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Complexity-Theoretic Aspects of Expanding Cellular Automata

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 11525)


The expanding cellular automata (XCA) variant of cellular automata is investigated and characterized from a complexity-theoretical standpoint. The respective polynomial-time complexity class is shown to coincide with , that is, the class of decision problems polynomial-time truth-table reducible to problems in . Corollaries on select XCA variants are proven: XCAs with multiple accept and reject states are shown to be polynomial-time equivalent to the original XCA model. Meanwhile, XCAs with diverse acceptance behavior are classified in terms of and the Turing machine polynomial-time class .

Parts of this paper have been submitted [13] in partial fulfillment of the requirements for the degree of Master of Science at the Karlsruhe Institute of Technology (KIT).

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Fig. 1.
Fig. 2.


  1. 1.

    That is, \(\delta (\ell ) = a\) for \(\ell (0) = r\) and vice-versa, where \(\delta \) and \(\ell \) are as in Definition 2.

  2. 2.

    This may be accomplished, for example, by using a bit counter in the cells’ states, and having cells wait for a step before transitioning to an accept or reject state if needed.

  3. 3.

    That is to say, each \(c_i\) corresponds to the so-called extended \((\tau - i)\)-neighborhood of a cell of A.

  4. 4.

    In the sense of evaluating to the same truth value under the respective interpretations (see Definition 1).

  5. 5.

    An allusion to the existential states of alternating Turing machines (ATMs).


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The author would like to thank Thomas Worsch for his mentoring, encouragement, and support during the writing of this paper. The author would also like to thank Dennis Hofheinz for pointing out a crucial mistake in a preliminary version of this paper as well as the anonymous referees for their valuable remarks and suggestions.

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Modanese, A. (2019). Complexity-Theoretic Aspects of Expanding Cellular Automata. In: Castillo-Ramirez, A., de Oliveira, P. (eds) Cellular Automata and Discrete Complex Systems. AUTOMATA 2019. Lecture Notes in Computer Science(), vol 11525. Springer, Cham.

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