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.

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.

The first actual implementation of a Turing-complete machine appeared in 1941: the program-controlled Z3 of Konrad Zuse, but the first machine explicitly designed to be Turing complete and widely appreciated as being universal was the 1946 ENIAC.

Turing is widely used in high schools in Ontario as an introduction to programming.

Researchers such as Turing and Konrad Zuse investigated the idea of using the computer's memory to hold the program as well as the data it was working on, but it was mathematician John von Neumann who became widely credited with defining that computer architecture, still used in almost all computers.

" The widely accepted Church – Turing thesis holds that any Turing-universal system can simulate any conceivable process that can be digitized, given enough time and memory.

He was highly influential in the development of computer science, giving a formalisation of the concepts of " algorithm " and " computation " with the Turing machine, which can be considered a model of a general purpose computer.

To establish that a function is computable by Turing machine, it is usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude " By the Church – Turing thesis " that the function is Turing computable ( equivalently partial recursive ).

The attribution of thought or thought processes to non-human objects and phenomena ( especially computers ) could be considered anthropomorphism, though such categorizations have been contested by such computer scientists as Alan Turing ( see Computing Machinery and Intelligence ).

Nowadays these are often considered as a single hypothesis, the Church – Turing thesis, which states that any function that is computable by an algorithm is a computable function.

A robot that easily passed the Turing test, and could do some things humans cannot, could be considered superhuman.

Others who have spent time at NPL include Harry Huskey, a computer pioneer ; Alan Turing, one of the fathers of modern digital computing who was largely responsible for the early ACE computer design ; Robert Watson-Watt, generally considered the inventor of radar, Oswald Kubaschewski, the father of computational materials thermodynamics and the numerical analyst James Wilkinson.

Having clarified the question, Turing turned to answering it: he considered the following nine common objections, which include all the major arguments against artificial intelligence raised in the years since his paper was first published.

The HP-41C still supports indirect addressing with which it is possible to implement a Universal Turing machine and therefore the programming model of the HP-41C can be considered Turing complete.

The HP-42S supports indirect addressing with which it is possible to implement a Universal Turing machine and therefore the programming model of the HP-42S can be considered Turing-complete.

In his 1977 Turing award lecture, Backus set forth what he considered to be the need to switch to a different philosophy in programming language design:

Independently, in 1988, Allen H. Brady considered the idea of two-dimensional Turing machines with an orientation and called them " TurNing machines ".

It differs from a Turing machine in that while the tape is initially considered to have unbounded length, only a finite contiguous portion of the tape, whose length is a linear function of the length of the initial input, can be accessed by the read / write head.

Representations of algorithms can be classed into three accepted levels of Turing machine description:

Structured programming, canonical structures: Per the Church-Turing thesis any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations Turing completeness requires only four instruction types — conditional GOTO, unconditional GOTO, assignment, HALT.

However, both Julius and Ethel wanted their children to be brought up in England, so they moved to Maida Vale, London, where Turing was born on 23 June 1912, as recorded by a blue plaque on the outside of the house of his birth, later the Colonnade Hotel.

A model of computation may be defined in terms of an abstract computer, e. g., Turing machine, and / or by postulating that certain operations are executed in unit time.

If the experiment were done in English, since Searle knows English, he would be able to take questions and give answers without any algorithms for English questions, and he would be affectively aware of what was being said and the purposes it might serve: Searle passes the Turing test of answering the questions in both languages, but he's only conscious of what he's doing when he speaks English.

As a third issue, philosophers who dispute the validity of the Turing test may feel that it is possible, at least in principle, for verbal report to be dissociated from consciousness entirely: a philosophical zombie may give detailed verbal reports of awareness in the absence of any genuine awareness.

The origins of cognitive thinking such as computational theory of mind can be traced back as early as Descartes in the 17th century, and proceeding up to Alan Turing in the 1940s and ' 50s.

They generate exactly all languages that can be recognized by a Turing machine.

Note that this is different from the recursive languages which can be decided by an always-halting Turing machine.

The languages described by these grammars are exactly all languages that can be recognized by a linear bounded automaton ( a nondeterministic Turing machine whose tape is bounded by a constant times the length of the input.

Since all problems in NP can be reduced to this problem it follows that for all problems in NP we can construct a non-deterministic Turing machine that decides the complement of the problem in polynomial time, i. e., NP is a subset of co-NP.

Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus as the formal representation of algorithms.

In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936, i. e. as " sequences of digits interpreted as decimal fractions " between 0 and 1: