2. Problem (hard) Solution CNF Satisfied assignment Encoding FINITE DOMAIN PROBLEM SOLVING Model Constraint Model Direct Constraint / Bits relation lost Large CNF CSP solving Model Solution Translate Decoding SAT solving

3. Constraint Model Simplified CNF Encoding CNF Simplify Simplified Model Encoding OPTIMIZED SAT ENCODING CNF’’ Problems: Constraint / Bits relation lost Large CNF CNF’ Partial Evaluation Tools such as: SatELite, ReVivAl Based on Unit Propagation and Resolution. Simplified Model’ Encoding Partial Evaluation using Equi-Propagation

4. OUR APPROACH Constraint ( C 1, B 1 ) … M = Constraint ( C 1, B 1 ) Constraint ( C’ 3, B‘ 3 ) Constraint ( C’ n, B’ n ) … M’ = φ = Simplify CSP techniques Encoding Boolean techniques Constraint ( C 2, B 2 ) Constraint ( C 3, B 3 ) Constraint ( C n, B n ) Constraint ( C 2, B 2 ) φ1φ1 φ' 3 φ' n … Equi-Propagation Standard encodings

5. OUTLINE  Modeling Finite Domain CSP  Equi-Propagation  Experimentation  Conclusions

6. MODELING FINITE DOMAIN CSP representing numbers (integers) BinaryUnaryOrder encoding x i ↔ (X ≥ i) (X = 3) = [1,1,1,0,0] Direct encoding x i ↔ (X = i) (X = 3) = [0,0,0,1,0,0] SMALL

7. WHY ORDER ENCODING ? X ij X ≥ iX < j 1 0 good for representing ranges X u v i good for arbitrary sets good for arithmetic operations with constants: + 3 = * 3 = div 3 = aaabbbcccacbacbcf111cbacbafedg ab c d efg b=c e=f=g

8. x= -y, x=y, x=0, x=1 single EQUI-PROPAGATION Equi-propagation is the process of inferring new equational consequences from a constraint in the model (and other existing equational information). x can now be removed from all constraints.

9. diff(U1,U2) U1 { 0..4 } U1 = [x1,x2,x3,x4] U2 { 0..4 } U2 = [y1,y2,y3,y4] Xs = [a,d,b,x3,y2,c,y4] EQUI-PROPAGATION EXAMPLE sumBits(Xs)=3Constraints …… Simplify U1 { 1,3 } U1 = [1,x2,x2,0] U2 { 1,3 } U2 = [1,y2,y2,0] Xs = [a,1,b,x2,y2,c,0] U1 { 1,3 } U1 = [1,x2,x2,0] U2 { 1,3 } U2 = [1,-x2,-x2,0] Xs = [a,1,b,x2,-x2,c,0] Partial Data: φsumBits([a,b,c])=1φConstraints …… diff(U1,U2) 1≤U1≤3 and U1 ≠ 2 1≤U2≤3 and U2 ≠ 2 d = 1 Learned: Simplify Encoding x1=1, x3=x2, x4=0 y1=1, y3=y2, y4=0 d = 1 U1 ≠ U2 (y2= -x2) … sumBits([a,b,c])=1 Constraints

10. EQUI-PROPAGATION  A complete equi-propagator for a constraint can be implemented using binary decision diagrams (BDDs) and can be evaluate in polynomial time. When C(X1,…,Xk) a constraint about (fixed) “k” integers with n bits each, the BDD representing it is of size O(n^k)  Global constraints (such as allDiff) implemented using Ad-Hoc rules.  There is a strong connection between Simplify and Encoding because Simplify done on the “encoding bits” and might change the encoding accordingly.

11. Kakuro QCP / Sudoku BIBD Nonograms Graph Crossing N-Queens Magic Square MAS SCM / MCM System Diagnostic Protein folding

12. BALANCED INCOMPLETE BLOCK DESIGNS (BIBD) Definition: a 5-tuple of positive integers and require to partition v distinct objects into b blocks such that each block contains k different objects, exactly r objects occur in each block, and every two distinct objects occur in exactly l blocks. Variables: B 11, …, B bv Domains: B ij {0,1} Constraints: each row constraint: sum(B i1,…,B iv ) = r each column constraint: sum(B 1i,…,B bi ) = k each two rows constraint: sum(B i1 *B j1, …,B iv *B jv )= l BIBD b=10 v=6

13. We can swap between two rows or two columns to generate another valid solution. BALANCED INCOMPLETE BLOCK DESIGNS (BIBD) k r r- ll

14. BIBD – BEE VS SUGAR Sugar (v1.14.7) (SymB)BEE (SymB)instance SAT (sec.) CNF size (clauses) generate (sec.) SAT (sec.) CNF size (clauses) compile (sec.) 8.99160783039.361.234941311.34 13.24248813612.011.736985791.65 36.43275311311.7413.6012119413.73 0.01371631.910.0000.02 1.8754008923.580.39815630.56 2.2662377364.810.561094420.81 0.429338816.020.06245940.10 --∞0.171130530.34 8.5256900742.651.33920590.64 ∞4660864.248.461160160.51 (CSP to CNF Encoder) Faster SAT solving time Smaller CNF Faster time to generate CNF

15. BIBD – BEE VS SATELITE SatELite (SymB)BEE (SymB)instance SAT (sec.) CNF size (clauses) preprocs (sec.) SAT (sec.) CNF size (clauses) compile (sec.) 1.655661911.271.234941311.34 2.188025761.671.736985791.65 5.1813971882.7313.6012119413.73 0.0000.010.0000.02 0.20795421.020.39815630.56 0.351052421.140.561094420.81 0.05238281.20.06245940.10 0.1411186910.450.171130530.34 8.93976231.011.33920590.64 ∞1341460.648.461160160.51 (CNF-Level preprocessor) Solving time About the same size Preprocess time

16. CONCLUSIONS  When encoding CSP model to SAT holding both representation for each constraint gives the ability to apply simplify techniques from both worlds on each constraint.  Apply complete CNF simplification on each constraint is possible in polynomial time.  By using the Equi-Propagation technique we generates a small optimized CNF which than can be simplify using CNF-Level simplifications. Constraint Model Simplified CNF EncodingCNF Simplified Model Encoding CNF’’ CNF’ Partial Evaluation Simplified Model Encoding Partial Evaluation using Equi-Propagation Constraint ( C1, φ 1 ) … M = Constraint ( C1, φ 1 ) Constraint ( C’3, φ‘ 3 ) Constraint ( C’n, φ’ n ) … M’ = Simplify CSP techniques Boolean techniques Constraint ( C2, φ 2 ) Constraint ( C3, φ 3 ) Constraint ( Cn, φ n ) Constraint ( C2, φ 2 ) Equi-Propagation

17. Questions ?

18. BIBD – BEE + SATELITE SatELite (SymB)BEE (SymB)instance SAT (sec.) CNF size (clauses) preprocs (sec.) SAT (sec.) CNF size (clauses) compile (sec.) 0.834875231.431.234888911.34 1.396905961.861.736923611.65 4.1812014393.1613.6012037583.73 0.00-- 00.02 0.11629690.840.39649680.56 0.20840571.000.56873330.81 0.05240071.150.06249120.10 0.1310793611.400.171091210.34 1.87854260.921.33863310.64 2.311153410.668.461154210.51 (CNF-Level preprocessor)

19. MINION v0.10BEE (SymB)instance SymB+ (sec.) SymB (sec.) [M’06] (sec.) SAT (sec.) CNF size (clauses) compile (sec.) 0.381.120.471.234941311.34 0.421.360.541.736985791.65 0.521.770.6613.6012119413.73 0.150.671.260.0000.02 0.311.4212.220.39815630.56 0.3513.40107.430.561094420.81 0.311.37∞0.06245940.10 0.351.71∞0.171130530.34 0.92∞∞1.33920590.64 75.87∞∞8.461160160.51 BIBD – BEE VS MINION (Constraint solver) Modeling control

20. NONOGRAMS Definition: an nXm board of cells to color black or white and given clues per row and column of a board. A clue is a number sequence indicating blocks of cells to be colored black. Variables: B 11, …, B nm R 11,…R 1k,…,R m1,…R mp C 11,…C 1q,…,C n1,…,C nr Domains: B ij {0,1} R mo {0,..,n} C no {0,..,m} Constraints: block constraint: block(R ij,R ij +,[B 1i,…,B ni ]) space constraint: block(R ij +,R ij+1,[-B 1i,…,-B ni ]) no overlap constraint: leq(R ij + +1, R ij+1 )

21. NONOGRAMS There are dedicates solvers such as Jan Wolter's pbnsolve (http://webpbn.com/pbnsolve.html)http://webpbn.com/pbnsolve.html Ben-Gurion University Solver (http://www.cs.bgu.ac.il/~benr/nonograms/)http://www.cs.bgu.ac.il/~benr/nonograms/ BEE is faster than the dedicated solvers on the hard puzzles. 5,000 random 30x30 puzzles Selected human puzzles