Contextual Equivalence for Signal Flow Graphs

We extend the signal flow calculus—a compositional account of the classical signal flow graph model of computation—to encompass affine behaviour, and furnish it with a novel operational semantics. The increased expressive power allows us to define a canonical notion of contextual equivalence, which we show to coincide with denotational equality. Finally, we characterise the realisable fragment of the calculus: those terms that express the computations of (affine) signal flow graphs.

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

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

[3]  Dan R. Ghica,et al.  Categorical semantics of digital circuits , 2016, 2016 Formal Methods in Computer-Aided Design (FMCAD).

[4]  Nobuko Yoshida,et al.  On Reduction-Based Process Semantics , 1995, Theor. Comput. Sci..

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

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

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

[8]  James H. Morris,et al.  Lambda-calculus models of programming languages. , 1969 .

[9]  Robin Milner,et al.  A Calculus of Communicating Systems , 1980, Lecture Notes in Computer Science.

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

[11]  Rocco De Nicola,et al.  Testing Equivalences for Processes , 1984, Theor. Comput. Sci..

[12]  Filippo Bonchi,et al.  The Calculus of Signal Flow Diagrams I: Linear relations on streams , 2017, Inf. Comput..

[13]  Dan R. Ghica,et al.  A structural and nominal syntax for diagrams , 2017, QPL.

[14]  Filippo Bonchi,et al.  Graphical Affine Algebra , 2019, 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[15]  Aleks Kissinger,et al.  Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning , 2017 .

[16]  Filippo Bonchi,et al.  Full Abstraction for Signal Flow Graphs , 2015, POPL.

[17]  Jan J. M. M. Rutten.  Rational Streams Coalgebraically , 2008, Log. Methods Comput. Sci..

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

[19]  Filippo Bonchi,et al.  Diagrammatic algebra: from linear to concurrent systems , 2019, Proc. ACM Program. Lang..

[20]  Stefan Milius.  A Sound and Complete Calculus for Finite Stream Circuits , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[21]  Brendan Fong,et al.  Corelations are the prop for extraspecial commutative Frobenius monoids , 2016, 1601.02307.

[22]  Joachim Kock,et al.  Frobenius Algebras and 2-D Topological Quantum Field Theories , 2004 .

[23]  C. A. R. Hoare,et al.  Communicating sequential processes , 1978, CACM.

[24]  Fabio Zanasi,et al.  The Algebra of Partial Equivalence Relations , 2016, MFPS.

[25]  Fabio Zanasi,et al.  Interacting Hopf Algebras: the theory of linear systems , 2018, ArXiv.

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

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

[28]  Roberto Bruni,et al.  Some algebraic laws for spans , 2001, Electron. Notes Theor. Comput. Sci..

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

[30]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[31]  Robin Milner,et al.  Barbed Bisimulation , 1992, ICALP.

[32]  Dominic R. Verity,et al.  ∞-Categories for the Working Mathematician , 2018 .

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

[34]  BonchiFilippo,et al.  Full Abstraction for Signal Flow Graphs , 2015 .

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

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

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