Is it always true that operators defined at the same level in the production rule, have same precedence ?

Question

If I have the grammar :

E->E*F | F+E |F

F->id ,

Now here although + and * are defined at the same level , with different associativity but this grammar produces 2 different parse trees for the string id+id*id , so here we can't make any generalization regarding precedence of the operators although they are defined at the same level .

So what conclusion can be drawn from this ?

Comments

This grammar is ambiguous because there are two left most derivations for the given string(an ambiguous grammar is a context free grammar for which there exists a string that can have more than one leftmost derivation or parse tree).

Yes, here we can't make any generalization regarding precedency, because both the operator are at the same level.

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Above Grammar is Operator precedence grammar. And in operator precedence grammar when the grammar is Ambiguous we use the general mathematical precedences.

Answer 1

The grammar is ambiguous since there are two leftmost derivations for the string id+id*id.

* and + have same precedence .as they are on same level.

* is left associativity and + is right associativity.

Comments

_{when the grammar is ambiguous , how can u define the precedence of the grammar ?}

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Why can't An ambiguous grammar not have precedence?

+ and * As they are on same level so they are of same precedence. + and * having same precedence is the reason for ambiguity . The compiler will get confuse whether to evaluate + or * first.

An ambiguous grammar can be converted to ambiguous :-

The example has two grammars:

Ambiguous:

E → E + E | E ∗ E | (E) | a

Unambiguous:

E → E + T | T
T → T ∗ F | F
F → (E) | a

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We can define precedence of operators in an ambiguous grammar because we can use Operator-Precedence Parser(https://en.wikipedia.org/wiki/Operator-precedence_parser) to parse some of these ambiguous grammar that follows Operator rule, i.e., no two operands should be together and there must be an operator between them.

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@Akhilesh Singla

Where is that given in that wiki link?

Answer 2

• * and + have same precedence .
• * is left associativity and + is right associativity.
•  grammar is ambiguous

Comments

_{when the grammar is ambiguous , how can u define the precedence of the grammar ?}

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+ and * at same level then we can say both have same presedence.

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Is this answer correct?

@radha gogia

What's the correct answer for this now?

Did you verified till now?

Answer 3

Yes as + and * are at same level so you can say they have the same precedence.In fact you need to have precedence of operators even for ambiguous grammar only then you can convert it to unambiguous. We use precedence and associativity for converting ambiguous to unambiguous.