Step by Step solution about How to resolve the algorithm Parsing/Shunting-yard algorithm step by step in the Mathematica/Wolfram Language programming language

The provided Wolfram program rpn takes an input string representing an arithmetic expression in infix notation and returns the equivalent expression in Reverse Polish notation (RPN). RPN, also known as postfix notation, is a mathematical notation where operators are written after their operands.

Here's a detailed explanation of the code:

1. The rpn function is defined with a single argument str, which is expected to be an infix arithmetic expression.

2. Inside the function, the expression string str is split into a list of individual characters using StringSplit.

3. Three data structures are initialized:

   • stack: A stack is used to store operators temporarily.
   • out: A list is used to store operands and operators in RPN format.
   • next: A variable is used to hold the current character being processed from the input string.

4. The main processing loop continues as long as the in list (the split input string) is not empty. In each iteration, the first character (next) from in is processed.

5. The code uses Which to handle different cases:

   • If next is a digit, it's appended to the out list as an operand.
   • If next is a letter (assuming variables or constants), it's appended to the stack as an operand.
   • If next is a comma (,), it triggers a loop to pop operators from the stack and append them to the out list until an opening parenthesis '(' is encountered. This ensures that operators with higher precedence are evaluated before lower precedence ones.
   • If next is a caret ('^'), it's appended to the stack.
   • If next is an asterisk (''), it triggers a loop to pop multiplication or division operators from the stack and append them to the out list until a lower precedence operator (addition, subtraction, or opening parenthesis) is encountered. Then, the '' is appended to the stack.
   • The same logic is applied for division ('/'), addition ('+'), and subtraction ('-').
   • If next is an opening parenthesis ('('), it's appended to the stack.
   • If next is a closing parenthesis (')'), it triggers a loop to pop operators from the stack and append them to the out list until an opening parenthesis is encountered. Then, the closing parenthesis is discarded.
   • If any other characters are encountered (e.g., spaces), they are ignored.

6. After processing all characters in the input string, any remaining operators in the stack are popped and appended to the out list.

7. Finally, the out list, which contains the RPN expression, is converted to a string using StringRiffle and returned.

8. The code includes an example usage where it converts the expression "3 + 4 * 2 / ( 1 - 5 ) ^ 2 ^ 3" to RPN and prints the result.

rpn[str_] := 
  StringRiffle[
   ToString /@ 
    Module[{in = StringSplit[str], stack = {}, out = {}, next}, 
     While[in != {}, next = in[[1]]; in = in[[2 ;;]]; 
      Which[DigitQ[next], AppendTo[out, next], LetterQ[next], 
       AppendTo[stack, next], next == ",", 
       While[stack[[-1]] != "(", AppendTo[out, stack[[-1]]]; 
        stack = stack[[;; -2]]], next == "^", AppendTo[stack, "^"], 
       next == "*", 
       While[stack != {} && MatchQ[stack[[-1]], "^" | "*" | "/"], 
        AppendTo[out, stack[[-1]]]; stack = stack[[;; -2]]]; 
       AppendTo[stack, "*"], next == "/", 
       While[stack != {} && MatchQ[stack[[-1]], "^" | "*" | "/"], 
        AppendTo[out, stack[[-1]]]; stack = stack[[;; -2]]]; 
       AppendTo[stack, "/"], next == "+", 
       While[stack != {} && 
         MatchQ[stack[[-1]], "^" | "*" | "/" | "+" | "-"], 
        AppendTo[out, stack[[-1]]]; stack = stack[[;; -2]]]; 
       AppendTo[stack, "+"], next == "-", 
       While[stack != {} && 
         MatchQ[stack[[-1]], "^" | "*" | "/" | "+" | "-"], 
        AppendTo[out, stack[[-1]]]; stack = stack[[;; -2]]]; 
       AppendTo[stack, "-"], next == "(", AppendTo[stack, "("], 
       next == ")", 
       While[stack[[-1]] =!= "(", AppendTo[out, stack[[-1]]]; 
        stack = stack[[;; -2]]]; stack = stack[[;; -2]]; 
       If[StringQ[stack[[-1]]], AppendTo[out, stack[[-1]]]; 
        stack = stack[[;; -2]]]]]; 
     While[stack != {}, AppendTo[out, stack[[-1]]]; 
      stack = stack[[;; -2]]]; out]];
Print[rpn["3 + 4 * 2 / ( 1 - 5 ) ^ 2 ^ 3"]];