SAT BASED ALGORITHMIC APPROACH FOR SUDOKU PUZZLE

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1 International Journal of Computer Engineering & Technology (IJCET) Volume 9, Issue 6, November-December 2018, pp , Article ID: IJCET_09_06_005 Available online at Journal Impact Factor (2016): (Calculated by GISI) ISSN Print: and ISSN Online: IAEME Publication SAT BASED ALGORITHMIC APPROACH FOR SUDOKU PUZZLE Deepika Rai Assistant Professor, School of Computer Science, IT Devi Ahilya Vishwavidyalaya, Indore, India N.S. Chaudhari Professor, Computer Science & Engineering, IIT, Indore, India Maya Ingle Sr. System Analyst, School of Computer Science and IT, Devi Ahilya Vishwavidyalaya, Indore, India ABSTRACT In theoretical computer science, Sudoku puzzle has always grappled its importance since it is an NP-complete problem. Most of the existing algorithms are exponential for large instances of Sudoku puzzle. In this paper, we present an efficient approach for Sudoku puzzle in which constraints of Sudoku puzzle are encoded into SAT clauses and subsequently encoded into 3-SAT clauses. 3-SAT formulation of Sudoku puzzle (n n) is very advantageous to obtain the solution of Sudoku puzzle using SAT solver. In this way, we design two algorithms SP 2 SAT and SP 2 3SAT that generate SAT clauses and 3-SAT clauses respectively in polynomial time. Algorithm I: SP 2 SAT generates total [ * (3n 2 3n + 2) + m] SAT clauses corresponding to Sudoku puzzle of size (n n). On the other hand, Algorithm II: SP 2 3SAT creates total [ * (3n 2 n 4) + m] 3-SAT clauses corresponding to SAT clauses. Our approach provides an efficient way to obtain the solution of Sudoku puzzle as it produces fewer number of 3-SAT clauses than existing approach. Key words: Conjunctive Normal Form, NP-Complete Problem, Polynomial Time, 3- SAT, Sudoku Puzzle. Cite this Article: Deepika Rai, N.S. Chaudhari and Maya Ingle, SAT Based Algorithmic Approach for Sudoku Puzzle. International Journal of Computer Engineering and Technology, 9(6), 2018, pp editor@iaeme.com

2 Deepika Rai, N.S. Chaudhari and Maya Ingle 1. INTRODUCTION Many problems such as software and hardware verification, automatic test-pattern generation, graph problems, combinational equivalence checking, scheduling, planning in artificial intelligence etc. have been represented in terms of SAT, in last two decades [1]. In recent years, several difficult problems in diversified domains are encoded into SAT to obtain the solution using SAT solver. As an example, n-fraction puzzle has been encoded as a SAT problem and obtained the solution of two open instances of this puzzle by using SAT solver [2]. Planning problem can also be modeled as SAT which is more accurate approach than deduction approach. SAT approach provides more flexible framework for constraint based planning system [3]. Also, detection of Hardware Trojan Horses (HTH) is done by Automatic Test Pattern Generation (ATPG) technique which is based on Genetic Algorithm and Boolean Satisfiability. This technique has been achieved higher detection than existing logic testing based Trojan detection technique [4]. On the other hand, large number of NP-complete problems are reduced into 3-SAT in polynomial time and solved by SAT solver. Some of these are channel assignment problem in cellular network, graph coloring problem, determining cliques and vertex cover for particular graph etc. [5][6][7]. Additionally, Sudoku puzzle (n n) is a numerical combinatorial puzzle which is also an NP-complete problem. There exist many approaches for the solution of Sudoku puzzle such as stochastic search, genetic algorithms, backtracking algorithms, integer linear programming etc. Aforementioned approaches can be used to solve Sudoku puzzle only for small value of n [8][9][10]. Due to NP-completeness of Sudoku puzzle, it can be represented in terms of 3-SAT to acquire the solution. There exists an exact 3-SAT encoding of Sudoku puzzle that generates [4 * (n 4 2n 2 + m)] 3-SAT clauses [11]. However, there is a scope to obtain an improved polynomial 3- SAT encoding of Sudoku puzzle of size (n n) based on constraint reduction. In this paper, we encode Sudoku puzzle of size (n n) into SAT clauses and subsequently convert into 3-SAT clauses. In Section 2, we present formulations for SAT and 3-SAT reductions of Sudoku puzzle (n n) on the basis of constraint reduction. The reductions furnish a crucial framework for converting one problem instance to other. In Section 3, we design two algorithms SP 2 SAT and SP 2 3SAT for the generation of clauses. Algorithm I: SP 2 SAT is used to obtain SAT clauses corresponding to Sudoku puzzle of size (n n) and Algorithm II: SP 2 3SAT converts the SAT clauses generated from Algorithm I, into 3-SAT clauses. In Section 4, we describe the execution results of both the algorithms SP 2 SAT and SP 2 3SAT for various Sudoku puzzles as well as highlight the number of 3-SAT clauses generated in polynomial 3-SAT reduction of Sudoku puzzle. Finally, we discuss the conclusion of our proposed work with suggesting future work in Section SAT CLAUSE GENERATION FOR SUDOKU PUZZLE (N N) We propose the constraint based 3-SAT encoding of Sudoku puzzle (n n). In this perspective, we review all the constraints of Sudoku puzzle and encode the constraints that are sufficient to describe Sudoku puzzle. The sufficient constraints in our case are categorized into two variants namely; Atleast Constraint for Cell (ACC) and Atmost Constraint for Row, Column and Block (ACRCB). ACC ensures that each cell of Sudoku puzzle (n n) contains atleast one number. ACRCB ensures that each number appears atmost once in each Row, Column and each Block of size ( n n). In other words, different numbers are to be filled in every cells of a row (column or block). Atmost constraint for Cell and atleast constraint for Row, Column and Block are redundant and not included in our SAT representation. Now, we transform these constrains into 3-SAT clauses to obtain the solution of Sudoku puzzle using SAT solver. In this context, let us assume all cells are represented as C(i, j), and a Boolean variable x i,j,k where 1 i, j, k n. Boolean variable x i,j,k is true if i th row and j th 39 editor@iaeme.com

3 SAT Based Algorithmic Approach for Sudoku Puzzle column of puzzle contains a number k otherwise it is false. In other words, x i,j,k is true if C(i, j) = k, otherwise it is false. There exist n possibilities to fill a particular cell, therefore SAT encoding requires n variables for each cell of Sudoku puzzle (n n). Now, we generate the 3- SAT clauses for both ACC and ACRCB constraints Atleast Constraint for Cell (ACC) According to this constraint, atleast one number is to be filled in each cell of Sudoku puzzle (n n). In other words, no one cell of Sudoku puzzle is to be remains blank. Thus, SAT representation of ACC constraint for n rows and n columns of a Sudoku puzzle of size (n n), as given by- ( x i,j,k ) (1) It is clear from eq. (1), one SAT clause is generated for each cell of Sudoku puzzle (n n). Consequently, n 2 clauses are produced for all cells and all are of length n. Now, we convert the obtained SAT clauses into 3-SAT clauses using well defined non recursive method [11]. Using this method, a SAT clause of k literals (k>3) is converted into (k-2) clauses of 3 distinct literals as well as this conversion is required (k-3) additional variables. Generated SAT clauses from eq. (1), are transformed into 3-SAT clauses using aforementioned method. As a result, [n 2 *(n 2)] 3-SAT clauses are obtained and [n 2 * (n 3)] additional variables required for conversion of SAT clauses to 3-SAT clauses. Now, we encode ACRCB into 3-SAT clauses as follows: 2.2. Atmost Constraint for Row, Column and Block (ACRCB) According to this constraint, each number appears in each row (each column or each block) of Sudoku puzzle without any repetition. Mathematically, if a number in a particular cell of a i th row of Sudoku puzzle is C(i, j) = k then C(i, m) k. In other words, if a cell in a particular row contains k then no other cell in same row contains k. Generalize SAT representation of ACRCB is given by: ( x i,j,k x i,m,k ) (2) In this way, (n 2 * n * (n 1)/2) SAT clauses are obtained for all n rows. Similarly, there is no repetition of any number in each column and each block of Sudoku puzzle and (n 3 * (n 1)/2) clauses are generated for all n columns and all n blocks. Therefore, total 3* [n 3 * (n 1) / 2] SAT clauses are attained for ACRCB. It is clear from eq. (2), all generated SAT clauses are of length 2 that are already in 3-SAT form. There is no need of any additional variables and hence, the number of SAT clauses and 3-SAT clauses generated for ACRCB are same. On the other hand, there exist m pre-assigned numbers in a given Sudoku puzzle (n n) that generate m unit clauses. Thus, total number of SAT clauses (say, C ) and 3-SAT clauses (say, C_3 ) generated for ACC and ACRCB are as follows: C = n 2 + [3* (n 3 * (n 1) / 2)] + m i.e. C = * [3n 2 3n + 2] + m and, C_3 = [n 2 * (n 2)] + [3 * (n 3 * (n 1) / 2)] + m i.e. C_3 = * [3n 2 n 4] + m Thus, we achieve a polynomial reduction of Sudoku puzzle into 3-SAT clauses. Value of C and C_3 shows that only [n 2 * (n 3)] clauses are increased in transformation of SAT clauses into 3-SAT clauses. On the basis of our formulation, we present two algorithms that generate SAT clauses and 3-SAT clauses for ACC and ACRCB in polynomial time editor@iaeme.com

4 Deepika Rai, N.S. Chaudhari and Maya Ingle 3. ALGORITHMIC APPROACH FOR 3SAT CLAUSE GENERATION We design two algorithms namely; Algorithm I: SP 2 SAT and Algorithm II: SP 2 3SAT. Algorithm I: SP 2 SAT generates all SAT clauses corresponding to constraints of Sudoku puzzle (n n). Input to the Algorithm I is the size of Sudoku puzzle and output file contains all SAT clauses corresponding to Sudoku puzzle in DIMACS form. On the other hand, Algorithm II: SP 2 3SAT converts all SAT clauses generated from Algorithm I into 3-SAT clauses in DIMACS form. Now, we describe our algorithms in informal and formal way as follows: 3.1. Informal Description Here, we describe the informal description of Algorithm I: SP 2 SAT and Algorithm II: SP 2 3SAT. For the input which is the size of Sudoku puzzle, Algorithm I: SP 2 SAT generates SAT clauses as output. We initialize ACC_clause and ACRCB_clause for storing a clause for ACC and ACRCB respectively. Also, we define a file SP 2 SAT.txt to store SAT clauses and a variable count, to count the number of SAT clauses. Variables i and j represent the row number and column number respectively and k represents the particular value in the cell. Loop for i and j are continuing according to constraints of Sudoku puzzle. In Algorithm II: SP 2 3SAT, input is a file SP 2 SAT.txt which is the output of Algorithm I. We define a variable k that counts the number of literals of a clause which is read from file SP 2 SAT.txt. Algorithm II: SP 2 3SAT generates 3-SAT clauses as output. We initialize a file SP 2 3SAT.txt to store 3-SAT clauses corresponding to SAT clauses obtained from Algorithm I. Now, we present the formal description of Algorithm I and Algorithm II are as follows: 3.2. Formal Description Now, we present a formal description of our algorithms SP 2 SAT and SP 2 3SAT. Algorithm I: SP 2 SAT is used for SAT clause generation according to constraints of Sudoku puzzle and Algorithm II: SP 2 3SAT helps to reduce SAT clauses into 3-SAT clauses. Algorithm I: // SP 2 SAT// //Input to the algorithm is the size of Sudoku puzzle, generate the SAT clauses corresponding to constraints of Sudoku puzzle// main ( ) //algorithm begins { STEP I: /* Input the size of Sudoku puzzle */ Input (n); STEP 2: /* Initialization of a file to store SAT clauses */ SP 2 SAT.txt = NULL; STEP 3: /* Generation of SAT clauses for ACC and write into SP 2 SAT.txt */ count 1 = NULL; // Initialize count 1 to store the number of clauses for ACC for (i = 1; i<= n; i++) do for (j = 1; j<= n; j++) do for (k = 1; j<= n; j++) do ACC_clause = (X 1,1,1 X 1,1,2 X 1,1,3 X 1, 1, k ); // A clause for each Cell count 1 = count 1 + 1; // count number of clauses SP 2 SAT.txt = ACC_clause; // write clause to output file 41 editor@iaeme.com

5 SAT Based Algorithmic Approach for Sudoku Puzzle STEP 4: /* Generation of SAT clauses for ACRCB */ count 2 = NULL; // Initialize count 2 to store the number of clauses for ACRCB STEP 4.1: /* Generation of SAT clauses for each Row */ for ( i = 1; i<=n ;i++) do for (k = i; k<= n; k++) do for ( j = 1; j<= n; j++) do // Print clause for: Cell C(i, j) and C(i, m) cannot contains k simultaneously ACRCB_clause = ( X i, j, k X i,m,k ) ; count 2 = count 2 + 1; // count number of clauses SP 2 SAT.txt = ACRCB_clause; // write clause to output file STEP 4.2: /* Generation of SAT clauses for each Column */ for (j = 1; i<= n; i++) do for (k=1; k<=n; k++) do for (i = 1; j<=n; j++) // Print clause for: Cell C(i, j) and C(m, j) cannot contains k simultaneously ACRCB_clause = ( X i, j, k X m, j, k ) ; count 2 = count 2 + 1; //count number of clauses SP 2 SAT.txt = ACRCB_clause; // write clause to output file STEP 4.3: /* Generation of SAT clauses for each Block. */ for (i = 1; i<=n; i++) do for (j = 1; j<=n; j++) do for (j = 1; j<=n; j++) do // Print clause for: Cell (i, j ) and Cell C(l, m) of same block cannot contains k simultaneously ACRCB_clause = ( X i, j, k X l,m, k ) ; count 2 =count 2 + 1; // count number of clauses SP 2 SAT.txt = ACRCB_clause; // write clause to output file STEP 5: /* Creation of output file and Calculate total number of clauses */ Output (SP 2 SAT.txt); 42 editor@iaeme.com

6 Deepika Rai, N.S. Chaudhari and Maya Ingle // Initialize count to store the total count of SAT clauses for constraints of Sudoku puzzle count = NULL; count = count 1 + count 2 ; } //end of main ( ) Algorithm II: //SP 2 3SAT// // Input to the algorithm is the file SP 2 SAT.txt and generate 3-SAT clauses// main ( ) //algorithm begins { STEP I: /* Input is the file generated from SP 2 SAT algorithm*/ READ (SP 2 SAT.txt); STEP 2: /* Initialize a file to store 3-SAT clauses */ SP 2 3SAT.txt = NULL; STEP 3: /*Read each SAT clause (say SP 2 SAT), count number of literals (say, k) and convert it into 3-SAT clause (say, SP 2 3SAT) if not.*/ if (k= = 1 2 3) // Clauses are already in 3-SAT begin SP 2 3SAT.txt = SP 2 SAT; // copy clause as it is in output file end if else if (k>3) // convert a SAT clause into 3-SAT clause begin SP 2 3SAT = (X 1,1,1 X 1,1,2 Y 1 ) ( Y 1 X 1,1,3 Y 2 ) ( Y 2 X 1,1,4 Y 3 ) ( Y n-3 X 1,1, (n-1) X 1,1,n ); SP 2 3SAT.txt = SP 2 3SAT; end if STEP 4: /* Convert all SAT clauses to 3-SAT clauses and obtained output file */ Repeat step 3 till the end of file; Output (SP 2 3SAT.txt); } //end of main ( ) 4. EXPERIMENTAL RESULTS We have implemented both algorithms SP 2 SAT and SP 2 3SAT in Java that generate SAT and 3-SAT clauses in DIMACS format. Implementation of Algorithm I: SP 2 SAT provides the SAT clauses in DIMACS format stored in a file. The obtained file is the input of Algorithm II: SP 2 3SAT that generates DIMACS file for 3-SAT clauses. We have executed both the algorithms on Sudoku puzzle of different sizes. As a result, we obtained the number of SAT clauses and 3-SAT clauses that are shown in Table 1. Implementation of Algorithm I and Algorithm II generate [ * (3n 2 3n + 2) + m] number of SAT clauses and [ * (3n 2 n 4) + m] 3-SAT clauses respectively in polynomial time. The experiments have been performed on Intel core i3 at 2.00 GHZ with 4 GB RAM. Implementation of Algorithm II: SP 2 3SAT 43 editor@iaeme.com

7 SAT Based Algorithmic Approach for Sudoku Puzzle shows that polynomial 3-SAT reduction of Sudoku puzzle requires only [n 2 * (n 3)] additional variables. It is observed that increases in number of clauses due to SAT to 3-SAT transformation is also [n 2 * (n 3)]. Hence, our approach is helpful to speed up the solution since complexity of SAT formula is dependent on number of clauses as well as number of variables in a clause. 5. CONCLUSIONS We have encoded Sudoku puzzle of size (n n) into 3-SAT clauses in polynomial time. We have used only sufficient constraints of Sudoku puzzle and define only two constraints namely; ACC and ACRCB. For the generation of SAT and 3-SAT clauses according to ACC and ACRCB, we designed algorithms SP 2 SAT and SP 2 3SAT respectively. As a result, we have obtained total number of 3-SAT clauses as well as number of additional variables required for polynomial 3-SAT reduction. Hence, we conclude that size of 3-SAT formula obtained by Algorithm II: SP 2 3SAT is O(n 4 ). Also, our 3-SAT reduction of Sudoku puzzle is polynomial in time as well as our algorithm generates better results for 3-SAT clauses. A direction for future work is to optimize the obtained number of 3-SAT clauses using some preprocessing techniques. Table 1 Results Obtained from Execution of Algorithm I and Algorithm II Size of Sudoku Puzzle (n n) No. of SAT Clauses C Elapsed Time for SAT Clause Generation (in seconds) No. of 3-SAT Clauses C_3 Elapsed Time for 3-SAT Clause Generation (in seconds) REFERENCES [1] Joao Marques-Silva, Practical applications of Boolean Satisfiability, International Workshop on Discrete Event Systems, IEEE, 2008, pp [2] Michael Codish, A SAT Encoding for the n-fractions Problem, Archives of Discrete Mathematics, Cornell University Library, [3] Stephen M.Majercik, APPSSAT: Approximate probabilistic planning using stochastic satisfiability, International Journal of Approximate Reasoning, Elsevier, 45(2), 2007, pp [4] Saha S., Chakraborty R.S., Nuthakki S.S., Anshul, Mukhopadhyay D., Improved Test Pattern Generation for Hardware Trojan Detection Using Genetic Algorithm and Boolean Satisfiability, Cryptographic Hardware and Embedded Systems-CHES, Lecture Notes in Computer Science, Springer, 9293, 2015, pp [5] Prakash C. Sharma and Narendra S. Chaudhari, A Graph Coloring approach for channel assignment in cellular network via propositional satisfiability, International Conference on 44 editor@iaeme.com

8 Deepika Rai, N.S. Chaudhari and Maya Ingle Emerging Trends in Networks and Computer Communications (ETNCC), IEEE, 2011, pp [6] Dahale N., Chaudhari, N.S. Ingle M., GCP constraint based SAT clause generation in polynomial time, International Journal of Advance Research in Computer Science, 7(2), 2016, pp [7] N. Dahale, N.S. Chaudhari, M. Ingle, Determining Vertex Cover Using Polynomial Encoding of 3sat, VNSGU Journal of Science and Technology, 4(1), 2015, pp [8] S. Chatterjee, S. Paladhi, R. Chakraborty, A comparative study on the performance characteristics of Sudoku solving algorithms, IOSR Journal of Computer Engineering (IOSR-JCE), 16(5), 2014, pp [9] Perez, Meir, Tshilidzi Marwala, Stochastic optimization approaches for solving Sudoku, Archives of Neural and Evolutionary Computing, Cornell University Library, [10] Bartlett, Andrew C., Amy N. Langville, An Integer Programming Model for the Sudoku Problem, Journal of Online Mathematics and its Applications, 8, Article ID 1798, [11] Rai Deepika, Chaudhari N.S., Ingle M., Polynomial 3-SAT reduction of Sudoku puzzle, International Journal of Advanced Research in Computer Science, 9(3), 2018, pp editor@iaeme.com