{-# LANGUAGE GADTSyntax #-}
import Data.Char (isSpace, isDigit)

Module 05: The Arith language: pretty-printing

• Write your team names here:

• You may again choose whoever you want to start as the driver. Write your choice here:

Arith syntax and semantics

The Arith language is represented by the following abstract syntax:

data Arith1 where
  Lit1 :: Integer -> Arith1
  Add  :: Arith1 -> Arith1 -> Arith1
  Sub  :: Arith1 -> Arith1 -> Arith1
  Mul  :: Arith1 -> Arith1 -> Arith1
  deriving (Show)

arithExample :: Arith1
arithExample = Add (Mul (Lit1 4) (Lit1 5)) (Lit1 2)

arithExample2 :: Arith1
arithExample2 = Mul (Lit1 4) (Add (Lit1 5) (Lit1 2))

(We are using the name Arith1 to avoid a name clash, since later we will use a more refined version called Arith.) The semantics of an Arith expression is an integer: Lit1 values represent themselves, Add represents addition, Sub subtraction, and Mul multiplication. For example, arithExample evaluates to (4 × 5) + 2 = 22, and arithExample2 evaluates to 4 × (5 + 2) = 28.

• Write an interpreter called interpArith1 for Arith1 expressions.

As concrete syntax for Arith, we use standard mathematical notation and standard conventions about operator precedence. For example, "4*5+2" is concrete syntax for arithExample, since by convention multiplication has higher precedence than addition. If we want concrete syntax to represent arithExample2, we have to use parentheses: "4*(5+2)".

• Write a pretty-printer prettyArith1 which turns Arith abstract syntax into valid concrete syntax. At this point, you should try to make your pretty-printer as simple as possible rather than try to produce the best output possible. (Hint: something like "((4*5)+2)" is perfectly valid concrete syntax, even though it has unnecessary parentheses.)

• How might you go about altering your pretty printer to omit needless parentheses? Write down some ideas here.

A better pretty-printer

• ROTATE ROLES and write the name of the new driver here:

data Op where
  Plus  :: Op
  Minus :: Op
  Times :: Op
  deriving (Show, Eq)

data Arith where
  Lit :: Integer -> Arith
  Bin :: Op -> Arith -> Arith -> Arith
  deriving (Show)

data Associativity where
  L :: Associativity
  R :: Associativity
  deriving (Show, Eq)

type Precedence = Int

-- 4 * (5 + 2)
expr1 :: Arith
expr1 = Bin Times (Lit 4) (Bin Plus (Lit 5) (Lit 2))

-- 44 - (7 * (1 + 2) - 3)
expr2 :: Arith
expr2 = Bin Minus (Lit 44) (Bin Minus (Bin Times (Lit 7) (Bin Plus (Lit 1) (Lit 2))) (Lit 3))

• Compare Arith and Arith1. How is Arith different/more general than Arith1?

• Write an interpreter interpArith for Arith expressions. (Hint: try writing a separate function interpOp :: Op -> (Int -> Int -> Int) to factor out the behavior of different operators.)

• Write functions assoc :: Op -> Associativity and prec :: Op -> Precedence to return the associativity and precedence of each operator. Addition, multiplication, and subtraction are all left-associative by convention. Addition and subtraction should have the same precedence level, with multiplication at a higher level (typically larger numbers represent higher precedence, that is, ``stickier glue’’).

• Now write a function prettyPrec :: Precedence -> Associativity -> Arith -> String. The three arguments to prettyPrec represent:

  • The precedence level of the parent operator
  • Whether the current expression is a left or right child of its parent
  • The expression to be pretty-printed

  Given these inputs, it should print out a properly parenthesized version of the expression, with parentheses surrounding the entire expression only if they are needed. Remember that parentheses are needed when (and only when):

  • The precedence of the parent operator is higher than the precedence of the operator at the root of the current expression, OR
  • The precedence of the parent operator is equal to the precedence of the root operator, and the associativity of the root operator is the opposite of which side of its parent it is on.

• Write a function prettyArith :: Arith -> String which works by calling prettyPrec with appropriate starting arguments.

• Now go back and add an exponentiation operator to the language. Exponentiation is right-associative and has higher precedence than multiplication. Be sure to update both the interpreter and pretty-printer appropriately.