Fast Iterative Arrays with Restricted Inter-cell Communication: Constructions and Decidability

  • Martin Kutrib
  • Andreas Malcher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)


Iterative arrays (IAs) are one-dimensional arrays of interconnected interacting finite automata with sequential input mode. We investigate IAs which work in real time and whose inter-cell communication is bounded by some constant number of bits not depending on the number of states. It is known [13] that such IAs can recognize rather complicated unary languages with a minimum amount of communication, namely one-bit communication, in real time. Some examples are the languages \(\{a^{2^n} \mid n \ge 1\}\), \(\{a^{n^2} \mid n \ge 1\}\), and {a p |p is prime}. Here, we consider non-unary languages and it turns out that the non-unary case is quite different. We present several real-time constructions for certain non-unary languages. For example, the languages {a n b n |n ≥1}, {a n (b n ) m |n,m ≥1}, and {a n ba m b(ba) n .m |n,m ≥1} are recognized in real time by 1-bit IAs. Moreover, it is shown that real-time 1-bit IAs can, in some sense, add and multiply integer numbers. Furthermore, closure properties and decidability questions of communication restricted IAs are investigated. Due to the constructions provided, non-closure results as well as undecidability results can be shown. It turns out that emptiness is still undecidable for 1-bit IAs despite their restricted communication. Thus, also the questions of finiteness, infiniteness, inclusion, and equivalence are undecidable.


Cellular Automaton Communication Cell Regular Language Closure Property Input Symbol 
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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Martin Kutrib
    • 1
  • Andreas Malcher
    • 2
  1. 1.Institut für InformatikUniversität GiessenGiessenGermany
  2. 2.Institut für InformatikJohann Wolfgang Goethe UniversitätFrankfurt am MainGermany

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