Importance of this material

1
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?

2
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?

3
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

4
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

5
I) Search Problems

• (countably) infinite range
• the mapping may be a relation

6
Examples

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

9
7
II) Function Problems

• (countably) infinite range
• mapping is a function
• unique solution per input

8
Examples

• Sorting
• Multiplication Problem
• Input 2 integers x and y
• Output xy

2,5
9
Examples

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

9
10
III) Decision Problems

• Output is yes or no
• range Yes, No
• unique solution
• only one of Yes/No is correct

11
Examples

• Decision sorting
• Input list of numbers
• Yes/No question Is list in nondecreasing order?

12
Examples

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

13
Examples

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

14
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

15
Table Lookup Program

• string x
• cin gtgt x
• switch x
• case Kona cout ltlt 3 break
• case Eric cout ltlt 31 break
• case Cliff cout ltlt 36 break
• default cout ltlt Illegal input\n

16
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

17
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

18
Program A

• integer x, max, j
• cin gtgt x
• max 1
• for (j2 jltx j)
• if (A(x,j)) // using A as a procedure
• max j
• return max

19
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

20
Set Membership Problem

• Setting
• Universe U
• set S subset of U
• Input
• Element x in U
• Yes/No Question
• Is x in S?

21
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

22
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

23
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

24
Decision multiplication problem

• Alphabet S1 0,1,2,3,4,5,6,7,8,9,
• 71025 NO instance
• 3412 YES instance
• Binary Alphabet S2 0,1
• Encode alphabet S1
• 11 characters in S1
• 4 bits per character of S1
• Encoding of 3412
• 001110110100101100010010

25
Decision divisor problem

• Fill in

26
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?

27
Decision Addition Problem
S
113
11113
112
751
347
1111111
Yes
No

• Illegal strings easily recognized
• Thus two problems essentially identical

28
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

29
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