Automated Reasoning for Classifying Finite Algebras Simon Colton Computational Bioinformatics Laboratory Imperial College, London

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

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

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

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

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 abcd aabcd bbacd ccbad ddbca abcd aabcd bbdca ccbad dabcd

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

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

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

Example Discriminating Concept First one: –Idempotent element appearing twice on the diagonal

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)

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

Example Decision Tree

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

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

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)

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…

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

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

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.

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