Computer Science 456/656
Automata and Formal Languages
What is a Language?
The formal definition is that a language is a set of
strings over a given alphabet.
An alphabet is defined to be a finite set whose members
are called symbols, but there is no definition of
symbol. Thus, anything could be called a symbol.
Examples of languages.

English, Mandarin Chinese, and other spoken and written languages. We call
these "natural languages." We do not study natural languages in this
course, yet certain concepts that are in this course, such as contextfree
grammars, were developed in an effort to analyze natural languages.
Serious effort is being made to understand the structure of natural
languages, and anyone doing serious research in that area must
have extensive knowledge of the subject matter of this course.

Formal mathematical languages, such as decimal numeration that you learned
as a small child, binary numeration, the language of algebraic expressions
you learned in school, or the language of formal logic you learned in PHI 109.

Pascal, C++, and other programming languages. These languages incorporate
the complexity of formal mathematical languages and much more. I will assume
that you are familiar with at least one programming language and have written
computer programs.

The genetic code. All living creatures, as far as we know, utilize a language
whose symbols are molecular groups, and which is written in DNA or RNA.

01 problems. This is probably a surprise use of the word "language" to
most of you. Most people usually think that a language must always be used
for communication, as are all of the above examples. But there is no
requirement that a language be useful for anything.
A 01 problem is a usually thought of as a class of questions where
the answer is always either "no" or "yes." For example, we would usually
think of the "Addition Problem" as all questions where we are asked if the sum
of two numbers is equal to a third number, such as, "Is it true that 2+3=5?"
or "Is it true that 2+2=6?" Such individual questions would be called
"instances" of the problem.
To think of a given 01 problem as a language, we must first encode
all instances using some alphabet, and then define the language to be
all such encodings where the answer is "yes." For example, all instances
of the addition problem can be encoded using 12 symbols, the digits, the
plus sign, and the equals sign, and the language L_{add} can
be defined to be all those strings where the equation is true. For example,
the strings "2+3=5" and "439+21=460" are members of L_{add}, while
the strings "2+2=6" and "+34=+" are not, in the first case, because it is
the encoding of an instance where the answer is "no," and in the second case
because it is not the encoding of any instance.
The box at the top of page 33 of Hopcroft, Motwani, and Ullman discusses
this important use of languages.