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

3 SAT Basics (2/3) A set of propositional variables and Conjnctive ormal Form (CF) 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 SAT Basics (3/3) 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 4/23

5 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 USATISFIABLE; 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); } }

6 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

7 SAT Solvers Most SAT solvers receives DIMACS CF formlas Dozens of indstry-strength SAT solvers available MiniSAT PicoSAT SAT4J borg-sat clasp sathys tts 7/23

8 8 Solving Varios Problems sing SAT Solver C Program Encoding Encoding 2 Optimal Path Planning Encoding 3 Sdok Pzzle CF SAT Formla SAT Solver Latin Sqare Problem Traveling Salesmen Probelm Encoding n

9 9 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.

10 0 Sdok as SAT Problem symbol table model Sdok Encoder CF SAT Solver SAT? yes Decoder no o soltion fond Soltion fond

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

12 2 Encoding Kowledge compilation into a target langage problem knowlege CF 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 CF A nmber appears once in each block 9 A pre-assigned nmber facts CF

13 3 Variables Each cell has one nmber from.. [,]= or [,]=2 or or [,]= Each cell needs boolean variables to consider all cases Total nmber of variables 3 v 2 3 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

14 4 Cell Rle CF A nmber appears once in each cell There is at least one nmber in each cell ),, ( v c r Cell v c r d (definedness) There is at most one nmber in each cell (niqeness) )),, ( ),, (( j i v v v c r v c r v c r Cell i j i

15 5 Row Rle CF A nmber appears once in each row Each nmber appears at least once in each row Row d r ( v c r, c, v) (definedness) Each nmber appears at most once in each (niqeness) row Row r v c c c (( r, ci, v) ( r, c j, v)) i j i

16 6 Colmn Rle CF A nmber appears once in each colmn Each nmber appears at least once in each colmn Col d c ( v r r, c, v) (definedness) Each nmber appears at most once in each colmn Col c (niqeness) v r r r (( ri, c, v) ( rj, c, v)) i j i

17 7 Block Rle CF A nmber appears once in each block Each nmber appears at least once in each block (definedness) Block d sb r offs sb sb sb c v r c ( roffs * sb r, coffs * sb c, v) offs Each nmber appears at most once in each block (niqeness) Block (( r sb r offs offs ( r sb c * sb offs offs * sb v r ( r mod ( c mod cr sb), c offs sb), c * sb ( r offs * sb mod sb), v) ( c mod sb), v))

18 8 Pre-Assigned Fact CF 3 A pre-assigned nmber As a constant; the nmber is never changed It can be represented as a nit clase Assigned k i {( r, c, a) a [ r, c] a} where k is a nmber of pre - assigned nmbers

19 9 Previos Encodings Minimal encoding [Lynce & Oaknine, 2006] Cell d Row Col Block sfficient to characterize the pzzle Cell d Block Cell d Block Row d Row Assigned Assigned Extended encoding [Lynce & Oaknine, 2006] Cell d Cell Row Col Col minimal encoding with redndant clases Efficient encoding [Weber, 2005] d Block Col Assigned between minimal encoding and extended encoding

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

21 Experimental Reslts 2 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

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 o soltion fond Soltion fond 64x64 easy stack stack stack 8x8 easy stack stack stack