Finite State Automata and Determinisation
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1 Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016
2 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Non-deterministi Finite Stte Automt (nf) 4 Regulr Expressions 5 Deterministi Finite Stte Automt (df) 6 Deterministion Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
3 Lnguges fs nf re df Deterministion 3 Alphets An lphet Σ is set of tokens used y lnguge Σ = {0, 1} is the lphet for inry Σ = {., 0, 1,..., 9} is the lphet for deiml numer Σ = {mro, polo} is the lphet for the ommunitions protool in the gme Mro Polo 1 Wht lnguge(s) is this the lphet for? Σ = {x, 0, 1,..., 9, A, B,..., F} 1 A gme involving lling out Mro! nd Polo!, with n optionl swimming pool Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
4 Lnguges fs nf re df Deterministion 4 Strings A omintion 2 of elements from Σ is lled string 0, , 111 re ll strings of Σ = {0, 1} The set Σ is the infinite set of ll omintions of the elements of Σ For the inry lphet, Σ = {0, 1}, Σ is 3 Σ = {, (0), (1), (00), (01), (10), (11), (000), (001),...} 2 Oviously, ontiguous ordered tuple of elements, not set: 01 is different string from 10 3 Here the empty set will do to represent null or empty string Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
5 Lnguges fs nf re df Deterministion 5 Lnguges A lnguge L is set of items from Σ whih re deemed to e vlid (L Σ ) For hexdeiml, only the strings whih egin with 0x nd hve t lest one more non- x token re deemed vlid 0xF x0 0x 1xA (the empty string) A lnguge in this ontext is not set of rules, syntx, grmmr nd ll: it s just set of vlid strings Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
6 Lnguges fs nf re df Deterministion 6 Finite Stte Automt Finite Stte Automt re representtions of self-ontined finite set of sttes, with rules tht govern movement mong those sttes. Finite Stte Automt (fss) exist everywhere They re used to illustrte the possile sttes some proess n e in how the proess moves etween these sttes Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
7 Lnguges fs nf re df Deterministion 7 Finite Stte Automt E.g. stte mhine for PIN ode entry on your moile 4 : 0-9 nel witing 0-9 witing nel nel witing 0-9 witing 0-9 nel hek pssword! 4 How mny people do you know who sy PIN Numer? Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
8 Lnguges fs nf re df Deterministion 8 Finite Stte Automt 0-9 nel witing 0-9 witing nel nel witing 0-9 witing 0-9 nel hek pssword! fss hve nodes Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
9 Lnguges fs nf re df Deterministion 8 Finite Stte Automt 0-9 nel witing 0-9 witing nel nel witing 0-9 witing 0-9 nel hek pssword! fss hve nodes fss hve edges Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
10 Lnguges fs nf re df Deterministion 8 Finite Stte Automt 0-9 nel witing 0-9 witing nel nel witing 0-9 witing 0-9 nel hek pssword! fss hve nodes fss hve edges fs edges (trnsitions) hve lels Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
11 Lnguges fs nf re df Deterministion 8 Finite Stte Automt 0-9 nel witing 0-9 witing nel nel witing 0-9 witing 0-9 nel hek pssword! fss hve nodes fss hve edges fs edges (trnsitions) hve lels fss hve strt node (entry point) Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
12 Lnguges fs nf re df Deterministion 8 Finite Stte Automt 0-9 nel witing 0-9 witing nel nel witing 0-9 witing 0-9 nel hek pssword! fss hve nodes fss hve edges fs edges (trnsitions) hve lels fss hve strt node (entry point) fss hve epting nodes, denoted with the doule outline Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
13 Lnguges fs nf re df Deterministion 9 Finite Stte Automt 0-9 nel witing 0-9 witing nel nel witing 0-9 witing 0-9 nel hek pssword! You re lwys on one of the nodes, i.e., in one of the sttes. You n trvel etween the nodes following the direted edges You n only trvel long n edge if you see the lel tht is on the edge Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
14 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
15 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q 3 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
16 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q 3 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
17 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
18 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q q 0 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
19 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q q 0 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
20 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q q 0 0x0 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
21 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q q 0 0x0 q 0 q 1 q 2 q 3 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
22 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q q 0 0x0 q 0 q 1 q 2 q 3 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
23 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q q 0 0x0 q 0 q 1 q 2 q 3 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
24 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q q 0 0x0 q 0 q 1 q 2 q 3 q 0 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
25 Lnguges fs nf re df Deterministion 10 Pttern Reognition using fss 0-9,A-F 0 x 0-9,A-F Wlk the following strings through the fs. Are they epted? String Pth Aepts? 0xF9 q 0 q 1 q 2 q 3 q q 0 0x0 q 0 q 1 q 2 q 3 q 0 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
26 Lnguges fs nf re df Deterministion 11 Non-deterministi Finite Stte Automt There re two min types of fss Deterministi Non-deterministi Wht we hve een looking t so fr re exmples of non-deterministi finite [stte] utomt (nf) E.g. nf for reognising hexdeiml numer: 0-9,A-F 0 x 0-9,A-F Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
27 Lnguges fs nf re df Deterministion 12 Properties of nfs Must hve strt node No restritions on the edge lels Allowed speil ε edge lels (we ll ome k to this) Allowed to e on multiple sttes simultneously non-determinism Bsilly n NFA is FSA tht isn t fully defined: in DFA (Deterministi Finite (Stte) Automton of ourse), everything hs to e determined: if there s n error ondition tht hs to e explined, there n t e ny miguity, nd so on. Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
28 Lnguges fs nf re df Deterministion 13 Non-determinism exmple Is the string epted y the nf? Pth: Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
29 Lnguges fs nf re df Deterministion 13 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
30 Lnguges fs nf re df Deterministion 13 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 1 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
31 Lnguges fs nf re df Deterministion 13 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 1 } {q 3 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
32 Lnguges fs nf re df Deterministion 13 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 1 } {q 3 } Aepted: Yes,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
33 Lnguges fs nf re df Deterministion 14 Non-determinism exmple Is the string epted y the nf? Pth: Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
34 Lnguges fs nf re df Deterministion 14 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
35 Lnguges fs nf re df Deterministion 14 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 0, q 2 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
36 Lnguges fs nf re df Deterministion 14 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 0, q 2 } {q 1, q 3 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
37 Lnguges fs nf re df Deterministion 14 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 0, q 2 } {q 1, q 3 } Aepted: Yes,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
38 Lnguges fs nf re df Deterministion 15 Non-determinism exmple Is the string epted y the nf? Pth: Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
39 Lnguges fs nf re df Deterministion 15 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
40 Lnguges fs nf re df Deterministion 15 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 0, q 2 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
41 Lnguges fs nf re df Deterministion 15 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 0, q 2 } {q 0, q 2 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
42 Lnguges fs nf re df Deterministion 15 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 0, q 2 } {q 0, q 2 } {q 0, q 1, q 2 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
43 Lnguges fs nf re df Deterministion 15 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 0, q 2 } {q 0, q 2 } {q 0, q 1, q 2 } {q 1, q 3 } Aepted:,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
44 Lnguges fs nf re df Deterministion 15 Non-determinism exmple Is the string epted y the nf? Pth: {q 0 } {q 0, q 2 } {q 0, q 2 } {q 0, q 1, q 2 } {q 1, q 3 } Aepted: Yes,,, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
45 Lnguges fs nf re df Deterministion 16 ε trnsitions nfs n hve speil trnsition lel ε ε llows trnsition without onsuming ny input tokens ϵ q4 In English, wht pttern does this nf therefore llow us to mth? Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
46 Lnguges fs nf re df Deterministion 16 ε trnsitions nfs n hve speil trnsition lel ε ε llows trnsition without onsuming ny input tokens ϵ q4 In English, wht pttern does this nf therefore llow us to mth? Any string strting with, followed y one or more instnes of, followed y Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
47 Lnguges fs nf re df Deterministion 17 Your turn Woo! Construt nf whih epts only inry numers with n even numer of zero s. Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
48 Lnguges fs nf re df Deterministion 17 Your turn Woo! Construt nf whih epts only inry numers with n even numer of zero s Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
49 Lnguges fs nf re df Deterministion 17 Your turn Woo! Construt nf whih epts only inry numers with n even numer of zero s As ove, exept for one s insted of zero s. Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
50 Lnguges fs nf re df Deterministion 17 Your turn Woo! Construt nf whih epts only inry numers with n even numer of zero s As ove, exept for one s insted of zero s Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
51 Lnguges fs nf re df Deterministion 18 Your turn Construt nf whih epts only inry numers with n even numer of zero s OR n even numer of one s. Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
52 Lnguges fs nf re df Deterministion 18 Your turn Construt nf whih epts only inry numers with n even numer of zero s OR n even numer of one s. 1 1 ϵ 0 0 ϵ q4 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
53 Lnguges fs nf re df Deterministion 19 Regulr Expressions (re) You rememer the power of regulr expressions? These things n sve the world! regex s rule.... nd it turns out they re muh etter interpreted nd delt with Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
54 Lnguges fs nf re df Deterministion 20 Regulr Expressions s nfs Regulr expressions n esily e represented using nfs We n group regulr expressions into 4 different omponents Chrter single hrter: // Contention two djent expressions: // Union two OR d expressions: / / Kleene str zero or more repetitions: /*/ /( d*)*e/ n e viewed s ont ont str union e ont str d Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
55 Lnguges fs nf re df Deterministion 21 And it relly does mtter Knowing the right wy to do regulr expressions, using good omputer siene, mens tht you don t hve to mke mjor errors leding to relly relly d performne, in hrdly-used it of softwre, like sy, Perl. This is the time tken to mth the sequene? n n ginst n where the supersript represents string repets, so? 2 2 =?? Soure: Russ Cox / rs@swth.om 5 5 For gret disussion of this, see Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
56 Lnguges fs nf re df Deterministion 22 re to nf: Chrter // Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
57 Lnguges fs nf re df Deterministion 23 re to nf: Contention // + eomes ϵ Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
58 Lnguges fs nf re df Deterministion 24 re to nf: Union / / + eomes q4 ϵ ϵ ϵ ϵ q5 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
59 Lnguges fs nf re df Deterministion 25 re to nf: Kleene Str /*/ eomes ϵ ϵ ϵ ϵ Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
60 Lnguges fs nf re df Deterministion 26 An exmple onversion /( d*)*e/ eomes ϵ ϵ ϵ ϵ q4 ϵ q5 q6 ϵ ϵ d ϵ q7 q8 ϵ q9 0 ϵ ϵ 1 ϵ 2 ϵ 3 e 4 ϵ Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
61 Lnguges fs nf re df Deterministion 27 DFAs vs NFAs How do dfs differ from nfs? Deterministi (not non-deterministi) No ε edges Every edge from node must hve unique lel (n t e in multiple sttes) Every node must hve n outwrd edge for eh token of the lphet (it is ompletely desried) Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
62 Lnguges fs nf re df Deterministion 28 dfs vs. nfs E.g. df for reognising hexdeiml numer: 0-9,A-F 0 x 1-9,A-F,x 0-9,A-F 0-9,A-F q4 x x 0-9,A-F,x q5 Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
63 Lnguges fs nf re df Deterministion 29 Why do we need dfs? Wht s wrong with nfs? The non-deterministi spet of nfs mkes them d for modern-dy omputtion Computers nnot effiiently perform the e in multiple sttes t one In df, you n only e in one stte t time Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
64 Lnguges fs nf re df Deterministion 30 Your turn dfs n e hrder to mnully onstrut thn nfs, due to their restritions. Construt df over Σ = {, } epting strings ending in. Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
65 Lnguges fs nf re df Deterministion 30 Your turn dfs n e hrder to mnully onstrut thn nfs, due to their restritions. Construt df over Σ = {, } epting strings ending in. Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
66 Lnguges fs nf re df Deterministion 31 nf? = df At first glne, it would pper tht nfs re more powerful thn dfs. Why? Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
67 Lnguges fs nf re df Deterministion 31 nf? = df At first glne, it would pper tht nfs re more powerful thn dfs. Why? multiple edges with the sme lel oming off node ε edges ility to e in multiple sttes t one Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
68 Lnguges fs nf re df Deterministion 31 nf? = df At first glne, it would pper tht nfs re more powerful thn dfs. Why? multiple edges with the sme lel oming off node ε edges ility to e in multiple sttes t one In 1959, Rin nd Sott proved tht nfs nd dfs hve the sme expressive power The proof is surprisingly simple: we need to show tht for eh NFA there is DFA, nd vie-vers, tht ept preisely the sme lnguges. Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
69 Lnguges fs nf re df Deterministion 32 Deterministion Algorithm This lgorithm onverts ny nf into n equivlent df ϵ 0 0 ϵ q into 0,1 {,,} 0,1 {,q4} 0 1 {,} 0 0 {q4} 1 {} Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
70 Lnguges fs nf re df Deterministion 33 Move nd Epsilon-Closure Move M(sttes, token) Given set of sttes nd n input token, wht set of sttes do you end up t Epsilon-Closure EC(sttes) Given set of sttes, wht set of sttes do you get y expnding ll ε trnsitions, ϵ, M({q 0, q 1 }, ) = {q 2, q 3 } EC({q 2, q 3 }) = {q 0, q 2, q 3 } Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
71 Lnguges fs nf re df Deterministion 34 Algorithm Ide We need to remove the onept of non-determinism in the df Ahieved y reting stte in the df for every possile set of sttes in the nf Q: If the nf hs n sttes, wht s the mximum numer of sttes its orresponding df ould hve? Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
72 Lnguges fs nf re df Deterministion 34 Algorithm Ide We need to remove the onept of non-determinism in the df Ahieved y reting stte in the df for every possile set of sttes in the nf Q: If the nf hs n sttes, wht s the mximum numer of sttes its orresponding df ould hve? A: P(n) = 2 n Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
73 Lnguges fs nf re df Deterministion 35 Pseudoode Require: nf df new DFA ojet df.strt EC(nf.strt) todo [df.strt] while todo = do sttes = pop the next item off todo for ll σ Σ NFA \ {ε} do s = EC (M(sttes, σ)) if s = then Add the edge sttes Add s to todo if it s new end if end for end while return df σ s to df Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
74 Lnguges fs nf re df Deterministion 36 Exmple, todo {0} ϵ, Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
75 Lnguges fs nf re df Deterministion 36 Exmple, todo {0} ϵ, {} Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
76 Lnguges fs nf re df Deterministion 36 Exmple ϵ,, todo {0} {0, 2} {1} {,} {} {} Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
77 Lnguges fs nf re df Deterministion 36 Exmple ϵ,, todo {0} {0, 2} {1} {0, 1, 2} {1, 3} {,} {,,} {,} {} {} Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
78 Lnguges fs nf re df Deterministion 36 Exmple ϵ,, todo {0} {0, 2} {1} {0, 1, 2} {1, 3} {3} {} {,} {,,} {,}, {} {} Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
79 Lnguges fs nf re df Deterministion 36 Exmple ϵ,, todo {0} {0, 2} {1} {0, 1, 2} {1, 3} {3} {0, 2, 3} {} {,} {,,} {,,} {,}, {} {} Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
80 Lnguges fs nf re df Deterministion 36 Exmple ϵ,, todo {0} {0, 2} {1} {0, 1, 2} {1, 3} {3} {0, 2, 3} {} {,} {,,} {,,} {,},, {} {} Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
81 Lnguges fs nf re df Deterministion 36 Exmple ϵ,, todo {0} {0, 2} {1} {0, 1, 2} {1, 3} {3} {0, 2, 3} {} {,} {,,} {,,} {,},, {} {},,,, {} Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
82 Lnguges fs nf re df Deterministion 36 Exmple ϵ,, todo {0} {0, 2} {1} {0, 1, 2} {1, 3} {3} {0, 2, 3}, {} {,} {,,} {,,} {,},, {} {},,,, {} Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
83 Lnguges fs nf re df Deterministion 37 Finish End of presenttion. Tim Dworn Finite Stte Automt nd Deterministion Jnury, 2016
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