Introduction to Constraint Programming in Copris

Introduction to Constraint Programming in Copris
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1.Introduction to Constraint Programming in Copris Naoyuki Tamura 2018-03-21

2.Contents 1 Introduction 2 2 Installation 2 3 First Step 2 3.1 Using REPL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.1.1 Defining CSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.1.2 Searching solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Writing script files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 Small Examples 5 4.1 A Coin Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.1.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 Subset Sum Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.3 Magic Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.3.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.4 Graph Coloring Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.4.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.5 Sudoku . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.5.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.6 Bin Packing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.6.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.7 Knapsack Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.7.1 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.8 4-Queens Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 Short Summary of Classes and Methods 11 5.1 Integer Variable Objects (Var objects) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2 Term Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.3 Constraint Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.4 CSP Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.4.1 Declaring Integer Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.4.2 Adding Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.5 Searching Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.6 Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.6.1 Switching SAT Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.6.2 Using PB Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.6.3 Using SMT Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.6.4 Using JSR-331 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.6.5 Importing XCSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.6.6 Exporting/Importing Sugar CSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.6.7 Using Multiple CSPs+Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.6.8 Aviod Using Copris DSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1

3. Introduction to Constraint Programming in Copris 1 Introduction This document describes an introduction to Constraint Programming by using Copris which is a DSL (Domain Specific Language) for constraint programming embedded in Scala language. 2 Installation 1. Install Java runtime version 1.8 or later 2. Install Scala version 2.11 3. Check the version of your Scala $ scala -version Scala code runner version 2.11.8 -- Copyright 2002-2016, LAMP/EPFL Copris can run on Scala version 2.11 (not on other versions). 4. Download Copris package 5. Unzip Copris package 3 First Step 3.1 Using REPL Let us consider the following example. You have some of 1 cent, 5 cents, and 10 cents coins. The number of coins is 15 in total and at least one for each. The total sum of the coins is 90 cents. What are the numbers of each coin? The problem can be formalized as the following CSP (Constraint Satisfaction Problem). x ∈ {1 .. 15} y ∈ {1 .. 15} z ∈ {1 .. 15} x + y + z = 15 x + 5y + 10z = 90 Let us solve the problem in Copris. Move to the “˜build˜” folder of Copris, and start Copris as follows. $ scala -cp copris-all-v2-2-8.jar 3.1.1 Defining CSP After you get the input prompt of Scala, you can directly define your CSP within Scala REPL. scala> import jp.kobe_u.copris._ scala> import jp.kobe_u.copris.dsl._ scala> int(’x, 1, 15) scala> int(’y, 1, 15) scala> int(’z, 1, 15) scala> add(’x + ’y + ’z === 15) scala> add(’x + ’y * 5 + ’z * 10 === 90) • The first line imports the class names provided by Copris. • The second line imports the methods provided by Copris DSL. • The third line declares the integer variable x (its lower and upper bounds are 1 and 15 respectively). The notation starts from single quotation mark denotes a Symbol object in Scala, and it is implicitly converted to the integer variable object (Var object) of Copris by Copris DSL. • The fourth and fifth lines declare the integer variables y and z in the same way. 2

4. Introduction to Constraint Programming in Copris • The sixth line adds the constraint x + y + z = 15. Note that === is used for the equality. The method add adds the constraint to the CSP object. • The seventh line adds the constraint x + 5y + 10z = 90. You can not use the notation 5 * ’y. Use Num(5) * ’y when you want to write an integer constant prior. The variable ‘csp’ refers the defined CSP object. scala> csp res: CSP = CSP(Vector(x, y, z),Vector(),Map(...),Vector(...)) The CSP object consists of a sequence of integer variables ‘variables’, a sequence of Boolean variables ‘bools’, a map of integer variable domains ‘dom’, and a sequence of constraints ‘constraints’. scala> csp.variables res: IndexedSeq[Var] = Vector(x, y, z) scala> csp.bools res: IndexedSeq[Bool] = Vector() scala> csp.dom res: Map[Var,Domain] = Map(x -> Domain(1,15), y -> Domain(1,15), z -> Domain(1,15)) scala> csp.constraints res: IndexedSeq[Constraint] = Vector(Eq(Add(Add(x,y),z),15), Eq(Add(Add(x,Mul(y,5)),Mul(z,10)),90)) The CSP can be displayed by ‘show’ command. scala> show int(x,1,15) int(y,1,15) int(z,1,15) Eq(Add(Add(x,y),z),15) Eq(Add(Add(x,Mul(y,5)),Mul(z,10)),90) • Here, ‘Add’ means addition, ‘Mul’ means multiplication,and ‘Eq’ means equality. CSP object is implemented as a mutable object for which additions of variables and constraints are allowed. 3.1.2 Searching solutions The ‘find’ method searches the first solution of the defined CSP. scala> find res: Boolean = true The result ‘true’ means there exists a solution. The CSP solution of CSP is stored in the ‘solution’ variable. scala> solution res: Solution = Solution(Map(x -> 5, y -> 3, z -> 7),Map()) A Solution object consists of a map representing an assignment for integer variables (Var objects) and a map representing an assignment for Boolean variables (Bool objects). scala> solution.intValues res: Map[Var,Int] = Map(x -> 5, y -> 3, z -> 7) scala> solution.boolValues res: Map[Bool,Boolean] = Map() A value assigned for each variable can be obtained by ‘solution’ method. scala> solution(’x) res: Int = 5 scala> solution(’y) res: Int = 3 scala> solution(’z) res: Int = 7 The ‘findNext’ method searches the next solution. scala> findNext res: Boolean = false 3

5. Introduction to Constraint Programming in Copris The result ‘false’ means there exist no more solutions. And the ‘solution’ has null value. scala> solution res: Solution = null You can confirm the extra condition by ‘show’ method. scala> show int(x,1,15) int(y,1,15) int(z,1,15) Eq(Add(Add(x,y),z),15) Eq(Add(Add(x,Mul(y,5)),Mul(z,10)),90) Not(And(And(Eq(x,5),Eq(y,3),Eq(z,7)),And())) The extra condition can be removed by ‘cancel’ method. scala> cancel scala> show int(x,1,15) int(y,1,15) int(z,1,15) Eq(Add(Add(x,y),z),15) Eq(Add(Add(x,Mul(y,5)),Mul(z,10)),90) A list of solutions can be obtained by ‘solutions’ method. scala> solutions.foreach(println) Solution(Map(x -> 5, y -> 3, z -> 7),Map()) You need to remove extra conditions by ‘cancel’ method when you want to get all solutions after ‘findNext’ is executed. 3.2 Writing script files CSP can be defined as a script file of Scala. The next shows the script file of the above example (Example-csp.scala). 1: import jp.kobe_u.copris._ 2: import jp.kobe_u.copris.dsl._ 3: 4: int(’x, 1, 15) 5: int(’y, 1, 15) 6: int(’z, 1, 15) 7: add(’x + ’y + ’z === 15) 8: add(’x + ’y * 5 + ’z * 10 === 90) The script file can be executed by the ‘:load’ command. scala> :load Example-csp.scala Loading Example-csp.scala... ..... scala> find res: Boolean = true scala> solution res: Solution = Solution(Map(x -> 5, y -> 3, z -> 7),Map()) If you want to reload the script file after you modify it, use ‘init’ method to initialize the CSP and the Solver before loading. scala> init scala> :load Example-csp.scala We will use ‘init’ method in the head of the script hereafter. 3.2.1 Exercise 1. Try the example for 105 cents. 4

6. Introduction to Constraint Programming in Copris 4 Small Examples 4.1 A Coin Problem First example is a coin problem. You have the following Euro coins (148 cents in total): • four 20 cents coins, • four 10 cents coins, • four 5 cents coins, and • four 2 cents coins. Is there a combination for 93 cents in total? The problem can be formalized as the following CSP (Constraint Satisfaction Problem). Each variable xi represents the number of i cents coins. x20 ∈ {0 .. 4} x10 ∈ {0 .. 4} x5 ∈ {0 .. 4} x2 ∈ {0 .. 4} 20x20 + 10x10 + 5x5 + 2x2 = 93 The CSP file can be written as follows. (Coin-csp.scala). 1: import jp.kobe_u.copris._ 2: import jp.kobe_u.copris.dsl._ 3: 4: init 5: int(’x(20), 0, 4) 6: int(’x(10), 0, 4) 7: int(’x(5), 0, 4) 8: int(’x(2), 0, 4) 9: add(’x(20) * 20 + ’x(10) * 10 + ’x(5) * 5 + ’x(2) * 2 === 93) In Copris, you can use indices as used in ’x(20). Indices can be integers or character strings (or any Scala objects). It is also possible to use multiple indices, such as ’x(1,"a"). 4.1.1 Exercise 1. How many solutions are there for the above problem? • (Answer) You can find it as follows. scala> cancel scala> solutions.size scala> cancel scala> solutions.foreach(println) 2. How many solutions are there when the number of coins is three? • (Answer) You can find it as follows. scala> :load Coin-csp.scala scala> add(’x(20) <= 3) scala> add(’x(10) <= 3) scala> add(’x(5) <= 3) scala> add(’x(2) <= 3) scala> find 5

7. Introduction to Constraint Programming in Copris 4.2 Subset Sum Problem Is there a subset of {2, 3, 5, 8, 13, 21, 34} which sums to 50? This problem is known as the Subset Sum Problem (Wikipedia:Subset sum problem). The subset sum problem is NP-complete. This problem can be formalized as the following CSP (Constraint Satisfaction Problem). x2 ∈ {0, 1} x3 ∈ {0, 1} x5 ∈ {0, 1} x8 ∈ {0, 1} x13 ∈ {0, 1} x21 ∈ {0, 1} x34 ∈ {0, 1} 2x2 + 3x3 + 5x5 + 8x8 + 13x13 + 21x21 + 34x34 = 50 The CSP file can be written as follows. (SubsetSum-csp.scala). 1: import jp.kobe_u.copris._ 2: import jp.kobe_u.copris.dsl._ 3: 4: def define(sum: Int) { 5: init 6: boolInt(’x(2)) 7: boolInt(’x(3)) 8: boolInt(’x(5)) 9: boolInt(’x(8)) 10: boolInt(’x(13)) 11: boolInt(’x(21)) 12: boolInt(’x(34)) 13: add(’x(2)*2 + ’x(3)*3 + ’x(5)*5 + ’x(8)*8 + ’x(13)*13 + ’x(21)*21 + ’x(34)*34 === sum) 14: } The ‘boolInt’ method declares a 0-1 integer variable, and boolInt(x) is equivalent to int(x, 0, 1). In the above program, the function define(sum: Int) is declared to define the CSP for the given summation. scala> :load SubsetSum-csp.scala scala> define(50) scala> find 4.2.1 Exercise 1. How the solution will be when the sum is 40? • (Answer) You can find it as follows. scala> define(40) scala> find 4.3 Magic Square Arrange the numbers from 1 to 9 in 3 × 3 matrix such that the sum of each row, each column, and each diagonal becomes 15. This arrangement is called a Magic Square (Wikipedia:Magic square). This problem can be formalized as the following CSP. xij ∈ {1 .. 9} (i = 0 .. 2, j = 0 .. 2) Alldifferent(x00 , x01 , . . . , x22 ) xi0 + xi1 + xi2 = 15 (i = 0 .. 2) x0j + x1j + x2j = 15 (j = 0 .. 2) x00 + x11 + x22 = 15 x02 + x11 + x20 = 15 6

8. Introduction to Constraint Programming in Copris Here, ‘Alldifferent’ means a constraint where the given arguments are mutually different. That is, Alldifferent(x1 , x2 , . . . , xn ) is equivalent to xi ̸= xj (for all i < j). The CSP file can be written as follows. (MagicSquare3-csp.scala). 1: import jp.kobe_u.copris._ 2: import jp.kobe_u.copris.dsl._ 3: 4: init 5: int(’x(0,0), 1, 9); int(’x(0,1), 1, 9); int(’x(0,2), 1, 9) 6: int(’x(1,0), 1, 9); int(’x(1,1), 1, 9); int(’x(1,2), 1, 9) 7: int(’x(2,0), 1, 9); int(’x(2,1), 1, 9); int(’x(2,2), 1, 9) 8: add(Alldifferent( 9: ’x(0,0), ’x(0,1), ’x(0,2), 10: ’x(1,0), ’x(1,1), ’x(1,2), 11: ’x(2,0), ’x(2,1), ’x(2,2) 12: )) 13: add(’x(0,0) + ’x(0,1) + ’x(0,2) === 15) 14: add(’x(1,0) + ’x(1,1) + ’x(1,2) === 15) 15: add(’x(2,0) + ’x(2,1) + ’x(2,2) === 15) 16: add(’x(0,0) + ’x(1,0) + ’x(2,0) === 15) 17: add(’x(0,1) + ’x(1,1) + ’x(2,1) === 15) 18: add(’x(0,2) + ’x(1,2) + ’x(2,2) === 15) 19: add(’x(0,0) + ’x(1,1) + ’x(2,2) === 15) 20: add(’x(0,2) + ’x(1,1) + ’x(2,0) === 15) 4.3.1 Exercise 1. How many solutions of 3 × 3 magic square? • (Answer) You can find it as follows. scala> solutions.size 2. What constraints should be added to break symmetries of rotation and reflection? • (Answer) You can find it as follows. scala> cancel scala> add(’x(0,0) < ’x(0,2)) scala> add(’x(0,0) < ’x(2,0)) scala> add(’x(0,0) < ’x(2,2)) scala> add(’x(0,2) < ’x(2,0)) scala> solutions.foreach(println) 3. Write a CSP of n × n magic square in Copris. • (Answer) See MagicSquare-csp.scala. 4.4 Graph Coloring Problem For each vertex of the following graph, assign the numbers from 1 to 3 such that the adjacent vertices have different numbers. This problem is known as a Graph Coloring Problem (GCP) (Wikipedia:Graph coloring). In the above graph, you can color vertices as v1 = 1, v2 = 2, v3 = 3, and v4 = 1. In addition, there are no such coloring with two colors. 7

9. Introduction to Constraint Programming in Copris This problem can be formalized as the following CSP. v1 ∈ {1, 2, 3} v2 ∈ {1, 2, 3} v3 ∈ {1, 2, 3} v4 ∈ {1, 2, 3} v1 ̸= v2 v1 ̸= v3 v2 ̸= v3 v2 ̸= v4 v3 ̸= v4 The CSP file can be written as follows. (GCP-csp.scala). 1: import jp.kobe_u.copris._ 2: import jp.kobe_u.copris.dsl._ 3: 4: def define(edges: Set[(Int,Int)], colors: Int) { 5: init 6: val nodes = for ((i,j) <- edges; k <- Seq(i,j)) yield k 7: for (i <- nodes) 8: int(’v(i), 1, colors) 9: for ((i,j) <- edges) 10: add(’v(i) =/= ’v(j)) 11: } 12: 13: val graph1 = Set((1,2), (1,3), (2,3), (2,4), (3,4)) scala> :load GCP-csp.scala scala> define(graph1, 3) scala> find 4.4.1 Exercise 1. How to define the CSP when the adjacent vertices have numbers at least two apart? (Hint: the absolute value of x is written as Abs(x) in Copris) 4.5 Sudoku Next, we try to solve Sudoku (Wikipedia:Sudoku). The CSP file can be written as follows. (Sudoku-csp.scala). 1: import jp.kobe_u.copris._ 2: import jp.kobe_u.copris.dsl._ 3: 4: def define(board: Seq[Seq[Int]]) { 5: val n = 9; val m = 3 6: init 7: for (i <- 0 until n; j <- 0 until n) 8: int(’x(i,j), 1, n) 9: for (i <- 0 until n) 10: add(Alldifferent((0 until n).map(j => ’x(i,j)))) 11: for (j <- 0 until n) 12: add(Alldifferent((0 until n).map(i => ’x(i,j)))) 13: for (i <- 0 until n by m; j <- 0 until n by m) { 14: val xs = for (di <- 0 until m; dj <- 0 until m) 15: yield ’x(i+di,j+dj) 16: add(Alldifferent(xs)) 17: } 18: for (i <- 0 until n; j <- 0 until n) 19: if (board(i)(j) > 0) 20: add(’x(i,j) === board(i)(j)) 21: } 22: 23: val sudoku_fujiwara = Seq( 24: Seq(0, 0, 0, 0, 0, 0, 0, 0, 0), 25: Seq(0, 4, 3, 0, 0, 0, 6, 7, 0), 26: Seq(5, 0, 0, 4, 0, 2, 0, 0, 8), 8

10. Introduction to Constraint Programming in Copris 27: Seq(8, 0, 0, 0, 6, 0, 0, 0, 1), 28: Seq(2, 0, 0, 0, 0, 0, 0, 0, 5), 29: Seq(0, 5, 0, 0, 0, 0, 0, 4, 0), 30: Seq(0, 0, 6, 0, 0, 0, 7, 0, 0), 31: Seq(0, 0, 0, 5, 0, 1, 0, 0, 0), 32: Seq(0, 0, 0, 0, 8, 0, 0, 0, 0) 33: ) scala> :load Sudoku-csp.scala scala> define(sudoku_fujiwara) scala> find 4.5.1 Exercise 1. Try to solve problems in http://puzzle.gr.jp. 4.6 Bin Packing Problem Four persons will carry eight bags. Weight of each bag is 3.3kg, 6.1kg, 5.8kg, 4.1kg, 5.0kg, 2.1kg, 6.0kg, and 6.4kg (38.8kg in total). Can you assign bags to persons so that each person carries at most 11kg? This problem is known as a Bin Packing Problem (Wikipedia:Bin packing problem). To formalize this problem as CSP, we consider a matrix where each row corresponds to a person, and each column corresponds to a baggage. Each components of the matrix is eight 0 or 1. Such matrix is called an incidence matrix. Consider the constraints for each row and column of the incidence matrix. Note that you need to specify the weights in 100g unit for example since real numbers are not allowed in Copris. 4.6.1 Exercise 1. What is the constraint for each row? 2. What is the constraint for each column? 3. Describe the CSP in Copris. • (Answer) See BinPacking-csp.scala. 4. How the solutions will be when the maximum weight is 10kg? • (Answer) You can find it as follows. scala> :load BinPacking-csp.scala scala> ex1(max = 110) 4.7 Knapsack Problem There are five items, and the price and the value for each item is given by the following table. Is there a combination of items where the total price is at most 15, and the total value is at least 19? Item Price Value Item0 3 4 Item1 4 5 Item2 5 6 Item3 7 8 Item4 9 10 The CSP file can be written as follows. (Knapsack-csp.scala). 1: import jp.kobe_u.copris._ 2: import jp.kobe_u.copris.dsl._ 3: 4: def define(maxCost: Int, minValue: Int, costs: Seq[Int], values: Seq[Int]) { 9

11. Introduction to Constraint Programming in Copris 5: init 6: val n = costs.size 7: for (i <- 0 until n) 8: boolInt(’x(i)) 9: val c = for (i <- 0 until n) yield ’x(i) * costs(i) 10: add(Add(c) <= maxCost) 11: val v = for (i <- 0 until n) yield ’x(i) * values(i) 12: add(Add(v) >= minValue) 13: } 14: 15: def ex1(maxCost: Int = 15, minValue: Int = 19) { 16: val costs = Seq(3, 4, 5, 7, 9) 17: val values = Seq(4, 5, 6, 8, 10) 18: define(maxCost, minValue, costs, values) 19: if (find) { 20: val items = (0 until costs.size).filter(i => solution(’x(i)) == 1) 21: println(items) 22: } 23: } When the upper bound of the total price is given, the problem to find a combination of items with the maximum total value is called a Knapsack Problem (Wikipedia:Knapsack problem). In these problems, we need to find a solution where the value of the specified variable is maximized (or minimized). Such problems are called a Constraint Optimization Problem (COP). You can solve COP in Copris by specifying the objective variable by ‘maximize’ (or ‘minimize’) method. The optimum solution can be found by ‘findOpt’ method. The CSP file can be modified as follows. val v = for (i <- 0 until n) yield ’x(i) * values(i) int(’value, 0, values.sum) add(’value === Add(v)) maximize(’value) 4.7.1 Exercise 1. Find a solution with maximum total value. • (Answer) See KnapsackOpt-csp.scala. scala> :load KnapsackOpt-csp.scala scala> ex1() 4.8 4-Queens Problem Place four chess Queens on 4 × 4 chessboard such that no two queens attack each other. In general, the problem of placing n Queens on n×n chessboard is called an n-Queens problem (Wikipedia:Eight queens puzzle). When we use 0-1 variable xij for each row i and column j, the 4-queens problem can be formalized as follows. xij ∈ {0, 1} (i = 0 .. 3, j = 0 .. 3) xi0 + xi1 + xi2 + xi3 = 1 (i = 0 .. 3) x0j + x1j + x2j + x3j = 1 (j = 0 .. 3) x01 + x10 ≤ 1 x02 + x11 + x20 ≤ 1 x03 + x12 + x21 + x30 ≤ 1 x13 + x22 + x31 ≤ 1 x23 + x32 ≤ 1 x02 + x13 ≤ 1 x01 + x12 + x23 ≤ 1 x00 + x11 + x22 + x33 ≤ 1 x10 + x21 + x32 ≤ 1 x20 + x31 ≤ 1 10

12. Introduction to Constraint Programming in Copris A constraint on the summation of 0-1 variables is called a Boolean cardinality constraint. Copris can automatically encode Boolean cardinality constraints to SAT. However, we try to implement the encoding explicitly in the program. There are a number of encodings of Boolean cardinality constraints proposed. Here, we use a pair-wise encoding which will be the simplest way. The CSP file can be written as follows (QueensBC-csp.scala). 1: import jp.kobe_u.copris._ 2: import jp.kobe_u.copris.dsl._ 3: 4: def addAtLeastOne(xs: Seq[Term]) { 5: add(Or(xs.map(x => x.?))) 6: } 7: def addAtMostOne(xs: Seq[Term]) { 8: for (Seq(x1,x2) <- xs.combinations(2)) 9: add(! (x1.? && x2.?)) 10: } 11: def addExactOne(xs: Seq[Term]) { 12: addAtLeastOne(xs) 13: addAtMostOne(xs) 14: } 15: 16: def define(n: Int) { 17: init 18: for (i <- 0 until n; j <- 0 until n) 19: boolInt(’x(i,j)) 20: for (i <- 0 until n) { 21: val xs = for (j <- 0 until n) yield ’x(i,j) 22: addExactOne(xs) 23: } 24: for (j <- 0 until n) { 25: val xs = for (i <- 0 until n) yield ’x(i,j) 26: addExactOne(xs) 27: } 28: for (s <- 1 to 2*n-3) { 29: val xs = for (i <- 0 until n; j <- 0 until n; if s == i+j) yield ’x(i,j) 30: addAtMostOne(xs) 31: } 32: for (s <- 2-n to n-2) { 33: val xs = for (i <- 0 until n; j <- 0 until n; if s == i-j) yield ’x(i,j) 34: addAtMostOne(xs) 35: } 36: } Here, x.? is equivalent to Ge(x,1). scala> :load QueensBC-csp.scala scala> val n = 4 scala> define(n) scala> find scala> for (i <- 0 until n) println((0 until n).map(j => solution(’x(i,j))).mkString(" ")) You might want to add the following implicit conversions. implicit def term2constraint(x: Term) = x.? implicit def symbol2constraint(s: Symbol) = Var(s.name).? Then, you can write constraints without ‘?’ methods as follows. boolInt(’x) boolInt(’y) add(’x + ’y <= 1) add(’x || ’y) However, these implicit conversions may be a cause of undesired (and unexpected) translation. Care should be taken to use them. 5 Short Summary of Classes and Methods This section describes a short summary of classes and methods provided by Copris. 11

13. Introduction to Constraint Programming in Copris 5.1 Integer Variable Objects (Var objects) Var creates an integer variable object. The argument specifies its name. scala> val x = Var("x") x: jp.kobe_u.copris.Var = x When no arguments are given, a new anonymous integer variable object is created. scala> var z = Var() z: jp.kobe_u.copris.Var = _I1 New Var object is created by applying an index to an exisiting Var object. Integers or character strings can be used as indices, and multiple indices can be given. However, integer variable objects of Copris can not be used as indices. scala> x(1) res: jp.kobe_u.copris.Var = x(1) scala> x("a") res: jp.kobe_u.copris.Var = x(a) scala> x(1, "a") res: jp.kobe_u.copris.Var = x(1,a) Symbol of Scala is implicitly converted to a Var object. scala> ’x(1) res: jp.kobe_u.copris.Var = x(1) Var is a subclass of Term class described later. 5.2 Term Objects Term object is either a Var object or one of the following object. • Num(Int) Num object represents a number. scala> Num(314) res: jp.kobe_u.copris.Num = 314 • Abs(T ) Abs object represents an absolute value. scala> Abs(’x) res: jp.kobe_u.copris.Abs = Abs(x) • Neg(T ) Neg object represents a unary minus. Unary operator - can be used also. scala> - ’x res: jp.kobe_u.copris.Neg = Neg(x) • Add(T1, . . . , Tn), Add(Seq(T1, . . . , Tn)) Add object represents an addition. Binary operator + can be used also. scala> ’x + ’y res: jp.kobe_u.copris.Add = Add(x,y) • Sub(T1, . . . , Tn), Sub(Seq(T1, . . . , Tn)) Sub object represents a subtraction. Binary operator - can be used also. scala> ’x - ’y res: jp.kobe_u.copris.Sub = Sub(x,y) • Mul(T1, . . . , Tn), Mul(Seq(T1, . . . , Tn)) Mul object represents a multiplication. The multiplier should be an integer constant. Binary operator * can be used also. scala> ’x * 2 res: jp.kobe_u.copris.Mul = Mul(x,2) • Div(T1, T2 ), Mod(T1, T2 ) Div and Mod operators represent a division and remainder operations respectively. The divisor should be an integer constant. Binary operators / and % can be used also. 12

14. Introduction to Constraint Programming in Copris scala> ’x / 3 res: jp.kobe_u.copris.Div = Div(x,3) • Max(T1, . . . , Tn), Max(Seq(T1, . . . , Tn)), Min(T1, . . . , Tn), Min(Seq(T1, . . . , Tn)) Max and Min object represent a maximum and minimum operations respectively. scala> Max(’x+1, ’y+2) res: jp.kobe_u.copris.Max = Max(Add(x,1),Add(y,2)) • If(C, T1, T2 ) If object represetns an ‘if’ expression which is equal to T1 when C is true, and equal to T2 otherwise. scala> If(’x > 0, Num(1), Num(0)) res: jp.kobe_u.copris.If = If(Gt(x,0),1,0) 5.3 Constraint Objects • Eq(T1, T2 ), Ne(T1, T2 ), Le(T1, T2 ), Lt(T1, T2 ), Ge(T1, T2 ), Gt(T1, T2 ) Eq, Ne, Le, Lt, Ge, and Gt objects represent a comparison. Binary operators ===, =/=, <=, <, >=, and > can be used also. scala> ’x === ’y res: jp.kobe_u.copris.Eq = Eq(x,y) Expression “˜x.?˜” and “˜x.!˜” are equivalent to “˜x ¿= 1˜” and “˜x ¡= 0˜” respectively. scala> ’x.? res: jp.kobe_u.copris.Eq = Ge(x,1) scala> ’x.! res: jp.kobe_u.copris.Le = Le(x,0) • Alldifferent(T1, . . . , Tn), Alldifferent(Seq(T1, . . . , Tn)) Alldifferent object represents an alldifferent constraint. • Not(C ) Not object represents a negation. Unary operator ! can be used also. • And(C1, . . . , Cn), And(Seq(C1, . . . , Cn)) And object represents a conjunction (logical AND). Binary operator && can be used also. • Or(C1, . . . , Cn), Or(Seq(C1, . . . , Cn)) Or object represents a disjunction (logical OR). Binary operator || can be used also. • Imp(C1, C2 ), Iff(C1, C2 ) Imp and Iff objects represent an implication and equivalence respectively. Binary operators ==> and <==> can be used also. • Xor(C1, C2 ) Xor object represents an exclusive or. Binary operator ^ can be used also. 5.4 CSP Objects CSP objects represents a Constraint Satisfaction Problem. When you import jp.kobe u.copris.dsl. , the default CSP object can be referred by a variable csp. 5.4.1 Declaring Integer Variables Integer variable can be declared by int method. Its domain can be given by specifying the lower and upper bounds. scala> int(’x, 1, 10) res: jp.kobe_u.copris.Var = x When it is a constant, one value is given. scala> int(’y, 5) res: jp.kobe_u.copris.Var = y When the domain is non-contiguous, a set can be specified as its domain. 13

15. Introduction to Constraint Programming in Copris scala> int(’p(1), Set(2,3,5,7)) res: jp.kobe_u.copris.Var = p(1) scala> int(’p(2), Set(3,7,5,2)) res: jp.kobe_u.copris.Var = p(2) 0-1 variable is declared by boolInt method. boolInt(x) is equivalent to int(x, 0, 1). scala> boolInt(’b) res: jp.kobe_u.copris.Var = b Domain of an integer variable can be shown by csp.dom method. scala> csp.dom(’x) res: jp.kobe_u.copris.Domain = Domain(1,10) scala> csp.dom(’p(1)) res: jp.kobe_u.copris.Domain = Domain(TreeSet(2, 3, 5, 7)) 5.4.2 Adding Constraints Constraint can be added by add method. scala> add(’x === ’y * 2) scala> add(’x === ’p(1) + ’p(2)) scala> add(’b.? <==> (’p(1) < ’p(2))) Current integer variable declarations and constraints can be shown by show method. scala> show int(x,1,10) int(y,5,5) int(p(1),Domain(TreeSet(2, 3, 5, 7))) int(p(2),Domain(TreeSet(2, 3, 5, 7))) int(b,0,1) Eq(x,Mul(y,2)) Eq(x,Add(p(1),p(2))) Iff(Ge(b,1),Lt(p(1),p(2))) CSP file in Sugar format can be created by dump method. scala> dump("output.csp") CNF file in DIMACS format can be created by dump method with “cnf” option. scala> dump("output.cnf", "cnf") 5.5 Searching Solutions The find method searches the first solution. scala> find res: Boolean = true When the result is ‘true’, there exists a solution, and when it is ‘false’, there exists no solutions. The solution found is stored in solution variable. scala> solution res: jp.kobe_u.copris.Solution = Solution(Map(b -> 0, p(2) -> 5, y -> 5, x -> 10, p(1) -> 5),Map()) The ‘find’ method searches a solution as follows. • CSP objected is encoded to SAT, and CNF file is created. • A SAT solver (Sat4j is used as a default solver) is used to find a solution for the generated CNF file. • A solution of the CNF (if any) is decoded to a solution of CSP. The name of CNF file generated by Copris can be checked by ‘info’ command. The file is automatically deleted after exiting Scala. scala> info res: Map[String,String] = Map(satFile -> /tmp/sugar3235357395765834639.cnf) The value of an integer variable can be obtained by solution method. 14

16. Introduction to Constraint Programming in Copris scala> solution(’x) res: Int = 10 scala> solution(’x, ’p(1), ’p(2)) res: Seq[Int] = ArrayBuffer(10, 5, 5) The findNext method searches the next solution. scala> findNext res: Boolean = true scala> solution res: jp.kobe_u.copris.Solution = Solution(Map(b -> 0, p(2) -> 3, y -> 5, x -> 10, p(1) -> 7),Map()) The ‘findNext’ method searches the next solution as follows. • An extra condition of negating current solution is added to the CNF file. • A SAT solver is used to find a solution. • A solution of the CNF (if any) is decoded to a solution of CSP. solutions method returns the iterator of all solutions. scala> cancel scala> solutions.foreach(println) Solution(Map(b -> 0, p(2) -> 5, y -> 5, x -> 10, p(1) -> 5),Map()) Solution(Map(b -> 0, p(2) -> 3, y -> 5, x -> 10, p(1) -> 7),Map()) Solution(Map(b -> 1, p(2) -> 7, y -> 5, x -> 10, p(1) -> 3),Map()) To find an optimum solution, specify the objective variable by maximize (or minimize) method, and use findOpt method. scala> cancel scala> maximize(’p(2)) scala> findOpt res: Boolean = true scala> solution res: jp.kobe_u.copris.Solution = Solution(Map(b -> 1, p(2) -> 7, y -> 5, x -> 10, p(1) -> 3),Map()) 5.6 Others 5.6.1 Switching SAT Solver SAT solver can be switched by use method. scala> use(new sugar.Solver(csp, new sugar.SatSolver2("/usr/local/bin/minisat"))) scala> use(new sugar.Solver(csp, new sugar.SatSolver1("/usr/local/bin/clasp"))) Use SatSolver2 for a SAT solver taking two argument files (such as MiniSat), and use SatSolver1 for a SAT solver taking one argument file (such as Clasp). 5.6.2 Using PB Solver You can use PB (Pseudo-Boolean) solver as follows. scala> use(new pb.Solver(csp, new pb.PbSolver("/usr/local/bin/clasp"), "binary")) 5.6.3 Using SMT Solver You can use SMT solver supporting QF_LIA theory (such as Z3) by ‘use’ command. scala> use(new smt.Solver(csp, new smt.SmtSolver("/usr/local/bin/z3"))) SMT file can be saved by ‘dump’ command. scala> dump("file.smt2") 5.6.4 Using JSR-331 Solver You can use JSR-331 solver by ‘use’ command. scala> use(new jsr331.Solver(csp)) You need to put implementation jar files (e.g. jsr331.choco.jar and choco-solver-2.1.5-20120603-with-sources.ja in lib/jsr331/ directory. $ scala -cp build/copris-all-v2-2-8.jar:../lib/jsr331/"*" JSR-331 solver implementations are available at http://openrules.com/jsr331/downloads.htm. 15

17. Introduction to Constraint Programming in Copris 5.6.5 Importing XCSP You can import XCSP file as follows. scala> init scala> (new loader.XCSPLoader(csp)).load("file.xml") 5.6.6 Exporting/Importing Sugar CSP You can export/import CSP by Sugar CSP format. scala> dump("file.csp") scala> init scala> (new loader.SugarLoader(csp)).load("file.csp") 5.6.7 Using Multiple CSPs+Solvers Multiple CSPs (with multiple Solvers) can be simultaneously used in the following way. CSPs and Solvers of copris1 and copris2 are independent. 1: import jp.kobe_u.copris._ 2: import jp.kobe_u.copris.dsl._ 3: 4: val copris1 = new sugar.Sugar() 5: val copris2 = new sugar.Sugar() 6: using(copris1) { 7: int(’x, 1, 4) 8: add(’x < 3) 9: } 10: using(copris2) { 11: int(’x, 1, 4) 12: add(’x >= 3) 13: } 14: copris1.solutions.foreach(println) 15: copris2.solutions.foreach(println) 5.6.8 Aviod Using Copris DSL If you don’t want to import Copris DSL methods (that is, to avoid import jp.kobe_u.copris.dsl._), you need to describe as follows (ExampleNoDSL-csp.scala). 1: import jp.kobe_u.copris._ 2: 3: val copris = new sugar.Sugar() 4: val x = copris.int(Var("x"), 1, 15) 5: val y = copris.int(Var("y"), 1, 15) 6: val z = copris.int(Var("z"), 1, 15) 7: copris.add(x + y + z === 15) 8: copris.add(x + y * 5 + z * 10 === 90) 9: 10: if (copris.find) { 11: println(copris.solution) 12: } 16