Chapter 4: Computation
Chapter 4
Computation
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• Computation in general
• Hilbert’s Program: Is mathematics
• complete,
• consistent and
• decidable? (Entscheidungsproblem)
• Answers
• Goedel’s theorem
• Turing’s machine
Chapter 4: Computation
Topics ahead
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•1 + 1 → 2
•The Universe is a
computer that computes
its own future in real time
•We’ll look at a middle
ground.
Chapter 4: Computation
Computation
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Chapter 4: Computation
David Hilbert, 1862-1943
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• In contrast to Emil du Bois-Reymond, a
German physiologist,
• Ignoramus et ignorabimus.
• We do not know, we shall not know.
• Hilbert’s famous quote, and epitaph, was
• Wir müssen wissen. Wir werden wissenm
• We must know. We will know.
Chapter 4: Computation
Hilbert believed nature was
solvable.
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Completeness
Can every mathematical
statement be proved or
disproved from a finite set
of axioms?
Chapter 4: Computation
Hilbert’s Program I/III
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Axioms
Are a priori truths,
statements which we assert
to be true at the beginning.
Euclid’s geometry has five.
Chapter 4: Computation
What are axioms
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Consistency
Can only true statements
be proved?
Chapter 4: Computation
Hilbert’s Program II/III
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Decidability
Is there an algorithm or
procedure what can
determine if any
proposition is true in a
finite number of steps?
Chapter 4: Computation
Hilbert’s Program III/III
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Chapter 4: Computation
Kurt Godel, 1906-1978
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• If arithmetic is consistent then there are
true statements about arithmetic which
cannot be proved.
• Mitchell’s example:
• This statement is not provable
• If false, then a false statement can be
proved (really bad news).
• If true, then a true statement cannot be
proved.
Chapter 4: Computation
Godel’s Theorem
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Chapter 4: Computation
The Go-To Book
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Chapter 4: Computation
Georg Cantor, 1845-1918
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Godel mapped the integers one-to-one onto the set
of true statements.
𝟏𝟐𝟑𝟗𝟖𝟒𝟕𝟓𝟔𝟎𝟑𝟐𝟗𝟔𝟖𝟑𝟕𝟒𝟓 …
𝟎𝟗𝟑𝟖𝟔𝟕𝟑𝟏𝟐𝟑𝟖𝟒𝟗𝟔𝟎𝟒𝟑𝟕𝟐 …
𝟖𝟒𝟓𝟔𝟎𝟗𝟏𝟐𝟑𝟗𝟒𝟕𝟓𝟑𝟔𝟐𝟖𝟑𝟎 …
Chapter 4: Computation
Cantor’s Diagonal Argument
Construct a new statement by adding 1 to the
appropriate digit
𝟐𝟎𝟔 … … … … … … … … … … . .
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• Completeness
• Godel proves otherwise
• Consistency
• Still good (essential)
• Decidability
• Waiting for Turing…
Chapter 4: Computation
Hilbert’s Program
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Chapter 4: Computation
Theoretical and practical
contributions to the development of
computer science and computing
machinery.
• Turing test and artificial intelligence
• LU decomposition
• mathematical biology
• design of realizable computers
Chapter 4: Computation
Many Contributions to Computing
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Chapter 4: Computation
The Bendix G-15
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tape
reader
rules and
state
Chapter 4: Computation
A Turing machine
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The Turing machine filled
Hilbert’s requirement for a
“definite procedure” that
could, he hoped, determine
whether any mathematical
statement was true or false.
Chapter 4: Computation
Turing’s Definite Procedure
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Let M be a Turing machine and I its
input.
Assertion: M running on I will reach a
halt state after a finite number of
steps.
Consequence: There exists a Turing
machine, H, which can examine M and
I and (in a finite number of steps)
determine if M would halt on I.
Chapter 4: Computation
The Halting Problem
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If we can design such an H it would
have the property that H(M,I) would
produce either “yes” or “no” in finite
time for any M and I.
Example of a non-halting machine:
For any input, move one cell to the
right.
Does H exist?
Chapter 4: Computation
The Halting Problem (cont.)
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If we assume H exists we can find a logical
contradiction.
Therefore no such H exists.
Therefore there is no definite procedure
for solving the Halting Problem.
Therefore there is no definite procedure
for proving any mathematical statement
true or false in a finite number of steps.
Chapter 4: Computation
Turing showed…
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1. Defined the “definite procedure”
of Hilbert’s Program.
2. Turing machine laid the foundation
for the development of digital
computers.
3. Showed that there are limits to
what can be computed.
Chapter 4: Computation
Turing’s accomplishments
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•Godel
•Turing
Chapter 4: Computation
Codas
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