Arden's Theorem is a fundamental result in the theory of regular languages and automata. It provides a method for solving equations involving regular expressions.
Statement: Let P and Q be two regular expressions over an alphabet Σ. If P does not contain the empty string (ε), then the equation R = Q + RP has a unique solution R = QP*.
Application: This theorem is particularly useful for converting finite automata to regular expressions by solving a set of equations. It helps eliminate recursive definitions and express the language accepted by an automaton as a regular expression.
Rice's Theorem is a fundamental result in computability theory that deals with the decidability of properties of recursively enumerable languages (or Turing machines).
Statement: Any non-trivial semantic property of the language recognized by a Turing machine is undecidable. A property is non-trivial if it holds for some recursively enumerable languages but not for others, and it is semantic if it depends only on the language recognized by the machine, not on its implementation.
Significance: This theorem establishes fundamental limitations on what can be algorithmically determined about programs. For example, it is undecidable whether a given Turing machine accepts a specific string, whether its language is empty, whether it is finite, or whether two machines accept the same language. Rice's Theorem shows that most interesting questions about program behavior cannot be solved algorithmically in general.