Hostname: page-component-77f85d65b8-g4pgd Total loading time: 0 Render date: 2026-04-18T09:55:05.040Z Has data issue: false hasContentIssue false
  • English
  • Français

Traces for coalgebraic components

Published online by Cambridge University Press:  25 March 2011

ICHIRO HASUO  and
BART JACOBS
ICHIRO HASUO
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan and PRESTO Research Promotion Program, Japan Science and Technology Agency Email: ichiro@kurims.kyoto-u.ac.jp
BART JACOBS
Affiliation:
Institute for Computing and Information Sciences, Radboud University Nijmegen, P.O.Box 9010, 6500 GL Nijmegen, the Netherlands Email: bart@cs.ru.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, state-based modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy's tube diagrams, which are an extension of standard string diagrams for monoidal categories, to represent and manipulate component diagrams. The microcosm principle then yields a canonical ‘inner’ traced monoidal structure on the category of resumptions (elements of final coalgebras/components). This generalises an observation by Abramsky, Haghverdi and Scott.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Special Issue 2: Coalgebraic Logic , April 2011 , pp. 267 - 320
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abramsky, S. (2009) Coalgebras, Chu spaces, and representations of physical systems. Available at http://arxiv.org/0910.3959.Google Scholar
Abramsky, S., Haghverdi, E. and Scott, P. (2002) Geometry of interaction and linear combinatory algebras. Mathematical Structures in Computer Science 12 (5)625–665.CrossRefGoogle Scholar
Abramsky, S. and Jagadeesan, R. (1994) Games and full completeness for multiplicative linear logic. Journ. Symb. Logic 59 (2)543–574.CrossRefGoogle Scholar
Arbab, F. (2004) Reo: a channel-based coordination model for component composition. Mathematical Structures in Computer Science 14 (3)329–366.CrossRefGoogle Scholar
Asada, K. and Hasuo, I. (2010) Categorifying computations into components via arrows as profunctors. In: Jacobs, B., Niqui, M., Rutten, J. and Silva, A. (eds.) Coalgebraic Methods in Computer Science (CMCS 2010). Electronic Notes in Theoretical Computer Science 264 25–45.CrossRefGoogle Scholar
Baez, J. C. and Dolan, J. (1998) Higher dimensional algebra III: n-categories and the algebra of opetopes. Adv. Math 135 145–206.CrossRefGoogle Scholar
Baier, C., Sirjani, M., Arbab, F. and Rutten, J. J. M. M. (2006) Modeling component connectors in Reo by constraint automata. Science of Comput. Progr. 61 (2)75–113.CrossRefGoogle Scholar
Barbosa, L. S. (2001) Components as Coalgebras, Ph.D. thesis, Univ. Minho.Google Scholar
Barbosa, L. S. (2003) Towards a calculus of state-based software components. Journ. of Universal Comp. Sci. 9 (8)891–909.Google Scholar
Barr, M. (1993) Terminal coalgebras in well-founded set theory. Theoretical Computer Science 114 (2)299–315. (Corrigendum in Theoretical Computer Science 124 189–192.)CrossRefGoogle Scholar
Borceux, F. (1994) Handbook of Categorical Algebra, Encyclopedia of Mathematics 50–52, Cambridge University Press.Google Scholar
Cockett, J. R. B. and Seely, R. A. G. (1999) Linearly distributive functors. Journal of Pure and Applied Algebra 143 (1-3) 155–203.CrossRefGoogle Scholar
Goldblatt, R. (1992) Logics of Time and Computation, second revised edition, CSLI Lecture Notes 7, Stanford.Google Scholar
Haghverdi, E. (2000) A Categorical Approach to Linear Logic, Geometry of Proofs and Full Completeness, Ph.D. thesis, University of Ottawa.Google Scholar
Hasegawa, M. (1999) Models of Sharing Graphs: A Categorical Semantics of Let and Letrec, Distinguished Dissertations in Computer Science, Cambridge University Press.CrossRefGoogle Scholar
Hasegawa, M. (2004) The uniformity principle on traced monoidal categories. Publ. RIMS, Kyoto Univ. 40 (3)991–1014.Google Scholar
Hasuo, I., Heunen, C., Jacobs, B. and Sokolova, A. (2009) Coalgebraic components in a many-sorted microcosm. In: Kurz, A. and Tarlecki, A. (eds.) Conference on Algebra and Coalgebra in Computer Science (CALCO 2009). Springer-Verlag Lecture Notes in Computer Science 5728 64–80.CrossRefGoogle Scholar
Hasuo, I., Jacobs, B. and Sokolova, A. (2007) Generic trace semantics via coinduction. Logical Methods in Comp. Sci. 3 (4:11).Google Scholar
Hasuo, I., Jacobs, B. and Sokolova, A. (2008) The microcosm principle and concurrency in coalgebra. In: Foundations of Software Science and Computation Structures. Springer-Verlag Lecture Notes in Computer Science 4962 246–260.CrossRefGoogle Scholar
Hughes, J. (2000) Generalising monads to arrows. Science of Comput. Progr. 37 (1–3)67–111.CrossRefGoogle Scholar
Jacobs, B. (1999) Categorical Logic and Type Theory, North Holland.Google Scholar
Jacobs, B. (2010) From coalgebraic to monoidal traces. In Jacobs, B., Niqui, M., Rutten, J. and Silva, A. (eds.) Coalgebraic Methods in Computer Science (CMCS 2010). Electronic Notes in Theoretical Computer Science 264 125–140.CrossRefGoogle Scholar
Jacobs, B., Heunen, C. and Hasuo, I. (2009) Categorical semantics for arrows. Journ. Funct. Progr. 19 (3-4)403–438.CrossRefGoogle Scholar
Jacobs, B. and Rutten, J. J. M. M. (1997) A tutorial on (co)algebras and (co)induction. EATCS Bulletin 62 222–259.Google Scholar
Janelidze, G. and Kelly, M. (2001) A note on actions of a monoidal category. Theory and Applications of Categories 9 (4)61–91. (Also available at www.tac.mta.ca/tac/volumes/9/n4/9-04abs.html.)Google Scholar
Joyal, A. and Street, R. (1991) The geometry of tensor calculus, I. Adv. Math 88 55–112.CrossRefGoogle Scholar
Joyal, A., Street, R. and Verity, D. (1996) Traced monoidal categories. Math. Proc. Cambridge Phil. Soc. 119 (3)425–446.CrossRefGoogle Scholar
Katis, P., Sabadini, N. and Walters, R. F. C. (1997) Bicategories of processes. Journ. of Pure and Appl. Algebra 115 (2)141–178.CrossRefGoogle Scholar
Kelly, G. M. and Street, R. (1974) Review of the elements of 2-categories. In Kelly, G. M. (ed.) Proc. Sydney Category Theory Seminar 1972/1973. Springer-Verlag Lecture Notes in Computer Science 420 75–103.Google Scholar
Klin, B. (2009) Bialgebraic methods and modal logic in structural operational semantics. Information and Computation 207 (2)237–257.CrossRefGoogle Scholar
Krstić, S., Launchbury, J. and Pavlović, D. (2001) Categories of processes enriched in final coalgebras. In: Honsell, F. and Miculan, M. (eds.) Foundations of Software Science and Computation Structures. Springer-Verlag Lecture Notes in Computer Science 2030 303–317.CrossRefGoogle Scholar
Kurz, A. and Pattinson, D. (2005) Coalgebraic modal logic of finite rank. Mathematical Structures in Computer Science 15 (3)453–473.CrossRefGoogle Scholar
MacLane, S. Lane, S. (1998) Categories for the Working Mathematician, second edition, Springer-Verlag.Google Scholar
McCurdy, M. B. (2010) The Tannaka-representation adjunction for weak bialgebras and separable Frobenius monoidal functors. (Forthcoming Ph.D. thesis, Macquarie University.)Google Scholar
Melliès, P.-A. (2006) Functorial boxes in string diagrams. In: Ésik, Z. (ed.) CSL. Springer-Verlag Lecture Notes in Computer Science 4207 1–30.CrossRefGoogle Scholar
Milner, R. (1975) Processes: a mathematical model of computing agents. In: Logic Colloq. '73, North-Holland157–173.Google Scholar
Milner, R. (1980) A Calculus of Communicating Systems. Springer-Verlag Lecture Notes in Computer Science 92.CrossRefGoogle Scholar
Moggi, E. (1991) Notions of computation and monads. Information and Computation 93 (1)55–92.CrossRefGoogle Scholar
Pattinson, D. (2003) An introduction to the theory of coalgebras. Course notes for NASSLLI.Google Scholar
Penrose, R. (1971) Applications of negative dimensional tensors. In: Welsh, D. (ed.) Combinatorial Mathematics and its Applications, Academic Press 221–244.Google Scholar
Rutten, J. J. M. M. (2000) Universal coalgebra: a theory of systems. Theoretical Computer Science 249 3–80.CrossRefGoogle Scholar
Rutten, J. J. M. M. (2006) Algebraic specification and coalgebraic synthesis of Mealy automata. Electronic Notes in Theoretical Computer Science 160 305–319.CrossRefGoogle Scholar
Selinger, P. (2011) A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics. Springer-Verlag Lecture Notes in Physics 813 289–355.CrossRefGoogle Scholar
Simpson, A. K. and Plotkin, G. D. (2000) Complete axioms for categorical fixed-point operators. In: LICS '00 Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society 30–41.Google Scholar
Sokolova, A. (2005) Coalgebraic Analysis of Probabilistic Systems, Ph.D. thesis, Techn. Univ. Eindhoven.Google Scholar
Szyperski, C. (1998) Component Software, Addison-Wesley.Google Scholar
Turi, D. and Plotkin, G. (1997) Towards a mathematical operational semantics. In: LICS '97 Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science IEEE Computer Society 280–291.Google Scholar