Finite Automata and Regular Languages

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F I N I T E & R E G U L A R A U T O M A T A L A N G U A G E S @tomstuart / London JS / 2014-01-27

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deterministic ﬁnite automata nondeterministic ﬁnite automata regular languages regular expressions

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DETERMINISTIC FINITE AUTOMATA

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a little imaginary machine with several possible states

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1 2 3

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always has a current state

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1 2 3 1

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changes state when it reads input

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, 1 2 3 1 a b b a a b

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1 2 3 a b b , a a b a b bb aa 1

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2 3 a b b , a a b a b bb aa 2 1

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1 2 3 a b b , a a b a b bb aa 2

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1 3 a b b , a a b a b bb aa 3 2

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1 2 3 a b b , a a b a b bb aa 3

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1 2 3 a b b , a a b a b bb aa 3

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1 3 a b b , a a b a b bb aa 3 2

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some states are "accept states"

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, 1 2 3 1 a b b a a b

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1 2 3 a b b , a a b a b bb aa 1

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2 3 a b b , a a b a b bb aa 2 1

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1 2 3 a b b , a a b a b bb aa 2

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1 3 a b b , a a b a b bb aa 3 2

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1 2 3 a b b , a a b a b bb aa 3

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1 2 3 a b b , a a b a b bb aa 3

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1 3 a b b , a a b a b bb aa 3 2

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"accepts" or "rejects" a string

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1 3 a b b , a a b a b bb aa 3 2 ✔

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, 1 2 3 1 a b b a a b

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a a 1 2 3 a b b , a a b b b bb 1

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a a 1 2 3 1 a b , a a b b b bb b

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a a 1 2 3 1 a b , a a b b b bb b

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a a 1 2 3 1 a b , a a b b b bb b

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a a 1 2 3 a b b , a a b b b bb 2

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a a 1 2 3 a b b , a a b b b bb 2

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a a 1 2 3 a b b , a a b b b bb 2

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a a 1 2 3 a b b , a a b b b bb ✘ 2

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bbbbaa baabba ✔ ✘ ab baba aaaab a baaaa b bbbbbb abbbb bbbbab bba ✔ ✔ ✔ ✔ ✔ ✘ ✘ ✘ ✘ ✘

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1 2 3 a b b , a a b 1

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1 2 3 a b b , a a b 2

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1 2 3 a b b , a a b 3

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1 2 3 a b b , a a b accepts strings containing "ab"

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NONDETERMINISTIC FINITE AUTOMATA

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accept strings whose third character is "b"

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1 3 4 5 2 a,b a,b a b a,b a,b

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accept strings whose third-from-last character is "b"

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1 3 4 2 b a,b a,b a,b

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accept strings whose length is a multiple of two or three

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1 2 a a

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1 2 a a 3 a

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1 2 a a 4 3 a a a

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4 5 6 a a a 1 2 3 a a

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nondeterministic ﬁnite automata can choose between different rules and change state without reading input

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but there's an algorithm to convert any nondeterministic ﬁnite automaton into a deterministic one

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1 3 4 2 b a,b a,b a,b

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1 1,2,4 1,3,4 1,3 b a a 1,2 1,4 1,2,3,4 1,2,3 b a b a b b a b a b a b a

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REGULAR LANGUAGES

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1 2 3 a b b , a a b

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bbbbaa baabba ✔ ✘ ab baba aaaab a baaaa b bbbbbb abbbb bbbbab bba ✔ ✔ ✔ ✔ ✔ ✘ ✘ ✘ ✘ ✘

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baabba ab baba aaaab abbbb bbbbab this is part of the language that the machine recognises

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some languages are ﬁnite, others are inﬁnite

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these machines can only recognise regular languages

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some regular languages: • the strings "one", "two" and "three" • all strings that contain "ab" • all strings whose third character is "b" • all strings whose third-from-last character is "b" • all strings that start and end with "a"

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some non-regular languages: • all strings that are palindromes • all strings that contain equal numbers of "a" and "b" characters • all strings of brackets that are balanced • all strings that are valid JSON • all strings that contain equal numbers of "a", "b" and "c" characters • all strings whose length is prime • all strings that contain a halting JavaScript program

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( () ✔ ✘ (()) (()(())) ()()()()() (( (() ) (((())))) ((((())))) (()((()))) (()))() ✔ ✔ ✔ ✔ ✔ ✘ ✘ ✘ ✘ ✘

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0 2 3 1 ( ) ( ) ( )

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REGULAR EXPRESSIONS

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/dog/ /cat/ /dogs?/ /dog|cat/ /do*g/

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/a/

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/ab/

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a b followed by

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a b

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a b

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a b b a

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a b b a

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a b b a

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a b b a

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a b b a

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a b b a ✔

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followed by

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No content

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No content

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/abc/

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c followed by a b

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c a b

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a b c

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/a|b/

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a b or

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a b

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a b a

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a b a

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a b a

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a b a

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a b a ✔

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a b

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a b b

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a b b

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a b b

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a b b

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a b b ✔

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No content

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/a*/

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a zero or more times

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zero or more times

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No content

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/(a(b|c))*/

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/(a(b|c))*/

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b c or

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b c

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/(a(b|c))*/

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a followed by c b

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a c b

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/(a(b|c))*/

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zero or more times a c b

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a c b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b

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b a c b a a ac b ✔

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/a+/ = /aa*/ /a?/ = /a|/

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every regular expression can be turned into an nondeterministic ﬁnite automaton

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every nondeterministic ﬁnite automaton can be turned into a regular expression

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regular expressions are a compact notation for nondeterministic ﬁnite automata