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CS-208Theory of Computation

Slides have been modified by Shaukat Wasi – CS@DSU

Human• A machine that

– recognizes one/multiple languages

– performs useful work when given instructions in the recognized language (s)

– has a capability of processing the instructions/input to• Solve a set of problems [NOT ALL Problems]

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Human Capabilities• To judge the capabilities of humans (in

general) with respect to:– What problems can be solved by humans?

• Can’t experiments with all humans…

• We would like to develop an abstract model of humans– Processing Model (Abstract Machine) [IPO]– A set of problems

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Problems solvable by Humans

• Recognition of a Language– A Problem (itself)

• Hence– Language = Problem

• Is language ‘x’ recognizable by the machine?• Is problem ‘x’ solvable by the machine?

• We (in this course) will focus mainly on this simple Problem (w.r.t Computers)

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Theory of Computation• Theory of computation is the branch that

deals with how efficiently problems can be solved on a Model of Computation, using an Algorithm.

• The field is divided into three major branches: – Automata Theory and language , – Computability Theory,– Computational Complexity Theory [Efficiency],

• Which are linked by the question: "What are the fundamental capabilities and limitations of computers?" Courtesy: Costas Busch / Jeff Ullman 5

Theory of Computation - Terms• A model of computation is the

definition of the set of allowable operations used in computation and their respective costs.

• What is Algorithm????

• Automata Theory is the study of Abstract Machines and Automata [Self Acting Machine]

• Formal Language: recognized by an automaton

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CPU memory

A widely accepted model of computation

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CPUinput

output

Program memory

temporary memory

The different components

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CPUinput

outputProgram memory

temporary memory

3)( xxf

Pseudo code for f(x) or Sequence of Steps????

Example:

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CPUinput

outputProgram memory

temporary memory

3)( xxf

compute xx

compute xx 2

Example:

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CPU

input

outputProgram memory

temporary memory

3)( xxf

compute xx

compute xx 2

2x

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CPU

input

outputProgram memory

temporary memory 3)( xxf

compute xx

compute xx 2

2x

42*2 z82*)( zxf

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CPU

input

outputProgram memory

temporary memory 3)( xxf

compute xx

compute xx 2

2x

42*2 z82*)( zxf

8)( xf

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Automaton [IPO]

CPUinput

output

Program memory

temporary memory

Automaton

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Automaton [Representation]

input

output

temporary memory

Automaton

statetransition

CPU + ProgramMem = States + Transitions

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Different Kinds of AutomataAutomata are distinguished by the temporary memory

• Finite Automata: no temporary memory

• Pushdown Automata: stack

• Turing Machines: random access memory

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Memory affects computational power:

More flexible memoryresults to

The solution of more computationalproblems

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input

output

temporary memory

Finite Automaton

Finite Automaton

Example: Elevators, Vending Machines, Lexical Analyzers (small computing power)

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input

output

Stack

PushdownAutomaton

Pushdown Automaton

Example: Parsers for Programming Languages (medium computing power)

Push, PopTemp.memory

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input

output

Random Access Memory

TuringMachine

Turing Machine

Examples: Any Algorithm (highest known computing power)

Temp.memory

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Finite Automata

PushdownAutomata

TuringMachine

Power of Automata

Less power More powerSolve more computational problems

Simple problems

More complexproblems

Hardestproblems

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Turing Machine is the most powerful known computational model

Question: can Turing Machines solve all computational problems?

Answer: NO(there are unsolvable problems)

QUERIES!!!

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