What is the significance of the turing machine

To us it seems that it is the best introduction to the subject, and we refer the reader to this superior piece of expository writing. We formalize Turing's description as follows: A Turing machine consists of a finite program, called the finite control , capable of manipulating a linear list of cells , called the tape , using one access pointer, called the head. These rules determine, from the current state of the finite control and the symbol contained in the cell under scan, the operation to be performed next and the state to enter at the end of the next operation execution.

For now, we assume that there are no two distinct quadruples that have their first two elements identical, the device is deterministic.

In this case we say that the device halts. In the following we need the notion of a 'self-delimiting' code of a binary string. If the Turing machine halts for all inputs, then the function computed is defined for all arguments and we call it total computable.

The class of algorithmically computable numerical functions in the intuitive sense coincides with the class of partial computable functions. It is possible to give an effective computable one-to-one pairing between natural numbers and Turing machines. This is called an effective enumeration. One way to do this is to encode the table of rules of each Turing machine in binary, in a canonical way. We order the resulting binary strings lexicographically according to increasing length.

For the contemporary reader there should be nothing mysterious in the concept of a general-purpose computer which can perform any computation when supplied with an appropriate program. The surprising thing is that a general-purpose computer can be very simple: Marvin Minsky has been shown that four tape symbols and seven states suffice easily in the above scheme.

The last reference contains an excellent discussion of Turing machines, their computations, and related machines. Obviously, all computable sets are computably enumerable. The following sets are computable: i the set of odd integers; ii the set of natural numbers; iii the empty set; iv the set of primes; v every finite set; vi every set with a finite complement.

The following theorem due to Turing in , formalizes this discussion. For convenience these formulas are taken to be finite binary strings. Invariably, the formulas are specified in such a way that an effective procedure exists that decides which strings are formulas and which strings are not. The formulas are the objects of interest of the theory and constitute the meaningful statements. With each theory we associate a set of true formulas and a set of provable formulas.

A theory is axiomatizable if it can be effectively enumerated. A theory is decidable if it is a recursive set. Hence, soundness implies consistency. A particularly important example of an axiomatizable theory is Peano arithmetic , which axiomatizes the standard elementary theory of the natural numbers. Church and S. The notion of computable functions can be extended from integer functions to real valued functions of rational arguments, and to weaker types of computability, using the framework explained above.

The extension to negative arguments and values is straightforward. A function is called semicomputable if it is either upper semicomputable or lower semicomputable or both. The total function versions are defined similarly. The idea is that a semicomputable function can be approximated from one side by a computable function with rational values, but we may never know how close we are to the real value. A computable real function can be approximated to any degree of precision by a computable function with rational values.

An example of a lower semicomputable function that is not computable. But we have seen above that it is not. Prominent examples are the Kolmogorov complexity function that is upper semicomputable but not computable and hence not lower semicomputable , and the universal algorithmic probability function that is lower semicomputable but not computable and hence not upper semicomputable. These are the fundamental notions in M. Li and P. Vitanyi and, among others, A. Nies These results are usually achieved by relying on other equivalent models of computability such as, for instance, tag systems.

The same result was achieved independently by Church a, b using a different kind of formal device which is logically equivalent to a Turing machine see Sec. The result went very much against what Hilbert had hoped to achieve with his finitary and formalist program. The true reason why Comte could not find an unsolvable problem, lies in my opinion in the assertion that there exists no unsolvable problem.

Instead of the stupid Ignorabimus, our solution should be: We must know. We shall know. Note that the solvability Hilbert is referring to here concerns solvability of mathematical problems in general and not just mechanically solvable. It is shown however in Mancosu et al.

There are two main methods:. The notion of reducibility has its origins in the work of Turing and Post who considered several variants of computability Post ; Turing The concept was later appropriated in the context of computational complexity theory and is today one of the basic concepts of both computability and computational complexity theory Odifreddi ; Sipser First of all, one needs a formalism which captures the notion of computability.

Turing proposed the Turing machine formalism to this end. A second step is to show that there are problems that are not computable within the formalism. To achieve this, a uniform process U needs to be set-up relative to the formalism which is able to compute every computable number. One can then use some form of diagonalization in combination with U to derive a contradiction.

Such machines were identified by Turing as circle-free. All other machines are called circular machines. A number n which is the D. The problem to decide for every number n whether or not it is satisfactory.

The proof of the uncomputability of CIRC? Hence, it relies for its construction on the universal Turing machine and a hypothetical machine that is able to decide CIRC?

Based on the uncomputability of CIRC? Turing shows that the Entscheidungsproblem is not decidable. This is achieved by showing:. A popular proof of HALT? Assume that HALT? More particularly, we have:. A popular but quite informal variant of this proof was given by Christopher Strachey in the context of programming Strachey As is clear from Sections 1.

One can use a quintuple or quadruple notation; one can have different types of symbols or just one; one can have a two-way infinite or a one-way infinite tape; etc. Several other less obvious modifications have been considered and used in the past.

These modifications can be of two kinds: generalizations or restrictions. This adds to the robustness of the Turing machine definition. It was Shannon who proved that for any Turing machine T with n symbols there is a Turing machine with two symbols that simulates T Shannon He also showed that for any Turing machine with m states, there is a Turing machine with only two states that simulates it.

In Moore , it was mentioned that Shannon proved that non-erasing machines can compute what any Turing machine computes. This result was given in a context of actual digital computers of the 50s which relied on punched tape and so, for which, one cannot erase. It was Wang who published the result Wang It was shown by Minsky that for every Turing machine there is a non-writing Turing machine with two tapes that simulates it.

Instead of one tape one can consider a Turing machine with multiple tapes. This turned out the be very useful in several different contexts. For instance, Minsky, used two-tape non-writing Turing machines to prove that a certain decision problem defined by Post the decision problem for tag systems is non-Turing computable Minsky They used multitape machines because they were considered to be closer to actual digital computers.

Another variant is to consider Turing machines where the tape is not one-dimensional but n -dimensional. This variant too reduces to the one-dimensional variant. An apparently more radical reformulation of the notion of Turing machine is that of non-deterministic Turing machines. As explained in 1.

Next to these, Turing also mentions the idea of choice machines for which the next state is not completely determined by the state and symbol pair.

Instead, some external device makes a random choice of what to do next. Non-deterministic Turing machines are a kind of choice machines: for each state and symbol pair, the non-deterministic machine makes an arbitrary choice between a finite possibly zero number of states. Thus, unlike the computation of a deterministic Turing machine, the computation of a non-deterministic machine is a tree of possible configuration paths. One way to visualize the computation of a non-deterministic Turing machine is that the machine spawns an exact copy of itself and the tape for each alternative available transition, and each machine continues the computation.

Notice the word successfully in the preceding sentence. In this formulation, some states are designated as accepting states and when the machine terminates in one of these states, then the computation is successful, otherwise the computation is unsuccessful and any other machines continue in their search for a successful outcome.

The addition of non-determinism to Turing machines does not alter the extent of Turing-computability. Non-deterministic Turing machines are an important model in the context of computational complexity theory.

Weak Turing machines are machines where some word over the alphabet is repeated infinitely often to the left and right of the input. Semi-weak machines are machines where some word is repeated infinitely often either to the left or right of the input.

These machines are generalizations of the standard model in which the initial tape contains some finite word possibly nil. They were introduced to determine smaller universal machines. Watanabe was the first to define a universal semi-weak machine with six states and five symbols Watanabe Recently, a number of researchers have determined several small weak and semi-weak universal Turing machines e. Besides these variants on the Turing machine model, there are also variants that result in models which capture, in some well-defined sense, more than the Turing -computable functions.

There are various reasons for introducing such stronger models. This is a very basic question in the philosophy of computer science. The existing computing machines at the time Turing wrote his paper, such as the differential analyzer or desk calculators, were quite restricted in what they could compute and were used in a context of human computational practices Grier It has the following restrictions Gandy ; Sieg :.

If that would have been the case, he would not have considered the Entscheidungsproblem to be uncomputable.

This results in versions of the physical Church-Turing thesis. More particularly, like Turing, Gandy starts from a basic set of restrictions of computation by discrete mechanical devices and, on that basis, develops a new model which he proved to be reducible to the Turing machine model. Others have proposed alternative models for computation which are inspired by the Turing machine model but capture specific aspects of current computing practices for which the Turing machine model is considered less suited.

One example here are the persistent Turing machines intended to capture interactive processes. These and other related proposals have been considered by some authors as reasonable models of computation that somehow compute more than Turing machines.

It is the latter kind of statements that became affiliated with research on so-called hypercomputation resulting in the early s in a rather fierce debate in the computer science community, see, e. By consequence, many consider it as a thesis or a definition.

The thesis would be refuted if one would be able to provide an intuitively acceptable effective procedure for a task that is not Turing-computable. This far, no such counterexample has been found. Other independently defined notions of computability based on alternative foundations, such as recursive functions and abacus machines have also been shown to be equivalent to Turing computability.

These equivalences between quite different formulations indicate that there is a natural and robust notion of computability underlying our understanding.

Given this apparent robustness of our notion of computability, some have proposed to avoid the notion of a thesis altogether and instead propose a set of axioms used to sharpen the informal notion. For each of these models it was proven that they capture the Turing computable functions.

Note that the development of the modern computer stimulated the development of other models such as register machines or Markov algorithms. More recently, computational approaches in disciplines such as biology or physics, resulted in bio-inspired and physics-inspired models such as Petri nets or quantum Turing machines. A discussion of such models, however, lies beyond the scope of this entry.

For more information, see the entry on recursive functions. In the context of recursive function one uses the notion of recursive solvability and unsolvability rather than Turing computability and uncomputability. This terminology is due to Post However, the logical system proposed by Church was proven inconsistent by his two PhD students Stephen C.

There are three operations or rules of conversion. Around —21 Emil Post developed different but related types of production systems in order to develop a syntactical form which would allow him to tackle the decision problem for first-order logic.

One of these forms are Post canonical systems C which became later known as Post production systems.

The symbols g are a kind of metasymbols: they correspond to actual sequences of letters in actual productions. The symbols P are the operational variables and so can represent any sequence of letters in a production.

Any set of finite sequences of words that can be produced by a canonical system is called a canonical set. AIT therefore constrains and clarifies what it is that practical algorithms can do. To give another example, it clarifies what you can know about the power of compression algorithms, which essentially exploit lack of randomness in the same way that AIT defines lack of randomness.

What I suspect is that in the past there would have been slower progress in understanding randomness in practical senses, and in file compression algorithms, without the clear conceptual framework provided by basic results and concepts in AIT--even though those results and concepts have no direct practical applications.

That is speculation on my part, but there can be subtle and nonobvious but important ways that ideas influence the development of knowledge, and I suspect this is one area where that is true. For the record, I am at the level of an advanced beginning student of AIT who has also studied practical random number generation a bit, with a long history with various aspects of CS.

There may very well be practical applications of AIT of which I am unaware. I have seen no evidence of that, though. My primary area of expertise has more to do with the "development of knowledge" point I made in the preceding paragraph.

Maybe it's also worth noting concerning Turing oracle machines--machines that are provided with a superior source of information to draw upon, an "oracle"--that a mathematician involved with Turing-machine related areas, Robert Soare, has suggested in a journal article that online databases are analogous to oracles.

I still don't think that has practical applications, however, and I'm not sure the analogy is anything more than just an analogy. Check Decidability If TM cannot solve a problem in countable time then there could not be any algorithm which could solve that problem That is the problem is undecidable.

For a decision problem if its TM halt in countable time for all finite length inputs then we can say that the problem could be solved by an algorithm in countable time. Suppose we found that the problem is decidable. Then out target become how efficiently we can solve it. Design and Implement Algorithm for Practical Machines TM helps to propagate idea of algorithm in other practical machines.

Turing machines are good mind exercise with little practical use. There is no harm in not having one. All applications of a Turing machine are either intuitive or a matter of religion because they cannot be proved or refuted. Sign up to join this community. The best answers are voted up and rise to the top.

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Here are some points to ponder: Numbers are "just symbols": In college, one of my assignments was to build a TM to add two decimal numbers.