formal language and automata theory unit 2

Formal Languages & Automata Theory
Department of Computer Science & Engineering
G. Pullaiah College of Engineering and Technology

Regular Expressions
and
Finite State Automata
UNIT-II

Introduction
• Regular expressions are equivalent to Finite State
Automata in recognizing regular languages, the first
step in the Chomsky hierarchy of formal languages
• The term regular expressions is also used to mean
the extended set of string matching expressions
used in many modern languages
– Some people use the term regexp to distinguish this use
• Some parts of regexps are just syntactic extensions
of regular expressions and can be implemented as a
regular expression – other parts are significant
extensions of the power of the language and are not
equivalent to finite automata

Concepts and Notations
• Set: An unordered collection of unique elements
S1 = { a, b, c } S2 = { 0, 1, …, 19 } empty set:
membership: x S union: S1  S2 = { a, b, c, 0, 1, …, 19 }
universe of discourse: U subset: S1  U
complement: if U = { a, b, …, z }, then S1' = { d, e, …, z } = U - S1
• Alphabet: A finite set of symbols
– Examples:
• Character sets: ASCII, ISO-8859-1, Unicode
• = { a, b } 2= { Spring, Summer, Autumn, Winter }
• String: A sequence of zero or more symbols from an alphabet
– The empty string: 

Concepts and Notations
• Language: A set of strings over an alphabet
– Also known as a formal language; may not bear any resemblance to a
natural language, but could model a subset of one.
– The language comprising all strings over an alphabet is written as:
*
• Graph: A set of nodes (or vertices), some or all of which may be
connected by edges.
– An example: – A directed graph example:
1
3
2 a
b
c

Regular Expressions
• A regular expression defines a regular
language over an alphabet :
–  is a regular language: //
– Any symbol from is a regular language:
 = { a, b, c} /a/ /b/ /c/
– Two concatenated regular languages is a
regular language:
 = { a, b, c} /ab/ /bc/ /ca/

Regular Expressions
• Regular language (continued):
– The union (or disjunction) of two regular
languages is a regular language:
 = { a, b, c} /ab|bc/ /ca|bb/
– The Kleene closure (denoted by the Kleene star: *)
of a regular language is a regular language:
 = { a, b, c} /a*/ /(ab|ca)*/
– Parentheses group a sub-language to override
operator precedence (and, we’ll see later, for
“memory”).

Finite Automata
• Finite State Automaton
a.k.a. Finite Automaton, Finite State Machine, FSA or FSM
– An abstract machine which can be used to
implement regular expressions (etc.).
– Has a finite number of states, and a finite
amount of memory (i.e., the current state).
– Can be represented by directed graphs or
transition tables

Finite-state Automata (1/23)
• Representation
– An FSA may be represented as a directed graph; each
node (or vertex) represents a state, and the edges (or
arcs) connecting the nodes represent transitions.
– Each state is labeled.
– Each transition is labeled with a symbol from the
alphabet over which the regular language represented
by the FSA is defined, or with , the empty string.
– Among the FSA’s states, there is a start state and at
least one final state (or accepting state).

q0 q1 q2 q3 q4  = { a, b, c }
a b c a
transition final state
start state
state
• Representation (continued)
– An FSA may also be
represented with a
state-transition table.
The table for the
above FSA:
Input
State a b c
0 1  
1  2 
2   3
3 4  
4   

• Given an input string, an FSA will either
accept or reject the input.
– If the FSA is in a final (or accepting) state after
all input symbols have been consumed, then the
string is accepted (or recognized).
– Otherwise (including the case in which an input
symbol cannot be consumed), the string is
rejected.

q0 q1 q2 q3 q4
 = { a, b, c }
a b c a
Input
State a b c
0 1  
1  2 
2   3
3 4  
4   
a b c a
c c b a
a b c a c
IS1:
IS2:
IS3:

• An FSA defines a regular language over an
alphabet :
–  is a regular language:
– Any symbol from is a regular language:
 = { a, b, c}
– Two concatenated regular languages is a regular
language:
 = { a, b, c}
q0
b
q0
q1
q0
b q1 q0
c q1
q1
c q2
q0
b

• regular language (continued):
– The union (or disjunction) of two regular languages is a
regular language:
 = { a, b, c}
– The Kleene closure (denoted by the Kleene star: *) of a
regular language is a regular language:
 = { a, b, c}
q0
b q1 q0
c q1
q2
c q3
q0
b
q1


q0
b q1


• Determinism
– An FSA may be either deterministic (DFSA or DFA)
or non-deterministic (NFSA or NFA).
• An FSA is deterministic if its behavior during recognition
is fully determined by the state it is in and the symbol to
be consumed.
– I.e., given an input string, only one path may be taken through the
FSA.
• Conversely, an FSA is non-deterministic if, given an input
string, more than one path may be taken through the FSA.
– One type of non-determinism is -transitions, i.e. transitions
which consume the empty string (no symbols).

• An example NFA:
q0 q1 q2 q3 q4
 = { a, b, c }
a b c a


c
State
Input
a b c 
0 1   
1  2  2
2   3,4 
3 4   
4    
– The above NFA is equivalent to the regular
expression /ab*ca?/.

• String recognition with an NFA:
– Backup (or backtracking): remember choice
points and revisit choices upon failure
– Look-ahead: choose path based on
foreknowlege about the input string and
available paths
– Parallelism: examine all choices simultaneously

• Recognition as search
– Recognition can be viewed as selection of the
correct path from all possible paths through an
NFA (this set of paths is called the state-space)
– Search strategy can affect efficiency: in what
order should the paths be searched?
• Depth-first (LIFO [last in, first out]; stack)
• Breadth-first (FIFO [first in, first out]; queue)
• Depth-first uses memory more efficiently, but may
enter into an infinite loop under some circumstances

• Conversion of NFAs to DFAs
– Every NFA can be expressed as a DFA.
/ab*ca?/
q0 q1 q2 q3 q4
 = { a, b, c }
a b c a


c
State
Input
a b c 
0 1   
1  2  2
2   3,4 
3 4   
4F    
New
State State
Input
a b c
0' 0 1  
1' 1  2 {3,4}
2' 2  2 {3,4}
3'F {3,4}F 4  
4'F 4F   
5    
q0' q1' q2' q3' q4' q5
a,b,c
a,b,c
a
c
b
a b,c
b
a
c
a
b,c
Subset
construction

• DFA minimization
– Every regular language has a unique minimum-state DFA.
– The basic idea: two states s and t are equivalent if for every
string w, the transitions T(s, w) and T(t, w) are both either
final or non-final.
– An algorithm:
• Begin by enumerating all possible pairs of both final or both non-
final states, then iteratively removing those pairs the transition pair
for which (for any symbol) are either not equal or are not on the list.
The list is complete when an iteration does not remove any pairs
from the list.
• The minimum set of states is the partition resulting from the unions
of the remaining members of the list, along with any original states
not on the list.

• The minimum-state DFA for the DFA
converted from the NFA for /ab*ca?/,
without the “failure” state (labeled “5”), and
with the states relabeled to the set Q = { q0",
q1", q2", q3" }:
q0" q1" q2" q3"
a
c
b
a

Finite Automata with Output
• Finite Automata may also have an output
alphabet and an action at every state that may
output an item from the alphabet
• Useful for lexical analyzers
– As the FSA recognizes a token, it outputs the
characters
– When the FSA reaches a final state and the token
is complete, the lexical analyzer can use
• Token value – output so far
• Token type – label of the output state

RegExps
– The extended use of regular expressions is in many modern
languages:
• Perl, php, Java, python, …
– Can use regexps to specify the rules for any set of possible
strings you want to match
• Sentences, e-mail addresses, ads, dialogs, etc
– ``Does this string match the pattern?'', or ``Is there a match
for the pattern anywhere in this string?''
– Can also define operations to do something with the
matched string, such as extract the text or substitute for it
– Regular expression patterns are compiled into a executable
code within the language

Regular Expressions
• Regexp syntax is a superset of the notation required to
express a regular language.
– Some examples and shortcuts:
1. /[abc]/ = /a|b|c/ Character class; disjunction
2. /[b-e]/ = /b|c|d|e/ Range in a character class
3. /[012015]/ = /n|r/ Octal characters; special
escapes
4. /./ = /[x00-xFF]/ Wildcard; hexadecimal
characters
5. /[^b-e]/ = /[x00-af-xFF]/ Complement of character class
6. /a*/ /[af]*/ /(abc)*/ Kleene star: zero or more
7. /a?/ = /a|/ /(ab|ca)?/ Zero or one
8. /a+/ /([a-zA-Z]1|ca)+/ Kleene plus: one or more
9. /a{8}/ /b{1,2}/ /c{3,}/ Counters: exact repeat
quantification

Regular Expressions
• Anchors
– Constrain the position(s) at which a pattern may match
– Think of them as “extra” alphabet symbols, though they
actually consume  (the zero-length string):
– /â/ Pattern must match at beginning of string
– /a$/ Pattern must match at end of string
– /bword23b/ “Word” boundary: /[a-zA-Z0-9_][â-zA-
Z0-9_]/
or /[â-zA-Z0-9_][a-zA-Z0-9_]/
– /B23B/ “Word” non-boundary

Regular Expressions
• Escapes
– A backslash “” placed before a character is said to “escape”
(or “quote”) the character. There are six classes of escapes:
1. Numeric character representation: the octal or hexadecimal
position in a character set: “012” = “xA”
2. Meta-characters: The characters which are syntactically meaningful
to regular expressions, and therefore must be escaped in order to
represent themselves in the alphabet of the regular expression: “[]
(){}|^$.?+*” (note the inclusion of the backslash).
3. “Special” escapes (from the “C” language):
newline: “n” = “xA” carriage return: “
r” = “xD”
tab: “t” = “x9” formfeed: “
f” = “xC”

Regular Expressions
• Escapes (continued)
– Classes of escapes (continued):
4. Aliases: shortcuts for commonly used character classes. (Note that the
capitalized version of these aliases refer to the complement of the alias’s
character class):
– whitespace: “s” = “[ trnfv]”
– digit: “d” = “[0-9]”
– word: “w” = “[a-zA-Z0-9_]”
– non-whitespace: “S” = “[^ trnf]”
– non-digit: “D” = “[^0-9]”
– non-word: “W” = “[^a-zA-Z0-9_]”
5. Memory/registers/back-references: “1”, “2”, etc.
6. Self-escapes: any character other than those which have special
meaning can be escaped, but the escaping has no effect: the character
still represents the regular language of the character itself.

Regular Expressions
• Memory/Registers/Back-references
– Many regular expression languages include a
memory/register/back-reference feature, in which sub-
matches may be referred to later in the regular expression,
and/or when performing replacement, in the replacement
string:
• Perl: /(w+)s+1b/ matches a repeated word
• Python: re.sub(”(thes+)the(s+|b)”,”1”,string)
removes the second of a pair of ‘the’s
– Note: finite automata cannot be used to implement the
memory feature.

Regular Expression Examples
Character classes and Kleene symbols
[A-Z] = one capital letter
[0-9] = one numerical digit
[st@!9] = s, t, @, ! or 9
[A-Z] matches G or W or E
does not match GW or FA or h or fun
[A-Z]+ = one or more consecutive capital letters
matches GW or FA or CRASH
[A-Z]? = zero or one capital letter
[A-Z]* = zero, one or more consecutive capital letters
matches on eat or EAT or I
so, [A-Z]ate
matches Gate, Late, Pate, Fate, but not GATE or gate
and [A-Z]+ate
matches: Gate, GRate, HEate, but not Grate or grate or
STATE
and [A-Z]*ate
matches: Gate, GRate, and ate, but not STATE, grate or
Plate

Regular Expression Examples (cont’d)
[A-Za-z] = any single letter
so [A-Za-z]+
matches on any word composed of only letters,
but will not match on “words”: bi-weekly , yes@SU or IBM325
they will match on bi, weekly, yes, SU and IBM
a shortcut for [A-Za-z] is w, which in Perl also includes _
so (w)+ will match on Information, ZANY, rattskellar and jeuvbaew
s will match whitespace
so (w)+(s)(w+) will match real estate or Gen Xers

Some longer examples:
([A-Z][a-z]+)s([a-z0-9]+)
matches: Intel c09yt745 but not IBM series5000
[A-Z]w+sw+sw+[!]
matches: The dog died!
It also matches that portion of “ he said, “ The dog died! “
[A-Z]w+sw+sw+[!]$
matches: The dog died!
But does not match “he said, “ The dog died! “ because the $
indicates end of Line, and there is a quotation mark before the end of
the line
(w+ats?s)+
parentheses define a pattern as a unit, so the above expression will
match:
Fat cats eat Bats that Splat

Conclusion
• Both regular expressions and finite-state automata
represent regular languages.
• The basic regular expression operations are: concatenation,
union/disjunction, and Kleene closure.
• The regular expression language is a powerful pattern-
matching tool.
• Any regular expression can be automatically compiled into
an NFA, to a DFA, and to a unique minimum-state DFA.
• An FSA can use any set of symbols for its alphabet,
including letters and words.

formal language and automata theory unit 2

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