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4.4 Universal Turing Transducers

A Representation for Turing Transducers

A Universal Turing Transducer

Programs are written to instruct computing machines on how to solve given problems. Aprogram P is considered to be executable by a computing machine A if A can, when given P andany x for P, simulate any computation of P on input x.

In many cases, a single computing machine can execute more than one program, and thus can beprogrammed to compute different functions. However, it is not clear from the previousdiscussion just how general a computing machine can be. Theorem4.4.1below, together withChurch's thesis, imply that there are machines that can be programmed to compute anycomputable function. One such example is the computing machine D, which consists of a

"universal" Turing transducer U and of a translator T, which have the following characteristics(see Figure4.4.1).

Figure 4.4.1A programmable computing machine D.

U is a deterministic Turing transducer that can execute any given deterministic Turing transducerM. That is, U on any given (M, x) simulates the computation of M on input x (see the proof ofTheorem4.4.1).

T is a deterministic Turing transducer whose inputs are pairs (P, x) of programs P written insome fixed programming language, and inputs x for P. T on a given input (P, x) outputs xtogether with a deterministic Turing transducer M that is equivalent to P. In particular, if P is adeterministic Turing transducer (i.e., a program written in the "machine" language), then T is atrivial translator that just outputs its input. On the other hand, if P is a program written in a

higher level programming language, then T is a compiler that provides a deterministic Turingtransducer M for simulating P.

When given an input (P, x), the computing machine D provides the pair to T, and then it feedsthe output (M, x) of T to U, to obtain the desired output of P on input x.

Definitions A universal Turing transducerU is a deterministic Turing transducer that on anygiven pair (M, x), of a deterministic Turing transducer M and of an input x for M, simulates the
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behavior of M on x. Inputs that do not have the form (M, x) are rejected by U. Universal Turingmachines are defined similarly.

It should be noted that a pair (M, x) is presented to a universal Turing transducer in encodedform, and that the output of the universal Turing transducer is the encoding of the output of M on

input x. For convenience, the mentioning of the encoding is omitted when no confusion arises.Moreover, unless otherwise stated, a "standard" binary representation is implicitly assumed forthe encodings.


In what follows, a string is said to be astandard binary representation of a Turing transducer M= if it is equal to E(M), where E is defined recursively in the followingway.

a. E(M) = E(F)01E( ).b.

E(F) = E(p1) E(pk) for some ordering {p1, . . . , pk} of the states of F.c. E(B) = 0 is the binary representation of the blank symbol.

d. E( ) = E( 1)01E( 2)01 01E( r)01 for some ordering { 1, . . . , r} of the transitionrules of .

e. E( ) = E(q)E(a)E(b1) E(bm)E(p)E(d0)E(c1)E(d1) E(cm) E(dm)E( ) for each = (q, a,b1, . . . , bm, p, d0, c1, d1, . . . , cm, dm, ) in .

f. E(d) = 011 for d = -1, E(d) = 0111 for d = 0, E(d) = 01111 for d = +1, and E( ) = 0 foran output = .

g. E(qi) = 01i+2 for each state qi in Q, and some ordering q0, . . . , qs of the states of Q. Notethat the order assumes the initial state q0 to be the first.

h. E(ei) = 01i+1 for each symbol ei in ( {, $}) - {B} and some order {e1, . . . , et}in which e1 = and e2 = $.

Intuitively, we see that E provides a binary representation for the symbols in the alphabets of theTuring transducer, a binary representation for the states of the Turing transducer, and a binaryrepresentation for the possible heads movements. Then it provides a representation for asequence of such entities, by concatenating the representations of the entities. The string 01 isused as separator for avoiding ambiguity.

By definition, a given Turing transducer can have some finite number of standard binaryrepresentations. Each of these representations depends on the order chosen for the states in Q, theorder chosen for the symbols in ( {, $}) - {B}, the order chosen for the states in F, andthe order chosen for the transition rules in . On the other hand, different Turing transducers canhave identical standard binary representations if they are isomorphic, that is, if they are equalexcept for the names of their states and the symbols in their alphabets.

Example 4.4.1 If M is the Turing transducer whose transition diagram is given in Figure4.1.3,then E(M) can be the standard binary representation
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where E(q0) = 011, E(q1) = 0111, E(q2) = 01111, E(q3) = 011111, E(q4) = 0111111, E(B) = 0,E() = 011, E($) = 0111, E(a) = 01111, . . .

0 and 00 are examples of binary strings that are not standard binary representations of any Turingtransducer.

The string

represents a Turing transducer with one accepting state and four transition rules. Only the firsttransition rule has a nonempty output. The Turing transducer has one auxiliary work tape.

E(M)01E(x$) is assumed to be the standard binary representation of (M, x), with E(x$) =E()E(a1) E(an)E($) when x = a1 an.


The proof of the following result provides an example of a universal Turing transducer.

Theorem 4.4.1 There exists a universal Turing transducer U.

Proof U can be a two auxiliary-work-tape Turing transducer similar to M2 in the proof of

Proposition4.3.1. Specifically, U starts each computation by checking that its input is a pair (M,x) of some deterministic Turing transducer M = and of some input x forM (given in standard binary representation). If the input is not of such a form, then U halts in anonaccepting configuration. However, if the input is of such a form, U simulates a computationof M on x.

U, like M2, uses two auxiliary work tapes for keeping track of the content of the auxiliary worktapes of M. However, U also uses the auxiliary work tapes for keeping track of the states and theinput head locations of M. Specifically, the universal Turing transducer U records aconfiguration (uqv, u1qv1, . . . , umqvm, w) of M by storing #E(q)#|u|#E(u1)#E(v1)##E(um)#E(vm)# in an auxiliary work tape, and storing E(w) in the output tape.

To determine the transition rule (q, a, b1, . . . , bm, p, d0, c1, d1, . . . , cm, dm, ) that M uses in asimulated move, U extracts the state q and the symbols a, b1, . . . , bm. U records the stringE(q)E(a)E(b1) E(bm) in the auxiliary work tape that does not keep the configuration of M that isin effect. Then U determines p, d0, c1, d1, . . . , cm, dm, by searching E(M) for the substring thatfollows a substring of the form 01E(q)E(a)E(b1) E(bm).

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4.4 Universal Turing Transducers



Programs are written to instruct computing machines on how to solve given problems. Aprogram P is considered to be executable by a computing machine A if A can, when given P andany x for P, simulate any computation of P on input x.

In many cases, a single computing machine can execute more than one program, and thus can beprogrammed to compute different functions. However, it is not clear from the previousdiscussion just how general a computing machine can be. Theorem4.4.1below, together withChurch's thesis, imply that there are machines that can be programmed to compute anycomputable function. One such example is the computing machine D, which consists of a

"universal" Turing transducer U and of a translator T, which have the following characteristics(see Figure4.4.1).

Figure 4.4.1A programmable computing machine D.

U is a deterministic Turing transducer that can execute any given deterministic Turing transducerM. That is, U on any given (M, x) simulates the computation of M on input x (see the proof ofTheorem4.4.1).

T is a deterministic Turing transducer whose inputs are pairs (P, x) of programs P written insome fixed programming language, and inputs x for P. T on a given input (P, x) outputs xtogether with a deterministic Turing transducer M that is equivalent to P. In particular, if P is adeterministic Turing transducer (i.e., a program written in the "machine" language), then T is atrivial translator that just outputs its input. On the other hand, if P is a program written in a

higher level programming language, then T is a compiler that provides a deterministic Turingtransducer M for simulating P.

When given an input (P, x), the computing machine D provides the pair to T, and then it feedsthe output (M, x) of T to U, to obtain the desired output of P on input x.

Definitions A universal Turing transducerU is a deterministic Turing transducer that on anygiven pair (M, x), of a deterministic Turing transducer M and of an input x for M, simulates the
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behavior of M on x. Inputs that do not have the form (M, x) are rejected by U. Universal Turingmachines are defined similarly.

It should be noted that a pair (M, x) is presented to a universal Turing transducer in encodedform, and that the output of the universal Turing transducer is the encoding of the output of M on

input x. For convenience, the mentioning of the encoding is omitted when no confusion arises.Moreover, unless otherwise stated, a "standard" binary representation is implicitly assumed forthe encodings.


In what follows, a string is said to be astandard binary representation of a Turing transducer M= if it is equal to E(M), where E is defined recursively in the followingway.

a. E(M) = E(F)01E( ).b.

E(F) = E(p1) E(pk) for some ordering {p1, . . . , pk} of the states of F.c. E(B) = 0 is the binary representation of the blank symbol.

d. E( ) = E( 1)01E( 2)01 01E( r)01 for some ordering { 1, . . . , r} of the transitionrules of .

e. E( ) = E(q)E(a)E(b1) E(bm)E(p)E(d0)E(c1)E(d1) E(cm) E(dm)E( ) for each = (q, a,b1, . . . , bm, p, d0, c1, d1, . . . , cm, dm, ) in .

f. E(d) = 011 for d = -1, E(d) = 0111 for d = 0, E(d) = 01111 for d = +1, and E( ) = 0 foran output = .

g. E(qi) = 01i+2 for each state qi in Q, and some ordering q0, . . . , qs of the states of Q. Notethat the order assumes the initial state q0 to be the first.

h. E(ei) = 01i+1 for each symbol ei in ( {, $}) - {B} and some order {e1, . . . , et}in which e1 = and e2 = $.

Intuitively, we see that E provides a binary representation for the symbols in the alphabets of theTuring transducer, a binary representation for the states of the Turing transducer, and a binaryrepresentation for the possible heads movements. Then it provides a representation for asequence of such entities, by concatenating the representations of the entities. The string 01 isused as separator for avoiding ambiguity.

By definition, a given Turing transducer can have some finite number of standard binaryrepresentations. Each of these representations depends on the order chosen for the states in Q, theorder chosen for the symbols in ( {, $}) - {B}, the order chosen for the states in F, andthe order chosen for the transition rules in . On the other hand, different Turing transducers canhave identical standard binary representations if they are isomorphic, that is, if they are equalexcept for the names of their states and the symbols in their alphabets.

Example 4.4.1 If M is the Turing transducer whose transition diagram is given in Figure4.1.3,then E(M) can be the standard binary representation
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where E(q0) = 011, E(q1) = 0111, E(q2) = 01111, E(q3) = 011111, E(q4) = 0111111, E(B) = 0,E() = 011, E($) = 0111, E(a) = 01111, . . .

0 and 00 are examples of binary strings that are not standard binary representations of any Turingtransducer.

The string

represents a Turing transducer with one accepting state and four transition rules. Only the firsttransition rule has a nonempty output. The Turing transducer has one auxiliary work tape.

E(M)01E(x$) is assumed to be the standard binary representation of (M, x), with E(x$) =E()E(a1) E(an)E($) when x = a1 an.


The proof of the following result provides an example of a universal Turing transducer.

Theorem 4.4.1 There exists a universal Turing transducer U.

Proof U can be a two auxiliary-work-tape Turing transducer similar to M2 in the proof of

Proposition4.3.1. Specifically, U starts each computation by checking that its input is a pair (M,x) of some deterministic Turing transducer M = and of some input x forM (given in standard binary representation). If the input is not of such a form, then U halts in anonaccepting configuration. However, if the input is of such a form, U simulates a computationof M on x.

U, like M2, uses two auxiliary work tapes for keeping track of the content of the auxiliary worktapes of M. However, U also uses the auxiliary work tapes for keeping track of the states and theinput head locations of M. Specifically, the universal Turing transducer U records aconfiguration (uqv, u1qv1, . . . , umqvm, w) of M by storing #E(q)#|u|#E(u1)#E(v1)##E(um)#E(vm)# in an auxiliary work tape, and storing E(w) in the output tape.

To determine the transition rule (q, a, b1, . . . , bm, p, d0, c1, d1, . . . , cm, dm, ) that M uses in asimulated move, U extracts the state q and the symbols a, b1, . . . , bm. U records the stringE(q)E(a)E(b1) E(bm) in the auxiliary work tape that does not keep the configuration of M that isin effect. Then U determines p, d0, c1, d1, . . . , cm, dm, by searching E(M) for the substring thatfollows a substring of the form 01E(q)E(a)E(b1) E(bm).

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