ArithmeticExpr-Parser

Introduction

Parsers are systems which read, analyze & evaluate input data conforming to some formal grammar wikipedia link. Below I present an Arithmetic Expression parser which evaluates an expression comprising of symbols [ + - * / ( ) and numbers ]

Installation

The parser is encapsulated as the class ArithExprParser with tokenize & evaluate methods.

To use as a calculator

After cloning the repo, it can built into executable ‘ArithExprParser’ as shown below.

From the working directory run the below command
g++ -std=c++14 -I./include ./src/*.cpp -o ArithExprParser

This will build the executable ‘ArithExprParser’. Just run the executable to start evaluating expressions.

To use inside your code

If you want to use it inside your code, just include the header file ‘ArithExprParser.h’ and link ‘ArithExprParser.cpp’ into your build. A sample usage is shown below

#include <iostream>
#include <ArithExprParser.h>
int main()
{
    std::string expr;
    ArithExprParser parser;
    while(true){
        std::cout<<"Enter Expression or type 'Q' to quit:";
        std::cin>>expr;
        if(expr == "Q")
            break;
        parser.tokenize(expr);
        std::cout<<"Result = "<<parser.evaluate()<<std::endl;
    }
}

Description

The order of operations performed to evaluate an arithmetic expression is usually referred to by acronym of BODMAS(Bracket, Order, Division, Multiplication, Addition, Subtraction) or PEMDAS(Paranthesis, Exponents, Multiplication, Division, Addition, Subraction). Here we implement this order by formalizing it into a grammar. Grammars are the classical solution to parsing, as they formally define the structure of the input. Formal grammars are a much more detailed topic and for those interested further pls refer to this wikipedia page Formal Grammars

ArithmeticExpr-Parser consists of 2 sub-systems:-

1. Tokenizer
2. Evaluator

Tokenizer

In order to parse any input expression, we need to breakdown the input into its basic symbols/tokens. In our case the basic tokens of arithmetic expressions are the usual symbols [ + - * / ( ) and numbers ]. For differentiating between - and -sign we follow the below rules

• ‘-‘ can be an operator or a qualifier -ve numbers
• ‘+’ sign is not allowed as a qualifer(its always implicit)
• successive signs are not allowed Ex:- 2—3, 2+-3 are invalid exprs. Use paranthesis to separate them like 2-(-3), 2+(-3)
• if ‘-‘ is preceded by ‘*’ or ‘/‘ or ‘(‘ or ‘’ its a qualifier
• otherwise ‘-‘ is an operator

Evaluator

Once tokens are available, the evaluator should follow the order of operations(BODMAS) to evaluate the tokens into the final result. We formalize BODMAS order into the below grammar.

Grammar for evaluating arithmetic expressions:-

 * Expression:
 *       term //should end with a term
 *       Expression : + : term
 *       Expression : - : term    
 * term:
 *       primary //should end with a primary
 *       term : * : primary
 *       term : / : primary
 * primary:  
 *       float 
 *       ( expression ) 
 *

The above grammar should be interpreted like this:-

1. Expression is a
   • term OR
   • expression ‘+’ term OR
   • expression ‘-‘ term
2. term is a
   • primary OR
   • term ‘*’ primary OR
   • term ‘/‘ primary
3. primary is a
   • number(floating point) OR
   • expression enclosed in paranthesis

The above structure is naturally recursive and hence we can define functions for each of expression, term & primary. Another advantage of defining the order into a grammar is it is highly intuitive.

The reasoning behind the grammar definition is :-

1. The order of evaluation of an expression is left to right(as + & - have same priority). Hence expression is defined as beginning with itself.
2. An expression consisting of ( & / ) operators has higher-evaluation priority than expression, hence we define a separate unit called term. A term is also evaluated from left to right(as & / have same priority), hence term starts with itself.
3. Any expression enclosed in paranthesis has higher-evaluation priority than term, hence we defined a separate unit called primary for ( expression ) and numbers. This is the base case.

An example evaluation order is shown below for a sample expression.

Input expression:- 2.3 + 3*(1+3)

Its evaluation is broken into the below steps:-

• 2.3 + 3 * (1+3) : Rule 2 of expression: expression + term
  • 2.3 : Rule 1 of expression: term
    • 2.3 : Rule 1 of term : primary
      • 2.3 : Rule 1 of primary : number (call returns)
  • 3 (1+3) : Rule 2 of term: term primary
    • 3 : Rule 1 of term: primary
      • 3 : Rule 1 of primary: number (call returns)
    • (1+3) : Rule 2 of primary : expression enclosed in paranthesis
      • 1+3 : Rule 2 of expression : expression + term
        • 1 : Rule 1 of expression : term
          • 1 : Rule 1 of term : primary
            • 1 : Rule 1 of primary : number (call returns)
        • 3 : Rule 1 of term : primary
          • 3 : Rule 1 of primary : number (call returns)

Conclusions

Arithmetic expression evaluation shows a simple application of grammars. Grammars are used almost universally for parsing. I plan to explore grammars in more detail in future by showing a markup language parser.