Automata and Computability (Undergraduate Texts in Computer by Dexter C. Kozen

By Dexter C. Kozen

This textbook presents undergraduate scholars with an creation to the fundamental theoretical versions of computability, and develops the various model's wealthy and sundry constitution. the 1st a part of the e-book is dedicated to finite automata and their homes. Pushdown automata offer a broader type of types and let the research of context-free languages. within the last chapters, Turing machines are brought and the ebook culminates in analyses of powerful computability, decidability, and Gödel's incompleteness theorems. scholars who have already got a few adventure with trouble-free discrete arithmetic will locate this a well-paced first path, and a couple of supplementary chapters introduce extra complex concepts.

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Intuitively, we can think of the machine in the leftmost state as guessing, every time it sees a 1, whether that 1 is the fifth letter from the right. It might be and it might not be-the machine doesn't know, and there is no way for it to tell at that point. If it guesses that it is not, then it goes around the loop again. If it guesses that it is, then it commits to that guess by moving to the second state, an irrevocable decision. Now it must verify that its guess was correct; this is the purpose of the tail of the automaton leading to the accept state.

Our notation is borrowed from Hopcroft and Ullman [60]. Lecture 5 Nondeterministic Finite Automata Nondeterm inism Nondetermini8m is an important abstraction in computer science. It refers to situations in which the next state of a computation is not uniquely determined by the current state. Nondeterminism arises in realUfe when there is incomplete information about the state or when there are external forces at work that can affect the course of a computation. For example, the behavior of a process in a distributed system might depend on messages from other processes that arrive at unpredictable times with unpredictable contents.

We show below that if A and B are regular, then so are Au B, An B, and '" A. We'll show later that AB and A * are also regular. The Product Construction Assume that A and B are regular. Then there are automata Ml M2 = (Ql> 1:,151 ,81, Fl)' = (Q2, 1:, 152 , 82, F2) with L(Ml) = A and L(M2) = B. To show that An B is regular, we will build an automaton Ma such that L(Ma) = An B. Intuitively, Ma will have the states of Ml and M2 encoded somehow in its states. On input x E 1:*, it will simulate Ml and M2 simultaneously on X, accepting iff both Ml and M2 would accept.

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