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Concept
Ambiguous grammar
Summary
In computer science, 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. Every non-empty context-free language admits an ambiguous grammar by introducing e.g. a duplicate rule. A language that only admits ambiguous grammars is called an inherently ambiguous language. Deterministic context-free grammars are always unambiguous, and are an important subclass of unambiguous grammars; there are non-deterministic unambiguous grammars, however.
For computer programming languages, the reference grammar is often ambiguous, due to issues such as the dangling else problem. If present, these ambiguities are generally resolved by adding precedence rules or other context-sensitive parsing rules, so the overall phrase grammar is unambiguous. Some parsing algorithms (such as (Earley or GLR parsers) can generate sets of parse trees (or "parse forests") from strings that are syntactically ambiguous.
The simplest example is the following ambiguous grammar (with start symbol A) for the trivial language that consists of only the empty string:
A → A | ε
...meaning that the nonterminal A can be derived to either itself again, or to the empty string. Thus the empty string has leftmost derivations of length 1, 2, 3, and indeed of any length, depending on how many times the rule A → A is used.
This language also has an unambiguous grammar, consisting of a single production rule:
A → ε
...meaning that the unique production can produce only the empty string, which is the unique string in the language.
In the same way, any grammar for a non-empty language can be made ambiguous by adding duplicates.
The regular language of unary strings of a given character, say 'a' (the regular expression a*), has the unambiguous grammar:
A → aA | ε
...but also has the ambiguous grammar:
A → aA | Aa | ε
These correspond to producing a right-associative tree (for the unambiguous grammar) or allowing both left- and right- association. This is elaborated below.
Official source
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