Gödel’s Incompleteness Theoremand the Birth of the Computer

Christos H. Papadimitriou

UC Berkeley

CS294, Lecture 1

Outline

• The Foundational Crisis in Math (1900 – 31)

• How it Led to the Computer (1931 – 46)

• And to P vs NP (1946 – 72)

CS294, Lecture 1

The prehistory of computation

Pascal’sCalculator

1650

Babbage & Ada, 1850 the analytical engine

Jacquard’s looms1805

CS294, Lecture 1

Trouble in Math

Non-euclideangeometries

Cantor, 1880: sets and infinity

∞

CS294, Lecture 1

Logic!

• Boole’s logic is inadequate (how do you say “for all integers x”?)

• Frege introduces First-Order Logic

• And writes a two-volume opus on the foundations of Arithmetic (~1890 – 1900)

CS294, Lecture 1

The quest for foundations

Hilbert, 1900:“We must know,

we can knowwe shall know!”

CS294, Lecture 1

The two quests

An axiomaticsystem that comprises

all of Mathematics

A machinethat finds

a proof forevery theorem

CS294, Lecture 1

The Paradox and the Book

• Russell discovers in 1901 the paradox about “the set of all sets that don’t contain themselves”

• And writes with Whitehead the Principia, trying to restore set theory and logic (1902-1911)

• Result is highly unsatisfactory (but inspires what comes next)

CS294, Lecture 1

20 years later: the disaster

Gödel 1931The Incompleteness Theorem“sometimes, we cannot know”Theorems that have no proof

CS294, Lecture 1

Recall the two quests

Find an axiomaticsystem that comprises

all of Mathematics

?

Find a machinethat finds aproof for

every theorem

CS294, Lecture 1

Also impossible?

but what is a machine?

CS294, Lecture 1

The mathematical machines (1934 – 37)

Post

Kleene

Church

Turing

CS294, Lecture 1

Universal Turing machine

Powerful and crucial ideawhich anticipates software

…and radical too:dedicated machines

were favored at the time

CS294, Lecture 1

“If it should turn out that the basic logicsof a machine designed for the numericalsolution of differential equations coincidewith the logics of a machine intended tomake bills for a department store, I wouldregard this as the most amazing coincidencethat I have ever encountered”

Howard Aiken, 1939

CS294, Lecture 1

In a world without Turing…

WELCOME TO THE COMPUTER STORE!

First Floor: Web browsers, e-mailers

Second Floor: Database engines, Word processors

Third Floor: Accounting computers, Business machines

Basement: Game engines, Video and Music computers

SPECIAL TODAY: All number crunchers 40% off!

CS294, Lecture 1

And finally…

von Neumann 1946EDVAC and report

CS294, Lecture 1

Johnny come lately

• von Neumann and the Incompleteness Theorem• “Turing has done good work on the theories of almost

periodic functions and of continuous groups” (1939)• Zuse (1936 – 44) , Turing (1941 – 52), Atanasoff/Berry

(1937 – 42), Aiken (1939 – 45), etc.• The meeting at the Aberdeen, MD train station• The “logicians” vs the “engineers” at UPenn• Eckert, Mauchly, Goldstine, and the First Draft

CS294, Lecture 1

Madness in their method?the painful human story

G. CantorD. Hilbert

K. Gödel

E. Post

A. M. TuringJ. Von Neumann

G. Frege

CS294, Lecture 1

Theory of Computation since Turing:Efficient algorithms

• Some problems can be solved in polynomial time (n, n log n, n2, n3, etc.)

• Others, like the traveling salesman problem and Boolean satisfiability, apparently cannot (because they involve exponential search)

• Important dichotomy (von Neumann 1952, Edmonds 1965, Cobham 1965, others)

CS294, Lecture 1

Polynomial algorithms deliver Moore’s Law to the world

• A 2n algorithm for SAT, run for 1 hour:

1956 1966 1976 1986 1996 2006

n = 15 n = 23 n = 31 n = 38 n = 45 n = 53

An n or n log n algorithm n3 n7

× 100 every decade × 5 × 2

CS294, Lecture 1

NP-completenessCook, Karp, Levin (1971 – 73)

• Efficiently solvable problems: P

• Exponential search: NP

• Many common problems capture the full power of exponential search: NP-complete

• Arguably the most influential concept to come out of Computer Science

• Is P = NP? Fundamental open question

CS294, Lecture 1

Intellectual debt to Gödel/Turing?

• Negative results are an important intellectual tradition in Computer Science (and Logic too)

• The Incompleteness Theorem and Turing’s halting problem are the archetypical negative results

• The Gödel letter (discovered 1992)

CS294, Lecture 1

CS294, Lecture 1

CS294, Lecture 1

Recall: Hilbert’s Quest

axioms+

conjecture

always answers “yes/no”

Turing’s halting problem

CS294, Lecture 1

Gödel’s revision

axioms+

conjecture

if there is a proof of length n it finds it in time k n

(this is trivial,just try all proofs)

CS294, Lecture 1

Hilbert’s last stand

• Gödel asked von Neumann in the 1956 letter:

“Can this be done in time n ?

n 2 ?

n c ?”

• This would still mechanize Mathematics…

CS294, Lecture 1

Surprise!

• Gödel’s question is equivalent to

“P = NP”• He seems to be optimistic about it…

CS294, Lecture 1

So…

• Hilbert’s foundations quest and the Incompleteness Theorem have started an intellectual Rube Goldberg that eventually led to the computer

• Some of the most important concepts in today’s Computer Science, including P vs NP, owe a debt to that tradition

CS294, Lecture 1

And this is the story we tell in…

CS294, Lecture 1

LOGICOMIX: A graphic novel of reason, madness and the birth of the computer

By Apostolos Doxiadis and Christos PapadimitriouArt: Alecos Papadatos and Annie Di Donna

Bloomsbury, 2008