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 Inputs Outputs

6 Examples Divisor –Input: Integer n –Output: An integral divisor of n Inputs Outputs

7 II) Function Problems (countably) infinite range mapping is a function –unique solution per input Inputs Outputs

8 Examples Sorting Multiplication Problem –Input: 2 integers x and y –Output: xy Inputs Outputs 2,5 10

9 Examples Maximum divisor problem –Input: Integer n –Output: size of maximum divisor of n smaller than n Inputs Outputs

10 III) Decision Problems Output is yes or no –range = {Yes, No} unique solution –only one of Yes/No is correct Inputs Outputs Yes No

11 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)

12 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)

13 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)

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 >> x; switch x { case “Kona”: cout << 3; break; case “Eric”: cout << 31; break; case “Cliff”: cout << 36; break; default: cout << “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 >> x; max = 1; for (j=2; j<x; 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  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;

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 Illegal strings easily recognized –Thus two problems essentially identical ** YesNo 3;4;7 1;1;3 ;;1111;3 1;1;2 7;5;1 111;1111 Illegal

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