1. Foundations of Constraint Processing, Spring 2008 April 9, 2008 nFCi1 Foundations of Constraint Processing CSCE421/821, Spring 2008: www.cse.unl.edu/~choueiry/S08-421-821/ Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 123B choueiry@cse.unl.edu Tel: +1(402)472-5444 FC for Non-Binary CSPs

2. Foundations of Constraint Processing, Spring 2008 April 9, 2008 nFCi2 Recommended reading (short, preliminary version) On forward checking for non-binary constraint satisfaction, by Christian Bessière, Pedro Meseguer, Eugene C Freuder, Javier Larrosa, CP 1999On forward checking for non-binary constraint satisfaction (longer, more complete version) On forward checking for non-binary constraint satisfaction, by Christian Bessière, Pedro Meseguer, Eugene C Freuder, Javier Larrosa, AIJ 2002On forward checking for non-binary constraint satisfaction

3. Foundations of Constraint Processing, Spring 2008 April 9, 2008 nFCi3 Non-binary FC Definitions Example Properties

4. Foundations of Constraint Processing, Spring 2008 April 9, 2008 nFCi4 nFC0 and nFC1 C c,1 : Constraints involving the current variable and exaclty one future variable  C c,1 : Set of constraint projections involving the current variable and exactly one future variable nFC0: Apply AC on each constraint in C c,1 (one pass) nFC1: Apply AC to each constraint in C c,1 and  C c,1 (one pass)

5. Foundations of Constraint Processing, Spring 2008 April 9, 2008 nFCi5 nFC2, nFC3, nFC4, nFC5 C c,f : set of constraints involving the current variable and at least one future variable C p,f : set of constraints involving at least one past variable and at least one future variable nFC2: Apply AC to each constraint in C c,f nFC3: Make C c,f arc-consistent nFC4: Apply AC to each constraint in C p,f nFC5: Make C p,f arc-consistent

6. Foundations of Constraint Processing, Spring 2008 April 9, 2008 nFCi6 Non-binary FC Definitions Example Properties

7. Foundations of Constraint Processing, Spring 2008 nFCi7 Filtering for x  a nFC0 does no filtering nFC1 applies AC on  c1 {x,y}, {x.z}, and and  c3 on {x,y} and {x,w}. It removes c from D(y), b from D(w) nFC2 applies AC on c1, then on c3. Same pruning as nFC1. A different ordering of constraints yields different filtering a, b, c x y z u v w c1 c2 c3 c1c2c3 xyzxyzxyz aaaaaaaaa abcabbabc acbccc X  a

8. Foundations of Constraint Processing, Spring 2008 nFCi8 Filtering for x  a nFC3 applies AC on {c1,c3}. Same filtering as nfC2, but also, removes b from D(z). nFC4 applied AC on c1 then c3 (like nFC2). Same filtering as nFC2 because x is the first variable. nfC5 yields the same filtering as nFC3 because x is the first variable. a, b, c x y z u v w c1 c2 c3 c1c2c3 xyzxyzxyz aaaaaaaaa abcabbabc acbccc X  a

9. Foundations of Constraint Processing, Spring 2008 nFCi9 Filtering for u  a nFC0 does no filtering nFC1 applies AC on  c1 on {u,v}, {u,w}. It removes c from D(v), c from D(w) nFC2 applies AC on c2. It removes b and c from D(v) and c from D(w) a, b, c x y z u v w c1 c2 c3 c1c2c3 xyzxyzxyz aaaaaaaaa abcabbabc acbccc X  a

10. Foundations of Constraint Processing, Spring 2008 nFCi10 Filtering for u  a nFC3 applies AC on {c2}. Same filtering as nFC2 nFC4 applies AC on c1, c2, then c3. It removes b from D(y) and D(z), b and c from D(v) and c from D(w). nfC5 does AC on {c1, c2, c3}. It removes b from D(y), c from D(z), b and c from D(v), and c from D(w) a, b, c x y z u v w c1 c2 c3 c1c2c3 xyzxyzxyz aaaaaaaaa abcabbabc acbccc u  a

11. Foundations of Constraint Processing, Spring 2008 April 9, 2008 nFCi11 Non-binary FC Definitions Example Properties

12. Foundations of Constraint Processing, Spring 2008 April 9, 2008 nFCi12 Filtering effectiveness  Let  (nFCi, k) be the set of (x,a) where a is removed from D(x)  (nFC0,k)   (nFC1,k)   (nFC2,k)  (nFC2,k)   (nFC3,k)   (nFC5,k)  (nFC2,k)   (nFC4,k)   (nFC5,k)

13. Foundations of Constraint Processing, Spring 2008 April 9, 2008 nFCi13 Nodes visited by nFCi nodes(nFC2,k)  nodes(nFC1,k)  nodes(nFC0,k) nodes(nFC5k)  nodes(nFC3,k)  nodes(nFC2,k) nodes(nFC5,k)  nodes(nFC4,k)  nodes(nFC2,k) nFC1 visits exactly the same nodes as FC+ on the hidden variable representation