By Dexter C. Kozen

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

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One might think that adding nondeterminism might increase expressive power, but in fact for finite automata it does not: in terms of the sets accepted, nondeterministic finite automata are no more powerful than deterministic ones. In other words, for every nondeterministic finite automaton, there is a deterministic one accepting the same set. However, nondeterministic machines may be exponentially more succinct. Nondeterministic Finite Automata A nondeterministic finite automaton (NFA) is one for which the next state is not necessarily uniquely determined by the current state and input symbol.

For u:j:. f { 0:"" - al E + ... + ale + E if k if k ~ 1, = o. X-{q} .... " where 8 is a start state and accepted by M. 19), it is best to try to choose one that disconnects the automaton as much as possible. 3 53 Let's convert the automaton o~: ~r to an equivalent regular expression. The set accepted by this automaton will be represented by the inductively defined regular expression o:{p,q,r} pp , since p is the only start and the only accept state. Removing the state q (we can choose any state we like here), we can take o:{p,q,r} - o:{p,r} pp pp + o:{p,r} (o:{p,r})*o:{p,r} pq qq qp' Looking at the automaton, the only paths going from p to p and staying in the states {p, r} are paths going around the single loop labeled 0 from p to p some finite number of times; thus we can take ",{p,r} "'pp - 0* • By similar informal reasoning, we can take o:{p,r} pq o:{p,r} qq =0*1 , = E+ 01 + 000*1 = E + O( E+ 00*)1 = E+OO*l, Q~V} = 000*.

The automaton is said to accept x if at least one computation path on input x starting from at least one start state leads to an accept state. The automaton is said to reject x if no computation path on input x from any start state leads to an accept state. Another way of saying this is that x is accepted iff there exists a path with label x from some start state to some accept state. Again, there is no mechanism for determining which state to start in or which of the possible next moves to take in response to an input symbol.