1. Generating Implied Constraints via Proof Planning Alan Frisch, Ian Miguel, Toby Walsh Dept of CS University of York EPSRC funded project GR/N16129

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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)

9. 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)

10. 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!

11. Method base §eliminate var(s) reducing constraint arity §introduce auxiliary vars §symmetry breaking §linearize constraints §all-different method §summation method

12. Eliminate method §Generalization of Gaussian elimination §PRESS methods may be useful: l attract l collect l isolate

13. 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

14. Implementation §Prolog §Based on CLAM-Lite §Input from ESRA or OPL?

15. Credits §Brahim Hnich §Julian Richardson l modified PRESS to deal with inequalities §EPSRC

16. 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!