IST 4 Information and Logic

Transcription

1 IST 4 Information and Logic

2 mon tue wed thr fri sun T = today 3 M oh x= hw#x out oh M 7 oh oh 2 M2 oh oh x= hw#x due 24 oh oh 2 oh = office hours oh oh T M2 8 3 oh midterms oh oh Mx= MQx out 5 oh 3 4 oh Mx= MQx due 22 oh oh oh oh oh 5 oh 5 oh

3 MQ2 Everyone has a gift! Due TODAY, 5/2/27, by pm Please PDF lastname-firstname.pdf to istta4@paradise.caltech.edu

4 Information System Memory Intention Languages Evolution External Memory Algorizms

5 Building Blocks finite number of building blocks à infinitely many descriptions DNA natural languages

6 Building Blocks Separation finite number of building blocks à infinitely many descriptions Separation A between syntax and semantics Separation C between algorithms and implementation Separation B between what is represented and reality, feasibility, time, space,...

7 The appearance of life is the first Information Megamorphosis DNA ~3.7 Billion ya The appearance of the human brain is the second Information Megamorphosis Spoken languages ~6Kya

8 Written languages ~5,ya Babylonians The language of numbers positional number systems mathematics our number sense is 3

9 Formal languages Greeks ~2,5ya Pythagoras BC Axioms Theorems Euclid,3BC Proofs

10 Formal languages Greeks Aristotle BC Logic Syllogism Inference... ~2,5ya our logical sense is 3 People that are wise are Babylonians Leibniz was wise Leibniz was a Babylonian

11 Algorizms Algorizmi 78-85AD ~ya Fibonacci 7-25AD Algorizms for everything!! ~3ya Gottfried Leibniz

12 Formal languages for ideas Gottfried Leibniz Let us calculate without further ado, to see who is right" Let s Google it!! Let s Leibniz it!!

13 Algorizms and syntax boxes Gottfried Leibniz instead of progression by tens, I have for many years used the most simple of all, which goes by two...

14 Gottfried Leibniz The Binary d d2 majority c 2 symbol adder c s a b m parity magic box finite universality

15 Gottfried Leibniz The Leibniz challenges : we need a language for... We need a language for...

16 ~2 years after Aristotle... George Boole No perfection but a lot of inspiration Calculus for logic,847 Shannon 96-2 and Calculus for syntax boxes Boole was a Babylonian...

17 Boolean Algebra Huntington 94; concise set of axioms Edward Huntington April 26, Undergrad and Masters at Harvard PhD at the U of Strasbourg, Germany (9) Professor at Harvard until 94 Huntington was a Greek...

18 The Algebra (Boolean Calculus) Algebraic system: set of elements B, two binary operations + and B has at least two elements ( and ) If the following axioms are true then it is a Boolean Algebra: A. identity A2. complement A3. commutative A4. distributive

19 Properties of an Axiomatic System consistent consistent complete independent

20 Complete: Every true statement in the math theory can be derived using the axioms Can we prove EVERYTHING? Can we build EVERYTHING?

21 A simpler question: Is everything countable? Can we prove EVERYTHING?

22 Are infinite length binary strings countable? Proof by contradiction: Assume that it is countable and reach a contradiction

23 Cantor Diagonal argument George Cantor, 89 Are infinite length binary strings countable? ?... Idea: Complement the diagonal This string is binary and is not counted, contradiction!

24 Is everything countable? NO Can we prove EVERYTHING?

25 Consistent: No contradictions in the math theory Complete: Every true statement in the math theory can be derived using the axioms Kurt Gödel April 28, : For any axiomatic system that is powerful enough to describe the arithmetic of the natural numbers: If the system is consistent, it cannot be complete In a consistent system there are statements that are not provable... The key idea: represent the axiomatic system using numbers, use the diagonal argument of Cantor

26 A simple example Can we prove EVERYTHING?

27 6, 3,, 5, 6, 8, 4, 2,, 34, 7, 52, 26, 3, 4, 2,, 5, 6, 8, 4, 2, Source: wikipedia

28 Does it always reach? Other options? 6, 3,, 5, 6, 8, 4, 2,, 34, 7, 52, 26, 3, 4, 2,, 5, 6, 8, 4, 2, n even n odd Source: wikipedia

29 Which number in -5 has the longest sequence to reach? 6, 8, 4, 2, 3, 4, 2,, 5, 6, 8, 4, 2, 7, 52, 26, 3, 4, 2,, 5, 6, 8, 4, 2, Source: wikipedia

30 Which number in -5 has the longest sequence to reach? 9, 28, 4, 7, 22,, 34, 7, 52, 26, 3, 4, 2,, 5, 6, 8, 4, 2, 7, 22,, 34, 7, 52, 26, 3, 4, 2,, 5, 6, 8, 4, 2, The number 9, a sequence with 2 numbers Source: wikipedia

31 Lothar Collatz 9-99 The Collatz conjecture (937): For every starting value m, the sequence always reaches Empirical evidence: Verified up to some large number (29): True False Prove that it is impossible to decide if the conjecture is true or false

32 Lothar Collatz 9-99 The Collatz conjecture (937): For every starting value m, the sequence always reaches A generalization:

33 The Collatz conjecture (937): For every starting value m, the sequence always reaches Open problem... This generalization is undecidable, J. Conway, 972 Undecidable: Given a function f, does the Collatz sequence reach, for all n>? Undecidable even if p = 648 is fixed We can prove that it is impossible to decide if true or false

34 Euclid,3BC Languages:: possible and impossible There are theorems that cannot be proved Turing Algorizmi 78-85AD There are problems that cannot be solved by an Gödel algorizm Cantor There are objects that cannot be counted

35 Boolean Algebra is: consistent independent complete

36 Boolean Algebra Proving theorems You have to see a giraffe to believe it exists Intuition is not natural it comes with practice

37 Proving theorems Intuition is not natural it comes with practice

38 If I satisfy the axioms then I am a Boolean Algebra You do not need to see it to believe it exists!

39 Boolean Algebra - algebra You can see this one

40 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND xy OR(x,y) xy AND(x,y) iff both x and y are iff both x and y are Is it a Boolean Algebra?

41 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND The following axioms are obviously true:??? A. identity A2. complement A3. commutative A4. distributive

42 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND A. identity a + = a a x = a xy OR(x,y) xy AND(x,y)

43 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND A. identity a + = a a x = a + = + = xy OR(x,y) xy AND(x,y)

44 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND A. identity a + = a a x = a x = x = xy OR(x,y) xy AND(x,y)

45 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND A2. complement a + a = a x a = xy OR(x,y) xy AND(x,y)

46 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND A2. complement a + a = a x a = + = + = x = x = xy OR(x,y) xy AND(x,y) a complement of a complement of

47 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND A3. commutative a + b = b + a a x b = b x a xy OR(x,y) xy AND(x,y)

48 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND A3. commutative a + b = b + a a x b = b x a + = + x = x + = + x = x + = + x = x xy OR(x,y) + = + x = x xy AND(x,y)

49 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND A4. distributive a + (b x c) = (a + b) x (a + c) a x (b + c) = (a x b) + (a x c) xy OR(x,y) xy AND(x,y)

50 - Boolean Algebra Boolean Algebra: set of elements B={,}, two binary operations OR and AND A4. distributive a + (b x c) = (a + b) x (a + c) + ( x ) = ( + ) x ( + ) + ( x ) = ( + ) x ( + ) xy OR(x,y) We can check all the cases... xy AND(x,y)

51 Now, to our first Boolean proof

52 Self Absorption ME-MYSELF&I Lemma : Proof: xy OR(x,y) Two-valued Boolean Algebra: set of elements B={,}, two binary operations OR and AND Is the lemma true? xy AND(x,y)

53 Self Absorption ME-MYSELF&I Lemma : Proof: A A2 A4 A2 A Q

54 Self Absorption ME-MYSELF&I Lemma : Proof: We only proved that Need to prove Ideas?

55 Boolean Algebra Duality

56 Duality Theorem : Any identity that is true in a Boolean algebra, is also true if + and are interchanged, and and are interchanged.

57 ME-MYSELF&I Lemma : Proof: if + and. are interchanged, and and are interchanged A A2 A4 A2 A

58 Duality Theorem : Any identity that is true in a Boolean algebra, is also true if + and are interchanged, and and are interchanged. Proof:????

59 Theorem : Any identity that is true algebra, is also true if + and. are interchanged, and and are interchanged. It is a syntax machine: It is true for the axioms!

60 Back to the Axioms Q: Is the complement unique / well defined?

61 Boolean Algebra One way to say NO

62 One Way to Theorem : Each element of a Boolean Algebra has exactly one complement. Proof: Say No! L: Self Absorption Warm-up: First we will prove that an element is not self-complement Assume that:?? By Lemma : However by A2:

63 One Way to Say No! L: Self Absorption Theorem : Each element of a Boolean Algebra has exactly one complement. Proof: Warm-up: First we will prove that an element is not self-complement Assume that: By duality: By Lemma : However by A2: Contradiction! and are distinct Q

64 One Way to Say No! Theorem : Each element of a Boolean Algebra has exactly one complement. Proof: We proved that an element is not self-complement Next will prove that the complement is unique

65 One Way to Say No! Proof: Need to prove that the complement is unique By contradiction: Assume an element has two distinct complements A A2 A4 A3 A2

66 One Way to Say No! Proof: Need to prove that the complement is unique By contradiction: Assume an element has two distinct complements A A2 A4 A3 A4 A2 A A2

67 One Way to Say No! Proof: Need to prove that the complement is unique By contradiction: Assume an element has two distinct complements A A2 A4 A3 A2 A2 A3 A4 A2 A Contradiction! Q

68 So far True for any Boolean Algebra T: duality principle T: one complement per element L: Self Absorption

69 Quiz time

70 Quiz #5 min - Boolean Algebra: set of elements B={,} two binary operations OR and AND xy OR(x,y) xy AND(x,y) Prove that the following statement is true for a - Boolean algebra: