Abstract
Turing Tumble is a toy gravity-fed mechanical computer (similar to the classic , but including additional types of pieces such as gears), in which marbles roll down a board, along paths determined by the locations of ramps, toggles and gears, which are placed by the “programmer,” and by their current states, which are altered by the passing marbles. Aaronson proved that a decision problem (viz., will any marbles reach the sink?) is -Complete, i.e., equivalent to evaluating comparator circuits, and posed the question of what additional functionality would raise the machine’s computational power beyond , speculating that a capability for toggles to affect one another’s states (which gears happen to provide) might suffice. This turns out to be so: we show, though a simple reduction from a variant of the circuit value problem ( ), that the decision problem is -Complete. The two models also differ in complexity when exponentially (or unboundedly) many marbles are permitted: while remains in , becomes -Complete.
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Notes
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Aaronson describes the actual as including some such interaction, but this appears (based on the instruction manual [8]) to be in reference to the effects of the Clear switch and the CF1 toggle, which send the marble to an underground passageway, affecting the pieces above. Since the marbles that Clear and CF1 send underground only affect the pieces they pass directly under, however, this arguably should not count as interaction between pieces.
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Technically speaking, in the actual product, nothing physically prevents gears from being placed on regular pegs as well, but there is no loss of generality in assuming that gear bits, rather than gears, will be placed on those pegs.
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More formally, gear components can be defined equivalently as follows. Consider the induced subgraph of the solid (undirected) grid graph that is induced by the gears and gear bits, i.e., each edge corresponds to a gear and a gear bit that are rectilinearly adjacent. Then a gear component is a component of this graph.
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Our is somewhat in the spirit of [17]’s Sequential NorCVP.
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Acknowledgements
This work was supported in part by NSF award INSPIRE-1547205, and by the Sloan Foundation via a CUNY Junior Faculty Research Award.
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Johnson, M.P. (2019). Turing Tumble Is P(SPACE)-Complete. In: Heggernes, P. (eds) Algorithms and Complexity. CIAC 2019. Lecture Notes in Computer Science(), vol 11485. Springer, Cham. https://doi.org/10.1007/978-3-030-17402-6_23
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