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

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, qreject the start, accept and reject states. qaccept  qreject Computation & Complexity/Y. Moses

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

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

States as Memory of Symbols For each  there exists a state q. M that returns n of 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

M that Computes the Right Cyclic 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

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

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

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

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

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

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

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, qreject the start, accept and rejects states. qaccept  qreject Computation & Complexity/Y. Moses

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

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

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 1 1 _ _ _ ... Computation & Complexity/Y. Moses

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

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

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

Computation & Complexity/Y. Moses M accept w: 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

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

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

Computation & Complexity/Y. Moses M decides L: If M accepts all wL and M rejects all wL. That is: For every wL M accept w. For every wL M reject w. M is a Decider TM: If M Halts on all inputs. (never loop) Computation & Complexity/Y. Moses

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

Computation & Complexity/Y. Moses 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

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

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

TM for Deciding the Language C={aibjck|i x j=k and i,j,k>0} M3 on 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 a and the same num. of b 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

The Element Distinctness Problem E={#x1#x2#…#xn| xi{0,1}* and xixj ij} M4 is a TM that accept E: it compares each pair of xi and 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

Computation & Complexity/Y. Moses 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

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

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

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

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

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

Computation & Complexity/Y. Moses M recognize L: If M accepts all wL That is: For every wL M accept w. For every wL M reject w or M does not halt. Computation & Complexity/Y. Moses

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

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

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

Computation & Complexity/Y. Moses 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