Formal Language Definitions

(Greek letters are written out so this may be read as plain text.)

Contents

• Symbol
• Alphabet
• String or Word
• Language, Formal Language
• Regular Language
• Grammar, Formal Grammar
• Regular Grammar
• Context Free Language, CFL
• Greibach Normal Form, GNF
• Chomsky Normal Form
• CYK Algorithm
• Recursive Languages, Sets
• Recursively Enumerable Languages, Sets
• Chomsky Hierarchy of Grammars, Languages
• P and NP Classes of Languages
• Symbol

A character, glyph, mark.
An abstract entity that has no meaning by itself, often called uninterpreted.
Letters from various alphabets, digits and special characters are
the most commonly used symbols.

Alphabet

A finite set of symbols.
An alphabet is often denoted by sigma, yet can be given any name.
B = {0, 1}  Says B is an alphabet of two symbols, 0 and 1.
C = {a, b, c}  Says C is an alphabet of three symbols, a, b and c.
Sometimes space and comma are in an alphabet while other times they
are meta symbols used for descriptions.

String also called a Word

A finite sequence of symbols from an alphabet.
01110 and 111 are strings from the alphabet B above.
aaabccc and b are strings from the alphabet C above.
A null string is a string with no symbols, usually denoted by epsilon.
The null string has length zero.
The null string is usually denoted epsilon.
Vertical bars around a string indicate the length of a string expressed as
a natural number. For example  |00100| = 5, |aab| = 3, | epsilon | = 0

Formal Language, also called a Language

A set of strings from an alphabet.
The set may be empty, finite or infinite.
There are many ways to define a language. See definitions below.
There are many classifications for languages. See definitions below.

Because a language is a set of strings, the words language and set
are often used interchangeably in talking about formal languages.

L(M) is the notation for a language defined by a machine  M.
The machine  M  accepts a certain set of strings, thus a language.

L(G) is the notation for a language defined by a grammar  G.
The grammar  G  recognizes a certain set of strings, thus a language.

M(L) is the notation for a machine that accepts a language.
The language  L  is a certain set of strings.

G(L) is the notation for a grammar that recognizes a language.
The language  L  is a certain set of strings.

The union of two languages is a language.  L = L1 union L2
The intersection of two languages is a language. L = L1 intersect L2
The complement of a language is a language. L = sigma* - L1
The difference of two languages is a language. L = L1 - L2

Regular Language, Regular Expression, regular

A set of strings from an alphabet.
The set may be empty, finite or infinite.

The building blocks of regular languages are symbols, concatenation of
symbols to make strings (words), set union of strings and Kleene
closure (denoted as *, also called the Kleene star, it should be
typed as a superscript but this is plain text.)

Informally, we use a syntax for regular expressions.
using sigma as the set {0, 1} (an alphabet of two symbols)
01110 is a string starting with the symbol  0  and then concatenating
1, then 1, then 1, and finally concatenating 0. No punctuation is used
between symbols or strings that are concatenated.
(01+10) is a union of the two strings 01 and 10. The set {01, 10}
(00+11)* is the Kleene closure of the union of 0 concatenated with 0 and
1 concatenated with 1.

The Kleene closure (star) is defined as the concatenation of
none, one, two, or any countable number strings it applies to.
Examples of Kleene star:
1*  is the set of strings {epsilon, 1, 11, 111, 1111, 11111, etc. }
This set is infinite.

(1100)* is the set of strings {epsilon, 1100, 11001100, 110011001100, etc. }

(00+11)* is the set of strings {epsilon, 00, 11, 0000, 0011, 1100, 1111,
000000, 000011, 001100, etc. }
note how the union ( + symbol) allows all possible choices
of ordering when used with the Kleene star.

(0+1)* is all possible strings of zeros and ones, often written as
sigma * where sigma = {0, 1}

(0+1)* (00+11) is all strings of zeros and ones that end with either
00 or 11.  Note that concatenation does not have an operator
symbol.

(w)+  is a shorthand for (w)(w)*   w is any string or expression and the
superscript plus, + ,  means one or more copies of w are in the set defined
by this expression.

Formally, a regular language is defined on the bottom of page 28.

Some versions of grep implement the regular expression search by
simulating a Finite Automata.
Note that grep uses a different syntax and uses a subset of ASCII
characters for symbols.

Grammar, a formal grammar

A grammar is defined as G = (V, T, P, S) where:
V is a set of symbols called variables, typically S, A, B, ...
T is a set of symbols called terminal, typically 0, 1, a, b, ...
P is a set of productions
S is the starting or goal variable from V

The productions P are of the form:
A -> w

Where A is a variable
w is any concatenation of variables and terminals

An example grammar is G = (V, T, P , S) where  V = { S, A }  T = { 0, 1 }
and the productions, P , are:

S -> 0A | 0
A -> 10A

This grammar corresponds to the regular expression  0(10)* which in turn
corresponds to the  deterministic finite automata
shown in  Figure DFA1 .

Regular Grammar

A grammar is defined as G = (V, T, P, S) where:
V is a set of symbols called variables, v1, v2, ... ,vn
T is a set of symbols called terminal, t1, t2, ,,, ,tm
P is a set of productions
S is the starting or goal variable from V

The productions P are of the form:
A -> w
A -> wB

Where A and B are variables
w is any combination terminals, may be empty string

Any regular grammar can be converted to an equivalent DFA, NFA,
regular language or regular expression.

Context Free Language, CFL

A set of strings from an alphabet.
The set may be empty, finite or infinite.

A grammar G = (V, T, P, S) with the productions restricted to:

coming soon

yacc and bison can process context free grammars and have the ability
to handle some context sensitive grammars as well.

Context Free Languages are related to Push Down Automata.

Greibach Normal Form, GNF

A grammar G = (V, T, P, S) with the productions, P, restricted to the form:
variable -> terminal variable(s)

A -> a alpha   where A is a variable in V, a is a terminal in T and
alpha is none, one or more variables in V.

Chomsky Normal Form, CNF

A grammar G = (V, T, P, S) with the productions restricted to the forms:
variable -> variable variable
variable -> terminal

A -> B C   A, B and C are variables in V and exactly two variables are on the right
A -> a     A is a variable in V and a is exactly one terminal symbol in T.

CYK Algorithm, Cocke-Younger-Kasami

Given a CFG  G=(V, T, P, S) and a string x in T*, is x in L(G)?
The CYK Algorithm uses dynamic programming (from 441 algorithms course)
to provide a |x| cubed time algorithm to answer this question.

see  p139-140 in text book

Recursive Languages, recursive sets

The languages, sets, accepted by Turing machines and unrestricted grammars.

Recursively enumerable sets, r.e. languages

The sets, languages, that can be generated (enumerated) where all strings
in the set, language, of a given length can be generated.

Usually the enumeration is strings of length 1, then strings of length 2,
and so fourth.  Of course there may be no strings for some lengths.

In some cases, generate Sigma ^ n and pass each string to a machine or
grammar.  The accepted strings are the strings of length n.

There is no requirement that the strings be generated in lexical or any
other order.

If both a set and its complement are recursively enumerable, then the
set is recursive.

Chomsky Hierarchy of Grammars / Languages

Type 0, Grammars that generate  r.e. sets
and characterize the  r.e.  languages
Unrestricted grammars of the form  G = (V, T, P ,S)
The restriction is removed from the form of the productions.

Type 1, Grammars that characterize context sensitive languages.
Type 2, Grammars that characterize context free languages.
Type 3, Grammars that characterize regular languages.

Note: Type 0 grammars can have infinite loops in the parser for the
grammar when a string not in the grammar is input to the parser.

P and NP classes of languages

A class of languages is a set of languages that share some characteristic.
Since a language is a set of strings from a finite alphabet, a class of
languages is a set of sets.

The language class P is the set of languages for which there exists a
deterministic Turing machine that accepts each language in a number of
transitions bounded by a fixed polynomial in the length of the input string.

Start with a "standard" Turing machine with a finite state control that
is deterministic, the TM has a transition table with one entry for each
(state,input-symbol) pair.

Consider a specific language that has strings of various lengths. Let the
length of any string in the language be denoted n. If there exists a fixed
polynomial in n, e.g. c * n^r for some fixed constant c and some fixed
constant r, such that the Turing machine accepts or rejects all strings in
the specific language in c * n^r moves, transitions, then that specific
language is in the set P.

The language class NP is different from the language class P in two ways:
1) NP languages have Turing machine with a nondeterministic finite state
control. Nondeterministic does not mean random.
2) NP languages have a Turing machine that does not have to reject a
string in any prescribed number of moves.

Note: Without the time restriction, bounded number of moves, any
nondeterministic Turing machine has an equivalent deterministic
Turing machine. It is believed that the language class P is not
equivalent to the language class NP but this belief is not yet proven.

Other Classes of Languages