Truth and How to See It

Truthand 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” ?

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

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

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: • 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 0 0 1  0  1  1      

NAND   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

Truth and How to See It