Gurevich: "... Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage, an algorithm is a computational process defined by a Turing machine ".

There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables ( see more at finite state machine, state transition table and control table ), as flowcharts ( see more at state diagram ), or as a form of rudimentary machine code or assembly code called " sets of quadruples " ( see more at Turing machine ).

But Minsky shows ( as do Melzak and Lambek ) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment / replacement / substitution, and HALT.

Turing is widely considered to be the father of computer science and artificial intelligence.

We could, alternatively, choose an encoding for Turing machines, where an encoding is a function which associates to each Turing Machine M a bitstring < M >.

If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.

This is formalised by a human-assisted Turing machine.

The complexity of executing an algorithm with a human-assisted Turing machine is given by a pair, where the first element represents the complexity of the human's part and the second element is the complexity of the machine's part.

The complexity of solving the following problems with a human-assisted Turing machine is:

This extreme growth can be exploited to show that f, which is obviously computable on a machine with infinite memory such as a Turing machine and so is a computable function, grows faster than any primitive recursive function and is therefore not primitive recursive.

Among the famous mathematicians and cryptanalysts working there, the most influential and the best-known in later years was Alan Turing who is widely credited with being " The Father of Computer Science ".

In computational complexity theory, BPP, which stands for bounded-error probabilistic polynomial time is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1 / 3 for all instances.

This class is defined for a quantum computer and its natural corresponding class for an ordinary computer ( or a Turing machine plus a source of randomness ) is BPP.

" is too loaded with spurious connotations to be meaningful ; but he proposed to replace all such questions with a specific operational test, which has become known as the Turing test.

The Turing test is commonly cited in discussions of artificial intelligence as a proposed criterion for machine consciousness ; it has provoked a great deal of philosophical debate.

For example, Daniel Dennett and Douglas Hofstadter argue that anything capable of passing the Turing test is necessarily conscious, while David Chalmers argues that a philosophical zombie could pass the test, yet fail to be conscious.

Another way of putting the argument is to say computational computer programs can pass the Turing test for processing the syntax of a language, but that semantics cannot be reduced to syntax in the way Strong AI advocates hoped: processing semantics is conscious and intentional because we use semantics to consciously produce meaning by what we say.

He mentions the beats only once '', when he refers to their having revived through mere power and abandonment and the unwillingness to, commit death in life some idea of a decent equivalent between verbal expression and actual experience,, but the entire narrative, is written in the tiresome vocabulary `` of '' that lost `` and '' dying cause, `` and in the '' `` sprung syntax that is supposed to supplant, our mother, tongue.

In the event the total of rupees accruing to the Government of the United States of America as a consequence of sales made pursuant to this Agreement is different from the rupee equivalent of $1,276 million, the amounts available for the purposes specified in paragraph 1, Article 2, will be adjusted proportionately.

In these readings, the double bass is either kept discreetly in the background, or it is dressed in clown's attire -- the musical equivalent of a bull in a china shop.

The cross-sectional area of the cylinders is determined and then the volume of the individual cylinders is computed by multiplying the area by the stroke length, which is the equivalent of the length of the cylinders.

It is well to bear in mind that gasoline will cost from $.80 to $.90 for the equivalent of a United States gallon and while you might prefer a familiar Ford, Chevrolet or even a Cadillac, which are available in some countries, it is probably wiser to choose the smaller European makes which average thirty, thirty-five and even forty miles to the gallon.

For example, when the film is only four minutes old, Neitzbohr refers to a small, Victorian piano stool as `` Wilhelmina '', and we are thereupon subjected to a flashback that informs us that this very piano stool was once used by an epileptic governess whose name, of course, was Doris ( the English equivalent, when passed through middle-Gaelic derivations, of Wilhelmina ).

It is convenient to classify a child's onset ages and completion ages as `` advanced '', `` moderate '' ( modal ), or `` delayed '' according to whether the child's age equivalent `` dots '' appeared to the left of, upon, or to the right of the appropriate short transverse line.

when it represents only itself and on which is its complement ( so that go on is semantically equivalent to board ), on has stronger stress than go does.

When I have instructions to leave is equivalent in meaning to I have instructions that I am to leave this place, dominant stress is ordinarily on leave.

When the same sequence is equivalent in meaning to I have instructions which I am to leave, dominant stress is ordinarily on instructions.

In the first of these sentences if by is the complement of come and Tuesday is an adjunct of time equivalent to on Tuesday, there will be strong stress on by in the spoken language ; ;

In the second sentence if drinking water is a gerundial clause and without drinking water is roughly equivalent in meaning to unless I drink water, there will be stronger stress on water than on drinking ; ;

but if drinking is a gerundial noun modifying water and without drinking water is equivalent to without water for drinking, there will be stronger stress on drinking than on water.

In the Steiners have busy lives without visiting relatives only context can indicate whether visiting relatives is equivalent in meaning to paying visits to relatives or to relatives who are visiting them, and in I looked up the number and I looked up the chimney only the meanings of number and chimney make it clear that up is syntactically a second complement in the first sentence and a preposition followed by its object in the second.

The Councilman, who is the Administration floor leader, also criticized Bernard L. Werner, public works director, for `` halting snow operations '' on Tuesday night after the Sunday storm.

In the computer science subfield of algorithmic information theory, a Chaitin constant ( Chaitin omega number ) or halting probability is a real number that informally represents the probability that a randomly constructed program will halt.

Although there are infinitely many halting probabilities, it is common to use the letter Ω to refer to them as if there were only one.

Each halting probability is a normal and transcendental real number which is not computable, which means that there is no algorithm that enumerates its digits.

The domain is always Turing equivalent to the halting problem.

Let the program for which the halting problem is to be solved be N bits long.

Because many outstanding problems in number theory, such as Goldbach's conjecture are equivalent to solving the halting problem for special programs ( which would basically search for counter-examples and halt if one is found ), knowing enough bits of Chaitin's constant would also imply knowing the answer to these problems.

But as the halting problem is not generally solvable, and therefore calculating any but the first few bits of Chaitin's constant is not possible, this just reduces hard problems to impossible ones, much like trying to build an oracle machine for the halting problem would be.

It is from this interpretation that halting probabilities take their name.

In this way, Ω < sub > F </ sub > represents the probability that a randomly selected infinite sequence of 0s and 1s begins with a bit string ( of some finite length ) that is in the domain of F. It is for this reason that Ω < sub > F </ sub > is called a halting probability.

Not every set that is Turing equivalent to the halting problem is a halting probability.

No halting probability is computable.

The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n. Since the halting problem is undecidable, Ω can not be computed.