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Automated Reasoning for Classifying Finite AlgebrasPowerPoint Presentation

Automated Reasoning for Classifying Finite Algebras

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### Automated Reasoning forClassifying 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

- Which can be used to classify algebras of that size
- 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

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

- Verify MACE’s results

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

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

- One possibility:
- Determine families and parameterisations of the family members
- Use the counting abilities of HR
- May be difficult for first order provers

- Use the counting abilities of HR

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……..?

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