Consistency for Quantified Constraint Satisfaction Problems Peter - - PowerPoint PPT Presentation

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Nov 30, 2023 454 likes •671 views

Consistency for Quantified Constraint Satisfaction Problems Peter Nightingale Talk structure Finite domain QCSP Connect-4 Consistency notions WQGAC WQGAC-Schema Comparing consistencies Summary Finite domain QCSP

SLIDE 1

Consistency for Quantified Constraint Satisfaction Problems Peter Nightingale

SLIDE 2

Talk structure

Finite domain QCSP – Connect-4
Consistency notions
WQGAC
WQGAC-Schema
Comparing consistencies
Summary

SLIDE 3

Finite domain QCSP

Connect-4 endgame

1 2 3 4 5 6 7

∃red1∀black1∃red2∀ black2∃red3: redwinsred1,black1,red2,black2,red3

SLIDE 4

red1 black1 red2 red2 2 2 1,3..7 black2 SAT 5 2 red3 red3 4 1..3,5..7 SAT SAT 5 4

Finite domain QCSP

Example strategy

1 2 3 4 5 6 7

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red1 black1 red2 red2 2 2 1,3..7 black2 SAT 5 2 red3 red3 4 1..3,5..7 SAT SAT 5 4

Finite domain QCSP

Example strategy

1 2 3 4 5 6 7

SLIDE 6

red1 black1 red2 red2 2 1,3..7 black2 SAT 5 2 red3 red3 4 1..3,5..7 SAT SAT 5 4

Finite domain QCSP

Example strategy

1 2 3 4 5 6 7

SLIDE 7

red1 black1 red2 red2 2 1,3..7 black2 SAT 5 2 red3 red3 4 1..3,5..7 SAT SAT 4 4

Finite domain QCSP

Example strategy

1 2 3 4 5 6 7

SLIDE 8

red1 black1 red2 red2 2 1,3..7 black2 SAT 5 2 red3 red3 4 1..3,5..7 SAT SAT 4 4

Finite domain QCSP

Example strategy

1 2 3 4 5 6 7

SLIDE 9

red1 black1 red2 red2 2 1,3..7 black2 SAT 5 2 red3 red3 4 1..3,5..7 SAT SAT 4 4

Finite domain QCSP

Example strategy

1 2 3 4 5 6 7

SLIDE 10

Talk structure

Finite domain QCSP – Connect-4
Consistency notions
WQGAC
WQGAC-Schema
Comparing consistencies
Summary

SLIDE 11

Consistency notions

Hasse diagram
Ordered by strength

– Then constraint arity

Ternary Boolean constraints

Bordeaux and Monfroy

QAC

Stergiou and Mamoulis

WQGAC

(this work)

Ternary interval constraints

Bordeaux and Monfroy

AC GAC Local inconsistency

Bordeaux, Cadoli and Mancini

SLIDE 12

Talk structure

Finite domain QCSP – Connect-4
Consistency notions
WQGAC
WQGAC-Schema
Comparing consistencies
Summary

SLIDE 13

WQGAC

With GAC each value has a supporting

tuple

With WQGAC each value has a supporting

tuple for each combination of values of inner universals

∃a ∀ b∃c:a⇔b∧c

a b c 0 0 0 0 0 1 0 1 0 1 1 1 Supporting a=0:

SLIDE 14

WQGAC

With GAC each value has a supporting

tuple

With WQGAC each value has a supporting

tuple for each combination of values of inner universals ∃a∀ b∃c:a⇔b∧c a b c 0 0 0 0 0 1 0 1 0 1 1 1 Supporting a=1:

SLIDE 15

WQGAC-Schema

Based on GAC-Schema (Bessière and

Régin)

Time: O(n2dn)
Space: O(n2du+1)
Generalization of GAC-Schema
Multidirectional

SLIDE 16

Talk structure

Finite domain QCSP – Connect-4
Consistency notions
WQGAC
WQGAC-Schema
Comparing consistencies
Summary

SLIDE 17

Comparing consistencies

Consistency QAC on the hidden variable encoding GAC WQGAC B,C & M inconsistency Inference none none 1,3,5..7 pruned from grey1 1,3,5..7 pruned from grey1 1,3,6,7 pruned from grey2 1,3,7 pruned from grey3 Resources used 0.046s, checked 15.2% of all 75 tuples.

SLIDE 18

Comparing consistencies

WQGAC weak

– For each value, set of supporting tuples – May not be part of one strategy

a=0 supported by: a b c 0 0 0 0 1 1 1 0 1 1 1 0

∀ a∃b ∀c∈{0,1}

Value of b is different

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Summary

Reasonably powerful algorithm for local

reasoning in finite domain QCSP

Future work

– Tuple/tree mismatch – Different support structure

SLIDE 20

Thank you

Any questions?