Definition of Algorithm

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Definition of algorithm

An algorithm is deterministic automaton for accomplishing a goal

which, given an initial state, will terminate in a defined end-state.

The efficiency of implementation of the algorithm depends upon

speed, size, resources consumption. We will discuss definitions,

classifications and the history.

Contents

What is an algorithm?

Definitions of algorithms

Blass and Gurevich

Boolos and Jeffrey

Knuth

Markov

Minsky

Stone

Some issues

Expressing algorithms

Must an algorithm halt?

Algorithm analysis

Simple example: multiplication

References

What is analgorithm?

No agreed-to definition of "algorithm" exists.

A simple definition:A set of instructions for solving a problem.

The algorithm is either implemented by a program or simulated by

a program. Algorithms often have steps that iterate (repeat ) or

require decisions such as logic or comparison.

An very simple example of an algorithm is multiplying two

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numbers: on first computers with limited processors, this was

accomplished by a routine that in a number of loop based on the

first number adds the second number. The algorithm translates a

method into computer commands.

Algorithms are essential to the way computers process information,

because a computer program is essentially an algorithm that tells

the computer what specific steps to perform (in what specificorder) in order to carry out a specified task, such as calculating

employees' paychecks or printing students' report cards. Thus, an

algorithm can be considered to be any sequence of operations

which can be performed by a Turing-complete system. Authors

who assert this thesis include Savage (1987) and Gurevich (2000):

...Turing's informal argument in favor of his thesis justifies

a stronger thesis: every algorithm can be simulated by a

Turing machine" ...according to Savage [1987], analgorithm is a computational process defined by a Turing

machine.

Typically, when an algorithm is associated with processing

information, data is read from an input source or device, written to

an output sink or device, and/or stored for further processing.

Stored data is regarded as part of the internal state of the entity

performing the algorithm. In practice, the state is stored in a data

structure.

For any such computational process, the algorithm must be

rigorously defined: specified in the way it applies in all possible

circumstances that could arise. That is, any conditional steps must

be systematically dealt with, case-by-case; the criteria for each

case must be clear (and computable).

Because an algorithm is a precise list of precise steps, the order of

computation will almost always be critical to the functioning of the

algorithm. Instructions are usually assumed to be listed explicitly,and are described as starting 'from the top' and going 'down to the

bottom', an idea that is described more formally byflow of

control.

So far, this discussion of the formalization of an algorithm has

assumed the premises of imperative programming. This is the most

common conception, and it attempts to describe a task in discrete,

'mechanical' means. Unique to this conception of formalized
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algorithms is the assignment operation, setting the value of a

variable. It derives from the intuition of 'memory' as a scratchpad.

For some alternate conceptions of what constitutes an algorithm

see functional programming and logic programming.

Definitions of algorithms

Blass and Gurevich

Blass and Gurevich describe their work as evolved from

consideration of Turing machines, Kolmogorov-Uspensky

machines (KU machines), Schnhage machine (storage

modification machine SMM), and pointer machines (linking

automata) as defined by Knuth. The work of Gandy and Markov

are also described as influential precursors.

Gurevich offers a 'strong' definition of an algorithm (that is

summarized here):

The Turing's informal argument in favor of his thesis

justifies a stronger thesis: every algorithm can be simulated

by a Turing machine. In practice, it would be ridiculous.

Can one generalize Turing machines so that any algorithm,

never mind how abstract, can be modeled by a generalized

machine? But suppose such generalized Turing machines

exist. What would their states be?A first-order structure. Aparticular small instruction set suffices in all cases.

Computation could be an evolution of the state, could be

nondeterministic, interact with its environment, could be

parallel and multi-agent, could have dynamic semantics.

The two underpinings of thier work are: Turing's thesis

andthe notion of first order structure of Tarski.

The above phrase computation as an evolution of the state

differs markedly from the definition of Knuth and Stone, the"algorithm" as a Turing machine program. Rather, it corresponds

to what Turing called the complete configuration, and includes

both the current instruction (state) andthe status of the tape.

Kleene (1952) shows an example of a tape with 6 symbols on it,

all other squares are blank, and how to "Gdelize" its combined

table-tape status.

Boolos and Jeffrey


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Their definition is:

"Explicit instructions for determining the nth member of

a set, for arbitrary finite n. Such instructions are to be

given quite explicitly, in a form in which they could be

followed by a computing machine, or by a human who

is capable of carrying out only very elementary

operations on symbols."

Knuth

Knuth (1968, 1973) has given a list of five properties that are

widely accepted as requirements for an algorithm:

1. Finiteness: "An algorithm must always terminate after a

fini te number of steps"

2. Definiteness: "Each step of an algorithm must be preciselydefined; the actions to be carried out must be rigorously

and unambiguously specified for each case"

3. Input: "...quantities which are given to it initially before

the algorithm begins. These inputs are taken from specified

sets of objects"

4. Output: "...quantities which have a specified relation to the

inputs"

5. Effectiveness: "... all of the operations to be performed in

the algorithm must be sufficiently basic that they can inprinciple be done exactly and in a finite length of time by a

man using paper and pencil"

Knuth offers as an example the Euclidean algorithm for

determining the greatest common divisor of two natural numbers.

Knuth admits that, while his description of an algorithm may be

intuitively clear, it lacks formal rigor, since it is not exactly clear

what "precisely defined" means, or "rigorously and

unambiguously specified" means, or "sufficiently basic", and soforth. He makes an effort in this direction in his first volume where

he defines in detailwhat he calls the "machine language" for his

"mythical MIX... the world's first polyunsaturated computer".

Many of the algorithms in his books are written in the MIX

language. He also uses tree diagrams, flow diagrams and state

diagrams.

Markov


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A. A. Markov (1954) provided the following definition of

algorithm:

1. In mathematics, "algorithm" is commonly understood to

be an exact prescription, defining a computational process,

leading from various initial data to the desired result....

The following three features are characteristic ofalgorithms and determine their role in mathematics:

a) the precision of the prescription, leaving no place to

arbitrariness, and its universal comprehensibility -- the

definiteness of the algorithm;

b) the possibility of starting out with initial data, which

may vary within given limits -- the generality of the

algorithm;

c) the orientation of the algorithm toward obtaining some

desired result, which is indeed obtained in the end withproper initial data -- the conclusiveness of the algorithm.

He admitted that this definition "does not pretend to mathematical

precision". His 1954 monograph was his attempt to define

algorithm more accurately; he saw his resulting definition -- his

"normal" algorithm -- as "equivalent to the concept of a recursive

function". His definition included four major components:

1. Separate elementary steps, each of which will be performed

according to one of the given substitution rules.2. Steps of local nature (the algorithm won't change more than

a certain number of symbols).

3. The scheme of the algorithm is a list of rules for the

substitution formulas.

4. A means to distinguish a concluding substitution (a final

state).

In his Introduction Markov observed that the entire significance for

mathematics of efforts to define algorithm more precisely wouldbe in connection with the problem of a constructive foundation for

mathematics.

Minsky

Minsky (1967) asserts that an algorithm is an effective procedure

and replaces further in his text algorithm by effective procedure.

The term is also used by Knuth. Here is its definition of effective


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procedure:

A set of rules which tell us, from moment to moment,

precisely how to behave.

But he recognizes that this is subject to a criticism:

The interpretation of the rules is left to depend on someperson or agent.

He made a refinement: to specify, along with the statement of the

rules, the details of the mechanism that is to interpret them. To

avoid the cumbersome process of having to do this over again for

each individual procedure he hopes to identify a reasonably

uniform family of rule-obeying mechanisms. Here is his

formulation:

(1) a language in which sets of behavioral rules are to beexpressed, and

(2) a single machine which can interpret statements in the

language and thus carry out the steps of each specified

process.

In the end, though, he still worries that "there remains a subjective

aspect to the matter. Different people may not agree on whether a

certain procedure should be called effective".

But Minsky is undeterred. He immediately introduces "Turing's

Analysis of Computation Process". He quotes what he calls

Turing's thesis:

Any process which could naturally be called an effective

procedure can be realized by a Turing machine.

(This is also called Church's thesis).

After an analysis of "Turing's Argument" he observes that

equivalence of many intuitive formulations of Turing, Church,

Kleene, Post, and Smullyan "leads us to suppose that there is

really here an objective or absolute notion".

Stone

Stone (1972) and Knuth (1968, 1973) were professors at Stanford

University at the same time so it is not surprising if there are

similarities in their definitions:


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To summarize ... we define an algorithm to be a set of rules

that precisely defines a sequence of operations such that

each rule is effective anddefinite and such that the

sequence terminates in a finite time.

Stone is noteworthy because of his detailed discussion of what

constitutes an effective rule for his robot, or person-acting-as-

robot, must have some information and abilities within them, and ifnot the information and the ability must be provided in "the

algorithm":

For people to follow the rules of an algorithm, the rules

must be formulated so that they can be followed in a robot-

like manner, that is, without the need for thought...

however, i f the instructions [to solve the quadratic

equation, his example] are to be obeyed by someone who

knows how to perform arithmetic operations but does notknow how to extract a square root, then we must also

provide a set of rules for extracting a square root in order

to satisfy the definition of algorithm.

(...) not all instructions are acceptable, because they may

require the robot to have abilities beyond those that we

consider reasonable.

He gives the example of a robot confronted with the question: "Is

Henry VIII is a King of England", print 1 if yes and 0 if no, but therobot has not been previously provided with this information. And

worse, if the robot is asked if Aristotle was a King of England and

the robot only had been provided with five names, it would not

know how to answer. Thus:

An intuitive definition of an acceptable sequence of

instructions is one in which each instruction is precisely

defined so that the robot is guaranteed to be able to obey it.

After providing us with his definition, Stone introduces the Turingmachine model and states that the set of five-tupes that are the

machine's instructions is an algorithm... known as a Turing

machine program. Puis he says that a computation of a Turing

machine is describedby stating:

1. The tape alphabet

2. The form in which the parameters are presented on the

tape


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3. The initial state of the Turing machine

4. The form in which answers will be represented on the

tape when the Turing machine halts

5. The machine program.

This is in the spirit of Blass and Gurevich.

Some issues

Expressing algorithms

Algorithms can be expressed in many kinds of notations:

- Natural language expressions of algorithms tend to be verbose

and ambiguous, and are rarely used for complex or technical

algorithms.

- Pseudocode and flowcharts are structured ways to express

algorithms that avoid the ambiguities, while remaining independentof a particular implementation language.

- Programming languages are primarily intended for expressing

algorithms in a form that can be executed by a computer, but are

often used as a way to define or document algorithms.

Must an algorithm halt?

Some writers restrict the definition ofalgorithm to procedures that

eventually finish. Others, as Kleene, include procedures that could

run forever without stopping. Such a procedure has been called a

"computational method" by Knuth or "calculation procedure or

algorithm" by Kleene. However, Kleene notes that such a method

must eventually exhibit "some object".

Minksy (1967) makes the observation that, if an algorithm hasn't

"terminated" then how can we answer the question: "Will it

terminate with the correct answer?'"

Thus the answer is: undecidable. We can never know, nor can wedo an analysis beforehand to find out. The analysis of algorithms

for their likelihood of termination is called "Termination analysis".

Algorithm analysis

The terms "analysis of algorithms" was coined by Knuth. Most

people who implement algorithms want to know how much of a

particular resource, such as time or storage, is required for the


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execution. Methods have been developed for the analysis of

algorithms to obtain such quantitative answers.

The analysis and study of algorithms is one discipline of computer

science, and is often practiced abstractly (without the use of a

specific programming language or hardware). But the Scriptol code

is portable, simple and abstract enough for such analysis.

Simple example: multiplication

int multiply(int x, int y)

int sum = 0

while y > 0

sum + x

let y - 1

return sum

int a = 5

int b = 7

print a,"x", b, "=", multiply(a, b)

References

Martin Davis. The Undecidable: Basic Papers On

Undecidable Propostions, Unsolvable Problems and

Computable Functions. New York: Raven Press, 1965.

Davis gives commentary before each article.Yuri Gurevich. Sequential Abstract State Machines

Capture Sequential Algorithms, ACM Transactions on

Computational Logic, Vol 1, no 1 (July 2000), pages 77-111.

Includes bibliography of 33 sources.

A. A. Markov. Theory of algorithms. Imprint Moscow,

Academy of Sciences of the USSR, 1954 . Original title:

Teoriya algerifmov.

Marvin Minsky. Computation: Finite and Infinite

Machines, First, Prentice-Hall, Englewood Cliffs, NJ, 1967.

Harold S. Stone.Introduction to Computer Organization

and Data Structures, 1972, McGraw-Hill, New York.

Cf in particular the first chapter titled:Algorithms, Turing

Machines, and Programs.

See also:

The list of algorithms.

Algorithms
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not getting it clearly

Momirao555

fine and thanks for.........

azad

please can you details information in finitness,

difinitness, effectiveness and input, output

azad
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