Efficiency
Without loss of generality, the input of Turing machine can be assumed to be in the alphabet {0, 1}; any other finite alphabet can be encoded over {0, 1}. The behavior of a Turing machine M is determined by its transition function. This function can be easily encoded as a string over the alphabet {0, 1} as well. The size of the alphabet of M, the number of tapes it has, and the size of the state space can be deduced from the transition function’s table. The distinguished states and symbols can be identified by their position, e.g. the first two states can by convention be the start and stop states. Consequently, every Turing machine can be encoded as a string over the alphabet {0, 1}. Additionally, we convene that every invalid encoding maps to a trivial Turing machine that immediately halts, and that every Turing machine can have an infinite number of encodings by padding the encoding with an arbitrary number of (say) 1's at the end, just like comments work in a programming language. It should be no surprise that we can achieve this encoding given the existence of a Gödel number and computational equivalence between Turing machines and μ-recursive functions. Similarly, our construction associates to every binary string α, a Turing machine Mα.
Starting from the above encoding, in 1966 F. C. Hennie and R. E. Stearns showed that given a Turing machine Mα that halts on input x within N steps, then there exists a multi-tape universal Turing machine that halts on inputs α, x (given on different tapes) in CN log N, where C is a machine-specific constant that does not depend on the length of the input x, but does depend on M's alphabet size, number of tapes, and number of states. Effectively this is an O(N log N) simulation.
Read more about this topic: Universal Turing Machine
Famous quotes containing the word efficiency:
““Never hug and kiss your children! Mother love may make your children’s infancy unhappy and prevent them from pursuing a career or getting married!” That’s total hogwash, of course. But it shows on extreme example of what state-of-the-art “scientific” parenting was supposed to be in early twentieth-century America. After all, that was the heyday of efficiency experts, time-and-motion studies, and the like.”
—Lawrence Kutner (20th century)
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—Samuel Goldwyn (1882–1974)
“Nothing comes to pass in nature, which can be set down to a flaw therein; for nature is always the same and everywhere one and the same in her efficiency and power of action; that is, nature’s laws and ordinances whereby all things come to pass and change from one form to another, are everywhere and always; so that there should be one and the same method of understanding the nature of all things whatsoever, namely, through nature’s universal laws and rules.”
—Baruch (Benedict)