Introduction to ω-autamata

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1 Fridy 25 th Jnury, 2013

2 Outline From finite word utomt ω-regulr lnguge ω-utomt Nondeterministic Models Deterministic Models Two Lower Bounds Conclusion Discussion Synthesis Preliminry

3 From finite word utomt We ssume ll re fmilir with NFA, DFA, regulr lnguges nd the reltionships mong them.

4 From finite word utomt We ssume ll re fmilir with NFA, DFA, regulr lnguges nd the reltionships mong them. In CS we need to formlize the infinite behviors for non-terminting systems.

5 From finite word utomt We ssume ll re fmilir with NFA, DFA, regulr lnguges nd the reltionships mong them. In CS we need to formlize the infinite behviors for non-terminting systems. How does these behviors be described?

6 From finite word utomt We ssume ll re fmilir with NFA, DFA, regulr lnguges nd the reltionships mong them. In CS we need to formlize the infinite behviors for non-terminting systems. How does these behviors be described? Wht re the new models (utomt) which cn ccept these behviors?

7 ω-regulr lnguge Let ω denote the set of non-negtive integers, i.e. ω = {0, 1, 2,...}. And let REG be the clss of regulr lnguges. Definition (ω-regulr lnguge) The ω-kleene closure of the clss of regulr lnguge (short for ω-regulr lnguge) hs the form L = 1 i k U iv i k with k ω nd U i, V i REG.

8 ω-regulr lnguge Let ω denote the set of non-negtive integers, i.e. ω = {0, 1, 2,...}. And let REG be the clss of regulr lnguges. Definition (ω-regulr lnguge) The ω-kleene closure of the clss of regulr lnguge (short for ω-regulr lnguge) hs the form L = 1 i k U iv i k with k ω nd U i, V i REG. Exmple Let Σ = {, b}, then n finite trce ξ Σ ω cn be {} ω, ({}{, b}) ω, {}{b}{, b} ω nd etc.

9 ω-regulr lnguge Let ω denote the set of non-negtive integers, i.e. ω = {0, 1, 2,...}. And let REG be the clss of regulr lnguges. Definition (ω-regulr lnguge) The ω-kleene closure of the clss of regulr lnguge (short for ω-regulr lnguge) hs the form L = 1 i k U iv i k with k ω nd U i, V i REG n 2m' m'' 1m

10 ω-utomt Definition An ω-utomton A = (Q, Σ, δ, q I, Acc) where Q is the set of sttes; Σ is finite lphbet; δ is the trnsition funtion: δ : Q Σ 2 Q for nondeterministic model, nd δ : Q Σ Q for deterministic model. q I is the initil stte; Acc is the cceptnce component (to be explicted)

11 ω-utomt Definition An ω-utomton A = (Q, Σ, δ, q I, Acc) where Q is the set of sttes; Σ is finite lphbet; δ is the trnsition funtion: δ : Q Σ 2 Q for nondeterministic model, nd δ : Q Σ Q for deterministic model. q I is the initil stte; Acc is the cceptnce component (to be explicted) A run of A on n ω-word α = Σ ω is n infinite stte sequence ρ = ρ(0)ρ(1)... Q ω such tht: ρ(0) = q I ; ρ(i) δ(ρ(i 1), i ) for nondeterministic model, nd ρ(i) = δ(ρ(i 1), i ) for deterministic model.

12 Comments The ω-utomt is introduced to ccept the clss of ω-regulr lnguges.

13 Comments The ω-utomt is introduced to ccept the clss of ω-regulr lnguges. How cn the ω-utomt identify their ccepting infinite words? n 2m' m'' 1m

14 Comments The ω-utomt is introduced to ccept the clss of ω-regulr lnguges. How cn the ω-utomt identify their ccepting infinite words? n 2m' m'' 1m 1 By one stte in cycle. (Büchi) 31 32

15 Comments The ω-utomt is introduced to ccept the clss of ω-regulr lnguges. How cn the ω-utomt identify their ccepting infinite words? n 2m' m'' 32 1m 1 By one stte in cycle. (Büchi) By ll sttes in cycle. (Muller)

16 Comments The ω-utomt is introduced to ccept the clss of ω-regulr lnguges. How cn the ω-utomt identify their ccepting infinite words? n 2m' m'' 32 1m 1 By one stte in cycle. (Büchi) By ll sttes in cycle. (Muller) By some sttes in nd some not in cycle. (Rbin/Streett)

17 Nondeterministic Models (1) Büchi Acceptnce

18 Nondeterministic Models (1) Büchi Acceptnce Definition A is clled Büchi utomton if Acc = F nd F stisfies: 1. F Q, nd 2. A word α Σ ω is ccepted by A iff there is run ρ on α such tht Inf (ρ) F.

19 Nondeterministic Models (1) Büchi Acceptnce Definition A is clled Büchi utomton if Acc = F nd F stisfies: 1. F Q, nd 2. A word α Σ ω is ccepted by A iff there is run ρ on α such tht Inf (ρ) F. Muller Acceptnce

20 Nondeterministic Models (1) Büchi Acceptnce Definition A is clled Büchi utomton if Acc = F nd F stisfies: 1. F Q, nd 2. A word α Σ ω is ccepted by A iff there is run ρ on α such tht Inf (ρ) F. Muller Acceptnce Definition A is clled Muller utomton if Acc = F nd F stisfies: 1. F 2 Q, nd 2. A word α Σ ω is ccepted by A iff there is run ρ on α such tht Inf (ρ) F.

21 Nondeterministic Models (2) Rbin Acceptnce

22 Nondeterministic Models (2) Rbin Acceptnce Definition A is clled Rbin utomton if Acc = Ω nd Ω stisfies: 1. Ω = {(E 1, F 1 ),..., (E k, F k )} with E i, F i Q, nd 2. A word α Σ ω is ccepted by A iff there is run ρ on α such tht (Inf (ρ) E i = ) (Inf (ρ) F i ) for some (E i, F i ).

23 Nondeterministic Models (2) Rbin Acceptnce Definition A is clled Rbin utomton if Acc = Ω nd Ω stisfies: 1. Ω = {(E 1, F 1 ),..., (E k, F k )} with E i, F i Q, nd 2. A word α Σ ω is ccepted by A iff there is run ρ on α such tht (Inf (ρ) E i = ) (Inf (ρ) F i ) for some (E i, F i ). Streett Acceptnce

24 Nondeterministic Models (2) Rbin Acceptnce Definition A is clled Rbin utomton if Acc = Ω nd Ω stisfies: 1. Ω = {(E 1, F 1 ),..., (E k, F k )} with E i, F i Q, nd 2. A word α Σ ω is ccepted by A iff there is run ρ on α such tht (Inf (ρ) E i = ) (Inf (ρ) F i ) for some (E i, F i ). Streett Acceptnce Definition A is clled Streett utomton if Acc = Ω nd Ω stisfies: 1. Ω = {(E 1, F 1 ),..., (E k, F k )} with E i, F i Q, nd 2. A word α Σ ω is ccepted by A iff there is run ρ on α such tht (Inf (ρ) E i ) (Inf (ρ) F i = ) for every (E i, F i ). (or if Inf (ρ) F i then Inf (ρ) E i )

25 Nondeterministic Models (3) Prity Acceptnce Consider specil cse for Rbin ccepting pirs: E 1 F 1 E 2 F 2... E n F n.

26 Nondeterministic Models (3) Prity Acceptnce Definition A is clled Prity utomton if Acc = c nd c stisfies: 1. c : Q {1,..., k} (k ω), nd 2. A word α Σ ω is ccepted by A iff there is run ρ on α such tht min{c(q) q Inf (ρ)} is even.

27 Nondeterministic Models (4) Trnsformtion Büchi Rbin Prity Muller Streett

28 Nondeterministic Models (4) Trnsformtion How to trnslte Büchi utomt to Muller utomt? Theorem Let A = (Q, Σ, δ, q I, F ) be Buchi utomton. Define the Muller utomton A = (Q, Σ, δ, q I, F) with F = {G 2 Q G F }. Then L(A) = L(A ).

29 Nondeterministic Models (4) Trnsformtion How to trnslte Büchi utomt to Muller utomt? Theorem Let A = (Q, Σ, δ, q I, F ) be Buchi utomton. Define the Muller utomton A = (Q, Σ, δ, q I, F) with F = {G 2 Q G F }. Then L(A) = L(A ). Proof. (Sketch). ( ) Consider ξ = ω 0 ω 1... be n ccepting run of A. It will run ccross set of S Q infinitely such tht S F. So S F, i.e. ξ cn be ccepted by A ; ( ) If ξ is ccepted by A, then there is S F such tht ξ will run ccross S infinitely. Since S F, thus it cn be lso be ccepted by A.

30 Nondeterministic Models (4) Trnsformtion How to trnslte Rbin/Streett utomt to Muller utomt? Theorem Let A = (Q, Σ, δ, q I, Ω) be Rbin utomton, respectively Streett utomton. Define the Muller utomton A = (Q, Σ, δ, q I, F) with F = {G 2 Q (E, F ) Ω G E = G F }, respectively with F = {G 2 Q (E, F ) Ω G E G F = }. Then L(A) = L(A ).

31 Nondeterministic Models (4) Trnsformtion How to trnslte Prity utomt to Rbin utomt, nd vice vers? Theorem Let A = (Q, Σ, δ, q I, c) be Prity utomton with c : Q {0,..., k}. Define the Rbin utomton A = (Q, Σ, δ, q I, Ω) with Ω = {(E 0, F 0 ),..., (E r, F r )} with r = k/2, E i = {q Q c(q) < 2i} nd F i = {q Q c(q) 2i}. Then L(A) = L(A ).

32 Nondeterministic Models (4) Trnsformtion How to trnslte Muller utomt to Büchi utomt? Theorem Let A = (Q, Σ, δ, q I, F) be Muller utomton. Define A = (Q, Σ, δ, q I, F ) be the Büchi utomton with: Q = Q G F (G 2G ); δ (q 1, ) = 1. δ(q 1, ) if q 1 Q nd every q 2 δ(q 1, ) is not in F; 2. δ(q 1, ) {(q 2, )} if q 1 Q nd there exists q 2 δ(q 1, ) nd q 2 F; 3. {(q 2, S {q 2})} if q 1 = (q1, S) nd q1 G q2 G for some G F 4. if q 1 = (q1, S) nd there is no G F such tht q1 G s well s every q 2 δ(q 1, ) is lso in G. F = {(q G, )}; Then L(A) = L(A ).

33 Nondeterministic Models (4) Trnsformtion How to trnslte Muller utomt to Rbin/Streett utomt? How to trnslte Rbin utomt to Streett utomt, nd vice vers?

34 Exmple 1 strt s 0 s 1 ξ = ( ) ω

35 Exmple 1 strt s 0 Büchi: F = {s 1 }; s 1

36 Exmple 1 strt s 0 s 1 Büchi: F = {s 1 }; Muller: F = {{s 1 }};

37 Exmple 1 strt s 0 s 1 Büchi: F = {s 1 }; Muller: F = {{s 1 }}; Rbin: Ω = {(, {s 1 })};

38 Exmple 1 strt s 0 s 1 Büchi: F = {s 1 }; Muller: F = {{s 1 }}; Rbin: Ω = {(, {s 1 })}; Streett: Ω = {({s 0 }, )}??

39 Exmple 1 strt s 0 s 1 Büchi: F = {s 1 }; Muller: F = {{s 1 }}; Rbin: Ω = {(, {s 1 })}; Streett: Ω = {({s 0 }, )}?? Prity: c(s 0 ) = 1; c(s 1 ) = 2??

40 Exmple (cont ) strt s 0 s 1 ξ = ( ) ω

41 Exmple (cont ) strt s 0 s 1 Büchi: F = {s 1 }?? ;

42 Exmple (cont ) strt s 0 s 1 Büchi: F = {s 1 }?? ; Muller: F = {{s 1 }}?? ;

43 Exmple (cont ) strt s 0 s 1 Büchi: F = {s 1 }?? ; Muller: F = {{s 1 }}?? ; Rbin: Ω = {(, {s 1 })}??;

44 Exmple (cont ) strt s 0 s 1 Büchi: F = {s 1 }?? ; Muller: F = {{s 1 }}?? ; Rbin: Ω = {(, {s 1 })}??; Streett: Ω = {({s 0 }, )}??

45 Exmple (cont ) strt s 0 s 1 Büchi: F = {s 1 }?? ; Muller: F = {{s 1 }}?? ; Rbin: Ω = {(, {s 1 })}??; Streett: Ω = {({s 0 }, )}?? Prity: c(s 0 ) = 1; c(s 1 ) = 2??

46 Deterministic Models (1) Büchi Condition becomes weker 1 strt s 0 s 1

47 Deterministic Models (1) Büchi Condition becomes weker 1 strt s 0 strt s 0 s 1 s 0, s 1

48 Deterministic Models (1) Büchi Condition becomes weker 1 strt s 0 strt s 0 s 1 s 0, s 1 Why subset construction not work here?

49 Deterministic Models (2) Trnsformtion Büchi Rbin Prity Muller Streett

50 Two Lower Bounds 1. The lower bound for trnsformtion from nondeterministic B chi utomt to deterministic Rbin utomt is 2 O(nlogn) ; (Sfr Construction)

51 Two Lower Bounds 1. The lower bound for trnsformtion from nondeterministic B chi utomt to deterministic Rbin utomt is 2 O(nlogn) ; (Sfr Construction) 2. The lower bound for trnsformtion from deterministic Streett utomt to deterministic Rbin utomt is 2 O(nlogn) ;

52 Conclusion All the nondeterministic ω utomt ccept the sme clss of ω regulr lnguges. Deterministic Büchi utomt is weker thn other deterministic ω utomt. The bound from nondeterministic utomt to deterministic ones is 2 nlogn.

53 Discussion Questions?

54 Synthesis Preliminry Definition (Synthesis) Given Specifiction S, is there progrm (system) P such tht P = S? If so then we cll S is relizble.

55 Synthesis Preliminry Definition (Synthesis) Given Specifiction S, is there progrm (system) P such tht P = S? If so then we cll S is relizble. We focus here only on the specifiction written by LTL, then wht is the frmework?

56 Frmework LTL Tbleu (Exp) NBW

57 Frmework LTL Tbleu (Exp) NBW NBW Sfr (Exp) DRW

58 Frmework LTL Tbleu (Exp) NBW NBW Sfr (Exp) DRW DRW Exp on pirs DRT

59 Frmework LTL Tbleu (Exp) NBW NBW Sfr (Exp) DRW DRW Exp on pirs DRT NBW nonempty? REA

60 Exmple FGp 1 strt s 0 p s 1 p

61 Exmple FGp 1 p strt s 0 strt s 0 p p p s 1 p s 1 p

62 Exmple GFp 1 strt s 0 p 1 s 1 p

63 Exmple GFp 1 p strt s 0 strt s 0 p 1 p p s 1 p s 1 p

64 Frmework LTL Tbleu (Exp) NBW NBW Sfr (Exp) DRW DRW Exp on pirs DRT NBW nonempty? REA

65 Discussion Questions?

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