Turing tape

A universal Turing machine uses an arbitrarily long tape as a potentially infinite memory storage device. Instead, for his proof, Conway used Minsky s idea that a potentially infinite memory can also be obtained by storing arbitrarily large numbers in memory registers. The idea is sketched in figure 3.85. 

Notice that only the last class of unrestricted languages requires a full universal computer (i.e. Turing Machine)-, the other classes require progressively simpler kinds of computers. Each one of these four automata act as a kind of black-box into which is fed a tape of symbols, sequentially, one cell at a time. During each cycle, the black-box reads the symbol at the appropriate cell, responds to that... 

Margolus (margfiOb] generalizes Feynman s formalism - which applies to strictly serial computation - to describe deterministic parallel quantum computation in one dimension. Each row in Margolus model is a tape of a Turing Machine, and adjacent Turing Machines can communicate when their tapes arc located at the same coordinate. Extension of the formalism to more than one dimension remains an open problem. 

The Turing machine is one of the key abstractions used in modern computability theory. It is a mathematical model of a device that changes its internal state and reads from, writes on, and moves a potentially infinite tape, all in accordance with its present state. The model of the Turing machine played an important role in the conception of the modern digital computer. 

THEOREM 6.1 It is not partially decidable whether a Turing machine diverges on the blank initial tape. 

An infinite tape w is ultimately periodic if w = uvw. .. for finite tapes u,v with v e, i.e. if w can be written as a finite (possibly enpty) initial string u followed by arbitrarily many repetitions of a nonempty finite string v. We need the next result which relates Turing machines and two-tape acceptors with special attention to behavior on ultimately periodic input tapes. 

THEOREM 6.2 For each Turing machine T, one can construct a two-tape one-way deterministic finite state accepter Mp such that if D = (w,w) w input tape, finite or infinite ... 

If Turing irachine T diverges on the initially blank tape, and is given... 

This was because in our construction of Mj, from Turing machine T, we forced Hp to halt on (t,t) unless t encoded an infinite computation of T on the blank initial tape and by placing extra s in the encodement we made sure that such a tape t could never be ultimately periodic. Hence we can assume that M halts on (t,t) for t ultimately periodic. 

For every Turing machine T one can construct a total recursive function g such that g(n) = n if T started on the initially balnk tape does not stop within n steps and g(n) = n + 1 if T started on the initially blank tape stops within n steps. This function is the identity if and only if T diverges on the initially blank tape. Hence it is not partially decidable whether a total recursive function computes the identity this settles (10). ... 

Just as we showed that for every Turing machine T there is a two-tape one-way deterministic finite state acceptor Mj such that T halts on the initially blank tape if and only if La(M, ) O D = , so it can be shown that far any recursively enumerable language L Q and a marker not in E, there is a deterministic two-tape one-way finite state acceptor M with input vocabulary E U, 0,1 such that ... 

In cryptology, probabilistic algorithms are usually represented as deterministic Turing machines with an additional input tape, the so-called random tape. The random tape contains a (potentially infinite) sequence of random bits. Each bit on this tape is read exactly once. The content (or each finite subsequence) is supposed to be uniformly distributed. If A is a probabilistic algorithm, A(i) denotes the probability space on the outputs if A is run on input i (i.e., with i on the input tape — i does not include the content of the random tape). To cover non-terminating computations, the output space is augmented by an element t. The probabilistic function computed by A is not distinguished from A in the notation. 

If the number of entities in a system and the connection structure is fixed in the protocol, each output of a state-transition function can be a tuple of values that represent the outputs on different channels. The individual output sets should be augmented by a special element nojoutput. With Turing machines, one can either use output tuples, too, or a separate tape for each simplex channel. 

Proof First we notice that Algorithm 5.1 does not illustrate the point of this theorem to implement Algorithm 5.1 on a multitape Turing machine in the natural way would require nlogn tape. The reason is the following an entry has to store elements in the... 

Next, there must be a program of instructions. Presently, we know of one such program, and it is physically represented by the enzyme DNA polymerase. This enzyme moves along the input DNA strand and produces a complimentary DNA strand a C yields a G, a G yields a C, a T yields an A, and an A yields a T. This is not very complicated, but does illustrate the Turing principle. The original DNA strand (input tape) is translated into a complementary DNA strand (output tape) by means of some translation mechanism (DNA polymerase). 

Let us start this section with a kind of description of our automata, i.e., how it could work. Our aim is to reach easily the next occurrence of a given letter. To do so, first, let us divide the tape into as many parts as the cardinality of the alphabet (about a similar way as a one-tape Turing machine can simulate a multi-tape Turing machine [10]). See Fig. 2 as well. In each division exactly one of the letters is used there is a bijection from U to the parts of the tape. 

Alan Mathison Turing (1912-1954), British mathematician, defined a device Turing machine") that consists of a read/write head that scans a 1-D tape divided into squares, each of which contains a 0 or a 1. The behavior of the machine is completely characterized by its current state, the content of the square it is just reading, and a table of instructions. 

The way it works is as following the head starts from the leftmost part of the tape, reads the symbol which is written in the first cell, takes an action, and moves to the next cell. The procedure is repeated until the machine eventually concludes the computation and halts. Let us give a very simple illustrative example a Turing Machine to perform the addition operation 3 + 5. In order to do so, we must first define the symbols to represent these numbers on the tape. We will adopt the following representation 3 = and 5 =. So, the expected result of our calculation is 8 =. The input state is simply the initial two blocks of s separated by a blank, which we will represent by a small box . So, our alphabet has only two symbols , . ... 

Turing machines, invented by Alan Turing in 1936, are extremely simple computers that consist of a finite-state compute head that can move back and forth on an infinite one-dimensional memory tape. Turing showed that these machines are universal in the sense that they can perform any computation that can be performed by any other mechanical device—there is no fundamental need to use a more complicated kind of computer ... 

The Turing machine has a tape that is divided into cells (squares) the tape extends infinitely to the right and to the left. Each cell of the tape can hold a symbol. The tape initially contains the input—a finite string of symbols in the... 

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