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Information and Logic

The Algebra of Google

S is the set of all the words in English

A web page is an element in the algebra

The Second Edition of the Oxford English Dictionary contains
full entries for W? words in current use, and
M? obsolete words

The Second Edition of the Oxford English Dictionary contains
full entries for 171,476 words in current use, and
47,156 obsolete words.

1/n  words are nouns, 1/v verbs,
and 1/a adjectives

A search is finding all the subsets (web pages) that
consistent with the subset defined by the query

A query is an element in the algebra

Examples of Boolean Algebras

• 0-1 ( two valued ) Boolean algebra
OR / AND

Arithmetic Boolean algebras (Boolean integers)
lcm / gcd

• Algebra of subsets = any Boolean algebra
union / intersection

An Amazing Theorem

Re presentation Theorem (Stone 1936):

Every finite Boolean algebra is isomorphic to a Boolean
algebra of subsets of some finite set S.

Algebra 1   Algebra 2
elements elements
ope rations /elements operations/elements

Marshall Stone

Marshall Stone
1903-1989

Proved in 1936
90AB = years After Boole
The Boolean Syntax invented in 1847
has a unique representative semantic!!!
Marshall entered Harvard in 1919
intending to continue his studies at
Harvard law school ; fell in love with
Mathematics
,
and the rest is history…
Harlan Fiske Stone
12th Chief Justice of the US
1941-1946
Marshall had a passion for travel. He began
traveling when he was young and was
on a trip to India when he died....

Arithmetic Boolean Algebras
Isomorphic to Algebra of Subsets

The set of divisors of a Boolean integer
{1,2,3,6}
The operations : lcm and gcd
The special elements: 1 and 6

Isomorphic to:

Arithmetic / Subsets

An Amazing Theorem

Representation Theorem (Stone 1936):
Every finite Boolean algebra is isomorphic to a Boolean
algebra of subsets of some finite set S.

Two Boolean Algebras with m elements are isomorphic

Every Boolean algebra has 2k elements

Provides intuition beyond the axioms:
We can \'naturally\' reason about results in Boolean algebra

Algebra of Subsets

Venn Diagram
John Venn
1834 - 1923

 

graphical representation
of all possible subsets

Correcting Errors

Three erasures are not correctable...

Is one error correctable?

The First nontrivial Code

Richard Hamming
1915-1988

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