Reversible cellular automaton (English Wikipedia)

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Wolfram (2002), p. 1018. Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8, MR 1920418
Toffoli & Margolus (1990). Toffoli, Tommaso; Margolus, Norman (1990), "Invertible cellular automata: a review", Physica D: Nonlinear Phenomena, 45 (1–3): 229–253, Bibcode:1990PhyD...45..229T, doi:10.1016/0167-2789(90)90185-R, MR 1094877.
Blanchard, Devaney & Keen (2004), p. 38: "The shift map is without doubt the fundamental object in symbolic dynamics." Blanchard, Paul; Devaney, Robert L.; Keen, Linda (2004), "Complex dynamics and symbolic dynamics", in Williams, Susan G. (ed.), Symbolic Dynamics and its Applications, Proceedings of Symposia in Applied Mathematics, vol. 60, Providence, RI: American Mathematical Society, pp. 37–60, doi:10.1090/psapm/060/2078845, MR 2078845.
Boykett (2004). Boykett, Tim (2004), "Efficient exhaustive listings of reversible one dimensional cellular automata", Theoretical Computer Science, 325 (2): 215–247, doi:10.1016/j.tcs.2004.06.007, MR 2086738.
Wolfram (2002), p. 1093. Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8, MR 1920418
Sutner (1991). Sutner, Klaus (1991), "De Bruijn graphs and linear cellular automata" (PDF), Complex Systems, 5: 19–30, MR 1116419.
Kari (2005). Kari, Jarkko (2005), "Reversible cellular automata" (PDF), Developments in Language Theory: 9th International Conference, DLT 2005, Palermo, Italy, July 4–8, 2005, Proceedings, Lecture Notes in Computer Science, vol. 3572, Springer-Verlag, pp. 2–23, doi:10.1007/11505877_5, MR 2187250.
Margolus (1984); Vichniac (1984); Wolfram (1984). Margolus, Norman (1984), "Physics-like models of computation", Physica D: Nonlinear Phenomena, 10 (1–2): 81–95, Bibcode:1984PhyD...10...81M, doi:10.1016/0167-2789(84)90252-5, MR 0762656. Reprinted in Wolfram, Stephen (1986), Theory and Applications of Cellular Automata, Advanced series on complex systems, vol. 1, World Scientific, pp. 232–246, Bibcode:1986taca.book.....W, and in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 83–104. Vichniac, Gérard Y. (1984), "Simulating physics with cellular automata", Physica D: Nonlinear Phenomena, 10 (1–2): 96–115, Bibcode:1984PhyD...10...96V, doi:10.1016/0167-2789(84)90253-7, MR 0762657. Wolfram, Stephen (1984), "Cellular automata as models of complexity" (PDF), Nature, 311 (5985): 419–424, Bibcode:1984Natur.311..419W, doi:10.1038/311419a0, S2CID 4237923.
Toffoli & Margolus (1987), Section 14.2, "Second-order technique", pp. 147–149. Wolfram (2002), pp. 437ff. McIntosh (2009). Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8. Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8, MR 1920418 McIntosh, Harold V. (2009), "12. Reversible cellular automata", One Dimensional Cellular Automata, Luniver Press, pp. 205–246.
Toffoli & Margolus (1990), section 5.3, "Conserved-landscape permutations", pp. 237–238. Toffoli, Tommaso; Margolus, Norman (1990), "Invertible cellular automata: a review", Physica D: Nonlinear Phenomena, 45 (1–3): 229–253, Bibcode:1990PhyD...45..229T, doi:10.1016/0167-2789(90)90185-R, MR 1094877.
In the one-dimensional case, several of these equivalences were already presented, in the language of dynamical systems rather than cellular automata, by Hedlund (1969), Theorem 4.1. For higher dimensions, see Richardson (1972) and Di Gregorio & Trautteur (1975). Hedlund, G. A. (1969), "Endomorphisms and automorphisms of the shift dynamical systems", Mathematical Systems Theory, 3 (4): 320–375, doi:10.1007/BF01691062, MR 0259881, S2CID 21803927. Richardson, D. (1972), "Tessellations with local transformations", Journal of Computer and System Sciences, 6 (5): 373–388, doi:10.1016/S0022-0000(72)80009-6, MR 0319678. Di Gregorio, S.; Trautteur, G. (1975), "On reversibility in cellular automata", Journal of Computer and System Sciences, 11 (3): 382–391, doi:10.1016/S0022-0000(75)80059-6, MR 0392201.
Myhill (1963). Myhill, John (1963), "The converse of Moore's Garden-of-Eden theorem", Proceedings of the American Mathematical Society, 14 (4): 685–686, doi:10.2307/2034301, JSTOR 2034301, MR 0155764. Reprinted in Burks, Arthur W. (1970), Essays on Cellular Automata, University of Illinois Press, pp. 204–205.
Richardson (1972). Richardson, D. (1972), "Tessellations with local transformations", Journal of Computer and System Sciences, 6 (5): 373–388, doi:10.1016/S0022-0000(72)80009-6, MR 0319678.
Hedlund (1969). Hedlund, G. A. (1969), "Endomorphisms and automorphisms of the shift dynamical systems", Mathematical Systems Theory, 3 (4): 320–375, doi:10.1007/BF01691062, MR 0259881, S2CID 21803927.
Moraal (2000). Moraal, Hendrik (2000), "Graph-theoretical characterization of invertible cellular automata", Physica D: Nonlinear Phenomena, 141 (1–2): 1–18, Bibcode:2000PhyD..141....1M, doi:10.1016/S0167-2789(00)00020-8, MR 1764165.
Culik cites a 1979 automata theory textbook for this result, but see Béal et al. (2003) for more recent developments on efficiently testing whether a transducer defines a function. Béal, Marie-Pierre; Carton, Olivier; Prieur, Christophe; Sakarovitch, Jacques (2003), "Squaring transducers: an efficient procedure for deciding functionality and sequentiality", Theoretical Computer Science, 292 (1): 45–63, doi:10.1016/S0304-3975(01)00214-6, MR 1964625.
Neither Amoroso & Patt (1972) nor Culik (1987) state their algorithms' time complexities explicitly, but Sutner (1991) does, and this bound can also be found e.g. in Czeizler & Kari (2007). Culik, Karel II (1987), "On invertible cellular automata" (PDF), Complex Systems, 1 (6): 1035–1044, MR 0931401. Sutner, Klaus (1991), "De Bruijn graphs and linear cellular automata" (PDF), Complex Systems, 5: 19–30, MR 1116419. Czeizler, Eugen; Kari, Jarkko (2007), "A tight linear bound on the synchronization delay of bijective automata", Theoretical Computer Science, 380 (1–2): 23–36, doi:10.1016/j.tcs.2007.02.052, MR 2330639.
Kari (1992); Czeizler (2004); Czeizler & Kari (2007). Kari, Jarkko (1992), "On the inverse neighborhoods of reversible cellular automata", Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology, Springer-Verlag, pp. 477–495, doi:10.1007/978-3-642-58117-5_29, MR 1226709. Czeizler, Eugen (2004), "On the size of the inverse neighborhoods for one-dimensional reversible cellular automata", Theoretical Computer Science, 325 (2): 273–284, doi:10.1016/j.tcs.2004.06.009, MR 2086740. Czeizler, Eugen; Kari, Jarkko (2007), "A tight linear bound on the synchronization delay of bijective automata", Theoretical Computer Science, 380 (1–2): 23–36, doi:10.1016/j.tcs.2007.02.052, MR 2330639.
Wolfram (2002), pp. 454–457. Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8, MR 1920418
Boykett (2004). See Hillman (1991) and Seck Tuoh Mora et al. (2005) for closely related work on the enumeration of width-2 reversible cellular automata. Boykett, Tim (2004), "Efficient exhaustive listings of reversible one dimensional cellular automata", Theoretical Computer Science, 325 (2): 215–247, doi:10.1016/j.tcs.2004.06.007, MR 2086738. Hillman, David (1991), "The structure of reversible one-dimensional cellular automata", Physica D: Nonlinear Phenomena, 52 (2–3): 277–292, Bibcode:1991PhyD...52..277H, doi:10.1016/0167-2789(91)90128-V, MR 1128996. Seck Tuoh Mora, Juan Carlos; Chapa Vergara, Sergio V.; Juárez Martínez, Genaro; McIntosh, Harold V. (2005), "Procedures for calculating reversible one-dimensional cellular automata" (PDF), Physica D: Nonlinear Phenomena, 202 (1–2): 134–141, Bibcode:2005PhyD..202..134S, doi:10.1016/j.physd.2005.01.018, MR 2131890.
Hattori & Takesue (1991); Fukś (2007). Hattori, Tetsuya; Takesue, Shinji (1991), "Additive conserved quantities in discrete-time lattice dynamical systems", Physica D: Nonlinear Phenomena, 49 (3): 295–322, Bibcode:1991PhyD...49..295H, doi:10.1016/0167-2789(91)90150-8, MR 1115865. Fukś, Henryk (2007), "Remarks on the critical behavior of second order additive invariants in elementary cellular automata", Fundamenta Informaticae, 78 (3): 329–341, arXiv:nlin/0502037, Bibcode:2005nlin......2037F, MR 2346870.
Boykett, Kari & Taati (2008). Boykett, Tim; Kari, Jarkko; Taati, Siamak (2008), "Conservation laws in rectangular CA" (PDF), Journal of Cellular Automata, 3 (2): 115–122, MR 2394641, archived from the original (PDF) on 2015-09-30.
Pomeau (1984); Takesue (1990); Capobianco & Toffoli (2011). Pomeau, Y. (1984), "Invariants in cellular automata", Journal of Physics A: Mathematical and General, 17 (8): L415–L418, Bibcode:1984JPhA...17L.415P, doi:10.1088/0305-4470/17/8/004, MR 0750565. Takesue, Shinji (1990), "Relaxation properties of elementary reversible cellular automata", Cellular automata: theory and experiment (Los Alamos, NM, 1989), Physica D: Nonlinear Phenomena, 45 (1–3): 278–284, Bibcode:1990PhyD...45..379K, doi:10.1016/0167-2789(90)90195-U, MR 1094882. Capobianco, Silvio; Toffoli, Tommaso (2011), "Can anything from Noether's theorem be salvaged for discrete dynamical systems?", Proceedings of the 10th International Conference on Unconventional Computation (UC 2011), Lecture Notes in Computer Science, vol. 6714, Springer-Verlag, pp. 77–88, arXiv:1103.4785, doi:10.1007/978-3-642-21341-0_13, S2CID 42541816.
Fredkin & Toffoli (1982). Fredkin, Edward; Toffoli, Tommaso (1982), "Conservative logic", International Journal of Theoretical Physics, 21 (3–4): 219–253, Bibcode:1982IJTP...21..219F, doi:10.1007/BF01857727, MR 0657156, S2CID 37305161. Reprinted in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 47–82.
Kari (2005, 2009) Kari, Jarkko (2005), "Reversible cellular automata" (PDF), Developments in Language Theory: 9th International Conference, DLT 2005, Palermo, Italy, July 4–8, 2005, Proceedings, Lecture Notes in Computer Science, vol. 3572, Springer-Verlag, pp. 2–23, doi:10.1007/11505877_5, MR 2187250. Kari, Jarkko (2009), "Structure of reversible cellular automata", Unconventional Computation: 8th International Conference, UC 2009, Ponta Delgada, Portugal, September 7–11, 2009, Proceedings, Lecture Notes in Computer Science, vol. 5715, Springer-Verlag, p. 6, Bibcode:2009LNCS.5715....6K, doi:10.1007/978-3-642-03745-0_5, MR 2539690.
Meyer (1996); Schumacher & Werner (2004); Shepherd, Franz & Werner (2006); Nagaj & Wocjan (2008). Meyer, David A. (1996), "From quantum cellular automata to quantum lattice gases", Journal of Statistical Physics, 85 (5–6): 551–574, arXiv:quant-ph/9604003, Bibcode:1996JSP....85..551M, doi:10.1007/BF02199356, MR 1418805, S2CID 661940. Schumacher, B.; Werner, R. F. (2004), Reversible quantum cellular automata, arXiv:quant-ph/0405174, Bibcode:2004quant.ph..5174S. Shepherd, D. J.; Franz, T.; Werner, R. F. (2006), "A universally programmable quantum cellular automaton", Physical Review Letters, 97 (2): 020502, arXiv:quant-ph/0512058, Bibcode:2006PhRvL..97b0502S, doi:10.1103/PhysRevLett.97.020502, PMID 16907423, S2CID 40900768. Nagaj, Daniel; Wocjan, Pawel (2008), "Hamiltonian quantum cellular automata in one dimension", Physical Review A, 78 (3): 032311, arXiv:0802.0886, Bibcode:2008PhRvA..78c2311N, doi:10.1103/PhysRevA.78.032311, S2CID 18879990.

archive.org

Toffoli & Margolus (1987), section 12.8.2, "Critters", pp. 132–134; Margolus (1999); Marotta (2005). Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8. Margolus, Norman (1999), "Crystalline computation", in Hey, Anthony J. G. (ed.), Feynman and Computation, Perseus Books, pp. 267–305, arXiv:comp-gas/9811002, Bibcode:1998comp.gas.11002M. Marotta, Sebastian M. (2005), "Living in Critters' world", Revista Ciências Exatas e Naturais, 7 (1), archived from the original on 2012-03-19.
Toffoli & Margolus (1987), Section 14.5, "Partitioning technique", pp. 150–153; Schiff (2008), Section 4.2.1, "Partitioning Cellular Automata", pp. 115–116. Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8. Schiff, Joel L. (2008), Cellular Automata: A Discrete View of the World, Wiley, ISBN 978-0-470-16879-0.
Toffoli & Margolus (1987), Chapter 12, "The Margolus Neighborhood", pp. 119–138. Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8.
Toffoli & Margolus (1987), Section 14.2, "Second-order technique", pp. 147–149. Wolfram (2002), pp. 437ff. McIntosh (2009). Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8. Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8, MR 1920418 McIntosh, Harold V. (2009), "12. Reversible cellular automata", One Dimensional Cellular Automata, Luniver Press, pp. 205–246.
Toffoli & Margolus (1987), Chapter 16, "Fluid dynamics", pp. 172–184. Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8.
Toffoli & Margolus (1987), Chapter 17.2, "Ising systems", pp. 186–190. Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8.
Toffoli & Margolus (1987), Section 12.8.3, "Asynchronous computation", pp. 134–136. Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8.

arxiv.org

Toffoli & Margolus (1987), section 12.8.2, "Critters", pp. 132–134; Margolus (1999); Marotta (2005). Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8. Margolus, Norman (1999), "Crystalline computation", in Hey, Anthony J. G. (ed.), Feynman and Computation, Perseus Books, pp. 267–305, arXiv:comp-gas/9811002, Bibcode:1998comp.gas.11002M. Marotta, Sebastian M. (2005), "Living in Critters' world", Revista Ciências Exatas e Naturais, 7 (1), archived from the original on 2012-03-19.
Miller & Fredkin (2005). Miller, Daniel B.; Fredkin, Edward (2005), "Two-state, reversible, universal cellular automata in three dimensions", Proceedings of the 2nd Conference on Computing Frontiers (CF '05), New York, NY, USA: ACM, pp. 45–51, arXiv:nlin/0501022, doi:10.1145/1062261.1062271, ISBN 1-59593-019-1, S2CID 14082792.
Miller & Fredkin (2012). Miller, Daniel B.; Fredkin, Edward (2012), Circular motion of strings in cellular automata, and other surprises, arXiv:1206.2060, Bibcode:2012arXiv1206.2060M.
Hattori & Takesue (1991); Fukś (2007). Hattori, Tetsuya; Takesue, Shinji (1991), "Additive conserved quantities in discrete-time lattice dynamical systems", Physica D: Nonlinear Phenomena, 49 (3): 295–322, Bibcode:1991PhyD...49..295H, doi:10.1016/0167-2789(91)90150-8, MR 1115865. Fukś, Henryk (2007), "Remarks on the critical behavior of second order additive invariants in elementary cellular automata", Fundamenta Informaticae, 78 (3): 329–341, arXiv:nlin/0502037, Bibcode:2005nlin......2037F, MR 2346870.
Pomeau (1984); Takesue (1990); Capobianco & Toffoli (2011). Pomeau, Y. (1984), "Invariants in cellular automata", Journal of Physics A: Mathematical and General, 17 (8): L415–L418, Bibcode:1984JPhA...17L.415P, doi:10.1088/0305-4470/17/8/004, MR 0750565. Takesue, Shinji (1990), "Relaxation properties of elementary reversible cellular automata", Cellular automata: theory and experiment (Los Alamos, NM, 1989), Physica D: Nonlinear Phenomena, 45 (1–3): 278–284, Bibcode:1990PhyD...45..379K, doi:10.1016/0167-2789(90)90195-U, MR 1094882. Capobianco, Silvio; Toffoli, Tommaso (2011), "Can anything from Noether's theorem be salvaged for discrete dynamical systems?", Proceedings of the 10th International Conference on Unconventional Computation (UC 2011), Lecture Notes in Computer Science, vol. 6714, Springer-Verlag, pp. 77–88, arXiv:1103.4785, doi:10.1007/978-3-642-21341-0_13, S2CID 42541816.
Meyer (1996); Schumacher & Werner (2004); Shepherd, Franz & Werner (2006); Nagaj & Wocjan (2008). Meyer, David A. (1996), "From quantum cellular automata to quantum lattice gases", Journal of Statistical Physics, 85 (5–6): 551–574, arXiv:quant-ph/9604003, Bibcode:1996JSP....85..551M, doi:10.1007/BF02199356, MR 1418805, S2CID 661940. Schumacher, B.; Werner, R. F. (2004), Reversible quantum cellular automata, arXiv:quant-ph/0405174, Bibcode:2004quant.ph..5174S. Shepherd, D. J.; Franz, T.; Werner, R. F. (2006), "A universally programmable quantum cellular automaton", Physical Review Letters, 97 (2): 020502, arXiv:quant-ph/0512058, Bibcode:2006PhRvL..97b0502S, doi:10.1103/PhysRevLett.97.020502, PMID 16907423, S2CID 40900768. Nagaj, Daniel; Wocjan, Pawel (2008), "Hamiltonian quantum cellular automata in one dimension", Physical Review A, 78 (3): 032311, arXiv:0802.0886, Bibcode:2008PhRvA..78c2311N, doi:10.1103/PhysRevA.78.032311, S2CID 18879990.

books.google.com

Blanchard, Devaney & Keen (2004), p. 38: "The shift map is without doubt the fundamental object in symbolic dynamics." Blanchard, Paul; Devaney, Robert L.; Keen, Linda (2004), "Complex dynamics and symbolic dynamics", in Williams, Susan G. (ed.), Symbolic Dynamics and its Applications, Proceedings of Symposia in Applied Mathematics, vol. 60, Providence, RI: American Mathematical Society, pp. 37–60, doi:10.1090/psapm/060/2078845, MR 2078845.

buffalo.edu

copper.math.buffalo.edu

Kari (2005). Kari, Jarkko (2005), "Reversible cellular automata" (PDF), Developments in Language Theory: 9th International Conference, DLT 2005, Palermo, Italy, July 4–8, 2005, Proceedings, Lecture Notes in Computer Science, vol. 3572, Springer-Verlag, pp. 2–23, doi:10.1007/11505877_5, MR 2187250.
Kari (2005, 2009) Kari, Jarkko (2005), "Reversible cellular automata" (PDF), Developments in Language Theory: 9th International Conference, DLT 2005, Palermo, Italy, July 4–8, 2005, Proceedings, Lecture Notes in Computer Science, vol. 3572, Springer-Verlag, pp. 2–23, doi:10.1007/11505877_5, MR 2187250. Kari, Jarkko (2009), "Structure of reversible cellular automata", Unconventional Computation: 8th International Conference, UC 2009, Ponta Delgada, Portugal, September 7–11, 2009, Proceedings, Lecture Notes in Computer Science, vol. 5715, Springer-Verlag, p. 6, Bibcode:2009LNCS.5715....6K, doi:10.1007/978-3-642-03745-0_5, MR 2539690.

complex-systems.com

Sutner (1991). Sutner, Klaus (1991), "De Bruijn graphs and linear cellular automata" (PDF), Complex Systems, 5: 19–30, MR 1116419.
Neither Amoroso & Patt (1972) nor Culik (1987) state their algorithms' time complexities explicitly, but Sutner (1991) does, and this bound can also be found e.g. in Czeizler & Kari (2007). Culik, Karel II (1987), "On invertible cellular automata" (PDF), Complex Systems, 1 (6): 1035–1044, MR 0931401. Sutner, Klaus (1991), "De Bruijn graphs and linear cellular automata" (PDF), Complex Systems, 5: 19–30, MR 1116419. Czeizler, Eugen; Kari, Jarkko (2007), "A tight linear bound on the synchronization delay of bijective automata", Theoretical Computer Science, 380 (1–2): 23–36, doi:10.1016/j.tcs.2007.02.052, MR 2330639.

doi.org

Toffoli & Margolus (1990). Toffoli, Tommaso; Margolus, Norman (1990), "Invertible cellular automata: a review", Physica D: Nonlinear Phenomena, 45 (1–3): 229–253, Bibcode:1990PhyD...45..229T, doi:10.1016/0167-2789(90)90185-R, MR 1094877.
Blanchard, Devaney & Keen (2004), p. 38: "The shift map is without doubt the fundamental object in symbolic dynamics." Blanchard, Paul; Devaney, Robert L.; Keen, Linda (2004), "Complex dynamics and symbolic dynamics", in Williams, Susan G. (ed.), Symbolic Dynamics and its Applications, Proceedings of Symposia in Applied Mathematics, vol. 60, Providence, RI: American Mathematical Society, pp. 37–60, doi:10.1090/psapm/060/2078845, MR 2078845.
Boykett (2004). Boykett, Tim (2004), "Efficient exhaustive listings of reversible one dimensional cellular automata", Theoretical Computer Science, 325 (2): 215–247, doi:10.1016/j.tcs.2004.06.007, MR 2086738.
Kari (2005). Kari, Jarkko (2005), "Reversible cellular automata" (PDF), Developments in Language Theory: 9th International Conference, DLT 2005, Palermo, Italy, July 4–8, 2005, Proceedings, Lecture Notes in Computer Science, vol. 3572, Springer-Verlag, pp. 2–23, doi:10.1007/11505877_5, MR 2187250.
Margolus (1984); Vichniac (1984); Wolfram (1984). Margolus, Norman (1984), "Physics-like models of computation", Physica D: Nonlinear Phenomena, 10 (1–2): 81–95, Bibcode:1984PhyD...10...81M, doi:10.1016/0167-2789(84)90252-5, MR 0762656. Reprinted in Wolfram, Stephen (1986), Theory and Applications of Cellular Automata, Advanced series on complex systems, vol. 1, World Scientific, pp. 232–246, Bibcode:1986taca.book.....W, and in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 83–104. Vichniac, Gérard Y. (1984), "Simulating physics with cellular automata", Physica D: Nonlinear Phenomena, 10 (1–2): 96–115, Bibcode:1984PhyD...10...96V, doi:10.1016/0167-2789(84)90253-7, MR 0762657. Wolfram, Stephen (1984), "Cellular automata as models of complexity" (PDF), Nature, 311 (5985): 419–424, Bibcode:1984Natur.311..419W, doi:10.1038/311419a0, S2CID 4237923.
Toffoli & Margolus (1990), section 5.3, "Conserved-landscape permutations", pp. 237–238. Toffoli, Tommaso; Margolus, Norman (1990), "Invertible cellular automata: a review", Physica D: Nonlinear Phenomena, 45 (1–3): 229–253, Bibcode:1990PhyD...45..229T, doi:10.1016/0167-2789(90)90185-R, MR 1094877.
Miller & Fredkin (2005). Miller, Daniel B.; Fredkin, Edward (2005), "Two-state, reversible, universal cellular automata in three dimensions", Proceedings of the 2nd Conference on Computing Frontiers (CF '05), New York, NY, USA: ACM, pp. 45–51, arXiv:nlin/0501022, doi:10.1145/1062261.1062271, ISBN 1-59593-019-1, S2CID 14082792.
In the one-dimensional case, several of these equivalences were already presented, in the language of dynamical systems rather than cellular automata, by Hedlund (1969), Theorem 4.1. For higher dimensions, see Richardson (1972) and Di Gregorio & Trautteur (1975). Hedlund, G. A. (1969), "Endomorphisms and automorphisms of the shift dynamical systems", Mathematical Systems Theory, 3 (4): 320–375, doi:10.1007/BF01691062, MR 0259881, S2CID 21803927. Richardson, D. (1972), "Tessellations with local transformations", Journal of Computer and System Sciences, 6 (5): 373–388, doi:10.1016/S0022-0000(72)80009-6, MR 0319678. Di Gregorio, S.; Trautteur, G. (1975), "On reversibility in cellular automata", Journal of Computer and System Sciences, 11 (3): 382–391, doi:10.1016/S0022-0000(75)80059-6, MR 0392201.
Myhill (1963). Myhill, John (1963), "The converse of Moore's Garden-of-Eden theorem", Proceedings of the American Mathematical Society, 14 (4): 685–686, doi:10.2307/2034301, JSTOR 2034301, MR 0155764. Reprinted in Burks, Arthur W. (1970), Essays on Cellular Automata, University of Illinois Press, pp. 204–205.
Richardson (1972). Richardson, D. (1972), "Tessellations with local transformations", Journal of Computer and System Sciences, 6 (5): 373–388, doi:10.1016/S0022-0000(72)80009-6, MR 0319678.
Hedlund (1969). Hedlund, G. A. (1969), "Endomorphisms and automorphisms of the shift dynamical systems", Mathematical Systems Theory, 3 (4): 320–375, doi:10.1007/BF01691062, MR 0259881, S2CID 21803927.
Moraal (2000). Moraal, Hendrik (2000), "Graph-theoretical characterization of invertible cellular automata", Physica D: Nonlinear Phenomena, 141 (1–2): 1–18, Bibcode:2000PhyD..141....1M, doi:10.1016/S0167-2789(00)00020-8, MR 1764165.
Culik cites a 1979 automata theory textbook for this result, but see Béal et al. (2003) for more recent developments on efficiently testing whether a transducer defines a function. Béal, Marie-Pierre; Carton, Olivier; Prieur, Christophe; Sakarovitch, Jacques (2003), "Squaring transducers: an efficient procedure for deciding functionality and sequentiality", Theoretical Computer Science, 292 (1): 45–63, doi:10.1016/S0304-3975(01)00214-6, MR 1964625.
Neither Amoroso & Patt (1972) nor Culik (1987) state their algorithms' time complexities explicitly, but Sutner (1991) does, and this bound can also be found e.g. in Czeizler & Kari (2007). Culik, Karel II (1987), "On invertible cellular automata" (PDF), Complex Systems, 1 (6): 1035–1044, MR 0931401. Sutner, Klaus (1991), "De Bruijn graphs and linear cellular automata" (PDF), Complex Systems, 5: 19–30, MR 1116419. Czeizler, Eugen; Kari, Jarkko (2007), "A tight linear bound on the synchronization delay of bijective automata", Theoretical Computer Science, 380 (1–2): 23–36, doi:10.1016/j.tcs.2007.02.052, MR 2330639.
Kari (1992); Czeizler (2004); Czeizler & Kari (2007). Kari, Jarkko (1992), "On the inverse neighborhoods of reversible cellular automata", Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology, Springer-Verlag, pp. 477–495, doi:10.1007/978-3-642-58117-5_29, MR 1226709. Czeizler, Eugen (2004), "On the size of the inverse neighborhoods for one-dimensional reversible cellular automata", Theoretical Computer Science, 325 (2): 273–284, doi:10.1016/j.tcs.2004.06.009, MR 2086740. Czeizler, Eugen; Kari, Jarkko (2007), "A tight linear bound on the synchronization delay of bijective automata", Theoretical Computer Science, 380 (1–2): 23–36, doi:10.1016/j.tcs.2007.02.052, MR 2330639.
Boykett (2004). See Hillman (1991) and Seck Tuoh Mora et al. (2005) for closely related work on the enumeration of width-2 reversible cellular automata. Boykett, Tim (2004), "Efficient exhaustive listings of reversible one dimensional cellular automata", Theoretical Computer Science, 325 (2): 215–247, doi:10.1016/j.tcs.2004.06.007, MR 2086738. Hillman, David (1991), "The structure of reversible one-dimensional cellular automata", Physica D: Nonlinear Phenomena, 52 (2–3): 277–292, Bibcode:1991PhyD...52..277H, doi:10.1016/0167-2789(91)90128-V, MR 1128996. Seck Tuoh Mora, Juan Carlos; Chapa Vergara, Sergio V.; Juárez Martínez, Genaro; McIntosh, Harold V. (2005), "Procedures for calculating reversible one-dimensional cellular automata" (PDF), Physica D: Nonlinear Phenomena, 202 (1–2): 134–141, Bibcode:2005PhyD..202..134S, doi:10.1016/j.physd.2005.01.018, MR 2131890.
Hattori & Takesue (1991); Fukś (2007). Hattori, Tetsuya; Takesue, Shinji (1991), "Additive conserved quantities in discrete-time lattice dynamical systems", Physica D: Nonlinear Phenomena, 49 (3): 295–322, Bibcode:1991PhyD...49..295H, doi:10.1016/0167-2789(91)90150-8, MR 1115865. Fukś, Henryk (2007), "Remarks on the critical behavior of second order additive invariants in elementary cellular automata", Fundamenta Informaticae, 78 (3): 329–341, arXiv:nlin/0502037, Bibcode:2005nlin......2037F, MR 2346870.
Pomeau (1984); Takesue (1990); Capobianco & Toffoli (2011). Pomeau, Y. (1984), "Invariants in cellular automata", Journal of Physics A: Mathematical and General, 17 (8): L415–L418, Bibcode:1984JPhA...17L.415P, doi:10.1088/0305-4470/17/8/004, MR 0750565. Takesue, Shinji (1990), "Relaxation properties of elementary reversible cellular automata", Cellular automata: theory and experiment (Los Alamos, NM, 1989), Physica D: Nonlinear Phenomena, 45 (1–3): 278–284, Bibcode:1990PhyD...45..379K, doi:10.1016/0167-2789(90)90195-U, MR 1094882. Capobianco, Silvio; Toffoli, Tommaso (2011), "Can anything from Noether's theorem be salvaged for discrete dynamical systems?", Proceedings of the 10th International Conference on Unconventional Computation (UC 2011), Lecture Notes in Computer Science, vol. 6714, Springer-Verlag, pp. 77–88, arXiv:1103.4785, doi:10.1007/978-3-642-21341-0_13, S2CID 42541816.
Fredkin & Toffoli (1982). Fredkin, Edward; Toffoli, Tommaso (1982), "Conservative logic", International Journal of Theoretical Physics, 21 (3–4): 219–253, Bibcode:1982IJTP...21..219F, doi:10.1007/BF01857727, MR 0657156, S2CID 37305161. Reprinted in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 47–82.
Kari (2005, 2009) Kari, Jarkko (2005), "Reversible cellular automata" (PDF), Developments in Language Theory: 9th International Conference, DLT 2005, Palermo, Italy, July 4–8, 2005, Proceedings, Lecture Notes in Computer Science, vol. 3572, Springer-Verlag, pp. 2–23, doi:10.1007/11505877_5, MR 2187250. Kari, Jarkko (2009), "Structure of reversible cellular automata", Unconventional Computation: 8th International Conference, UC 2009, Ponta Delgada, Portugal, September 7–11, 2009, Proceedings, Lecture Notes in Computer Science, vol. 5715, Springer-Verlag, p. 6, Bibcode:2009LNCS.5715....6K, doi:10.1007/978-3-642-03745-0_5, MR 2539690.
Meyer (1996); Schumacher & Werner (2004); Shepherd, Franz & Werner (2006); Nagaj & Wocjan (2008). Meyer, David A. (1996), "From quantum cellular automata to quantum lattice gases", Journal of Statistical Physics, 85 (5–6): 551–574, arXiv:quant-ph/9604003, Bibcode:1996JSP....85..551M, doi:10.1007/BF02199356, MR 1418805, S2CID 661940. Schumacher, B.; Werner, R. F. (2004), Reversible quantum cellular automata, arXiv:quant-ph/0405174, Bibcode:2004quant.ph..5174S. Shepherd, D. J.; Franz, T.; Werner, R. F. (2006), "A universally programmable quantum cellular automaton", Physical Review Letters, 97 (2): 020502, arXiv:quant-ph/0512058, Bibcode:2006PhRvL..97b0502S, doi:10.1103/PhysRevLett.97.020502, PMID 16907423, S2CID 40900768. Nagaj, Daniel; Wocjan, Pawel (2008), "Hamiltonian quantum cellular automata in one dimension", Physical Review A, 78 (3): 032311, arXiv:0802.0886, Bibcode:2008PhRvA..78c2311N, doi:10.1103/PhysRevA.78.032311, S2CID 18879990.

harvard.edu

ui.adsabs.harvard.edu

Toffoli & Margolus (1990). Toffoli, Tommaso; Margolus, Norman (1990), "Invertible cellular automata: a review", Physica D: Nonlinear Phenomena, 45 (1–3): 229–253, Bibcode:1990PhyD...45..229T, doi:10.1016/0167-2789(90)90185-R, MR 1094877.
Toffoli & Margolus (1987), section 12.8.2, "Critters", pp. 132–134; Margolus (1999); Marotta (2005). Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8. Margolus, Norman (1999), "Crystalline computation", in Hey, Anthony J. G. (ed.), Feynman and Computation, Perseus Books, pp. 267–305, arXiv:comp-gas/9811002, Bibcode:1998comp.gas.11002M. Marotta, Sebastian M. (2005), "Living in Critters' world", Revista Ciências Exatas e Naturais, 7 (1), archived from the original on 2012-03-19.
Margolus (1984); Vichniac (1984); Wolfram (1984). Margolus, Norman (1984), "Physics-like models of computation", Physica D: Nonlinear Phenomena, 10 (1–2): 81–95, Bibcode:1984PhyD...10...81M, doi:10.1016/0167-2789(84)90252-5, MR 0762656. Reprinted in Wolfram, Stephen (1986), Theory and Applications of Cellular Automata, Advanced series on complex systems, vol. 1, World Scientific, pp. 232–246, Bibcode:1986taca.book.....W, and in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 83–104. Vichniac, Gérard Y. (1984), "Simulating physics with cellular automata", Physica D: Nonlinear Phenomena, 10 (1–2): 96–115, Bibcode:1984PhyD...10...96V, doi:10.1016/0167-2789(84)90253-7, MR 0762657. Wolfram, Stephen (1984), "Cellular automata as models of complexity" (PDF), Nature, 311 (5985): 419–424, Bibcode:1984Natur.311..419W, doi:10.1038/311419a0, S2CID 4237923.
Toffoli & Margolus (1990), section 5.3, "Conserved-landscape permutations", pp. 237–238. Toffoli, Tommaso; Margolus, Norman (1990), "Invertible cellular automata: a review", Physica D: Nonlinear Phenomena, 45 (1–3): 229–253, Bibcode:1990PhyD...45..229T, doi:10.1016/0167-2789(90)90185-R, MR 1094877.
Miller & Fredkin (2012). Miller, Daniel B.; Fredkin, Edward (2012), Circular motion of strings in cellular automata, and other surprises, arXiv:1206.2060, Bibcode:2012arXiv1206.2060M.
Moraal (2000). Moraal, Hendrik (2000), "Graph-theoretical characterization of invertible cellular automata", Physica D: Nonlinear Phenomena, 141 (1–2): 1–18, Bibcode:2000PhyD..141....1M, doi:10.1016/S0167-2789(00)00020-8, MR 1764165.
Boykett (2004). See Hillman (1991) and Seck Tuoh Mora et al. (2005) for closely related work on the enumeration of width-2 reversible cellular automata. Boykett, Tim (2004), "Efficient exhaustive listings of reversible one dimensional cellular automata", Theoretical Computer Science, 325 (2): 215–247, doi:10.1016/j.tcs.2004.06.007, MR 2086738. Hillman, David (1991), "The structure of reversible one-dimensional cellular automata", Physica D: Nonlinear Phenomena, 52 (2–3): 277–292, Bibcode:1991PhyD...52..277H, doi:10.1016/0167-2789(91)90128-V, MR 1128996. Seck Tuoh Mora, Juan Carlos; Chapa Vergara, Sergio V.; Juárez Martínez, Genaro; McIntosh, Harold V. (2005), "Procedures for calculating reversible one-dimensional cellular automata" (PDF), Physica D: Nonlinear Phenomena, 202 (1–2): 134–141, Bibcode:2005PhyD..202..134S, doi:10.1016/j.physd.2005.01.018, MR 2131890.
Hattori & Takesue (1991); Fukś (2007). Hattori, Tetsuya; Takesue, Shinji (1991), "Additive conserved quantities in discrete-time lattice dynamical systems", Physica D: Nonlinear Phenomena, 49 (3): 295–322, Bibcode:1991PhyD...49..295H, doi:10.1016/0167-2789(91)90150-8, MR 1115865. Fukś, Henryk (2007), "Remarks on the critical behavior of second order additive invariants in elementary cellular automata", Fundamenta Informaticae, 78 (3): 329–341, arXiv:nlin/0502037, Bibcode:2005nlin......2037F, MR 2346870.
Pomeau (1984); Takesue (1990); Capobianco & Toffoli (2011). Pomeau, Y. (1984), "Invariants in cellular automata", Journal of Physics A: Mathematical and General, 17 (8): L415–L418, Bibcode:1984JPhA...17L.415P, doi:10.1088/0305-4470/17/8/004, MR 0750565. Takesue, Shinji (1990), "Relaxation properties of elementary reversible cellular automata", Cellular automata: theory and experiment (Los Alamos, NM, 1989), Physica D: Nonlinear Phenomena, 45 (1–3): 278–284, Bibcode:1990PhyD...45..379K, doi:10.1016/0167-2789(90)90195-U, MR 1094882. Capobianco, Silvio; Toffoli, Tommaso (2011), "Can anything from Noether's theorem be salvaged for discrete dynamical systems?", Proceedings of the 10th International Conference on Unconventional Computation (UC 2011), Lecture Notes in Computer Science, vol. 6714, Springer-Verlag, pp. 77–88, arXiv:1103.4785, doi:10.1007/978-3-642-21341-0_13, S2CID 42541816.
Fredkin & Toffoli (1982). Fredkin, Edward; Toffoli, Tommaso (1982), "Conservative logic", International Journal of Theoretical Physics, 21 (3–4): 219–253, Bibcode:1982IJTP...21..219F, doi:10.1007/BF01857727, MR 0657156, S2CID 37305161. Reprinted in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 47–82.
Kari (2005, 2009) Kari, Jarkko (2005), "Reversible cellular automata" (PDF), Developments in Language Theory: 9th International Conference, DLT 2005, Palermo, Italy, July 4–8, 2005, Proceedings, Lecture Notes in Computer Science, vol. 3572, Springer-Verlag, pp. 2–23, doi:10.1007/11505877_5, MR 2187250. Kari, Jarkko (2009), "Structure of reversible cellular automata", Unconventional Computation: 8th International Conference, UC 2009, Ponta Delgada, Portugal, September 7–11, 2009, Proceedings, Lecture Notes in Computer Science, vol. 5715, Springer-Verlag, p. 6, Bibcode:2009LNCS.5715....6K, doi:10.1007/978-3-642-03745-0_5, MR 2539690.
Meyer (1996); Schumacher & Werner (2004); Shepherd, Franz & Werner (2006); Nagaj & Wocjan (2008). Meyer, David A. (1996), "From quantum cellular automata to quantum lattice gases", Journal of Statistical Physics, 85 (5–6): 551–574, arXiv:quant-ph/9604003, Bibcode:1996JSP....85..551M, doi:10.1007/BF02199356, MR 1418805, S2CID 661940. Schumacher, B.; Werner, R. F. (2004), Reversible quantum cellular automata, arXiv:quant-ph/0405174, Bibcode:2004quant.ph..5174S. Shepherd, D. J.; Franz, T.; Werner, R. F. (2006), "A universally programmable quantum cellular automaton", Physical Review Letters, 97 (2): 020502, arXiv:quant-ph/0512058, Bibcode:2006PhRvL..97b0502S, doi:10.1103/PhysRevLett.97.020502, PMID 16907423, S2CID 40900768. Nagaj, Daniel; Wocjan, Pawel (2008), "Hamiltonian quantum cellular automata in one dimension", Physical Review A, 78 (3): 032311, arXiv:0802.0886, Bibcode:2008PhRvA..78c2311N, doi:10.1103/PhysRevA.78.032311, S2CID 18879990.

jstor.org

Myhill (1963). Myhill, John (1963), "The converse of Moore's Garden-of-Eden theorem", Proceedings of the American Mathematical Society, 14 (4): 685–686, doi:10.2307/2034301, JSTOR 2034301, MR 0155764. Reprinted in Burks, Arthur W. (1970), Essays on Cellular Automata, University of Illinois Press, pp. 204–205.

leidenuniv.nl

pub.math.leidenuniv.nl

Boykett, Kari & Taati (2008). Boykett, Tim; Kari, Jarkko; Taati, Siamak (2008), "Conservation laws in rectangular CA" (PDF), Journal of Cellular Automata, 3 (2): 115–122, MR 2394641, archived from the original (PDF) on 2015-09-30.

nih.gov

pubmed.ncbi.nlm.nih.gov

Meyer (1996); Schumacher & Werner (2004); Shepherd, Franz & Werner (2006); Nagaj & Wocjan (2008). Meyer, David A. (1996), "From quantum cellular automata to quantum lattice gases", Journal of Statistical Physics, 85 (5–6): 551–574, arXiv:quant-ph/9604003, Bibcode:1996JSP....85..551M, doi:10.1007/BF02199356, MR 1418805, S2CID 661940. Schumacher, B.; Werner, R. F. (2004), Reversible quantum cellular automata, arXiv:quant-ph/0405174, Bibcode:2004quant.ph..5174S. Shepherd, D. J.; Franz, T.; Werner, R. F. (2006), "A universally programmable quantum cellular automaton", Physical Review Letters, 97 (2): 020502, arXiv:quant-ph/0512058, Bibcode:2006PhRvL..97b0502S, doi:10.1103/PhysRevLett.97.020502, PMID 16907423, S2CID 40900768. Nagaj, Daniel; Wocjan, Pawel (2008), "Hamiltonian quantum cellular automata in one dimension", Physical Review A, 78 (3): 032311, arXiv:0802.0886, Bibcode:2008PhRvA..78c2311N, doi:10.1103/PhysRevA.78.032311, S2CID 18879990.

scottaaronson.com

See also "A Physically Universal Cellular Automaton", Shtetl-Optimized, Scott Aaronson, June 26, 2014.

semanticscholar.org

api.semanticscholar.org

Margolus (1984); Vichniac (1984); Wolfram (1984). Margolus, Norman (1984), "Physics-like models of computation", Physica D: Nonlinear Phenomena, 10 (1–2): 81–95, Bibcode:1984PhyD...10...81M, doi:10.1016/0167-2789(84)90252-5, MR 0762656. Reprinted in Wolfram, Stephen (1986), Theory and Applications of Cellular Automata, Advanced series on complex systems, vol. 1, World Scientific, pp. 232–246, Bibcode:1986taca.book.....W, and in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 83–104. Vichniac, Gérard Y. (1984), "Simulating physics with cellular automata", Physica D: Nonlinear Phenomena, 10 (1–2): 96–115, Bibcode:1984PhyD...10...96V, doi:10.1016/0167-2789(84)90253-7, MR 0762657. Wolfram, Stephen (1984), "Cellular automata as models of complexity" (PDF), Nature, 311 (5985): 419–424, Bibcode:1984Natur.311..419W, doi:10.1038/311419a0, S2CID 4237923.
Miller & Fredkin (2005). Miller, Daniel B.; Fredkin, Edward (2005), "Two-state, reversible, universal cellular automata in three dimensions", Proceedings of the 2nd Conference on Computing Frontiers (CF '05), New York, NY, USA: ACM, pp. 45–51, arXiv:nlin/0501022, doi:10.1145/1062261.1062271, ISBN 1-59593-019-1, S2CID 14082792.
In the one-dimensional case, several of these equivalences were already presented, in the language of dynamical systems rather than cellular automata, by Hedlund (1969), Theorem 4.1. For higher dimensions, see Richardson (1972) and Di Gregorio & Trautteur (1975). Hedlund, G. A. (1969), "Endomorphisms and automorphisms of the shift dynamical systems", Mathematical Systems Theory, 3 (4): 320–375, doi:10.1007/BF01691062, MR 0259881, S2CID 21803927. Richardson, D. (1972), "Tessellations with local transformations", Journal of Computer and System Sciences, 6 (5): 373–388, doi:10.1016/S0022-0000(72)80009-6, MR 0319678. Di Gregorio, S.; Trautteur, G. (1975), "On reversibility in cellular automata", Journal of Computer and System Sciences, 11 (3): 382–391, doi:10.1016/S0022-0000(75)80059-6, MR 0392201.
Hedlund (1969). Hedlund, G. A. (1969), "Endomorphisms and automorphisms of the shift dynamical systems", Mathematical Systems Theory, 3 (4): 320–375, doi:10.1007/BF01691062, MR 0259881, S2CID 21803927.
Pomeau (1984); Takesue (1990); Capobianco & Toffoli (2011). Pomeau, Y. (1984), "Invariants in cellular automata", Journal of Physics A: Mathematical and General, 17 (8): L415–L418, Bibcode:1984JPhA...17L.415P, doi:10.1088/0305-4470/17/8/004, MR 0750565. Takesue, Shinji (1990), "Relaxation properties of elementary reversible cellular automata", Cellular automata: theory and experiment (Los Alamos, NM, 1989), Physica D: Nonlinear Phenomena, 45 (1–3): 278–284, Bibcode:1990PhyD...45..379K, doi:10.1016/0167-2789(90)90195-U, MR 1094882. Capobianco, Silvio; Toffoli, Tommaso (2011), "Can anything from Noether's theorem be salvaged for discrete dynamical systems?", Proceedings of the 10th International Conference on Unconventional Computation (UC 2011), Lecture Notes in Computer Science, vol. 6714, Springer-Verlag, pp. 77–88, arXiv:1103.4785, doi:10.1007/978-3-642-21341-0_13, S2CID 42541816.
Fredkin & Toffoli (1982). Fredkin, Edward; Toffoli, Tommaso (1982), "Conservative logic", International Journal of Theoretical Physics, 21 (3–4): 219–253, Bibcode:1982IJTP...21..219F, doi:10.1007/BF01857727, MR 0657156, S2CID 37305161. Reprinted in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 47–82.
Meyer (1996); Schumacher & Werner (2004); Shepherd, Franz & Werner (2006); Nagaj & Wocjan (2008). Meyer, David A. (1996), "From quantum cellular automata to quantum lattice gases", Journal of Statistical Physics, 85 (5–6): 551–574, arXiv:quant-ph/9604003, Bibcode:1996JSP....85..551M, doi:10.1007/BF02199356, MR 1418805, S2CID 661940. Schumacher, B.; Werner, R. F. (2004), Reversible quantum cellular automata, arXiv:quant-ph/0405174, Bibcode:2004quant.ph..5174S. Shepherd, D. J.; Franz, T.; Werner, R. F. (2006), "A universally programmable quantum cellular automaton", Physical Review Letters, 97 (2): 020502, arXiv:quant-ph/0512058, Bibcode:2006PhRvL..97b0502S, doi:10.1103/PhysRevLett.97.020502, PMID 16907423, S2CID 40900768. Nagaj, Daniel; Wocjan, Pawel (2008), "Hamiltonian quantum cellular automata in one dimension", Physical Review A, 78 (3): 032311, arXiv:0802.0886, Bibcode:2008PhRvA..78c2311N, doi:10.1103/PhysRevA.78.032311, S2CID 18879990.

stephenwolfram.com

Margolus (1984); Vichniac (1984); Wolfram (1984). Margolus, Norman (1984), "Physics-like models of computation", Physica D: Nonlinear Phenomena, 10 (1–2): 81–95, Bibcode:1984PhyD...10...81M, doi:10.1016/0167-2789(84)90252-5, MR 0762656. Reprinted in Wolfram, Stephen (1986), Theory and Applications of Cellular Automata, Advanced series on complex systems, vol. 1, World Scientific, pp. 232–246, Bibcode:1986taca.book.....W, and in Adamatzky, Andrew, ed. (2002), Collision-Based Computing, Springer-Verlag, pp. 83–104. Vichniac, Gérard Y. (1984), "Simulating physics with cellular automata", Physica D: Nonlinear Phenomena, 10 (1–2): 96–115, Bibcode:1984PhyD...10...96V, doi:10.1016/0167-2789(84)90253-7, MR 0762657. Wolfram, Stephen (1984), "Cellular automata as models of complexity" (PDF), Nature, 311 (5985): 419–424, Bibcode:1984Natur.311..419W, doi:10.1038/311419a0, S2CID 4237923.

unicentro.br

web01.unicentro.br

Toffoli & Margolus (1987), section 12.8.2, "Critters", pp. 132–134; Margolus (1999); Marotta (2005). Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8. Margolus, Norman (1999), "Crystalline computation", in Hey, Anthony J. G. (ed.), Feynman and Computation, Perseus Books, pp. 267–305, arXiv:comp-gas/9811002, Bibcode:1998comp.gas.11002M. Marotta, Sebastian M. (2005), "Living in Critters' world", Revista Ciências Exatas e Naturais, 7 (1), archived from the original on 2012-03-19.

uwe.ac.uk

eprints.uwe.ac.uk

Boykett (2004). See Hillman (1991) and Seck Tuoh Mora et al. (2005) for closely related work on the enumeration of width-2 reversible cellular automata. Boykett, Tim (2004), "Efficient exhaustive listings of reversible one dimensional cellular automata", Theoretical Computer Science, 325 (2): 215–247, doi:10.1016/j.tcs.2004.06.007, MR 2086738. Hillman, David (1991), "The structure of reversible one-dimensional cellular automata", Physica D: Nonlinear Phenomena, 52 (2–3): 277–292, Bibcode:1991PhyD...52..277H, doi:10.1016/0167-2789(91)90128-V, MR 1128996. Seck Tuoh Mora, Juan Carlos; Chapa Vergara, Sergio V.; Juárez Martínez, Genaro; McIntosh, Harold V. (2005), "Procedures for calculating reversible one-dimensional cellular automata" (PDF), Physica D: Nonlinear Phenomena, 202 (1–2): 134–141, Bibcode:2005PhyD..202..134S, doi:10.1016/j.physd.2005.01.018, MR 2131890.

web.archive.org

Toffoli & Margolus (1987), section 12.8.2, "Critters", pp. 132–134; Margolus (1999); Marotta (2005). Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8. Margolus, Norman (1999), "Crystalline computation", in Hey, Anthony J. G. (ed.), Feynman and Computation, Perseus Books, pp. 267–305, arXiv:comp-gas/9811002, Bibcode:1998comp.gas.11002M. Marotta, Sebastian M. (2005), "Living in Critters' world", Revista Ciências Exatas e Naturais, 7 (1), archived from the original on 2012-03-19.
Boykett, Kari & Taati (2008). Boykett, Tim; Kari, Jarkko; Taati, Siamak (2008), "Conservation laws in rectangular CA" (PDF), Journal of Cellular Automata, 3 (2): 115–122, MR 2394641, archived from the original (PDF) on 2015-09-30.

wolframscience.com

Wolfram (2002), p. 1018. Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8, MR 1920418
Wolfram (2002), p. 1093. Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8, MR 1920418
Toffoli & Margolus (1987), Section 14.2, "Second-order technique", pp. 147–149. Wolfram (2002), pp. 437ff. McIntosh (2009). Toffoli, Tommaso; Margolus, Norman (1987), Cellular Automata Machines: A New Environment for Modeling, MIT Press, ISBN 978-0-262-20060-8. Wolfram, Stephen (2002), A New Kind of Science, Wolfram Media, ISBN 1-57955-008-8, MR 1920418 McIntosh, Harold V. (2009), "12. Reversible cellular automata", One Dimensional Cellular Automata, Luniver Press, pp. 205–246.