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.Comment: In Proceedings PAR 2010, arXiv:1012.455

A Type and Scope Safe Universe of Syntaxes with Binding: Their Semantics and Proofs

Almost every programming language’s syntax includes a notion of binder and corresponding bound occurrences, along with the accompanying notions of α-equivalence, capture avoiding substitution, typing contexts, runtime environments, and so on. In the past, implementing and reasoning about programming languages required careful handling to maintain the correct behaviour of bound variables. Modern programming languages include features that enable constraints like scope safety to be expressed in types. Nevertheless, the programmer is still forced to write the same boilerplate over again for each new implementation of a scope safe operation (e.g., renaming, substitution, desugaring, printing, etc.), and then again for correctness proofs. We present an expressive universe of syntaxes with binding and demonstrate how to (1) implement scope safe traversals once and for all by generic programming; and (2) how to derive properties of these traversals by generic proving. Our universe description, generic traversals and proofs, and our examples have all been formalised in Agda and are available in the accompanying material

Exploring the regular tree types

Abstract. In this paper we use the Epigram language to define the universe of regular tree types—closed under empty, unit, sum, product and least fixpoint. We then present a generic decision procedure for Epigram’s in-built equality at each type, taking a complementary approach to that of Benke, Dybjer and Jansson [7]. We also give a generic definition of map, taking our inspiration from Jansson and Jeuring [21]. Finally, we equip the regular universe with the partial derivative which can be interpreted functionally as Huet’s notion of ‘zipper’, as suggested by McBride in [27] and implemented (without the fixpoint case) in Generic Haskell by Hinze, Jeuring and Löh [18]. We aim to show through these examples that generic programming can be ordinary programming in a dependently typed language.

Exploring the regular tree types

Abstract. In this paper we use the Epigram language to define the universe of regular tree types—closed under empty, unit, sum, product and least fixpoint. We then present a generic decision procedure for Epigram’s in-built equality at each type, taking a complementary approach to that of Benke, Dybjer and Jansson [7]. We also give a generic definition of map, taking our inspiration from Jansson and Jeuring [21]. Finally, we equip the regular universe with the partial derivative which can be interpreted functionally as Huet’s notion of ‘zipper’, as suggested by McBride in [26] and implemented (without the fixpoint case) in Generic Haskell by Hinze, Jeuring and Löh [18]. We aim to show through these examples that generic programming can be ordinary programming in a dependently typed language.