M. M. Fokkinga and E. Meijer. Program calculation properties of continuous algebras. Technical Report CS-R9104, CWI, Amsterdam, Jan. 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to type theory (where calculational elegance rather than a type inference algorithm is the goal) it has become standard to adopt (variations of) the Hagino or, as we prefer to call it, F algebra paradigm of type definition. Good explanations are given by Malcolm [2] and Fokkinga and Meijer [14]. Although we ourselves do not adopt directly the F algebra paradigm, it provides a stepping stone to the real fixpoint approach introduced later on in this chapter. We give now a brief review of the F algebra approach: it is not intended as a tutorial; readers not familiar with this this ....

....adopt directly the F algebra paradigm, it provides a stepping stone to the real fixpoint approach introduced later on in this chapter. We give now a brief review of the F algebra approach: it is not intended as a tutorial; readers not familiar with this this paradigm are directed to [2] and [14]. As in most modern functional languages, types in the F algebra style are built via constructor functions. For example, the type of cons lists A over some base type A may be built from a null ary function nil 2 A 11 and a binary function cons 2 A A Theta A. Implicit is that the type A ....

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Maarten Fokkinga and Erik Meijer. Program calculation properties of continuous algebras. Appears in [21].

....of a fixpoint for a functor F is a type A for which FA is isomorphic to A, in the sense that there exist two functions in : FA A and out : A FA that are each other s inverses. A functor is called locally continuous if its mapping on functions is itself continuous. A standard fixpoint theorem [10, 5] states that every locally continuous functor F on cpos and continuous functions has a unique (up to isomorphism) fixpoint A for which the identity function id : A A is the unique solution to the recursive equation f = in Delta Ff Delta out. This unique fixpoint is called the least fixpoint of ....

M. Fokkinga and E. Meijer. Program calculation properties of continuous algebras. Technical Report 91-4, Centre for Mathematics and Computer Science (CWI), Amsterdam, 1991.

....b List a List b scanr f e [ e] scanr f e (a:x) f a (head x ) x where x = scanr f e x (Here, a:x is the non empty list with head a and tail x, and the function head satis es head (a:x) a. The notation is essentially that of Haskell [20] For example, scanr ( 0 [1,2,3] [6,5,3,0] (Here, is the addition operator passed as an argument, and [1,2,3] a list with three elements. Rightwards accumulation, of course, distributes information in the opposite direction: scanl : b a b) b List a List b scanl f e [ e] scanl f e (a:x) e : scanl f (f e a) ....

....passed as an argument, and [1,2,3] a list with three elements. Rightwards accumulation, of course, distributes information in the opposite direction: scanl : b a b) b List a List b scanl f e [ e] scanl f e (a:x) e : scanl f (f e a) x For example, scanl ( 0 [1,2,3] [0,1,3,6] To make the analogy with trees clearer, we will rst adapt the standard list accumulations to operate on a type of non empty lists: data Plist a = Wrap a ConsP a (Plist a) Informally, a non empty list is either a singleton, Wrap a, or an element pre xed onto another non empty list, ConsP a ....

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Maarten M. Fokkinga and Erik Meijer. Program calculation properties of continuous algebras. Technical Report CS-R9104, CWI, Amsterdam, January 1991.

.... Section 4 to provide universal types for call by value and call by name versions of the untyped lambda calculus; and finally, in Section 5, we compare our system to several recent proposals advocating a categorical style of program construction and manipulation based on iteration and coiteration [MFP91, FM91, Mei92, Kie93]. 2 The language Figure 1 presents a convenient formulation of the syntax of our base language with finite sums and products. As usual, a typing judgment Gamma . M : oe means that the term M has type oe, given the context Gamma (a list of free variables and their types) Figure 2 lists the ....

....been significantly influenced by their suggested style of programming, which reflects our intuitions about how the structure of data should guide the structure of programs. In the language of the Squiggol group, the functions defined using it are catamorphisms and those using gen are anamorphisms ([FM91, MFP91, Mei92]) A hylomorphism is a combination of these concepts which first uses an anamorphism to build up what Meijer refers to as a call tree , and then uses a catamorphism to reduce this tree to a final result (compare the Fibonacci example of the introduction) A requirement for this is that the ....

M.M. Fokkinga and E. Meijer. Program calculation properties of continuous algebras. Technical Report CS-R9104, CWI, January 1991.

....the natural operations over them. More general views of how to use such structures in programming, particularly in cases where computations may not terminate (which causes inductive and coinductive types to coincide) may be found in Kieburtz [Kie93] and some of the papers from the Squiggol group [FM91, MFP91]. Acknowledgements The ideas in this note were originally hashed out in discussion sessions with Dave Schmidt, John Hatcliff, and Jayant DeSouza at Kansas State University. The reference to the papers by Barstow and Merritt was provided by Richard Lorentz after a preliminary version of this ....

M.M. Fokkinga and E. Meijer. Program calculation properties of continuous algebras. Technical Report CS-R9104, CWI, January 1991.

.... combinator over pointed types is constructed in Section 3, based on the fixpoint object of Crole and Pitts [CP92] then in Section 4, we compare our system to several recent proposals advocating a categorical style of program construction and manipulation based on iteration and coiteration [MFP91, FM91, Mei92, Kie93]. 2 The language Figure 1 presents a convenient formulation of the syntax of our base language with finite sums and products. As usual, a typing judgment Gamma . M : oe means that the term M has type oe, given the context Gamma (a list of free variables and their types) Figure 2 lists the ....

....been significantly influenced by their suggested style of programming, which reflects our intuitions about how the structure of data should guide the structure of programs. In the language of the Squiggol group, the functions defined using it are catamorphisms and those using gen are anamorphisms([FM91, MFP91, Mei92]) A hylomorphism is a combination of these concepts which first uses an anamorphism to build up what Meijer refers to as a call tree , and then uses a catamorphism to reduce this tree to a final result (compare the Fibonacci example of the introduction) A requirement for this is that the ....

Maarten M. Fokkinga and Erik Meijer. Program calculation properties of continuous algebras. Technical Report CS-R9104, CWI, January 1991.

....45 t, we may form the three recursive types t: oe, t: oe, and aet: oe. An interesting fact in CPO is that the last two will always be isomorphic. If oe is unconditionally pointed, then all three coincide. This result is related to recent work of Barr, Freyd, and Fokkinga and Meijer [2, 14, 13], although it was arrived at independently. 4.1 Syntax of the language ae In this section we will describe the syntax for the types and terms of ae as an extension to the language of the previous chapter. We start by adding two type constructors: the lifted type oe , which ....

....conditions on F ; he refers to such functors as algebraically compact . For the specific case of CPO, Barr s theorem proves that all of our pointed functors are algebraically compact, therefore F = F . This is one part of our result below, which we obtained independently. Fokkinga and Meijer [13] also establish a similar result, although since CHAPTER 4. RETRACTIVE TYPES AND NON TERMINATION 61 they only work with pointed types they find that the initial and terminal fixed points always correspond (for suitably continuous functors) We have already described how to construct F and F in ....

M.M. Fokkinga and E. Meijer. Program calculation properties of continuous algebras. Technical Report CS-R9104, CWI, January 1991.

....identified several laws for specific data types (most notably finite 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 defined data types [17, 12]. Recently the group of Backhouse [1] has extended the calculus to a relational framework, thus covering indeterminancy. Independently, Paterson [21] has developed a calculus of functional programs similar in contents but very dissimilar in appearance (like many Australian animals) to the work ....

....i.e. all monofunctors defined using the above basic functors or any of the map functors introduced in x5. Then there exists a type L and two strict functions in F 2 LF L and out F 2 L LF (omitting subscripts whenever possible) which are each others inverse and even id = in F out) [6, 23, 16, 24, 30, 12]. We let F denote the pair (L; in) and say that it is the least fixed point of F . Since in and out are each others inverses we have that LF is isomorphic to L, and indeed L is upto isomorphism a fixed point of F. For example taking XL = 1 j AkX, we have that (A; in) L defines the ....

Maarten Fokkinga and Erik Meijer. Program calculation properties of continuous algebras. Technical Report 91-4, CWI, 1991.

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M. M. Fokkinga and E. Meijer. Program calculation properties of continuous algebras. Technical Report CS-R9104, CWI, Amsterdam, Jan. 1991.

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Fokkinga, M. M., Meijer, E.: Program Calculation Properties of Continuous Algebras, Technical Report CS--R9104, CWI, Amsterdam, January 1991, Available online at http://wwwhome.cs.utwente.nl/ ~fokkinga/#detail_0000003528.

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M. M. Fokkinga, E. Meijer. Program calculation properties of continuous algebras. Technical report CS-R9104, Computer Science/Department of Algorithmics and Architecture, CWI. January 1991.

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Maarten Fokkinga and Erik Meijer. Program calculation properties of continuous algebras. Technical Report CS-R9104, CWI, Amsterdam, January 1991.

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