Iterative Arrays with Finite Inter-cell Communication

Abstract

Iterative arrays whose internal inter-cell communication is quantitatively restricted are investigated. The quantity of communication is measured by counting the number of uses of the links between cells. In particular, iterative arrays are studied where the maximum number of communications per cell occurring in accepting computations is drastically bounded by a constant number. Additionally, the iterative arrays have to work in realtime. We study the computational capacity of such devices. One main result is that a strict and dense hierarchy with respect to the constant number of communications exists. Due to their very restricted communication, the question arises whether the usually studied decidability problems such as, for example, emptiness, finiteness, inclusion, or equivalence become decidable for such devices. However, by reduction of Hilbert's tenth problem it can be shown that all such decidability questions remain undecidable.

Iterative arrays with finite inter-cell communication

“…It is shown in Kutrib and Malcher (2019) by a reduction of Hilbert's tenth problem that emptiness is undecidable for realtime MCðconstÞ-IAs. However, this proof does not provide a precise constant, since the number of communications per cell in the construction depends on the polynomial given in Hilbert's tenth problem.…”

“…We would like to note that a preliminary version of this paper has been presented in Kutrib and Malcher (2019). Here, we have provided the full proofs of the results in Sect.…”

“…3. Moreover, the undecidability results in Kutrib and Malcher (2019) have been obtained by a reduction of Hilbert's tenth problem. Here, a different reduction is chosen which leads to more precise results in Sect.…”

“…Acknowledgements The authors would like to thank one anonymous reviewer of the conference version (Kutrib and Malcher 2019) who gave us the valuable hint to consider the simulation of counter machines by cellular automata given in Carton et al (2018). Additionally, the authors gratefully acknowledge the comments and suggestions given by the anonymous referees of the journal submission which improved the presentation of the paper.…”

Graphic modeling in Distributed Autonomous and Asynchronous Automata (DA3)

“…Only the previous state of the cell and its neighbors' states are taken into account. Cellular automata with reduced communication for the purpose of distributed implementation are presented in [50].…”