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Introduction to theory

of computation Tom Carterhttp://astarte.csustan.edu/˜ tom/SFI-CSSS

Complex Systems Summer School

June, 20051

Our general topics:←

?Symbols, strings and languages?Finite automata?Regular expressions and languages?Markov models?Context free grammars and languages?Language recognizers and generators?The Chomsky hierarchy?Turing machines?Computability and tractability?Computational complexity?References2

The quotes?

?No royal road?Mathematical certainty?I had a feeling once about Mathematics?Terminology (philosophy and math)?RewardsTo topics←3

Introduction←

What follows is an extremely abbreviated look

at some of the important ideas of the general areas of automata theory, computability, and formal languages. In various respects, this can be thought of as the elementary foundations of much of computer science.

The area also includes a wide variety of tools,

and general categories of tools . . .4

Symbols, strings and

languages← •The classical theory of computation traditionally deals with processing an input string of symbols into an output string of symbols. Note that in the special case where the set of possible output strings is just{'yes", 'no"}, (often abbreviated{T, F}or{1, 0}), then we can think of the string processing as string (pattern) recognition.

We should start with a few definitions.

The first step is to avoid defining the

term 'symbol" - this leaves an open slot to connect the abstract theory to the world . . . We define:1.Analphabetis a finite set of symbols.5

2.Astringover an alphabet A is a finite

ordered sequence of symbols from A.

Note that repetitions are allowed. The

length of a string is the number of symbols in the string, with repetitions counted. (e.g.,|aabbcc|= 6)3.The empty string, denoted by?, is the (unique) string of length zero. Note that the empty string?is not the same as the empty set∅.4.If S and T are sets of strings, then ST ={xy|x?S andy?T}5.Given an alphabet A, we define A 0={?} A n+1= AAn A n=0An6.AlanguageL over an alphabet A is a subset of A ?. That is, L?A?.6 •We can define the natural numbers,N, as follows:

We let

0 =∅

1 ={∅}

2 ={∅,{∅}}

and in general n+ 1 ={0,1,2,...,n}. Then N={0,1,2,...}.•Sizes of sets and countability:

1.Given two sets S and T, we say that

they are the same size (|S|=|T|) if there is a one-to-one onto function f: S→T.2.We write|S| ≤ |T|if there is a one-to-one (not necessarily onto) functionf: S→T.7

3.We write|S|<|T|if there is a

one-to-one functionf: S→T, but there does not exist any such onto function.4.We call a set S

(a)Finite if|S|<|N|(b)Countable if|S| ≤ |N|(c)Countably infinite if|S|=|N|(d)Uncountable if|N|<|S|.5.Some examples:

(a)The set of integers Z={0,1,-1,2,-2,...}is countable.(b)The set of rational numbers

Q={p/q|p,q?Z,q?= 0}is

countable.8 (c)If S is countable, then so is SxS, the cartesian product of S with itself, and so is the general cartesian product S nfor anyn <∞.(d)For any nonempty alphabet A, A ?is countably infinite.

Exercise: Verify each of these

statements.6.Recall that thepower setof a set S is the set of all subsets of S:

P(S) ={T|T?S}.

We then have the fact that for any set

S, |S|<|P(S)|.

Pf: First, it is easy to see that

|S| ≤ |P(S)| since there is the one-to-one function f: S→P(S) given byf(s) ={s}for s?S.9

On the other hand, no function

f: S→P(S) can be onto. To show this, we need to exhibit an element of

P(S) that is not in the image off. For

any givenf, such an element (which must be a subset of S) is R f={x?S|x /?f(x)}.

Now suppose, for contradiction, that

there is somes?S withf(s) = Rf.

There are then two possibilities: either

s?f(s) = Rfors /?f(s) = Rf. Each of these leads to a contradiction:

Ifs?f(s) = Rf, then by the definition

of R f,s /?f(s). This is a contradiction.

Ifs /?f(s) = Rf, then by the definition

of R f,s?Rf=f(s). Again, a contradiction.quotesdbs_dbs2.pdfusesText_3