Promotional Transformation on Monadic Programs

Monads, proposed by Moggi [16] of their use in structuring denotational descriptions and then popularized by Wadler[21], are becoming an increasingly important tool for structural functional programming[8, 10, 11]. The reason is that monads provide a uniform framework for describing a wide range of programming language features including, for example, state, I/O, continuations, exceptions, parsing and non-determinism, without leaving the framework of a purely functional language. Programs using monads, which will be called monadic programs hereafter, can be structured once again over data type for facilitating program transformation [5, 9, 15]. It is known that each data type comes equipped with a catamorphism[13, 14] (i.e., a generalized fold[18]) which satis es several laws that are very useful for program transformation. Fokkinga[5] derived a su cient condition under which there is also a kind of so-called monadic catamorphism which satisfy similar laws and can thus be used for transformation of monadic programs. Apart from the theoretical study, very recently, Meijer and Jeuring[15] discussed the simultaneous use of catamorphisms and monads in practical functional programming. They convincingly demonstrated through many examples that by programming in this style resulting programs are of an astonishing clarity and conciseness. Categorically speaking, monadic catamorphisms are the lifting of normal catamorphisms to the Kleisli category[5]. This is also the approach taken in the Algebraic Design Language (ADL) [9]. Despite its mathematical elegance, Fokkinga's theory contains an assumption on monad that is not valid for several known monads, in particular it is not valid for the state monad. Therefore, for these monads, the whole reasoning of Fokkinga's doesn't make sense (see the conclusion in [5]). Meijer and Jeuring[15] solved this problem informally in their case study of deriving an e cient abstract G-machine from an initial naive monadic program related to the state monad; they de ned their speci c monadic catamorphism in terms of a normal catamorphism and found some transformation rules for the calculation of monadic programs based on the fusion laws (i.e., promotion theorem) for the normal catamorphism. However, a general study on direct transformation of monadic catamorphisms was not given.