theory of computation (fall 2014): euclid's gift to computer science
TRANSCRIPT
Theory of Computation
Euclid's Gift to Computer Science
Vladimir Kulyukin
www.vkedco.blogspot.comwww.vkedco.blogspot.com
Outline● Interesting Facts from History of Numbers● Several Concepts & Theorems of Number Theory
Well-Ordering Principle
Prime Divisibility
Euclid Numbers
Euclid's 2nd Theorem
Euclid's Gift to Computer Science
● Computing Primes is Primitive Recursive
History of Numbers
Are Arabic Numbers Really Arabic?
Origin of Decimal Numbers
“The world owes the decimal notation to the Hindus, and arithmetic as a practical science would have been impossible without the decimal notation. The Arabs first learned that notation from the Hindus and introduced it into Europe.”
Romesh Chunder Dutt. “A History of Civilization in Ancient India Based on Sanskrit Literature – Rationalistic Age” (1,000 BC – 242 BC)”
Nagari Script Numerals and Their Modern Decimal Equivalents
1 2 3 4 5 6 7 8 9 0
Sulva Sutra (8-th Century B.C.)● Sulva Sutra is a Sanskrit document that dates back to the 8th century B.C.● The document handles the question of how to find the value of a diagonal in relation to the side of the square?● Quote: “Increase the measure by its third part, and this third by its own forth, less the thirty-fourth part of that fourth.”● In other words, the square root of 2 can be computed as:
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Several Concepts & Theorems of Number Theory
Two Number Theory Concepts
● We will need the following two concepts from number theory to prove that computing prime numbers is primitive recursive:
Well-Ordering PrinciplePrime Divisibility: Every number has a prime divisor
Well-Ordering Principle
● The well-ordering principle states that every non-empty set of natural numbers has a smallest element
● In axiomatic set theory, the set of natural numbers is defined as the set that contains 0 and is closed under the successor operation
● The set of natural numbers {n | {0, …, n} is well-ordered} contains all natural numbers
Prime Divisibility
p|Ap
A
such that number prime a exists
there,1number naturalany For
Proof
1. Assume that the statement is not true2. Then there exists a non-empty set K of natural numbers, each of which is
greater than 1 and has no prime divisors3. By the Well-Ordering Principle, K has a smallest element s4. s cannot be a prime, because it would have itself as a prime divisor5. Thus, s is a composite number such that s = ab, where 1 < a < s and 1 < b < s6. But, then a and b must have prime divisors 7. Thus, s has the same prime divisors as a and b8. But this contradicts 2 and hence 1
A Number Theory Factoid
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Euclid Numbers: Definition
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Euclid’s 2nd Theorem (Proposition IX.20 in Euclid’s Elements)
There are infinitely many primes.
Euclid’s 2nd Theorem Reformulated
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or prime new aeither is 1...532
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Proof
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Euclid’s 2nd Theorem: Another Formulation
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Proof
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divisor prime a havemust it But then composite. is Then
prime. anot is that Assume done. are weprime, a is If
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Looking for the Next Prime: Lower & Upper Bounds
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for the 1 Hence, .an greater th is after
primenext thebecause ,1!,11
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Examples
121,61!,17
7,41!,15
3,31!,13
2,11!,12
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223
112
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Euclid's Gift to Computer Science● What does Euclid's Theorem give to computer scientists?● It gives us the strict lower and upper bounds where we are
guaranteed to find the next prime● The existence of the strict upper bound implies the applicability of
bound minimalization● The applicability of bounded minimalization implies the primitive
recursiveness of computing primes● Why? Because we have shown that checking the primality of a
number is primitive recursive
Computing Prime Numbers
Computing N-th Prime is Primitive Recursive
recursive primitive is that Show np
Recurrences
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&Primemin 2.
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Formalizing Recurrences
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Formalizing Recurrences
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References & Reading Suggestions
● Ch. 3, Computability, Complexity, and Languages, 2nd Edition, by Davis, Weyuker, Sigal
● Romesh Chunder Dutt. A History of Civilization in Ancient India Based on Sanskrit Literature – Rationalistic Age (1,000 BC – 242 BC)