The Category Theoretic Solution of Recursive Program Schemes

This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the category-theoretic version of the classical area of algebraic semantics. The overall assumptions needed are small indeed: working only in categories with "enough final coalgebras" we show how to formulate, solve, and study recursive program schemes. Our general theory is algebraic and so avoids using ordered, or metric structures. Our work generalizes the previous approaches which do use this extra structure by isolating the key concepts needed to study substitution in infinite trees, including second-order substitution. As special cases of our interpreted solutions we obtain the usual denotational semantics using complete partial orders, and the one using complete metric spaces. Our theory also encompasses implicitly defined objects which are not usually taken to be related to recursive program schemes. For example, the classical Cantor two-thirds set falls out as an interpreted solution (in our sense) of a recursive program scheme.

[1]  Jirí Adámek,et al.  Elgot Algebras: (Extended Abstract) , 2006, MFPS.

[2]  Jirí Adámek.  On a Description of Terminal Coalgebras and Iterative Theories , 2003, CMCS.

[3]  A. R. D. Mathias,et al.  NON‐WELL‐FOUNDED SETS (CSLI Lecture Notes 14) , 1991 .

[4]  Stefan Milius.  Completely iterative algebras and completely iterative monads , 2005, Inf. Comput..

[5]  Jirí Adámek,et al.  On the Greatest Fixed Point of a Set Functor , 1995, Theor. Comput. Sci..

[6]  Jirí Adámek,et al.  On tree coalgebras and coalgebra presentations , 2004, Theor. Comput. Sci..

[7]  Z. Ésik,et al.  Iteration Theories: The Equational Logic of Iterative Processes , 1993 .

[8]  Christoph Lüth,et al.  Dualising Initial Algebras , 2003, Math. Struct. Comput. Sci..

[9]  Bruno Courcelle,et al.  Fundamental Properties of Infinite Trees , 1983, Theor. Comput. Sci..

[10]  Peter Aczel,et al.  A Coalgebraic View of Infinite Trees and Iteration , 2001, CMCS.

[11]  Michael Barr,et al.  Terminal Coalgebras in Well-Founded Set Theory , 1993, Theor. Comput. Sci..

[12]  Stefan Milius.  On Iteratable Endofunctors , 2002, CTCS.

[13]  Jirí Adámek,et al.  Banach's Fixed-Point Theorem as a base for data-type equations , 1994, Appl. Categorical Struct..

[14]  Maurice Nivat,et al.  The metric space of infinite trees. Algebraic and topological properties , 1980, Fundam. Informaticae.

[15]  J. Adámek,et al.  Automata and Algebras in Categories , 1990 .

[16]  Stephen L. Bloom,et al.  All Solutions of a System of Recursion Equations in Infinite Trees and Other Contraction Theories , 1983, J. Comput. Syst. Sci..

[17]  C. C. Elgot.  Monadic Computation And Iterative Algebraic Theories , 1982 .

[18]  James Worrell,et al.  On the final sequence of a finitary set functor , 2005, Theor. Comput. Sci..

[19]  D. Harrison,et al.  Vicious Circles , 1995 .

[20]  Christoph Lüth,et al.  Solving Algebraic Equations Using Coalgebra , 2003, RAIRO Theor. Informatics Appl..

[21]  Irène Guessarian,et al.  Algebraic semantics , 1981, Lecture Notes in Computer Science.

[22]  Peter Aczel,et al.  Non-well-founded sets , 1988, CSLI lecture notes series.

[23]  Jirí Adámek,et al.  Free iterative theories: a coalgebraic view , 2003, Mathematical Structures in Computer Science.

[24]  Stefan Milius,et al.  Terminal coalgebras and free iterative theories , 2006, Inf. Comput..

[25]  Peter Aczel,et al.  Infinite trees and completely iterative theories: a coalgebraic view , 2003, Theor. Comput. Sci..

[26]  Lawrence S. Moss.  Uniform Functors on Sets , 2006, Essays Dedicated to Joseph A. Goguen.

[27]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[28]  Stefan Milius,et al.  On coalgebra based on classes , 2004, Theor. Comput. Sci..

[29]  Jirí Adámek,et al.  Iterative algebras at work , 2006, Mathematical Structures in Computer Science.

[30]  Pierre America,et al.  Solving Reflexive Domain Equations in a Category of Complete Metric Spaces , 1987, J. Comput. Syst. Sci..

[31]  J. van Leeuwen,et al.  Theoretical Computer Science , 2003, Lecture Notes in Computer Science.

[32]  Stefan Milius,et al.  The category-theoretic solution of recursive program schemes , 2006, Theor. Comput. Sci..

[33]  Lawrence S. Moss.  Parametric corecursion , 2001, Theor. Comput. Sci..

[34]  Ralph Matthes,et al.  Substitution in Non-wellfounded Syntax with Variable Binding , 2003, CMCS.

[35]  Christoph Lüth,et al.  Algebras, Coalgebras, Monads and Comonads , 2001, CMCS.

[36]  C. C. Elgot,et al.  On the algebraic structure of rooted trees , 1978 .

[37]  J. Lambek.  A fixpoint theorem for complete categories , 1968 .