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. 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. A Turing machine is a general example of a CPU that controls all data manipulation done by a computer. Turing machine can be halting as well as non halting and it depends on algorithm and input associated with the algorithm.
Halting means that the program on certain input will accept it and halt or reject it and halt and it would never go into an infinite loop.
Basically halting means terminating. So can we have an algorithm that will tell that the given program will halt or not. In terms of Turing machine, will it terminate when run on some machine with some particular given input string. The answer is no we cannot design a generalized algorithm which can appropriately say that given a program will ever halt or not? The only way is to run the program and check whether it halts or not. Skip to content.
While some of these correspondences may become clearer in this course, especially as we consider functional programming languages and techniques for programming language implementation, it is not a focus of the course. We are less concerned with the nature of this correspondence itself than with its implications.
I will use these words during the semester. It is important to remember what these words do and do not mean. Any function that can be computed by a program in one Turing-complete programming language can be computed by a program in every other Turing-complete programming language!
So why do we have more than one programming language if they are all equivalently powerful in the computability sense? The answer, which we will grow to understand more deeply this semester, is that, in the design of programming languages we typically care very much more about how a computation is expressed than about whether it can be expressed. Different sorts of programming languages are well-suited to different sorts of problems and poorly suited to others.
After all, would you like to build a web browser or Facebook using machine language the low-level instructions executed by computer hardware? We have claimed that uncomputable functions exist. Back when I was a PhD student, I needed a succinct way to summarize the Halting Problem, one of the core demonstrations of the limits of computation. There weren't a lot of online resources available at the time, so I wrote up this explanation.
It appears that people continue to find it useful after all these years the Internet Archive records a version from August !
Here are some sample problems:. A problem is decidable if it has a solution. If there is no algorithm that solves the problem in a finite amount of time, the problem is undecidable. Are all problems decidable? Given enough thought, can we always come up with a well-defined procedure that takes some input and answers a given question about it?
At the start of the 20th century, the belief was that this was true. Mathematicians there were no computer scientists back then! Using some very clever techniques, he showed that as soon as we devise a system that's sufficiently powerful and well-behaved to encompass mathematical reasoning, that system will necessarily contain a statement that we could never prove is true, even though it is true.
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