Uma escala para medir a dificuldade dos quebra-cabeças de Sudoku

In the last few years, Sudoku has become a popular game, deserving the attention of many researchers. Most Sudoku puzzles are classified as easy, average and hard according to their degree of difficulty. However when asked about the criteria used to classify them, there is no clear answer. This paper presents a PnP (paper-and-pencil) method to solve and measure the degree of difficulty of Sudoku puzzles. Contrary to other methods, this method can classify the puzzles automatically without human intervention. The scale used here was inspired on the type of scale used in the snow trails: green, blue, red and black. The Sudoku puzzle is transformed to SAT, and its level is set depending on the SAT problem being easy or hard to solve in SAT, which is equivalent to the puzzle being solved using an easy or hard PnP method. A classifier and a set of classified Sudoku problems are made available on the web.Resumo: Nos últimos anos, o Sudoku tornou-se um jogo muito popular, merecendo a atenção de muitos investigadores. A maior parte dos puzzles de Sudoku é classificada como simples, média ou difícil, consoante o seu grau de dificuldade. Contudo, quando questionados acerca do critério de classificação, não existe uma resposta clara. Este artigo apresenta um método manual para medir o grau de dificuldade dos puzzles de Sudoku. Ao contrário de outros métodos, este método classifica os puzzles automaticamente, sem intervenção humana. A escala utilizada é inspirada na escala utilizada nas pistas de neve: verde, azul, vermelho e negro. O puzzle de Sudoku é transformado num problema de SAT e o nível de dificuldade é dado pela resolução do problema de SAT, o qual é equivalente ao da resolução manual. O classificador de dificuldade e um conjunto de puzzles de Sudoku classificados estão disponíveis na Web.peerreviewe

Réduction et Encodage des Contraintes Ensemblistes en SAT

On the one hand, Constraint Satisfaction Problems(CSP) are a declarative and expressive approach for mo-deling problems. On the other hand, propositional sa-tisfiability problem (SAT) solvers can handle huge SATinstances up to millions of variables and clauses. In thisarticle, we present an approach for taking advantageof both CSP modeling and SAT solving. Our techniqueconsists in expressively modeling set constraint problemsas CSPs that are automatically treated by some reduc-tion rules to remove values that do not participate inany solution. These reduced CSPs are then encoded into”good” SAT instances that can be solved by standardSAT solvers. We illustrate our technique on various well-known problems such as Sudoku, the Social Golfer pro-blem, and the Sports Tournament Scheduling problem.Our technique is simpler, more expressive, and less error-prone than direct SAT modeling. The SAT instances thatwe automatically generate are rather small (even w.r.t.direct-written SAT instances for the Social Golfer pro-blem [18]) and can efficiently be solved up to huge ins-tances. Moreover, the reduction phase enables to pushback the limits and treat even bigger problems.D’un cˆot ́e, les probl`emes de satisfaction decontraintes (CSP) procurent une m ́ethode d ́eclarative etexpressive pour mod ́eliser les probl`emes. D’un autre cˆot ́e,les solveurs pour les probl`emes de satisfiabilit ́e de for-mules logique propositionnelle (SAT) peuvent manipulerdes instances ́enormes jusqu’`a des millions de clauses etvariables. Dans cet article, nous pr ́esentons une approcheb ́en ́eficiant de la mod ́elisation CSP et de la r ́esolutionSAT. Notre technique consiste `a mod ́eliser, de fa ̧con ex-pressive, des probl`emes de contraintes ensemblistes enen CSP qui sont ensuite automatiquement r ́eduits afinde retirer les valeurs des variables qui ne participent `aaucune solution. Ces CSPs r ́eduits sont ensuite encod ́esen de “bonnes” instances SAT qui peuvent ˆetre r ́esoluespar des solveurs SAT standards. Nous illustrons notretechnique par divers probl`emes standards : le Sudoku, leSocial Golfer Problem et le Sports Tournament Schedu-ling Problem.Notre technique est plus simple, plus expressive etmoins sensible aux erreurs qu’une mod ́elisation directeen SAT. De plus, les instances SAT automatiquementg ́en ́er ́ees sont g ́en ́eralement plus petites que celles direc-tement ́ecrite pour un probl`eme particulier (comme parexemple pour le Social Golfer Problem [18]) et peuventˆetre ́evalu ́ees efficacement mˆeme pour des instances ́enormes. Enfin, la phase de r ́eduction nous permet derepousser les limites et de traiter des probl`emes encoreplus gros

Solving of constraint satisfaction problems by reduction to SAT

Mnogi realni problemi se danas mogu predstaviti u obliku problema zadovoljenja ogranicenja (CSP) i zatim rešiti nekom od mnogobrojnih tehnika za rešavanje ovog problema. Jedna od tehnika podrazumeva svođenje na problem SAT, tj. problem iskazne zadovoljivosti. Promenljive i ogranicenja problema CSP se prevode (kodiraju) u SAT instancu, ona se potom rešava pomocu modernih SAT rešavaca i rešenje se, ako postoji, prevodi u rešenje problema CSP. Glavni cilj ove teze je unapređenje rešavanja problema CSP svođenjem na SAT. Razvijena su dva nova hibridna kodiranja (prevođenja u SAT formulu) koja kombinuju dobre strane postojecih kodiranja. Dat je dokaz korektnosti jednog od kodiranja koji do sada nije postojao u literaturi. Razvijen je sistem meSAT koji omogucava svođenje problema CSP na SAT pomocu cetiri osnovna i dva hibridna kodiranja, kao i rešavanje problema CSP svođenjem na dva problema srodna problemu SAT, SMT i PB. Razvijen je portfolio za automatski odabir kodiranja/rešavaca za ulaznu instancu koju je potrebno rešiti i pokazano je da je razvijeni portfolio uporediv sa najefikasnijim savremenim pristupima. Prikazan je novi pristup zasnovan na kratkim vremenskim ogranicenjima sa ciljem da se znacajno smanji vreme pripreme portfolija. Pokazano je da se ovim pristupom dobijaju rezultati konkurentni onima koji se dobijaju korišcenjem standardnog vremena za pripremu. Izvršeno je poređenje nekoliko tehnika mašinskog ucenja, sa ciljem da se ustanovi koja od njih je pogodnija za pristup sa kratkim treniranjem. Prikazan je jedan realan problem, problem raspoređivanja kontrolora leta, kao i tri njegova modela. Veliki broj metoda rešavanja i raznovrsnih rešavaca je upotrebljeno za rešavanje ovog problema. Razvijeno je više optimizacionih tehnika koje imaju za cilj pronalaženje optimalnih rešenja problema. Pokazuje se da je najefikasnija hibridna tehnika koja kombinuje svođenje na SAT i lokalnu pretragu. Razmotren je i problem sudoku, kao i postojece tehnike rešavanja sudoku zagonetki vecih dimenzija od 9 x 9. Pokazuje se da je u rešavanju ovih zagonetki najefikasnije vec postojece svođenje na SAT. Unapređen je postojeci algoritam za generisanje velikih sudoku zagonetki. Pokazano je da jednostavna pravila preprocesiranja dodatno unapređuju brzinu generisanja sudokua.Many real-world problems can be modeled as constraint satisfaction problems (CSPs) and then solved by one of many available techniques for solving these problems. One of the techniques is reduction to SAT, i.e. Boolean Satisfiability Problem. Variables and constraints of CSP are translated (encoded) to SAT instance, that is then solved by state-of-the-art SAT solvers and solution, if exists, is translated to the solution of the original CSP. The main aim of this thesis is to improve CSP solving techniques that are using reduction to SAT. Two new hybrid encodings of CSPs to SAT are presented and they combine good sides of the existing encodings. We give the proof of correctness of one encoding that did not exist in literature. We developed system meSAT that enables reduction of CSPs to SAT by using 4 basic and 2 hybrid encodings. The system also enables solving of CSPs by reduction to two problems related to SAT, SMT and PB. We developed a portfolio for automated selection of encoding/solver to be used on some new instance that needs to be solved. The developed portfolio is comparable with the state-of-the-art portfolios. We developed a hybrid approach based on short solving timeouts with the aim of significantly reducing the preparation time of a portfolio. By using this approach, we got results comparable to the ones obtained by using preparation time of usual length. We made comparison between several machine learning techniques with the aim to find out which one is the best suited for the short training approach. The problem of assigning air traffic controllers to shifts is described and three models of this problem are presented. We used a large number of different solving methods and a diverse set of solvers for solving this problem. We developed optimization techniques that aim to find optimal solutions of the problem. A hybrid technique combining reduction to SAT and local search is shown to be the most efficient one. We also considered sudoku puzzles and the existing techniques of solving the puzzles of greater size than 9x9. Amongst the used techniques, the existing reduction to SAT is the most efficient in solving these puzzles. We improved the existing algorithm for generating large sudoku puzzles. It is shown that simple preprocessing rules additionally improve speed of generating large sudokus

Solving Sudoku with Membrane Computing

Sudoku is a very popular puzzle which consists on placing several numbers in a squared grid according to some simple rules. In this paper we present an efficient family of P systems which solve sudokus of any order verifying a specific property. The solution is searched by using a simple human-style method. If the sudoku cannot be solved by using this strategy, the P system detects this drawback and then the computations stops and returns No. Otherwise, the P system encodes the solution and returns Yes in the last computation step.Ministerio de Ciencia e Innovación TIN2008-04487-EMinisterio de Ciencia e Innovación TIN2009–13192Junta de Andalucía P08-TIC-0420