Automated Theory Formation
for Tutoring Tasks
in Pure Mathematics
Simon Colton, Roy McCasland,
Alan Bundy, Toby Walsh
Tools for Maths Teachers
 Setting exercises is important in
mathematics education
 Automated tools available for
 High
school mathematics and below
 No tool available for University level
 Use ATF via the HR program
 As
an aid to setting exercises
 Example in group theory
The HR Program
 Automated theory formation
 Concepts, conjectures, proofs, counters
 Concept formation
 10 production rules
 Conjecture making
 Find patterns empirically
 Settling conjectures
 Using Otter and MACE (and others)
 See CADE system description
Group theory exercises
 In most text books/courses:
 Determining
subgroups
 Negative results useful in learning process
 Example: centre of a group
 Elements
which commute with all others
 Student has to show:
Closure, associativity, identity, inverse
 Or: non-empty and a & b in Z  a * b-1 in Z

Approach using HR
 Use HR’s concept formation
 Find different types of element
 Flag those forming a subgroup empirically
 Use Otter
 To show that the subset forms a subgroup
 Use MACE
 To find a counterexample group
 Use (human) teacher
 Interpret results as exercises
Improvements to HR #1
Embed Algebra PR
 Designed to be domain independent
 Find embedded algebraic structures
 In algebras, graphs, integers
 Any arity four concept
 Triples of subobjects possibly form algebra
 User sets algebras to look for
 HR abstracts subobjects into Cayley table
 MACE used to check axioms
 HR checks isomorphism with previous
 MACE not asked to search (efficient)
Improvements to HR #2
“Reactive” search
 Heuristic search  Best first search
 BFS often better after delay
 Some PRs should be used sparingly
 E.g., disjunction, instantiation
 Other PRs should be used when poss
 E.g., embed algebra rule
 HR’s reactive search: after each step:
 Java code fragment read by HR
 Different to the paper
Experimental Setup
 Groups up to order 8 (14 groups)
 Reaction to new element type
 Force use of embed-algebra
 Flag concepts forming non-trivial subgroups
 10,000 theory formation step
 No proving or counterexample finding (fast)
 Any promising subgroup types
 Further investigate with Otter and MACE
 Pentium 4 (2.0 Ghz) under Windows XP
Suggestions for Using Results
 If subgroup property is proved
 Ask
student to prove this
 If known counterexamples
 Ask
student to determine smallest
 Ask student to identify classes of group
 Ask student to characterise all groups
  Caveat 
 These
problems may be too difficult
Results
 301 seconds to finish search
 330 concepts
 17 element types found
 10
produced subgroups empirically
 8 were non-trivial
 Look now at two element types in detail
Concept g93
 [a,b] : all c (exists d (d*c=b & c*d=b))
 Actually defines centre of group (paper)
 HR often comes up with usual definition
 Empirically true, can we prove it?
 Otter employed for three tests:
 Closure of: identity, inverse, multiplication
 0.2 secs, 25 secs, 55 secs
 Obvious interpretation for tutorial
 Nice to see historically interesting result
Concept g43
 [a,b] : exists c (c*c = b)
 Diagonal
elements on Cayley table
 Groups up to order 8: forms a subgroup
 Fairly certain not in general case
 Suggested
trying to disprove first
 Passed MACE: axioms, g43 definition
 And
multiplicative closure property
 143 seconds later, MACE produced:
Smallest(?) “Bad” group for
concept g43
* |
0
1
2
3
4
5
6
7
8
9 10 11
--+-----------------------------------0 | 0 1 2 3 4 5 6 7 8 9 10 11
1 | 1 0 3 2 5 4 7 6 9 8 11 10
2 | 2 3 4 5 0 1 8 9 10 11 6 7
3 | 3 2 5 4 1 0 9 8 11 10 7 6
4 | 4 5 0 1 2 3 10 11 6 7 8 9
5 | 5 4 1 0 3 2 11 10 7 6 9 8
6 | 6 7 10 11 8 9 1 0 5 4 3 2
7 | 7 6 11 10 9 8 0 1 4 5 2 3
8 | 8 9 6 7 10 11 3 2 1 0 5 4
9 | 9 8 7 6 11 10 2 3 0 1 4 5
10 | 10 11 8 9 6 7 5 4 3 2 1 0
11 | 11 10 9 8 7 6 4 5 2 3 0 1
Possible Use of Concept g43
 Could ask student to find MACE’s c-ex.
 Alternatively:
 Ask
for a subclass of groups with property
 (Example Abelian groups)
 Ask for a characterisation (honors)
 See Appendix A of paper
 For
10 tutorial questions which arose
Other ways to use results
 Some subsets of elements are contained
 In other subsets of elements
 Both sets identified by HR (and the conjecture)
 Intuition of students
 Mathematicians use minimal hypotheses
 HR sometimes produces conjectures:


With non-minimal and/or convoluted hypotheses
Useful for students to prove theorems with nonminimal or convoluted hypotheses
b*c=d & d*b=c & d*d=b  inv(c)=d
Conclusions and Future Work
 Performed an initial feasibility study


HR, Otter, MACE help with maths tutorials
Example in group theory

Novel questions arose from non-human results
 Possible to use HR semi-automatically
 Maybe
HR used as a tool in future