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Counting is Easy | | Joel Seiferas
; Paul Vitanyi
; | | Date: |
18 Oct 2001 | | Journal: | J. Seiferas and P.M.B. Vitanyi, Counting is easy, J. Assoc. Comp. Mach. 35 (1988), pp. 985-1000 | | Subject: | Computational Complexity; Data Structures and Algorithms ACM-class: E.1, F.1.1., F.2.2, G.2.1, E.2, E.4 | cs.CC cs.DS | | Affiliation: | University of Rochester) and Paul Vitanyi (CWI and University of Amsterdam | | Abstract: | For any fixed $k$, a remarkably simple single-tape Turing machine can simulate $k$ independent counters in real time. Informally, a counter is a storage unit that maintains a single integer (initially 0), incrementing it, decrementing it, or reporting its sign (positive, negative, or zero) on command. Any automaton that responds to each successive command as a counter would is said to simulate a counter. (Only for a sign inquiry is the response of interest, of course. And zeroness is the only real issue, since a simulator can readily use zero detection to keep track of positivity and negativity in finite-state control. In this paper we describe a remarkably simple real-time simulation, based on just five simple rewriting rules, of any fixed number $k$ of independent counters. On a Turing machine with a single, binary work tape, the simulation runs in real time, handling an arbitrary counter command at each step. The space used by the simulation can be held to $(k+epsilon) log_2 n$ bits for the first $n$ commands, for any specified $epsilon > 0$. | | Source: | arXiv, cs.CC/0110038 | | Services: | Forum | Review | PDF | Favorites | |
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