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Putting it together: the computational complexity of designing robot controllers and environments for distributed construction

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Abstract

Creating target structures through the coordinated efforts of teams of autonomous robots (possibly aided by specific features in their environments) is a very important problem in distributed robotics. Many specific instances of distributed robotic construction teams have been developed manually. An important issue is whether automated controller design algorithms can both quickly produce robot controllers and guarantee that teams using these controllers will build arbitrary requested target structures correctly; this task may also involve specifying features in the environment that can aid the construction process. In this paper, we give the first computational and parameterized complexity analyses of several problems associated with the design of robot controllers and environments for creating target structures. These problems use a simple finite-state robot controller model that moves in a non-continuous deterministic manner in a grid-based environment. Our goal is to establish whether algorithms exist that are both fast and correct for all inputs and if not, under which restrictions such algorithms are possible. We prove that none of these problems are efficiently solvable in general and remain so under a number of plausible restrictions on controllers, environments, and target structures. We also give the first restrictions relative to which these problems are efficiently solvable and discuss what theoretical solvability and unsolvability results derived relative to the problems examined here mean for real-world construction using robot teams.

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Notes

  1. Our conception of FSR determinism is actually very different than the traditional definition of determinism for finite-state automata (Hopcroft et al. 2001, Section 2.2), in which automata can sense a single symbol at a time and each symbol maps to at most one state change (and, in the case of transducers, an associated action). Such automata themselves are deterministic by virtue of the structure of their state transition functions. FSR, on the other hand, can sense and be enabled by arbitrary patterns of squares within radius r of their current position. The number of such patterns that can be encountered is both exceptional large and fluid, as the sensed environment can change as the FSR and other FSR on its team move and/or change the environment. Therefore, an individual FSR cannot itself be deterministic; rather, the operation of that FSR can only be deterministic in the context of a particular FSR team operating in a particular environment.

  2. Note that this corresponds to the simplest possible type of collision avoidance policy, i.e., no collision avoidance at all.

  3. Proofs of all results stated in this section are given in the online supplementary material.

  4. All polynomial-time intractability results in this section hold relative to the \(P \ne NP\) conjecture, which is widely believed to be true (Fortnow 2009; Garey and Johnson 1979).

  5. This result holds relative to both the \(P \ne NP\) conjecture mentioned in Footnote 1 and the \(P = BPP\) conjecture, the latter of which is also widely believed to be true (Clementi et al. 1998; Wigderson 2007).

  6. Proofs of all results stated in this section are given in the online supplementary material.

  7. Each fp-intractability result in this section holds relative to one of the conjectures \(P \ne NP\) or \(FPT \ne W[1]\) (see the online supplementary material for details). Both of these conjectures are widely believed to be true (Downey and Fellows 2013; Fortnow 2009; Garey and Johnson 1979).

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Acknowledgements

The authors would like to thank the anonymous reviewers for many detailed comments and suggestions that helped to significantly improve the presentation and content of this paper. TW was supported by National Science and Engineering Research Council (NSERC) Discovery Grant 228104-2015.

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Authors and Affiliations

  1. Department of Computer Science, Memorial University of Newfoundland, St. John’s, NL, A1B 3X5, Canada

    Todd Wareham

  2. Department of Computer Science and Department of Electrical and Computer Engineering, Memorial University of Newfoundland, St. John’s, NL, A1B 3X5, Canada

    Andrew Vardy

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  1. Todd Wareham
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Correspondence to Todd Wareham.

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Wareham, T., Vardy, A. Putting it together: the computational complexity of designing robot controllers and environments for distributed construction. Swarm Intell 12, 111–128 (2018). https://doi.org/10.1007/s11721-017-0152-7

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