Truth and How to See It

CS-113

Gene Itkis

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

The Truth

Truth

• Truth - αλήθεια (alethia)

– Un-hiddenness, un-concealness

• Proof: “uncovering the truth”,

“making truth self-evident” ?

Creation (almost) ex nihilo

Hmm…

11

10

1

1

10

10

10

11

1

10/111

On What You See

Tiger

Trust not thine eyes!

When on lion’s cage you see a sign “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 : aI, bN };

– 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,bN . a+b N – Existential: = “for some”, “there exists”

aN bN . a·b=1

– FALSE

a≠0Q bQ . a·b=1

– TRUE

AND (2b2b)

: or , e.g. x,S . (xS) (xS) : and, e.g. aN bN . 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}– AB={x: xA xB}

: set intersection, e.g. {1,2,3}{2,4}={2}– A B={x: xA xB}

: (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

00

1

Input

1

1 0

Output

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

NAND

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 1a b = …homeworka b = … homework