WHAT IS THE VALUE OF A COMPUTER PROOF?

By Sara Billey University of Washington August 8, 2014

Mathfest 2014

http://en.wikipedia.org/wiki/Roman_surface

Goals: • Research: Is there a computer assisted proof of that statement?

•  Many people study/develop theorem proving systems, proof

certificates, formal verification methods. These improve reliability and shorten human verification time. New reduction lemmas and classification theorems are invented to meet space and time constraints.

• Teaching/Learning: How can we add computer assisted proofs to the undergraduate curriculum?

•  Study computer assisted proofs as a method of proof. Collect the best examples of computer assisted proofs to teach students.

Outline 1.  Human proofs vs computer-assisted proofs. 2.  The first major computer proof: 4 Color Theorem 3.  Hales’ proof of Kepler’s conjecture 4.  Zeilberger’s proof of Conway’s Lost Cosmological Thm 5.  Classifying smooth Schubert varieties 6.  Philosophical Questions

Definition of “Proof”

Definition of “Proof” • Proof: An argument or evidence establishing the truth of a statement.

• From Bing: •  Definitions of proof (n) •  proof [ proof ] •  conclusive evidence: evidence or an argument that serves to establish a fact or

the truth of something •  test of something: a test or trial of something to establish whether it is true •  state of having been proved: the quality or condition of having been proved •  Synonyms: resistant, resilient, impervious, immune

Statement There exists an infinite number of prime numbers. True or False?

Classic Human Proof Thm: There exists an infinite number of prime numbers. Proof: Assume there exists only N distinct primes: 1<p1 = 2 < p2=3 < p3 < …< pN. Set K= p1 p2 p3 … pN + 1. K has a prime factorization say K= q1 q2 q3 …qM. Subtracting q1 q2 q3 …qM - p1 p2 p3 … pN =1 (*). If q1 = pj for some j then the left side of (*) is divisible by pj with no

remainder. But the right side of (*) is 1 so it is not divisible by pj >1 with no remainder. Contradiction!

Q.E.D.

Examples of Computer Proof Use symbolic algebra package like Maple, Mathematica or

Sage. • Simplify polynomials •  Trigonometric identities •  Integrals •  Finite sums Go to Maple Demo.

Examples of Computer Proof The WZ-Method: A perfect example of computer proof. Given a hypergeometric series, does it have a closed form? Example: Reference: “A = B” by Petkovšek, Wilf and Zeilberger

(1997) AK Peters or available online.

Examples of Computer Proof The Four Color Theorem

Statement •  “Every map of states/countries/counties etc can be

colored using 4 colors such that no two adjacent states are given the same color. “

•  True or False?

• Caveats: No two states touch at isolated points. Each state is connected.

A 4-coloring of the states

http://en.wikipedia.org/wiki/File:Map_of_USA_with_state_names.svg

Experiment Try constructing a map for yourself which requires 5 colors.

Statement “Every map of states/countries/counties etc can be

colored using 4 colors such that no two adjacent states are given the same color. “

History: •  1852: Conjectured to be true by Francis Guthrie

(cartographer or botanist). •  Francis Guthrie -> Fredrick Guthrie -> Augustus De Morgan

-> Arthur Cayley

History •  1852: Conjectured to be true by Francis Guthrie •  1878: Cayley published Guthrie’s conjecture. •  1879: Kempe published a proof. •  1880: Tait published a proof. •  1890: Heawood pointed out a flaw with Kempe’s proof! •  1891: Petersen pointed out a flaw with Tait’s proof! • …. Many proofs appear and get rejected. But much

progress was made along the way. The field of graph theory was born into mathematics.

•  1976 : Appel and Haken publish a highly controversial computer assisted proof. NY Times refuses to mention it.

Reformulation •  Instead of coloring maps, the problem was generalized to

coloring planar graphs.

• Replace each state on the map with a bold dot. • Connect the dots representing two states if and only if

they are adjacent on the map by path on the paper.

•  This construction gives vertices and edges of a graph.

Draw a path between every pair vertices representing adjacent states which Only passes through those two states. Each path is an edge of the graph.

Delete the map. What remains is a planar graph.

A graph G=(V,E) is a set of vertices V and a subset of all pairs of vertices E. G is planar if all the edges can be drawn in the plane without crossing each other.

Four Color Theorem “Every vertex in a planar graph can be assigned a color distinct from all of its neighbors using at most 4 colors.”

To date, there is no human only proof.

Five Color Theorem (Kempe 1879) “Every vertex in a planar graph can be assigned a color

distinct from all of its neighbors using at most 5 colors.” Lemma: Every minimal counterexample contains a vertex

with exactly 5 neighbors (called a 5-star). Lemma: Any graph containing a 5-star cannot be a minimal counterexample.

If every vertex had at least 6 neighbors then N=Number of vertices E = Number of edges 6N ≤ 2E ≤ 6N-3 since G is planar and N>3.

Five Color Theorem (Kempe 1879) Lemma: Every minimal counterexample contains a vertex

with exactly 5 neighbors (called a 5-star). Proof: If there is a vertex v in G with 4 or fewer neighbors,

then there exists a 5-coloring G-v and it extends to G.

Five Color Theorem Lemma: Any graph containing a 5-star cannot be a minimal counterexample. Proof: Say G is a minimal counterexample containing a

vertex v with 5 neighbors. Then G-v can be 5-colored. We want to show that this coloring can be used to obtain a 5-coloring of G.

Thus, G was not a counterexample. Contradiction.

Five Color Theorem Given a 5-coloring of G-v, extend to a 5-coloring of G:

Five Color Theorem Given a 5-coloring of G-v, extend to a 5-coloring of G:

If A and B are in different components of the orange/purple graph, flip colors on A’s component.

If A and B are in the same component of the orange/purple subgraph, flip colors on the blue/red subgraph inside the path connecting A and B. Because G is planar, new coloring

If the neighbors of v use at most 4 colors, then color v one of the unused colors.

A

B

A

B

Four Color Theorem “Every vertex in a planar graph can be assigned a color distinct from all of its neighbors using at most 4 colors.”

1976 : Appel and Haken publish a highly controversial computer assisted proof.

Good ideas in the Appel-Haken Proof • Outline: Assume G is a counterexample to the 4CT with a

minimal number of vertices. • Reducibility: AH give a finite list of 1478 configurations of

graphs. Each one of these cannot appear in G because if it did, they gave rules to replace G by a smaller graph that would still be planar and require more than 4 colors.

Good ideas in the Appel-Haken Proof • Unavoidability: Every minimal counterexample to the 4CT

must contain one of the configurations on the list.

• A configuration is a small neighborhood in a graph. AH prove they only need to look at the second neighbors of each vertex and they bound the number of neighbors in each configuration. There are only a finite number of such graphs.

Good ideas in the Appel-Haken Proof Outline: Assume G is a counterexample to the 4CT with a

minimal number of vertices. • Reducibility: AH give a finite list of 1478 configurations in

graphs which cannot appear in G. • Unavoidability: G must contain one of the configurations on

the list.

Question: What can you conclude about G?

Answer: there always exists a 4 coloring of any planar graph.

Good ideas in the Appel-Haken Proof • Question: How do we find a 4-coloring?

• AH gave an algorithm for finding a 4-coloring in G. Guaranteed to succeed if a 4-coloring exists. If N is the number of vertices, it runs in time proportional to N5.

Controversy over Computer Proof •  Imagine back to 1976 when the 4 Color Theorem was first

proved by Appel and Haken ..

PDP-8 Computer built around 1970

•  Assembly Language (from Wikipedia: http://en.wikipedia.org/wiki/PDP-8) •  ******************************************************************************* •  This complete PDP-8 assembly language program outputs "Hello, world!" to the teleprinter. •  *10 / Set current assembly origin to address 10, •  STPTR, STRNG-1 / An auto-increment register (one of eight at 10-17) •  •  *200 / Set current assembly origin to program text area •  HELLO, CLA CLL / Clear AC and Link again (needed when we loop back from tls) •  TAD I Z STPTR / Get next character, indirect via PRE-auto-increment address from the zero page •  SNA / Skip if non-zero (not end of string) •  HLT / Else halt on zero (end of string) •  TLS / Output the character in the AC to the teleprinter •  TSF / Skip if teleprinter ready for character •  JMP .-1 / Else jump back and try again •  JMP HELLO / Jump back for the next character •  STRNG, 310 / H •  345 / e •  354 / l •  354 / l •  357 / o •  254 / , •  240 / (space) •  367 / w •  357 / o •  362 / r •  354 / l •  344 / d •  241 / ! •  0 / End of string •  $HELLO /DEFAULT TERMINATOR

Controversy over Computer Proof Appel and Haken Proof (1976). • Human part of the proof is over 1000 pages long. • No one else has ever been able to verify it. • Many typos were found. • Computer portion of the proof is written in assembly

language and no one else has programmed it. •  1478 graphs had to be coded by hand. Question: Are you convinced they have a proof?

History 1996: “A New Proof of the Four Color Theorem” published by Robertson, Sanders, Seymour, and Thomas

based on the same outline. • Human part of the proof is about 20 pages long. • Computer portion of the proof was written in C. • Several other people have independently programmed it. • No graphs had to be input by hand. • Only 633 configurations used.

Question: Are you convinced they have a proof?

History 1996: “A New Proof of the Four Color Theorem” Published by Robertson, Sanders, Seymour, and Thomas

based on the same outline.

• Algorithm: RSST also give an algorithm to find a 4-coloring of a planar graph that takes about n2 seconds on a graph with n vertices.

•  2005: Georges Gonthier gave a formal proof verification of the 4CT. No human proof required!

Some of the 633 Configurations

Examples of Computer Proof Thomas Hales proof of Kepler’s conjecture.

Harriot Investigation (ca 1600) What is the best way to pack cannon balls in space so they

are as densely packed as possible? Thomas Harriot was the assistant to a ship captain. He

corresponded with Johannes Kepler.

Hexagonal Close Packing

http://upload.wikimedia.org/wikipedia/commons/8/8e/Close-packed_spheres.jpg

Kepler’s Conjecture What is the best way to pack cannon balls in space so they

are as close as possible? Conjecture: The portion of space filled by cannonballs in

the densest possible packing is given by the hexagonal close packing and has density

.

Proof of Kepler Conjecture •  1953: Toth showed that the problem could be reduced to a

finite check of about 5000 cases (decidable inequalities!).

•  1992: Thomas Hales began using linear programming to check these cases along with Samuel Ferguson.

•  1996 : Hales announced the proof was complete.

•  2005: Hales’ paper was published in Annals of Math after being reviewed by a committee of 12 referees who said they were 99% certain it was correct.

Honeycomb Theorem • Honeycomb conjecture (Varro 36 BC): Any partition of

the plane into equal area regions has perimeter at least that of the regular hexagonal honeycomb tiling.

•  THOMAS C. HALES, The honeycomb conjecture, Disc. Comp. Geom. 25 (2001) 1–22.

•  Finding an optimal partition of 3-dimensions into equal area regions is an unsolved problem.

Examples of Computer Proof Zeilberger’s proof of Conway’s Lost Cosmological Thm: Conway’s “Look and Say” sequence:

1 11 21

1211 111221

Question: If Ln is the length of the sequence on the ith

step, what is Ln+1/Ln as n approaches infinity?

Examples of Computer Proof Question: If Ln is the length of the sequence on the ith

step, what is Ln+1/Ln as n approaches infinity? Answer: Ln+1/Ln à 1.30357726903… Conway’s Constant Somehow the proof was lost until …. In 1997, Doron Zeilberger gave a computer assisted proof. His proof relies on a halting problem: from each atomic

word constructed so far, construct all ways of building further, until the empty set remains.

Examples of Computer Proof Classifying smooth Schubert varieties in generalized flag

manifolds for all Lie types.

http://en.wikipedia.org/wiki/Roman_surface

Inspiration

•  Theorem(Lakshmibai-Sandhya 1990).For a permutation w in Sn, the Schubert variety X(w) in SL(n)/B is nonsingular if and only if w avoids the patterns 3412 and 4231.

• Ex: X(654321) has no singularities. X(516243) has singularities.

http://en.wikipedia.org/wiki/Roman_surface

Inspiration

• G = semisimpleconnected complex Lie group • B = Borel subgroup of G • w = element of the Weyl group for G • Bw = orbit of a rep of w under left B multiplication.

•  Theorem (Billey-Postnikov 2005). The Schubert variety X(w)=Bw can be completely characterized by projections in at most 4 dimensions corresponding to the root systems of types A3, B3, C3, D4, G2.

Computer Assisted Proof • Proof outline: Reduce each Lie type to checking a finite

number of cases for projections up to dimension 8.

Type E8 required •  about 40 hours on 100 computers, • many reduction lemmas, •  only one chance to try it. Miraculously, the proof finished in time!

Inspiration Quote from original referee report rejecting the paper:

•  “These results are original (to the best of my knowledge) and quite interesting. They represent substantial improvements on earlier criteria. But their proof is not very enlightening.”

Philosophical Question What is the value of a computer proof? • We get a new result which we can build on!

• We learn one more method of using computers to prove theorems.

• Every computer proof with no known human proof contains a miracle! Those miracles are important lemmas which inspire further research.

Philosophical Question What is the value of a computer proof? • They have the potential to increase reliability.

• They can be easier for humans to read.

• They sharpen out computing skills.

• There are many practical applications of formal verification of computer programs and proofs in airplane design, medical equipment, parallel computing, etc.

A halting problem

Problem: Find all graph types corresponding with rank 5 starred strong tableaux under cloning.

Human part of the proof is 50 pages long. Computer part ran for 1 year on 8 computers, but it didn’t finish. See Billey-Assaf 2012.

Difficult Questions What if a computer program or hardware has a bug? What evidence of computer verification is sufficient for a proof?

How can we insure computer assisted proofs are “reproducible”?

Does a referee need to reprogram a computer assisted proof in order to recommend it?

Risk Assessment • There are issues to resolve about presentation and validation of computer proofs.

• There are pitfalls with human proofs too.

Summary Computers can be very helpful proving theorems about… • Algebraic identities. • Geometric density inequalities. •  Finite calculations. • Halting problems.

And what else? Please join the conversation! It is time to embrace computer proofs as a powerful

technique and incorporate them in our math classes.

Lots more • Origami: Can you fold this? See “Geometric Folding

Algorithms” by Demaine and O’Rourke. (2007)

• Automatic Theorem Checking: Is this human proof correct? See “How to Write a Proof” by Leslie Lamport in American Mathematical Monthly 102, 7 (1993) . And the follow up “How to Write a 21st Century Proof” (2012).

• Game Theory: Does this game have a winning strategy?

See history of Connect Four in Wikipedia.

Upcoming Event

“Computational Aspects of Algebra, Geometry, and Combinatorics”

Today 1:00pm-5:30pm

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