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
\bigwedge_i p_i \bigwedge_j c_j \models q
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 |
\geq1
\geq1
\geq1
\geq1
\leq1
\leq1
\leq1
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 |
\geq1
\geq1
\geq1
\geq1
\leq1
\leq1
\leq1
Level 1 explanation sequence:
r_1 \wedge r_2 \wedge r_3 \wedge c_1 \wedge c_2 \wedge c_3 \models \overline{x_{41}} \wedge \overline{x_{42}} \wedge \overline{x_{43}}
\overline{x_{41}} \wedge \overline{x_{42}} \wedge \overline{x_{43}} \wedge r_4 \models \emptyset
(\overline{x_{41}}),
(\overline{x_{42}}),
(\overline{x_{43}})
"learned" clauses:
| x11 | x12 | x13 |
| x21 | x22 | x23 |
| x31 | x32 | x33 |
| x41 | x42 | x43 |
\geq1
\geq1
\geq1
\geq1
\leq1
\leq1
\leq1
Level 2 explanation sequence of
x_{41} \wedge c_1 \models \overline{x_{11}} \wedge \overline{x_{21}} \wedge \overline{x_{31}}
(\overline{x_{41}} \vee \overline{x_{11}}),
(\overline{x_{41}} \vee \overline{x_{21}}),
(\overline{x_{41}} \vee \overline{x_{31}})
decomposition clauses:
r_1 \wedge r_2 \wedge r_3 \wedge c_1 \wedge c_2 \wedge c_3 \models \overline{x_{41}}
| x11 | x12 | x13 |
| x21 | x22 | x23 |
| x31 | x32 | x33 |
| x41 | x42 | x43 |
\geq1
\geq1
\geq1
\geq1
\leq1
\leq1
\leq1
x_{41} \wedge c_1 \models \overline{x_{11}} \wedge \overline{x_{21}} \wedge \overline{x_{31}}
(\overline{x_{41}} \vee \overline{x_{11}}),
(\overline{x_{41}} \vee \overline{x_{21}}),
(\overline{x_{41}} \vee \overline{x_{31}})
decomposition clauses:
\overline{x_{11}} \wedge \overline{x_{21}} \wedge r_1 \wedge r_2 \wedge c_2 \wedge c_3 \models \overline{x_{32}} \wedge \overline{x_{33}}
(x_{11} \vee x_{21} \vee \overline{x_{32}}),
(x_{11} \vee x_{21} \vee \overline{x_{33}})
"learned" clauses:
Level 2 explanation sequence of
r_1 \wedge r_2 \wedge r_3 \wedge c_1 \wedge c_2 \wedge c_3 \models \overline{x_{41}}
| x11 | x12 | x13 |
| x21 | x22 | x23 |
| x31 | x32 | x33 |
| x41 | x42 | x43 |
\geq1
\geq1
\geq1
\geq1
\leq1
\leq1
\leq1
x_{41} \wedge c_1 \models \overline{x_{11}} \wedge \overline{x_{21}} \wedge \overline{x_{31}}
\overline{x_{31}} \wedge \overline{x_{32}} \wedge \overline{x_{33}} \wedge r_3 \models \emptyset
(\overline{x_{41}} \vee \overline{x_{11}}),
(\overline{x_{41}} \vee \overline{x_{21}}),
(\overline{x_{41}} \vee \overline{x_{31}})
decomposition clauses:
\overline{x_{11}} \wedge \overline{x_{21}} \wedge r_1 \wedge r_2 \wedge c_2 \wedge c_3 \models \overline{x_{32}} \wedge \overline{x_{33}}
(x_{11} \vee x_{21} \vee \overline{x_{32}}),
(x_{11} \vee x_{21} \vee \overline{x_{33}})
"learned" clauses:
Level 2 explanation sequence of
r_1 \wedge r_2 \wedge r_3 \wedge c_1 \wedge c_2 \wedge c_3 \models \overline{x_{41}}
| x11 | x12 | x13 |
| x21 | x22 | x23 |
| x31 | x32 | x33 |
| x41 | x42 | x43 |
\geq1
\geq1
\geq1
\geq1
\leq1
\leq1
\leq1
Level 3 explanation sequence of
x_{32} \wedge c_2 \models \overline{x_{12}} \wedge \overline {x_{22}}
(\overline{x_{32}} \vee \overline{x_{12}}),
(\overline{x_{32}} \vee \overline{x_{22}})
decomposition clauses:
\overline{x_{11}} \wedge \overline{x_{21}} \wedge r_1 \wedge r_2 \wedge c_2 \wedge c_3 \models \overline{x_{32}}
| x11 | x12 | x13 |
| x21 | x22 | x23 |
| x31 | x32 | x33 |
| x41 | x42 | x43 |
\geq1
\geq1
\geq1
\geq1
\leq1
\leq1
\leq1
x_{32} \wedge c_2 \models \overline{x_{12}} \wedge \overline {x_{22}}
(\overline{x_{32}} \vee \overline{x_{12}}),
(\overline{x_{32}} \vee \overline{x_{22}})
decomposition clauses:
Level 3 explanation sequence of
\overline{x_{11}} \wedge \overline{x_{21}} \wedge r_1 \wedge r_2 \wedge c_2 \wedge c_3 \models \overline{x_{32}}
\overline{x_{11}} \wedge \overline{x_{12}} \wedge r_1 \models x_{13}
\overline{x_{21}} \wedge \overline{x_{22}} \wedge r_2 \models x_{23}
(x_{11} \vee x_{12} \vee x_{13}),
(x_{21} \vee x_{22} \vee x_{23})
decomposition clauses:
| x11 | x12 | x13 |
| x21 | x22 | x23 |
| x31 | x32 | x33 |
| x41 | x42 | x43 |
\geq1
\geq1
\geq1
\geq1
\leq1
\leq1
\leq1
x_{32} \wedge c_2 \models \overline{x_{12}} \wedge \overline {x_{22}}
(\overline{x_{32}} \vee \overline{x_{12}}),
(\overline{x_{32}} \vee \overline{x_{22}})
decomposition clauses:
Level 3 explanation sequence of
\overline{x_{11}} \wedge \overline{x_{21}} \wedge r_1 \wedge r_2 \wedge c_2 \wedge c_3 \models \overline{x_{32}}
\overline{x_{11}} \wedge \overline{x_{12}} \wedge r_1 \models x_{13}
\overline{x_{21}} \wedge \overline{x_{22}} \wedge r_2 \models x_{23}
(x_{11} \vee x_{12} \vee x_{13}),
(x_{21} \vee x_{22} \vee x_{23})
decomposition clauses:
x_{13} \wedge x_{23} \wedge c_3 \models \emptyset
- 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
