1 / 23

CSE 105 Theory of Computation

CSE 105 Theory of Computation. Alexander Tsiatas Spring 2012. Theory of Computation Lecture Slides by Alexander Tsiatas is licensed under a Creative Commons Attribution- NonCommercial - ShareAlike 3.0 Unported License. Based on a work at http://peerinstruction4cs.org.

callum
Download Presentation

CSE 105 Theory of Computation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CSE 105Theory of Computation Alexander Tsiatas Spring 2012 Theory of Computation Lecture Slides by Alexander Tsiatas is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Based on a work at http://peerinstruction4cs.org. Permissions beyond the scope of this license may be available at http://peerinstruction4cs.org.

  2. At last we reach: Turing Machines

  3. Turing Machine Model Draw a picture here

  4. PDA’s vs. Turing Machines PDA Turing Machine Processes input string in any order; it starts out on the tape Can move tape head to any point on the tape Deterministic Halts whenever q_accept or q_reject is reached “Blank” symbol included • Processes input string left to right • Only memory access is the top of the stack • Non-deterministic • Halts when all input is read • No “blank” symbol Both models: infinite memory

  5. Important Turing Machine detail • There is ONE way for a Turing Machine to accept a string. • Visit state qaccept; the machine halts and accepts. • There are TWO ways for a Turing Machine to not accept a string. • Visit state qreject; the machine halts and rejects. • Infinite loop. • The LANGUAGE of a Turing machine is: the set of strings that are accepted by a Turing machine. • All strings that are NOT accepted are NOT in the language.

  6. Turing Machine Formal Description • In the TM transition function δ: Q x Γ -> Q x Γ x {L,R}, a given input of the transition function is: • A current state, a character read, and whether we came from the left or right • A current state, a character read • A current state, a character to write • A destination state, a character read • None of the above or more than one of the above

  7. Turing Machine Formal Description • In the TM transition function δ: Q x Γ -> Q x Γ x {L,R}, a given output of the transition function is: • A current state, a character read, and whether we came from the left or right • A current state, a character to write, and whether we should next go left or right • A destination state, a character read, and whether we should write ‘L’ or ‘R’ • A destination state, a character to write, and whether we should next go left or right • None of the above or more than one of the above

  8. Turing Machine configurations • Suppose we have at TM s.t. Γ={a,b,c,d,_}, Q = {qx | 1≤x≤10} U {qacc,qrej}. Suppose the current configuration is xbq1ay. What does this mean? • The TM is in state q1, the tape contains “xbay”, and the tape head is on the “b”. • The TM is in state q1, the tape contains “xbay”, and the tape head is on the “a”. • The TM is in state q1, the tape contains “xbq1ay”, and we don’t know where the tape head is.

  9. Executing a Transition, “yields” • Suppose we have at TM s.t. Γ={a,b,c,d,_}, Q = {qx | 1≤x≤10} U {qacc,qrej}, and the transition function includes rules δ(q1,a) = (q3,b,R) and δ(q1,b) = (q3,a,L). We also have strings x,y in Γ*. Which configuration does the current configuration, xbq1ay, yield? • xq3bay • xbbq3y • xbaq3by • xq3aay • None of the above or more than one of the above

  10. Executing a Transition, “yields” • Suppose we have at TM s.t. Γ={a,b,c,d,_}, Q = {q1, q2, q3, q4,qacc,qrej}, and the transition function includes rules δ(q2,c) = (q3,d,R) and δ(q2,d) = (q3,c,L). We also have strings u,v,x,y in Γ*. Which configuration does the current configuration, xcq2dy, yield? • xq3ddy • xcdq3y • xq3ccy • xq3ccdy • None of the above or more than one of the above

  11. More Transition Function • (a) TRUE • (b) FALSE x0qrej11y yields x01q11y can never happen in any Turing Machine

  12. Turing machines give you these: DECIDABLE AND TURING-RECOGNIZABLE LANGUAGES

  13. So a language is not context-free. What is it? • L = {1^{2n} | n ≥ 0}. • Not context-free. • Exercise: prove it! • So there is no PDA or CFG with L as its language. • What about Turing Machines? What about this stuff??? Context-Free Regular

  14. L = {1^{2n} | n ≥ 0} WHOA, that looks complicated. Let’s break this down.

  15. L = {1^{2n} | n ≥ 0} • How could you check to see if a number is a power of 2? • Check to see if it’s even • Keep on dividing by 2; if you get an odd number, it’s not a power of 2 • Keep on dividing by 2; if you get an odd number > 2, it’s not a power of 2 • Keep on dividing by 2; if you get down to 0, it’s a power of 2

  16. L = {1^{2n} | n ≥ 0} No 1’s Even 1’s One 1 Odd 1’s STEP 1: Scan the tape left to right, cross off every other 1, keep track if the number of 1’s is even, odd, or equal to 1.

  17. L = {1^{2n} | n ≥ 0} No 1’s Even 1’s One 1 Odd 1’s STEP 2: If you’re left with just one 1, halt and accept!

  18. L = {1^{2n} | n ≥ 0} No 1’s Even 1’s One 1 Odd 1’s STEP 3: If you’re left with an odd number of 1’s, but more than one, halt and reject!

  19. L = {1^{2n} | n ≥ 0} No 1’s Even 1’s One 1 Odd 1’s STEP 4: Send the tape head back to the left end of the tape.

  20. L = {1^{2n} | n ≥ 0} No 1’s Even 1’s One 1 Odd 1’s STEP 5: Go back to STEP 1.

  21. L = {1^{2n} | n ≥ 0} • This is a “high-level” description of a Turing Machine: • STEP 1: Scan the tape left to right, cross off every other 1, keep track if the number of 1’s is even, odd, or equal to 1. • STEP 2: If you’re left with just one 1, halt and accept! • STEP 3: If you’re left with an odd number of 1’s, but more than one, halt and reject! • STEP 4: Send the tape head back to the left end of the tape. • STEP 5: Go back to STEP 1. • On Homework and Exams: give high-level descriptions. • This should be easier than drawing the diagram, specifying the transition details, etc.

  22. L = {1^{2n} | n ≥ 0} • This language L is: • Decidable • Turing-recognizable • Decidable AND Turing-recognizable • Neither decidable nor Turing-recognizable

  23. So a language is not context-free. What is it? What’s out here…? After Exam 2. • Decidable: there is a TM that always halts on all inputs. • Turing-recognizable: there is a TM that always halts for accepted strings, but MAY NOT HALT on rejected strings. Turing-recognizable Decidable Context-Free Regular

More Related