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

- 6 Downloads

## Abstract

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.

## Keywords

Bijective Invertible Injective Surjective One-to-one Onto Cayley graphs Hedlund Block representation Lattice-gas automaton Reversible cellular automata## Notes

### Acknowledgements

This work has been funded by the ANR-12-BS02-007-01 TARMAC grant, the ANR-10-JCJC-0208 CausaQ grant, and the John Templeton Foundation, grant ID 15619, the STICAmSud project 16STIC05 FoQCoSS. The authors acknowledge enlightening discussions with Gilles Dowek, Emmanuel Jeandel and Bruno Martin. This work has been partially done when PA was delegated at Inria Nancy Grand Est, in the project team Carte.

## References

- Arrighi P, Dowek G (2012) Causal graph dynamics. In: Proceedings of ICALP 2012, Warwick, July 2012, LNCS, vol 7392, pp 54–66CrossRefGoogle Scholar
- Arrighi P, Dowek G (2013) Causal graph dynamics (long version). Inf Comput 223:78–93CrossRefGoogle Scholar
- Arrighi P, Martiel S (2012) Generalized Cayley graphs and cellular automata over them. In: Proceedings of GCM 2012, Bremen, September 2012. Pre-print. arXiv:1212.0027, pp 129–143Google Scholar
- Arrighi P, Nesme V (2011) A simple block representation of Reversible Cellular Automata with time-symmetry. In: 17th international workshop on cellular automata and discrete complex systems, (AUTOMATA 2011), Santiago de Chile, November 2011Google Scholar
- Arrighi P, Martiel S, Nesme V (2018) Cellular automata over generalized Cayley graphs. Math Struct Comput Sci 18:340–383 arXiv:1212.0027MathSciNetCrossRefGoogle Scholar
- Arrighi P, Martiel S, Perdrix S (2015) Block representation of reversible causal graph dynamics. In: Proceedings of FCT 2015, Gdansk, Poland, August 2015, Springer, pp 351–363Google Scholar
- Arrighi P, Martiel S, Perdrix S (2016) Reversible causal graph dynamics. In: Proceedings of International Conference on Reversible Computation, RC 2016, Bologna, Italy, July 2016, LNCS 9720, Springer, pp 73–88Google Scholar
- Arrighi P, Nesme V, Werner R (2010) Unitarity plus causality implies localizability. J Comput Syst Sci 77:372–378 QIP 2010 (long talk)MathSciNetCrossRefGoogle Scholar
- Bennett CH (1973) Logical reversibility of computation. IBM J Res Dev 17(6):525–532MathSciNetCrossRefGoogle Scholar
- Boehm P, Fonio HR, Habel A (1987) Amalgamation of graph transformations: a synchronization mechanism. J Comput Syst Sci 34(2–3):377–408MathSciNetCrossRefGoogle Scholar
- Chalopin J, Das S, Widmayer P (2013) Deterministic symmetric rendezvous in arbitrary graphs: overcoming anonymity, failures and uncertainty. In: Search theory, Springer, pp 175–195CrossRefGoogle Scholar
- Danos V, Laneve C (2004) Formal molecular biology. Theor Comput Sci 325(1):69–110MathSciNetCrossRefGoogle Scholar
- Durand-Lose J (2001) Representing reversible cellular automata with reversible block cellular automata. Discret Math Theor Comput Sci 145:154zbMATHGoogle Scholar
- Ehrig H, Lowe M (1993) Parallel and distributed derivations in the single-pushout approach. Theor Comput Sci 109(1–2):123–143MathSciNetCrossRefGoogle Scholar
- Gromov M (1999) Endomorphisms of symbolic algebraic varieties. J Eur Math Soc 1(2):109–197MathSciNetCrossRefGoogle Scholar
- Hamma A, Markopoulou F, Lloyd S, Caravelli F, Severini S, Markstrom K, Brouder C, Mestre Â, JAD F.P, Burinskii A, et al (2009) A quantum Bose-Hubbard model with evolving graph as toy model for emergent spacetime. Arxiv preprint arXiv:0911.5075Google Scholar
- Hasslacher B, Meyer DA (1998) Modelling dynamical geometry with lattice gas automata. In: Expanded version of a talk presented at the seventh international conference on the discrete simulation of fluids held at the University of OxfordGoogle Scholar
- Hedlund GA (1969) Endomorphisms and automorphisms of the shift dynamical system. Math Syst Theory 3:320–375MathSciNetCrossRefGoogle Scholar
- Kari J (1991) Reversibility of 2D cellular automata is undecidable. In: Cellular automata: theory and experiment, vol 45. MIT Press, pp 379–385Google Scholar
- Kari J (1996) Representation of reversible cellular automata with block permutations. Theory Comput Syst 29(1):47–61MathSciNetzbMATHGoogle Scholar
- Kari J (1999) On the circuit depth of structurally reversible cellular automata. Fundam Inf 38(1–2):93–107MathSciNetzbMATHGoogle Scholar
- Klales A, Cianci D, Needell Z, Meyer DA, Love PJ (2010) Lattice gas simulations of dynamical geometry in two dimensions. Phys Rev E 82(4):046705MathSciNetCrossRefGoogle Scholar
- Löwe M (1993) Algebraic approach to single-pushout graph transformation. Theor Comput Sci 109(1–2):181–224MathSciNetCrossRefGoogle Scholar
- Maignan L, Spicher A (2015) Global graph transformations. In: Proceedings of the 6th international workshop on graph computation models, L’Aquila, Italy, July 20, 2015, pp 34–49Google Scholar
- Papazian C, Remila E (2002) Hyperbolic recognition by graph automata. In: Automata, languages and programming: 29th international colloquium, ICALP 2002, Málaga, Spain, July 8-13, 2002: proceedings, vol 2380. Springer, pp 330Google Scholar
- Sorkin R (1975) Time-evolution problem in Regge calculus. Phys Rev D 12(2):385–396MathSciNetCrossRefGoogle Scholar
- Taentzer G (1996) Parallel and distributed graph transformation: formal description and application to communication-based systems. PhD thesis, Technische Universitat BerlinGoogle Scholar
- Taentzer G (1997) Parallel high-level replacement systems. Theor Comput Sci 186(1–2):43–81MathSciNetCrossRefGoogle Scholar
- Tomita K, Kurokawa H, Murata S (2002) Graph automata: natural expression of self-reproduction. Phys D Nonlinear Phenom 171(4):197–210MathSciNetCrossRefGoogle Scholar