Truth and How to See It CS-113 Gene Itkis

The Truth Do you solemnly swear to tell the truth, the whole truth and nothing but the truth, so help you G*d?

Truth Truth - αλήθεια (alethia) Un-hiddenness, un-concealness Proof: “uncovering the truth”, “making truth self-evident” ?

Creation (almost) ex nihilo Hmm… Creation (almost) ex nihilo 11 11 10 10 1 10/11 1 1 1 10 10

On What You See Tiger When on lion’s cage you see a sign “Tiger” – Trust not thine eyes! Tiger

As long as it is done right !

CS as problem solving What is the most famous/grand question answered by a computer: The Great Question of Life, the Universe and Everything

Universal algorithm (ISO) Input the PROBLEM Solve the PROBLEM Output the ANSWER

The Universe U={ “objects” } Popular (Sub-)Universes: Integers: I ={0,1,-1,2,…}; Natural numbers: N ={1,2,…}; Rationals: Q ={a/b : aI, bN }; Reals: R

Computers are dumb! People are nice: Computers are not: Understanding will try to understand what you really meant fill in some gaps identify and correct some of your mistakes Forgiving provide some error-correction Computers are not: “do what I mean not what I say” never works your mistake is its command

Conclusion Must be extra precise in what you say Must prove that what you say is correct Must build in your own error-detection and error-correction (if/when things do go wrong – e.g., when assumptions turn out to be false)

Everything Quantifiers: Universal:  = “for every”, “for all” a,bN . a+b N Existential:  = “for some”, “there exists” aN bN . a·b=1 FALSE a≠0Q bQ . a·b=1 TRUE

AND (2b2b)  : or , e.g. x,S . (xS)  (xS)  : and, e.g. aN bN . a·b=b  b/a=b  : negation, e.g. claim C . C  C : set union, e.g. {1,2,3}{2,4}={1,2,3,4} AB={x: xA  xB}  : set intersection, e.g. {1,2,3}{2,4}={2} A  B={x: xA  xB}  : (proper) subset, e.g. {2}{2,4}  : subset or equal, e.g. set S . (S  S)  (  S)

Implications  : implies, A  B (“A implies B” or “if A then B”) “A  B” = “A  B” E.g. if pigs can fly then …

Circuits Output Input  1  1  1      

 Universal Gate: NAND  a = a NAND 1 a  b = ( a NAND b ) = 1 NAND (a NAND b) a  b = … homework Any Boolean function (truth table) can be expressed in terms of a circuit of AND (), OR () and NOT () gates  it can also be expressed using only NAND gates

XOR : Exclusive OR  : Exclusive OR (a or b but not both) also a  b= (a+b mod 2) 0  0 = 1  1 = 0 1  0 = 0  1 = 1 a = a  1 a  b = …homework a  b = … homework