Abstract
We introduce \(\mathbb{IH}\), a sound and complete graphical theory of vector subspaces over the field of polynomial fractions, with relational composition. The theory is constructed in modular fashion, using Lack’s approach to composing PROPs with distributive laws.
We then view string diagrams of \(\mathbb{IH}\) as generalised stream circuits by using a formal Laurent series semantics. We characterize the subtheory where circuits adhere to the classical notion of signal flow graphs, and illustrate the use of the graphical calculus on several examples.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Arbab, F.: Reo: a channel-based coordination model for component composition. Mathematical Structures in Computer Science 14, 329–366 (2004)
Baez, J.C.: Network theory (2014), http://math.ucr.edu/home/baez/networks/
Baez, J.C., Erbele, J.: Categories in control. CoRR, abs/1405.6881 (2014), http://arxiv.org/abs/1405.6881
Barnabei, M., Guerrini, C., Montefusco, L.: Some algebraic aspects of signal processing. Linear Algebra and its Applications 284(1-3), 3–17 (1998)
Basold, H., Bonsangue, M., Hansen, H.H., Rutten, J.: (co)algebraic characterizations of signal flow graphs. In: van Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds.) Panangaden Festschrift. LNCS, vol. 8464, pp. 124–145. Springer, Heidelberg (2014)
Berstel, J., Reutenauer, C.: Rational series and their languages. EATCS Monographs on Theoretical Computer Science 12 (1988)
Bonchi, F., Sobociński, P., Zanasi, F.: Interacting bialgebras are Frobenius. In: Muscholl, A. (ed.) FOSSACS 2014. LNCS, vol. 8412, pp. 351–365. Springer, Heidelberg (2014)
Bonchi, F., Sobociński, P., Zanasi, F.: Interacting Hopf algebras. CoRR, abs/1403.7048 (2014), http://arxiv.org/abs/1403.7048
Bruni, R., Lanese, I., Montanari, U.: A basic algebra of stateless connectors. Theor. Comput. Sci. 366, 98–120 (2006)
Bruni, R., Melgratti, H.C., Montanari, U., Sobociński, P.: Connector algebras for C/E and P/T nets’ interactions. Log. Meth. Comput. Sci. 9(16) (2013)
Coecke, B., Duncan, R., Kissinger, A., Wang, Q.: Strong complementarity and non-locality in categorical quantum mechanics. In: LiCS 2012, pp. 245–254 (2012)
Katis, P., Sabadini, N., Walters, R.F.C.: Span(Graph): an algebra of transition systems. In: Johnson, M. (ed.) AMAST 1997. LNCS, vol. 1349, pp. 322–336. Springer, Heidelberg (1997)
Lack, S.: Composing PROPs. Theor. App. Categories 13(9), 147–163 (2004)
Lahti, B.P.: Signal Processing and Linear Systems. Oxford University Press (1998)
Mac Lane, S.: Categorical algebra. Bull. Amer. Math. Soc. 71, 40–106 (1965)
Mason, S.J.: Feedback Theory: I. Some Properties of Signal Flow Graphs. Massachusetts Institute of Technology, Research Laboratory of Electronics (1953)
Milius, S.: A sound and complete calculus for finite stream circuits. In: LICS (2010)
Rutten, J.J.M.M.: A tutorial on coinductive stream calculus and signal flow graphs. Theor. Comput. Sci. 343(3), 443–481 (2005)
Selinger, P.: Dagger compact closed categories and completely positive maps (extended abstract). Electr. Notes Theor. Comput. Sci. 170, 139–163 (2007)
Selinger, P.: A survey of graphical languages for monoidal categories. arXiv:0908.3347v1 [math.CT] (2009)
Sobociński, P.: Representations of petri net interactions. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 554–568. Springer, Heidelberg (2010)
Willems, J.C.: The behavioural approach to open and interconnected systems. IEEE Contr. Syst. Mag. 27, 46–99 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bonchi, F., Sobociński, P., Zanasi, F. (2014). A Categorical Semantics of Signal Flow Graphs. In: Baldan, P., Gorla, D. (eds) CONCUR 2014 – Concurrency Theory. CONCUR 2014. Lecture Notes in Computer Science, vol 8704. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44584-6_30
Download citation
DOI: https://doi.org/10.1007/978-3-662-44584-6_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44583-9
Online ISBN: 978-3-662-44584-6
eBook Packages: Computer ScienceComputer Science (R0)