Application of Propositional Logic - How to Solve Sudoku? Moonzoo Kim

Transcription

1 Application of Propositional Logic - How to Solve Sdok? Moonzoo Kim

2 SAT Basics (/2) SAT = Satisfiability = Propositional Satisfiability Propositional Formla NP-Complete problem We can se SAT solver for many NP-complete problems Hamiltonian path 3 coloring problem SAT problem Traveling sales man s problem Recent interest as a verification engine SAT UNSAT 2/23

3 SAT Basics (2/2) A set of propositional variables and Conjnctive Normal Form (CNF) clases involving variables (x v x 2 v x 3 ) (x 2 v x v x 4 ) x, x 2, x 3 and x 4 are variables (tre or false) Literals: Variable and its negation x and x A clase is satisfied if one of the literals is tre x =tre satisfies clase x =false satisfies clase 2 Soltion: An assignment/interpretation/model that satisfies all clases 3/23

4 Intro. to Logic CS402 DIMACS format for CNF The file can start with comments, that is lines beginning with the character c. Right after the comments, there is the line p cnf <nbvar> <nbclases> indicating that the instance is in CNF format; <nbvar> is the exact nmber of variables appearing in the file; <nbclases> is the exact nmber of clases contained in the file. It is garanteed that each variable between and nbvar appears at least once in a clase. Then the clases follow. Each clase is a seqence of distinct non-nll nmbers between -nbvar and nbvar ending with 0 on the same line. Positive nmbers denote the corresponding variables. Negative nmbers denote the negations of the corresponding variables. A clase is not allowed to contain the opposite literals i and -i simltaneosly c c start with comments c c p cnf

5 Example DIMACS SAT Format Ex. (x x 2 x 3 ) (x 2 x x 4 ) p cnf Model/ soltion x x 2 x 3 x 4 Formla T T T T T T T T F T T T F T T T T F F T T F T T T T F T F F T F F T T T F F F F F T T T T F T T F T F T F T F F T F F F F F T T T F F T F T F F F T T F F F F T 5/23

6 DPLL (Davis-Ptnam-Logemann-Lveland) Algorithm /* The Qest for Efficient Boolean Satisfiability Solvers * by L.Zhang and S.Malik, Compter Aided Verification 2002 */ DPLL(a formla Á, assignment) { necessary = dedction(á, assignment); new_asgnment = nion(necessary, assignment); if (is_satisfied(á, new_asgnment)) retrn SATISFIABLE; else if (is_conflicting(á, new_asgnmnt)) retrn UNSATISFIABLE; var = choose_free_variable(á, new_asgnmnt); asgn = nion(new_asgnmnt, assign(var, )); if (DPLL(Á, asgn) == SATISFIABLE) retrn SATISFIABLE; else { asgn2 = nion (new_asgnmnt, assign(var,0)); retrn DPLL (Á, asgn2); } }

7 Example {p r} { p q r} {p r} p=t p=f {T r} { T q r} {T r} {F r} { F q r} {F r} SIMPLIFY { q,r} {r} { r} SIMPLIFY {} SIMPLIFY

8 SAT Solvers Most SAT solvers receives DIMACS CNF formlas Dozens of indstry-strength SAT solvers available MiniSAT PicoSAT SAT4J borg-sat clasp sathys tts 8/23

9 9 Solving Varios Problems sing SAT Solver C Program Encoding Encoding 2 Optimal Path Planning Encoding 3 Sdok Pzzle CNF SAT Formla SAT Solver Latin Sqare Problem Traveling Salesmen Probelm Encoding n

10 0 What is Sdok? Problem Given a problem, the objectvie is to find a satisfying assignment w.r.t. Sdok rles. Soltion Sodok rles There is a nmber in each cell. A nmber appears once in each row. A nmber appears once in each colmn. A nmber appears once in each block.

11 Sdok as SAT Problem symbol table model Sdok Encoder CNF SAT Solver SAT? yes Decoder no No soltion fond Soltion fond

12 2 Previos Encodings symbol table model Sdok Encoder CNF SAT Solver SAT? yes Decoder Minimal encoding [Lynce & Oaknine, 2006] Extended encoding [Lynce & Oaknine, 2006] Efficient encoding [Weber, 2005]

13 3 Encoding Kowledge compilation into a target langage problem knowlege CNF Knowlede abot Sdok A nmber appears once in each cell A nmber appears once in each row A nmber appears once in each col rles CNF A nmber appears once in each block 9 A pre-assigned nmber facts CNF

14 4 Variables Each cell has one nmber from..n [,]= or [,]=2 or or [,]=N Each cell needs N boolean variables to consider all cases Total nmber of variables N 3 v 2 3 N Boolean variable name as a triple (r,c,v) (i.e., x rcv ) iff [r,c] = v (r,c,v) (i.e., x rcv ) iff [r,c] v r c

15 5 Cell Rle CNF A nmber appears once in each cell There is at least one nmber in each cell (definedness) Cell ( r, c, N N N d = r= c= v= v There is at most one nmber in each cell (niqeness) ) Cell (( r, c, v ) ( r, c, v N N N N = r= c= v = v = v + i j i j i ))

16 6 Row Rle CNF A nmber appears once in each row Each nmber appears at least once in each row Row N d = r= = N N ( v= c r, c, v) (definedness) Each nmber appears at most once in each (niqeness) row Row N N N v c c c (( r, ci, v) ( r, c j, v)) N = r= = = = + i j i

17 7 Colmn Rle CNF A nmber appears once in each colmn Each nmber appears at least once in each colmn Col = N N v= r ( r, c, v) N d c= = (definedness) Each nmber appears at most once in each colmn Col N = c= v= r = r = r + (niqeness) N N N (( r, c, v) ( r, c, v)) i j i i j

18 8 Block Rle CNF A nmber appears once in each block Each nmber appears at least once in each block (definedness) Block sbn d = r = = sbn N sbn sbn c = v = r = c (( roffs ) * sbn + r,( coffs ) * sbn c, v ) + offs offs where sbn =3 for the above example Each nmber appears at most once in each block (niqeness) Block ((( r offs (( r sbn sbn N N N = r = c = v = r = c = r + offs offs offs ) * sbn + ( r mod ) * sbn + ( c mod sbn ),( c offs sbn ),( c offs ) * sbn + ( r mod sbn ), v ) ) * sbn + ( c mod sbn ), v ))

19 9 Pre-Assigned Fact CNF 3 A pre-assigned nmber As a constant; the nmber is never changed It can be represented as a nit clase Assigned {( r, c, a) [ r, c] k = i = a N = a } where k is a nmber of pre - assigned nmbers

20 20 Previos Encodings Minimal encoding [Lynce & Oaknine, 2006] φ = Cell d Row Col Block sfficient to characterize the pzzle φ = Cell d Block Cell d Block Row d Row Assigned Col Assigned Extended encoding [Lynce & Oaknine, 2006] minimal encoding with redndant clases Efficient encoding [Weber, 2005] φ = Cell d Cell Row Col d Block Col Assigned between minimal encoding and extended encoding

21 2 Exponential Growth in Clases size minimal efficient extended 9x x x x x x x Nmber of clases 90,000,000 80,000,000 70,000,000 60,000,000 50,000,000 40,000,000 30,000,000 20,000,000 0,000,000 0 minimal efficient extended 9x9 6x6 25x25 36x36 49x49 64x64 8x8 size

22 Experimental Reslts 22 minimal encoding efficient encoding extended encoding size level vars clases time vars clases time vars clases time 9x9 easy x9 hard x6 easy x6 hard x25 easy x25 hard time time x36 easy time time x36 hard time time x49 easy time time x64 easy stack stack stack 8x8 easy stack stack stack

23 Experimental Reslts 23 minimal encoding efficient encoding extended encoding size level vars clases time vars clases time vars clases time 9x9 easy x9 hard x6 easy x6 hard x25 easy x25 hard time time x36 easy time time x36 hard time time x49 easy time time No soltion fond Soltion fond 64x64 easy stack stack stack 8x8 easy stack stack stack