Algebra of programming in Agda: Dependent types for relational program derivation

Abstract

Relational program derivation is the technique of stepwise refining a relational specification to a program by algebraic rules. The program thus obtained is correct by construction. Meanwhile, dependent type theory is rich enough to express various correctness properties to be verified by the type checker. We have developed a library, AoPA (Algebra of Programming in Agda), to encode relational derivations in the dependently typed programming language Agda. A program is coupled with an algebraic derivation whose correctness is guaranteed by the type system. Two non-trivial examples are presented: an optimisation problem and a derivation of quicksort in which well-founded recursion is used to model terminating hylomorphisms in a language with inductive types.

Relational algebraic ornaments

“…The relational framework of this paper heavily borrows techniques from the AoPA library [18]. AoPA deals with non-dependently typed programs only, whereas to work with indexed datatypes we need to move to indexed families of relations; to work with the ornamental universe we parametrise the relational fold with a description, making it fully datatype-generic, whereas AoPA has only specialised versions for lists and binary trees; we defined min · Λ as a single operator (which happens to be the shrinking operator proposed by Mu and Oliveira [17]) to avoid the struggle with predicativity that AoPA had.…”

“…As demonstrated by the AoPA library [18], they can be faithfully formalised with preorder reasoning combinators in Agda and used to discharge the fusion conditions of algOrn-fusion-⊆ and algOrn-fusion-⊇. Hence we get two conversions, one of type Vec A n → (n m) → BVec A m which relaxes a vector of length n to a bounded vector whose length is bounded above by some m that is at least n, and the other of type BVec A m → Σ [ n : Nat ] Vec A n × (n m) which converts a bounded vector whose length is at most m to a vector of length precisely n and guarantees that n is at most m.…”

“…In this section, we first introduce and formalise some basic notions in relational program derivation [4] by importing and generalising a small part of the AoPA library [18]. We then introduce relational algebraic ornamentation, which acts as a bridge between the two worlds of internalist programming and relational program derivation.…”

“…We import a part of the AoPA library [18] -which provides facilities for encoding relational derivations in Agda -to the ornament framework [7,12,15] and generalise McBride's algebraic ornamentation to the relational setting (Section 3). (The ornament framework is briefly recapped in Section 2.)…”

Program Calculation in Coq

“…Our work is much related to AoPA [16], a library to support encoding of relational derivations in the dependently typed programming language Agda. This follows the line of research for bridging the gap between dependent types and practical programming [17][18][19].…”