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.
A special class of canonical forms defined by Post are normal systems. Any set of finite sequences of words that can be produced by a normal system is called a normal set. Post production systems became important formal devices in computer science and, more particularly, formal language theory Davis ; Pullum Post also defined a specific terminology for his formulation 1 in order to define the solvability of a problem in terms of formulation 1.
These notions are applicability, finiteprocess, 1-solution and 1-given. Roughly speaking these notions assure that a decision problem is solvable with formulation 1 on the condition that the solution given in the formalism always terminates with a correct solution. Turing is today one of the most celebrated figures of computer science.
Many consider him as the father of computer science and the fact that the main award in the computer science community is called the Turing award is a clear indication of that Daylight This was strengthened by the Turing centenary celebrations from , which were largely coordinated by S.
Barry Cooper. However, recent historical research shows also that one should treat the impact of Turing machines with great care and that one should be careful in retrofitting the past into the present. Today, the Turing machine and its theory are part of the theoretical foundations of computer science. It is a standard reference in research on foundational questions such as:. It is also one of the main models for research into a broad range of subdisciplines in theoretical computer science such as: variant and minimal models of computability, higher-order computability, computational complexity theory , algorithmic information theory, etc.
This significance of the Turing machine model for theoretical computer science has at least two historical roots. First of all, there is the continuation of the work in mathematical logic from the s and s by people like Martin Davis—who is a student of Post and Church—and Kleene. Both Davis and Kleene published a book in the s on these topics Kleene ; Davis which soon became standard references not just for early computability theory but also for more theoretical reflections in the late s and s on computing.
Secondly, one sees that in the s there is a need for theoretical models to reflect on the new computing machines, their abilities and limitations and this in a more systematic manner. It is in that context that the theoretical work already done was picked up. It are these more theoretical developments that contributed to the establishment of computational complexity theory in the s.
Of course, besides Turing machines, other models also played and play an important role in these developments. Still, within theoretical computer science it is mostly the Turing machine which remains the model, even today. In several accounts, Turing has been identified not just as the father of computer science but as the father of the modern computer.
Roughly speaking this means the storage of instructions and data in the same memory allowing the manipulation of programs as data. This argument is then strengthened by the fact that Turing was also involved with the construction of an important class of computing devices the Bombe used for decrypting the German Enigma code and later proposed the design of the ACE Automatic Computing Engine which was explicitly identified as a kind of physical realization of the universal machine by Turing himself:.
Some years ago I was researching on what might now be described as an investigation of the theoretical possibilities and limitations of digital computing machines. Turing Based on that research it is clear that claims about Turing being the inventor of the modern computer give a distorted and biased picture of the development of the modern computer.
At best, he is one of the many who made a contribution to one of the several historical developments scientific, political, technological, social and industrial which resulted, ultimately, in our concept of the modern computer. In the s then the universal Turing machine starts to become an accepted model in relation to actual computers and is used as a tool to reflect on the limits and potentials of general-purpose computers by both engineers, mathematicians and logicians.
More particularly, with respect to machine designs, it was the insight that only a few number of operations were required to built a general-purpose machine which inspired in the s reflections on minimal machine architectures. He called this machine a universal computer.
The description given by Turing of a universal computer is not unique. Many computers, some of quite modest complexity, satisfy the requirements for a universal computer. Frankel Of course, by minimizing the machine instructions, coding or programming became a much more complicated task. And indeed, one sees that with these early minimal designs, much effort goes into developing more efficient coding strategies. It is here that one can also situate one historical root of making the connection between the universal Turing machine and the important principle of the interchangeability between hardware and programs.
Today, the universal Turing machine is by many still considered as the main theoretical model of the modern computer especially in relation to the so-called von Neumann architecture.
Of course, other models have been introduced for other architectures such as the Bulk synchronous parallel model for parallel machines or the persistent Turing machine for modeling interactive problems. The idea that any general-purpose machine can, in principle, be modeled as a universal Turing machine also became an important principle in the context of automatic programming in the s Daylight In the machine design context it was the minimizing of the machine instructions that was the most important consequence of that viewpoint.
This is introduced in several forms in the s by people like John W. Thus, also in the context of programming, the universal Turing machine starts to take on its foundational role in the s Daylight Whereas the Turing machine is and was a fundamental theoretical model delimiting what is possible and not on the general level, it did not have a real impact on the syntax and semantics of programming languages. In fact, Turing machines were often regarded as machine models rather than as a model for programming:.
Turing machines are not conceptually different from the automatic computers in general use, but they are very poor in their control structure. It is sufficient that computable functions be represented somehow by symbolic expressions, e. However, a practical theory of computation must be applicable to particular algorithms. McCarthy Thus one sees that the role of the Turing machine for computer science should be situated rather on the theoretical level: the universal machine is today by many still considered as the model for the modern computer while its ability to mimic machines through its manipulation of programs-as-data is one of the basic principles of modern computing.
Moreover, its robustness and naturalness as a model of computability have made it the main model to challenge if one is attacking versions of the so-called physical Church-Turing thesis. Turing machines are more powerful than any device that can actually be built, but they can be simulated both in software and hardware. There are many Turing machine simulators available. Here are three software simulators that use different technologies to implement simulators using your browser.
Here is an application that you can run on the desktop no endorsement of these programs is implied. Church-Turing Thesis computability and complexity computational complexity theory recursive functions Turing, Alan. This is known as Turing's thesis.
Enter Alonzo Church Over the years, all serious attempts to give precise yet intuitively satisfactory definitions of a notion of effective procedure what Church called effectively calculable function in the widest possible sense have turned out to be equivalentto define essentially the same class of processes. In his original paper, Turing established the equivalence of his notion of effective procedure with his automatic machine a-machine now called a Turing machine.
The Church-Turing thesis states that a function on the positive integers is effectively calculable if and only if it is computable. An informal accumulation of the tradition in S. Kleene has transformed it to the Computability thesis : there is an objective notion of effective computability independent of a particular formalization.
The informal arguments Turing sets forth in his paper are as lucid and convincing now as they were then. 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.
For example, a Turing machine is said to recognise a sequence of symbols written on the tape if it is started on the tape and halts in a special state called a final state.
What is interesting about this idea is that there are sequences that a Turing machine can recognise that a finite state machine, i. For example, a finite state machine can recognise a sequence that has three As followed by three Bs e. But only a Turning machine can recognise a sequence that has an arbitrary number of As followed by the same number of Bs.
That is a Turing machine is more powerful than a finite state machine because it can count. This at first seems very odd but a little thought reveals it to be quite reasonable, obvious even! To discover if the number of Bs is the same as the number of As you have to do something that is equivalent to counting them, or at least matching them up, and given there can be arbitrarily many input A symbols even before you ever see a B you need unlimited storage to count or match them up.
The finite state machine doesn't have unlimited storage but the Turing machine does in the form of its tape! The machine doesn't, in fact, need an infinite tape, just one that isn't limited in length and this is a subtle difference. Imagine if you will that Turing machines are produced with a very long but finite tape and if the machine ever runs out of space you can just order some more tape.
This is the difference between the tape actually being infinitely long or just unlimited…. We could also start arguing that the Turing machine is not really of any importance because it is a very feeble machine but there are two reasons for not dismissing it. The first is that Alan Turing thought up the machine as a model for computation. He reasoned that any computation that could be performed by a human involved writing down intermediate results, reading them back and carrying out actions that depend only on what has been read and the current state of things.
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