Full Abstraction for Signal Flow Graphs

Network theory uses the string diagrammatic language of monoidal categories to study graphical structures formally, eschewing specialised translations into intermediate formalisms. Recently, there has been a concerted research focus on developing a network theoretic approach to signal flow graphs, which are classical structures in control theory, signal processing and a cornerstone in the study of feedback. In this approach, signal flow graphs are given a relational denotational semantics in terms of formal power series. Thus far, the operational behaviour of such signal flow graphs has only been discussed at an intuitive level. In this paper we equip them with a structural operational semantics. As is typically the case, the purely operational picture is too concrete -- two graphs that are denotationally equal may exhibit different operational behaviour. We classify the ways in which this can occur and show that any graph can be realised -- rewritten, using the graphical theory, into an executable form where the operational behavior and the denotation coincides.

[1]  A. Joyal,et al.  The geometry of tensor calculus, I , 1991 .

[2]  Jan J. M. M. Rutten,et al.  A tutorial on coinductive stream calculus and signal flow graphs , 2005, Theor. Comput. Sci..

[3]  P. Selinger.  A Survey of Graphical Languages for Monoidal Categories , 2009, 0908.3347.

[4]  Pawel Sobocinski,et al.  Nets, Relations and Linking Diagrams , 2013, CALCO.

[5]  John C. Baez,et al.  Categories in Control , 2014, 1405.6881.

[6]  Samson Abramsky,et al.  What are the fundamental structures of concurrency? We still don't know! , 2006, APC 25.

[7]  Herbert J. Carlin,et al.  Network theory , 1964 .

[8]  Samuel J. Mason,et al.  Feedback Theory-Some Properties of Signal Flow Graphs , 1953, Proceedings of the IRE.

[9]  Dusko Pavlovic,et al.  Monoidal computer I: Basic computability by string diagrams , 2012, Inf. Comput..

[10]  Roberto Bruni,et al.  On Hierarchical Graphs: Reconciling Bigraphs, Gs-monoidal Theories and Gs-graphs , 2014, Fundam. Informaticae.

[11]  Filippo Bonchi,et al.  Interacting Hopf Algebras , 2014, ArXiv.

[12]  Bob Coecke,et al.  Interacting Quantum Observables , 2008, ICALP.

[13]  Nicoletta Sabadini,et al.  Span(Graph): A Categorial Algebra of Transition Systems , 1997, AMAST.

[14]  G. M. Kelly,et al.  Coherence for compact closed categories , 1980 .

[15]  Marcello M. Bonsangue,et al.  (Co)Algebraic Characterizations of Signal Flow Graphs , 2014, Horizons of the Mind.

[16]  Yves Lafont,et al.  Towards an algebraic theory of Boolean circuits , 2003 .

[17]  Stephen P. Borgatti,et al.  Network Theory , 2013 .

[18]  Marcelo P. Fiore,et al.  The Algebra of Directed Acyclic Graphs , 2013, Computation, Logic, Games, and Quantum Foundations.

[19]  Jamie Vicary,et al.  Abstract structure of unitary oracles for quantum algorithms , 2014, QPL.

[20]  Aleks Kissinger,et al.  Strong Complementarity and Non-locality in Categorical Quantum Mechanics , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[21]  B. Fong,et al.  Transfer of status qualifying dissertation : a compositional approach to control theory , 2013 .

[22]  Dusko Pavlovic,et al.  Monoidal computer II: Normal complexity by string diagrams , 2014, ArXiv.

[23]  Filippo Bonchi,et al.  A Categorical Semantics of Signal Flow Graphs , 2014, CONCUR.

[24]  J. Willems.  The Behavioral Approach to Open and Interconnected Systems , 2007, IEEE Control Systems.

[25]  A. Carboni,et al.  Cartesian bicategories I , 1987 .

[26]  Roberto Bruni,et al.  A basic algebra of stateless connectors , 2006, Theor. Comput. Sci..

[27]  S. Maclane,et al.  Categorical Algebra , 2007 .

[28]  Nicoletta Sabadini,et al.  Representing Place/Transition Nets in Span(Graph) , 1997, AMAST.

[29]  Dan R. Ghica,et al.  Diagrammatic Reasoning for Delay-Insensitive Asynchronous Circuits , 2013, Computation, Logic, Games, and Quantum Foundations.

[30]  B. P. Lathi.  Signal Processing And Linear Systems , 1998 .

[31]  Filippo Bonchi,et al.  Interacting Bialgebras Are Frobenius , 2014, FoSSaCS.