String Diagram Rewrite Theory III: Confluence with and without Frobenius

In this paper we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorically as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewrite systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure by critical pair analysis. For the latter, we introduce the new notion of path joinability for critical pairs, which enables finitely many joins of a critical pair to be lifted to an arbitrary context in spite of the strong non-local constraints placed on rewriting in a generic symmetric monoidal theory.

[1]  Hartmut Ehrig,et al.  Parallelism of Manipulations in Multidimensional Information Structures , 1976, MFCS.

[2]  Fabio Gadducci,et al.  Confluence of Graph Rewriting with Interfaces , 2017, ESOP.

[3]  Hartmut Ehrig,et al.  Introduction to the Algebraic Theory of Graph Grammars (A Survey) , 1978, Graph-Grammars and Their Application to Computer Science and Biology.

[4]  Pawel Sobocinski,et al.  A Programming Language for Spatial Distribution of Net Systems , 2014, Petri Nets.

[5]  Fabio Gadducci,et al.  String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure , 2021, ArXiv.

[6]  Hartmut Ehrig,et al.  Local Confluence for Rules with Nested Application Conditions , 2010, ICGT.

[7]  Roberto Bruni,et al.  A Connector Algebra for P/T Nets Interactions , 2011, CONCUR.

[8]  Annegret Habel,et al.  Double-pushout graph transformation revisited , 2001, Mathematical Structures in Computer Science.

[9]  Detlef Plump,et al.  Confluence up to Garbage , 2020, ICGT.

[10]  Andrea Corradini.  On the Definition of Parallel Independence in the Algebraic Approaches to Graph Transformation , 2016, STAF Workshops.

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

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

[13]  Fabio Gadducci,et al.  Adhesivity Is Not Enough: Local Church-Rosser Revisited , 2011, MFCS.

[14]  Fabio Gadducci,et al.  Synthesising CCS bisimulation using graph rewriting , 2009, Inf. Comput..

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

[16]  Gérard Huet,et al.  On the Uniform Halting Problem for Term Rewriting Systems , 1978 .

[17]  Pawel Sobocinski,et al.  Adhesive and quasiadhesive categories , 2005, RAIRO Theor. Informatics Appl..

[18]  Leen Lambers,et al.  Initial Conflicts for Transformation Rules with Nested Application Conditions , 2020, ICGT.

[19]  Barbara König,et al.  Conditional Reactive Systems , 2011, FSTTCS.

[20]  Aleks Kissinger,et al.  Quantomatic: A proof assistant for diagrammatic reasoning , 2015, CADE.

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

[22]  Pawel Sobocinski.  Deriving process congruences from reaction rules , 2004 .

[23]  Fabio Gadducci,et al.  An inductive view of graph transformation , 1997, WADT.

[24]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

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

[26]  Hartmut Ehrig,et al.  Deriving Bisimulation Congruences in the DPO Approach to Graph Rewriting , 2004, FoSSaCS.

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

[28]  Samuel Mimram,et al.  Towards 3-Dimensional Rewriting Theory , 2014, Log. Methods Comput. Sci..

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

[30]  John Power,et al.  The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads , 2007, Computation, Meaning, and Logic.

[31]  Friedrich Otto,et al.  Finite complete rewriting systems and the complexity of the word problem , 1984, Acta Informatica.

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

[33]  Detlef Plump,et al.  Hypergraph rewriting: critical pairs and undecidability of confluence , 1993 .

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

[35]  Andrea Corradini,et al.  On the Essence of Parallel Independence for the Double-Pushout and Sesqui-Pushout Approaches , 2018, Graph Transformation, Specifications, and Nets.

[36]  Peter Padawitz,et al.  New Results on Completeness and Consistency of Abstract Data Types , 1980, MFCS.

[37]  Detlef Plump.  Checking Graph-Transformation Systems for Confluence , 2010, Electron. Commun. Eur. Assoc. Softw. Sci. Technol..

[38]  Paolo Rapisarda,et al.  A categorical approach to open and interconnected dynamical systems , 2015, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[39]  Paliath Narendran,et al.  On Ground-Confluence of Term Rewriting Systems , 1990, Inf. Comput..

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

[41]  Fabio Gadducci,et al.  String Diagram Rewrite Theory I: Rewriting with Frobenius Structure , 2020, ArXiv.

[42]  Vladimiro Sassone,et al.  Reactive systems over cospans , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[43]  Fabio Gadducci.  Graph rewriting for the pi-calculus , 2007, Math. Struct. Comput. Sci..

[44]  Fabio Gadducci,et al.  Rewriting modulo symmetric monoidal structure , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

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

[46]  Samuel Mimram,et al.  Computing Critical Pairs in 2-Dimensional Rewriting Systems , 2010, RTA.

[47]  Hartmut Ehrig,et al.  Adhesive High-Level Replacement Categories and Systems , 2004, ICGT.