Swapping Lemmas for Regular and Context-Free Languages with Advice

Chomsky studied regular languages and context-free languages to develop his theory of formal languages. These languages are generated by restricted forms of grammars and also characterized by finite-state automata. Karp and Lipton examined roles of supplemental information given besides original inputs, under the term of "advice," which depends only on the size of the inputs. We study the power and limitation of such advice, when it is given to automata. A standard pumping lemma in formal language theory is, however, of no use in order to prove that a given language is not regular with advice. We develop its substitution, called a swapping lemma for regular languages, to prove the non-regularity of the language with advice. For context-free languages, we also present a similar form of swapping lemma, which serves as a technical tool to show that certain languages are not context-free with advice.