What does it mean for a grammar to be LR(0)?
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...smXm where s0 is the initial state, the only one present at the beginning of the computation on the stack, while sm is the state at the top of the stack. Each Xi is a grammatical symbol while each si 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' -&gt; .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.