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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 l scheduling, assignment, routing, … §Constraints is BIG business l US: i2 is worth $4B l Europe: ILOG $200M l UK: Parc Technologies >$15M

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

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Why use proof planning? §Many possible logical consequences l preconditions can restrict us to those likely to be useful §Methods can act at high level l complex rewriting, simplification, … §Cleanliness l 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 = 10*B+C EF = 10*E+F HI = 10*H+I

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Fractions puzzle §Symmetry method A/BC D/EF G/HI §Eliminate method 3A/BC 1 3G/HI 1 §Linearize method 3*A 10*B+C 3*G 10*H+I

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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 3*A 10*B+C will prune immediately

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Fractions puzzle §Can also generate (implied) bounds 3*G 10*H+I §Bounds consistency gives: 3*G 11 §all-different method gives: 3*G 12 bound is unary implied constraint (but sadly no tighter as both give G 4)

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Prof. Smart’s safe §Again from Oz tutorial §Find sequence of non-zero digits with: x4-x6 = x7 x1*x2*x3 = x8+x9 x2+x3+x6 < x8 x9 < x8 x i =/= i all-different(x1,..x9)

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Prof. Smart’s safe §all-different method gives: x2+x3+x6 6 §eliminate method eliminates x2+x3+x6 (or transitivity method?): 6 < x8 §node consistency on x i =/= 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: l attract l collect l isolate

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Proof planning §PRESS’s waterfall probably not adequate l fractions: eliminate then all-different l safe: all-different then eliminate §Even with strong preconditions to methods, some implied constraints will need to be pruned 3*G 12 no tighter than 3*G 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 l 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 PRESS’s methods §Come to CIAO-2002 to see how we do!

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