CMSC 330: Organization of Programming Languages. DFAs, and NFAs, and Regexps (Oh my!)
|
|
- Horace Bond
- 5 years ago
- Views:
Transcription
1 CMSC 330: Orgniztion of Progrmming Lnguges DFAs, nd NFAs, nd Regexps (Oh my!) CMSC330 Spring 2018
2 Types of Finite Automt Deterministic Finite Automt (DFA) Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt (NFA) My hve mny sequences of steps for ech string Accepts if ny pth ends in finl stte t end of string More compct thn DFA Ø But more expensive to test whether string mtches CMSC 330 Spring
3 Quiz 1: Which DFA mtches this regexp? b(b +b?) A. B. C. 0 b 1 b 2 b 3 b 4 b 0 1 b b b 1 02,b D. None of the bove CMSC 330 Spring
4 Quiz 1: Which DFA mtches this regexp? b(b +b?) A. B. C. 0 b 1 b 2 b 3 b 4 b 0 1 b b b 1 02,b D. None of the bove CMSC 330 Spring
5 Compring DFAs nd NFAs NFAs cn hve more thn one trnsition leving stte on the sme symbol DFAs llow only one trnsition per symbol I.e., trnsition function must be vlid function DFA is specil cse of NFA CMSC 330 Spring
6 Compring DFAs nd NFAs (cont.) NFAs my hve trnsitions with empty string lbel My move to new stte without consuming chrcter ε e-trnsition DFA trnsition must be lbeled with symbol DFA is specil cse of NFA CMSC 330 Spring
7 DFA for ( b)*bb CMSC 330 Spring
8 NFA for ( b)*bb b Hs pths to either S0 or S1 Neither is finl, so rejected bbbb Hs pths to different sttes One pth leds to S3, so ccepts string CMSC 330 Spring
9 NFA for (b b)* b Hs pths to sttes S0, S1 bb Hs pths to S0, S1 Need to use ε-trnsition CMSC 330 Spring
10 Compring NFA nd DFA for (b b)* DFA NFA CMSC 330 Spring
11 NFA Acceptnce Algorithm Sketch When NFA processes string s NFA must keep trck of severl current sttes Ø Due to multiple trnsitions with sme lbel Ø ε-trnsitions If ny current stte is finl when done then ccept s Exmple After processing Ø NFA my be in sttes S1 S2 S3 S1 S2 ε S3 CMSC 330 Spring
12 Forml Definition A deterministic finite utomton (DFA) is 5-tuple (Σ, Q, q 0, F, δ) where Σ is n lphbet Q is nonempty set of sttes q 0 Î Q is the strt stte F Q is the set of finl sttes δ : Q x Σ Q specifies the DFA's trnsitions Ø Wht's this definition sying tht δ is? A DFA ccepts s if it stops t finl stte on s CMSC 330 Spring
13 Forml Definition: Exmple Σ = {0, 1} Q = {S0, S1} q 0 = S0 F = {S1} input stte symbol δ 0 1 S0 S0 S1 S1 S0 S1 or s { (S0,0,S0),(S0,1,S1),(S1,0,S0),(S1,1,S1) } CMSC 330 Spring
14 Nondeterministic Finite Automt (NFA) An NFA is 5-tuple (Σ, Q, q 0, F, δ) where Σ, Q, q0, F s with DFAs δ Q x (Σ È {ε}) x Q specifies the NFA's trnsitions S1 S2 ε Exmple S3 Σ = {} Q = {S1, S2, S3} q 0 = S1 F = {S3} δ = { (S1,,S1), (S1,,S2), (S2,ε,S3) } An NFA ccepts s if there is t lest one pth vi s from the NFA s strt stte to finl stte CMSC 330 Spring
15 Relting REs to DFAs nd NFAs Regulr expressions, NFAs, nd DFAs ccept the sme lnguges! DFA cn reduce NFA cn trnsform cn reduce RE CMSC 330 Spring
16 Reducing Regulr Expressions to NFAs Gol: Given regulr expression A, construct NFA: <A> = (Σ, Q, q 0, F, δ) Remember regulr expressions re defined recursively from primitive RE lnguges Invrint: F = 1 in our NFAs Ø Recll F = set of finl sttes Will define <A> for bse cses: σ, ε, Where σ is symbol in Σ And for inductive cses: AB, A B, A* CMSC 330 Spring
17 Reducing Regulr Expressions to NFAs Bse cse: σ σ <σ> = ({σ}, {S0, S1}, S0, {S1}, {(S0, σ, S1)} ) CMSC 330 Spring
18 Reduction Bse cse: ε <ε> = (, {S0}, S0, {S0}, ) Bse cse: < > = (, {S0, S1}, S0, {S1}, ) CMSC 330 Spring
19 Reduction: Conctention Induction: AB <A> <B> <A> = (Σ A, Q A, q A, {f A }, δ A ) <B> = (Σ B, Q B, q B, {f B }, δ B ) CMSC 330 Spring
20 Reduction: Conctention Induction: AB <A> <B> <A> = (Σ A, Q A, q A, {f A }, δ A ) <B> = (Σ B, Q B, q B, {f B }, δ B ) <AB> = (Σ A È Σ B, Q A È Q B, q A, {f B }, δ A È δ B È {(f A,ε,q B )} ) CMSC 330 Spring
21 Reduction: Union Induction: A B <A> = (Σ A, Q A, q A, {f A }, δ A ) <B> = (Σ B, Q B, q B, {f B }, δ B ) CMSC 330 Spring
22 Reduction: Union Induction: A B <A> = (Σ A, Q A, q A, {f A }, δ A ) <B> = (Σ B, Q B, q B, {f B }, δ B ) <A B> = (Σ A È Σ B, Q A È Q B È {S0,S1}, S0, {S1}, δ A È δ B È {(S0,ε,q A ), (S0,ε,q B ), (f A,ε,S1), (f B,ε,S1)}) CMSC 330 Spring
23 Reduction: Closure Induction: A* <A> = (Σ A, Q A, q A, {f A }, δ A ) CMSC 330 Spring
24 Reduction: Closure Induction: A* <A> = (Σ A, Q A, q A, {f A }, δ A ) <A*> = (Σ A, Q A È {S0,S1}, S0, {S1}, δ A È {(f A,ε,S1), (S0,ε,q A ), (S0,ε,S1), (S1,ε,S0)}) CMSC 330 Spring
25 Quiz 2: Which NFA mtches *? A. B. C. D. CMSC 330 Spring
26 Quiz 2: Which NFA mtches *? A. B. C. D. CMSC 330 Spring
27 Quiz 3: Which NFA mtches b*? A. B. C. D. CMSC 330 Spring
28 Quiz 3: Which NFA mtches b*? A. B. C. D. CMSC 330 Spring
29 Reduction Complexity Given regulr expression A of size n... Size = # of symbols + # of opertions How mny sttes does <A> hve? Two dded for ech, two dded for ech * O(n) Tht s pretty good! CMSC 330 Spring
30 Reducing NFA to DFA DFA cn reduce NFA cn reduce RE CMSC 330 Spring
31 Reducing NFA to DFA NFA my be reduced to DFA By explicitly trcking the set of NFA sttes Intuition Build DFA where Ø Ech DFA stte represents set of NFA current sttes Exmple S1 S2 ε S3 S1 S1, S2, S3 NFA DFA CMSC 330 Spring
32 Algorithm for Reducing NFA to DFA Reduction pplied using the subset lgorithm DFA stte is subset of set of ll NFA sttes Algorithm Input Ø NFA (Σ, Q, q 0, F n, δ) Output Ø DFA (Σ, R, r 0, F d, d) Using two subroutines Ø ε-closure(d, p) (nd ε-closure(d, S)) Ø move(d, p, ) (nd move(d, S, )) CMSC 330 Spring
33 ε-trnsitions nd ε-closure ε We sy p q If it is possible to go from stte p to stte q by tking only e-trnsitions in δ If $ p, p 1, p 2, p n, q Î Q such tht Ø {p,ε,p 1 } Î δ, {p 1,ε,p 2 } Î δ,, {p n,ε,q} Î δ ε-closure(δ, p) Set of sttes rechble from p using ε-trnsitions lone Ø Set of sttes q such tht p ε q ccording to δ Ø ε-closure(δ, p) = {q p ε q in δ } ε Ø ε-closure(δ, Q) = { q p Î Q, p q in δ } Notes Ø ε-closure(δ, p) lwys includes p Ø We write ε-closure(p) or ε-closure(q) when δ is cler from context CMSC 330 Spring
34 ε-closure: Exmple 1 Following NFA contins ε S1 S2 ε S2 S3 ε S1 S3 ε ε Ø Since S1 S2 nd S2 S3 ε ε S1 S2 S3 ε-closures ε-closure(s1) = ε-closure(s2) = ε-closure(s3) = { S3 } ε-closure( { S1, S2 } ) = { S1, S2, S3 } { S2, S3 } { S1, S2, S3 } È { S2, S3 } CMSC 330 Spring
35 ε-closure: Exmple 2 Following NFA contins ε S1 S3 ε S3 S2 ε S1 S2 ε ε Ø Since S1 S3 nd S3 S2 b S1 S2 S3 ε ε ε-closures ε-closure(s1) = ε-closure(s2) = ε-closure(s3) = { S1, S2, S3 } { S2 } { S2, S3 } ε-closure( { S2,S3 } ) = { S2 } È { S2, S3 } CMSC 330 Spring
36 ε-closure Algorithm: Approch Input: NFA (Σ, Q, q 0, F n, δ), Stte Set R Output: Stte Set R Algorithm Let R = R Repet Let R = R Let R = R È {q p Î R, (p, e, q) Î d} Until R = R // strt sttes // continue from previous // new ε-rechble sttes // stop when no new sttes This lgorithm computes fixed point see note linked from project description CMSC 330 Spring
37 ε-closure Algorithm Exmple Clculte ε-closure(d,{s1}) R R ε ε S1 S2 S3 {S1} {S1} {S1} {S1, S2} {S1, S2} {S1, S2, S3} Let R = R Repet Let R= R Let R = R È {q p Î R, (p, e, q) Î d} Until R = R {S1, S2, S3} {S1, S2, S3} CMSC 330 Spring
38 Clculting move(p,) move(δ,p,) Set of sttes rechble from p using exctly one trnsition on Ø Set of sttes q such tht {p,, q} Î δ Ø move(δ,p,) = { q {p,, q} Î δ } Ø move(δ,q,) = { q p Î Q, {p,, q} Î δ } i.e., cn lift move() to strt from set of sttes Q Notes: Ø move(δ,p,) is Ø if no trnsition (p,,q) Î δ, for ny q Ø We write move(p,) or move(r,) when δ cler from context CMSC 330 Spring
39 move(,p) : Exmple 1 Following NFA Σ = {, b } Move move(s1, ) = move(s1, b) = move(s2, ) = move(s2, b) = move(s3, ) = move(s3, b) = { S2, S3 } Ø Ø { S3 } Ø Ø b S1 S2 S3 move({s1,s2},b) = { S3 } CMSC 330 Spring
40 move(,p) : Exmple 2 Following NFA Σ = {, b } Move move(s1, ) = move(s1, b) = move(s2, ) = move(s2, b) = move(s3, ) = move(s3, b) = { S2 } { S3 } { S3 } Ø Ø Ø S1 S2 S3 b move({s1,s2},) = {S2,S3} ε CMSC 330 Spring
41 NFA DFA Reduction Algorithm ( subset ) Input NFA (Σ, Q, q 0, F n, δ), Output DFA (Σ, R, r 0, F d, d ) Algorithm Let r 0 = e-closure(δ,q 0 ), dd it to R While $ n unmrked stte r Î R Mrk r For ech Î S Let E = move(δ,r,) Let e = e-closure(δ,e) If e Ï R Let R = R È {e} Let d = d È {r,, e} Let F d = {r $ s Î r with s Î F n } // DFA strt stte // process DFA stte r // ech stte visited once // for ech letter // sttes reched vi // sttes reched vi e // if stte e is new // dd e to R (unmrked) // dd trnsition r e // finl if include stte in F n CMSC 330 Spring
42 NFA DFA Exmple 1 Strt = e-closure(δ,s1) = { {S1,S3} } R = { {S1,S3} } r Î R = {S1,S3} move(δ,{s1,s3},) = {S2} Ø e = e-closure(δ,{s2}) = {S2} Ø R = R È {{S2}} = { {S1,S3}, {S2} } Ø d = d È {{S1,S3},, {S2}} move(δ,{s1,s3},b) = Ø b S1 S2 S3 {1,3} NFA DFA {2} ε CMSC 330 Spring
43 NFA DFA Exmple 1 (cont.) R = { {S1,S3}, {S2} } r Î R = {S2} move(δ,{s2},) = Ø move(δ,{s2},b) = {S3} Ø e = e-closure(δ,{s3}) = {S3} Ø R = R È {{S3}} = { {S1,S3}, {S2}, {S3} } Ø d = d È {{S2}, b, {S3}} b S1 S2 S3 {1,3} NFA DFA {2} ε b {3} CMSC 330 Spring
44 NFA DFA Exmple 1 (cont.) R = { {S1,S3}, {S2}, {S3} } r Î R = {S3} Move({S3},) = Ø Move({S3},b) = Ø Mrk {S3}, exit loop F d = {{S1,S3}, {S3}} Ø Since S3 Î F n Done! b S1 S2 S3 {1,3} NFA DFA {2} ε b {3} CMSC 330 Spring
45 Quiz 4: Which DFA is equiv to this NFA? NFA: b S0 S1 S2 ε A. S0 S1 b S1, S2 b B. C. S0 S1 S2, S0 S0 S1 b S2, S0 D. None of the bove CMSC 330 Spring b
46 Quiz 4: Which DFA is equiv to this NFA? NFA: b S0 S1 S2 ε A. S0 S1 b S1, S2 b B. C. S0 S1 S2, S0 S0 S1 b S2, S0 D. None of the bove CMSC 330 Spring b
47 Actul Answer b S0 S1 S2 NFA: ε S0 S1 b S2, S0 b S1, S0 CMSC 330 Spring
48 Subset Algorithm s Fixed Point Input: NFA (Σ, Q, q 0, F, δ) Output: DFA M Algorithm Let q 0 = ε-closure(δ, q 0 ) Let F = {q 0 } if q 0 F, or otherwise Let M = (Σ, {q 0 }, q 0, F, ) DFA // strting pproximtion of Repet Let M = M // current DFA pprox For ech q Î sttes(m), Î Σ // for ech DFA stte q nd letter Let s = ε-closure(δ, move(δ, q, )) // new subset from q Let F = {s} if s F, or otherwise, // subset contins finl? M = M È (, {s},, F, {(q,, s)}) // updte DFA Until M = M // reched fixed point CMSC 330 Spring
49 Redux: DFA to NFA Exmple 1 q ' ε-closure(δ,s1) = {S1,S3} F = {{S1,S3}} since {S1,S3} {S3} NFA b S1 S2 S3 {1,3} M = { Σ, {{S1,S3}}, {S1,S3}, {{S1,S3}}, } Q q 0 ' F δ' DFA ε CMSC 330 Spring
50 Redux: DFA to NFA Exmple 1 (cont) M = { Σ, {{S1,S3}}, {S1,S3}, {{S1,S3}}, } q = {S1, S3} = s = {S2} Ø since move(δ,{s1, S3},) = {S2} Ø nd e-closure(δ,{s2}) = {S2} F = Ø Ø Since {S2} {S3} = where s = {S2} nd F = {S3} b S1 S2 S3 {1,3} CMSC 330 Spring NFA DFA M = M È ( { S2}},, { S1,S3},,{S2 } { Σ, {{S1,S3},{S2}}, {S1,S3}, {{S1,S3}}, {({S1,S3},,{S2})} } {2} Q q 0 ' F δ' ε
51 Redux: DFA to NFA Exmple 1 (cont) M = { Σ, {{S1,S3},{S2}}, {S1,S3}, {{S1,S3}}, {({S1,S3},,{S2})} } q = {S2} = b s = {S3} Ø since move(δ,{s2},b) = {S3} Ø nd e-closure(δ,{s3}) = {S3} F = {{S3}} Ø Ø Since {S3} {S3} = {S3} where s = {S3} nd F = {S3} {1,3} M = M È { S3 {{S3}} { S2 b S3 } NFA b S1 S2 S3 DFA { Σ, {{S1,S3},{S2},{S3}}, {S1,S3}, {{S1,S3},{S3}}, {({S1,S3},,{S2}), ({S2},b,{S3})} } Q q 0 ' F δ' CMSC 330 Spring {2} ε b {3}
52 Anlyzing the Reduction Cn reduce ny NFA to DFA using subset lg. How mny sttes in the DFA? Ech DFA stte is subset of the set of NFA sttes Given NFA with n sttes, DFA my hve 2 n sttes Ø Since set with n items my hve 2 n subsets Corollry Ø Reducing NFA with n sttes my be O(2 n ) CMSC 330 Spring
53 Reducing DFA to RE DFA cn reduce NFA cn trnsform cn trnsform RE CMSC 330 Spring
54 Reducing DFAs to REs Generl ide Remove sttes one by one, lbeling trnsitions with regulr expressions When two sttes re left (strt nd finl), the trnsition lbel is the regulr expression for the DFA CMSC 330 Spring
55 Other Topics Minimizing DFA Hopcroft reduction Complementing DFA Implementing DFA CMSC 330 Spring
56 Minimizing DFAs Every regulr lnguge is recognizble by unique minimum-stte DFA Ignoring the prticulr nmes of sttes In other words For every DFA, there is unique DFA with minimum number of sttes tht ccepts the sme lnguge b S1 S2 S3 c CMSC 330 Spring S1 c S3 S2 b
57 J. Hopcroft, An n log n lgorithm for minimizing sttes in finite utomton, 1971 Minimizing DFA: Hopcroft Reduction Intuition Look to distinguish sttes from ech other Ø End up in different ccept / non-ccept stte with identicl input Algorithm Construct initil prtition Ø Accepting & non-ccepting sttes Itertively split prtitions (until prtitions remin fixed) Ø Split prtition if members in prtition hve trnsitions to different prtitions for sme input Two sttes x, y belong in sme prtition if nd only if for ll symbols in Σ they trnsition to the sme prtition Updte trnsitions & remove ded sttes CMSC 330 Spring
58 Splitting Prtitions No need to split prtition {S,T,U,V} All trnsitions on led to identicl prtition P2 Even though trnsitions on led to different sttes S P1 X P2 T Y U V Z CMSC 330 Spring
59 Splitting Prtitions (cont.) Need to split prtition {S,T,U} into {S,T}, {U} Trnsitions on from S,T led to prtition P2 Trnsition on from U led to prtition P3 S P1 X P2 P4 U T Z Y b P3 CMSC 330 Spring
60 Resplitting Prtitions Need to reexmine prtitions fter splits Initilly no need to split prtition {S,T,U} After splitting prtition {X,Y} into {X}, {Y} we need to split prtition {S,T,U} into {S,T}, {U} P4 S U T P1 CMSC 330 Spring b P3 X b P2 Y
61 Minimizing DFA: Exmple 1 DFA b S T R Initil prtitions b Split prtition CMSC 330 Spring
62 Minimizing DFA: Exmple 1 DFA Initil prtitions Accept { R } = P1 b P2 P1 Reject { S, T } = P2 Split prtition? Not required, minimiztion done move(s,) move(t,) = T P2 = T P2 b S T R b move(s,b) move (T,b) = R P1 = R P1 CMSC 330 Spring
63 Minimizing DFA: Exmple 2 b S T R b CMSC 330 Spring
64 Minimizing DFA: Exmple 2 DFA P2 b S T R P1 Initil prtitions Accept Reject { R } { S, T } = P1 = P2 Split prtition? Yes, different prtitions for B move(s,) move(t,) = T P2 = T P2 b P3 move(s,b) move (T,b) DFA lredy miniml = T P2 = R P1 CMSC 330 Spring
65 Complement of DFA Given DFA ccepting lnguge L How cn we crete DFA ccepting its complement? Exmple DFA Ø Σ = {,b} CMSC 330 Spring
66 Complement of DFA Algorithm Add explicit trnsitions to ded stte Chnge every ccepting stte to non-ccepting stte & every non-ccepting stte to n ccepting stte Note this only works with DFAs Why not with NFAs? CMSC 330 Spring
67 Implementing DFAs (one-off) It's esy to build progrm which mimics DFA cur_stte = 0; while (1) { symbol = getchr(); switch (cur_stte) { cse 0: switch (symbol) { cse '0': cur_stte = 0; brek; cse '1': cur_stte = 1; brek; cse '\n': printf("rejected\n"); return 0; defult: printf("rejected\n"); return 0; } brek; } } cse 1: switch (symbol) { cse '0': cur_stte = 0; brek; cse '1': cur_stte = 1; brek; cse '\n': printf("ccepted\n"); return 1; defult: printf("rejected\n"); return 0; } brek; defult: printf("unknown stte; I'm confused\n"); brek; CMSC 330 Spring
68 Implementing DFAs (generic) More generlly, use generic tble-driven DFA given components (Σ, Q, q 0, F, d) of DFA: let q = q 0 while (there exists nother symbol s of the input string) q := d(q, s); if q Î F then ccept else reject q is just n integer Represent d using rrys or hsh tbles Represent F s set CMSC 330 Spring
69 Running Time of DFA How long for DFA to decide to ccept/reject string s? Assume we cn compute d(q, c) in constnt time Then time to process s is O( s ) Ø Cn t get much fster! Constructing DFA for RE A my tke O(2 A ) time But usully not the cse in prctice So there s the initil overhed But then processing strings is fst CMSC 330 Spring
70 Summry of Regulr Expression Theory Finite utomt DFA, NFA Equivlence of RE, NFA, DFA RE NFA Ø Conctention, union, closure NFA DFA Ø e-closure & subset lgorithm DFA Minimiztion, complement Implementtion CMSC 330 Spring
Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt
More informationCMSC 330: Organization of Programming Languages
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 CMSC 330 1 Types of Finite Automt Deterministic Finite Automt (DFA) Exctly one sequence of steps for ech string All exmples so fr Nondeterministic
More informationTypes of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. Comparing DFAs and NFAs (cont.) Finite Automata 2
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt
More informationNFA DFA Example 3 CMSC 330: Organization of Programming Languages. Equivalence of DFAs and NFAs. Equivalence of DFAs and NFAs (cont.
NFA DFA Exmple 3 CMSC 330: Orgniztion of Progrmming Lnguges NFA {B,D,E {A,E {C,D {E Finite Automt, con't. R = { {A,E, {B,D,E, {C,D, {E 2 Equivlence of DFAs nd NFAs Any string from {A to either {D or {CD
More informationTheory of Computation Regular Languages. (NTU EE) Regular Languages Fall / 38
Theory of Computtion Regulr Lnguges (NTU EE) Regulr Lnguges Fll 2017 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of Finite Automt A finite utomton hs finite set of control
More informationTheory of Computation Regular Languages
Theory of Computtion Regulr Lnguges Bow-Yw Wng Acdemi Sinic Spring 2012 Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring 2012 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of
More informationNon-Deterministic Finite Automata. Fall 2018 Costas Busch - RPI 1
Non-Deterministic Finite Automt Fll 2018 Costs Busch - RPI 1 Nondeterministic Finite Automton (NFA) Alphbet ={} q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 2 Nondeterministic Finite Automton (NFA) Alphbet
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 22 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 22 September 2017 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationNFAs and Regular Expressions. NFA-ε, continued. Recall. Last class: Today: Fun:
CMPU 240 Lnguge Theory nd Computtion Spring 2019 NFAs nd Regulr Expressions Lst clss: Introduced nondeterministic finite utomt with -trnsitions Tody: Prove n NFA- is no more powerful thn n NFA Introduce
More informationNondeterminism. Nondeterministic Finite Automata. Example: Moves on a Chessboard. Nondeterminism (2) Example: Chessboard (2) Formal NFA
Nondeterminism Nondeterministic Finite Automt Nondeterminism Subset Construction A nondeterministic finite utomton hs the bility to be in severl sttes t once. Trnsitions from stte on n input symbol cn
More informationAUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton
25. Finite Automt AUTOMATA AND LANGUAGES A system of computtion tht only hs finite numer of possile sttes cn e modeled using finite utomton A finite utomton is often illustrted s stte digrm d d d. d q
More informationConvert the NFA into DFA
Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:
More informationAnatomy of a Deterministic Finite Automaton. Deterministic Finite Automata. A machine so simple that you can understand it in less than one minute
Victor Admchik Dnny Sletor Gret Theoreticl Ides In Computer Science CS 5-25 Spring 2 Lecture 2 Mr 3, 2 Crnegie Mellon University Deterministic Finite Automt Finite Automt A mchine so simple tht you cn
More informationNon Deterministic Automata. Linz: Nondeterministic Finite Accepters, page 51
Non Deterministic Automt Linz: Nondeterministic Finite Accepters, pge 51 1 Nondeterministic Finite Accepter (NFA) Alphbet ={} q 1 q2 q 0 q 3 2 Nondeterministic Finite Accepter (NFA) Alphbet ={} Two choices
More informationLexical Analysis Finite Automate
Lexicl Anlysis Finite Automte CMPSC 470 Lecture 04 Topics: Deterministic Finite Automt (DFA) Nondeterministic Finite Automt (NFA) Regulr Expression NFA DFA A. Finite Automt (FA) FA re grph, like trnsition
More informationFinite Automata. Informatics 2A: Lecture 3. Mary Cryan. 21 September School of Informatics University of Edinburgh
Finite Automt Informtics 2A: Lecture 3 Mry Cryn School of Informtics University of Edinburgh mcryn@inf.ed.c.uk 21 September 2018 1 / 30 Lnguges nd Automt Wht is lnguge? Finite utomt: recp Some forml definitions
More informationJava II Finite Automata I
Jv II Finite Automt I Bernd Kiefer Bernd.Kiefer@dfki.de Deutsches Forschungszentrum für künstliche Intelligenz Finite Automt I p.1/13 Processing Regulr Expressions We lredy lerned out Jv s regulr expression
More informationRegular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene-*
Regulr Expressions (RE) Regulr Expressions (RE) Empty set F A RE denotes the empty set Opertion Nottion Lnguge UNIX Empty string A RE denotes the set {} Alterntion R +r L(r ) L(r ) r r Symol Alterntion
More informationRegular expressions, Finite Automata, transition graphs are all the same!!
CSI 3104 /Winter 2011: Introduction to Forml Lnguges Chpter 7: Kleene s Theorem Chpter 7: Kleene s Theorem Regulr expressions, Finite Automt, trnsition grphs re ll the sme!! Dr. Neji Zgui CSI3104-W11 1
More informationChapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1
Chpter Five: Nondeterministic Finite Automt Forml Lnguge, chpter 5, slide 1 1 A DFA hs exctly one trnsition from every stte on every symol in the lphet. By relxing this requirement we get relted ut more
More informationChapter 2 Finite Automata
Chpter 2 Finite Automt 28 2.1 Introduction Finite utomt: first model of the notion of effective procedure. (They lso hve mny other pplictions). The concept of finite utomton cn e derived y exmining wht
More informationCS 275 Automata and Formal Language Theory
CS 275 Automt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Problem (II) Chpter II.6.: Push Down Automt Remrk: This mteril is no longer tught nd not directly exm relevnt Anton Setzer (Bsed
More informationLet's start with an example:
Finite Automt Let's strt with n exmple: Here you see leled circles tht re sttes, nd leled rrows tht re trnsitions. One of the sttes is mrked "strt". One of the sttes hs doule circle; this is terminl stte
More informationDeterministic Finite Automata
Finite Automt Deterministic Finite Automt H. Geuvers nd J. Rot Institute for Computing nd Informtion Sciences Version: fll 2016 J. Rot Version: fll 2016 Tlen en Automten 1 / 21 Outline Finite Automt Finite
More informationCS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University
CS415 Compilers Lexicl Anlysis nd These slides re sed on slides copyrighted y Keith Cooper, Ken Kennedy & Lind Torczon t Rice University First Progrmming Project Instruction Scheduling Project hs een posted
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More
More informationCS375: Logic and Theory of Computing
CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tble of Contents: Week 1: Preliminries (set lgebr, reltions, functions) (red Chpters 1-4) Weeks
More informationNFAs continued, Closure Properties of Regular Languages
Algorithms & Models of Computtion CS/ECE 374, Fll 2017 NFAs continued, Closure Properties of Regulr Lnguges Lecture 5 Tuesdy, Septemer 12, 2017 Sriel Hr-Peled (UIUC) CS374 1 Fll 2017 1 / 31 Regulr Lnguges,
More informationMinimal DFA. minimal DFA for L starting from any other
Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA
More informationFinite Automata-cont d
Automt Theory nd Forml Lnguges Professor Leslie Lnder Lecture # 6 Finite Automt-cont d The Pumping Lemm WEB SITE: http://ingwe.inghmton.edu/ ~lnder/cs573.html Septemer 18, 2000 Exmple 1 Consider L = {ww
More informationCS:4330 Theory of Computation Spring Regular Languages. Equivalences between Finite automata and REs. Haniel Barbosa
CS:4330 Theory of Computtion Spring 208 Regulr Lnguges Equivlences between Finite utomt nd REs Hniel Brbos Redings for this lecture Chpter of [Sipser 996], 3rd edition. Section.3. Finite utomt nd regulr
More informationNon-deterministic Finite Automata
Non-deterministic Finite Automt Eliminting non-determinism Rdoud University Nijmegen Non-deterministic Finite Automt H. Geuvers nd T. vn Lrhoven Institute for Computing nd Informtion Sciences Intelligent
More informationNondeterminism and Nodeterministic Automata
Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationConverting Regular Expressions to Discrete Finite Automata: A Tutorial
Converting Regulr Expressions to Discrete Finite Automt: A Tutoril Dvid Christinsen 2013-01-03 This is tutoril on how to convert regulr expressions to nondeterministic finite utomt (NFA) nd how to convert
More informationNon-deterministic Finite Automata
Non-deterministic Finite Automt From Regulr Expressions to NFA- Eliminting non-determinism Rdoud University Nijmegen Non-deterministic Finite Automt H. Geuvers nd J. Rot Institute for Computing nd Informtion
More informationState Minimization for DFAs
Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid
More informationMore on automata. Michael George. March 24 April 7, 2014
More on utomt Michel George Mrch 24 April 7, 2014 1 Automt constructions Now tht we hve forml model of mchine, it is useful to mke some generl constructions. 1.1 DFA Union / Product construction Suppose
More informationNon Deterministic Automata. Formal Languages and Automata - Yonsei CS 1
Non Deterministic Automt Forml Lnguges nd Automt - Yonsei CS 1 Nondeterministic Finite Accepter (NFA) We llow set of possible moves insted of A unique move. Alphbet = {} Two choices q 1 q2 Forml Lnguges
More informationFABER Formal Languages, Automata and Models of Computation
DVA337 FABER Forml Lnguges, Automt nd Models of Computtion Lecture 5 chool of Innovtion, Design nd Engineering Mälrdlen University 2015 1 Recp of lecture 4 y definition suset construction DFA NFA stte
More information1.4 Nonregular Languages
74 1.4 Nonregulr Lnguges The number of forml lnguges over ny lphbet (= decision/recognition problems) is uncountble On the other hnd, the number of regulr expressions (= strings) is countble Hence, ll
More informationDesigning finite automata II
Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of
More informationCHAPTER 1 Regular Languages. Contents
Finite Automt (FA or DFA) CHAPTE 1 egulr Lnguges Contents definitions, exmples, designing, regulr opertions Non-deterministic Finite Automt (NFA) definitions, euivlence of NFAs nd DFAs, closure under regulr
More informationCS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018
CS 301 Lecture 04 Regulr Expressions Stephen Checkowy Jnury 29, 2018 1 / 35 Review from lst time NFA N = (Q, Σ, δ, q 0, F ) where δ Q Σ P (Q) mps stte nd n lphet symol (or ) to set of sttes We run n NFA
More information5.1 Definitions and Examples 5.2 Deterministic Pushdown Automata
CSC4510 AUTOMATA 5.1 Definitions nd Exmples 5.2 Deterministic Pushdown Automt Definitions nd Exmples A lnguge cn be generted by CFG if nd only if it cn be ccepted by pushdown utomton. A pushdown utomton
More informationFirst Midterm Examination
Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does
More informationGNFA GNFA GNFA GNFA GNFA
DFA RE NFA DFA -NFA REX GNFA Definition GNFA A generlize noneterministic finite utomton (GNFA) is grph whose eges re lele y regulr expressions, with unique strt stte with in-egree, n unique finl stte with
More informationFormal Languages and Automata
Moile Computing nd Softwre Engineering p. 1/5 Forml Lnguges nd Automt Chpter 2 Finite Automt Chun-Ming Liu cmliu@csie.ntut.edu.tw Deprtment of Computer Science nd Informtion Engineering Ntionl Tipei University
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationLecture 6 Regular Grammars
Lecture 6 Regulr Grmmrs COT 4420 Theory of Computtion Section 3.3 Grmmr A grmmr G is defined s qudruple G = (V, T, S, P) V is finite set of vribles T is finite set of terminl symbols S V is specil vrible
More informationFinite-State Automata: Recap
Finite-Stte Automt: Recp Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute of Science, Bnglore. 09 August 2016 Outline 1 Introduction 2 Forml Definitions nd Nottion 3 Closure under
More informationNFAs continued, Closure Properties of Regular Languages
lgorithms & Models of omputtion S/EE 374, Spring 209 NFs continued, losure Properties of Regulr Lnguges Lecture 5 Tuesdy, Jnury 29, 209 Regulr Lnguges, DFs, NFs Lnguges ccepted y DFs, NFs, nd regulr expressions
More informationCSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science
CSCI 340: Computtionl Models Kleene s Theorem Chpter 7 Deprtment of Computer Science Unifiction In 1954, Kleene presented (nd proved) theorem which (in our version) sttes tht if lnguge cn e defined y ny
More informationFormal languages, automata, and theory of computation
Mälrdlen University TEN1 DVA337 2015 School of Innovtion, Design nd Engineering Forml lnguges, utomt, nd theory of computtion Thursdy, Novemer 5, 14:10-18:30 Techer: Dniel Hedin, phone 021-107052 The exm
More informationNon-Deterministic Finite Automata
Non-Deterministic Finite Automt http://users.comlb.ox.c.uk/luke. ong/teching/moc/nf2up.pdf 1 Nondeterministic Finite Automton (NFA) Alphbet ={} q1 q2 2 Alphbet ={} Two choices q1 q2 3 Alphbet ={} Two choices
More information1.3 Regular Expressions
56 1.3 Regulr xpressions These hve n importnt role in describing ptterns in serching for strings in mny pplictions (e.g. wk, grep, Perl,...) All regulr expressions of lphbet re 1.Ønd re regulr expressions,
More informationWhere did dynamic programming come from?
Where did dynmic progrmming come from? String lgorithms Dvid Kuchk cs302 Spring 2012 Richrd ellmn On the irth of Dynmic Progrmming Sturt Dreyfus http://www.eng.tu.c.il/~mi/cd/ or50/1526-5463-2002-50-01-0048.pdf
More informationCS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)
CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts
More informationAutomata and Languages
Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive
More informationFundamentals of Computer Science
Fundmentls of Computer Science Chpter 3: NFA nd DFA equivlence Regulr expressions Henrik Björklund Umeå University Jnury 23, 2014 NFA nd DFA equivlence As we shll see, it turns out tht NFA nd DFA re equivlent,
More informationHarvard University Computer Science 121 Midterm October 23, 2012
Hrvrd University Computer Science 121 Midterm Octoer 23, 2012 This is closed-ook exmintion. You my use ny result from lecture, Sipser, prolem sets, or section, s long s you quote it clerly. The lphet is
More informationClosure Properties of Regular Languages
of Regulr Lnguges Dr. Neil T. Dntm CSCI-561, Colordo School of Mines Fll 2018 Dntm (Mines CSCI-561) Closure Properties of Regulr Lnguges Fll 2018 1 / 50 Outline Introduction Closure Properties Stte Minimiztion
More informationLexical Analysis Part III
Lexicl Anlysis Prt III Chpter 3: Finite Automt Slides dpted from : Roert vn Engelen, Florid Stte University Alex Aiken, Stnford University Design of Lexicl Anlyzer Genertor Trnslte regulr expressions to
More informationCISC 4090 Theory of Computation
9/6/28 Stereotypicl computer CISC 49 Theory of Computtion Finite stte mchines & Regulr lnguges Professor Dniel Leeds dleeds@fordhm.edu JMH 332 Centrl processing unit (CPU) performs ll the instructions
More informationAssignment 1 Automata, Languages, and Computability. 1 Finite State Automata and Regular Languages
Deprtment of Computer Science, Austrlin Ntionl University COMP2600 Forml Methods for Softwre Engineering Semester 2, 206 Assignment Automt, Lnguges, nd Computility Smple Solutions Finite Stte Automt nd
More informationCMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014
CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA
More informationCHAPTER 1 Regular Languages. Contents. definitions, examples, designing, regular operations. Non-deterministic Finite Automata (NFA)
Finite Automt (FA or DFA) CHAPTER Regulr Lnguges Contents definitions, exmples, designing, regulr opertions Non-deterministic Finite Automt (NFA) definitions, equivlence of NFAs DFAs, closure under regulr
More informationFirst Midterm Examination
24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet
More information3 Regular expressions
3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll
More information12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016
CS125 Lecture 12 Fll 2016 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple
More informationToday s Topics Automata and Languages
Tody s Topics Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn DFA to Regulr Expression GFNA DFAèGNFA GNFA è RE DFA è RE Exmples 2 DFA è RE NFA DFA -NFA REX GNFA 3 Definition
More informationAutomata and Languages
Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Tody s Topics DFA to Regulr Expression GFNA DFAèGNFA GNFA è RE DFA è RE Exmples 2 DFA è RE NFA DFA -NFA REX GNFA 3 Definition
More informationWorked out examples Finite Automata
Worked out exmples Finite Automt Exmple Design Finite Stte Automton which reds inry string nd ccepts only those tht end with. Since we re in the topic of Non Deterministic Finite Automt (NFA), we will
More information11.1 Finite Automata. CS125 Lecture 11 Fall Motivation: TMs without a tape: maybe we can at least fully understand such a simple model?
CS125 Lecture 11 Fll 2016 11.1 Finite Automt Motivtion: TMs without tpe: mybe we cn t lest fully understnd such simple model? Algorithms (e.g. string mtching) Computing with very limited memory Forml verifiction
More informationTuring Machines Part One
Turing Mchines Prt One Wht problems cn we solve with computer? Regulr Lnguges CFLs Lnguges recognizble by ny fesible computing mchine All Lnguges Tht sme drwing, to scle. All Lnguges The Problem Finite
More informationTuring Machines Part One
Turing Mchines Prt One Hello Hello Condensed Condensed Slide Slide Reders! Reders! Tody s Tody s lecture lecture consists consists lmost lmost exclusively exclusively of of nimtions nimtions of of Turing
More informationLecture 08: Feb. 08, 2019
4CS4-6:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny
More informationDeterministic Finite-State Automata
Deterministic Finite-Stte Automt Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute of Science, Bnglore. 12 August 2013 Outline 1 Introduction 2 Exmple DFA 1 DFA for Odd number of
More information12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2014
CS125 Lecture 12 Fll 2014 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple
More information1 From NFA to regular expression
Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work
More informationFormal Language and Automata Theory (CS21004)
Forml Lnguge nd Automt Forml Lnguge nd Automt Theory (CS21004) Khrgpur Khrgpur Khrgpur Forml Lnguge nd Automt Tle of Contents Forml Lnguge nd Automt Khrgpur 1 2 3 Khrgpur Forml Lnguge nd Automt Forml Lnguge
More informationLecture 09: Myhill-Nerode Theorem
CS 373: Theory of Computtion Mdhusudn Prthsrthy Lecture 09: Myhill-Nerode Theorem 16 Ferury 2010 In this lecture, we will see tht every lnguge hs unique miniml DFA We will see this fct from two perspectives
More informationThis lecture covers Chapter 8 of HMU: Properties of CFLs
This lecture covers Chpter 8 of HMU: Properties of CFLs Turing Mchine Extensions of Turing Mchines Restrictions of Turing Mchines Additionl Reding: Chpter 8 of HMU. Turing Mchine: Informl Definition B
More informationSpeech Recognition Lecture 2: Finite Automata and Finite-State Transducers
Speech Recognition Lecture 2: Finite Automt nd Finite-Stte Trnsducers Eugene Weinstein Google, NYU Cournt Institute eugenew@cs.nyu.edu Slide Credit: Mehryr Mohri Preliminries Finite lphet, empty string.
More informationSome Theory of Computation Exercises Week 1
Some Theory of Computtion Exercises Week 1 Section 1 Deterministic Finite Automt Question 1.3 d d d d u q 1 q 2 q 3 q 4 q 5 d u u u u Question 1.4 Prt c - {w w hs even s nd one or two s} First we sk whether
More informationCS 330 Formal Methods and Models
CS 330 Forml Methods nd Models Dn Richrds, George Mson University, Spring 2017 Quiz Solutions Quiz 1, Propositionl Logic Dte: Ferury 2 1. Prove ((( p q) q) p) is tutology () (3pts) y truth tle. p q p q
More information1 Nondeterministic Finite Automata
1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
More informationFinite Automata Part Three
Finite Automt Prt Three Hello Hello Wonderful Wonderful Condensed Condensed Slide Slide Reders! Reders! The The first first hlf hlf of of this this lecture lecture consists consists lmost lmost exclusively
More informationAutomata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ω-automata. Ralf Huuck.
Outline Automt Theory 101 Rlf Huuck Introduction Finite Automt Regulr Expressions ω-automt Session 1 2006 Rlf Huuck 1 Session 1 2006 Rlf Huuck 2 Acknowledgement Some slides re sed on Wolfgng Thoms excellent
More informationNormal Forms for Context-free Grammars
Norml Forms for Context-free Grmmrs 1 Linz 6th, Section 6.2 wo Importnt Norml Forms, pges 171--178 2 Chomsky Norml Form All productions hve form: A BC nd A vrile vrile terminl 3 Exmples: S AS S AS S S
More informationMyhill-Nerode Theorem
Overview Myhill-Nerode Theorem Correspondence etween DA s nd MN reltions Cnonicl DA for L Computing cnonicl DFA Myhill-Nerode Theorem Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute
More informationa,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1
CS4 45- Determinisitic Finite Automt -: Genertors vs. Checkers Regulr expressions re one wy to specify forml lnguge String Genertor Genertes strings in the lnguge Deterministic Finite Automt (DFA) re nother
More information1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.
York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech
More informationMidterm 1 Practice. CS 350 Fall 2018 gilray.org/classes/fall2018/cs350/
Midterm 1 Prctice CS 350 Fll 2018 gilry.org/clsses/fll2018/cs350/ 1 Midterm #1: Thursdy, Septemer 27! Bring less stuff, if possile. Keep ny gs under the tle. You my hve out: pencil, pen, nd/or erser. My
More informationFORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY. FLAC (15-453) - Spring L. Blum
15-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY THE PUMPING LEMMA FOR REGULAR LANGUAGES nd REGULAR EXPRESSIONS TUESDAY Jn 21 WHICH OF THESE ARE REGULAR? B = {0 n 1 n n 0} C = { w w hs equl numer of
More informationHomework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)
More informationɛ-closure, Kleene s Theorem,
DEGefW5wiGH2XgYMEzUKjEmtCDUsRQ4d 1 A nice pper relevnt to this course is titled The Glory of the Pst 2 NICTA Resercher, Adjunct t the Austrlin Ntionl University nd Griffith University ɛ-closure, Kleene
More informationCS5371 Theory of Computation. Lecture 20: Complexity V (Polynomial-Time Reducibility)
CS5371 Theory of Computtion Lecture 20: Complexity V (Polynomil-Time Reducibility) Objectives Polynomil Time Reducibility Prove Cook-Levin Theorem Polynomil Time Reducibility Previously, we lernt tht if
More informationSpeech Recognition Lecture 2: Finite Automata and Finite-State Transducers. Mehryar Mohri Courant Institute and Google Research
Speech Recognition Lecture 2: Finite Automt nd Finite-Stte Trnsducers Mehryr Mohri Cournt Institute nd Google Reserch mohri@cims.nyu.com Preliminries Finite lphet Σ, empty string. Set of ll strings over
More information19 Optimal behavior: Game theory
Intro. to Artificil Intelligence: Dle Schuurmns, Relu Ptrscu 1 19 Optiml behvior: Gme theory Adversril stte dynmics hve to ccount for worst cse Compute policy π : S A tht mximizes minimum rewrd Let S (,
More information