Not Enough Points Is Enough

Abstract

Models of the untyped λ-calculus may be defined either as applicative structures satisfying a bunch of first order axioms, known as "λ-models", or as (structures arising from) any reflexive object in a cartesian closed category (ccc, for brevity). These notions are tightly linked in the sense that: given a λ-model A, one may define a ccc in which A (the carrier set) is a reflexive object; conversely, if U is a reflexive object in a ccc C, having enough points, then C(½, U ) may be turned into a λ-model. It is well known that, if C does not have enough points, then the applicative structure C(½, U ) is not a λ-model in general. This paper: (i) shows that this mismatch can be avoided by choosing appropriately the carrier set of the λ-model associated with U ; (ii) provides an example of an extensional reflexive object D in a ccc without enough points: the Kleisli-category of the comonad "finite multisets" on Rel; (iii) presents some algebraic properties of the λ-model associated with D by (i) which make it suitable for dealing with non-deterministic extensions of the untyped λ-calculus.

Essential and relational models

“…In this section, we describe a class of λ-models based on the category of sets and relations, defined in Bucciarelli et al (2007). The reader already acquainted with this class of models L. Paolini, M. Piccolo and S. Ronchi Della Rocca 14 could skip this section, which we introduced to make the paper self-contained.…”

“…The reader already acquainted with this class of models L. Paolini, M. Piccolo and S. Ronchi Della Rocca 14 could skip this section, which we introduced to make the paper self-contained. All the proofs of the cited results are in Bucciarelli et al (2007), so for sake of facility we will use the same notations. A category with terminal object 1 has enough points when for all f, g :…”

“…Therefore, authors of Bucciarelli et al (2007) used an alternative technique presented in Selinger (2002) in order to build a λ-model starting from a λ-algebra.…”

“…The conclusion of (var) must have the shape x i : σ x i : σ. Since we treat the canonical bijection between M f (S 1 ) × M f (S 2 ) and M f (S 1 &S 2 ) as an equality (as done in Bucciarelli et al (2007)), we have that…”

“…In fact, the family of essential models we define provides a logical description of a class of models formalised in Bucciarelli et al (2007) in MRel, a category of sets and relations. The proposed Essential and relational models 3 non-standard interpretation of terms in the essential λ-models reflects the fact that MRel has not 'enough points'.…”