DLOGTIME
In computational complexity theory, DLOGTIME is the complexity class of all computational problems solvable in a logarithmic amount of computation time on a deterministic Turing machine. It must be defined on a random-access Turing machine, since otherwise the input tape is longer than the range of cells that can be accessed by the machine. It is a very weak model of time complexity: no random-access Turing machine with a smaller deterministic time bound can access the whole input.[1]
DLOGTIME machines are rarely used as-is. Instead, they are usually used as part of a theoretical construct for studying reducibility. They cannot be used to produce output that is longer than logarithmic length. However, they can be used to indirectly produce such an output, in the following sense: Given any log-length binary address, it produces the corresponding bit of the output. Since a polynomial-sized integer has logarithmic-length binary representation, this is possible in DLOGTIME.
Examples
[edit]DLOGTIME includes problems relating to verifying the length of the input,[1] for example the problem "Is the input of even length?", which can be solved in logarithmic time using binary search.
Applications
[edit]DLOGTIME-uniformity is important in circuit complexity. Specifically, a boolean circuit family can be regarded as an abstract language. Each circuit in the family is a word in the language, and each circuit outside the family is not a word in the language.
A family is DLOGTIME-uniform iff it, as a language, can be recognized by a random-access Turing machine in DLOGTIME.[1][2]
References
[edit]- ^ a b c Johnson, David S. (1990), "A catalog of complexity classes", Handbook of theoretical computer science, Vol. A, Elsevier, Amsterdam, pp. 67–161, MR 1127168. See in particular p. 140.
- ^ Allender, Eric; Gore, Vivek (1993), "On strong separations from AC0", Advances in computational complexity theory (New Brunswick, NJ, 1990), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 13, Amer. Math. Soc., Providence, RI, pp. 21–37, MR 1255326. See in particular p. 23.