Easy lambda-terms are not always simple

Abstract

Abstract. A closed λ-term M is easy if, for any other closed term N , the lambda theory generated by M = N is consistent. Recently, it has been introduced a general technique to prove the easiness of λ-terms through the semantical notion of simple easiness. Simple easiness implies easiness and allows to prove consistency results via construction of suitable filter models of λ-calculus living in the category of complete partial orderings: given a simple easy term M and an arbitrary closed term N , it is possible to build (in a canonical way) a non-trivial filter model which equates the interpretation of M and N . The question whether easiness implies simple easiness constitutes Problem 19 in the TLCA list of open problems. In this paper we negatively answer the question providing a non-empty co-r.e. (complement of a recursively enumerable) set of easy, but not simple easy, λ-terms.

Minimal lambda-theories by ultraproducts

“…In Section 2.4.1 we indicate how some of the most known classes of webbed models are recovered as particular instances of i-models (more details for Filter Models are in [12]). Along these lines the notion of partial i-web generalizes those of partial pair [6] (related to graph models) as well as the notions of partial webs of the other types.…”

“…If the EATS satisfies condition ( * ), then the function φ : P f (A) × A → A given by φ (a, α) = ( a) → α is a b-morphism, and hence an i-web A = (A , φ ). The corresponding filter model is exactly the i-model A + (see [12] for the details).…”

On sets of terms having a given intersection type

“…Since their introduction, intersection types have played a role of increasing prominence in programming languages research. From completeness of type assignment [BCD83], to characterization of strongly normalizing (and weakly normalizing) terms [DG03], to syntactic presentations of domain models [AC98], including graph models and filter models [Roc18], to classifications of certain classes of easy terms [CS12], to using types to count resources [KV17], and many others -the theory of intersection types is now a well-established research field of type theory.…”

Graph easy sets of mute lambda terms

“…Studying C-easiness gives insights on the expressive power of C. Concerning filter lambda models [8], for instance, it had been conjectured [3] that they have full expressive power for easy terms, in the sense that any easy term was conjectured to be filter easy. Carraro and Salibra [20] showed that this is not the case: there exists a co-r.e. set of easy terms that are not filter easy.…”