Arithmetic expression grammar

Question

I'm trying to create an appropriate grammer for some arithmetic expressions. The valid tokens for my expressions are the following:

'+', '-', '/', '*' , and '**' for powers. The expressions could also contain symbols and functions. The functions can take multiple arguments, some of which might be optional. From what little I remember from expression parsing, I must come up with a grammar that is not left-recursive and also preserves operator associativity.

So here is what I came up with, after a little bit of searching (though not sure about associativity):

E = T Eopt
Eopt = '+' T Eopt | '-' T Eopt | ε 
T = F Topt
Topt = '*' F Topt | '/' F Topt | ε
F = Number | '(' E ')' 

which can be found in many textbooks. What changes does the above grammar need, so that it can acommodate the power token ('**') and both symbols and functions?

Please do not point me to flex/yacc etc. Thank you.

Answer 1

You're almost there. Changes begin at F:

E = T Eopt
Eopt = '+' T Eopt | '-' T Eopt | ε 
T = F Topt
Topt = '*' F Topt | '/' F Topt | ε
F = P Fopt
Fopt = '**' P Fopt | ε
P = Number | '(' E ')'

This assumes your tokenizer is distinguishing between * and ** .