How is this grammar ambiguous?

Question

I'm writing a simple expression parser in Jison. Here's my grammar:

{
    "operators": [
        ["left", "+", "-"],
        ["left", "*", "/", "%"]
    ],
    "bnf": {
        "program": [
            ["statement EOF", "return $1;"]
        ],
        "statement": [
            ["expression NEWLINE", "$$ = $1 + ';';"]
        ],
        "expression": [
            ["NUMBER",                       "$$ = yytext;"],
            ["expression binary expression", "$$ = $1 + $2 + $3;"]
        ],
        "binary": [
            ["+",              "$$ = ' + ';"],
            ["-",              "$$ = ' - ';"],
            ["*",              "$$ = ' * ';"],
            ["/",              "$$ = ' / ';"],
            ["%",              "$$ = ' % ';"],
            ["binary NEWLINE", "$$ = $1;"]
        ]
    }
}

When I try to run it it gives me the following error:

Conflict in grammar: multiple actions possible when lookahead token is + in state
13
- reduce by rule: expression -> expression binary expression
- shift token (then go to state 8)
Conflict in grammar: multiple actions possible when lookahead token is - in state
13
- reduce by rule: expression -> expression binary expression
- shift token (then go to state 9)
Conflict in grammar: multiple actions possible when lookahead token is * in state
13
- reduce by rule: expression -> expression binary expression
- shift token (then go to state 10)
Conflict in grammar: multiple actions possible when lookahead token is / in state
13
- reduce by rule: expression -> expression binary expression
- shift token (then go to state 11)
Conflict in grammar: multiple actions possible when lookahead token is % in state
13
- reduce by rule: expression -> expression binary expression
- shift token (then go to state 12)

States with conflicts:
State 13
  expression -> expression binary expression . #lookaheads= NEWLINE + - * / %
  expression -> expression .binary expression
  binary -> .+
  binary -> .-
  binary -> .*
  binary -> ./
  binary -> .%
  binary -> .binary NEWLINE

However it still produces the correct output in the end. For example 2 + 3 * 5 / 7 % 11 is correctly translated to 2 + 3 * 5 / 7 % 11;.

The way I see it my grammar appears to be unambiguous, so why is Jison complaining?

Update: As @icktoofay explained it's an operator associativity problem. By parsing an operator as a non-terminal symbol operator precedence and associativity information is lost. Hence I solved the problem as follows:

{
    "operators": [
        ["left", "+", "-"],
        ["left", "*", "/", "%"]
    ],
    "bnf": {
        "program": [
            ["statement EOF", "return $1;"]
        ],
        "statement": [
            ["expression NEWLINE", "$$ = $1 + ';';"]
        ],
        "expression": [
            ["NUMBER",                          "$$ = yytext;"],
            ["expression + expression",         "$$ = $1 + ' + ' + $3;"],
            ["expression - expression",         "$$ = $1 + ' - ' + $3;"],
            ["expression * expression",         "$$ = $1 + ' * ' + $3;"],
            ["expression / expression",         "$$ = $1 + ' / ' + $3;"],
            ["expression % expression",         "$$ = $1 + ' % ' + $3;"],
            ["expression + NEWLINE expression", "$$ = $1 + ' + ' + $4;"],
            ["expression - NEWLINE expression", "$$ = $1 + ' - ' + $4;"],
            ["expression * NEWLINE expression", "$$ = $1 + ' * ' + $4;"],
            ["expression / NEWLINE expression", "$$ = $1 + ' / ' + $4;"],
            ["expression % NEWLINE expression", "$$ = $1 + ' % ' + $4;"]
        ]
    }
}

That being said this grammar only allows one optional newline to follow a binary operator. How do I rewrite it so as to allow an arbitrary number of newlines to follow a binary operator? Also there must be some way in which I don't have to write 2 rules for each operator.

Answer 1

I'm not entirely familiar with Jison, but it looks like you're defining a rule that looks like this:

expression ::= number;
expression ::= expression binary expression;

Consider the expression 1 - 2 - 3. That could be interpreted as (1 - 2) - 3 or 1 - (2 - 3). Which is it? Your grammar is ambiguous. Normal mathematical rules say it should be left-associative. You need to make your grammar reflect that:

expression ::= number;
expression ::= expression binary number;