8 research outputs found

    Some properties of one-pebble Turing machines with sublogarithmic space

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    AbstractThis paper investigates some aspects of the accepting powers of deterministic, nondeterministic, and alternating one-pebble Turing machines with spaces between loglogn and logn. We first investigate a relationship between the accepting powers of two-way deterministic one-counter automata and deterministic (or nondeterministic) one-pebble Turing machines, and show that they are incomparable. Then we investigate a relationship between nondeterminism and alternation, and show that there exists a language accepted by a strongly loglogn space-bounded alternating one-pebble Turing machine, but not accepted by any weakly o(logn) space-bounded nondeterministic one-pebble Turing machine. Finally, we investigate a space hierarchy, and show that for any one-pebble (fully) space constructible function L(n)⩽logn, and for any function L′(n)=o(L(n)), there exists a language accepted by a strongly L(n) space-bounded deterministic one-pebble Turing machine, but not accepted by any weakly L′(n) space-bounded nondeterministic one-pebble Turing machine

    Translation from Classical Two-Way Automata to Pebble Two-Way Automata

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    We study the relation between the standard two-way automata and more powerful devices, namely, two-way finite automata with an additional "pebble" movable along the input tape. Similarly as in the case of the classical two-way machines, it is not known whether there exists a polynomial trade-off, in the number of states, between the nondeterministic and deterministic pebble two-way automata. However, we show that these two machine models are not independent: if there exists a polynomial trade-off for the classical two-way automata, then there must also exist a polynomial trade-off for the pebble two-way automata. Thus, we have an upward collapse (or a downward separation) from the classical two-way automata to more powerful pebble automata, still staying within the class of regular languages. The same upward collapse holds for complementation of nondeterministic two-way machines. These results are obtained by showing that each pebble machine can be, by using suitable inputs, simulated by a classical two-way automaton with a linear number of states (and vice versa), despite the existing exponential blow-up between the classical and pebble two-way machines

    Tight bounds for undirected graph exploration with pebbles and multiple agents

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    We study the problem of deterministically exploring an undirected and initially unknown graph with nn vertices either by a single agent equipped with a set of pebbles, or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles Θ(logn)\Theta(\log n) bits of memory are necessary and sufficient to explore any graph with at most nn vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles, or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory Θ(loglogn)\Theta(\log \log n) pebbles are necessary and sufficient for exploration. We further prove that the same bound holds for the number of collaborating agents needed for exploration. For the upper bound, we devise an algorithm for a single agent with constant memory that explores any nn-vertex graph using O(loglogn)\mathcal{O}(\log \log n) pebbles, even when nn is unknown. The algorithm terminates after polynomial time and returns to the starting vertex. Since an additional agent is at least as powerful as a pebble, this implies that O(loglogn)\mathcal{O}(\log \log n) agents with constant memory can explore any nn-vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph of size nn is already Ω(loglogn)\Omega(\log \log n) when we allow each agent to have at most O(logn1ε)\mathcal{O}( \log n ^{1-\varepsilon}) bits of memory for any ε>0\varepsilon>0. This also implies that a single agent with sublogarithmic memory needs Θ(loglogn)\Theta(\log \log n) pebbles to explore any nn-vertex graph

    36th International Symposium on Theoretical Aspects of Computer Science: STACS 2019, March 13-16, 2019, Berlin, Germany

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    27th Annual European Symposium on Algorithms: ESA 2019, September 9-11, 2019, Munich/Garching, Germany

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    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum
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