta Böhm trees

“…Because the collection of sets of weakly meaningless terms is uncountable, we get an uncountable class of models which are not continuous (Severi and de Vries, 2005a). The infinitary lambda calculus λ ∞ β⊥η = (Λ ∞ ⊥ , −→ β⊥η ) sketched in the last but one row incorporates the η-rule (Severi and de Vries, 2002). This calculus captures the notion of η-Böhm tree, which can be described as the eta-normal form of a Böhm tree, and gives rise to an extensional model of the lambda calculus that has the same local structure as Coppo, Dezani and Zacchi's filter model D * ∞ (Coppo et al, 1987).…”

“…Figure 1 summarises the correspondences between the infinitary lambda calculi and the trees which have been studied so far. All these calculi include a notion of ⊥-reduction and they are all proved to be confluent and normalising before except for the one on the last row (Berarducci, 1996;Kennaway et al, 1995a;Kennaway et al, 1997;Kennaway and de Vries, 2003;Severi and de Vries, 2002;Severi and de Vries, 2011). From any infinitary lambda calculus which is confluent and normalising, we can construct a model of the finite lambda calculus by defining the interpretation of a term to be exactly the (infinite) normal form of that term (or equivalently the tree of that term).…”

Continuity and Discontinuity in Lambda Calculus

“…We will now briefly recall some notions and facts of infinite lambda calculus from our earlier work [9,10,12,16,15]. We assume familiarity with basic notions and notations from [2].…”

“…The variation comes from the choice of the set U and the strength of extensionality. Figure 1 summarises the infinitary lambda calculi studied so far [4,9,10,12,16,15]. An interesting aspect of infinitary lambda calculus is the possibility of capturing the notion of tree (such as Böhm and Lévy-Longo trees) as a normal form.…”

“…And if U is the set of terms without top head normal form to ⊥, we recover the Berarducci trees. The infinitary lambda calculus sketched in the one but last row incorporates the η-rule [16]. This calculus captures the notion of η-Böhm tree.…”