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The halting problem is a decision problem about properties of computer programs on a fixed Turing-complete model of computation, i.e., all programs that can be written in some given programming language that is general enough to be equivalent to a Turing machine. The problem is to determine, given a program and an input to the program, whether ...
The Turing test does not test for highly intelligent behaviours, such as the ability to solve difficult problems or come up with original insights. In fact, it specifically requires deception on the part of the machine: if the machine is more intelligent than a human being it must deliberately avoid appearing too intelligent.
An example: Suppose machine H has tested 13472 numbers and produced 5 satisfactory numbers, i.e. H has converted the numbers 1 through 13472 into S.D's (symbol strings) and passed them to D for test. As a consequence H has tallied 5 satisfactory numbers and run the first one to its 1st "figure", the second to its 2nd figure, the third to its ...
One of the most interesting aspects of the busy beaver game is that, if it were possible to compute the functions Σ(n) and S(n) for all n, then this would resolve all mathematical conjectures which can be encoded in the form "does <this Turing machine> halt". [5] For example, a 27-state Turing machine could check Goldbach's conjecture for each ...
For example, Turing's 1936 paper (p. 263) observed that since the halting problem for each Turing machine is equivalent to a first-order logical formula of form , the problem () is undecidable. The precise boundaries are known, sharply:
The success of the Church–Turing thesis prompted variations of the thesis to be proposed. For example, the physical Church–Turing thesis states: "All physically computable functions are Turing-computable." [54]: 101 The Church–Turing thesis says nothing about the efficiency with which one model of computation can simulate another.
For example, let U be the set of natural numbers and S the set of prime numbers. The corresponding decision problem corresponds to primality testing. A function problem consists of a function f from a set U to a set V. An instance of the problem is to compute, given an element u in U, the corresponding element f(u) in V.
The Visual Turing Test (VTT) unlike the Turing test has a query engine system which interrogates a computer vision system in the presence of a human co-ordinator. It is a system that generates a random sequence of binary questions specific to the test image, such that the answer to any question k is unpredictable given the true answers to the ...