Evidenced Frames: A Unifying Framework Broadening Realizability Models

Abstract

Constructive foundations have for decades been built upon realizability models for higher-order logic and type theory. However, traditional realizability models have a rather limited notion of computation, which only supports non-termination and avoids many other commonly used effects. Work to address these limitations has typically overlaid structure on top of existing models, such as by using powersets to represent non-determinism, but kept the realizers themselves deterministic. This paper alternatively addresses these limitations by making the structure underlying realizability models more flexible. To this end, we introduce evidenced frames: a general-purpose framework for building realizability models that support diverse effectful computations. We demonstrate that this flexibility permits models wherein the realizers themselves can be effectful, such as λ-terms that can manipulate state, reduce non-deterministically, or fail entirely. Beyond the broader notions of computation, we demonstrate that evidenced frames form a unifying framework for (realizability) models of higher-order dependent predicate logic. In particular, we prove that evidenced frames are complete with respect to these models, and that the existing completeness construction for implicative algebras-another foundational framework for realizability-factors through our simpler construction. As such, we conclude that evidenced frames offer an ideal domain for unifying and broadening realizability models.

Stateful Realizers for Nonstandard Analysis

“…Since for any S ∈ S the last term belongs to S, we can conclude by anti-reduction that λe S 1 → (S 2 ⊃ S). Proposition 4.20 implies, in particular, that our interpretation also induces a tripos and a topos, by following the method described in [CMT21]. In the following sections, we pay attention to nonstandard reasoning principles for which we can define universal realizers, as these are the evidences for our interpretation (as shown by Proposition 4.20).…”

“…Before studying the properties of this interpretation, we shall connect it with the usual algebraic tools to deal with realizability interpretation, in order to better emphasize its structure and peculiarities. In recent work, Cohen et al have been introducing a new framework to capture the algebraic structure of realizability interpretations, which they named evidenced frames [CMT21]. These have the benefit of being generic enough to easily encompass effectful interpretation, while uniformly inducing triposes (and thus toposes), hence a model of higher-order logic.…”

“…This paper is an extended version of [DM21]. The main novelties here are in Section 4.4, where we establish a connection with evidenced frames [CMT21], and in Section 6, where we give a realizer for a nonstandard version of the Lesser Limited Principle of Omniscience (LLPO). We also now have a better understanding of why performing a quotient leads to some counter-intuitive properties (cf.…”