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CSC-4890 Introduction to the Theory of Computation

General Info about Course

Instructor Konstantin Busch

Book Introduction to the Theory of

Computation

Michael Sipser, 2nd edition

Course Goals

Provide computation Models

Analyze power of Models

Answer Intractability questions

What computational problems can each model solve?

Answer Time Complexity questions

How much time we need to solve the problems?

A widely accepted model of computation

memory

CPU

The different components of memory

temporary memory

input

CPU

output

Program memory

Example

temporary memory

input

CPU

output

Program memory

compute

compute

temporary memory

input

CPU

output

Program memory

compute

compute

temporary memory

input

CPU

output

Program memory

compute

compute

temporary memory

input

CPU

Program memory

output

compute

compute

Automaton

temporary memory

Automaton

input

CPU

output

Program memory

Automaton

temporary memory

Automaton

input

output

transition

state

CPUProgramMem States Transitions

Different Kinds of Automata

- Automata are distinguished by the temporary

memory - Finite Automata no temporary memory
- Pushdown Automata stack
- Turing Machines random access memory

Memory affects computational power

More flexible memory

results to

The solution of more computational problems

Finite Automaton

temporary memory

input

Finite Automaton

output

Example Elevators, Vending Machines,

Lexical Analyzers (small

computing power)

Pushdown Automaton

Temp. memory

Stack

Push, Pop

input

Pushdown Automaton

output

Example Parsers for Programming Languages

(medium computing power)

Turing Machine

Temp. memory

Random Access Memory

input

Turing Machine

output

Examples Any Algorithm

(highest known computing power)

Power of Automata

Simple problems

More complex problems

Hardest problems

Finite Automata

Pushdown Automata

Turing Machine

Less power

More power

Solve more computational problems

Turing Machine is the most powerful known

computational model

Question can Turing Machines solve

all computational problems?

Answer NO

(there are unsolvable problems)

Time Complexity of Computational Problems

P problems

(Polynomial time problems)

Solved in polynomial time

NP-complete problems

(Non-deterministic Polynomial time problems)

Believed to take exponential time to be solved

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