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CSE 105 THEORY OF COMPUTATION Fall 2016 http://cseweb.ucsd.edu/classes/fa16/cse105-abc/ T oday's learning goals Sipser Ch 1.4, 2.1 Introduction to T uring Machines Confjgurations and computations Deciders and Recognizers ??? ???

  1. CSE 105 THEORY OF COMPUTATION Fall 2016 http://cseweb.ucsd.edu/classes/fa16/cse105-abc/

  2. T oday's learning goals Sipser Ch 1.4, 2.1 • Introduction to T uring Machines • Confjgurations and computations • Deciders and Recognizers

  3. ??? ??? Context-free languages Context-free languages Regular Regular languages languages

  4. T uring machines • Unlimited input • Unlimited (read/write) memory • Unlimited time

  5. T uring machine computation • Read/write head starts at leftmost position on tape • Input string written on leftmost squares of tape, rest is blank • Computation proceeds according to transition function: • Given current state of machine, and current symbol being read • the machine • transitions to new state • writes a symbol to its current position (overwriting existing symbol) • moves the tape head L or R • Computation ends if and when it enters either the accept or the reject state.

  6. Formal defjnition of TM Is this a deterministic model? A) Yes B) No C) It depends on the state diagram D) It depends on the input E) I don’t know q reject ≠ q accept q reject ≠ q accept

  7. T uring Machine example ● L = {w#w | w in {a,b}*} (No CFG/PDA!) ● Informal description: Γ = {a,b,#,c,[_]} 1) If input is # 1) Move right skipping all c’s 2) If input is [_], then accept, else reject 2) Read a or b, store in internal state, and change it to c 3) Move right until read # 4) Move right skipping all c’s 5) Read a or b, matching the symbol read in step 1. 1) If mismatch → reject 2) If match, overwrite with c 6) Move left until read # 7) Move left until read c 8) Go back to 1).

  8. T uring Machine (formal) ● L = {w#w | w in {a,b}*} (No CFG/PDA!) ● Draw state diagram in JFLAP

  9. T uring Machine Computations ● Confjguration: all the information store by the system at any point during computation Internal state (Q) – T ape content (Γ*) – Position of tape head – ● Formally: (α,q,β) in Γ* x Q x Γ* q: internal state – α: tape content to the left of tape head – β: tape content from tape head to the right –

  10. TM Computations ● Fix TM M=(Q,Σ,Γ,δ,q s ,q a ,q r ) ● Computation of M on input w: Sequence of confjgurations – Start from initial confjguration C0=(ε,q s ,w) – Move from one confjguration C i to the next C i+1 according to δ – Until reaching halting confjguration (…,q a ,…) or (…,q r ,…) – ● Input is accepted if fjnal state is q a ● Input is rejected if fjnal state is q r

  11. TM Computations What is the maximum length of a computation? ● Fix TM M=(Q,Σ,Γ,δ,q s ,q a ,q r ) A) Same as input length ● Computation of M on input w: B) At most length of input Sequence of confjgurations – C) May be longer than input, Start from initial confjguration C 0 =(ε,q s ,w) – but it is always fjnite Move from one confjguration C i to the next C i+1 according to δ D) May be fjnite or infjnite – E) I don’t know Until reaching halting confjguration (…,q a ,…) or (…,q r ,…) – ● Input is accepted if fjnal state is q a ● Input is rejected if fjnal state is q r

  12. TM Computations Does M need to read the whole input w before halting? ● Fix TM M=(Q,Σ,Γ,δ,q s ,q a ,q r ) A) Yes, computation terminates ● Computation of M on input w: only after reading w Sequence of confjgurations – B) Only before accepting w Start from initial confjguration C0=(ε,q s ,w) C) No. M may accept or reject – without reading w entirely Move from one confjguration C i to the next C i+1 according to δ – D) I don’t know Until reaching halting confjguration (…,q a ,…) or (…,q r ,…) – ● Input is accepted if fjnal state is q a ● Input is rejected if fjnal state is q r

  13. Which of the following is true? TM Computations ● Fix TM M=(Q,Σ,Γ,δ,q s ,q a ,q r ) A) Computation may visit both qa and qr ● Computation of M on input w: B) Computation may visit q a Sequence of confjgurations – more than once Start from initial confjguration C0=(ε,q s ,w) – C)Computation may visit q b Move from one confjguration C i to the next C i+1 according to δ – more than once D) Computation may visit q s Until reaching halting confjguration (…,q a ,…) or (…,q r ,…) – ● Input is accepted if fjnal state is q a more than once E) None of the above ● Input is rejected if fjnal state is q r

  14. Language of a TM ● L(M) = {w | M accepts w} ● M may reject or loop on strings not in L(M) ● A language X is recognizable if X=L(M) for some TM M ● A TM M is a decider if M(w) halts on every input w ● A language X is decidable is X=L(M) for some decider M

  15. Language of a TM Which of the following is true? ● L(M) = {w | M accepts w} A) If X is decidable, then X is recognizable ● M may reject or loop on strings not in L(M) B) If X is recognizable, then X ● A language X is recognizable if X=L(M) for some is recognizable TM M C) If X is decidable, then X is decidable ● A TM M is a decider if M(w) halts on every input w D) A and C, but not C ● A language X is decidable is X=L(M) for some E) I can’t decide, I do not decider M recognize this question :(

  16. Reminders • Reading: Sipser Chapter 3 • HW5 due Monday Oct 31 • Review for Exam 2 • T uesday Nov 1 TBA • Exam 2 Wednesday Nov 2 • Seating chart: see Piazza • Study guide: see Piazza • Haskell 3 on CFG is out. Due Nov 7.

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