Learning Cycle Length through Finite Automata
AbstractWe study the space-and-time automaton-complexity of the CYCLE-LENGTH problem. The input is a periodic stream of bits whose cycle length is bounded by a known number n. The output, a number between 1 and n, is the exact cycle length. We also study a related problem, CYCLE-DIVISOR. In the latter problem the output is a large number that divides the cycle length, that is, a number k >> 1 that divides the cycle length, or (in case the cycle length is small) the cycle length itself. The complexity is measured in terms of the SPACE, the logarithm of the number of states in an automaton that solves the problem, and the TIME required to reach a terminal state. We analyze the worst input against a deterministic (pure) automaton, and against a probabilistic (mixed) automaton. In the probabilistic case we require that the probability of computing a correct output is arbitrarily close to one. We establish the following results: o CYCLE-DIVISOR can be solved in deterministic SPACE o(n), and TIME O(n). o CYCLE-LENGTH cannot be solved in deterministic SPACE X TIME smaller than (n^2). o CYCLE-LENGTH can be solved in probabilistic SPACE o(n), and TIME O(n). o CYCLE-LENGTH can be solved in deterministic SPACE O(nL), and TIME O(n/L), for any positive L
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Bibliographic InfoPaper provided by The Center for the Study of Rationality, Hebrew University, Jerusalem in its series Discussion Paper Series with number dp546.
Length: 8 pages
Date of creation: Apr 2010
Date of revision:
This paper has been announced in the following NEP Reports:
- NEP-ALL-2010-06-11 (All new papers)
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- Abraham Neyman, 2008.
"Learning Effectiveness and Memory Size,"
Levine's Working Paper Archive
122247000000001945, David K. Levine.
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