Time Magazine (1984)

Transcription

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2 Time Magazine (1984) Put the right kind of software into a computer, and it will do whatever you want it to. There may be limits on what you can do with the machines themselves, but there are no limits on what you can do with software.

3 Faster than a Busy Beaver

4 (define (c) (define (b n)...) (+ 1 (b (* N 5))))

5 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 4 Why Study Theory? Basic Computer Science Issues What is a computation? Are computers omnipotent? What are the fundamental capabilities and limitations of computers? Pragmatic Reasons Avoid intractable or impossible problems. Apply efficient algorithms when possible. Learn to tell the difference.

6 A 2-MINUTE PROOF OF THE 2nd-MOST IMPORTANT THEOREM OF THE 2nd MILLENUIUM by Doron Zeilberger Written: Oct. 4, 1998

7 Kurt Gödel (1931) The (First) Incompleteness Theorem The most important theorem of the second millennium No formal logic for arithmetic can prove everything that is true x n +y n =z n has a solution only if n=2

8 Proof 1. Imagine some program halt(p,x) that answers yes when p(x) halts and no otherwise. 2. Construct the program alan(p) = if halt(p,p) says yes then do nothing forever otherwise answer yes 3. Consider the question halt(alan,alan).

9 Proof 1. Imagine some program halt(p,x) that answers yes when p(x) halts and no otherwise. 2. Construct the program alan(alan) = if halt(alan,alan) says yes then do nothing forever otherwise answer yes 3. Consider the question halt(alan,alan).

10 Proof 1. Imagine some program halt(p,x) that answers yes when p(x) halts and no otherwise. 2. Construct the program alan(alan) = if halt(alan,alan) says yes then do nothing forever otherwise answer yes 3. Consider the question halt(alan,alan). If it answers yes, it halts, in fact alan(alan) loops forever

11 Proof 1. Imagine some program halt(p,x) that answers yes when p(x) halts and no otherwise. 2. Construct the program alan(alan) = if halt(alan,alan) says yes then do nothing forever otherwise answer yes 3. Consider the question halt(alan,alan). If it answers no, it loops, in fact alan(alan) halts with answer yes

12 Proof 1. Imagine some program halt(p,x) that answers yes when p(x) halts and no otherwise. 2. Construct the program alan(alan) = if halt(alan,alan) says yes then do nothing forever otherwise answer yes 3. Consider the question halt(alan,alan). If it never answers, then it s also no good.

13 Epimenides (-550) All Cretans are liars... One of their own poets has said so.

14 Socrates (-400) One thing that I know is that I know nothing

15 Eubulides (-350) This statement is false!

16 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 9 Some Other Undecidable Problems Does a program run forever? Is a program correct? Are two programs equivalent? Is a program optimal (time wise is it the fastest program solving a specific problem)? Does a polynomial equation with one or more variables and integer coefficients (e.g. 5x +15y +19xyz 2 =12) have an integer solution (Hilbert s 10th problem). Is a finitely-presented group trivial? Given a string, x, how compressible is it?

17 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 1 Critical CS Questions What is a computer? And What is a Computation?

18 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 1 Critical CS Questions What is a computer? And What is a Computation? real computers too complex for any theory

19 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 1 Critical CS Questions What is a computer? And What is a Computation? real computers too complex for any theory need manageable mathematical abstraction

20 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 1 Critical CS Questions What is a computer? And What is a Computation? real computers too complex for any theory need manageable mathematical abstraction idealized models: accurate in some ways, but not in all details

21 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 2 Finite Automata formal definition of finite automata

22 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 2 Finite Automata formal definition of finite automata deterministic vs. non-deterministic finite automata

23 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 2 Finite Automata formal definition of finite automata deterministic vs. non-deterministic finite automata regular languages

24 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 2 Finite Automata formal definition of finite automata deterministic vs. non-deterministic finite automata regular languages operations on regular languages

25 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 2 Finite Automata formal definition of finite automata deterministic vs. non-deterministic finite automata regular languages operations on regular languages regular expressions

26 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 3 Example: A One-Way Automatic Door front pad rear pad door open when person approaches

27 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 3 Example: A One-Way Automatic Door front pad rear pad door open when person approaches hold open until person clears

28 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 3 Example: A One-Way Automatic Door front pad rear pad door open when person approaches hold open until person clears don t open when someone standing behind door

29 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 4 The Automatic Door as DFA REAR BOTH NEITHER closed FRONT open FRONT REAR BOTH States: OPEN CLOSED NEITHER

30 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 4 The Automatic Door as DFA REAR BOTH NEITHER closed FRONT open FRONT REAR BOTH States: OPEN CLOSED NEITHER Sensor: FRONT: someone on front pad REAR: someone on rear pad BOTH: someone(s) on both pads NEITHER no one on either pad.

31 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 5 The Automatic Door as DFA DFA is Deterministic Finite Automata

32 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 5 The Automatic Door as DFA DFA is Deterministic Finite Automata REAR BOTH NEITHER closed FRONT open FRONT REAR BOTH NEITHER

33 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 5 The Automatic Door as DFA DFA is Deterministic Finite Automata REAR BOTH NEITHER closed FRONT open FRONT REAR BOTH NEITHER neither front rear both closed closed open closed closed open closed open open open

34 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 6 DFA: Informal Definition q 1 q q 2 3 0,1 The machine M 1 : states: q 1,q 2, and q 3.

35 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 6 DFA: Informal Definition q 1 q q 2 3 0,1 The machine M 1 : states: q 1,q 2, and q 3. start state: q 1 (arrow from outside ).

36 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 7 DFA: Informal Definition (cont.) q 1 q q 2 3 0,1 On an input string DFA begins in start state q 1 after reading each symbol, DFA makes state transition with matching label.

37 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 7 DFA: Informal Definition (cont.) q 1 q q 2 3 0,1 On an input string DFA begins in start state q 1 after reading each symbol, DFA makes state transition with matching label. After reading last symbol, DFA produces output: accept if DFA is an accepting state. reject otherwise.

38 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 8 Informal Definition - Example q 1 q q 2 3 0,1 What happens on input strings 1101

39 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 8 Informal Definition - Example q 1 q q 2 3 0,1 What happens on input strings

40 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 8 Informal Definition - Example q 1 q q 2 3 0,1 What happens on input strings

41 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 8 Informal Definition - Example q 1 q q 2 3 0,1 What happens on input strings In general?!

42 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 9 Informal Definition q 1 q q 2 3 This DFA accepts all input strings that end with a 1 0,1 all input strings that contain at least one 1, and end with an even number of 0 s no other strings

43 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 10 Languages and Alphabets An alphabet Σ is a finite set of letters. Σ = {a, b, c,..., z} the English alphabet. Σ = {α, β, γ,...,ζ} the Greek alphabet. Σ = {0, 1} the binary alphabet. Σ = {0, 1,...,9} the digital alphabet.

44 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 10 Languages and Alphabets An alphabet Σ is a finite set of letters. Σ = {a, b, c,..., z} the English alphabet. Σ = {α, β, γ,...,ζ} the Greek alphabet. Σ = {0, 1} the binary alphabet. Σ = {0, 1,...,9} the digital alphabet. The collection of all strings over Σ is denoted by Σ.

45 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 10 Languages and Alphabets An alphabet Σ is a finite set of letters. Σ = {a, b, c,..., z} the English alphabet. Σ = {α, β, γ,...,ζ} the Greek alphabet. Σ = {0, 1} the binary alphabet. Σ = {0, 1,...,9} the digital alphabet. The collection of all strings over Σ is denoted by Σ. For the binary alphabet, ε, 1, 0, , are all members of Σ.

46 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 11 Languages and Examples A language over Σ is a subset L Σ. For example Modern English. Ancient Greek. All prime numbers, written using digits. A = {w w has at most seventeen 0 s}. B = {0 n 1 n n 0}. C = {w w has an equal number of 0 s and 1 s}.

47 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 12 Languages and DFA Definition: L(M), the language of a DFA M, is the set of strings L that M accepts, L(M) =L.

48 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 12 Languages and DFA Definition: L(M), the language of a DFA M, is the set of strings L that M accepts, L(M) =L. Note that M may accept many strings, but M accepts only one language.

49 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 12 Languages and DFA Definition: L(M), the language of a DFA M, is the set of strings L that M accepts, L(M) =L. Note that M may accept many strings, but M accepts only one language. What language does M accept if it accepts no strings?

50 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 12 Languages and DFA Definition: L(M), the language of a DFA M, is the set of strings L that M accepts, L(M) =L. Note that M may accept many strings, but M accepts only one language. What language does M accept if it accepts no strings? A language is called regular if some deterministic finite automaton accepts it.

51 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 13 Formal Definitions A deterministic finite automaton (DFA) is a 5-tuple (Q, Σ, δ,q 0,F), where Q is a finite set called the states, Σ is a finite set called the alphabet, δ : Q Σ Q is the transition function, q 0 Q is the start state, and F Q is the set of accept states.

52 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 14 Back to M q 1 q q 2 3 0,1 M 1 =(Q, Σ, δ,q 1,F) where Q = {q 1,q 2,q 3 }, Σ = {0, 1},

53 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 14 Back to M q 1 q q 2 3 0,1 M 1 =(Q, Σ, δ,q 1,F) where Q = {q 1,q 2,q 3 }, Σ = {0, 1}, 0 1 the transition function δ is q 1 q 1 q 2 q 2 q 3 q 2 q 3 q 2 q 2

54 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 14 Back to M q 1 q q 2 3 0,1 M 1 =(Q, Σ, δ,q 1,F) where Q = {q 1,q 2,q 3 }, Σ = {0, 1}, 0 1 the transition function δ is q 1 is the start state, and F = {q 2 }. q 1 q 1 q 2 q 2 q 3 q 2 q 3 q 2 q 2

55 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 15 Another Example a s b a q 1 r 1 b b a a b b q 2 r 2 a

56 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 17 A Formal Model of Computation Let M = (Q, Σ, δ,q 0,F) be a DFA, and let w = w 1 w 2 w n be a string over Σ. We say that M accepts w if there is a sequence of states r 0,...,r n (r i Q) such that r 0 = q 0 δ(r i,w i+1 )=r i+1, 0 i<n r n F

57 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 18 The Regular Operations Let A and B be languages. The union operation: A B = {x x A or x B} The concatenation operation: A B = {xy x A and y B} The star operation: A = {x 1 x 2...x k k 0 and each x i A}

58 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 19 The Regular Operations Examples Let A= {good, bad} and B = {boy, girl}. Union A B = {good, bad, boy, girl} Concatenation A B = {goodboy, goodgirl, badboy, badgirl} Star A = {ε, good, bad, goodgood, goodbad, badbad, badgood,...}

59 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 20 Claim: Closure Under Union If A 1 and A 2 are regular languages, so is A 1 A 2.

60 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 20 Claim: Closure Under Union If A 1 and A 2 are regular languages, so is A 1 A 2. Approach to Proof: some M 1 accepts A 1 some M 2 accepts A 2 construct M that accepts A 1 A 2.

61 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 20 Claim: Closure Under Union If A 1 and A 2 are regular languages, so is A 1 A 2. Approach to Proof: some M 1 accepts A 1 some M 2 accepts A 2 construct M that accepts A 1 A 2. Attempted Proof Idea: first simulate M 1, and if M 1 doesn t accept, then simulate M 2.

62 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 20 Claim: Closure Under Union If A 1 and A 2 are regular languages, so is A 1 A 2. Approach to Proof: some M 1 accepts A 1 some M 2 accepts A 2 construct M that accepts A 1 A 2. Attempted Proof Idea: first simulate M 1, and if M 1 doesn t accept, then simulate M 2. What s wrong with this?

63 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 20 Claim: Closure Under Union If A 1 and A 2 are regular languages, so is A 1 A 2. Approach to Proof: some M 1 accepts A 1 some M 2 accepts A 2 construct M that accepts A 1 A 2. Attempted Proof Idea: first simulate M 1, and if M 1 doesn t accept, then simulate M 2. What s wrong with this? Fix: Simulate both machines simultaneously.

64 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 21 Closure Under Union: Correct Proof Suppose M 1 =(Q 1, Σ, δ 1,q 1,F 1 ) accepts L 1, and M 2 =(Q 2, Σ, δ 2,q 2,F 2 ) accepts L 2.

65 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 21 Closure Under Union: Correct Proof Suppose M 1 =(Q 1, Σ, δ 1,q 1,F 1 ) accepts L 1, and M 2 =(Q 2, Σ, δ 2,q 2,F 2 ) accepts L 2. Define M as follows (M will accept L 1 L 2 ): Q = Q 1 Q 2. Σ is the same.

66 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 21 Closure Under Union: Correct Proof Suppose M 1 =(Q 1, Σ, δ 1,q 1,F 1 ) accepts L 1, and M 2 =(Q 2, Σ, δ 2,q 2,F 2 ) accepts L 2. Define M as follows (M will accept L 1 L 2 ): Q = Q 1 Q 2. Σ is the same. For each (r 1,r 2 ) Q and a Σ, δ((r 1,r 2 ),a)=(δ 1 (r 1,a), δ 2 (r 2,a))

67 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 21 Closure Under Union: Correct Proof Suppose M 1 =(Q 1, Σ, δ 1,q 1,F 1 ) accepts L 1, and M 2 =(Q 2, Σ, δ 2,q 2,F 2 ) accepts L 2. Define M as follows (M will accept L 1 L 2 ): Q = Q 1 Q 2. Σ is the same. For each (r 1,r 2 ) Q and a Σ, δ((r 1,r 2 ),a)=(δ 1 (r 1,a), δ 2 (r 2,a)) q 0 =(q 1,q 2 )

68 Slides modified by Yishay mansour on a modification by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 21 Closure Under Union: Correct Proof Suppose M 1 =(Q 1, Σ, δ 1,q 1,F 1 ) accepts L 1, and M 2 =(Q 2, Σ, δ 2,q 2,F 2 ) accepts L 2. Define M as follows (M will accept L 1 L 2 ): Q = Q 1 Q 2. Σ is the same. For each (r 1,r 2 ) Q and a Σ, δ((r 1,r 2 ),a)=(δ 1 (r 1,a), δ 2 (r 2,a)) q 0 =(q 1,q 2 ) F = {(r 1,r 2 ) r 1 F 1 or r 2 F 2 }.