Final answer:
The statement is true: a language is context-free if and only if there exists a push-down automaton that recognizes it. Push-down automata utilize a stack to process the nested structures typical of context-free languages, like palindrome strings.
Step-by-step explanation:
The statement that a language L is context-free if and only if there is a push-down automaton M such that L = L(M) is true. A context-free language is a type of formal language which can be generated by a context-free grammar, and it can be recognized by a push-down automaton.
A push-down automaton is a computational model that makes use of a stack to store extra information. This capability to use the stack enables push-down automata to recognize context-free languages, which often have nested structures or patterns.
A context-free language such as palindrome strings (e.g., strings that read the same forward and backward like 'racecar') can be recognized by a push-down automaton which uses the stack to remember the first half of the string and then validate that the second half matches.
On the other hand, languages that require more computational power, such as those requiring comparison of two or more non-adjacent sections of a string, cannot be recognized by a push-down automaton and are not context-free. The equivalence between context-free languages and push-down automata is a fundamental concept in the theory of computation.