Relational algebraic ornaments

Abstract

Dependently typed programming is hard, because ideally dependently typed programs should share structure with their correctness proofs, but there are very few guidelines on how one can arrive at such integrated programs. McBride's algebraic ornamentation provides a methodological advancement, by which the programmer can derive a datatype from a specification involving a fold, such that a program that constructs elements of that datatype would be correct by construction. It is thus an effective method that leads the programmer from a specification to a dependently typed program. We enhance the applicability of this method by generalising algebraic ornamentation to a relational setting and bringing in relational algebraic methods, resulting in a hybrid approach that makes essential use of both dependently typed programming and relational program derivation. A dependently typed solution to the minimum coin change problem is presented as a demonstration of this hybrid approach. We also give a theoretically interesting "completeness theorem" of relational algebraic ornaments, which sheds some light on the expressive power of ornaments and inductive families.

Programming with ornaments

“…In addition, the relationship between BHeap M and BHeap N (at least in a nonindexed-first variant on the boolean index) turns out to be an algebraic ornamentation (McBride, 2011;Ko & Gibbons, 2013b), so we could get not only the forgetful function from BHeap N to BHeap M but also its converse:…”

“…Gap 4 (representation optimisation for algebraic ornamentation) None of the existing formulations of general algebraic ornamentation (McBride, 2011;Dagand & McBride, 2014;Ko & Gibbons, 2013b) takes representation optimisation into account -they all work by inserting some equality constraints as fields into descriptions, and cannot further classify constructors in an index-first manner, which happened for BHeap N . (Dagand & McBride (2014)'s reornaments, whilst being algebraic and achieving representation optimisation, are specialised only to the ornamental algebras.)…”

“…(Dagand & McBride (2014)'s reornaments, whilst being algebraic and achieving representation optimisation, are specialised only to the ornamental algebras.) Ko & Gibbons (2013b) noted that, in order to fully exploit the optimising power of index-first inductive families, we might need to switch to coalgebraic ornamentation (for inductive datatypes), which remains to be investigated.…”

“…Dagand & McBride (2014) then proposed functional ornaments, generalising ornaments to work also for functions. Algebraic ornamentation has also been generalised to a relational setting, and shown to work with relational calculation techniques (Ko & Gibbons, 2013b).…”

Partial type equivalences for verified dependent interoperability

“…In particular, two key ingredients of the ornamental toolbox are algebraic ornaments (McBride 2010) and relational ornaments (Ko and Gibbons 2013). From an inductive type and an algebra (over an endofunctor on Set for algebraic ornaments, over an endofunctor on Rel for relational ornaments), these ornaments construct an inductive family satisfying-by construction-the desired invariants.…”