Announcements. Final exam review session tomorrow in Gates 104 from 2:15PM 4:15PM Final Friday Four Square of the quarter!

Transcription

1 The Big Picture Problem Problem Set Set 99 due due in in the the box box up up front. front. Congrats Congrats on on finishing finishing the the last last problem problem set set of of the the quarter! quarter!

2 Announcements Problem Set 9 due right now. We'll release solutions right after lecture. Final exam review session tomorrow in Gates 104 from 2:15PM 4:15PM Final Friday Four Square of the quarter! Today at 4:15PM, Outside Gates Please evaluate this course on Axess! Your feedback really does make a difference!

3 The Big Picture

4 The Big Picture

5 Imagine what it must have been like to discover all of the results in this class.

6 Cantor's Theorem: S < ( S) Corollary: Unsolvable problems exist.

7 What problems are unsolvable?

8 First, we need to learn how to prove things. Otherwise, how can we know for sure that we're right about anything?

9 Now, we need to learn how to prove things about processes that proceed step-by-step. So let's learn induction.

10 We also should be sure we have some rules about reasoning itself. Let's add some logic into the mix.

11 Okay! So now we're ready to go! What problems are unsolvable?

12 Well, first we need a definition of a computer!

13 start 0 q 0 q q q 2

14 Cool! Now we have a model of a computer!

15 We're not quite sure what we can solve at this point, but that's okay for now. Let's call the languages we can capture this way the regular languages.

16 I wonder what other machines we can make?

17 0 start q q 1 q 2 ε q 3 0 q 4 0 q 5 1

18 Wow! Those new machines are way cooler than our old ones!

19 I wonder if they're more powerful?

20 start ε q 0 q q 1 q q 2 q 5 q 14 0 q 03 start q 2 0, 1 1 q 0 q q 1 q q 3 1

21 Wow! I guess not. That's surprising! So now we have a new way of modeling computers with finite memory!

22 I wonder how we can combine these machines together?

23 ε start ε ε

24 Cool! Since we can glue machines together, we can glue languages together as well.

25 How are we going to do that?

26 a (.a (.a )

27 Cool! We've got a new way of describing languages.

28 So what sorts of languages can we describe?

29 ε R 11 * R 12 R 11 R 22 R 12 start ε ε q s q 1 q 2 R 21 q f

30 Awesome! We got back the exact same class of languages.

31 It seems like all our models give us the same power! Did we get every language?

32 xw L yw L

33 Wow, I guess not.

34 But we did learn something cool: We have just explored what problems can be solved with finite computers.

35 So what else is out there?

36 Can we describe languages another way?

37 S ax X b C C Cc ε

38 Awesome!

39 So did we get every language yet?

40 Hmmm... guess not.

41 So what if we make our memory a little better?

42 , R 0 0, R 0 0, L 1 1, L Go to start 1, L Clear a 1 start, R, L 1, R q acc rej Check for 0 0, R Go to end 0 0, R 1 1, R, R q acc

43 Cool! Can we make these more powerful?

44 q rej, L start Go left 1 1, L Write = =, L Write 1 st 1 1, L Write 2 nd 1 1, L 1, L Guess 1s 1, L Write 2 nd 1 1, L Write 1 st 1, L Guess 1s 1, L, R run old TM =

45 Workspace Worklist of IDs To To simulate simulate the the NTM NTM N with with a DTM DTM D, D, we we construct construct D as as follows: follows: On On input input w, w, D converts converts w into into an an initial initial ID ID for for N starting starting on on w. w. While While D has has not not yet yet found found an an accepting accepting state: state: D finds finds the the next next ID ID for for N from from the the worklist. worklist. D copies copies this this ID ID once once for for each each possible possible transition. transition. D simulates simulates one one step step of of the the computation computation for for each each of of these these IDs. IDs. D copies copies these these IDs IDs to to the the back back of of the the worklist. worklist.

46 Wow! Looks like we can't get any more powerful. (The Church-Turing thesis says that this is not a coincidence!)

47 So why is that?

48 Space for M Space to simulate M's tape. U TM = TM On On input input M, M, w : w : Copy Copy M to to one one part part of of the the tape tape and and w to to another. another. Place Place a a marker marker in in w to to track track M's M's current current state state and and the the position position of of its its tape tape head. head. Repeat Repeat the the following: following: If If the the simulated simulated version version of of M has has entered entered an an accepting accepting state, state, accept. accept. If If the the simulated simulated version version of of M has has entered entered a a rejecting rejecting state, state, reject. reject. Otherwise: Otherwise: Consult Consult M's M's simulated simulated tape tape to to determine determine what what symbol symbol is is currently currently being being read. read. Consult Consult the the encoding encoding of of M to to determine determine how how to to simulate simulate the the transition. transition. Simulate Simulate that that transition. transition.

49 Wow! Our machines can simulate one another! This is a theoretical justification for why all these models are equivalent to one another.

50 So... can we solve everything yet?

51 Mw 0 Mw 1 Mw 2 Mw 3 Mw 4 Mw 5 M 0 Acc No No Acc Acc No M 1 Acc Acc Acc Acc Acc Acc M 2 Acc Acc Acc Acc Acc Acc M 3 No Acc Acc No Acc Acc M 4 Acc No Acc No Acc No M 5 No No Acc Acc No No No No No Acc No Acc

52 Oh great. Some problems are impossible to solve.

53 So is there just one problem we can't solve?

54 L D M A TM A TM RE A TM R

55 Okay... maybe we can't decide or recognize everything. Can we at least verify or refute everything?

56 A TM M EQ TM A TM M EQ TM

57 R CFL REG co-re RE

58 Wow. That's pretty deep.

59 So... what can we do efficiently?

60 P

61 N P NP

62 So... how are you two related again?

63 No clue.

64 But what do we know about them?

65 NP NP-Hard NPC P

66 What other mysteries remain in theoretical computer science?

67 A Whole World of Theory Awaits!

68 Theory is all about exploring, experimenting, and discovering. We've barely scratched the surface of theoretical computer science.

69 Theory is all about exploring, experimenting, and discovering. We've barely scratched the surface of theoretical computer science.

70 Your Questions

71 If we develop quantum computing that allows for completion of NP problems in the same amount of time as P problems, does this minimize the importance of the P NP problem?

72 What skills do you think are learned in CS courses versus in 'real world' job and internship experience? Going forward, what should we focus on in each?

73 In the problem set, it says that NP is a strict subset of R. What are some languages that are in R but not in NP?

74 I am really interested in the cocktail parties you keep mentioning, how do I get an invite?

75 What is your favorite algorithm and why?

76 Keith, will you go on a date with me?

77 Where to Go From Here

78 Applied Theoretical

79 Applied Theoretical CS154 Intro to Automata and Complexity Theory An in-depth treatment of automata, computability, and complexity. Emphasis on theoretical results in automata theory and complexity. Launching point for more advanced courses (CS254, CS354)

80 Applied Theoretical CS258 Intro to Programming Language Theory Explore questions of computability in terms of recursion and recursive functions. Excellent complement to the material in this course; highly recommended. Offered every other year; consider checking it out!

81 Applied Theoretical CS109 Intro to Probability for Computer Scientists Learn to embrace randomness. Use your newly acquired proof skills in an entirely different domain. See how computers can use statistics to learn patterns.

82 Applied Theoretical CS255 Intro to Cryptography Use hard problems to your advantage! Explore NP-hardness and its relation to cryptography. See how to design secure systems out of hard problems.

83 Applied Theoretical CS161 Design and Analysis of Algorithms Learn how to approach new problems and solve them efficiently. Learn how to deal with NP-hardness in the real world. Learn how to ace job interviews

84 Applied Theoretical CS143 Compilers Watch automata, grammars, undecidability, and NP-completeness come to life by building a complete working compiler from scratch. See just how much firepower you can get from all this material.

85 Applied Theoretical CS107 Computer Organization and Systems You don't need to be a theoretician to love computer science! If you want to learn how the machine works under the hood, look no further.

86

87 There are more problems to solve than there are programs to solve them.

88 Where We've Been Given this hard theoretical limit, what can we compute? What are the hardest problems we can solve? How powerful of a computer do we need to solve these problems? Of what we can compute, what can we compute efficiently? What tools do we need to reason about this? How do we build mathematical models of computation? How can we reason about these models?

89 What We've Covered Sets Graphs Proof Techniques Relations Functions Cardinality Induction Logic Pigeonhole Principle DFAs NFAs Regular Expressions Nonregular Languages CFGs Turing Machines R, RE, and co-re Unsolvable Problems Reductions Time Complexity P NP NP-Completeness

90 My Address:

91 Final Thoughts