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- Among the many styles and methodologies for the construction of computer programs the
- Squiggol style in our opinion deserves attention from the functional programming community.
- The overall goal of Squiggol is to calculate programs from their specication in the way a mathematician calculates solutions to dierential equations, or uses arithmetic to solve numerical
- It is not hard to state, prove and use laws for well-known operations such as addition, multiplication and |at the function level| composition. It is, however, quite hard to state, prove
- and use laws for arbitrarily recursively dened functions, mainly because it is dicult to refer to
- the recursion scheme in isolation. The algorithmic structure is obscured by using unstructured
- recursive denitions. We crack this problem by treating various recursion schemes as separate
- higher order functions, giving each a notation of its own independent of the ingredients with
- which it constitutes a recursively dened function.
- University of Nijmegen, Department of Informatics, Toernooiveld 6525 ED Nijmegen, e-mail:
- CWI, Amsterdam & University of Twente
- Imperial College, London
- This philosophy is similar in spirit to the `structured programming' methodology for imperative
- programming. The use of arbitrary goto's is abandoned in favour of structured control
- primitives such as conditionals and while-loops that replace xed patterns of goto's, so that reasoning about programs becomes feasible and sometimes even elegant. For functional programs
- the question is which recursion schemes are to be chosen as a basis for a calculus of programs.
- We shall consider several recursion operators that are naturally associated with algebraic type
- denitions. A number of general theorems are proven about these operators and subsequently
- used to transform programs and prove their correctness.
- Bird and Meertens [4, 18] have identied several laws for specic data types (most notably nite
- lists) using which they calculated solutions to various programming problems. By embedding
- the calculus into a categorical framework, Bird and Meertens' work on lists can be extended
- to arbitrary, inductively dened data types [17, 12]. Recently the group of Backhouse  has
- extended the calculus to a relational framework, thus covering indeterminancy.
- Independently, Paterson  has developed a calculus of functional programs similar in contents
- but very dissimilar in appearance (like many Australian animals) to the work referred to above.
- Actually if one pricks through the syntactic dierences the laws derived by Paterson are the
- same and in some cases slightly more general than those developped by the Squiggolers.
- This paper gives an extension of the theory to the context of lazy functional programming, i.e.,
- for us a type is an !-cpo and we consider only continuous functions between types (categorically,
- we are working in the category CPO). Working in the category SET as done by for example
- Malcolm  or Hagino  means that nite data types (dened as initial algebras) and innite
- data types (dened as nal co-algebras) constitute two dierent worlds. In that case it is not
- possible to dene functions by induction (catamorphisms) that are applicable to both nite and
- innite data types, and arbitrary recursive denitions are not allowed. Working in CPO has
- the advantage that the carriers of initial algebras and nal co-algebras coincide, thus there is a
- single data type that comprises both nite and innite elements. The price to
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