FINITE DOMAIN Constraint / Bits PROBLEM SOLVING relation lost - PowerPoint PPT Presentation

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

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

OUR APPROACH … Constraint Constraint Constraint Constraint M = ( C 1 , B 1 ) ( C 2 , B 2 ) ( C 3 , B 3 ) ( C n , B n ) Simplify Boolean techniques CSP techniques Equi-Propagation … M‟ = Constraint Constraint Constraint Constraint ( C‟ 3 , B„ 3 ) ( C‟ n , B‟ n ) ( C 1 , B 1 ) ( C 2 , B 2 ) Encoding Standard encodings … φ = φ 1 φ' 3 φ' n

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

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

WHY ORDER ENCODING ?  good for representing ranges X 1 0 X ≥ i i j X < j  good for arbitrary sets X u v i b=c e=f=g a b c d e f g  good for arithmetic operations with constants: a b c 1 1 1 a b c + 3 = a b c a a a b b b c c c * 3 = a b c d e f g c f div 3 =

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

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

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.

Graph Crossing BIBD MAS Kakuro B en-Gurion-University Protein E qui-propagation folding E ncoder N-Queens System Diagnostic Magic Square SCM / MCM QCP / Sudoku Nonograms

BALANCED INCOMPLETE BLOCK DESIGNS (BIBD) Definition: a 5-tuple of positive integers < v , b , r , k , l > 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. b =10 Variables: B 11 , …, B bv v =6 Domains: B ij ϵ {0,1} Constraints: BIBD <6,10,5,3,2> • 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

BALANCED INCOMPLETE BLOCK DESIGNS (BIBD) We can swap between two rows or two columns to generate another valid solution. r k l r- l < v , b , r , k , l >

BIBD – BEE VS SUGAR (CSP to CNF Encoder) instance BEE (SymB) Sugar (v1.14.7) (SymB) compile CNF size SAT generate CNF size SAT (sec.) (clauses) (sec.) (sec.) (clauses) (sec.) ∞ - - ∞ Faster time to generate CNF Smaller CNF Faster SAT solving time

BIBD – BEE VS SATELITE (CNF-Level preprocessor) instance BEE (SymB) SatELite (SymB) compile CNF size SAT preprocs CNF size SAT (sec.) (clauses) (sec.) (sec.) (clauses) (sec.) ∞ Preprocess time About the same size Solving time

BIBD – BEE VS MINION (Constraint solver) instance BEE (SymB) MINION v0.10 [M‟ 06] compile CNF size SAT SymB SymB+ (sec.) (clauses) (sec.) (sec.) (sec.) (sec.) ∞ ∞ ∞ ∞ ∞ ∞ Modeling control

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. Constraint Constraint Constraint Constraint … M = ( C1 , φ 1 ) ( C2 , φ 2 ) ( C3 , φ 3 ) ( Cn, φ n ) Simplify Boolean techniques CSP techniques Equi-Propagation Constraint Constraint Constraint Constraint … M‟ = ( C1 , φ 1 ) ( C2 , φ 2 ) ( C‟ 3 , φ„ 3 ) ( C‟n , φ‟ n )  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 Encoding CNF Model Simplified CNF‟ Partial Evaluation Encoding Model Simplified CNF‟‟ Partial Evaluation using Equi-Propagation Encoding Model Simplified CNF

Questions ?

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 ϵ {0,1} R mo ϵ {0,..,n} C no ϵ {0,..,m} Constraints: • block constraint: block(R ij ,R ij +<ij size>,[B 1i ,…, B ni ]) • space constraint: block(R ij +<ij size>,R ij+1 ,[-B 1i ,…, -B ni ]) • no overlap constraint: leq(R ij +<ij size>+1, R ij+1 )

NONOGRAMS There are dedicates solvers such as • Jan Wolter's pbnsolve (http://webpbn.com/pbnsolve.html) • Ben-Gurion University Solver (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