Arithmetic expressions grammar transformation

Question

In the article Parsing Expressions by Recursive Descent by Theodore Norvell (1999) the author starts with the following grammar for arithmetic expressions:

E --> E "+" E | E "-" E | "-" E | E "*" E | E "/" E | E "^" E | "(" E ")" | v

which is quite bad, because it's ambiguous and left-recursive. So he starts from removing the left recursion from it, and his result is as such:

E --> P {B P}
P --> v | "(" E ")" | U P
B --> "+" | "-" | "*" | "/" | "^"
U --> "-"

But I can't figure out how did he get to this result. When I try to remove the left recursion myself, I'm doing it the following way:

1. Firs, I group together the productions which doesn't have left recursion in one group, and other (left-recursive) in another group:

   E --> E "+" E | E "-" E | E "*" E | E "/" E | E "^" E     // L-recursive
   E --> v | "(" E ")" | "-" E

2. Next, I name them and factor for easier manipulations:

   E --> E B E  // L-recursive; B stands for "Binary operator"
   E --> P  // not L-recursive; P stands for "Primary Expression"
   P --> v | "(" E ")" | U E   // U stands for "Unary operator"
   B --> "+" | "-" | "*" | "/" | "^"
   P --> "-"

   Now I need to deal only with the first two productions, which are now easier to deal with.

3. I rewrite those first two productions by starting from the non-L-recursive production (which is simply P, the Primary expression) and following it by the optional Tail T, which I define as the rest of the original production less the first left-recursive nonterminal (that is, just B E) followed by the Tail T, or which could be empty:

   E --> P T
   T --> B E T |

   (note the empty alternative for the tail).

4. These two productions I can now rewrite in EBNF like this:

   E --> P {B E}

   which is nearly what the author get, but I have E instead of P there inside the zero-or-more repetition pattern (the Tail). The other productions I get quite the same as he have got:

   P --> v | "(" E ")" | U E
   B -> "+" | "-" | "*" | "/" | "^"
   U -> "-"

   but here too I have E instead of P in the first production for P.

So, my question is: What am I missing? What algebraic transformation on the syntax I need to proceed now to get the same exact form as the autor gets? I tried substitutions for E, but it only leads me into loops. I suspect that I need somehow to substitute P for E, but I don't know any legal transformation to justify it. Maybe you know what's the last missing step?

Answer 1

The missing step:

E --> P T
T --> B E T |

rewrite E in T:

E --> P T
T --> B P T T | 

Simplify T:

E --> P T
T --> B P T | 

Equivalent to:

E --> P T
T --> {B P}

And there you are.