Convert Arithmetic Expression Tree to string without unnecessary parenthesis?

Question

Assume I build a Abstract Syntax Tree of simple arithmetic operators, like Div(left,right), Add(left,right), Prod(left,right),Sum(left,right), Sub(left,right).

However when I want to convert the AST to string, I found it is hard to remove those unnecessary parathesis.

Notice the output string should follow the normal math operator precedence.

Examples:

Prod(Prod(1,2),Prod(2,3)) let's denote this as ((1*2)*(2,3)) make it to string, it should be 1*2*2*3

more examples:

(((2*3)*(3/5))-4) ==> 2*3*3/5 - 4

(((2-3)*((3*7)/(1*5))-4) ==> (2-3)*3*7/(1*5) - 4

(1/(2/3))/5 ==> 1/(2/3)/5 

((1/2)/3))/5 ==> 1/2/3/5

((1-2)-3)-(4-6)+(1-3) ==> 1-2-3-(4-6)+1-3

Answer 1

I find the answer in this question.

Although the question is a little bit different from the link above, the algorithm still applies.

The rule is: if the children of the node has lower precedence, then a pair of parenthesis is needed. If the operator of a node one of -, /, %, and if the right operand equals its parent node's precedence, it also needs parenthesis.

I give the pseudo-code (scala like code) blow:

def toString(e:Expression, parentPrecedence:Int = -1):String = {

    e match {

      case Sub(left2,right2) =>
        val p = 10
        val left = toString(left2, p)
        val right = toString(right, p + 1) // +1 !!
        val op = "-"
        lazy val s2 = left :: right :: Nil mkString op
        if (parentPrecedence > p )
          s"($s2)"
        else s"$s2"

      //case Modulus and divide is similar to Sub except for p

      case Sum(left2,right2) =>
        val p = 10
        val left = toString(left2, p)
        val right = toString(right, p) //
        val op = "-"
        lazy val s2 = left :: right :: Nil mkString op
        if (parentPrecedence > p )
          s"($s2)"
        else s"$s2"

      //case Prod is similar to Sum
      ....    
    }
  }

Answer 2

For a simple expression grammar, you can eliminate (most) redundant parentheses using operator precedence, essentially the same way that you parse the expression into an AST.

If you're looking at a node in an AST, all you need to do is to compare the precedence of the node's operator with the precedence of the operator for the argument, using the operator's associativity in the case that the precedences are equal. If the node's operator has higher precedence than an argument, the argument does not need to be surrounded by parentheses; otherwise it needs them. (The two arguments need to be examined independently.) If an argument is a literal or identifier, then of course no parentheses are necessary; this special case can easily be handled by making the precedence of such values infinite (or at least larger than any operator precedence).

However, your example includes another proposal for eliminating redundant parentheses, based on the mathematical associativity of the operator. Unfortunately, mathematical associativity is not always applicable in a computer program. If your expressions involving floating point numbers, for example, a+(b+c) and (a+b)+c might have very different values:

(gdb) p (1000000000000000000000.0 + -1000000000000000000000.0) + 2
$1 = 2
(gdb) p 1000000000000000000000.0 + (-1000000000000000000000.0 + 2)
$2 = 0

For this reason, it's pretty common for compilers to avoid rearranging the order of application of multiplication and addition, at least for floating point arithmetic, and also for integer arithmetic in the case of languages which check for integer overflow.

But if you do really want to rearrange based on mathematical associativity, you'll need an additional check during the walk of the AST; before checking precedence, you'll want to check whether the node you're visiting and its left argument use the same operator, where that operator is known to be mathematically associative. (This assumes that only operators which group to the left are mathematically associative. In the unlikely case that you have a mathematically associative operator which groups to the right, you'll want to check the visited node and its right-hand argument.)

If that condition is met, you can rotate the root of the AST, turning (for example) PROD(PROD(a,b),□)) into PROD(a,PROD(b,□)). That might lead to additional rotations in the case that a is also a PROD node.