Reducibility

1. Reducibility • A reduction is a way of converting one problem into another problem in such a way that a solution to the second problem can be used to solve the first problem.

2. HALTTM={<M,w>| M is a TM and M halts on input w} • Thm: HALTTM is un-decidable

3. Proof: • By contradiction, assume R is a TM that decides HALTTM . • With R, we construct a TM S to decide ATM: S= “ On input <M,w>, an encoding of M and w: 1. Run TM R on input <M,w> 2. If R rejects, REJECT 3. If R accepts, simulate M on w until it halts. 4. If M has accepted, ACCEPT; If M has rejected, REJECT. “

4. (Proof conti.) • If R decides HALTTM, then S decides ATM. But ATM is un-decidable, thus HALTTM is un-decidable.

5. ETM={<M>| M is a TM and L(M)=∅ } • Thm: ETM is un-decidable.

6. Proof: • Define Mw = “ On input x: 1. If x ≠ w , reject 2. If x = w , run M on input w and accept if M does ” • Assume that R decides ETM. • Construct S that decides ATM as follows: S = “ On input <M,w> 1. Use M and w to construct Mw 2. Run R on <Mw> 3. If R accepts, REJECT; if R rejects, ACCEPT. “ • But ATM is un-decidable, thus ETM is un-decidable .

7. REGULARTM = {<M>| M is a TM and L(M) is regular }. • Thm: REGULARTM is un-decidable.

8. Proof: • Assume R decides REGULARTM . • Construct TM S to decide ATM . S = “ On input <M,w>: 1. Construct the following TM Mw Mw = “ On input x : 1. If x has the form 0n1n ,ACCEPT. 2. If x does not have this form, run M on input w and ACCEPT if M accepts w. 2. Run R on input <Mw> 3. If R accepts, ACCEPT; If R rejects, REJECT. “

9. EQTM = {<M1, M2>| M1 and M2 are TMs and L(M1)=L(M2) } • Thm: EQTM is un-decidable.

10. Proof: • Let TM R decide EQTM and construct S to decide ETM as follows: S = “ On input <M>, where M is a TM: 1. Run R on input <M,M1>, where M1 rejects all inputs. 2. If R accepts, accept; otherwise reject. “ • If R decides EQTM , S decides ETM .

11. Reductions via computation histories • Def: M: Turing Machine w: an input string An accepting computation history for M on w is a sequence of configurations, C1, C2, …, Cl , where C1 is the start configuration of M on w, Cl is an accepting configuration of M, and each Ci legally follows from Ci-1.

12. Def: Linear Bounded Automaton is a type of TM where in the tape head is not permitted to move off the portion of the tape containing the input.

13. ALBA = {<M,w>| M is an LBA that accepts w} • Lemma: Let M be an LBA with q states and g symbols in the tape alphabet. There are exactly qngn distinct configurations of M for a tape of length n.

14. 1 2 3 n n-1 Proof: Head positions : n states : q tape contents : gn

15. ALBA = {<M,w> | M : LBA, M accepts w } • Thm: ALBA is decidable.

16. Proof: • Algorithm: L = “ On input <M,w>, where M is an LBA and w is a string. 1. Simulate M on w for qngn steps until it halts. 2. If M has halted, accept if it has accepted; and reject if it has rejected. If it has not halted, reject. “

17. ELBA = {<M>| M is an LBA where L(M)= ∅ } • Thm: ELBA is un-decidable.

18. Proof: • Suppose R decides ELBA . • Construct S that decides ATM . S = “ On input <M,w>, where M is a TM and w is a string : 1. Construct LBA B from M and w 2. Run R on <B>. 3. If R rejects, accept; if R accepts, reject. “ Recognizes all accepting computation histories for M and w B Ci ⊦ Ci+1 # # #

19. ALLCFG = {<G>| G is a CFG and L(G)= * } • Thm: ALLCFG is un-decidable.

20. A Simple un-decidable problem • Post correspondence problem(PCP) • Eg: : dominos match: • Make a list of dominos (repetitions permitted) so that the string we get by reading off the symbols on the top is the same as the string of symbols on the bottom.

21. An instance of the PCP is a collection P of dominos: a match is a sequence i1, i2, …, il , where • The problem is to determine whether P has a match. • PCP={<P>| P is an instance of the Post Correspondence Problem with a match.}

22. MPCP = {<P> | P is an instance of the Post Correspondence Problem with a match that starts with the first domino } • Thm: PCP is un-decidable.

23. Proof: • Let TM R decide the PCP and construct S deciding ATM . Let M = (Q, , , , q0, qaccept, qreject ) . • S constructs an instance of the PCP P that has a match iff M accepts w.

24. MPCP (Proof conti.) • Part 1: Put into P’ as the 1st domino . • Part 2: For every a,b∈ and every q,r ∈Q, if (q,a)=(r,b,R), put into P’ . • Part 3: For every a,b,c ∈ and every q,r ∈Q, if (q,a)=(r,b,L), put into P’ . • Part 4: For every a∈ , put into P’ .

25. (Proof conti.) • Eg.  ={0,1,2, }. w = 0100 Suppose (q0,0)=(q7,2,R), then Part 2: places into P’ . Part 4: places into P’ . # q00 1 0 0 # # q00 1 0 0 # 2q7 1 0 0 #

26. (Proof conti.) • Part 5: Put and into P’ . If (q7,1)=(q5,0,R), then is in P’ . # 2 q7 1 0 0 # ⋯ # 2 q7 1 0 0 # 2 0 q5 0 0 # If (q5,0)=(q9,2,L), then we have # 2 0 q5 0 0 # ⋯ # 2 0 q5 0 0 # 2 q9 0 2 0 #

27. (Proof conti.) • Part 6: for every a∈ , Put into P’ . #2 1 qaccept0 2 #⋯#qaccept # # ⋯ # 2 1 qaccept0 2 #2 1 qaccept 2 # ⋯ # qaccept ## • Part 7: Add # qaccept # # ⋯ # qaccept # #

28. (Proof conti.) • Convert P’ to P: Let u = u1u2…un Define *u = *u1*u2… *un u*= u1*u2*…*un* *u*=*u1*u2*…*un* P’ : P : ■ Must start with the first domino, and ends with an extra * on the top

29. Mapping Reducibility: (many- one reducibility) • Def: f : * *is a computable function if some TM M, on every input w, halts with just f(w) on its step.

30. Def: Language A is mapping reducible to B, written A ≤m B , if there is a computable function f : * * , where for every w, w ∈ A ⇔ f(w) ∈ B f is called the reduction of A to B A B f w f

31. Thm: If A ≤m B and B is decidable, then A is decidable.

32. Proof: • Let M be the decider for B, f be the reduction from A to B. N = “ On input w: 1. Compute f(w) 2. Run M on input f(w) and output whatever M outputs. “

33. Cor: If A ≤m B and A is un-decidable, then B is un-decidable. • Thm: If A ≤m B and B is Turing-recognizable, then A is Turing-recognizable. • Cor: If A ≤m B and A is not Turing-recognizable, then B is not Turing-recognizable.

34. Thm: EQTM is neither Turing-recognizable nor co- Turing-recognizable. __ Pf: We prove by showing ATM≤m EQTM and ATM≤m EQTM. Why? F=“On input <M, w>: 1. Construct M1 and M2. M1= “On any input: Reject.” M2= “On any input: Run M on w. If it accepts, accept.” 2. Output <M1, M2>.

35. G=“On input <M, w>: 1. Construct M1 and M2. M1= “On any input: Accept.” M2= “On any input: Run M on w. If it accepts, accept.” 2. Output <M1, M2>. G completes the reduction of ATM≤m EQTM.

36. Turing Reducibility (mapping reducibility?) • Def: An oracle for a language B is an external device that is capable of reporting whether any string w is a member of B. • Def: An oracle Turing machine is a modified Turing machine that has the additional capability of querying an oracle. MB : an oracle TM that has an oracle for B.

37. Eg: ETM : un-decidable. = “ On input <M>, where M is a TM: 1. Construct TM N: N = “ On any input: 1. Run M in parallel on all strings in * . 2. If M accepts any of these strings, accept. “ 2. Query the oracle to determine whether <N,0>∈ ATM . 3. If the oracle answers NO, accept; if YES, reject. “ decides ETM. Thus, ETM is decidable relative to ATM.

38. Def: Language A is Turing reducible to language B, written A ≤T B, if A is decidable relative to B. • Thm: If A ≤T B and B is decidable, then A is decidable.

39. Proof: • If B is decidable, then replace the oracle for B by an actual procedure(or TM) that decides B . • Thus, we may replace the oracle TM that decides A by an ordinary TM that decides A.