Feedback, trace and fixed-point semantics
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 36 (2002) no. 2, pp. 181-194.

We introduce a notion of category with feedback-with-delay, closely related to the notion of traced monoidal category, and show that the Circ construction of [15] is the free category with feedback on a symmetric monoidal category. Combining with the Int construction of Joyal et al. [12] we obtain a description of the free compact closed category on a symmetric monoidal category. We thus obtain a categorical analogue of the classical localization of a ring with respect to a multiplicative subset. In this context we define a notion of fixed-point semantics of a category with feedback which is seen to include a variety of classical semantics in computer science.

DOI : https://doi.org/10.1051/ita:2002009
Classification : 68Q55,  68Q70,  18D10
@article{ITA_2002__36_2_181_0,
     author = {Katis, P. and Sabadini, Nicoletta and Walters, Robert F. C.},
     title = {Feedback, trace and fixed-point semantics},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {181--194},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {2},
     year = {2002},
     doi = {10.1051/ita:2002009},
     zbl = {1050.68100},
     mrnumber = {1948768},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita:2002009/}
}
Katis, P.; Sabadini, Nicoletta; Walters, Robert F. C. Feedback, trace and fixed-point semantics. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 36 (2002) no. 2, pp. 181-194. doi : 10.1051/ita:2002009. http://www.numdam.org/articles/10.1051/ita:2002009/

[1] S. Abramsky, Retracing some paths in process algebras, Concur '96, edited by U. Montanari and V. Sassone. Springer-Verlag, Lecture Notes in Comput. Sci. 1119 (1996) 1-17

[2] M. Bartha, An algebraic model of synchronous systems. Inform. and Comput. 97 (1992) 97-131. | MR 1154208 | Zbl 0768.68098

[3] L. Bernatsky and Z. Ésik, Sematics of flowchart programs and the free Conway theories. RAIRO: Theoret. Informatics Applic. 32 (1998) 35-78. | Numdam | MR 1657511

[4] S.L. Bloom and Z. Ésik, Axiomatising schemes and their behaviours. J. Comput. System Sci. 31 (1985) 375-393. | MR 835132 | Zbl 0613.68013

[5] S.L. Bloom and Z. Ésik, Iteration Theories: The Equational Logic of Iterative Processes. Springer-Verlag, EATCS Monogr. Theoret. Comput. Sci. (1993). | MR 1295433 | Zbl 0773.03033

[6] S.L. Bloom and Z. Ésik, Matrix and matricial iteration theories, Part I. J. Comput. System Sci. 46 (1993) 381-408. | MR 1228813 | Zbl 0791.08006

[7] S.L. Bloom, N. Sabadini and R.F.C. Walters, Matrices, machines and behaviors. Appl. Categorical Structures 4 (1996) 343-360. | MR 1421175 | Zbl 0859.18005

[8] R.F. Blute, J.R.B. Cockett and R.A.G. Seely, Feedback for linearly distributive categories: Traces and fixpoints. J. Pure Appl. Algebra 154 (2000) 27-69. | MR 1787590 | Zbl 0964.03063

[9] A. Carboni and R.F.C. Walters, Cartesian Bicategories I. J. Pure Appl. Algebra 49 (1987) 11-32. | MR 920513 | Zbl 0637.18003

[10] J. Conway, Regular Algebra and Finite Machines. Chapman and Hall, London (1971). | Zbl 0231.94041

[11] C.C. Elgot, Monadic computation and iterative algebraic theories, edited by J.C. Shepherdson. North Holland, Amsterdam, Logic Colloquium 1973, Studies in Logic 80 (1975). | MR 413584 | Zbl 0327.02040

[12] C.C. Elgot, Matricial Theories. J. Algebra 42 (1976) 391-421. | MR 430017 | Zbl 0361.18004

[13] F. Gadducci, U. Montanari, P. Katis, N. Sabadini and R.F.C. Walters, Comparing Cospan-spans and Tiles via a Hoare-style process calculus. TOSCA Udine, Electron. Notes Theoret. Comput. Sci. 62 (2001) 152-171.

[14] M. Hasegawa, Models of Sharing Graphs: A categorical semantics of let and letrec, Ph.D. Thesis. Edinburgh (1997), Springer (1999). | MR 1696494 | Zbl 0943.68109

[15] A. Joyal, R. Street and D. Verity, Traced monoidal categories. Math. Proc. Camb. Phil. Soc. 119 (1996) 447-468. | MR 1357057 | Zbl 0845.18005

[16] A. Joyal and R. Street, Braided tensor categories. Adv. in Math. 102 (1993) 20-78. | MR 1250465 | Zbl 0817.18007

[17] W. Khalil and R.F.C. Walters, An imperative language based on distributive categories II. RAIRO: Theoret. Informatics Appl. 27 (1993) 503-522. | Numdam | MR 1258750 | Zbl 0806.18006

[18] P. Katis, N. Sabadini and R.F.C. Walters, Bicategories of processes. J. Pure Appl. Algebra 115 (1997) 141-178. | MR 1431159 | Zbl 0933.18008

[19] P. Katis, N. Sabadini and R.F.C. Walters, Span(Graph): A categorical algebra of transition systems, in Proc. Algebraic Methodology and Software Technology. Springer-Verlag, Lecture Notes in Comput. Sci. 1349 (1997) 307-321. | Zbl 0885.18004

[20] P. Katis, N. Sabadini and R.F.C. Walters, On the algebra of systems with feedback and boundary. Rend. Circ. Mat. Palermo (2) Suppl. 63 (2000) 123-156. | MR 1785817 | Zbl 1003.94051

[21] P. Katis, N. Sabadini and R.F.C. Walters, A formalization of the IWIM Model, in Proc. COORDINATION 2000, edited by A. Porto and G.-C. Roman. Springer-Verlag, Lecture Notes in Comput. Sci. 1906 (2000) 267-283.

[22] P. Katis, N. Sabadini and R.F.C. Walters, Recursion and concurrency, Invited talk, FICS 2001. Florence (2001).

[23] P. Katis and R.F.C. Walters, The compact closed bicategory of left adjoints. Math. Proc. Camb. Phil. Soc. 130 (2001) 77-87. | MR 1797732 | Zbl 0980.18002

[24] G.M. Kelly and M. Laplaza, Coherence for compact closed categories. J. Pure Appl. Algebra 19 (1980) 193-213. | MR 593254 | Zbl 0447.18005

[25] K. Krohn and J. Rhodes, Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines. Trans. Amer. Math. Soc. 116 (1965) 450-464. | MR 188316 | Zbl 0148.01002

[26] M. Nagata, Local rings. Interscience (1962). | MR 155856 | Zbl 0123.03402

[27] R. Penrose, Applications of negative dimensional torsors, edited by D.J.A. Welsh. Academic Press, New York, Comb. Math. Appl. (1971) 221-244. | MR 281657 | Zbl 0216.43502

[28] W.J. Rugh, Linear System Theory, Second Edition. Prentice Hall (1996). | MR 1211190 | Zbl 0892.93002

[29] J.J.M.M. Rutten, A calculus of transition systems (towards universal coalgebra), in Modal Logic and Process Algebra, a bisimulation perspective, edited by A. Ponse, M. de Rijke and Y. Venema. CSLI Publications, Standford, CSLI Lecture Notes 53 (1995) 231-256. | MR 1375710

[30] N. Sabadini, S. Vigna and R.F.C. Walters, A note on recursive functions. Math. Struct. Comput. Sci. 6 (1996) 127-139. | MR 1386615 | Zbl 0844.03025

[31] M.W. Shields, An introduction to Automata Theory. Blackwell Scientic Publications, Oxford (1987).

[32] A. Simpson and G. Plotkin, Complete axioms for categorical fixed-point operators, in Proc. 15th LICS (2000) 30-41. | MR 1946083

[33] Gh. Stefanescu, On flowchart theories I: The deterministic case. J. Comput. System Sci. 35 (1985) 163-191. | MR 910211 | Zbl 0628.68018

[34] G. Stefanescu, Network Algebra. Springer-Verlag (2000). | MR 1753209 | Zbl 0956.68002

[35] R.F.C. Walters, Categories and Computer Science. Carslaw Publications (1991), Cambridge University Press (1992). | MR 1204658 | Zbl 0789.18001