Quotienting the delay monad by weak bisimilarity

Abstract

Abstract. The delay datatype was introduced by Capretta [3] as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. It is a monad and it constitutes a constructive alternative to the maybe monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay monad quotiented by weak bisimilarity is still a monad. In this paper, we consider Hofmann's alternative approach [6] of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. We have fully formalized our results in the Agda dependently typed programming language.

Partiality, Revisited

“…In 2015, Andrej Bauer and the first-named author of the current paper suggested to use a similar approach to define a partiality monad without using countable choice, an idea which was mentioned by Chapman et al [8]. Here, we show that this is indeed possible.…”

“…In Sect. 4 we show that, assuming countable choice, our partiality monad is equivalent to (in bijective correspondence to) the one of Chapman et al [8]. We outline some applications of the partiality monad in Sect.…”

“…With an additional assumption, Chapman et al have managed to show that the partiality operator D(−)/∼ D is a monad [8]. This additional assumption is known as countable choice.…”

“…The type theory that we work in can be described as a fragment of the theory considered in the standard textbook on homotopy type theory [20] (henceforth referred to as the HoTT book), and is quite close to the theory considered by Chapman et al [8]. Details are given in Sect.…”

“…We assume that equality of functions is extensional, and that (strong) bisimilarity implies equality for coinductive types. Chapman et al [8] assume the axiom of uniqueness of identity proofs, UIP, for all small types (types in the lowest universe). UIP holds for a type A if, for any elements x, y : A, if we have equalities p, q : x = y, then we have p = q.…”