What does it mean for a grammar to be LR(0)?

Question

I am unsure what it means for a grammar to be $X$. More specifically, what it means for a grammar to be LR(0).

For part of an assignment I had to form the DFA for a grammar, which I had no issues with. And then I have been asked whether the grammar is LR(0). My guess is yes, but not because I understand exactly what it means. I would assume it isn't if, when creating the DFA I came across some strange behaviors that prevented the DFA making any sense (if you know what I mean/if I'm making any sense).

Answer 1

A bottom-up parsing technique, used for parsing most context-free grammars is LR(k): L indicates that the input is parsed from left to right, R indicates that in the parsing process a rightmost derivation is produced; the number k indicates how many lookahead symbols are used.

An LR parsing scheme consists of: input, output, stack, a parsing program and a parsing table divided into two parts (actions (shift, reduce) and goto).

The idea is that the parsing program reads characters from an input buffer one at a time and uses the stack to store strings of the form s₀X₁s₁X₂s₂...s_mX_m where s₀ is the initial state, the only one present at the beginning of the computation on the stack, while s_m is the state at the top of the stack. Each X_i is a grammatical symbol while each s_i is a symbol that identifies a state. Each state describes the information contained in the stack before it; the combination of the state at the top of the stack and the current input symbol is used to index the parsing table and to determine the parser's shift-reduce decisions.

Given a grammar, to prove that this is LR(0) builds the DFA of viable prefixes. The first prefix is always S' -> .S because we consider the grammar increased. Then follow the rules to build the LR(0) parsing table.

A grammar is LR(0) if each position in the parsing table LR(0) contains at most one element.

Furthermore, the LR(0) languages all enjoy the prefix property. So if a language does not have this property, we can immediately say that it is not LR(0).

If the grammar is ambiguous then it cannot be LR(k) for any k.