Types and Neural Networks

Generalized Decidability via Brouwer Trees

In the setting of constructive mathematics, we suggest and study a framework for decidability of properties, which allows for finer distinctions than just "decidable, semidecidable, or undecidable". We work in homotopy type theory and use Brouwer ordinals to specify the level of decidability of a property. In this framework, we express the property that a proposition is $α$-decidable, for a Brouwer ordinal $α$, and show that it generalizes decidability and semidecidability. Further generalizing known results, we show that $α$-decidable propositions are closed under binary conjunction, and discuss for which $α$ they are closed under binary disjunction. We prove that if each $P(i)$ is semidecidable, then the countable meet $\forall i\in \mathbb N. P(i)$ is $ω^2$-decidable, and similar results for countable joins and iterated quantifiers. We also discuss the relationship with countable choice. All our results are formalized in Cubical Agda.

Assistant Professor in Computer Science

Develop the next generation of CS and AI intelligence technology that powers science and society at TU Delft. Combine research with real technological impact while educating talented students. Job description Are you inspired by shaping the next generation of…

Readings shared April 9, 2026

The readings shared in Bluesky on 9 April 2026 are: ABC implies that Ramanujan's tau function misses almost all primes. ~ David Kurniadi Angdinata et als. #LeanProver #ITP #Math Do we need frontier m

A Graded Modal Dependent Type Theory with Erasure, Formalized

We present a graded modal type theory, a dependent type theory with grades that can be used to enforce various properties of the code. The theory has $Π$-types, weak and strong $Σ$-types, natural numbers, an empty type, and a universe, and we also extend the theory with weak and strong unit types and graded $Σ$-types. The theory is parameterized by a modality structure, a kind of partially ordered semiring, whose elements (grades) are used to track the usage of variables in terms and types. Different modalities are possible. We focus mainly on quantitative properties, in particular erasure: with the erasure modality one can mark function arguments as erasable. The theory is fully formalized in Agda. The formalization, which uses a syntactic Kripke logical relation at its core and is based on earlier work, establishes major meta-theoretic properties such as subject reduction, consistency, normalization, and decidability of definitional equality. We also prove a substitution theorem for grade assignment, and preservation of grades under reduction. Furthermore we study an extraction function that translates terms to an untyped $λ$-calculus and removes erasable content, in particular function arguments with the "erasable" grade. For a certain class of modalities we prove that extraction is sound, in the sense that programs of natural number type have the same value before and after extraction. Soundness of extraction holds also for open programs, as long as all variables in the context are erasable, the context is consistent, and erased matches are not allowed for weak $Σ$-types.

Readings shared April 4, 2026

The readings shared in Bluesky on 4 April 2026 are: Why Lean?. ~ Leonardo de Moura. #LeanProver #ITP A formalization of the Gelfond-Schneider theorem. ~ Michail Karatarakis, Freek Wiedijk. #LeanProve

Reseña de «A new paradigm for mathematical proof?»

En la conferencia «A new paradigm for mathematical proof?», Emily Riehl explora los desafíos actuales en la verificación de demostraciones matemáticas. Ejemplos como la conjetura de Kepler y el progra