Automated Reasoning for Classifying Finite Algebras

1. Automated Reasoning forClassifying Finite Algebras Simon Colton Computational Bioinformatics Laboratory Imperial College, London

2. Joint work with • Roy McCasland (Edinburgh) • Mathematical insights • Andreas Meier (Saarbrucken) • Theorem proving expertise • Volker Sorge (Birmingham) • ATP and CAS expertise • Truly collaborative • i.e., I may not be able to answer some questions 

3. Classification of Finite Algebras • Major driving force in mathematics • E.g., Kronecker’s 1870 classification of Abelian groups • Also, 1980 classification of finite simple groups • For loops and quasigroups, etc. • Large numbers of isomorphism/isotopy classes • E.g., 109 loops of size, 1441 quasigroups of size 5 • Computational approaches have been used • In a quantitative, rather than a qualitative way • E.g., existence of QGX quasigroups of certain sizes

4. The Task We Set Ourselves • Write a system which can… • Be given only a particular size and an algebraic specification (in terms of a set of axioms) • And produce a fully verified classification theorem • Which can be used to classify algebras of that size • Up to isomorphism • As a simple example • Given the axioms of group theory and the size 6 • Our system proves that groups of size six are either Abelian or non-Abelian up to isomorphism

5. The Tools We Used • Automated Reasoning: • Spass theorem prover • MACE-4 model generator • Omega proof planning system • Machine Learning: • HR automated theory formation system • C4.5 decision tree learner • Computer Algebra • Gap system

6. Why Machine Learning? • Why are these two algebras non-isomorphic? • Did you use deduction (only) to show this? • My problem with the term “automated reasoning” • Doesn’t include inductive reasoning

7. The HR System • Starts with minimal information • E.g., dividing two numbers, ring theory axioms • Produces a rich theory containing: • Examples, concepts, conjectures, proofs • 15 Generic production rules form concepts • 20+ Measures of interestingness • Drive a best-first search • Conjecture making performed empirically • Theorem proving/disproving by third party software • Usually Otter and MACE

8. Approach One • Use MACE (+isofilter) to produce: • A single example of each isomorphism class • Use HR to form a theory: • With a concept describing each class uniquely • Use Spass to: • Verify MACE’s results • That each example satisfies axioms • Every algebra is isomorphic to one of the classes • Verify HR’s results • That each example has the concept’s property • Prove that each concept is a classifier • Discriminant and isomorphism-class theorems are true

9. Approaches Two and Three • Same as approach 1 • But HR allowed to stop before it has found a classifying concept for each class • In many cases, this is necessary • Approach 2: use Prolog to combine concepts • Approach 3: use C4.5 to learn a decision tree • Problem: sometimes sub-optimal trees produced

10. Example Discriminating Concept • First one: • Idempotent element appearing twice on the diagonal

11. Difficulties and Lessons Learned • Difficulty 1: • MACE intermediate files > 4GB • Solution: don’t require generation of all isomorphism classes • Difficulty 1: • HR has trouble with more than 6 or 7 examples • Solution: only use HR to discriminate a few examples (pairs) • Difficulty 2: • Spass has trouble with sizes greater than 6 or 7 • (Partial) solution: use CAS to describe problem in terms of generators and relations (decrease potential mappings)

12. Approach Four (Bootstrapping) • Want fully automated decision tree process • See IJCAR’04 paper for full algorithm description • Step 1: MACE produces a non-isomorphic pair • Step 2: HR discriminates the pair • Step 3: Spass proves that some discriminants are actually classifiers • Step 4: For non-classifiers, use MACE to produce a non-iso pair which both have the property • If successful, go to step 2 • If not, use Spass to prove it’s a dead-end

13. Example Decision Tree

14. Nice Result in Group Theory(Produced by Approach 1) Class 1: -(exists b (-(inv(b)=b))) Class 2: exists b c (-(inv(b)=b) & c*c=b) Class 3: -(exists b (inv(b)=b & -(exists c d (commutator(d,c)=b))) Class 4: exists b c d (b*c=d & -(c*b=d) & inv(d)=d) Class 5: none of the above

15. In English… Groups of order 8 can be classified according to the self-inverse (inv(x)=x) elements they contain: they will either have: (i) all self inverse elements (ii) an element which squares to give a non-self inverse element (iii) no self-inverse elements which aren't also commutators (iv) a self inverse element which can be expressed as the product of two non-commutative elements (v) none of these properties

16. Classification Theorems Produced Using Approach 4 • Generated classifying theorems for • Groups of size 4 (#2), 6 (#2), 8 (#5) • Loops of size 4 (#2), 5 (#6), 6 (#109) • Quasigroups • Of size 3 (#5), 4 (#35), 5 (#1441) • Monoids of size 3 (#7) • QG4-quasigroups of size 5 (#4) • QG5-quasigroups of size 7 (#3)

17. Conclusions • Computers can help in classification tasks • In a qualitative, as well as quantitative way • Can produce fully verified classification theorems • Cannot be achieved by deduction alone • Our approach requires deduction (ATP), induction (ML), and symbolic manipulation (CAS) • Long live the Calculemus project!! • Application to model generation (please ask) • Results are not conclusive yet…

18. Future Work #1 • Improve the current system • By trying out different tools/methods • SEM, FINDER for model generation • SAT solvers for the ATP tasks • Progol (ILP) for machine learning tasks • First test: 68% success (HR was 96%) • Look at different domains • Possibly domains associated with Zariski spaces • Also look at isotopy as well as isomorphism

19. Future Work #2 • Produce general classification theorems • Analysis of trees produced so far • Important concepts, etc. • Generalise results over sizes • One possibility: • Use smaller size decision trees as seeds for the larger trees • Determine families and parameterisations of the family members • Use the counting abilities of HR • May be difficult for first order provers

20. Future Work #3 • Look at sub-algebra structures/mappings • E.g., centre of a group forms a subgroup • Look for more specific results than this • Look for algebras embedded within others • HR has abilities to do this • May be a tough problem for theorem proving • Build up an “Atlas” for loops & quasigroups • Start building more constructive classification results • E.g., using cross products, etc.

21. Future Work #4 • Find mathematical applications of this • Any help……..?