Mathematical Games σ-game, σ+-game and two-dimensional additive cellular automata

Rana Barua, S. Ramakrishnan

Research output: Contribution to journalArticle

29 Scopus citations

Abstract

The σ-game, introduced by Sutner, is a combinatorial game played on a graph G and is closely related to the σ-automaton first studied by Lindenmayer. A related game is the σ+-game. In this article, we study the σ-game (σ+-game) played on the rectangular grid {1,2,...,m} x {1,2,...,n}. We analyse the σ+-game by studying the divisibility properties of the polynomials pn(λ) which we have introduced here. (Similar polynomials were earlier studied by Sutner). We give a simple algorithm for finding the number of solutions for the σ+-game and also give a necessary and sufficient condition for the existence of a unique solution for the σ+-game, thus partially answering a question posed by Sutner. Further, we compute the number of solutions of the σ+-game when one of n, m is of the form 2k - 1. Finally, we look at the σ-game and the σ+-game played on cylinders and tori and give necessary and sufficient conditions for the existence of unique solutions for these games.

Original languageEnglish (US)
Pages (from-to)349-366
Number of pages18
JournalTheoretical Computer Science
Volume154
Issue number2
DOIs
StatePublished - Feb 5 1996

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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