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A universal machine U is a machine for which every other machine V there exists a total computable function h such that () = (()). An optimal machine is a universal machine that achieves the Kolmogorov complexity invariance bound , i.e. for every machine V , there exists c such that for all outputs x , if a V -program of length n outputs x ...
This machine was composed of four modified Triumphator calculators. [31] [32] [33] Leslie Comrie in 1928 described how to use the Brunsviga-Dupla calculating machine as a difference engine of second-order (15-digit numbers). [28] He also noted in 1931 that National Accounting Machine Class 3000 could be used as a difference engine of sixth-order.
An oracle machine can be conceived as a Turing machine connected to an oracle. The oracle, in this context, is an entity capable of solving some problem, which for example may be a decision problem or a function problem. The problem does not have to be computable; the oracle is not assumed to be a Turing machine or computer program.
In addition to the function Σ, Radó [1962] introduced another extreme function for Turing machines, the maximum shifts function, S, defined as follows: [4] s(M) = the number of shifts M makes before halting, for any M ∈ E n, S(n) = max{s(M) | M ∈ E n} = the largest number of shifts made by any halting n-state 2-symbol Turing machine.
It was based on Simon and Newell's theoretical work on logic machines. GPS was the first computer program that separated its knowledge of problems (rules represented as input data) from its strategy of how to solve problems (a generic solver engine). GPS was implemented in the third-order programming language, IPL. [2]
If there is an algorithm (say a Turing machine, or a computer program with unbounded memory) that produces the correct answer for any input string of length n in at most cn k steps, where k and c are constants independent of the input string, then we say that the problem can be solved in polynomial time and we place it in the class P. Formally ...
A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently.
For example, it's quite possible to reduce a difficult-to-solve NP-complete problem like the boolean satisfiability problem to a trivial problem, like determining if a number equals zero, by having the reduction machine solve the problem in exponential time and output zero only if there is a solution. However, this does not achieve much ...