Automated Theory Formation for Tutoring Tasks in Pure

Automated Theory Formation for Tutoring Tasks in Pure Mathematics Simon Colton, Roy Mc. Casland, Alan Bundy, Toby Walsh

Tools for Maths Teachers n Setting exercises is important in mathematics education n Automated tools available for n High school mathematics and below n No tool available for University level n Use ATF via the HR program n As an aid to setting exercises n Example in group theory

The HR Program n Automated theory formation n Concepts, conjectures, proofs, counters n Concept formation n 10 production rules n Conjecture making n Find patterns empirically n Settling conjectures n Using Otter and MACE (and others) n See CADE system description

Group theory exercises n In most text books/courses: n Determining subgroups n Negative results useful in learning process n Example: centre of a group n Elements which commute with all others n Student has to show: n Closure, associativity, identity, inverse n Or: non-empty and a & b in Z a * b-1 in Z

Approach using HR n Use HR’s concept formation n Find different types of element n Flag those forming a subgroup empirically n Use Otter n To show that the subset forms a subgroup n Use MACE n To find a counterexample group n Use (human) teacher n Interpret results as exercises

Improvements to HR #1 Embed Algebra PR n Designed to be domain independent n Find embedded algebraic structures n In algebras, graphs, integers n Any arity four concept n Triples of subobjects possibly form algebra n User sets algebras to look for n HR abstracts subobjects into Cayley table n MACE used to check axioms n HR checks isomorphism with previous n MACE not asked to search (efficient)

Improvements to HR #2 “Reactive” search n Heuristic search Best first search n BFS often better after delay n Some PRs should be used sparingly n E. g. , disjunction, instantiation n Other PRs should be used when poss n E. g. , embed algebra rule n HR’s reactive search: after each step: n Java code fragment read by HR n Different to the paper

Experimental Setup n Groups up to order 8 (14 groups) n Reaction to new element type n Force use of embed-algebra n Flag concepts forming non-trivial subgroups n 10, 000 theory formation step n No proving or counterexample finding (fast) n Any promising subgroup types n Further investigate with Otter and MACE n Pentium 4 (2. 0 Ghz) under Windows XP

Suggestions for Using Results n If subgroup property is proved n Ask student to prove this n If known counterexamples n Ask student to determine smallest n Ask student to identify classes of group n Ask student to characterise all groups n Caveat n These problems may be too difficult

Results n 301 seconds to finish search n 330 concepts n 17 element types found n 10 produced subgroups empirically n 8 were non-trivial n Look now at two element types in detail

Concept g 93 n [a, b] : all c (exists d (d*c=b & c*d=b)) n Actually defines centre of group (paper) n HR often comes up with usual definition n Empirically true, can we prove it? n Otter employed for three tests: n Closure of: identity, inverse, multiplication n 0. 2 secs, 25 secs, 55 secs n Obvious interpretation for tutorial n Nice to see historically interesting result

Concept g 43 n [a, b] : exists c (c*c = b) n Diagonal elements on Cayley table n Groups up to order 8: forms a subgroup n Fairly certain not in general case n Suggested trying to disprove first n Passed MACE: axioms, g 43 definition n And multiplicative closure property n 143 seconds later, MACE produced:

Smallest(? ) “Bad” group for concept g 43 * | 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 g 43 n Could ask student to find MACE’s c-ex. n Alternatively: n Ask for a subclass of groups with property n (Example Abelian groups) n Ask for a characterisation (honors) n See Appendix A of paper n For 10 tutorial questions which arose

Other ways to use results n Some subsets of elements are contained n In other subsets of elements n Both sets identified by HR (and the conjecture) n Intuition of students n Mathematicians use minimal hypotheses n HR sometimes produces conjectures: n n 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 n Performed an initial feasibility study HR, Otter, MACE help with maths tutorials n Example in group theory n n Novel questions arose from non-human results n Possible to use HR semi-automatically n Maybe HR used as a tool in future