Order Structures on Böhm-Like Models

Decomposing the Lattice of Meaningless Sets in the Infinitary Lambda Calculus

“…Checking this condition on some restricted set of terms will be enough provided the set U is closed under β-expansions. This criterion differs from the one in [15] on the fact that the common structure is now a Berarducci tree and not a skeleton.…”

“…We will now define the sets of meaningless terms that occur in Figure 1. To define these sets, we will first need to introduce new forms of terms analogous to the notions of head, weak head and top normal forms and define certain specific subsets of Λ ∞ containing the respective forms [15].…”

“…Although in the initial papers [6,8,10,5] on infinite lambda calculus only those three sample sets were presented as set of meaningless terms, these sets are not the only sets of meaningless terms. Only in the more recent papers [14,15,9] some aspects of the rich lattice of the sets of meaningless terms have been explored.…”

“…Because, despite the fact that R ∪ SIL itself is not meaningless, there are infinitely many sets of meaningless terms between R and R ∪ SIL. The other vertices in Figure 1 represent all sets of meaningless terms that decompose as the disjoint union of one or more of the basic sets R, SIL, IL, SA, HA and O [15].…”

The infinitary lambda calculus of the infinite eta Böhm trees

“…The infinitary lambda calculi λ ∞ β⊥ with a ⊥-rule parametric on a set of (weakly) meaningless terms encompasses the previous three cases ( Kennaway and de Vries, 2003;Severi and de Vries, 2011). This method to construct models of the lambda beta calculus is quite flexible as there is ample choice for the set of meaningless terms (Severi and de Vries, 2005a;Severi and de Vries, 2005b;Severi and de Vries, 2011). 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).…”