1. Truth and How to See It
CS-113
Gene Itkis

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

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

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

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

6. As long as it is done right !

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

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

9. 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

10. 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

11. 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)

12. 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

13. 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)

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

15. Circuits
Output
Input  1  1  1      

16.  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

17. 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