Hard limits on step-wise explanation sequences (SWES)
by contradiction
slides.com/jod/explanation_limits
Central hypothesis
A maximally nested SWES contains a clausal proof.
Consequence
There exist problems with short explanations that have an exponential (nested) SWES.
Intuition
Stepwise explanation is a simple propagation
where p_i are previously derived facts and q is a new fact implied by constraints c_j.
This propagation corresponds to a clause implied by the constraints c_j.
This clause can either be part of the clausal decomposition of some individual c_j, or a learned (?) clause by multiple c_j.
A look at my favorite example
x11 | x12 | x13 |
x21 | x22 | x23 |
x31 | x32 | x33 |
x41 | x42 | x43 |
4 pigeons, 3 holes
12 variables x_ij
4 row / pigeon constraints r_i
3 column / hole constraints c_i
x11 | x12 | x13 |
x21 | x22 | x23 |
x31 | x32 | x33 |
x41 | x42 | x43 |
Level 1 explanation sequence:
"learned" clauses:
x11 | x12 | x13 |
x21 | x22 | x23 |
x31 | x32 | x33 |
x41 | x42 | x43 |
Level 2 explanation sequence of
decomposition clauses:
x11 | x12 | x13 |
x21 | x22 | x23 |
x31 | x32 | x33 |
x41 | x42 | x43 |
decomposition clauses:
"learned" clauses:
Level 2 explanation sequence of
x11 | x12 | x13 |
x21 | x22 | x23 |
x31 | x32 | x33 |
x41 | x42 | x43 |
decomposition clauses:
"learned" clauses:
Level 2 explanation sequence of
x11 | x12 | x13 |
x21 | x22 | x23 |
x31 | x32 | x33 |
x41 | x42 | x43 |
Level 3 explanation sequence of
decomposition clauses:
x11 | x12 | x13 |
x21 | x22 | x23 |
x31 | x32 | x33 |
x41 | x42 | x43 |
decomposition clauses:
Level 3 explanation sequence of
decomposition clauses:
x11 | x12 | x13 |
x21 | x22 | x23 |
x31 | x32 | x33 |
x41 | x42 | x43 |
decomposition clauses:
Level 3 explanation sequence of
decomposition clauses:
- How many levels did we nest?
- 3 == #holes
- How many nested explanations would we generate?
- 3! == #holes!
- exponential in number of holes
- In general: as long as explanations only use clauses,
pigeonhole problem will have exponential SWES -
Polynomial formal explanations exist!
- symmetry argument
- argument per induction
- rational infeasibility argument
- cutting planes argument
- implies #holes >= #pigeons
- ?
Should / can we extend SWES to incorporate these?
Explanation limits
By Jo Devriendt
Explanation limits
- 430