Finite-State Automata: Recap
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1 Finite-Stte Automt: Recp Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute of Science, Bnglore. 09 August 2016
2 Outline 1 Introduction 2 Forml Definitions nd Nottion 3 Closure under oolen ops 4 Induction 5 NFA s
3 Exmple DFA 1 DFA for Odd numer of s How DFA works.
4 Exmple DFA 1 DFA for Odd numer of s e o How DFA works. Ech stte represents property of the input string red so fr: Stte e: Numer of s seen is even. Stte o: Numer of s seen is odd.
5 Exmple DFA 2 Accept strings over {0, 1} which hve even prity in ech length 4 lock. Accept Reject DFA for Even prity checker 0 1, e 0 2, e 0 3, e 0, e , o 0, 1 1, o 0 2, o 0 3, o 1
6 Exmple DFA 3 DFA for Strings over {, } tht contin the sustring
7 Exmple DFA 3 DFA for Strings over {, } tht contin the sustring, ɛ Ech stte represents property of the input string red so fr: Stte ɛ: Not seen nd no suffix in or. Stte : Not seen nd hs suffix. Stte : Not seen nd hs suffix. Stte : Seen.
8 Definitions nd nottion An lphet is finite set of symols or letters. Eg. A = {,, c} or Σ = {0, 1}. A string or word over n lphet A is finite sequence of letters from A. Eg. is string over {,, c}. Empty string denoted y ɛ. Set of ll strings over A denoted y A. Wht is the size or crdinlity of A?
9 Definitions nd nottion An lphet is finite set of symols or letters. Eg. A = {,, c} or Σ = {0, 1}. A string or word over n lphet A is finite sequence of letters from A. Eg. is string over {,, c}. Empty string denoted y ɛ. Set of ll strings over A denoted y A. Wht is the size or crdinlity of A? Infinite ut Countle: Cn enumerte in lexicogrphic order:. ɛ,,, c,,,...
10 Definitions nd nottion An lphet is finite set of symols or letters. Eg. A = {,, c} or Σ = {0, 1}. A string or word over n lphet A is finite sequence of letters from A. Eg. is string over {,, c}. Empty string denoted y ɛ. Set of ll strings over A denoted y A. Wht is the size or crdinlity of A? Infinite ut Countle: Cn enumerte in lexicogrphic order: ɛ,,, c,,,.... Opertion of conctention on words: String u followed y string v: written u v or simply uv. Eg. =.
11 Definitions nd nottion: Lnguges A lnguge over n lphet A is set of strings over A. Eg. for A = {,, c}: L = {c, }. L 1 = {ɛ,,,,,,,,...}. L 2 = {}. L 3 = {ɛ}. How mny lnguges re there over given lphet A?
12 Definitions nd nottion: Lnguges A lnguge over n lphet A is set of strings over A. Eg. for A = {,, c}: L = {c, }. L 1 = {ɛ,,,,,,,,...}. L 2 = {}. L 3 = {ɛ}. How mny lnguges re there over given lphet A? Uncountly infinite Use digonliztion rgument: ɛ L L L L L L L L
13 Definitions nd nottion: Lnguges Conctention of lnguges: L 1 L 2 = {u v u L 1, v L 2 }. Eg. {c, } {ɛ,, } = {c,, c,, c, }.
14 Definitions nd nottion: DFA A Deterministic Finite-Stte Automton A over n lphet A is structure of the form (Q, s, δ, F ) where Q is finite set of sttes s Q is the strt stte δ : Q A Q is the trnsition function. F Q is the set of finl sttes.
15 Definitions nd nottion: DFA A Deterministic Finite-Stte Automton A over n lphet A is structure of the form (Q, s, δ, F ) where Q is finite set of sttes s Q is the strt stte δ : Q A Q is the trnsition function. F Q is the set of finl sttes. Exmple of Odd s DFA: Here: Q = {e, o}, s = e, F = {o}, nd δ is given y: δ(e, ) = o, δ(e, ) = e, δ(o, ) = e, δ(o, ) = o. e o
16 Definitions nd nottion: Lnguge ccepted y DFA δ tells us how the DFA A ehves on given word u. Define δ : Q A Q s δ(q, ɛ) = q δ(q, w ) = δ( δ(q, w), ). Lnguge ccepted y A, denoted L(A), is defined s: L(A) = {w A δ(s, w) F }. Eg. For A = DFA for Odd s, L(A) = {,,,,,,,...}.
17 Regulr Lnguges A lnguge L A is clled regulr if there is DFA A over A such tht L(A) = L. Exmples of regulr lnguges: Odd s, strings tht don t end inside C-style comment, {}, ny finite lnguge. All lnguges over A Regulr Are there non-regulr lnguges?
18 Regulr Lnguges A lnguge L A is clled regulr if there is DFA A over A such tht L(A) = L. Exmples of regulr lnguges: Odd s, strings tht don t end inside C-style comment, {}, ny finite lnguge. All lnguges over A Regulr Are there non-regulr lnguges? Yes, uncountly mny, since Reg is only countle while clss of ll lnguges is uncountle.
19 Closure properties Clss of Regulr lnguges is closed under Complement, intersection, nd union. Conctention, Kleene itertion. Non-deterministic Finite-stte Automt (NFA) = DFA. All strings over A All lnguges over A Regulr L M
20 Closure under complementtion Ide: Flip finl sttes. Forml construction: Let A = (Q, s, δ, F ) e DFA over lphet A. Define B = (Q, s, δ, Q F ). Clim: L(B) = A L(A). Proof of clim L(B) A L(A). w L(B) = δ(s, w) (Q F ). = δ(s, w) F = w L(A) = w A L(A). L(B) A L(A).
21 Closure under intersection Product construction. Given DFA s A = (Q, s, δ, F ), B = (Q, s, δ, F ), define product C of A nd B: C = (Q Q, (s, s ), δ, F F ), where δ ((p, p ), ) = (δ(p, ), δ (p, )). Product construction exmple e o e, o, e, o, A B A B
22 Correctness of product construction Clim: L(C) = L(A) L(B). Proof of clim L(C) = L(A) L(B). L(C) L(A) L(B). w L(C) = δ ((s, s ), w) F F. = ( δ(s, w), δ (s, w)) F F (y suclim) = δ(s, w) F nd δ (s, w) F = w L(A) nd w L(B) = w L(A) L(B). L(C) L(A) L(B). Suclim: δ ((s, s ), w) = ( δ(s, w), δ (s, w)).
23 Closure under union Follows from closure under complement nd intersection since L 1 L 2 = L 1 L 2.
24 Closure under union Follows from closure under complement nd intersection since L 1 L 2 = L 1 L 2. Cn lso do directly y product construction: Given DFA s A = (Q, s, δ, F ), B = (Q, s, δ, F ), define C: C = (Q Q, (s, s ), δ, (F Q ) (Q F )), where δ ((p, p ), ) = (δ(p, ), δ(p, )). Union construction e o e, o, e, o, A B A B
25 Principle of Mthemticl Induction N = {0, 1, 2...} P(n): A sttement P out nturl numer n. Exmple: P(n) = n is even. P 1 (n) = Sum of the numers 1... n equls n(n + 1)/2. P 2 (n) = For ll w A, if length of w is n then δ ((s, s ), w) = ( δ(s, w), δ (s, w)). Principle of Induction If sttement P out nturl numers is true for 0 (i.e P(0) is true), nd, is true for n + 1 whenever it is true for n (i.e. P(n) = P(n + 1)) then P is true of ll nturl numers (i.e. For ll n, P(n) is true).
26 Proof of suclim Exercise: Prove the Suclim: using induction. δ ((s, s ), w) = ( δ(s, w), δ (s, w)).
27 Nondeterministic Finite-stte Automt (NFA) Allows multiple strt sttes. Allows more thn one trnsition from stte on given letter. Non-deterministic trnsitions q p r A word is ccepted if there is some pth on it from strt to finl stte.
28 Exmple NFA s NFA for contins s suword,, ɛ
29 NFA definition Mthemticl representtion of NFA A = (Q, S,, F ), where S Q, nd : Q A 2 Q. Define reltion p w q which sys there is pth from stte p to stte q lelled w. p ɛ p p u q iff there exists r Q such tht p u r nd q (r, ). Define L(A) = {w A s S, f F : s w f }. NFA DFA: Suset construction Exmple: determinize NFA for contins. Forml construction Correctness
30 Closure under conctention nd Kleene itertion Conctention of lnguges: L M = {u v u L, v M}. Kleene itertion of lnguge: where L = {ɛ} L L 2 L 3, L n = L L L (n times). = {w 1 w n ech w i L}.
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