Currently studying for an exam and looking through past papers when I came across this question.
Below is a grammar in EBNF that describes simple arithmetic expressions, like 1 + 2 * 3 - 4:
Expression = Operand, {Operator, Operand}; Operand = "1"|"2"|"3"|"4"|"5"|"6"|"7"|"8"|"9"; Operator = "+"|"-"|"*"|"/";(iv) Using this grammar, there are multiple ways of evaluating an expression like 1 + 2 * 3 - 4. Describe two of them, and explain what this means about the grammar provided. [2 marks]
To my understanding, ambiguous grammar means there is either more than one left-most or right-most derivation, which usually implies there is some ambiguity in the grammar's order of precendence. But there is no precedence here, and the recursion is linear.
Advice?
You have part of the answer in your question.
Yes; you have almost the correct definition of an ambiguous grammar. If one performs a left-most and a right most derivation of the grammar they should produce an identical parse tree.
Yes; You are almost correct when you think this implies a problem with the grammars order of precedence, and yes, this grammar does not have any. Therein lies the problem: The operators are all given the same precedence and thus the different derivations will result in different answers from evaluating the example.
We could reduce 1 + 2 * 3 - 4 into either:
(1+2) * (3-4)
1 + (2 * 3) - 4
1 + (2 * (3 - 4))
depending on how the precedence of the operators are treated.
If you draw out explicitly the left-most and right-most reductions and hence derive the parse trees it will be clearer. This is often what students are expected to do for full marks in an exam question like this. I will therefore leave this as a revision exercise.