Who cares about Euclidean geometry? Miroslav Olk Euclidean - - PowerPoint PPT Presentation

▶
who cares about euclidean geometry
SMART_READER_LITE
LIVE PREVIEW

Who cares about Euclidean geometry? Miroslav Olk Euclidean - - PowerPoint PPT Presentation

Apr 07, 2024 145 likes •344 views

Who cares about Euclidean geometry? Miroslav Olk Euclidean geometry Points, lines and circles in the plane Nice but Without applications Algorithmically decidable AITP Automated theorem proving Machine learning Me

slide-1
SLIDE 1

Who cares about Euclidean geometry?

Miroslav Olšák

slide-2
SLIDE 2

Euclidean geometry

  • Points, lines and circles in the plane
  • Nice but…

– Without applications – Algorithmically decidable

slide-3
SLIDE 3

AITP

  • Automated theorem proving
  • Machine learning
slide-4
SLIDE 4

Me

slide-5
SLIDE 5

A computer program exceeded world best mathematicians

B r e a k i n g n e w s !

slide-6
SLIDE 6

Mathematics

  • Abstraction
  • Reasoning
  • Problem solving
slide-7
SLIDE 7

Abstraction

A conceptual process where general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other

  • methods. (Wikipedia)
slide-8
SLIDE 8

Abstraction – Logician’s approach

  • Replacing constants with variables
  • Construction of lambda function (HOL)
  • ...
slide-9
SLIDE 9

Machine learning approach

These are dogs

Now I have an idea what a "dog" means

M L m a g i c

slide-10
SLIDE 10
slide-11
SLIDE 11

Elementary school

  • Arithmetic
  • Word problems
  • Euclidean geometry

– One abstraction level – Teaches reasoning

slide-12
SLIDE 12

Problem solving – benchmark

slide-13
SLIDE 13

International Mathematical Olympiad (IMO)

  • Prestigious competition for high school students
  • 4 “domains”

– Algebra – Number theory – Combinatorics – Euclidean geometry

  • National rounds

– Plenty of problems of various difficulty

  • Thousands for geometry
slide-14
SLIDE 14

One more point

  • People at IMO are really clever
slide-15
SLIDE 15

One more point

  • People at IMO are really clever
  • The plan

1) Solve geometry 2) Get AI generated problems to competitions 3) Get attention of talented people 4) Invite them into a project of solving entire IMO

Let's solve the problem solving

slide-16
SLIDE 16

Euclidean geometry

  • Problems from elementary school to IMO
  • Relatively simple → suitable for experiments
  • Concrete → suitable for image processing
  • We can attract IMO people
  • Without applications?

– Not the ultimate goal

  • Algorithmically decidable?

– Ignorable

slide-17
SLIDE 17

Suggestions

  • Data collection

– Prasolov, Chan, 106, 107, AOPS, … – ...

  • Reinforcement learning

– Guide by image / structure – Problems to prove – Constructional problems – ...

  • ...
slide-18
SLIDE 18

What do you think?