Categorical Fixed Point Calculus

ion theorem, these operators can be lifted to _+ and _for arbitrary domain cat-egories E ; i.e. _+ and _are binary operators on the category Fun . They are, in fact,the coproduct and the product in Fun : by the parameterised limit theorem it followsthat (F _+G;inlF ;G;inrF ;G ) is the coproduct of F and G , where (inlF ;G)x = inlF:x;G:x andsimilarly for inrF ;G . In the same way _is the binary product functor on Fun and alsothe adjoints of F _and _F can be de ned. The category Fun is thus a bicartesian,exponential category where, for instance,sumassF ;G;H 2 (F _+G) _+H = F _+(G _+H)is given bysumassF ;G;H = sumass (F 4G 4H) :That is,(sumassF ;G;H)x = sumassF:x;G:x;H:x :The abstraction theorem admits a similar result for and for map functors in general.Suppose F is a functor. De ne functor by a y = 1 + F:a y . Then we observethatF=fde nition and composition ga 7! (a )=fabstraction theorem g(G 7! Id _G)=f(Id _G):x = x G:x = 1+(F:x G:x)= (K:1 _+F _G):x , extensionality g(G 7! K:1 _+F _G) :If we now de ne the functor _on objects F by (G 7! K:1 _+F _G) we can reformulatethis observation asF = _F :(23)The list decomposition theorem now becomes a theorem in the functor category wherebyeach functor is replaced by its \dotted" version. That is, the list decomposition theoremconstructs an isomorphism decompF ;G satisfyingdecompF ;G 2 _F __(G __F ) = _(F _+G) :Moreover, we can now use abstraction to obtain the required isomorphism between the twofunctors rather than a collection of isomorphisms between objects.19 a;b 7! a (b a)=fIntroducing the functors Exl and Exr , where Exl :(a; b) = a andExr :(a; b) = b ga;b 7! Exl :(a; b) (Exr :(a; b) Exl :(a; b))=fabstraction g( Exl) _( (Exr _( Exl)))=fabstraction: (23) g_Exl __(Exr __Exl)=ftheorem 22decompExl ;Exr g_(Exr _+Exl)=fabstraction ga;b 7! (b+a) :If full details of the de nition of decompExl ;Exr are required then we would have to instan-tiate the witnesses in the statement of (22) in the following way:a;b := Exl ;Exr ;+; ; := _+; _; _and 1 := K:1 :Simpli cation would then yield the witness obtained earlier.AcknowledgementWe are grateful to the referees for pointing out Freyd's work to us.References[1] R. C. Backhouse, M. Bijsterveld, R. van Geldrop, and J.C.S.P. van der Woude. Cate-gory theory as coherently constructive lattice theory. Department of Mathematics andComputing Science, Eindhoven University of Technology. Working document. Avail-able via world-wide web at http://www.win.tue.nl/win/cs/wp/papers, 1995.[2] R.C. Backhouse and M. Bijsterveld. Category theory as coherently constructivelattice theory: an illustration. Technical report, Department of Computing Sci-ence, Eindhoven University of Technology, 1994. Available via world-wide web athttp://www.win.tue.nl/win/cs/wp/papers.20 [3] Patrick Cousot and Radhia Cousot. Systematic design of program analysis frame-works. In Conference Record of the Sixth Annual ACM Symposium on Principles ofProgramming Languages, pages 269{282, San Antonio, Texas, January 1979.[4] E.W. Dijkstra and C.S. Scholten. Predicate Calculus and Program Semantics.Springer-Verlag, Berlin, 1990.[5] Maarten M. Fokkinga. Calculate categorically! Formal Aspects of Computing, 4:673{692, 1992.[6] Peter Freyd. Algebraically complete categories. In G. Rosolini A. Carboni, M.C. Pedic-chio, editor, Category Theory, Proceedings, Como 1990, volume 1488 of Lecture Notesin Mathematics, pages 95{104. Springer-Verlag, 1990.[7] P.J. Freyd and A. Scedrov. Categories, Allegories. North-Holland, 1990.[8] Claudio A. Hermida and Bart Jacobs. An algebraic view of structural induction.Extended Abstract, May 1994.[9] J. Lambek. A xpoint theorem for complete categories. Mathematische Zeitschrift,103:151{161, 1968.[10] J. Lambek. Least xpoints of endofunctors of cartesian closed categories. MathematicalStructures in Computer Science, 3:229{257, 1993.[11] J. Lambek and P.J. Scott. Introduction to Higher Order Categorical Logic, volume 7of Studies in Advanced Mathematics. Cambridge University Press, 1986.[12] S. Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Textsin Mathematics. Springer-Verlag, 1971.[13] D.J. Lehman and M.B. Smyth. Algebraic speci cation of data types: A syntheticapproach. Math. Syst. Theory, 14(2):97{140, 1981.[14] G. Malcolm. Algebraic data types and program transformation. PhD thesis, GroningenUniversity, 1990.[15] G. Malcolm. Data structures and program transformation. Science of ComputerProgramming, 14(2{3):255{280, October 1990.[16] E.G. Manes and M.A. Arbib. Algebraic Approaches to Program Semantics. Texts andMonographs in Computer Science. Springer-Verlag, Berlin, 1986.[17] Eindhoven University of Technology Mathematics of Program Construction Group.Fixed point calculus. Information Processing Letters, 53(3):131{136, February 1995.[18] P. Wadler. Theorems for free! In 4'th Symposium on Functional Programming Lan-guages and Computer Architecture, ACM, London, September 1989.[19] G. Winskel. ??? ???, ???21