Traced premonoidal categories

Nick Benton; Martin Hyland

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2003)

  • Volume: 37, Issue: 4, page 273-299
  • ISSN: 0988-3754

Abstract

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Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a well-known theorem relating traces and Conway operators in cartesian categories.

How to cite

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Benton, Nick, and Hyland, Martin. "Traced premonoidal categories." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 37.4 (2003): 273-299. <http://eudml.org/doc/246065>.

@article{Benton2003,
abstract = {Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a well-known theorem relating traces and Conway operators in cartesian categories.},
author = {Benton, Nick, Hyland, Martin},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {traces; fixed point operators; premonoidal categories; recursion; monads; Freyd category; Conway operators in Cartesian categories},
language = {eng},
number = {4},
pages = {273-299},
publisher = {EDP-Sciences},
title = {Traced premonoidal categories},
url = {http://eudml.org/doc/246065},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Benton, Nick
AU - Hyland, Martin
TI - Traced premonoidal categories
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2003
PB - EDP-Sciences
VL - 37
IS - 4
SP - 273
EP - 299
AB - Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a well-known theorem relating traces and Conway operators in cartesian categories.
LA - eng
KW - traces; fixed point operators; premonoidal categories; recursion; monads; Freyd category; Conway operators in Cartesian categories
UR - http://eudml.org/doc/246065
ER -

References

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  1. [1] R.S. Bird, Using circular programs to eliminate multiple traversals of data. Acta Informatica 21 (1984) 239-250. Zbl0551.68017
  2. [2] P. Bjesse, K. Claessen, M. Sheeran and S. Singh, Lava: Hardware design in Haskell, in International Conference on Functional Programming (1998). 
  3. [3] S.L. Bloom and Z. Ésik, Axiomatizing schemes and their behaviors. J. Comput. Syst. Sci. 31 (1985) 375-393. Zbl0613.68013MR835132
  4. [4] S.L. Bloom and Z. Ésik, Iteration Theories. EATCS Monographs on Theoretical Computer Science, Springer-Verlag (1993). Zbl0773.03033MR1295433
  5. [5] V.E. Cazanescu and Gh. Stefanescu, A general result on abstract flowchart schemes with applications to the study of accessibility, reduction and minimization. Theor. Comput. Sci. 99 (1992) 1-63. Zbl0770.68037MR1168009
  6. [6] K. Claessen and D. Sands, Observable sharing for functional circuit description, in Asian Computing Science Conference (1999). 
  7. [7] J.H. Conway, Regular Algebra and Finite Machines. Chapman Hall (1971). Zbl0231.94041
  8. [8] R.L. Crole and A.M. Pitts, New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic. Inf. Comput. 98 (1992) 171-210. Zbl0763.03031MR1167030
  9. [9] L. Erkök, Value Recursion in Monadic Computations. Ph.D. thesis, Oregon Graduate Institute, OHSU (October 2002). Zbl1011.68017
  10. [10] L. Erkök, J. Launchbury and A. Moran, Semantics of value recursion for monadic input/output. RAIRO Theoret. Informatics Appl. 36 (2002) 155-180. Zbl1011.68017MR1948767
  11. [11] Z. Ésik, Axiomatizing iteration categories. Acta Cybernet. 14 (1999). Zbl0959.68059MR1682014
  12. [12] Z. Ésik, Group axioms for iteration. Inf. Comput. 148 (1999) 131-180 Zbl0924.68143MR1674307
  13. [13] D.P. Friedman and A. Sabry, Recursion is a computational effect. Technical Report 546, Computer Science Department, Indiana University (December 2000). 
  14. [14] C. Fuhrmann, A. Bucalo and A. Simpson, An equational notion of lifting monad. Theor. Comput. Sci. 294 (2003) 31-60. Zbl1022.18003MR1963656
  15. [15] J.-Y. Girard, Towards a geometry of interaction, in Categories in Computer Science and Logic, edited by J.W. Gray and A. Scedrov. Contemp. Math. 92 (1989) 69-108. Zbl0672.03039MR1003197
  16. [16] M. Hasegawa, Models of Sharing Graphs (A Categorical Semantics of Let and Letrec). Distinguished Dissertations in Computer Science, Springer-Verlag (1999). Zbl0943.68109MR1696494
  17. [17] M. Hasegawa, The uniformity principle on traced monoidal categories, edited by R. Blute and P. Selinger. Elsevier, Electronic Notes in Theor. Comput. Sci. (2003). Zbl1270.18019MR2074708
  18. [18] J. Hughes, Generalising monads to arrows. Sci. Comput. Program. 37 (2000) 67-112. Zbl0954.68034MR1766319
  19. [19] M. Hyland and A.J. Power, Symmetric monoidal sketches, in Proc. of the 2nd Conference on Principles and Practice of declarative Programming (2000) 280-288. Zbl1271.68185
  20. [20] A. Jeffrey, Premondoidal categories and a graphical view of programs. http://www.cogs.susx.ac.uk/users/alanje/premon/ (June 1998). 
  21. [21] A. Joyal, R. Street and D. Verity, Traced monoidal categories. Math. Proc. Cambridge Philoso. Soc. 119 (1996). Zbl0845.18005MR1357057
  22. [22] J. Launchbury and L. Erkök, Recursive monadic bindings, in International Conference on Functional Programming (2000). Zbl1321.68151
  23. [23] J. Launchbury, J.R. Lewis and B. Cook, On embedding a microarchitectural design language within Haskell. International Conference on Functional Programming (1999). 
  24. [24] J. Launchbury and S.L. Peyton Jones, State in Haskell. Lisp and Symbolic Computation 8 (1995) 293-341. 
  25. [25] E. Moggi, Notions of computation and monads. Inf. Comput. 93 (1991) 55-92. Zbl0723.68073MR1115262
  26. [26] E. Moggi and A. Sabry, An abstract monadic semantics for value recursion, in Proc. of the 2003 Workshop on Fixed Points in Computer Science (April 2003). Zbl1089.68029MR2098196
  27. [27] P.S. Mulry, Strong monads, algebras and fixed points, in Applications of Categories in Computer Science, edited by M.P. Fourman, P.T. Johnstone and A.M. Pitts. LMS Lecture Notes 177 (1992) 202-216. Zbl0797.18005MR1176965
  28. [28] R. Paterson, A new notation for arrows, in Proc. of the International Conference on Functional Programming. ACM Press (September 2001). Zbl1323.68147
  29. [29] R. Paterson, Arrows and computation, in The Fun of Programming, edited by J. Gibbons and O. de Moor. Palgrave (2003) 201-222. 
  30. [30] A.J. Power and E.P. Robinson, Premonoidal categories and notions of computation. Math. Struct. Comput. Sci. 7 (1997) 453-468. Zbl0897.18002MR1486319
  31. [31] A.J. Power and H. Thielecke, Closed Freyd and κ -categories. International Conference on Automata, Languages and Programming (1999). Zbl0938.03026MR1731522
  32. [32] A.K. Simpson and G.D. Plotkin, Complete axioms for categorical fixed-point operators, in Proc. of 15th Annual Symposium on Logic in Computer Science. IEEE Computer Society (2000). MR1946083
  33. [33] P. Wadler, The essence of functional programming, in Proc. of the 19th Symposium on Principles of Programming Languages. ACM (1992). 

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