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Lecture 2

• Topics– Importance of this material

• Fundamental Limitations

– Connecting Problems and Languages• Problems

– Search, function and decision problems

• Languages– Encodings of Problems

» Encode input instances of problem as strings

– Language Recognition Problem

» Is an input string in the language?

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Importance: Philosophy

• Phil. of life– What is the purpose of

life?

– What is achievable in life?

– What is NOT achievable in life?

• Phil. of computing– What is the purpose of

programming?

– What is solvable through programming?

– What is NOT solvable through programming?

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Importance: Science

• Physics– Study of fundamental

physical laws and phenomenon like gravity and electricity

• Engineering– Governed by physical

laws

• Our material– Study of fundamental

computational laws and phenomenon like undecidability and universal computers

• Programming– Governed by

computational laws

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Problem Types

• We classify problems based on– range (output set) size

• infinite

• binary

– mapping type• function

– unique output per input

• relation– more than one output per input

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I) Search Problems

• (countably) infinite range

• the mapping may be a relation

Inputs Outputs

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Examples

• Divisor– Input: Integer n– Output: An integral divisor of n

Inputs Outputs

9 13

9

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II) Function Problems

• (countably) infinite range

• mapping is a function– unique solution per input

Inputs Outputs

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Examples

• Sorting

• Multiplication Problem– Input: 2 integers x and y– Output: xy

Inputs Outputs

2,5 10

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Examples

• Maximum divisor problem– Input: Integer n– Output: size of maximum divisor of n smaller

than n

Inputs Outputs

9 13

9

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III) Decision Problems• Output is yes or no

– range = {Yes, No}

• unique solution– only one of Yes/No is correct

Inputs Outputs

Yes

No

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Examples

• Decision sorting– Input: list of numbers– Yes/No question: Is list in nondecreasing

order?

Inputs Outputs

Yes

No

(1,3,2,4)

(1,2,3,4)

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Examples

• Decision multiplication– Input: Three numbers x,y, z– Yes/No question: Is xy = z?

Inputs Outputs

Yes

No

(3,5,14)

(3,5,15)

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Examples

• Decision Divisor Problem– Input: Two numbers x and y– Yes/No question: Is y a divisor of x?

Inputs Outputs

Yes

No

(14,5)

(14,7)

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What about finite domain problems?

• Note all problems described above have infinite domains (input set)– size of range varies from 2 to countably infinite

• What happens if we had a problem with only a finite domain?– “Table lookup”

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Table Lookup Program

string x;

cin >> x;

switch x {

case “Kona”: cout << 3; break;

case “Eric”: cout << 31; break;

case “Cliff”: cout << 36; break;

default: cout << “Illegal input\n”;

}

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Generality of Decision Problems

• We now show that we can focus only on decision problems

– Having yes/no outputs is not limiting.– If we can solve a decision version of a problem,

we can typically solve the function and search versions of the same problem as well

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Example

• Decision divisor problem– Input: Integers x and y– Yes/No Question: Is y a divisor of x?– Assumption: Alg. A solves this problem

• Function divisor problem– Input: Integer x– Output: Largest divisor of x that is not x– Goal: Construct alg. A’ to solve this problem

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Program A’

integer x, max, j;

cin >> x;

max = 1;

for (j=2; j<x; j++)

if (A(x,j)) // using A as a procedure

max = j;

return max;

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Connecting Problems and Languages

• Start with any problem P

• Convert P to equivalent decision problem P’

• Convert P’ to equiv set membership problem SP– Reinterpret P’

• Convert SP to language recognition problem L– Encoding scheme

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Set Membership Problem

• Setting– Universe U– set S subset of U

• Input– Element x in U

• Yes/No Question– Is x in S?

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Decision Problems to Set Membership Problems

• Reformulate/interpret a decision problem as a set membership problem– Universe U: set of all input instances– Set S: either set of all YES or all NO input

instances

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Example

• Decision divisor problem– Input: Integers x and y– Yes/No Question: Is y a divisor of x?

• Set membership problem– Universe U = {(x,y) | x and y are integers}– Set S = {(x,y) | y divides x}

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Sets to Languages

• How do we represent an input instance of a set membership problem?– We use an encoding scheme

• Each input instance is mapped to a unique string over some finite alphabet

• Think ASCII

• Corresponding language L: the set of strings corresponding to yes instances

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Decision multiplication problem

• Alphabet 1= {0,1,2,3,4,5,6,7,8,9,;}

– 7;10;25 NO instance– 3;4;12 YES instance

• Binary Alphabet 2 = {0,1}

– Encode alphabet 1

• 11 characters in 1

• 4 bits per character of 1

– Encoding of 3;4;12• 001110110100101100010010

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Decision divisor problem

• Fill in

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Language recognition problem (LRP)

• Input Instance Description– Finite length string x in * (why finite?)

• Yes/No Question – Is input string x in language L?

• How does the language recognition problem for L relate to corresponding set membership problem?

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Decision Addition Problem

• Illegal strings easily recognized– Thus two problems essentially identical

*

Yes No

3;4;7

1;1;3 ;;1111;31;1;27;5;1

111;1111

Illegal

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Every decision problem can be formulated as an LRP

• First convert to set membership problem

• Then convert to language recognition problem using an encoding scheme– Illegal strings easily recognized

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Key Concepts

• Different types of problems– always countably infinite domain

• Decision problems– binary range = {YES, NO}– still general despite limited range

• Set membership problem• Language recognition problem

– encoding scheme