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--- pp:2023:scala:l06 [2023/04/01 17:11]
tpruteanu [6.2. Polymorphic expressions]
+++ pp:2023:scala:l06 [2023/04/08 07:49] (current)
pdmatei
@@ Line 1: / Line 1: @@
-====== Lab 06. Polymorphism ======
+====== Lab 6. Scala on steroids ======
-===== 6.1. Maps =====
+==== 5-Tic-Tac-Toe ====
-Maps are collections of **(key, value)** pairs. Keys should be unique, and every key is associated with a value. \\
-Some of the fundamental operations on maps include:
-  * retrieving / updating the value associated with a key
-  * adding a **(key, value)** pair
-  * removing a **(key, value)** pair
-You can find more information on maps on the Scala Docs (https://docs.scala-lang.org/overviews/collections/maps.html). \\
-<note important> maps are **immutable**, functions working with maps return a new updated version instead of modifing the map </note>
-Some examples with the most used functions can be found below.
-<hidden>
-<code scala>let map = Map(1 -> 2, 3 -> 4): Map[Int, Int]</code>
-  * Adding a (key, value) pair to a map
-<code scala>
-map + (5 -> 6)  // Map(1 -> 2, 3 -> 4, 5 -> 6)
-map + (3 -> 5)  // Map(1 -> 2, 3 -> 5)  -- if key exists, it updates the value
-</code>
-  * Removing the pair associated with a key
-<code scala>
-map - (3 -> 4)  // Map(1 -> 2)
-</code>
-  * Querying a value
-<code scala>
-map get 1  // return 2
-map get 3  // return 4
-map getOrElse (1, 0)  // return 2
-map getOrElse (5, 0)  // return 0 (if key doesn't exist, return provided value)
-map contains 1  // True
-map contains 5  // False
-</code>
-  * Higher-order functions
-<code scala>
-map mapValues (x => x + 5)  // Map(1 -> 7, 2 -> 9)
-map filterKeys (x => x <= 2)  // Map(1 -> 2)
-</code>
-</hidden>
-==== Exercises ====
+**Tic Tac Toe** is usually played on a 3x3 board, marking positions by each player in rounds. Our game is slightly different (usually called 5-in-a-row):
-We represent a gradebook as a map which holds, for each student (encoded as String), its grade (an Int). We implement a gradebook as a case class, as follows:
+  * it can be played on a square board of any size **larger or equal to 5**.
-<code scala>
+  * A player wins if it has marked a line, column or diagonal of **5 consecutive positions** in a row.
-case class Gradebook(book: Map[String,Int])
-</code>
-**6.1.1.** Add a method ''+'' to the class, which adds a new entry to the gradebook.
+Example of a winning position for ''X'' on a 5x5 board:
-<code scala>
+<code>
-def + (entry: (String, Int)): Gradebook = ???
+X...0
+X.0.
+..X0.
+...X.
+.0..X
 </code>
-**6.1.2.** Add a method ''setGrade'' which modifies the grade of a given student. Note that your method should return an updated gradebook, not modify the current one, since the latter is **immutable**.
+Example of a winning position for ''0'' on a 7x7 board:
-<code scala>
+<code>
-def setGrade(name: String, newGrade: Int): Gradebook = ???
+.X...X.
+...0...
+...0...
+.X.0..X
+..0..0
+...0...
+...X...
 </code>
-**6.1.3.** Add a method ''++'' which merges two gradebooks, if a student is in both gradebook, take the higher of the two grades.
+==== Encodings ====
-<code scala>
-def ++(other: Gradebook): Gradebook = {
-  // the best strategy is to first implement the update of an entry into an existing Map...
-  def updateBook(book: Map[String,Int], pair: (String,Int)): Map[String,Int] = ???
-  // and then use a fold to perform updates for all pairs of the current map.
-  ???
-}
-</code>
-**6.1.4. (!)** Add a method which returns a map, containing as key, each possible grade (from 1 to 10) and a value, the **number** of students having that grade. (Hint: follow the same strategy as for the previous exercise).
+  * In your project template, ''X'' is encoded as the **first** player (''One''), and ''0'', as ''Two''.
 <code scala>
-def gradeNo: Map[Int,Int] = ???
+trait Player {}
-</code>
+case object One extends Player {
+  override def toString: String = "X"
-===== 6.2. Polymorphic expressions =====
-In this section we will implement a very simple expression evaluation (which might be part of a programming language), and we will experiment with different features that will: \\
-  - make the code easier to use
-  - make the code more general, suitable for a broad range of scenarios (easier to extend)
-**NOTE**: In this process, we will have to rewrite and delete parts of the code from previous steps in order to improve it.
-Start with the following polymorphic type definition for a expression:
-<code scala>
-trait Expr[A] {
-  def eval(): A
 }
-</code>
+case object Two extends Player {
+  override def toString: String = "0"
-**6.2.1.** Implement case classes ''BoolAtom'', ''BoolAdd'' and ''BoolMult'', which evaluates ''Add'' as boolean //or// and ''Mult'' as boolean //and//.
-<code scala>
-case class BoolAtom(b: Boolean) extends Expr[Boolean] {
-  override def eval(): Boolean = b
 }
-case class BoolAdd(left: Expr[Boolean], right: Expr[Boolean]) extends Expr[Boolean] {
+case object Empty extends Player {
-  override def eval(): Boolean = ???
+  override def toString: String = "."
-}
-case class BoolMult(left: Expr[Boolean], right: Expr[Boolean]) extends Expr[Boolean] {
-  override def eval(): Boolean = ???
 }
 </code>
+  * A ''Board'' is encoded as a List of Lists of **positions** (i.e. a matrix), where a position can be ''One'', ''Two'' or ''Empty''. We make no distinction in the code between a position and a player, although ''Empty'' cannot be seen as a valid player. This makes the code slightly easier to write.
-The code structure can be easily deployable for other types, such as **Int**, **Double** etc, but it's inconvenient to be defining different types for each such case. In order to make our code more extensible in this sense, we can add a few ingredients. \\
-We will add a new case class ''Strategy'', which stores the operations that can be performed, in our case ''Add'' and ''Mult''. And we will have our ''eval'' function take a ''Strategy'' as a argument.
 <code scala>
-case class Strategy[A] (add: (A,A) => A, mult: (A,A) => A)
+type Line = List[Player]
+type BoardList = List[Line]
-trait Expr[A] {
+case class Board(b: BoardList) {
-   def eval(s: Strategy[A]): A
+  override def toString: String = ???
 }
 </code>
-**6.2.2.** Adjust the rest of your code with the new definitions (**NOTE**: ''eval'' should have a more general implementation now). Implement ''boolStrategy''.
+==== Tasks ====
-<code scala>
+The following functions have a given signature. However, it is up to the student to decide whether these will be methods of a class or just simple functions.
-val boolStrategy = Strategy[Boolean](
+**6.1.1.** Write a function which converts a string into a ''Board''. As a helper, you can use ''_.split( c )'' where c is a separator string, and ''_.toList''. The best solution is to use a combination of ''map'' calls with the above mentioned functions. A string is encoded exactly as in the examples shown above:
-    (left, right) => ,  // add implementation
+  * there are no whitespaces - empty positions are marked by the character '.'
-    (left, right) =>    // mult implementation
+  * lines are delimited by '\n' (the last line does not have a trailing '\n').
-)
-val expr1 = BoolMult(BoolAdd(BoolAtom(true), BoolAtom(false)), BoolAtom(false))
-val expr2 = BoolMult(BoolAdd(BoolAtom(true), BoolAtom(false)), BoolAtom(true))
-println(expr1.eval(boolStrategy))  // false
-println(expr2.eval(boolStrategy))  // true
-</code>
-**6.2.3.** If you implemented ''eval'' correctly at **6.3.2.**, you might notice that it doesn't rely on the ''Boolean'' type anymore. Add more general case classes ''Atom'', ''Add'' and ''Mult''.
 <code scala>
-case class Atom[A](a: A) extends Expr[A] {
+def makeBoard(s: String): Board = {
-    override def eval(f: Strategy[A]) = ???
+    def toPos(c: Char): Player =
-}
+      c match {
+        case 'X' => One
-case class Add[A](left: Expr[A], right: Expr[A]) extends Expr[A] {
+        case '0' => Two
-    override def eval(f: Strategy[A]): A = ???
+        case _ => Empty
-}
+      }
+    ???
-case class Mult[A](left: Expr[A], right: Expr[A]) extends Expr[A] {
-    override def eval(f: Strategy[A]): A = ???
 }
 </code>
-**6.2.4.** Currently, writing down expressions is cumbersome, due to the escalating number of parentheses. It would be advisable to allow building expressions using a friendlier syntax, such as: ''Atom(1) + Atom(2) * Atom(3)''. This is possible if we define the operations ''+'' and ''*'' as member functions. Then, for instance ''Atom(1) + Atom(2)'' could be translate from ''Atom(1).+(Atom(2))''. Where would it be most convenient for you - the programmer, to define functions ''+'' and ''*''? Define them! \\
+**6.1.2.** Write a function which checks if a position on the board is free. Recall that list indexing can be done using ''l(_)''. Positions are numbered from 0.
-\\
-**Hint:**
-<hidden>
-In Scala it is possible to define function implementations in traits. Define ''+'' and ''*'' accordingly.
-</hidden>
-\\
-So far, our expressions contain just values (of different sorts). Making a step towards a programming language (like we mentioned above), we would like to also introduce **variables** in our expressions. To do so, we need to **interpret** each variable as boolean or int etc. We define a **store**:
 <code scala>
-type Store[A] = Map[String, A]
+def isFree(x:Int, y:Int):Boolean = ???
 </code>
-to be a mapping from values of type ''String'' (variable names) to their values. \\
-**6.2.5.**
+**6.1.3.** Write a function which returns the //opponent// of a player:
-  * Modify ''eval'' function to also take a **store** as an argument, modify the code accordingly
+<code scala>
-  * Add a new **case class** ''Var'' which allows creating variables as expressions (e.g. ''Var("x") + Atom(1)'' should be a valid expression)
+def complement(p: Player): Player = ???
+</code>
-<hidden>
+**6.1.4.** We want to write a function which converts a board to a string, following the same strategy. Complete the ''toString'' in the Board class. Hint: instead of ''foldRight'', you can use ''reduce'' which works quite similarly, but without requiring an accumulator.
-You don't have to modify the expressions from **6.3.5**, you can use ''eval'' with just 1 parameter by overloading the method and calling the new ''eval'' with the empty Map.
-<code scala>
-def eval(s: Strategy[A]) = eval(s, Map[String, A]())
-</code>
-</hidden>
-\\
-**6.2.6.** We have not treated the case when the expression uses variables not found in the store. To do so, we change the type signature of eval to:
+**6.1.5.** Write a function which returns the //columns// of a board:
 <code scala>
-def eval (s: Strategy[A], store: Store[A]): Option[A]
+def getColumns: Board = ???
 </code>
-Modify your implementation accordingly. If an expression uses variables not present in the store, ''eval'' should return ''None''.
-===== 6.3. Polynomials =====
+**6.1.6.** Implement the following two functions for extracting the first and second diagonal, as lines, from a board. Hint: use for comprehensions.
-Consider a polynomial encoded as a map, where each present key denotes a power of **x**, and a value denotes its coefficient.
 <code scala>
-Map(2 -> 1, 1 -> 2, 0 -> 1)  // encodes x^2 + 2*x + 1
+def getFstDiag(): Line = ???
+def getSndDiag(): Line = ???
 </code>
-<code scala>
-case class Polynomial (terms: Map[Int,Int])
-</code>
-<hidden>
-You can override the ''toString'' method to to see your results in a more friendly format:
-<code scala>
-case class Polynomial (terms: Map[Int,Int]) {
-    override def toString: String = {
-        def printRule(x: (Int, Int)): String = x match {
-            case (0, coeff) => coeff.toString
-            case (1, coeff) => coeff.toString ++ "*x"
-            case (p, coeff) => coeff.toString ++ "*x^" ++ p.toString
-        }
-        terms.toList.sortWith(_._1 >= _._1)
+**6.1.7.** Implement the following functions for extracting diagonals above/below the first/second diagonal, as lines. It's not really necessary to make sure that at least 5 positions are available, for now. Hint: if one function must be implemented with element-by-element iteration, the three other can be implemented using each-other, as single-line calls.
-                      .map(printRule)
-                      .reduce(_ ++ " + " ++ _)
-    }
-}
-</code>
-</hidden>
-\\
-**6.3.1.** Add a method ''*'' which multiplies the polynomial by a given coeficient:
 <code scala>
-def * (n: Int): Polynomial = ???
+def getAboveFstDiag: List[Line] = ???
+def getBelowFstDiag: List[Line] = ???
+def getAboveSndDiag: List[Line] = ???
+def getBelowSndDiag: List[Line] = ???
 </code>
-**6.3.2.** Implement a method ''hasRoot'' which checks if a given integer is a root of the polynomial.
+**6.1.8.** Write a function which checks if a player is the winner. Hint: functions ''l.forall(_)'' and ''l.exists(_)'' may be very helpful, together with patterns.
 <code scala>
-def hasRoot(r: Int): Boolean = ???
+def winner(p: Player): Boolean = ???
 </code>
-**6.3.3.** Implement a method ''+'' which adds a given polynomial to this one (Hint: this operation is very similar to gradebook merging).
+**6.1.9.** Write a function which updates a position from the board, with a given player. The position need not be empty and you are not required to check this. Hint: re-use an inner aux-function together with ''take'' and ''drop''.
 <code scala>
-def + (p2: Polynomial): Polynomial = ???
+def update(p: Player)(ln: Int, col: Int) : Board = ???
 </code>
-**6.3.4.** Implement a method ''*'' which multiplies two polynomials.
+**6.1.10.** Write a function which generates all possible next-moves for any of the two players. A next-move consists in a new board, where the player-at-hand played his move.
 <code scala>
-def * (p2: Polynomial): Polynomial = ???
+def next(p: Player): List[Board] = ???
 </code>
-**6.3.5.** Implement ''polynomialStrategy'' and try to evaluate a few expressions with polynomials.
+==== Testing ====
-<hidden>
+Use the following board configurations to test your solutions:
-You can test your code with the following expression:
 <code scala>
-val p1 = Polynomial(Map(2 -> 1, 1 -> 2, 0 -> 1))
+val t1 =
-val p2 = Polynomial(Map(3 -> 1, 1 -> 3, 0 -> 2))
+  """X0X0X0
-val expr = (Atom(p1) + Atom(p2)) * Atom(p1)
+    |0X0X0X
+    |X0X0X0
+    |.XX0..
+    |X00...
+    |X0X0X0""".stripMargin
-println(expr.eval(polynomialStrategy))
+val t2 =
-// result: x^5 + 3*x^4 + 8*x^3 + 14*x^2 + 11*x + 3
+  """......
+    |......
+    |......
+    |.XX...
+    |.0000.
+    |......""".stripMargin
+val t3 =
+  """0X0X0.
+    |000.X0
+    |0.0X..
+    |0..0..
+    |0X..0X
+    |...X..""".stripMargin
 </code>
-</hidden>