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TuringMobile: a turing machine of oblivious mobile robots with limited visibility and its applications

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Abstract

In this paper we investigate the computational power of a set of mobile robots with limited visibility. At each iteration, a robot takes a snapshot of its surroundings, uses the snapshot to compute a destination point, and it moves toward its destination. Robots are punctiform and memoryless, they operate in \(\mathbb {R}^m\), they have local reference systems independent of each other, and are activated asynchronously by an adversarial scheduler. Moreover, robots are non-rigid, in that they may be stopped by the scheduler at each move before reaching their destination (but are guaranteed to travel at least a fixed unknown distance before being stopped). We show that despite these strong limitations, it is possible to arrange \(3m+3k\) of these weak entities in \(\mathbb {R}^m\) to simulate the behavior of a stronger robot that is rigid (i.e., it always reaches its destination) and is endowed with k registers of persistent memory, each of which can store a real number. We call this arrangement a TuringMobile. In its simplest form, a TuringMobile consisting of only three robots can travel in the plane and store and update a single real number. We also prove that this task is impossible with fewer than three robots. Among the applications of the TuringMobile, we focused on Near-Gathering (all robots have to gather in a small-enough disk) and Pattern Formation (of which Gathering is a special case) with limited visibility. Interestingly, our investigation implies that both problems are solvable in Euclidean spaces of any dimension, even if the visibility graph of the robots is initially disconnected, provided that a small amount of these robots are arranged to form a TuringMobile. In the special case of the plane, a basic TuringMobile of only three robots is sufficient.

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Notes

  1. Another model of swarm robotics, alternative to the LCM one, is the “continuous” model. Some examples are found in [5, 7, 25,26,27].

  2. The real RAM was introduced by Shamos in [29, Chapter 2.3] (see also [30, Chapter 1.4]). It is a modified RAM (i.e., Random-Access Machine, see [3, Chapter 1.2]) that can natively operate on real numbers.

  3. Nonetheless, the constant operands in a real RAM’s program cannot be arbitrary real numbers, but have to be integers.

  4. Observe that, in general, the machine cannot salvage its memory by encoding its contents in the registers: since its instruction set has only analytic functions, it cannot injectively map a tuple of arbitrary real numbers into a single real number.

  5. Note, in particular, that robots do not obstruct each others line of sight: two robots are able to see each other whenever they are at distance at most V, even if there is a third robot between them.

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Acknowledgements

The authors are grateful to the anonymous reviewers for greatly improving the quality of this paper with helpful comments and suggestions.

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Authors and Affiliations

  1. DIAG, Sapienza University of Rome, Via Ariosto 25, Roma, Italy

    Giuseppe A. Di Luna

  2. School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward Avenue, Ottawa, ON, Canada

    Paola Flocchini

  3. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, ON, Canada

    Nicola Santoro

  4. School of Information Science, JAIST, 1-1 Asahidai, Nomi City, Ishikawa Prefecture, 923-1211, Japan

    Giovanni Viglietta

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  1. Giuseppe A. Di Luna
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  2. Paola Flocchini
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  4. Giovanni Viglietta
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Correspondence to Giovanni Viglietta.

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Luna, G.A.D., Flocchini, P., Santoro, N. et al. TuringMobile: a turing machine of oblivious mobile robots with limited visibility and its applications. Distrib. Comput. 35, 105–122 (2022). https://doi.org/10.1007/s00446-021-00406-6

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