# AUTOMATA AND COMPUTABILITY

Language: English

Pages: 413

ISBN: 0387949070

Format: PDF / Kindle (mobi) / ePub

This textbook provides undergraduate students with an introduction to the basic theoretical models of computability, and develops some of the model's rich and varied structure. The first part of the book is devoted to finite automata and their properties. Pushdown automata provide a broader class of models and enable the analysis of context-free languages. In the remaining chapters, Turing machines are introduced and the book culminates in analyses of effective computability, decidability, and Gödel's incompleteness theorems. Students who already have some experience with elementary discrete mathematics will find this a well-paced first course, and a number of supplementary chapters introduce more advanced concepts.

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And take all the transitions of M and N as transitions of P. Make the start states of M the start states of P and the final states of N the final states of P. Finally, put f-transitions from all the final states of M to all the start states of N. Then L(P) = AB. Example 6.7 . Let A = {aa}, B = {bbl. Here are automata for A and B: ~ a .. •• b •• Here is the automaton you get by the construction above for AB: ~. a~. a •• E •• b~. b .® o If A is regular, then so is its asterate:.

Equivalent to one using only atomic patterns a E ~, e, fl, and operators +, " and *. Patterns using only these symbols are called regular ezpressions. Actually, as we have observed, even e is redundant, but we include it in the definition of regular expressions because it occurs so often. Our goal for this lecture and the next will be to show that the family of subsets of ~* represented by patterns is exactly the family of regular sets. Thus as a way of describing subsets of ~*, finite automata,.

Of Lecture 14 actually gives this automaton. Suppose M = (Q, E, 6, s, F) is a DFA for R that is already collapsedj that is, there are no inaccessible states, and the collapsing relation p ~ q ~ 't/z E E* (6(p,x) E F {:::::} 6(q,x) E F) is the identity relation on Q. Then the Myhill-Nerode relation sponding to M is exactly =R: =M corre- X=RY {:::::} 't/z E E* (zz E R {:::::} yz E R) definition of =R {:::::} 't/z E E* (6(s,xz) E F {:::::} 6(s,yz) E F) definition of acceptance {:::::} 't/z E E*.

Any finite set. For example, we might use the alphabet {O, 1, 2, ... ,9} if we are talking about decimal numbers; the set of all ASCII characters if talking about text; {O, I} if talking about bit strings. The only restriction is that the alphabet be finite. When speaking about an arbitrary finite alphabet abstractly, we usually denote it by the Greek letter E. We call elements of E letters or symbols and denote them by a, b, c, .... We usually do not care at all about the nature of the elements.

Operators of equal precedence, we will perform the operations from left to right. (The left-to-right convention corresponds to common informal usage, but some programming languages, such as APL, use the opposite convention.) Under these conventions, we only need parentheses when we want to depart from the default parse tree. For example, consider the following grammar for well-formed arithmetic expressions over constants 0,1, variables a, b, c, and operator symbols + (addition), binary -.