Generating Implied Constraints via Proof Planning

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Generating Implied Constraints via Proof Planning

• Alan Frisch, Ian Miguel, Toby Walsh
• Dept of CS
• University of York
• EPSRC funded project GR/N16129

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Motivation

• Constraints useful in many domains
• scheduling, assignment, routing,
• Constraints is BIG business
• US i2 is worth 4B
• Europe ILOG 200M
• UK Parc Technologies gt15M

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Motivation

• Implied constraints often crucial
• logically redundant
• but can reduce search dramatically
• Implied constraints added by hand
• can automated reasoning help?
• proof planning looks promising

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Why use proof planning?

• Many possible logical consequences
• preconditions can restrict us to those likely to
  be useful
• Methods can act at high level
• complex rewriting, simplification,
• Cleanliness
• logic v search

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Fractions puzzle

• From Oz tutorial
• Give 9 distinct non-zero digits (A-I) such that
• A/BC D/EF G/HI 1
• nb BC 10BC
• EF 10EF
• HI 10HI

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Fractions puzzle

• Symmetry method
• A/BC ? D/EF ? G/HI
• Eliminate method
• 3A/BC ? 1
• 3G/HI ? 1
• Linearize method
• 3A ? 10BC
• 3G ? 10HI

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Fractions puzzle

• Constraint solvers will delay non-linear
  constraints like
• A/BC D/EF G/HI 1
• until all 9 variables are ground
• (i.e. generate and test)
• Implied linear constraints like 3A ? 10BC will
  prune immediately

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Fractions puzzle

• Can also generate (implied) bounds
• 3G ? 10HI
• Bounds consistency gives
• 3G ? 11
• all-different method gives
• 3G ? 12
• bound is unary implied constraint (but sadly no
  tighter as both give G?4)

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Prof. Smarts safe

• Again from Oz tutorial
• Find sequence of non-zero digits with
• x4-x6 x7
• x1x2x3 x8x9
• x2x3x6 lt x8
• x9 lt x8
• xi / i
• all-different(x1,..x9)

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Prof. Smarts safe

• all-different method gives
• x2x3x6 ? 6
• eliminate method eliminates x2x3x6 (or
  transitivity method?)
• 6 lt x8
• node consistency on xi / i gives
• x8 7 or 9
• Only two out of nine values to try!

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Method base

• eliminate var(s) reducing constraint arity
• introduce auxiliary vars
• symmetry breaking
• linearize constraints
• all-different method
• summation method

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Eliminate method

• Generalization of Gaussian elimination
• PRESS methods may be useful
• attract
• collect
• isolate

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Proof planning

• PRESSs waterfall probably not adequate
• fractions eliminate then all-different
• safe all-different then eliminate
• Even with strong preconditions to methods, some
  implied constraints will need to be pruned
• 3G ? 12 no tighter than 3G ? 11

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Implementation

• Prolog
• Based on CLAM-Lite
• Input from ESRA or OPL?

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Credits

• Brahim Hnich
• Julian Richardson
• modified PRESS to deal with inequalities
• EPSRC

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Conclusions

• Implied constraints are simply logical
  consequences of initial model
• Proof planning looks promising for generating
  useful implied constraints
• We hope to re-use and adapt some of PRESSs
  methods
• Come to CIAO-2002 to see how we do!