1. Chapter 4 Computation
Chapter 4: Computation

2. Topics ahead Computation in general Hilbert’s Program: Is mathematics
complete,
consistent and
decidable? (Entscheidungsproblem)
Answers
Goedel’s theorem
Turing’s machine
Chapter 4: Computation

3. The Universe is a computer that computes its own future in real time
Computation
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

4. Quantum mechanics Number theory Hilbert space Hilbert’s Program
David Hilbert,
Quantum mechanics
Number theory
Chapter 4: Computation
Hilbert space
Hilbert’s Program

5. Hilbert believed nature was solvable.
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

6. Hilbert’s Program I/III
Completeness Can every mathematical statement be proved or disproved from a finite set of axioms?
Chapter 4: Computation

7. What are axioms
Axioms Are a priori truths, statements which we assert to be true at the beginning. Euclid’s geometry has five.
Chapter 4: Computation

8. Hilbert’s Program II/III
Consistency Can only true statements be proved?
Chapter 4: Computation

9. Hilbert’s Program III/III
Decidability Is there an algorithm or procedure what can determine if any proposition is true in a finite number of steps?
Chapter 4: Computation

10. Kurt Godel,
Chapter 4: Computation

11. Godel’s Theorem
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

12. The Go-To Book
Chapter 4: Computation

13. Georg Cantor,
Chapter 4: Computation

14. Cantor’s Diagonal Argument
Godel mapped the integers one-to-one onto the set of true statements. 𝟏𝟐𝟑𝟗𝟖𝟒𝟕𝟓𝟔𝟎𝟑𝟐𝟗𝟔𝟖𝟑𝟕𝟒𝟓… 𝟎𝟗𝟑𝟖𝟔𝟕𝟑𝟏𝟐𝟑𝟖𝟒𝟗𝟔𝟎𝟒𝟑𝟕𝟐… 𝟖𝟒𝟓𝟔𝟎𝟗𝟏𝟐𝟑𝟗𝟒𝟕𝟓𝟑𝟔𝟐𝟖𝟑𝟎… Construct a new statement by adding 1 to the appropriate digit 𝟐𝟎𝟔…………………………..
Chapter 4: Computation

15. Godel proves otherwise Consistency Still good (essential) Decidability
Hilbert’s Program
Completeness
Godel proves otherwise
Consistency
Still good (essential)
Decidability
Waiting for Turing…
Chapter 4: Computation

16. Chapter 4: Computation

17. Many Contributions to Computing
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

18. The Bendix G-15
Chapter 4: Computation

19. A Turing machine
tape
reader
Chapter 4: Computation
rules and
state

20. Turing’s Definite Procedure
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

21. The Halting Problem
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

22. The Halting Problem (cont.)
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

23. Turing showed…
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

24. Turing’s accomplishments
Defined the “definite procedure” of Hilbert’s Program.
Turing machine laid the foundation for the development of digital computers.
Showed that there are limits to what can be computed.
Chapter 4: Computation

25. Codas
Godel
Turing
Chapter 4: Computation