Reversible causal graph dynamics: invertibility, block representation, vertex-preservation

Causal Graph Dynamics extend Cellular Automata to arbitrary time-varying graphs of bounded degree. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We add a further physics-like symmetry, namely reversibility. In particular, we extend two fundamental results on reversible cellular automata, by proving that the inverse of a causal graph dynamics is a causal graph dynamics, and that these reversible causal graph dynamics can be represented as finite-depth circuits of local reversible gates. We also show that reversible causal graph dynamics preserve the size of all but a finite number of graphs.

[1]  Eric Rémila,et al.  Hyperbolic Recognition by Graph Automata , 2002, ICALP.

[2]  Vincent Nesme,et al.  Cellular automata over generalized Cayley graphs , 2017, Mathematical Structures in Computer Science.

[3]  Gilles Dowek,et al.  Causal Graph Dynamics , 2012, ICALP.

[4]  Simon Perdrix,et al.  Block Representation of Reversible Causal Graph Dynamics , 2015, FCT.

[5]  Michael Löwe,et al.  Algebraic Approach to Single-Pushout Graph Transformation , 1993, Theor. Comput. Sci..

[6]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[7]  G. A. Hedlund.  Endomorphisms and automorphisms of the shift dynamical system , 1969, Mathematical systems theory.

[8]  Hartmut Ehrig,et al.  Parallel and Distributed Derivations in the Single-Pushout Approach , 1993, Theor. Comput. Sci..

[9]  Jérôme Olivier Durand-Lose,et al.  Representing Reversible Cellular Automata with Reversible Block Cellular Automata , 2001, DM-CCG.

[10]  Jérémie Chalopin,et al.  Deterministic Symmetric Rendezvous in Arbitrary Graphs: Overcoming Anonymity, Failures and Uncertainty , 2013 .

[11]  Misha Gromov,et al.  Endomorphisms of symbolic algebraic varieties , 1999 .

[12]  J. Kari.  Representation of reversible cellular automata with block permutations , 1996, Mathematical systems theory.

[13]  Vincent Nesme,et al.  Unitarity plus causality implies localizability , 2007, J. Comput. Syst. Sci..

[14]  Annegret Habel,et al.  Amalgamation of Graph Transformations: A Synchronization Mechanism , 1987, J. Comput. Syst. Sci..

[15]  Vincent Nesme,et al.  Generalized Cayley Graphs and Cellular Automata over them , 2012, ArXiv.

[16]  R. Sorkin.  Time-evolution problem in regge calculus , 1975 .

[17]  Gabriele Taentzer,et al.  Parallel High-Level Replacement Systems , 1997, Theor. Comput. Sci..

[18]  K. Tomita,et al.  Graph automata: natural expression of self-reproduction , 2002 .

[19]  Cosimo Laneve,et al.  Formal molecular biology , 2004, Theor. Comput. Sci..

[20]  Vincent Nesme,et al.  A simple block representation of reversible cellular automata with time-symmetry , 2012, ArXiv.

[21]  Luidnel Maignan,et al.  Global Graph Transformations , 2015, GCM@ICGT.

[22]  David A. Meyer,et al.  Modeling dynamical geometry with lattice gas automata , 1998 .

[23]  D. Meyer,et al.  Lattice gas simulations of dynamical geometry in two dimensions. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Simone Severini,et al.  Quantum bose-hubbard model with an evolving graph as a toy model for emergent spacetime , 2009, 0911.5075.

[25]  Gabriele Taentzer,et al.  Parallel and distributed graph transformation - formal description and application to communication-based systems , 1996, Berichte aus der Informatik.

[26]  Jarkko Kari,et al.  On the Circuit Depth of Structurally Reversible Cellular Automata , 1999, Fundam. Informaticae.

[27]  Simon Perdrix,et al.  Reversible Causal Graph Dynamics , 2015, RC.