Truth and How to See It CS-113 Gene Itkis
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
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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