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7.4 Uniform Families of Circuits

"Table Look-Up" Circuits

"Unrolled Hardware" CircuitsUniform Families of Circuits

"Table Look-Up" Circuits

Families of circuits were introduced to help characterize the resources that problems require

from parallel machines. The following theorem implies that the families cannot serve such apurpose in their general form, because they can recognize languages that are not recursively

enumerable.

Theorem 7.4.1 Each language L in {0, 1}* is decidable by a family of circuits.

Proof Consider any language L in {0, 1}*, and any natural number n. Let Ln denote the set L{0, 1}

n. That is, Ln denotes the set of all the binary strings of length n in L.

For any given string w in Ln, a subcircuit cw with n input nodes can be constructed that accepts agiven input if and only if the input is equal to w. The language Ln is finite. As a result, the

subcircuits cw that correspond to the strings w in Ln can be merged, by letting them share the

input nodes and by OR ing their outputs. The obtained circuit cn determines the membership inLn by a table look-up technique.

Consequently, L is decidable by the family (c0, c1, c2, . . . ) of circuits.

Example 7.4.1 The circuit c3 in Figure7.4.1

Figure 7.4.1A "table look-up" circuit for the language L3 = {011, 100, 101}.

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decides the language L3 = {011, 100, 101} by a table look-up approach. For each string w in L3,

the circuit has a corresponding subcircuit cw that decides just the membership of the string.

The families of table look-up circuits in the proof of Theorem7.4.1have size complexity 2O(n)

and depth complexity O(n). These families do not reflect the complexity of deciding the

languages, because they assume the knowledge of which strings are in a given language andwhich are not. That is, the complexity involved in deciding the languages is shifted into the

complexity of constructing the corresponding families of circuits.

"Unrolled Hardware" Circuits

A circuit can be obtained to characterize a halting computation of a parallel machine by laying

down the portion of the hardware of that is involved in the computation. During the laying

down of the hardware, cycles can be avoided by unrolling the hardware. The depth of such a

circuit provides a measurement for the time that the computation requires, and the circuits sizeprovides an upper bound on the space the computation requires.

Example 7.4.2 The circuit in Figure7.4.2(b)

Figure 7.4.2(a) "Hardware." (b) Corresponding "unrolled hardware."

computes the function that the hardware of Figure7.4.2(a) computes in three units of time. It is

assumed that initially each input to a gate is either an input value or the constant 0.

In a similar way, one can also obtain a circuit cn that corresponds to all the halting computations

of on instances of length n, n 0. (The outputs for the inputs of a given length n are assumed

to be appended by a string of the form 10 0 to let them all have identical lengths.)Consequently, the approach implies a family C = (c0, c1, c2, . . . ) of circuits for each parallel

machine that halts on all inputs. Moreover, the families of circuits faithfully reflect thecomplexity of the parallel computations and can be effectively obtained from each such parallel

machine .

Uniform Families of Circuits

http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8001t7.4.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8001t7.4.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8001t7.4.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-9http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-9http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-10http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-10http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-10http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-9http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8001t7.4.1

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By the previous discussion, from each parallel machine that halts on all inputs, a circuits

constructor can be obtained to compute { (1n, cn) | n 0 }, where C = (c0, c1, c2, . . . ) is a family

of circuits that computes the same function as . The circuits constructor can be one thatprovides families of table look-up, unrolled hardware, or other types of circuits.

The interest here is in circuits constructors that preserve, in the families of circuits that theyconstruct, the complexity of the given parallel machines. Such constructors do not allow the shift

of complexity from the constructed families of circuits to the constructors. Moreover, they also

do not allow an unrealistic increase in the complexity of the constructed families of circuits.Circuits constructors with such characteristics are said to be uniform circuits constructors. A

family C = (c0, c1, c2, . . . ) of circuits is said to be uniform if a uniform circuits constructor can

compute the function { (1n, cn) | n 0 }.

Many characterizations have been offered for the uniform circuits constructors. The

characterization used here, which has been widely accepted, defines these conditions in terms ofa class of deterministic Turing transducers.

Definition A Turing transducer is said to be a uniform circuits constructorif it is an O(log

Z(n)) space-bounded, deterministic Turing transducer that computes { (1n, cn) | n 0 }, where C

= (c0, c1, c2, . . . ) is a family of circuits of size complexity Z(n). A family C = (c0, c1, c2, . . . ) of

circuits of size complexity Z(n) is said to be a uniform family of circuits if an O(log Z(n)) space-

bounded, deterministic Turing transducer can compute { (1n, cn) | n 0 }.

The characterization of uniform families of circuits is motivated by the unrolled hardwareapproach. With such an approach the circuits constructor needs O(log H(n) + log T(n)) = O(log

(H(n)T(n))) space, if the parallel machine has size complexity H(n) and time complexity T(n).

O(log H(n)) space is used for tracing through the hardware, and O(log T(n)) space is used for

tracing through time. H(n)T(n) is of a similar order of magnitude to the size Z(n) of the circuits.

Example 7.4.3 Consider the language L = { uu | u is in {0, 1}* }. Let Ln = L {0, 1}n

for n 0,that is, Ln denotes the set of all the binary strings of length n in L. The language Ln is decided by

the circuit cn in Figure7.4.3(a)

http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-10005r7.4.3http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-10005r7.4.3http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-10005r7.4.3http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-10005r7.4.3

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7.4 Uniform Families of Circuits

"Table Look-Up" Circuits

"Unrolled Hardware" CircuitsUniform Families of Circuits

"Table Look-Up" Circuits

Families of circuits were introduced to help characterize the resources that problems requirefrom parallel machines. The following theorem implies that the families cannot serve such a

purpose in their general form, because they can recognize languages that are not recursively

enumerable.

Theorem 7.4.1 Each language L in {0, 1}* is decidable by a family of circuits.

Proof Consider any language L in {0, 1}*, and any natural number n. Let Ln denote the set L

{0, 1}n

. That is, Ln denotes the set of all the binary strings of length n in L.

For any given string w in Ln, a subcircuit cw with n input nodes can be constructed that accepts a

given input if and only if the input is equal to w. The language Ln is finite. As a result, the

subcircuits cw that correspond to the strings w in Ln can be merged, by letting them share theinput nodes and by OR ing their outputs. The obtained circuit cn determines the membership in

Ln by a table look-up technique.

Consequently, L is decidable by the family (c0, c1, c2, . . . ) of circuits.

Example 7.4.1 The circuit c3 in Figure7.4.1

Figure 7.4.1A "table look-up" circuit for the language L3 = {011, 100, 101}.

decides the language L3 = {011, 100, 101} by a table look-up approach. For each string w in L3,

the circuit has a corresponding subcircuit cw that decides just the membership of the string.

http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk.html#Q2-80004-7http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk.html#Q2-80004-7http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q1-80004-8http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q1-80004-8http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q1-80004-9http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q1-80004-9http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q1-80004-10http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q1-80004-10http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-8http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-8http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8005r7.4.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8005r7.4.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8005r7.4.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8005r7.4.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-8http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q1-80004-10http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q1-80004-9http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q1-80004-8http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk.html#Q2-80004-7

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The families of table look-up circuits in the proof of Theorem7.4.1have size complexity 2O(n)

and depth complexity O(n). These families do not reflect the complexity of deciding the

languages, because they assume the knowledge of which strings are in a given language andwhich are not. That is, the complexity involved in deciding the languages is shifted into the

complexity of constructing the corresponding families of circuits.

"Unrolled Hardware" Circuits

A circuit can be obtained to characterize a halting computation of a parallel machine by laying

down the portion of the hardware of that is involved in the computation. During the layingdown of the hardware, cycles can be avoided by unrolling the hardware. The depth of such a

circuit provides a measurement for the time that the computation requires, and the circuits sizeprovides an upper bound on the space the computation requires.

Example 7.4.2 The circuit in Figure7.4.2(b)

Figure 7.4.2(a) "Hardware." (b) Corresponding "unrolled hardware."

computes the function that the hardware of Figure7.4.2(a) computes in three units of time. It is

assumed that initially each input to a gate is either an input value or the constant 0.

In a similar way, one can also obtain a circuit cn that corresponds to all the halting computations

of on instances of length n, n 0. (The outputs for the inputs of a given length n are assumedto be appended by a string of the form 10 0 to let them all have identical lengths.)

Consequently, the approach implies a family C = (c0, c1, c2, . . . ) of circuits for each parallel

machine that halts on all inputs. Moreover, the families of circuits faithfully reflect the

complexity of the parallel computations and can be effectively obtained from each such parallelmachine .

Uniform Families of Circuits

By the previous discussion, from each parallel machine that halts on all inputs, a circuits

constructor can be obtained to compute { (1n, cn) | n 0 }, where C = (c0, c1, c2, . . . ) is a family

http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8001t7.4.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8001t7.4.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8001t7.4.1http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-9http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-9http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-10http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-10http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-10http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-9002r7.4.2http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#Q2-80004-9http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-8001t7.4.1

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of circuits that computes the same function as . The circuits constructor can be one thatprovides families of table look-up, unrolled hardware, or other types of circuits.

The interest here is in circuits constructors that preserve, in the families of circuits that they

construct, the complexity of the given parallel machines. Such constructors do not allow the shift

of complexity from the constructed families of circuits to the constructors. Moreover, they alsodo not allow an unrealistic increase in the complexity of the constructed families of circuits.

Circuits constructors with such characteristics are said to be uniform circuits constructors. A

family C = (c0, c1, c2, . . . ) of circuits is said to be uniform if a uniform circuits constructor can

compute the function { (1n, cn) | n 0 }.

Many characterizations have been offered for the uniform circuits constructors. Thecharacterization used here, which has been widely accepted, defines these conditions in terms of

a class of deterministic Turing transducers.

Definition A Turing transducer is said to be a uniform circuits constructorif it is an O(log

Z(n)) space-bounded, deterministic Turing transducer that computes { (1

n

, cn) | n 0 }, where C= (c0, c1, c2, . . . ) is a family of circuits of size complexity Z(n). A family C = (c0, c1, c2, . . . ) ofcircuits of size complexity Z(n) is said to be a uniform family of circuits if an O(log Z(n)) space-

bounded, deterministic Turing transducer can compute { (1n, cn) | n 0 }.

The characterization of uniform families of circuits is motivated by the unrolled hardware

approach. With such an approach the circuits constructor needs O(log H(n) + log T(n)) = O(log

(H(n)T(n))) space, if the parallel machine has size complexity H(n) and time complexity T(n).O(log H(n)) space is used for tracing through the hardware, and O(log T(n)) space is used for

tracing through time. H(n)T(n) is of a similar order of magnitude to the size Z(n) of the circuits.

Example 7.4.3 Consider the language L = { uu | u is in {0, 1}* }. Let Ln = L {0, 1}

n

for n 0,that is, Ln denotes the set of all the binary strings of length n in L. The language Ln is decided by

the circuit cn in Figure7.4.3(a)

http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-10005r7.4.3http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-10005r7.4.3http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-10005r7.4.3http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-sevense4.html#80004-10005r7.4.3

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Figure

7.4.3

A circuit cn that, according to the case, checks whether an input of length n is of the

form uu. (a) n 0, and n is even. (b) n is odd. (c) n = 0.

if n is a nonzero even integer, by the circuit cn in Figure7.4.3(b) if n is an odd integer, and by the

circuit cn in Figure7.4.3(c) if n = 0.

The family (c0, c1, c2, . . . ) of circuits is of depth complexity D(n) = O(log n) and sizecomplexity Z(n) = O(n/2 + n/4 + + 1) = O(n). The family is uniform because the function { (1

n,

cn) | n 0 } is computable by a log Z(n) = O(log n) space-bounded, deterministic Turing

transducer.

The following thesis for parallel computations is stated in terms of uniform families of circuits.

As in the previous theses for sequential and probabilistic computations, only supportiveevidences can be provided to exhibit the correctness of the thesis.

The Parallel Computation Thesis A function can be mechanically computed by a parallelmachine of size complexity H(n) and time complexity T(n) only if it has a uniform family of

circuits of size complexity p(H(n)T(n)) and depth complexity p(T(n)), for some polynomial p( ).

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defines job satisfaction as a Pleasurable or positive emotional state resulting from the appraisal

of ones job or job experience, to the extent that persons job fulfill his dominant need and is

consistent with his expectations and values[2]

. There are three major theories of job satisfaction

viz,

i) Herzbergs Motivation-Hygiene Theory.

ii) Need Fulfillment Theory.

iii) Social Reference Group Theory.

Smith Kendall and Hulin have suggested that there are five job dimensions that represent the

most important characteristics of a job about which people have affective responses, these are:

i) The work itself: The extent to which the job provides the individual with the interestingtask, opportunities for learning and the chance of accept responsibility.

ii) Pay: The amount of financial remuneration that is received and the degree to which thisis viewed as equitable vis-a-vis that of others in the organization.

iii) Promotion opportunities: The chance for advancement in the hierarchy.

vi) Supervision: The ability is on the supervision to provide technical assistance and

behavioral support.

1v) Co-workers: The degree to which follow workers are technically proficient and socially

supportive.

However, a comprehensive approach requires that may additional factors be included before a

complete understanding of job satisfaction can be obtained. Such factors as the employees are,

health, temperament, desires and levels of aspiration should be considered.

Further, his family relationship, social status, recreational or purely social-contribute ultimately

to job satisfaction.