Automated Conjectures

1
Automated Conjectures

• Graffitis Conjectures in Mathematics, Chemistry,
  and Education

2
Philosophical Considerations

• Penroses Shadows of the Mind

3
Penroses Argument

• Computer programs cannot have human mathematical
  intuitions.
  
• For a computer, a belief must be synonymous
  with a proof.
  
• Inconsistency with Gödels incompleteness
  Theorem
  

4
Putnams Argument

• Mathematics is partially formal, partially
  empirical.
  
• In Quasi-empirical mathematics, proof plays a
  secondary role to observation.
  
• Penroses claim is an empirical proposition which
  can be tested.
  

5
Task of Automation

• Putnams test can be implemented by writing a
  conjecture-making program (sf)
  
• Conjectures without any doubt should be invented
  by computers (as in the Turing test)
  

6
Program vs. User Conjectures

• Attributing users conjectures to a program
• Is pointless and actually harmful because
• It denies the possibility of genuine
  conjecture-making programs

7
To Penroses Credit

• A daring, fairly clearly stated proposition and a
  claim of impossibility
  
• Proofs of impossibility are among the greatest
  achievements in mathematics
  
• Making the dispute about what computers can or
  cant do more specific (Most arguments in the
  AI Debate are controversial)
  
• An argument questioning exaggerated claims and
  against attributing human ideas to machines
  

8
Purpose of Automation

• To figure out what makes a good conjecture
• Penrose - Putnam version of the Turing test is a
  secondary consideration
  
• It is certainly not an issue of data mining,
  experimental, or computer assisted vs automated
  hypothetical mathematics

9
Intelligent Machinery and Mathematical Discovery

• Craig Larson, in GTN XLII (2002), reviews all
  previously published claims concerning
  conjecture-making programs
  
• In the follow-up paper, Larson describes an
  unknown effort of Hao Wang, a student and a
  historian of Gödel, a pioneer of ATP,
  philosopher, and logician working on decision
  problems.

10
The Earliest Program Wangs Attempt

• Hao Wang (1950s) first attempt at writing a
  conjecture making program (Larsons new paper)
  
• Unsuccessful program, but detailed description of
  the experiment and the only open admission of
  failure
  
• First person to state the problem Can a computer
  select an interesting conjecture
  

11
The Early Programs Lenats AM and Eurisco

• According to Lenat, AM rediscovered the Goldbach
  conjecture and the concept of primes
  
• Based on 250 heuristics
• Supposedly failed to make discoveries because of
  shortage of heuristics. The goal of Eurisco was
  to invent more heuristics.
  
• Received praise and criticism
• Inspired other discovery, or to be more precise,
  rediscovery programs

12

• Graffiti The Implementation

13
Format of Conjectures

• ? ?,
• where ? and ? are invariants or terms of
  invariants.
  
• Primarily graph theoretical conjectures, but most
  methods are domain-independent
  
• Some conjectures in elementary geometry and
  number theory
  

14
Heuristics for Removal of Trivially True
Conjectures

• IRIN and CNCL test for transitivity and
  cancellation
  
• BEAGLE tests similarity of ? and ?
• ECHO makes conjectures more specific
• DALMATIAN tests for contribution of new
  information

15
Beagle

• Rejects conjectures of the form ? ?,
• In which ? and ? contain similar invariants
• In general if the distance between trees
  represeting ? and ? is not
  too large

16
Example

• Dual degree of vertex v is the mean of degrees
  of neighbors of v
  
• average degree maximum degree
• maximum dual degree maximum degree
• average degree average dual
  degree
  
• maximum eigenvalue maximum dual degree
• ( Jim Shearer and independently Favaron, Maheo
• and Sacle, Written on the Wall, conj. 256)

17
Echo

• Rejects conjectures which can be generalized to a
  larger class of objects (background)
  
• Example The Euler characteristic formula is not
  interesting for fullerenes, because it is valid
  for all planar graphs

18
Dalmatian Heuristic

• Rejects conjectures which do not contribute new
  information
  
• Written jointly with DeLaVina in the early
  nineties

19
Early Versions of Graffiti

• Pre-Dalmation versions were plagued by trivially
  true or otherwise non-interesting conjectures,
  which had to be deleted by hand
  
• Some improvement with first heuristics IRIN and
  CNCL
  
• Situation was markedly improved with Beagle and
  Echo

20
DeLaVinas Graffiti.Pc

• Dalmatian version, initially developed only for
  educational purposes
  
• Recently made new interesting conjectures
• Rerun and reimplementation of early and inactive
  parts of Graffiti
  
• Almost completed Some History of Development of
  Graffiti (on her web page)

21
Pre-Dalmation Version
Some authors of papers inspired by conjectures of
the
eighties version

• Noga Alon
• Bela Bollobas
• Fan Chung
• Paul Erdös
• Daniel Kleitman

• Laszlo Lovász
• Janosz Pach
• Yuri Razborov
• Paul Seymour
• Joel Spencer

22
Example of Conjectures from this Period

• Average distance independence number
  (Fan Chung)
  
• Independence number residue
  (Favaron, Maheo, and Sacle)
  
• The number of positive eigenvalues the sum of
  positive eigenvalues
  (open, partially proved
  by myself in OCG 2)
• Chromatic number 1 rank (almost the same as
  an overlooked conjecture of Van Neufallen)

23
Rank Coloring Conjecture

• Thanks to application in communication
  complexity
  
• proposed by Lovász, the conjecture inspired work
  by
  
• Alon and Seymour
• Kotlov and Lovász
• Nissan and Widgersen
• Raz and Spieker
• Razborov

24
Educational Version (2000)

• Purpose to obtain conjectures which are simple
  and easy to prove or refute
  
• Students were asked to find simplest
  counterexample to conjectures and prove
  minimality
  

25
Graffiti in the Classroom

• First classes based on Graffiti
• Fall 2000, individual instruction with Jimmy
  Pritts an afficionado of Texas style . Later,
  two small graduate classes
  
• Ryan Pepper proved some cases of the Ramsey
  Theorem, before he knew the statement of the
  result.
  
• Classes taught based on Graffiti.PC (written by
  DeLaVina)
  
• 2001, DeLaVina, individual instruction with
  several students
  
• 2002, Brinkman, University of Bielefeld

26
Two Examples

• In every planar, connected, eulerian graph G,
• f r 1,
• where r is the number of repetitions in the Euler
  tour of G, and f is the number of faces of G.
  
• a ? n 1,
• where a is the independence, ? the clique
  number, and n the number of vertices.

27
Fully Automated Conjecture-making Programs

• Deterministic selection of counterexamples
• Counterexamples with which Graffiti is provided
  have a definite impact on its conjectures.
  
• A deterministic algorithm for selection of
  counterexamples eliminates the arbitrariness of
  the process.
  
• Result A deterministic algorithm for selection
  of conjectures

28
Fully Automated Conjecture-making Programs

• Let pi(n) p
• Graffiti makes conjectures about pi using graphs
  PRn with vertices 2..n, two be adjacent iff
  they are not relatively prime.

29
Dalmatian version of Graffiti

• runs in rounds, with each round terminating when
  the program believes (?) that it found a formula
  for pi(n) the number of primes not more than n
  . For example, the program may (as it once did)
  conjecture that
• pi(n) s,
• where s is the number of non-negative eigenvalues
  of PRn.

30
The Simplest Counterexample is n93

• And after the program is informed about it, it
  proceeds to the next round. The program obviously
  could figure this counterexample by itself,
  leading to a full automation of the process.
• Incidentally,
• pi s
• is true, and if one could show that if
• s pi
• is not too large, then this would imply the
  Riemann Hypothesis (Andrew Odlyzko).
  

31
Related Decision Problems Depending on
Invariants of PRn

• Will the program ever halt?
• Will the number of rounds be infinite?
• Is every integer a simplest counterexample to a
  round?
  
• (rounds usually involve several conjectures)
  
  

32
Interpretation of Conjectures

• Rounds usually involve several inequalities of
  the form ? ?i which are interpreted as
  quasi-equality
  
• For every object G (in the domain of
  consideration) there exists i such that
  
• ?(G) ?i(G)

33
Halting Problem

• If ? is NP-hard for a given class of objects K,
  and the remaining invariants are polynomially
  computable in K, the same or analogous questions
  are of interest.
• Concerning the halting problem, the program will
  run forever, yielding at least one false
  conjecture in each round (i.e., after the round
  terminates) unless
• P NP.

34
Fullerene and Benzenoid Version
35
Recent Conjectures - Pony Express

• Conjectures about Benzenoids
• 102 conjectures in a 10-hour run
• I could neither prove nor refute any out of hand
• 40 conjectures selected because of sorting
  patterns
  
• Described in Written on the Wall
  (WoW, conjectures 914-966) can be
  found on my web page

36
Sorting Patterns

• For a given conjecture,
• All objects in the database are sorted by the
  difference between the left and right side of the
  conjectured inequality
  
• In some cases, there are clear patterns objects
  with a given property appear at the beginning or
  end of the list
  

37
Sorting Patterns

• Sorting patterns about
• Stability (for fullerenes)
• Carcinogenicity (for benzenoids)

38
Stable Fullerenes Tend to be Good Expanders

• One of these conjectures suggested that stable
  fullerenes tend to be good expanders
  
• Supported by computations of Larson
• Conjecture itself proved by Dragan Stevanovic and
  Gilles Caporossi.
  

39
Stability - Expanding Hypothesis

• May be used to explain why fullerenes have no
  triangular nor quadrangular faces and actually
  even IPR hypothesis
  
• Other plausibility arguments are given in
• Toward Fully Automated Fragments of Graph
  Theory
  
• GTN, XLII

40
Stability and Independence Number of Fullerenes

• Another conjecture involving the independence
  number displayed a strong sorting pattern for
  stability
  
• 7 out of 8 classical examples of stable
  fullerenes obtained in bulk minimize their
  independence number (with respect to the number
  of atoms)
• The 8th stable fullerene is second on the list
  ranked by independence number
  

41
Stability of Fullerenes Benzenoids

• In a joint paper with Larson, it is shown that
  the independence number is a better predictor of
  stability than other criteria. (to appear in
  Chemical Physics Letters)
• One reason for developing the benzenoid version
  was to understand this phenomenon, which until
  recently was only a statistical pattern.

42
Stable Benzenoids also Minimize their
Independence Numbers

• The most stable benzenoids have perfect matchings
  (Kekule structures), and because for bipartite
  graphs,
  
• the independence number matching number n,
• it follows that like fullerenes they also tend to
  minimize the size of their maximum independent
  sets.
  

43
Benzenoid Stability-expanding Hypothesis

• Among benzenoids with perfect matchings, the most
  stable are those that have many Kekule
  structures, suggesting that, like fullerenes,
  they also tend to be good expanders.

44
Buckminsterfullerene