## P-completeness of cellular automaton Rule 110 (2006)

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Venue: | In International Colloquium on Automata Languages and Programming (ICALP), volume 4051 of LNCS |

Citations: | 20 - 7 self |

### BibTeX

@INPROCEEDINGS{Neary06p-completenessof,

author = {Turlough Neary and Damien Woods},

title = {P-completeness of cellular automaton Rule 110},

booktitle = {In International Colloquium on Automata Languages and Programming (ICALP), volume 4051 of LNCS},

year = {2006},

pages = {132--143},

publisher = {Springer}

}

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### Abstract

We show that the problem of predicting t steps of the 1D cellular automaton Rule 110 is P-complete. The result is found by showing that Rule 110 simulates deterministic Turing machines in polynomial time. As a corollary we find that the small universal Turing machines of Mathew Cook run in polynomial time, this is an exponential improvement on their previously known simulation time overhead.

### Citations

3836 |
J.D.: Introduction to automata theory, languages, and computation
- Hopcroft, Motwani, et al.
(Show Context)
Citation Context ...as been asked, either directly or indirectly, in a number of previous works (for example [14–16]). 2 Clockwise Turing machines A clockwise Turing machine is like a standard single-tape Turing machine =-=[17]-=- except for the following details: (i) the tape is assumed to be circular, (ii) the tape head moves only clockwise on the tape, (iii) the machine’s transition function is of the form f : Q×Σ → (Σ ∪ΣΣ)... |

496 | A New Kind of Science
- Wolfram
- 2002
(Show Context)
Citation Context ...llular automaton, and Matthew Cook [2, 3] gave four small universal Turing machines 3 that simulate its computation. Their size given as (number of states, number of symbols), are respectively (2, 5),=-=(3, 4)-=-,(4, 3) and (7, 2). In terms of program size these machines are a significant improvement on previous small universal Turing machine results [5–10]. However in terms of time complexity Cook’s machines... |

214 |
Statistical mechanics of cellular automata
- Wolfram
- 1983
(Show Context)
Citation Context ...of Rule 110 which is one of the simplest cellular automata. We show that the prediction problem for Rule 110 is P-complete. Rule 110 is a nearest neighbour, one dimensional, binary cellular automaton =-=[1]-=-. It is composed of a sequence of cells . . . p−1p0p1 . . . where each cell has a binary state pi ∈ {0, 1}. At timestep t + 1 the value of cell pi,t+1 = F (pi−1,t, pi,t, pi+1,t) is given by the synchr... |

75 |
Universality in elementary cellular automata
- Cook
(Show Context)
Citation Context ... in unary. Is cell pi in state 1 at time t? This problem is in P as a Turing machine simulates the cellular automaton in O(t 2 ) steps by repeatedly traversing from left to right. From Matthew Cook’s =-=[2]-=- result one infers a NC lower bound on the problem. Cook showed that Rule 110 simulates Turing machines via the following sequence of simulations Turing machine ↦→ 2-tag system ↦→ cyclic tag system ↦→... |

49 |
Universal computation in simple onedimensional cellular automata
- Lindgren, Nordahl
- 1990
(Show Context)
Citation Context ...ta (e.g. more states, neighbours or dimensions). For example prediction of cellular automata of dimension d � 1 with an arbitrary number of states is known to be P-complete [10]. Lindgren and Nordahl =-=[11]-=- show that prediction for one dimensional nearest neighbour cellular automata is P-complete for seven states and Ollinger’s result [12] improves this to six. If the update rule depends on the states o... |

27 | Small universal Turing machines
- Rogozhin
- 1996
(Show Context)
Citation Context ... cellular automaton, and Matthew Cook [2] gave four small universal Turing machines 3 that simulate its computation. Their size given as (number of states, number of symbols), are respectively (2, 5),=-=(3, 4)-=-,(4, 3) and (7, 2). In terms of program size these machines are a significant improvement on previous small universal Turing machine results [4–9]. However in terms of time complexity Cook’s machines ... |

21 | Majority-vote cellular automata, ising dynamics, and pcompleteness
- Moore
- 1997
(Show Context)
Citation Context ...ar automata is P-complete for seven states and Ollinger’s result [12] improves this to six. If the update rule depends on the states of five neighbours then four states are sufficient [11, 10]. Moore =-=[13]-=- shows that prediction of binary majority voting cellular automata is P-complete for dimension d � 3. On the other hand, the prediction problem for a variety 3 Cook’s small “universal Turing machines”... |

20 |
Universality of Tag Systems With P = 2
- Cocke, Minsky
- 1964
(Show Context)
Citation Context ... cellular automaton, and Matthew Cook [2] gave four small universal Turing machines 3 that simulate its computation. Their size given as (number of states, number of symbols), are respectively (2, 5),=-=(3, 4)-=-,(4, 3) and (7, 2). In terms of program size these machines are a significant improvement on previous small universal Turing machine results [4–9]. However in terms of time complexity Cook’s machines ... |

18 |
Size and structure of a Universal Turing Machine using tag systems
- Minsky
- 1962
(Show Context)
Citation Context ...n, and Matthew Cook [2] gave four small universal Turing machines 3 that simulate its computation. Their size given as (number of states, number of symbols), are respectively (2, 5),(3, 4),(4, 3) and =-=(7, 2)-=-. In terms of program size these machines are a significant improvement on previous small universal Turing machine results [4–9]. However in terms of time complexity Cook’s machines offer no improveme... |

18 | Predicting nonlinear cellular automata quickly by decomposing them into linear ones. Physica D: Nonlinear Phenomena
- Moore
- 1998
(Show Context)
Citation Context ...repeated words. This has no bearing on our P-completeness result as we require only a bounded initial configuration for Rule 110 prediction. 2 (2)sof linear and quasilinear cellular automata is in NC =-=[14, 15]-=-. The question of whether Rule 110 prediction is P-complete has been asked, either directly or indirectly, in a number of previous works (for example [14–16]). 2 Clockwise Turing machines A clockwise ... |

17 |
Three Small Universal Turing Machines
- Baiocchi
(Show Context)
Citation Context ...sional) cellular automaton, and Matthew Cook [2] gave four small universal Turing machines 3 that simulate its computation. Their size given as (number of states, number of symbols), are respectively =-=(2, 5)-=-,(3, 4),(4, 3) and (7, 2). In terms of program size these machines are a significant improvement on previous small universal Turing machine results [4–9]. However in terms of time complexity Cook’s ma... |

15 | Four fast universal Turing machines
- Neary, Woods
- 2009
(Show Context)
Citation Context ...ft move simulation is complete. A left move by M, when reading a blank at the leftmost tape end, is simulated using a similar strategy to that above. Proof details are to be found in a previous paper =-=[8]-=-. Lemma 1. Let M be a single-tape Turing machine that runs in time T (n). Then there is a clockwise Turing machine RM that simulates the computation of M in time O(T 2 (n)). In the next section we pro... |

14 | Non-abelian cellular automata
- Moore
- 1998
(Show Context)
Citation Context ...repeated words. This has no bearing on our P-completeness result as we require only a bounded initial configuration for Rule 110 prediction. 2 (2)sof linear and quasilinear cellular automata is in NC =-=[14, 15]-=-. The question of whether Rule 110 prediction is P-complete has been asked, either directly or indirectly, in a number of previous works (for example [14–16]). 2 Clockwise Turing machines A clockwise ... |

13 |
The Quest for Small Universal Cellular Automata
- Ollinger
- 2002
(Show Context)
Citation Context ...r of states is known to be P-complete [10]. Lindgren and Nordahl [11] show that prediction for one dimensional nearest neighbour cellular automata is P-complete for seven states and Ollinger’s result =-=[12]-=- improves this to six. If the update rule depends on the states of five neighbours then four states are sufficient [11, 10]. Moore [13] shows that prediction of binary majority voting cellular automat... |

9 | A universal Turing machine with 3 states and 9 symbols - Kudlek, Rogozhin - 2002 |

8 |
W.L.: Limits to Parallel Computation: PCompleteness Theory
- Greenlaw, Hoover, et al.
- 1995
(Show Context)
Citation Context ...ore general cellular automata (e.g. more states, neighbours or dimensions). For example prediction of cellular automata of dimension d � 1 with an arbitrary number of states is known to be P-complete =-=[10]-=-. Lindgren and Nordahl [11] show that prediction for one dimensional nearest neighbour cellular automata is P-complete for seven states and Ollinger’s result [12] improves this to six. If the update r... |

2 | Book review: A new kind of science - Aaronson |