1. 1 Lecture 10 Proving more specific problems are not recursive Reduction technique –Use subroutine theme to show that if one problem is unsolvable, so is a second problem –Need to clearly differentiate between use of program as a subroutine and a program being an input to another program

2. 2 Question 1 What can we conclude from the following scenario? –We prove/know language L 2 is recursive –We show that we can construct a program P 1 which decides L 1 using any program P 2 which decides language L 2 as a subroutine. –We conclude that Does this occur in “real-life programming”?

3. 3 Question 2 What can we conclude from the following scenario? –We prove/know language L 1 is not recursive –We show that we can construct a program P 1 which decides L 1 using any program P 2 which decides language L 2 as a subroutine. –We conclude that

4. 4 Rephrasing key step We show that we can construct a program P 1 which decides L 1 using any program P 2 which decides language L 2 as a subroutine. –This can be rephrased in the following 2 ways: If L 2 is recursive, then L 1 is recursive –Scenario in question 1 If L 1 is not recursive, then L 2 is not recursive –Scenario in question 2

5. 5 Illustration of “If L 1 is not recursive, then L 2 is not recursive” Set of all languages REC L1L1 L2L2 L2L2 L1L1 L1L1 L2L2 L1L1 L2L2

6. 6 What we will typically do Set of all languages REC L1L1 L2L2 If L 1 is not recursive, then L 2 is not recursive

7. 7 Proving “If L 2 is recursive, then L 1 is recursive” Assume L 2 is recursive Let P 2 be a program which decides L 2 Construct a program P 1 which decides L 1 using P 2 [subroutine theme] –We have no idea how P 2 decides L 2 We still treat P 2 as a black box –L 1 and L 2 are now SPECIFIC languages Thus, we can use properties of L 1 and L 2 in this construction Argue P 1 decides L 1

8. 8 Constructing P 1 from P 2 Many ways to construct P 1 from P 2 We focus on one method –Construct a program P 3 which computes a function f which we call a many-one reduction –P 3 (or f) transforms inputs to problem L 1 into inputs to problem L 2 so that yes inputs map to yes inputs no inputs map to no inputs

9. 9 Construction Overview P1P1 xYes/No Properties of the construction Input string x is transformed into a new string P 3 (x) x is an input to program P 1 (problem L 1 ) P 3 (x) is an input to program P 2 (problem L 2 ) We must give actual program P 3 P 3 will use specific knowledge about L 1 and L 2 We use P 2 as a black box routine P2P2 Y/NP3P3 P 3 (x)

10. 10 Many-one Reductions in Detail Properties of f and program P 3

11. 11 Properties of P 3 (f) P 3 must compute a many-one reduction function f:  * -->  * –For all x in  *, P 3 (x) must be defined That is P 3 must halt with some output P 3 (x) –For all x in  *, (x in L 1 ) iff (P 3 (x) in L 2 ) (x in L 1 ) --> (P 3 (x) in L 2 ) Not (x in L 1 ) --> Not (P 3 (x) in L 2 ) P2P2 P3P3 P1P1 x P 3 (x) Yes/No

12. 12 Yes->Yes and No->No ** L1L1 L1L1 ** L2L2 L2L2 P2P2 P3P3 P1P1 x P 3 (x) Yes/No

13. 13 Many-one Reduction If there is such a many-one reduction function f (and the corresponding program P 3 ), we say that L 1 many-one reduces to L 2 Notation –L 1 b m L 2 – b denotes L 1 is no harder than L 2 ** L1L1 L1L1 ** L2L2 L2L2

14. 14 Example L 1 is even length strings over {0,1} L 2 is odd length strings over {0,1} Tasks –Give a reduction function f which shows that L 1 b m L 2 –Give a corresponding program P 3 which computes f ** L1L1 L1L1 ** L2L2 L2L2

15. 15 Example 2 L 1 is {0,1} * L 2 is {0} Tasks –Give a reduction function f which shows that L 1 b m L 2 –Give a corresponding program P 3 which computes f ** L1L1 L1L1 ** L2L2 L2L2

16. 16 Constructing an example P 3

17. 17 Example L 1 is H Input –Program P’ –input y to program P’ Yes/No Question –Does P’ halt on y? L 2 Input –Program P’’ Yes/No question –Does L(P’’) = the set of even length strings?

18. 18 Construction overview again P1P1 xYes/No We are building a program P 1 to solve L 1 which, in this case, is the halting problem H P3P3 P 3 (x) P 1 will use P 3 as a subroutine, and we must explicitly construct P 3 using specific properties of L 1 and L 2 P2P2 Y/N P 1 will use P 2 as a subroutine, and we have no idea how P 2 accomplishes its task

19. 19 Keep this in mind Programs which are PART of program P 1 and thus “executed” when P 1 executes –Program P 3, an actual program we construct –Program P 2, an assumed program which solves problem L 2 Programs which are INPUTS/OUTPUTS of programs P 1, P 2, and P 3 and which are not “executed” when P 1 executes –Programs P’, P’’, and P’’’

20. 20 Analysis of L 2 L 2 –Input Program P’’ –Yes/No question Does L(P’’) = the set of even length strings? Program P 2 –Solves L 2 –We don’t know how Consider a program P such that L(P) = {} –P 2 rejects program P as input Consider the following program P’’ –If input z has even length then yes else no –L(P’’) = set of even length strings –P 2 accepts program P’’ as input

21. 21 Specifying P 3 Remember, we are solving H (L 1 ), not L 2 Input x to P 3 is an input to H –Program P’ –Input string y to program P’ Output P 3 (x) of P 3 is an input to problem L 2 –Program P’’’ Specification for P 3 –P’’’ = P 3 (x) = P 3 (P’,y) –L(P’’’) = L(P’’) = set of even length strings iff P’ halts on y P2P2 P3P3 P1P1 x P 3 (x) Yes/No

22. 22 How does P 3 work? Key –P 3 creates P’’’ by modifying program P’ to run y to “test” if P’ halts on y –P 3 then appends a program P’’ where P’’ has the right property Example P2P2 P3P3 P1P1 P’,y P’’’ Yes/No Input to P 3 Program P’ Variables Integer array A[10] Integer max,i; Instructions read input string into array A; max = A[0]; for (i=1;i<10;i++) IF A[i] > max THEN max = A[i]; return max; Input y 0;1;5;3;2;7;5;8;2;10 Output of P 3 Program P’’’ Variables Integer array A[10] Integer max,i; Instructions Initialize A to contain y; max = A[0]; for (i=1;i<10;i++) IF A[i] > max THEN max = A[i]; return max; If input z has even length THEN yes ELSE no

23. 23 Specification for program P’’’ –L(P’’’) = even length strings if and only if P’ halts on y –In actuality If P’ halts on y, L(P’’’) = L(P’’) = set of even length strings If P’ does not halt on y, P’’’ will loop on all strings –Thus L(P’’’) = {} Program P’’’ P2P2 P3P3 P1P1 P’,y P’’’ Yes/No /* Run P’’ on z */ If (z has even length) THEN yes ELSE no start P’’ Y/N z P’’’ Program P’’’ Let z denote the input string to P’’’ halt P’ y Run P’ on y

24. 24 If P’ loops on y, L(P’’’)={ } –P’’’ begins by running P’ on y –Suppose P’ does NOT halt on y P’’’ never even looks at input z; it just loops –Typically, L(P’’’) = { } means P’’’ is a no input to problem L 2 If P’ halts on y, L(P’’’) = L(P’’) –Note, P’’ could be ANY program we want –CHOOSE P’’ such that P’’ is a yes input to problem L 2 or if L(P’’’) = { } means P’’’ is a yes input to problem L 2, then choose P’’ such that P’’ is a no input to problem L 2 Key Step start P’’ Y/N z P’’’ halt P’ y

25. 25 P 3 illustrated P2P2 P3P3 P1P1 P’,y P’’’ Yes/No Program P’ Variables Integer array A[10] Integer max,i; Instructions read input string into array A; max = A[0]; for (i=1;i<10;i++) IF A[i] > max THEN max = A[i]; return max; Input y 0;1;5;3;2;7;5;8;2;10 Program P’’’ Variables Integer array A[10] Integer max,i; Instructions Initialize A to contain y; max = A[0]; for (i=1;i<10;i++) IF A[i] > max THEN max = A[i]; return max; If input z has even length THEN yes ELSE no P3P3 P’ string y P3P3 start P’’ Y/N z P’’’ halt P’ y

26. 26 Pseudocode for P 3 –Make P’ a procedure of P’’’ –Make the first line of the main procedure for P’’’ a call to P’ with y as the input P’(y) Ignore what P’ returns; only interested if P’ halts on y –Make P’’ a procedure of P’’’ –Make the second line of the main procedure for P’’’ a call to P’’ with the input z to P’’’ as the input P’’(z) Note P’’’ will return what P’’(z) returns P 3 in more detail P2P2 P3P3 P1P1 P’,y P’’’ Yes/No

27. 27 Summary Many-one Reductions –L 1 b m L 2 –Construct P 1 from P 2 Similarities with closure property constructions –Use P 2 as a subroutine Differences with closure property constructions –Use specific properties of L 1 and L 2 –P 3 is inside P 1 and it maps inputs of L 1 to inputs of L 2 –Often have programs as inputs and outputs