Turing machines: concepts, varieties and simulation, Turing Machines

Turing machines: concepts, varieties and simulation

The Turing machine (TM) is a finite state machine which controls one or more tapes, where at least one tape is of the unbounded length. TMs come in many distinct flavors and we utilize various conventions whenever it is convenient. This section considers deterministic TMs (DTM), that is, the next one as well non-deterministic TMs (NDTM). DTMs and NDTMs are equal in computing power, however not with the respect to computing performance.

Apart from distinction DTM versus NDTM, the biggest differences among different TM models have to do with the number and access characteristic of tapes, example: semi-infinite or doubly infinite tape; separate read only input tape; multi-tape machines; input alphabet varies from work tape alphabet. The minor differences include the accurate specification of all actions that might take place throughout a transition, like whether halting is a portion of transition, or needs a halting state. Such differences influence the complexity of computations and convenience of programming, however not the computational power of TMs in terms of what they can calculate, given unbounded time and memory. The version of a TM can simulate any other, with gain or loss of time or space complexity.

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The ‘standard’ TM model is deterministic with the single semi-infinite tape (it consists of one fixed end and is unbounded in other direction) employed for input, computation and output. This is defined by the given components:

M = (Q, A, f: Q x A -> Q x A x {L, R, -, H}, q0, F}.

Q: Finite state space; A: finite alphabet; f: transition function; q0: initial state; F ⊆ A accepting states.

{L, R, -, H}: tape actions: L = move left, R = move right, - stay put and optional H = halt.

When a tape action calls for the read or write head to drop off fixed end of the tape, this action is avoided.

Non-deterministic TM: Substitute f: Q x A -> Q x A x {L, R...} by f: Q x A -> 2 Q x A x {L, R...}.

The deterministic TM can simulate the non-deterministic TM N by breadth-first traversal of tree of all possible executions of N. Why not depth-first traversal?

Multi-tape TMs are stated analogously: The transition function of k-tape TM based on all k characters presently being scanned and specifies k separate tape actions.

Definition: The configuration of a TM M with tapes T1,..Tk is given by the content of each tapes, the position of all the read or write heads, and (internal) state of M. The configuration implicitly specifies the whole future of a TM.

Example of a simulation: The TM M1 with semi-infinite tape can simulate the TM M2 = (Q, A, f, q0) with doubly infinite tape. Fold M2’s tape at some random place, and let M1 read or write two squares of the folded tape at once, therefore expanding the alphabet from A to A x A. We employ an additional character ¢ to mark the end of folded tape. The internal state of M1 recalls whether M2’s current square is on the upper half or on lower half of the tape; at any moment, M1 efficiently reads or writes on merely one of the ‘half-tapes’, although formally it reads or writes on both.

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Given M2 = (Q, A, f, q0), construct M1 = (Q’, A’, f’, q0‘) as:

Q’ is the Cartesian product Q x {up, down}, that we abbreviate as Q’ = {ui, di | for each qi ∈ Q}, that is, each state qi of M produces a state ui (up) and a state di (down). A’ = A x A ∪ {¢}.

Transition function f’: Q’ x A’ -> Q’ x A’ x {L, R, -, H} explodes quadratically.

All transition qi, a -> qj, b, m of M2 produces 2 |A| transitions of the given form:

ui, (a, -)  -> uj, (b, -),  m,  di, (-, a)  -> dj, (-, b),  m-1

Here (a, -) signifies for all the symbols in A x A whose upper square comprises an a, that is, the lower square of any letter in A.

Analogously for (-, a).  m-1 represents the inverse motion of m, that is, L and R are each other’s inverse. Two additional transitions handle the U-turn at left end of the tape: ui, ¢ -> di, ¢, R;  di, ¢ -> ui, ¢, R
 
The initial state q0‘of M1 is either u0 or d0 based on the initial configuration of M2.

M1simulates M2 ‘in real time’, transition for transition, merely occasionally wasting the transition for a U-turn.

Most of the other simulations include a considerable loss of time, however for the aim of theory of computability; the only related issue regarding time is whether a computation terminates or does not. For the aim of complexity theory speed-up and slow-down do matter. However complexity theory frequently considers the polynomial-time slow down to be negligible. From this viewpoint, simulating a multi-tape TM on the single-tape TM is relatively cheap, as the given theorem states.

Theorem: The TM Mk with k tapes utilizing time t(n) and space s(n) can be simulated by the single-tape TM M1 by using time O(t2(n) ) and space  O(s(n)).

General comments:

The unbounded tape can be viewed in two distinct ways: A) At any moment, the tape is finite; whenever the read or write head moves away from an extendable tape end, a new square is add on, or B) The tape is really infinite. In second case, the alphabet A comprises a designated symbol ‘blank’ that is differentiated from all other symbols: blank is the only symbol which takes place infinitely frequently on the tape. Therefore, at any moment, the TM tape can be considered to be limited in length; however the tape might grow beyond any finite bound as computation continues. The graphic appearance of ‘blank’ differs: ‘ ‘, or Ø; in binary alphabet A = {0, 1}, we consider 0 to be ‘blank’.

TMs are generally interpreted either as computing some function, say f: N -> N from natural numbers to natural numbers or as accepting the language, that is, set of strings. For any computation, we should specify the accurate format of the input data, example: how integer arguments x1... xn of a function y = f(x1... xn) are written on tape; and the format of outcome y if and only if when the computation terminates. Integers are usually coded into strings by using binary or unary number representation. The representation selected affects the complexity of the computation, however not the feasibility in principle. As later illustrations show, simple TMs can transform from any radix representation to unary and vice-versa.

A TM which computes a function receives its (coded) input as initial content of its tape[s], and generates its output as final content of its tape[s], if it halts. When it does not halt, the function is undefined for this specific value (or partial function). The TM which always halts evaluates a total function.

The TM acceptor consists of halt states qa = accept, qr = reject, with three possible answers: accept, reject and loop.

The TM decider is an acceptor which always halts, that is, always responds with either reject or accept.

The definition of Turing machine is remarkably robust. For illustration, a 2-d or 3-d ‘tape’ doesn’t increase computational power (Turing: ‘the 2-dimensional character of paper is no necessary of computation’).

The 2-d infinite array of squares can be linearized according to any space-filling curve, like a spiral. The TM with an ordinary 1-d tape can simulate 2-d array by traversing the plane all along the spiral, while keeping track of longitude and latitude of its present place in the plane. This insensitivity to the access features of its unbounded storage is in marked contrast to PDAs, where second stack already raises power. If considering the time and space complexity of computation, then of course the details of TM definition matter a big deal.

As the consequence of equivalence of numerous versions of TMs, we will select various conventions to simplify specific illustrations. Only in case of well-known ‘TM competitions’, like finding ‘small’ universal TMs or the ‘Busy Beaver TM’, one should adhere to an accurate syntax and hence competing TMs can be somewhat assessed.

Universal TM, UTM: The universal TM U is an interpreter which reads the description <M> of any random TM M and faithfully functions operations on data D accurately as M does. For single-tape TMs, imagine that <M> is written at the starting of the tape, followed by D.

Minimal TMs:

i) The 2-symbol alphabet A = {0, 1} is sufficient: code any larger alphabet as bit strings.

ii) Claude Shannon: The 2-state fsm controller is sufficient!! Code ‘state information’ to a big alphabet.

iii) Small universal TMs: The complexity of a TM is computed by the ‘state-symbol product’ |Q| x |A|.
There are UTMs having state-symbol product of less than 30.

Illustrations:

Illustration: f(x) = 2x in unary notation (x ≥ 1): double the length of string of contiguous 1’s Configurations. The bold symbol points out; it is presently being scanned by the read-write head.

Initial: ..0 1 1 1 1 0 ..    The read or write head scans the right-most 1 of a string of contiguous 1s
Intermediate: ..0 1 1 1 0 1 1 0 0 ..   There are 2 strings of contiguous 1’s separated by the single 0.
1s have been erased at left, twice that number has been written at right.
Next step: .. 0 1 1 0 1 1 1 1 0 ..  The new string of 1s has been expanded by 1 at each end
Final:..0 1 1 1 1 1 1 1 1 0 .. String of 1s deleted, a new string twice as long has been made.

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Illustration: Duplicate a bit string

Initial: # 1 0 1 1 # #
Intermediate: # 1’ 0’ 1 1 # 1 0 #   (Some of the bits have been substituted by 1’ or 0’ and encompass been copied)
Final: # 1’ 0’ 1’ 1’ # 1 0 1 1 # (Must be followed by a clean up phase where 1’ is modified to 1).

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Illustration: Parentheses expressions: For a simple task like checking the syntactic rightness of parentheses expressions, the special purpose device PDA, with push-pop access to the stack, is more suitable. The PDA throws away a pair of matching parentheses as fast as it has been discovered and never appears at it again. On a tape, on other hand, ‘throwing away’ leaves a gap which somehow has to be filled, making the task of ‘memory management’ more complex.

The given TM M with alphabet {(,), Ø, #} successively substitutes a pair of matching parentheses by blanks Ø.

We suppose that the string of parentheses is bounded on left and right by #, and the read-write head begins scanning the left #. Afterward, the head runs back and forth across the string, avoiding all blanks which keep getting created. This runs towards the right looking for the leftmost ‘)’ and substitutes it by a blank; then runs left looking for first ‘(‘and substitutes it by a blank. As soon as the pair of outermost parentheses has been removed, M returns to the left boundary marker # to begin searching the other pair of parentheses.

Begin configuration: # <parentheses> #, that is, all the parentheses are compressed among the #s, and the r/w head scans the left #. The null string is as well considered an accurate p-expression

Accept configuration: #Ø Ø .. Ø #. Any transition not displayed explicitly beneath leads to a refuse state.

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Illustration: Conversion of integers from binary notation to the unary

There are numerous ways to attack any programming problem. We can examine a given binary integer.. b2 b1 b0 bit by bit, build up the string of 1s of successive length 1, 2, 4,.. by using the doubling TM of first illustration, and catenate such strings of 1s of length 2k where bk = 1. The 2-state TM beneath is a candidate for the simplest TM which does the job, thanks to its utilization of two tapes. The upper tape initially includes the input, a positive integer with most significant bit 1. This is as well employed as a counter to be decremented past 0 down to #11..1#, a number which can be interpreted as -1. At each decrement, an additional ‘1’ is written on output tape. The counter is decremented by modifying the tail 100..0  to  011..1.

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