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Complexity and Computability Theory I

Complexity and Computability Theory I. Lecture #14 Instructor: Rina Zviel-Girshin Lea Epstein. Formally: (Q, , , , q 0, q accept, q reject ). Q is the set of states.  is the input alphabet _   .  is the tape alphabet, _ and  .

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Complexity and Computability Theory I

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  1. Complexity and Computability Theory I Lecture #14 Instructor: Rina Zviel-Girshin Lea Epstein Computation & Complexity/Y. Moses

  2. Formally: (Q, , , , q0, qaccept, qreject) • Q is the set of states. •  is the input alphabet _. •  is the tape alphabet, _ and . • :Qx  Qxx{R,L} the transition function. • q0, qaccept, qrejectthe start, accept and reject states. qaccept qreject Computation & Complexity/Y. Moses

  3. s s s 1 s 2 s 3 4 5 qaccept Example of TM Computing a Function f1(w)=0|w| For every , (q0, )=(q0,0,R) (q0,_)=(q1,_,R). (q1,_)=(qaccept,_,L). Computation & Complexity/Y. Moses

  4. Computing the Function: f2(w)=w For every , (q0, )=(q0, ,R) (q0,_)=(q1,_,R). (q1,_)=(qaccept,_,L). Computation & Complexity/Y. Moses

  5. States as Memory of Symbols • For each  there exists a state q. • M that returns nof the input w= 12 ...n. • (q0, i)=(qi,#,R) mark left cell+ remember first cell. • (qi, j)=(qj,_,R) erase and remember the current cell. • (qi, _)=(q’i,_,L) end-of-tape + return the last cell. • (q’i, _)=(q’i,_,L) go backward with last cell. • (q’i, #)=(qaccept, i,R) detect left cell write right cell+ stop. Computation & Complexity/Y. Moses

  6. M that Computes the RightCyclic Transformation: f(12 ...n)= n12 ...n-1 • Remember the left cell (using a state), mark it with # and move Right. • Until right-end, _, write the previous cell value, remember the current value, and move right. • When reach the right end, remember the value by the state and move left until reach the first cell, #. • Write the last cell value in the first cell and stop. Computation & Complexity/Y. Moses

  7. B={w#w|w{0,1}*} Computation & Complexity/Y. Moses

  8. Summary • TM: formal definition (Q, , , , q0, qaccept, qreject) • TM: input. • TM: output. • Some examples. Computation & Complexity/Y. Moses

  9. A “Stay” Move Can use: (q0,_)=(q1,_,R) (q1,_)=(qaccept,_,L) • :Qx  Qxx{R,L,S} Computation & Complexity/Y. Moses

  10. Computing the Function: f2(w)=w For every , (q0, )=(q0, ,R) (q0,_)=(qaccept,_,S). Computation & Complexity/Y. Moses

  11. Computers and Intuitive Computation • Every computer has a set of “simple” steps that change the environment. • Can every function be computed by some computer? • Formally: Can every function f:{0,1}* :{0,1}* be computed? Computation & Complexity/Y. Moses

  12. Class 2 • TM Configurations • A Decider • A Language of a TM • More Examples of TM’s Computation & Complexity/Y. Moses

  13. Formally: (Q, , , , q0, qaccept, qreject) • Q is the set of states q0, qaccept, qrejectQ. •  is the input alphabet _  . •  is the tape alphabet, _ and . • :{Q\ qaccept, qreject }x  Qxx{R,L} the transition function. • q0, qaccept, qrejectthe start, accept and rejects states. qaccept qreject Computation & Complexity/Y. Moses

  14. q b b a c b b d a c _ _ u v Configuration • The tape. • The state. • The control location on the tape. • uqrejectv • u,v *u=bbac v=bbdac Computation & Complexity/Y. Moses

  15. q0 ... 0 0 1 0 1 _ _ _ Intial Configuration: q0w • The tape: w(the input)on the leftmost nsquares of the tape. Example: w=00101 • The rest of the tape is blank. • The head starts on the left most square. Computation & Complexity/Y. Moses

  16. Final Configuration: uqw • q {qaccept , qreject} • The tape contains uw: Example: u = 00 w=101, • The rest of the tape is blank. • The output is u. q 0 0 1 0 1 _ _ _ ... Computation & Complexity/Y. Moses

  17. qi C1: b b a c a b b b d a c _ _ u v C2: qj b b a c a c b b d a c _ _ u v C1yieldsC2 if C2can be “reached” from C1by the function. Computation & Complexity/Y. Moses

  18. Special Case: left-hand end • Right transition (qi,b)= (qj,c,R): qibvyieldscqjv • Left transition (qi,b)= (qj,c,L): qibv yieldsqjcv Computation & Complexity/Y. Moses

  19. A Configuration Sequence For a given TM M we denoteby C1* Cnthe existence of a set of configurations. Where Ci yields Ci+1 for i=1,…,n Computation & Complexity/Y. Moses

  20. Macceptw: If a sequence of configurations C1, C2,…,Cn exists where: 1.C1 is a start configuration with w. 2. Each Ci yields Ci+1 3.Cn is an accepting configuration. 1-2:C1 *Cn Computation & Complexity/Y. Moses

  21. Mrejectw: If a sequence of configurations C1, C2,…,Cn exists where: 1.C1 is a start configuration on input w. 2. Each Ciyields Ci+1 3.Cn is a rejecting configuration. 1-2:C1 *Cn Computation & Complexity/Y. Moses

  22. Three possible outcomes of a TM • Accept. • Reject. • Loop. not necessarily repeating the same steps. Computation & Complexity/Y. Moses

  23. MdecidesL: If Maccepts all wLandMrejectsallwL. That is: • For everywL M accept w. • For everywL M reject w. M is aDeciderTM: If M Halts on all inputs. (never loop) Computation & Complexity/Y. Moses

  24. The Language of TM M L(M) = the collections of strings that Maccepts. Computation & Complexity/Y. Moses

  25. Example The following TM M decides the language M=“on input string w 1. From left to right: cross off every other 0. 2. If in stage 1 a single 0 left: accept. an odd number of 0’s” reject. An even number: go to 1.” Computation & Complexity/Y. Moses

  26. Example: B={w#w|w{0,1}*} M1=“On input string w: • Scan and check that the input contains a single #. If not reject. • Zig-zag to check whether corresponding positions on either sides of # contain the same symbol. If not, reject. Cross off the checked symbols. • When all symbols to the left of # are crossed, check for remaining symbols to the right of #. If any symbol, reject. Otherwise, accept. Computation & Complexity/Y. Moses

  27. B={w#w|w{0,1}*} How are 0’s and 1’s decoded? Where are the reject states? Computation & Complexity/Y. Moses

  28. TM for Deciding the Language C={aibjck|i xj=kandi,j,k>0} M3on input w: 1. Scan to the right to check whether w{a* b* c*} if not, reject. 2. Return head to the left hand side. 3. Cross: one aand the same num. ofb and c. 4. Restore the crossed b and return to 3. accept if all a’s and c’s are crossed. Computation & Complexity/Y. Moses

  29. The Element Distinctness Problem E={#x1#x2#…#xn| xi{0,1}*andxixj ij} M4is a TM that accept E: it compares each pair of xiand xj. • At each “round” it marks a pair of the symbols # in an order that covers all possible pairs. • It compares the two strings that follow the marked symbols #. Computation & Complexity/Y. Moses

  30. Alphabet Duplication • How can we mark cells on the tape without changing their values? • Duplicate the alphabet of  such that for each  there is another symbol ’ . . Computation & Complexity/Y. Moses

  31. B1={w1#w2|wi{0,1}* & w1 w2} What do we have to change? B={w#w|w{0,1}*} Computation & Complexity/Y. Moses

  32. Turing-decidable Languages: • B={w#w|w{0,1}*} • C={aibjck|i xj=k and i,j,k>0} • E={#x1#x2#…#xn| xi{0,1}*andxixi ij} Computation & Complexity/Y. Moses

  33. Using a TM to Compute a Function. M is a TM that computes a (partial) function fM:*  *if there exist , *and qacceptQ s.t. q0w  *  qaccept. In this case fM(w)=. Computation & Complexity/Y. Moses

  34. A Function With a Number of Arguments • Use a special symbol on the tape to separate between the arguments. • w1&w2&…wnwhere &. • Examples of functions: • add(w1,w2) • mult(w1,w2) • max(w1,w2) Computation & Complexity/Y. Moses

  35. Can we distinguish a machine that is looping from one that is merely taking a long time? Computation & Complexity/Y. Moses

  36. MrecognizeL: If Maccepts all wL That is: • For everywL M acceptw. • For everywL M rejectw or M does not halt. Computation & Complexity/Y. Moses

  37. Turing-Recognizable Language A language L is Turing recognizableif there exists a TM M that recognizes it. That is L(M) for some TM M. Alsonamed recursively-enumerable Computation & Complexity/Y. Moses

  38. Turing-Recognizable Languages: • B={w#w|w{0,1}*} • C={aibjck|i xj=k and i,j,k>0} • E={#x1#x2#…#xn| xi{0,1}*andxixi ij} Computation & Complexity/Y. Moses

  39. Turing-recognizable Turing-decidable  Turing-decidable Turing-recognizable Computation & Complexity/Y. Moses

  40. Up to Here • Definition of a TM. • Configurations: initial, final, general, transitions . • Coding S (stay) transition. • Remember symbols by states. • Mark symbols. • Computing a function: input & output values. • Recognizing/deciding a Language. Computation & Complexity/Y. Moses

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