## Introduction

Most of the expression parsers I have seen are designed to parse and evaluate mathematical expressions. While I want to parse math expressions, I also wanted the ability to handle different types of expressions. I wanted a parser that would return a list of the base elements in an expression and let me apply my own evaluator to that list.

## Using the Code

In order to parse an expression, a user supplied tokenizer is required. The tokenizer provides two functions:

- It parses the expression into its base elements. These elements are of one of the following types:
**Literal **- A `string `

encoded in double or single quote marks
**Identifier **- A variable name to be resolved by the evaluator
**Constant **- An numeric or hex constant
**Function **- The name of a function to be invoked by the evaluator
**Argument **- The function argument delimiter, i.e. a comma
**GroupStart **- i.e. a left perenthesis
**GroupEnd **- i.e. a closing right perenthesis
**Operator **- An expression operator
**CondTrue **- Marks the start of elements to be evaluated when a condition is `true`

.

The **CondTrue **and **CondFalse **operators come from conditional expressions, i.e. `(a>b ? a : b) `

**CondFalse **- Marks the start of elements to be evaluated when a condition is `false `

**Assignment **- An assignment operator

The tokenizer is free to determine which type an element belongs to. The tokenizer also assigns the precedence to the operators. It also assigns the association of the operator. A operator is either Left or Right associative. Most operators are Left associative but the exponent, "**", and conditional, "?", operators are right associative.

- It provides a function to create an operator from a token element. An operator falls into one of the following groups:
**Arithmetic **- A mathematical operator, i.e. +, - or *
**Comparison **- A comparison operator, i.e. ==, > or <
**Logical **- A logical operator, i.e. && (AND) or || (OR)
**Bitwise **- A bitwise operator, i.e. & (AND), | (OR) or ^ (XOR)
**Conditional **- `True `

or `False `

element of a conditional expression, i.e. `(a>b ? ... : ...) `

**Assignment **- Assign result to a variable, i.e. =, +=, or -+

Once the expression has been tokenized, an RPN list elements is created using the "Shunting Yard" algorithm.

Below are some examples of expressions and the parsing results using the supplied "`MathParser`

" as the tokenizer.

x = a+b*c -a+(b+c)*2 a>b ? a-b : a+b a+min(x, y)
0: Identifier x 0: Identifier a 0: Identifier a 0: Identifier a
1: Identifier a 1: Operator - 1: Identifier b 1: Identifier x
2: Identifier b 2: Identifier b 2: Operator > 2: Identifier y
3: Identifier c 3: Identifier c 3: CondTrue ? 8 3: Function min 2
4: Operator * 4: Operator + 4: Identifier a 4: Operator +
5: Operator + 5: Constant 2 5: Identifier b
6: Assignment = 6: Operator * 6: Operator -
7: Operator + 7: CondFalse : 11
8: Identifier a
9: Identifier b
10: Operator +

What is shown above is the result of applying the "`ToString`

" method to the elements returned by the parser. The "`Identifier`

", "`Constant`

" and "`Operator`

" items are self explanatory. The "`CondTrue`

" and `CondFalse`

" elements act as an "`If`

, `Else`

" construct. The number on the "`CondTrue`

" element is the index of the first element to be evaluated if the condition is `false`

. The number of the "`CondFalse`

" element is the index of the first element following the conditional expression. The "`CondFalse`

" element in effect acts as a "`GoTo`

" the element specified. The number on the "`Function`

" element is the number of arguments to be passed to the named function.

The list of elements in an expression is generated by calling the parser. For example, from the test application:

. . .
ITokeniser parser = new MathParser();
List<RpnElement> tokens = Parser.ParseExpression(parser,rpnExpression.Text);
. . .

## Evaluating the Expression

The act of evaluating an expression involves the following steps:

- If the next element on the expression list is a "
**Literal**" or "**Constant**" then push its value onto the operand stack.
- If the next element is an "
**Identifier**" then get the current value of the identifier and push it on to the stack.
- If the next element is a function or an operator, then pop the required function arguments or the Left and Right sides of the operator from the stack. Then evaluate the function or the operator. Push the result onto the operand stack.
- Repeat until all elements in the expression list have been processed.
- When all elements have been processed, there should be one operand left on the operand stack, this is the result of the expression.

Below is the main loop of the sample evaluator included with the test application:

private RpnParser.RpnOperand RpnEval(List<RpnElement> tokens) {
Stack<RpnOperand> oprndStack = new Stack<RpnOperand>();
RpnOperand oprnd = null;
for (int nextElem = 0; nextElem < tokens.Count; ++nextElem) {
RpnElement token = tokens[nextElem];
switch (token.ElementType) {
case ElementType.Literal:
case ElementType.Constant:
oprnd = new RpnOperand((RpnToken)token);
break;
case ElementType.Identifier:
oprnd = EvalIdentifier((RpnToken)token);
break;
case ElementType.Operator:
RpnOperator oper = (RpnOperator)token;
RpnOperand lhs, rhs;
lhs = rhs = null;
if (oper.IsMonadic && oprndStack.Count >= 1) {
lhs = new RpnOperand(0);
rhs = oprndStack.Pop();
} else
if (oprndStack.Count >= 2) {
rhs = oprndStack.Pop();
lhs = oprndStack.Pop();
}
if (lhs == null)
StackError("operator", token.StrValue);
else
oprnd = EvalOperator(oper, lhs, rhs);
break;
case ElementType.CondTrue:
case ElementType.CondFalse:
oper = (RpnOperator)token;
if (oprndStack.Count < 1)
StackError("operator", token.StrValue);
if (oper.OpType == OperatorType.CondTrue) {
lhs = oprndStack.Pop();
if (lhs.NumValue == 0)
nextElem = oper.CondGoto - 1; } else
nextElem = oper.CondGoto - 1; continue;
case ElementType.Function:
RpnFunction func = (RpnFunction)token;
if (oprndStack.Count >= func.ArgCount) {
RpnOperand[] args = new RpnOperand[func.ArgCount];
for (int i = func.ArgCount - 1; i >= 0; --i)
args[i] = oprndStack.Pop();
if (evalCBox.Checked)
oprnd = EvalFunction(func, args);
} else
StackError("function", token.StrValue);
break;
case RpnParser.ElementType.Assignment:
if (oprndStack.Count > 1) {
rhs = oprndStack.Pop(); lhs = oprndStack.Pop(); oprnd = AssignOperator((RpnOperator)token, lhs, rhs);
} else
StackError("assignment", token.StrValue);
break;
}
oprndStack.Push(oprnd);
}
if (oprndStack.Count != 1)
StackError("result", "");
return oprndStack.Pop();
}
private void StackError(string token, string value) {
string text = String.Format("Insufficient operands on stack for {0}: {1}",
token, value);
throw new ApplicationException(text);
}

## Summary

Hopefully this article demonstrates a generalized expression parser, where the elements that make up an expression and the evaluation of that expression are controlled by the user. The only job of the parser is to present the elements in a standard, consistent manner.

## References

## History

- July 9, 2008 - Version 1 released
- July 11, 2008 - Removed signing requirement from project
- June 26, 2011 - Updated source code