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.

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.

A problem is # P-complete if and only if it is in # P, and every problem in # P can be reduced to it by a polynomial-time counting reduction, i. e. a polynomial-time Turing reduction relating the cardinalities of solution sets.

Equivalently, a problem is # P-complete if and only if it is in # P, and for any non-deterministic Turing machine (" NP machine "), the problem of computing its number of accepting paths can be reduced to this problem.

For example, the Boolean satisfiability problem can be reduced to the halting problem by transforming it to the description of a Turing machine that tries all truth value assignments and when it finds one that satisfies the formula it halts and otherwise it goes into an infinite loop.

Within the class P, however, polynomial-time reductions are inappropriate, because any problem in P can be polynomial-time reduced ( both many-one and Turing ) to almost any other problem in P. Thus, for classes within P such as L, NL, NC, and P itself, log-space reductions are used instead.

Turing suggests that Lovelace's objection can be reduced to the assertion that computers " can never take us by surprise " and argues that, to the contrary, computers could still surprise humans, in particular where the consequences of different facts are not immediately recognizable.

* Linear speedup theorem, that the space and time requirements of a Turing machine solving a decision problem can be reduced by a multiplicative constant factor.

That is, any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the problem of determining whether a Boolean formula is satisfiable.

The lab had successfully built a reduced version of Turing ’ s Automatic Computing Engine ( ACE ) the concept of which dated from 1945: the Pilot ACE.

The domain is always Turing equivalent to the halting problem.

* It is Turing equivalent to the halting problem and thus at level of the arithmetical hierarchy.

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

An alternate form of ( 2 ) – the machine successively prints all n of the digits on its tape, halting after printing the n < sup > th </ sup > – emphasizes Minsky's observation: ( 3 ) That by use of a Turing machine, a finite definition – in the form of the machine's table – is being used to define what is a potentially-infinite string of decimal digits.

Most general purpose functional programming languages allow unrestricted recursion and are Turing complete, which makes the halting problem undecidable, can cause unsoundness of equational reasoning, and generally requires the introduction of inconsistency into the logic expressed by the language's type system.

Interestingly, the halting paradox still applies to such machines ; although they determine whether particular Turing machines will halt on particular inputs, they cannot determine, in general, if machines equivalent to themselves will halt.

Similarly the question of whether a Turing machine T terminates on an initially empty tape ( rather than with an initial word w given as second argument in addition to a description of T, as in the full halting problem ) is still undecidable.

By a straightforward reduction to the halting problem it is possible to prove that ( for any Turing complete language ) finding all possible run-time errors in an arbitrary program ( or more generally any kind of violation of a specification on the final result of a program ) is undecidable: there is no mechanical method that can always answer truthfully whether a given program may or may not exhibit runtime errors.

This result dates from the works of Church, Gödel and Turing in the 1930s ( see the halting problem and Rice's theorem ).

The statement that the halting problem cannot be solved by a Turing machine is one of the most important results in computability theory, as it is an example of a concrete problem that is both easy to formulate and impossible to solve using a Turing machine.

A computer with access to an infinite tape of data is sometimes more powerful than a Turing machine, because the tape can in principle contain the solution to the halting problem, or some other undecidable problem.

Church and Turing then showed that the lambda calculus and the Turing machine used in Turing's halting problem were equivalent in capabilities, and subsequently demonstrated a variety of alternative " mechanical processes for computation.

Arslanov's completeness criterion states that the only recursively enumerable Turing degree that computes a fixed point free function is 0 ′, the degree of the halting problem.

The halting problem, which is the set of ( descriptions of ) Turing machines that halt on input 0, is a well known example of a noncomputable set.

Thus the halting problem is an example of a recursively enumerable set, which is a set that can be enumerated by a Turing machine ( other terms for recursively enumerable include computably enumerable and semidecidable ).

1936: Alan Turing proved that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist.

An example of a problem a Turing machine cannot solve is the halting problem.

Generalized Turing machines can solve the halting problem by evaluating a Specker sequence.

Also provably unsolvable are so-called undecidable problems, such as the halting problem for Turing machines.

The first theory about software was proposed by Alan Turing in his 1935 essay Computable numbers with an application to the Entscheidungsproblem ( decision problem ).

: Alan Turing ( writing 30 years before Searle presented his argument ) noted that people never consider the problem of other minds when dealing with each other.

Despite the language's intentionally obtuse and wordy syntax, INTERCAL is nevertheless Turing-complete: given enough memory, INTERCAL can solve any problem that a Universal Turing machine can solve.

He was appointed a lecturer in mathematics at Cambridge in 1927, where his 1935 lectures on the Foundations of Mathematics and Gödel's Theorem inspired Alan Turing to embark on his pioneering work on the Entscheidungsproblem ( decision problem ) using a hypothetical computing machine.

Newman subsequently arranged for Turing to visit Princeton where Alonzo Church was working on the same problem but using his Lambda calculus.

For example, P < sup > SAT </ sup > is the class of problems solvable in polynomial time by a deterministic Turing machine with an oracle for the Boolean satisfiability problem.

It might seem that the potentially infinite memory capacity is an unrealizable attribute, but any decidable problem solved by a Turing machine will always require only a finite amount of memory.

So in principle, any problem that can be solved ( decided ) by a Turing machine can be solved by a computer that has a bounded amount of memory.

During World War II, Turing worked for the Government Code and Cypher School ( GCCS ) at Bletchley Park, Britain's codebreaking centre.

His father, Julius Mathison Turing ( 1873 – 1947 ), was a member of an old aristocratic family of Scottish descent who worked for the Indian Civil Service ( the ICS ).

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 >.

This topic was further developed in the 1930s by Alonso Church and Alan Turing, who on the one hand gave two independent but equivalent definitions of computability, and on the other gave concrete examples for undecidable questions.

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.

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.

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.

:" A computable number one for which there is a Turing machine which, given n on its initial tape, terminates with the nth digit of that number on its tape.

* British mathematician Alan Turing created a theoretical model for a machine, now called a universal Turing machine, that could carry out calculations from inputs,

Other formalisms ( besides recursion, the λ-calculus, and the Turing machine ) have been proposed for describing effective calculability / computability.

Dirk van Dalen ( in Gabbay 2001: 284 ) gives the following example for the sake of illustrating this informal use of the Church – Turing thesis:

In 1997 he was awarded the Lemelson-MIT Prize of $ 500, 000, the world's largest single prize for invention and innovation, and the ACM Turing Award.

The basis for later theoretical computer science, in Alonzo Church and Alan Turing also grew directly out of this ' debate '.

He received the 1972 Turing Award for fundamental contributions to developing programming languages, and was the Schlumberger Centennial Chair of Computer Sciences at The University of Texas at Austin from 1984 until 2000.

* The Association for Computing Machinery's A. M. Turing Award ( 1972 )