Turing Machines (intro)
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1 CHAPTER 3 The Church-Turng Thess Contents Turng Machnes defntons, examples, Turng-recognzable and Turng-decdable languages Varants of Turng Machne Multtape Turng machnes, non-determnstc Turng Machnes, Enumerators, equvalence wth other models The defnton of Algorthm Hlbert s problems, termnology for descrbng Turng machnes Theory of Computaton, Feodor F. Dragan, Kent State Unversty 1 Turng Machnes (ntro) So far n our development of theory of computaton we have presented several models for computng devces Fnte automata are good models for devces that have a small amount of memory. Pushdown automata are good models for devces that have an unlmted memory that s usable only n the last n, frst out manner of a stack. We have shown that some very smple tasks are beyond the capabltes of these models. Now we wll consder a much more powerful model, frst proposed by Alan Turng n 1936, called the Turng Machne (TM). It s smlar to a fnte automaton but wth an unlmted and unrestrcted memory. TM s much more accurate model of a general purpose computer. It can do everythng that a real computer can do. But a TM also cannot solve certan problems. There are problems that are beyond the theoretcal lmts of computaton. Theory of Computaton, Feodor F. Dragan, Kent State Unversty 2 1
2 Turng Machnes (nformal) The Turng machne model uses an nfnte tape as ts unlmted memory. It has a head that can read and wrte symbols and move around on the tape. Intally the tape contans only the nput strng and s blank everywhere else. If TM needs to store nformaton, t may wrte ths nfo on the tape. To read the nformaton that t has wrtten, TM can move ts head back over t. The machne contnues computng untl t produces an output. The output accept and reect are obtaned by enterng desgnated acceptng and reectng states. If t does not enter an acceptng or a reectng state, t wll go on forever, never haltng. Schematc of a Turng Machne: control a a b b read-wrte head The dfferences between fnte automata and Turng machnes. A TM can both wrte on the tape and read from t. The read-wrte head can move both to the left and to the rght. The tape s nfnte. The specal states for reectng and acceptng take mmedate effect. nfnte tape Theory of Computaton, Feodor F. Dragan, Kent State Unversty 3 Example We want to desgn a TM M1 whch accepts f ts nput s a member of B B = { w# w : w {0,1}*}. Informal descrpton how the TM works on nput strng s. Scan the nput to be sure that t contans a sngle # symbol. If not, reect. Zg-zag across the tape to correspondng postons on ether sde of the # symbol to check on whether these postons contan the same symbol. If they do not, reect. Cross off symbols as they are checked to keep track of whch symbols correspond. When all symbols to the left of the # have been crossed off, check for any remanng symbols to the rght of the #. If any symbols reman, reect; otherwse, accept # M1 on nput # x11000# x11000# x x11000# x xx1000# x xxxxxx # xxxxxx... accept Theory of Computaton, Feodor F. Dragan, Kent State Unversty 4 2
3 Formal Defnton of TMs A Turng machne (TM) s specfed by a 7-tuple ( Q, Σ, Γδ, q0,, qreect ), where Q s a fnte set of states, Σ s a fnte nput alphabet not contanng, s a fnte tape alphabet, such that Γ, Σ Γ, s the transton functon, s the start state, s the accept state, and Γ δ : Q Γ Q Γ { L, R} q 0 Q Q q Q s the reect state, where q. accept q reect The heart of the defnton of a TM s the transton functon because t tells us how the machne gets from one step to the next. δ ( q, a) = ( r, b, L) means that when the machne s n a certan state q and head s over a tape square contanng a symbol a, the machne wrtes the symbol b replacng the a, and goes to state r. The thrd component s ether L or R and ndcates whether the head moves to the left or rght after wrtng. Theory of Computaton, Feodor F. Dragan, Kent State Unversty 5 How does a TM compute? Intally TM receves ts nput w w w2... w of the tape, and the rest of the tape s blank. * on the leftmost n squares The head starts on the leftmost square of the tape. Note that Σ does not contan the blank symbol, so the frst blank symbol appearng on the tape marks the end of the nput. Once TM starts, the computaton proceeds accordng to the rules descrbed by the transton functon. If TM ever tres to move ts head to the left off the left-hand end of the tape, the head stays n the same place for that move, even though the transton functon ndcates L. The computaton contnues untl t enters ether accept state or reect state at whch pont t halts. If nether occurs, TM goes on forever. = 1 n Σ control a a b b Theory of Computaton, Feodor F. Dragan, Kent State Unversty 6 3
4 Acceptance of Strngs and the Language of TM A confguraton C of the TM. q q70010 For a state q and two strngs u and v over the tape alphabet Γ we wrte uqv for the confguraton where the current state s q, the current tape contents s uv, and the current head locaton s the frst symbol of v. Let a, b, c Γ, u, v Γ*, q, q Q. We say that confguraton u q acv ua q bv yelds u ac q v f f δ ( q, b) = ( q, c, L) δ ( q, b) = ( q, c, R) Note that q cv q bv yelds c q v and that ua q s equvalent to ua q f δ ( q, b) = ( q, c, L) f δ ( q, b) = ( q, c, R) and we can handle ths as before. Theory of Computaton, Feodor F. Dragan, Kent State Unversty 7 Acceptance of Strngs and the Language of TM (cont.) The start confguraton of TM on nput w s q w. 0 In an acceptng confguraton the state s. In an reectng confguraton the state s q. reect Haltng confguratons A Turng machne TM accepts nput w f a sequence of confguratons C exsts where, 1 C2,..., C 1 s the start confguraton of TM on nput w, each C yelds C + 1 and C k s an acceptng confguraton. If L s a set of strngs that TM accepts, we say that L s the language of TM and wrte L=L(TM). We say TM recognzes L or TM accepts L. A language s Turng-recognzable f some TM recognzes t. For a TM three outcomes are possble on an nput: t may accept, reect or loop. Decders are TMs that always make a decson to accept or reect the nput. A language s Turng-decdable or smply decdable f t s accepted by a decder. C k Theory of Computaton, Feodor F. Dragan, Kent State Unversty 8 4
5 Example 1. A TM M2 whch decdes the language { 0 2 n A = : n 0}. Hgher-level descrpton. M2= On nput strng w: M 1. Sweep left to rght across the tape, crossng off every other If n stage 1 the tape contaned a sngle 0, accept. 3. If n stage 1 the tape contaned more than a sngle 0 and the number of 0 s was odd, reect. 4. Return the head to the left-hand end of the tape. 5. Go to stage 1. Formal descrpton. = ( Q, Σ, Γ, δ, q1,, q ) 2 reect Q = q1, q2, q3, q4, q5,, q Σ = {0} Γ = { 0, x, } { reect Start state Run M2 on nput 0000 and 000 } Theory of Computaton, Feodor F. Dragan, Kent State Unversty 9 q1 q reect 0 _, R q2 0 L x L q5 0 x, R _ L q3 q4 0 R 0 x, R Example 2. A TM M1 whch decdes the language B = { w# w: w {0,1}*}. For hgher-level descrpton see slde #4. Formal descrpton. M1 = ( Q, Σ, Γ, δ, q1,, qreect ) Σ = {0,1,# } Q = { q1,..., q14,, qreect} Γ = { 0,1,#, x, } 0,1 R 0,1 R 0,1, # L 0,1 R 1 x, R 1 _, R q1 0 _, R q3 q5 q7 q9 q11 _ L # R _ L q2 q4 q6 q8 q10 0,1 R 0,1 R 0,1, # L q reect 0,1 R 1 x, L 0 x, L 0,1, #, x L Theory of Computaton, Feodor F. Dragan, Kent State Unversty 10 q12 0 x, R q14 q13 5
6 Example 3: A TM solvng the element unqueness problem. It s gven a lst of strngs over {0,1} separated by #s and ts ob s to accept f all strngs are dfferent. The language s E = {# x1 # x2#...# xl : x {0,1}* {1,..., l}, x x for }. TM M3 works by comparng x wth through, then by comparng 1 x2 xl x2 wth x through x, and so on. 3 l Hgher-level descrpton: M3= On nput w: 1. Place a mark on top of the leftmost tape symbol. If that symbol was blank, accept. If t was a #, contnue wth the next stage. Otherwse, reect. 2. Scan rght to the next # and place a second mark on top of t. If no # s encountered before a blank symbol, only was present, so accept. x 1 3. By zg-zaggng, compare the two strngs to the rght of the marked #s. If they are equal, reect. 4. Move the rghtmost of the two marks to the next # symbol to the rght. If no # symbol s encountered before a blank symbol, move the leftmost mark to the next # to ts rght and the rghtmost mark to the # after that. If no # s avalable for the rghtmost mark, all the strngs have been compared, so accept. 5. Go to stage 3.. (In the actual mplementaton, the machne has two dfferent symbols, # and #, n ts tape alphabet). Theory of Computaton, Feodor F. Dragan, Kent State Unversty 11 6
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