Final answer:
The condition that prevents the use of predictive parsing in a context-free grammar where one production derives an empty string is the introduction of left recursion. Left recursion occurs when a non-terminal can eventually derive itself as the leftmost symbol, leading to potential infinite loops for certain parsers.
Step-by-step explanation:
The question asks about a context-free grammar (CFG) and under what conditions the use of a specific production rule is prevented. In the given CFG, there's a production: S → α | β where α derives an empty string (often represented by epsilon, ε). If a production can derive an empty string, it means we have a nullable production. In context-free grammars, having nullable productions can potentially cause ambiguity or confusion in certain parsing strategies, especially predictive parsing. Predictive parsing relies on lookahead symbols to make decisions, and if a production leads to an empty string, it may not be possible to determine the lookahead symbol accurately.
Left recursion refers to a situation in context-free grammars where a non-terminal symbol in a production can eventually derive itself as the leftmost symbol in a sequence. This can create an infinite loop in certain types of parsers, such as top-down parsers, which are a subclass of predictive parsers. Therefore, if α derives an empty string or a string starting with the same non-terminal 'S', it may introduce left recursion to this CFG. Since the question prompt is incomplete, we can indicate that left recursion, introduced when a non-terminal symbol can derive itself, is one condition that always prevents the use of predictive parsing, because predictive parsers cannot handle left-recursive grammars.