1. Nogood Recording for Static and Dynamic Constraint Satisfaction Problems Thomas Schiex, Gerard Verfaillie C.E.R.T.-O.N.E.R.A.(France)

2. Outline Introduction Definition Build nogoods during Backtrack search An example Conclusion

3. Introduction In dynamic environment, the knowledge is not defined once and for all, but may incrementally evolve between successive requests. Solution maintenance problem to successively find, when it exists, a new solution after each modification of the problem.

4. Introduction Knowledge may be modified 1.the enviorment 2.the user 3.a distributed environment most existing satisfaction algorithm -->rediscover many identical faliures at each satisfaction

5. Introduction Nogood recording 1.improve efficiently 2.compute an approximate description of the frontier of the space explored

6. Definition CSP X={x1,..,xn} D={d1,..,dn} C={c1,..,cm} goal: find an assignment of X,such that all constraint are satisfied.

7. Definition Constraint propagation(k-consistency) add an induced constraint to a CSP does not change its solution set and may enable some non globally consistent assignments to be discovered early.

8. Definition Dynamic CSP P = a sequence P0,..,Pi of static CSPs each one resulting from a change in the preceding one this change may be: 1.A restriction: a new constraint is impose on a subset of variables 2. A relaxation: a constraint is remove from the CSP

9. Definition The problem of solution maintenance in a DCSP = sequentially computing a solution for each of the CSP P0..,Pi... Some obvious optimizations are relevant 1.A|a solution of Pi = ( X,Ci)->is a solution of (X,Ci-c) 2.A|not a solution of Pi = ( X,Ci)->is also not a solution of (X,Ci+c)

10. Definition 3.if the CSP(X,Ci) is inconsistent, then the constraint ¬c is induced by the CSP(X,Ci-c) and thus may be added to (X,Ci-c) prior to search.

11. Building nogoods In usual backtrack algorithm, each leaf of the tree explored offers an opportunity to build a nogood. Property if an assignment A violates a constraint c, then (A,{c}) is a nogood

12. Building nogoods Corollary Let a be an assignment of the variables in XA, xs be an unassigned variable, and {A1,..,Am} be all the possible extensions of A along xs, using every possible value of d(xs). If (A1,J1),..,(Am,Jm) are nogoods, then (A, Ui Ji) is a nogood.

13. Using nogoods The nogood may serve three different purposes 1.backjump to the lowest variable -->dependency directed backtracking 2.Pruning branches-->improving efficiency 3.when solving a CSP, the constraint forbidding A may be added to the CSP before starting the search

14. An example

16. Conclusion The collected can bring vary important saving, especially on hard or over- constrained CSP. It enhance the efficiency.