James A. Foster

Formal Languages

 

We need a mathematical model of input and output, and a definition of a problem, in order to build general, abstract, mathematical models of computation. Our tool will be formal languages formal languages. We used the following definitions:

Alphabet

A finite set of characters (which are primitives)

Word (or string)

A finite sequence of zero or more characters from a given alphabet

Language

A set of words (may be infinite or finite)

For example, assume we're using the alphabet . The characters characters are: a and b. Some words words are: (the word with zero characters), aaabb, a, and aaaaaaaaaaaaaaaaa. Some languages languages are: (a set with no words), (the set with a single word, where the word has no characters), , and .

Here are some operations on words:

Catenation

The catenation of two words w and u, denoted , is the word consisting of all the characters in w followed by all those in u. Iterated catenation of a word, , represents k copies of w stuck together. (note: .)

Reversal

The reversal of a word, denoted , is all the characters in w in reverse order

Length

The length of a word, denoted , is the number of characters in w

Census

The number of times a particular character x occurs in word w is denoted

Here are some operations on languages:

Catenation

The catenation of two languages and , denoted , is the set . denotes the set . denotes .

Union, Intersection

These have the usual set-theoretic definitions.

Complement

. Complementation is defined as .

The power set power set of a set L is the set of all subsets of L. We will denote this . We distinguish between a language, and a set of languages by calling the latter a class class of languages.

Each formal language L has an associated decision problem decision problem. We call this the decision problem L and sometimes just the problem L. That is

Notice that the answer to any particular question about any decision problem is either ``yes'' or ``no''. This defines a natural partitioning of all words into those for which the answer is ``yes'', which is precisely L, and everything else. In this sense, any language is synonymous with the set of ``yes'' instances for it's decision problem. So, we will often use the terms formal language and problem problem interchangeably.

A decision problem is solveable solveable if there is an algorithm which solves it. That is, L is solveable if there is an algorithm which will output ``yes'' (or something equivalent) whenever given a string . Note that we do not specify what this sort of algorithm should do when .

Any problem which is not solvable is unsolvable unsolvable.

There are many interesting questions which we can now pose and answer with this terminology. For example:

	Given a language L and a mathematical model of computation, is L solvable using that model?
	Given two mathematical models, are there any languages whose decision problems are soveable with one, but not the other?
	Are there any languages whose decision problem is not solvable by any model of computation?
	Is there any model of computation which can solve all decision problems?

This approach characterizes "computation" as "solution of decision problems with algorithms". This seems highly artificial. We usually think of computation as a functional transformation of inputs into outputs. However, note that the problem of computing a function f(x)=f is very similar to recognizing the set of pairs in that if we could write a program to recognise this set, we could compute f(x) with the following algorithm:

So, this approach isn't as artificial as it seems. However, we will discuss the theory of computable functions later in the course.

Last Updated 01/13/2003 15:56 -0800