Recursive Domain Equations of Filter Models

Abstract

Abstract. Filter models and (solutions of) recursive domain equations are two different ways of constructing lambda models. Many partial results have been shown about the equivalence between these two constructions (in some specific cases). This paper deepens the connection by showing that the equivalence can be shown in a general framework. We will introduce the class of disciplined intersection type theories and its four subclasses: natural split, lazy split, natural equated and lazy equated. We will prove that each class corresponds to a different recursive domain equation. For this result, we are extracting the essence of the specific proofs for the particular cases of intersection type theories and making one general construction that encompasses all of them. This general approach puts together all these results which may appear scattered and sometimes with incomplete proofs in the literature.

Intersection Types for the lambda-mu Calculus

“…But then we need a detailed analysis of K(D) and K(C), keeping into account that D and C are both co-limits of certain chains of domains, and that their compact points are into one-to-one correspondence with the union of the compact points of the domains approximating D and C. This leads us to a mutually inductive definition of L D and L C and of ≤ D and ≤ C . In this way, we obtain an extension of the type theory used in [17], which is a natural equated intersection type theory in terms of [2] and hence is isomorphic to the inverse limit construction of a D ∞ λ-model (as an aside, we observe that this matches perfectly with Theorem 3.1 in [51]).…”

From Semantics to Types: the Case of the Imperative lambda-Calculus

“…Since [6] we know that a λ -model can be constructed by taking the filters of types in an intersection type system with subtyping. The relation among the filter-model and Scott's D ∞ construction has been subject to extensive studies, starting with [12] and continuing with [16,3,4,5]. In the meantime Abramsky's theory of domain logic in [1] provided a generalization of the same ideas to algebraic domains in the category of 2/3 SFP, of which ω-ALG is a (full) subcategory, based on Stone duality.…”

“…for all atomic α: these correspond to the axioms α = ω − → α in [6], which are responsible of obtaining a "natural equated" solution to the equation D = [D − → D] of Scott's model: see [5]. The λ imp filter model.…”

“…Alas, this cannot be done directly because of the recursive nature of the equation ( 5), but it can be obtained by mirroring the inverse limit construction, e.g. along the lines of [3,5]. Although possible in principle, such a construction requires lots of machinery from the theory of the solution of domain equations; instead we follow the shorter path to define the involved type theories below simply by mutual induction:…”

“…The theories Th D and Th S induce the filter domains F D and F S which satisfy equation(5).Proof. By Prop.…”