Deterministic Finite-State Automata
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1 Deterministic Finite-Stte Automt Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute of Science, Bnglore. 12 August 2013
2 Outline 1 Introduction 2
3 Exmple DFA 1 DFA for Odd number of s b b How DFA works.
4 Exmple DFA 1 DFA for Odd number of s b b e o How DFA works. Ech stte represents property of the input string red so fr: Stte e: Number of s seen is even. Stte o: Number of s seen is odd.
5 Exmple DFA 2 DFA for Contins the substring bb b, b b b Ech stte represents property of the input string red so fr:
6 Exmple DFA 2 DFA for Contins the substring bb b b b, b ɛ b bb Ech stte represents property of the input string red so fr: Stte ɛ: Not seen bb nd no suffix in or b. Stte : Not seen bb nd hs suffix. Stte b: Not seen bb nd hs suffix b. Stte bb: Seen bb.
7 Exmple DFA 3 Accept strings over {0, 1} which stisfy even prity in length 4 blocks. 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
8 Exmple DFA 4 Accept strings over {, b, /, } which don t end inside C-style comment. Scn from left to right till first /* is encountered; from there to next */ is first comment; nd so on. Accept b/ /bb nd b/ / /bb /. Reject b/ nd b/ / /bb/.
9 Exmple DFA 4 Accept strings over {, b, /, } which don t end inside C-style comment. Scn from left to right till first /* is encountered; from there to next */ is first comment; nd so on. Accept b/ /bb nd b/ / /bb /. Reject b/ nd b/ / /bb/. DFA for C-comment trcker /, /, / / out pbc in pec
10 Definitions nd nottion An lphbet is finite set of set of symbols or letters. Eg. A = {, b, c}, Σ = {0, 1}. A string or word over n lphbet A is finite sequence of letters from A. Eg. b is string over {, b, c}. Empty string denoted by ɛ. Set of ll strings over A denoted by A. Wht is the size or crdinlity of A?
11 Definitions nd nottion An lphbet is finite set of set of symbols or letters. Eg. A = {, b, c}, Σ = {0, 1}. A string or word over n lphbet A is finite sequence of letters from A. Eg. b is string over {, b, c}. Empty string denoted by ɛ. Set of ll strings over A denoted by A. Wht is the size or crdinlity of A? Infinite but Countble: Cn enumerte in lexicogrphic order:. ɛ,, b, c,, b,...
12 Definitions nd nottion An lphbet is finite set of set of symbols or letters. Eg. A = {, b, c}, Σ = {0, 1}. A string or word over n lphbet A is finite sequence of letters from A. Eg. b is string over {, b, c}. Empty string denoted by ɛ. Set of ll strings over A denoted by A. Wht is the size or crdinlity of A? Infinite but Countble: Cn enumerte in lexicogrphic order: ɛ,, b, c,, b,.... Opertion of conctention on words: String u followed by string v: written u v or simply uv. Eg. bb = bb.
13 Definitions nd nottion: Lnguges A lnguge over n lphbet A is set of strings over A. Eg. for A = {, b, c}: L = {bc, b}. L 1 = {ɛ, b,, bb, b, b, b, bbb,...}. L 2 = {}. L 3 = {ɛ}. How mny lnguges re there over given lphbet A?
14 Definitions nd nottion: Lnguges A lnguge over n lphbet A is set of strings over A. Eg. for A = {, b, c}: L = {bc, b}. L 1 = {ɛ, b,, bb, b, b, b, bbb,...}. L 2 = {}. L 3 = {ɛ}. How mny lnguges re there over given lphbet A? Uncountbly infinite Use digonliztion rgument: ɛ b b b bb b b bb bbb L L L L L L L L
15 Definitions nd nottion: Lnguges Conctention of lnguges: L 1 L 2 = {u v u L 1, v L 2 }. Eg. {bc, b} {ɛ,, bb} = {bc, b, bc, b, bcbb, bbb}.
16 Definitions nd nottion: DFA A Deterministic Finite-Stte Automton A over n lphbet 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.
17 Definitions nd nottion: DFA A Deterministic Finite-Stte Automton A over n lphbet 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 by: δ(e, ) = o, δ(e, b) = e, δ(o, ) = e, δ(o, b) = o. b e b o
18 Definitions nd nottion: Lnguge ccepted by DFA δ tells us how the DFA A behves on given word u. Define δ : Q A Q s δ(q, ɛ) = q δ(q, w ) = δ( δ(q, w), ). Lnguge ccepted by A, denoted L(A), is defined s: L(A) = {w A δ(s, w) F }. Eg. For A = DFA for Odd s, L(A) = {, b, b,, bb, bb, bb,...}.
19 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?
20 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, uncountbly mny, since Reg is only countble while clss of ll lnguges is uncountble.
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