Chapter 4 Computation Chapter 4: Computation

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

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

Quantum mechanics Number theory Hilbert space Hilbert’s Program David Hilbert, 1862-1943 Quantum mechanics Number theory Chapter 4: Computation Hilbert space Hilbert’s Program

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

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

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

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

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

Kurt Godel, 1906-1978 Chapter 4: Computation

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

The Go-To Book Chapter 4: Computation

Georg Cantor, 1845-1918 Chapter 4: Computation

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

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

Chapter 4: Computation

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

The Bendix G-15 Chapter 4: Computation

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

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

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

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

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

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

Codas Godel Turing Chapter 4: Computation