Introduction to ω-autamata
|
|
- Ernest Lang
- 5 years ago
- Views:
Transcription
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?
Probabilistic Model Checking Michaelmas Term Dr. Dave Parker. Department of Computer Science University of Oxford
Probbilistic Model Checking Michelms Term 2011 Dr. Dve Prker Deprtment of Computer Science University of Oxford Long-run properties Lst lecture: regulr sfety properties e.g. messge filure never occurs
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 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 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 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 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 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 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 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 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 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 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 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 informationLecture 9: LTL and Büchi Automata
Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationFormal Methods in Software Engineering
Forml Methods in Softwre Engineering Lecture 09 orgniztionl issues Prof. Dr. Joel Greenyer Decemer 9, 2014 Written Exm The written exm will tke plce on Mrch 4 th, 2015 The exm will tke 60 minutes nd strt
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 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 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 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 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 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 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 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 informationStreamed Validation of XML Documents
Preliminries DTD Document Type Definition References Jnury 29, 2009 Preliminries DTD Document Type Definition References Structure Preliminries Unrnked Trees Recognizble Lnguges DTD Document Type Definition
More informationAutomata, Games, and Verification
Automt, Gmes, nd Verifiction Prof. Bernd Finkbeiner, Ph.D. Srlnd University Summer Term 2015 Lecture Notes by Bernd Finkbeiner, Felix Klein, Tobis Slzmnn These lecture notes re working document nd my contin
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 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 informationGood-for-Games Automata versus Deterministic Automata.
Good-for-Gmes Automt versus Deterministic Automt. Denis Kuperberg 1,2 Mich l Skrzypczk 1 1 University of Wrsw 2 IRIT/ONERA (Toulouse) Séminire MoVe 12/02/2015 LIF, Luminy Introduction Deterministic utomt
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 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 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 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 informationCMSC 330: Organization of Programming Languages. DFAs, and NFAs, and Regexps (Oh my!)
CMSC 330: Orgniztion of Progrmming Lnguges DFAs, nd NFAs, nd Regexps (Oh my!) CMSC330 Spring 2018 Types of Finite Automt Deterministic Finite Automt (DFA) Exctly one sequence of steps for ech string All
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 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 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 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 informationSection: Other Models of Turing Machines. Definition: Two automata are equivalent if they accept the same language.
Section: Other Models of Turing Mchines Definition: Two utomt re equivlent if they ccept the sme lnguge. Turing Mchines with Sty Option Modify δ, Theorem Clss of stndrd TM s is equivlent to clss of TM
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 information80 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES. 2.6 Finite State Automata With Output: Transducers
80 CHAPTER 2. DFA S, NFA S, REGULAR LANGUAGES 2.6 Finite Stte Automt With Output: Trnsducers So fr, we hve only considered utomt tht recognize lnguges, i.e., utomt tht do not produce ny output on ny input
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 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 informationCS 267: Automated Verification. Lecture 8: Automata Theoretic Model Checking. Instructor: Tevfik Bultan
CS 267: Automted Verifiction Lecture 8: Automt Theoretic Model Checking Instructor: Tevfik Bultn LTL Properties Büchi utomt [Vrdi nd Wolper LICS 86] Büchi utomt: Finite stte utomt tht ccept infinite strings
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 informationSoftware Engineering using Formal Methods
Softwre Engineering using Forml Methods Propositionl nd (Liner) Temporl Logic Wolfgng Ahrendt 13th Septemer 2016 SEFM: Liner Temporl Logic /GU 160913 1 / 60 Recpitultion: FormlistionFormlistion: Syntx,
More informationOn Determinisation of History-Deterministic Automata.
On Deterministion of History-Deterministic Automt. Denis Kupererg Mich l Skrzypczk University of Wrsw YR-ICALP 2014 Copenhgen Introduction Deterministic utomt re centrl tool in utomt theory: Polynomil
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 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 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 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 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 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 informationLearning Moore Machines from Input-Output Traces
Lerning Moore Mchines from Input-Output Trces Georgios Gintmidis 1 nd Stvros Tripkis 1,2 1 Alto University, Finlnd 2 UC Berkeley, USA Motivtion: lerning models from blck boxes Inputs? Lerner Forml Model
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 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 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 informationCSC 473 Automata, Grammars & Languages 11/9/10
CSC 473 utomt, Grmmrs & Lnguges 11/9/10 utomt, Grmmrs nd Lnguges Discourse 06 Decidbility nd Undecidbility Decidble Problems for Regulr Lnguges Theorem 4.1: (embership/cceptnce Prob. for DFs) = {, w is
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 informationCDM Automata on Infinite Words
CDM Automt on Infinite Words 1 Infinite Words Klus Sutner Crnegie Mellon Universlity 60-omeg 2017/12/15 23:19 Deterministic Lnguges Muller nd Rin Automt Towrds Infinity 3 Infinite Words 4 As mtter of principle,
More informationGrammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages
5//6 Grmmr Automt nd Lnguges Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive
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 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 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 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 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 informationCS 311 Homework 3 due 16:30, Thursday, 14 th October 2010
CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w
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 informationScanner. Specifying patterns. Specifying patterns. Operations on languages. A scanner must recognize the units of syntax Some parts are easy:
Scnner Specifying ptterns source code tokens scnner prser IR A scnner must recognize the units of syntx Some prts re esy: errors mps chrcters into tokens the sic unit of syntx x = x + y; ecomes
More informationNondeterministic Biautomata and Their Descriptional Complexity
Nondeterministic Biutomt nd Their Descriptionl Complexity Mrkus Holzer nd Sestin Jkoi Institut für Informtik Justus-Lieig-Universität Arndtstr. 2, 35392 Gießen, Germny 23. Theorietg Automten und Formle
More information1 Structural induction, finite automata, regular expressions
Discrete Structures Prelim 2 smple uestions s CS2800 Questions selected for spring 2017 1 Structurl induction, finite utomt, regulr expressions 1. We define set S of functions from Z to Z inductively s
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 informationNondeterministic Automata vs Deterministic Automata
Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n
More information4 Deterministic Büchi Automata
Bernd Finkeiner Dte: April 26, 2011 Automt, Gmes nd Verifiction: Lecture 3 4 Deterministic Büchi Automt Theorem 1 The lnguge ( + ) ω is not recognizle y deterministic Büchi utomton. Assume tht L is recognized
More informationKleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem 2/16/15
Models of Comput:on Lecture #8 Chpter 7 con:nued Any lnguge tht e defined y regulr expression, finite utomton, or trnsi:on grph cn e defined y ll three methods We prove this y showing tht ny lnguge defined
More informationinput tape head moves current state
CPS 140 - Mthemticl Foundtions of CS Dr. Susn Rodger Section: Finite Automt (Ch. 2) (lecture notes) Things to do in clss tody (Jn. 13, 2004): ffl questions on homework 1 ffl finish chpter 1 ffl Red Chpter
More informationRegular languages refresher
Regulr lnguges refresher 1 Regulr lnguges refresher Forml lnguges Alphet = finite set of letters Word = sequene of letter Lnguge = set of words Regulr lnguges defined equivlently y Regulr expressions Finite-stte
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 informationLanguages & Automata
Lnguges & Automt Dr. Lim Nughton Lnguges A lnguge is sed on n lphet which is finite set of smols such s {, } or {, } or {,..., z}. If Σ is n lphet, string over Σ is finite sequence of letters from Σ, (strings
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 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 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 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 informationTypes 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 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 information