arXiv subjects

Nils Anders Danielsson

Publications and source records attributed to Nils Anders Danielsson.

4 recordsLinked to original sources

arXiv2026

Erased Postulates, Identity Types and Quotients

This text is concerned with the question of whether, in type theory with erasure annotations, one can postulate that some type is inhabited and still have a guarantee that a program will not get stuck. Previous work has provided such guarantees for consistent erased postulates, i.e. postulates that are restricted to be used in erased contexts. Here those guarantees are extended to type theory with identity types. Similar ideas provide a simple way to support quotient types: it is shown that one can let things like "the equivalence classes for two related values are equal" be erased postulates and have an eliminator that only computes for the equivalence class constructor, and still get a guarantee that programs will compute correctly. Another question is whether programs compute correctly if one is allowed to transport (cast) using erased identity proofs. It is shown that this is safe in the absence of quotients and postulates, and in the presence of quotients and erased postulates that can be implemented using equality reflection. However, unrestricted transports of this kind are not compatible with erased, postulated univalence. For that reason the text includes a study of the function []-cong, which encapsulates a limited form of transport for erased identity proofs. The text is accompanied by machine-checked Agda proofs.

cs.PL↗

arXiv2026

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.

cs.LO↗

arXiv2017

Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type

Capretta's delay monad can be used to model partial computations, but it has the "wrong" notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by the "right" notion of equality, weak bisimilarity. However, recent work by Chapman et al. suggests that it is impossible to define a monad structure on the resulting construction in common forms of type theory without assuming (instances of) the axiom of countable choice. Using an idea from homotopy type theory - a higher inductive-inductive type - we construct a partiality monad without relying on countable choice. We prove that, in the presence of countable choice, our partiality monad is equivalent to the delay monad quotiented by weak bisimilarity. Furthermore we outline several applications.

cs.LO↗

arXiv2010

Beating the Productivity Checker Using Embedded Languages

Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures that programs are productive, i.e. that every finite prefix of an infinite value can be computed in finite time. However, many productive programs are not guarded, and it can be nontrivial to put them in guarded form. This paper gives a method for turning a productive program into a guarded program. The method amounts to defining a problem-specific language as a data type, writing the program in the problem-specific language, and writing a guarded interpreter for this language.

cs.LO↗