February 8, 2019 CS21 Lecture 14

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February 8, 2019 CS21 Lecture 14




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February 8, 2019 CS21 Lecture 14 1 CS21 Decidability and Tractability Lecture 14 February 8, 2019

February 8, 2019 CS21 Lecture 14 2 Outline Post Correspondence Problem b eyond RE and co-RE Recursion Theorem G ö del Incompleteness Theorem

February 8, 2019 CS21 Lecture 14 3 Post Correspondence Problem many undecidable problems unrelated to TMs and automata classic example: Post Correspondence Problem PCP = {<(x 1, y 1), (x2 , y 2 ), …, ( x k , yk)> : xi, yi Σ* and there exists (a1, a2, …, an) for which xa1xa2…xan = ya1ya2…yan}  

February 8, 2019 CS21 Lecture 14 4 Post Correspondence Problem PCP = {<(x 1 , y 1 ), (x2, y 2 ), …, ( x k , yk)> : xi, yi Σ* and there exists (a1, a2, …, an) for which xa1xa2…xan = ya1ya2…yan}   x 1 y 1 x 2 y 2 x3y3 xkyk “tiles” x2y2 x1y1 x5y5 x2y2 x1y1 x3y3 x4y4 x2x1x5x2x1x3x4x4 = y2y1y5y2y1y3y4y4 x4y4 “match”

February 8, 2019 CS21 Lecture 14 5 Post Correspondence Problem Theorem : PCP is undecidable. Proof: reduce from ATM (i.e. show ATM ≤ m PCP) two step reduction makes it easier first, show A TM ≤ m MPCP(MPCP = “modified PCP”)next, show MPCP ≤m PCP

February 8, 2019 CS21 Lecture 14 6 Post Correspondence Problem MPCP = {< (x 1 , y1 ), (x2 , y 2 ), …, ( x k , yk)> : xi, yi Σ* and there exists (a1, a2, …, an) for which x1xa1xa2…xan = y1ya1ya 2 … y a n } Proof of MPCP ≤m PCP:notation: for a string u = u1u2u3…umu means the string u1u2u3u4…u mu means the string u1u2u3u4… umu means the string u1 u2u3u4…um 

February 8, 2019 CS21 Lecture 14 7 Post Correspondence Problem Proof of MPCP ≤ m PCP: given an instance (x1 , y 1 ), …, ( x k , yk) of MPCPproduce an instance of PCP:(x1, y1) , (x1, y1), (x2, y2), …, (xk, yk), (, ) YES maps to YES? given a match in original MPCP instance, can produce a match in the new PCP instance NO maps to NO? given a match in the new PCP instance, can produce a match in the original MPCP instance  

February 8, 2019 CS21 Lecture 14 8 Post Correspondence Problem YES maps to YES? given a match in original MPCP instance, can produce a match in the new PCP instance x 1 y 1 x 4 y 4 x 5 y 5 x 2 y 2 x 1y1 x3y3 x4y4 x4y4 x 1y1  x4Y4   x5y5   x2y2   x1y1   x3y3   x 4 y 4   x 4 y 4    

February 8, 2019 CS21 Lecture 14 9 Post Correspondence Problem NO maps to NO? given a match in the new PCP instance, can produce a match in the original MPCP instance x 1 y 1 x 4 y 4 x5 y 5 x 2 y 2 x 1 y1 x3y3 x4y4 x4y4 can’t match unless start with this tile symbols must align   can only appear at the end x1y1  x 4Y4  x5y5   x2y2   x 1 y 1   x 3 y 3   x 4 y 4   x 4 y 4    

February 8, 2019 CS21 Lecture 14 10 Post Correspondence Problem Theorem : PCP is undecidable. Proof: show ATM ≤ m MPCPMPCP = {< (x 1 , y 1 ) , (x2, y2), …, (xk, yk)> : xi, yi Σ* and there exists (a1, a2, …, an) for which x1xa1xa2…xan = y1y a 1 y a 2 … y an}show MPCP ≤m PCP 

February 8, 2019 CS21 Lecture 14 11 Post Correspondence Problem Proof of A TM ≤ m MPCP:given instance of A TM: <M, w>idea: a match will record an accepting computation history for M on input w start tile records starting configuration: add tile (#, #q 0 w 1 w2w3…wn#)##q0w1w2…wn# # #C 1 # =

February 8, 2019 CS21 Lecture 14 12 Post Correspondence Problem tiles for head motions to the right: for all a,b and all q, r Q with q q reject, if δ(q, a) = (r, b, R), add tile (qa, br)tiles for head motions to the left:for all a,b,c and all q, r Q with q qreject, if δ(q, a) = (r, b, L), add tile (cqa, rcb)  # # q 0 w 1 w 2…wn# ?? ?? ?? … = #C1##C1#C2# qabr cqarcb

February 8, 2019 CS21 Lecture 14 13 Post Correspondence Problem tiles for copying (not near head) for all a , add tile (a, a) tiles for copying # marker add tile (#, #) tiles for copying # marker and adding _ to end of tape add tile (#, _#)   ## q0w1w2…wn#? ? ? ? ? ? … = #C 1 # #C 1#C2# aa ## #_#

February 8, 2019 CS21 Lecture 14 14 Post Correspondence Problem tiles for deleting symbols to left of q accept for all a , add tile ( aq accept , qaccept)  ##uaqacceptv#?? ? ? … = # uaq accept v# # uaq acceptv#uqaccept v#aqaccept qaccept

February 8, 2019 CS21 Lecture 14 15 Post Correspondence Problem tiles for deleting symbols to right of q accept for all a , add tile ( q accept a , qaccept)  ##qacceptav#? ? ? ? … = # q accept av# # qacceptav#q acceptv#q acceptaqaccept

February 8, 2019 CS21 Lecture 14 16 Post Correspondence Problem tiles for completing the match for all a , add tile ( q accept ##, #)   # #qaccept#?? ? ? … = # q accept ## # q accept## qaccept###

February 8, 2019 CS21 Lecture 14 17 Post Correspondence Problem YES maps to YES? by construction, if M accepts w, there is a way to assemble the tiles to achieve this match: NO maps to NO? sketch: at any step if the “intended” next tile is not used, then it is impossible to recover and produce a match in the end (case analysis) #C 1 #C 2 #C 3 #...#Cm##C1#C2#C3#...#Cm# where #C 1 #C 2 #C 3 #...#C m# is an accepting computation history

February 8, 2019 CS21 Lecture 14 18 Post Correspondence Problem We have proved: Theorem : PCP is undecidable. by showing:ATM ≤ m MPCPMPCP ≤ m PCP conclude A TM ≤m PCP

February 8, 2019 CS21 Lecture 14 19 Beyond RE and co-RE We saw (by a counting argument) that there is some language that is neither RE or co-RE. We will prove this for a natural language:EQTM = {<M 1, M2 > : L(M 1 ) = L(M 2 )} Recall: ATM is undecidable, but REco-ATM is undecidable, but coRE Therefore, not in co-RETherefore, not in RE

February 8, 2019 CS21 Lecture 14 20 Beyond RE and co-RE Theorem : EQ TM is neither RE nor coRE.Proof: not RE:reduce from co-ATM (i.e. show co-ATM ≤ m EQ TM ) what should f(<M, w>) produce? not co-RE:reduce from ATM (i.e. show ATM ≤m EQTM)what should f(<M, w>) produce?

February 8, 2019 CS21 Lecture 14 21 Beyond RE and co-RE Proof (A TM ≤m EQTM )f(<M, w>) = <M1, M2 > described below: TM M 1 : on input x, accept TM M2: on input x, simulate M on input w accept if M accepts wYES maps to YES?<M, w> ATM L(M 1 ) = Σ * and L(M 2 ) = Σ* f(<M, w>) EQTM NO maps to NO?<M, w> ATM L(M1) = Σ* and L(M2) = Ø f(<M, w>) EQTM 

February 8, 2019 CS21 Lecture 14 22 Beyond RE and co-RE Proof ( co- ATM ≤m EQ TM)f(<M, w>) = <M1 , M 2 > described below: TM M 1 : on input x, rejectTM M2: on input x, simulate M on input w accept if M accepts wYES maps to YES?<M, w> co-ATM L(M 1 ) = Ø and L(M 2 ) = Ø f(<M, w>) EQTM NO maps to NO?<M, w> co-ATM L(M1) = Ø and L(M2) = Σ* f(<M, w>) EQTM 

February 8, 2019 CS21 Lecture 14 23 Summary regular languages context free languages all languages decidable RE {a n b n : n ≥ 0 } {a n b n c n : n ≥ 0 } some language HALT co-RE co-HALT PCP EQ TM

February 8, 2019 CS21 Lecture 14 24 The Recursion Theorem A very useful, and non-obvious, capability of Turing Machines: in the course of computation, can print out a description of itself! how is this possible? an example of a program that prints out self:Print two copies of the following, the 2nd one in quotes: “Print two copies of the following, the 2nd one in quotes:”

February 8, 2019 CS21 Lecture 14 25 The Recursion Theorem Why is this useful? Example: slick proof that A TM undecidableassume TM M decides ATMconstruct machine M’ as follows: on input x, obtain own description <M’> run M on input <M’, x> if M rejects, accept; if M accepts, reject. if M’ on input x: accepts, then M rejects <M’, x>, but then M’ does not accept! rejects, then M accepts <M’, x>, but then M’ accepts!

February 8, 2019 CS21 Lecture 14 26 The Recursion Theorem Lemma: there is a computable function q: Σ * → Σ* such that q(w) is a description of a TM P w that prints out w and then halts. Proof: on input w, construct TM P w that has w hard-coded into it; output <P w >

February 8, 2019 CS21 Lecture 14 27 The Recursion Theorem Warm-up: produce a TM SELF that prints out its own description. Two parts: Part A: output a description of Bpass control to B.Part B: prepend a description of A done

February 8, 2019 CS21 Lecture 14 28 The Recursion Theorem Part A: output a description of B pass control to B. Part B: prepend a description of A done A output <B> B read contents of tape apply q to it prepend* result to tape Recall: q(w) is a description of a TM P w that prints out w and then halts. Note: <A> = q(<B>) *combine with description on tape to produce a complete TM

February 8, 2019 CS21 Lecture 14 29 The Recursion Theorem watch closely as TM AB runs: A runs. Tape contents: <B> B runs. Tape contents: q(<B>)<B> ) <AB>AB is our desired machine SELF. A output <B> B read contents of tape apply q to it prepend result to tape Note: <A> = q(<B>) Recall: q(w) is a description of a TM P w that prints out w and then halts.

February 8, 2019 CS21 Lecture 14 30 The Recursion Theorem Theorem : Let T be a TM that computes fn: t: Σ* x Σ* → Σ * There is a TM R that computes the fn: r: Σ * → Σ* defined as r(w) = t(w, <R>).This allows “obtain own description” as valid step in TM programfirst modify TM so that it takes an additional input (that is own description); use at will

February 8, 2019 CS21 Lecture 14 31 The Recursion Theorem Theorem : Let T be a TM that computes fn: t: Σ* x Σ* → Σ * There is a TM R that computes the fn: r: Σ * → Σ* defined as r(w) = t(w, <R>).Proof outline: TM R has 3 partsPart A: output description of BTPart B: prepend description of APart “T”: run TM T

February 8, 2019 CS21 Lecture 14 32 The Recursion Theorem Proof details: TM R has 3 parts Part A: output description of BT <A> = q(<BT>) Part B: prepend description of A read contents of tape <BT>apply q to it q(<BT>) = <A> prepend to tape <ABT> Part “T”: run TM T 2 nd argument on tape is description of R


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