CNS 188a Computation Theory and Neural Systems. Monday and Wednesday 1:30-3:00 Moore 080

Transcription

1 CNS 88a Computation Theory and Neural Systems Monday and Wednesday :30-3:00 Moore 080 Lecturer: Shuki Bruck; 33 Moore office hours: Mon, Wed, 3-4pm TAs: Vincent Bohossian, Matt Cook; 3 Moore office hours: Mon, Tue, 8-9pm Secretary: Michelle Chen; 304 Moore

2 From Screws to Systems

3 The Lineage of BMW

4 C. Elegans Lineage total of 959 cells 302 nerve cells 3 cells are destined to die

5 C. Elegans Lineage Simple Questions Dealing with identity: How do cells remember what to do? Dealing with time: How do cells know when? No clock Dealing with order: How do cells coordinate their actions? total of 959 cells 302 nerve cells 3 cells are destined to die

6 Control via Stochastic Chemical Reactions A B C D E F G A G E G D F F E D D C B C B A k k k k k

7 Chemical Reactions Circuits A B k C 4 B C D E k2 k3 D F 2 3 F k4 D G E G k5 A 5

8 Chemical Reactions Circuits A B k C 4 B C D E k2 k3 D F 2 3 F k4 D G E G k5 A 5

9 Chemical Reactions Circuits A B k C 4 B C D E k2 k3 D F 2 3 F k4 D G E G k5 A 5

10 Chemical Reactions Circuits A B k C 4 B C D E k2 k3 D F 2 3 F k4 D G E G k5 A 5

11 Chemical Reactions Circuits A B k C 4 B C D E k2 k3 D F 2 3 F k4 D G E G k5 A 5

12 Chemical Reactions Circuits A B k C 4 B C D E k2 k3 D F 2 3 F k4 D G E G k5 A 5

13 Chemical Reactions Circuits A B k C 4 B C D E k2 k3 D F 2 3 F k4 D G E G k5 A 5

14

15 Descriptive Biology: Is It Enough?

16 A HUGE Gap between Ability to Design and Analyze Design Analysis x y S C z

17 Key to the Progress in Design: Abstractions in Information Systems Reasoning to Calculations to Physical Circuits Logical reasoning Boolean Calculus Circuits

18 Key to the Progress in Design: Abstractions in Information Systems Logic to Boolean Calculus to Physical Circuits Boole x y S Shannon C z 847 Connected Logic with Algebra Boolean Algebra Logical Calculation 938 Boolean Algebra to Electrical Circuits Logic Design

19 Text to Algebra George Boole, 854

20 Text to Algebra George Boole, 854

21 Shannon MSc Thesis, 938 Algebra to Circuits

22 The Algebra (Boolean Calculus) Boole, Jevons, Peirce, Schroder (8xx) Axiomatic System: Huntington (904) Algebraic system: set of elements B, two binary operations and B has at least two elements (0 and ) If the following axioms are true then it is a Boolean Algebra:

23 The Algebra (Boolean Calculus) Boole, Jevons, Peirce, Schroder (8xx) Axiomatic System: Huntington (904) Algebraic system: set of elements B, two binary operations and B has at least two elements (0 and ) If the following axioms are true then it is a Boolean Algebra: A. identity a 0 = a ; a = a A2. complement A3. commutative A4. distributive a a = ; a a = 0 a b = b a; a b = b a a b c = ( a b) ( a c); a ( b c) = a b a c

24 Two-valued Boolean Algebra Boolean Algebra: set of elements B={0,}, two binary operations OR and AND xy OR(x,y) xy AND(x,y)

25 Two-valued Boolean Algebra Boolean Algebra: set of elements B={0,}, two binary operations OR and AND The following axioms are obviously true: a 0 = a ; a = a a a = ; a a = 0 a b = b a; a b = b a a b c = ( a b) ( a c); a ( b c) = a b a c

26 Two-valued Boolean Algebra Boolean Algebra: set of elements B={0,}, two binary operations OR and AND A. identity a 0 = a ; a = a 0 0 = 0 0 = 0 = 0 = xy OR(x,y) xy AND(x,y)

27 Two-valued Boolean Algebra Boolean Algebra: set of elements B={0,}, two binary operations OR and AND A2. complement a a = ; a a = 0 0 = 0 = 0 = 0 0 = 0 xy OR(x,y) xy AND(x,y)

28 Two-valued Boolean Algebra Boolean Algebra: set of elements B={0,}, two binary operations OR and AND A3. commutative a b = b a; a b = b a 00 = 00 0 = 0 0 = 0 = 0 0 = = 0 0 = 0 =

29 Two-valued Boolean Algebra Boolean Algebra: set of elements B={0,}, two binary operations OR and AND A4. distributive a b c = ( a b) ( a c); a ( b c) = a b a c

30 Two-valued Boolean Algebra Boolean Algebra: set of elements B={0,}, two binary operations OR and AND A4. distributive a b c = ( a b) ( a c); a ( b c) = a b a c 00 0 = (00) (00) = 0 00 = (00) (0) = = (0) (00) = 0 0 = (0) (0) = 0 0 = (0) (0) = 0 = (0) () = 0 = () (0) = = () () =

31 Historical notes: Boolean Algebra Defined by Axioms (Postulates) Pre-Boole (6xx): Leibniz; universal language for reasoning? Beginning (8xx): Boole, Jevons, Peirce, Schroder, Whitehead Next step: Huntington 904; improved set of axioms Sheffer 93: Five axioms and one binary operation Huntington/Robbins 933 three axioms, conjecture Recent progress : McCune 996; Robbins conjecture proved!

32 Historical notes: Three Axioms

33 Back to the Axioms Q: is the complement unique / well defined?

34 One Way to Say No! Theorem : Each element of a Boolean Algebra has exactly one complement. Proof: First we will prove that an element is not self-complement

35 Self Absorption Lemma : Proof: A A2 A4 A2 A Q

36 SpinOut Keister William Keister ( ) was a pioneer in switching theory and design at Bell Labs. When he retired in 972, he was director of Bell Labs' Computing Technology Center at Holmdel, New Jersey. Mr. Keister began working in his spare time to prove that puzzles could be solved using Boolean algebra. U.S. Patent (972): SpinOut U.S. Patent (972): The Hexadecimal Puzzle

37 One Way to Say No! Theorem : Each element of a Boolean Algebra has exactly one complement. Proof: First we will prove that an element is not self-complement Assume that: By Lemma : However by A2: Contradiction! By A: By A2: Q

38 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

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

40 Back to the Axioms T: one complement per element L: aa = a Is?? Yes: by Principle of Duality

41 Duality Theorem 0: Any identity that is true in a Boolean algebra, is also true if and are interchanged, and 0 and are interchanged. Proof: It is true for the axioms!

42 Duality Lemma : Proof:

43 So far True for any Boolean Algebra T0: duality principle T: one complement per element L: aa = a

44 Non-Binary Boolean Algebras Boolean Integers (Bunitskiy 899) 2 x 3 x 5 = 30 2 x 3 x 7 = 42 Every prime in the prime factorization is a power of one The set of divisors of a Boolean integer {,2,3,5,6,0,5,30} The operations: lcm and gcd The special elements: and 30

45 Non-Binary Boolean Algebras The set {,2,3,5,6,0,5,30} The operations: lcm and gcd The special elements: and = lcm(6,5) =? 6 5 = gcd(6,5) =?

46 Non-Binary Boolean Algebras The set {,2,3,5,6,0,5,30} The operations: lcm and gcd The special elements: and = lcm(6,5) = = gcd(6,5) = 3

47 What is the Complement? The set {,2,3,5,6,0,5,30} The operations: lcm and gcd The special elements: and 30

48 CNS 88a Overview Boolean algebra as an axiomatic system Boolean functions and their representations using Boolean formulas and spectral methods Implementing Boolean functions with circuits of AON (AND, OR, NOT) gates and LT (Linear Threshold) gates Analyzing the complexity (size and depth) of circuits Relations (as opposed to functions) and their implementation in circuits Feedback and convergence in LT circuits