Lecture 9: LTL and Büchi Automata

Size: px
Start display at page:

Download "Lecture 9: LTL and Büchi Automata"

Transcription

1 Lecture 9: LTL nd Büchi Automt 1

2 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 the scope of property. Typicl scopes re: Glol: The property should hold on the whole pth (i.e. on ll suffixes of run). Before R: The property should hold efore the first ppernce of R (i.e. on ll suffixes efore the first suffix stisfying R). After Q: The property should hold fter the first ppernce of Q. 2

3 Between Q nd R: The property should hold in ll sequences in which the first suffix stisfies Q nd the lst one stisfies R. After Q until R: As efore, ut lso includes the sequences in which Q ppers, ut is never followed y n R. Note: In the following ptterns, scopes re interpreted in wy tht lwys includes the suffix t which the event triggering the scope hppens, ut excludes the suffix t which the event ending the scope hppens. 3

4 Scopes Glol Before R R R After Q Q Q Between Q nd R Q Q R Q R Q After Q until R Q Q R Q 4

5 LTL Property Ptterns: Asence Asence ptterns specify tht P is flse within the scope: Glol Before R After Q Between Q nd R After Q until R ( ( ( ( G P (F R) ( P U R) G(Q G P) G((Q R F R) ( P U R)) ( G((Q R) ( P W R)) 5

6 LTL Property Ptterns: Existence Existence ptterns specify tht P ecomes true within the scope: Glol Before R After Q Between Q nd R After Q until R ( ( ( ( F P R W (P R) (G Q) (F(Q F P)) G((Q R) ( R W (P R))) ( G((Q R) ( R U (P R))) 6

7 LTL Property Ptterns The property ptterns re useful to mke specifying esier: If desired property flls into one of those ptterns, the pttern cn e instntited for pproprite P, Q, nd R. The ptterns (nd the definition of scopes) from the previous slides were tken from: where you cn find mny more of them (lso for other temporl logics). There re lso other ptterns ville expressing: Universlity: P is true (dul of sence) Precedence: S precedes P Response: S responds to P Etc., etc. 7

8 Reltions etween Temporl nd Logicl Opertors X(φ 1 φ 2 ) X φ 1 X φ 2 X(φ 1 φ 2 ) X φ 1 X φ 2 X φ X φ F(φ 1 φ 2 ) F φ 1 F φ 2 F φ G φ G(φ 1 φ 2 ) G φ 1 G φ 2 G φ F φ (φ 1 φ 2 ) U ψ (φ 1 U ψ) (φ 2 U ψ) φ U (ψ 1 ψ 2 ) (φ U ψ 1 ) (φ U ψ 2 ) 8

9 Idempotence nd Recursion Lws F φ F F φ G φ G G φ φ U ψ φ U (φ U ψ) F φ φ X F φ G φ φ X G φ φ U ψ ψ (φ X(φ U ψ)) φ W ψ ψ (φ X(φ W ψ)) φ R ψ (φ ψ) (ψ X(φ R ψ)) 9

10 LTL Model Checking We now turn to the question how to check whether given Kripke structure stisfies given formul. In the context of temporl logics, this is clled model checking (i.e. checking whether ll runs re models of the given formul). Like in the cse of sfety properties, we follow n utomt-theoretic pproch: 1. We introduce new clss of utomt tht cn express LTL properties. 2. We show how to trnslte properties into these utomt. 3. We check whether the intersection of the system nd the utomton for the negtion of the property is empty. Literture: Clrke, Grumerg, Peled: Model Checking, MIT Press,

11 Büchi Automt: Definition A Büchi utomton is tuple such tht B = Σ, S, S 0,, F Σ S S 0 Q S Σ S F S is finite lphet, is finite set of sttes, re the initil sttes, is the trnsition reltion, nd re the ccepting sttes. So fr, Büchi utomt look exctly like finite utomt. However, they operte on infinite words, nd they hve different cceptnce condition (see next slide). 11

12 Büchi Automt: Acceptnce nd Lnguge Let B = Σ, S, S 0,, F e Büchi utomton. A run of B on n infinite word σ Σ ω is n infinite sequence of sttes ρ S ω such tht ρ(0) S 0, nd (ρ(i), σ(i), ρ(i + 1)) for ll i 0. We cll run ρ ccepting iff for infinitely mny indices i it holds tht ρ(i) F (i.e. ρ infinitely often visits ccepting sttes). A word σ Σ ω is ccepted y B iff there is n ccepting run on σ in B. The lnguge of B, denoted L(B) Σ ω is defined s the set of infinite words over Σ ccepted y B. 12

13 Büchi Automt: Exmples infinitely often q0 q1, infinitely often q0 q1 13

14 Opertions on Büchi Automt Like finite utomt, the lnguges ccepted y Büchi utomt re closed under oolen opertions. We will exmine the following opertions: Intersection of Büchi utomt B 1 nd B 2 : construct B with L(B) = L(B 1 ) L(B 2 ) Union of Büchi utomt B 1 nd B 2 : construct B with L(B) = L(B 1 ) L(B 2 ) Complementtion of Büchi utomton B 1 : construct B with L(B) = Σ \ L(B 1 ) Emptiness check: given B, check if L(B) = 14

15 Intersection of Büchi utomt The construction of the intersection utomton works little differently from the finite-stte cse. We need to check whether oth sets of ccepting sttes occur infinitely often. Ide: We crete two copies of the intersected stte spce. In the first copy, we check for occurrence of the first cceptnce set. In the second copy, we check for occurrence of the second cceptnce set. We jump ck nd forth etween the copies whenever we find n ccepting stte. Let B 1 = Σ, S 1, S 0 1, 1, F 1, B 2 = Σ, S 2, S 0 2, 2, F 2. We define the intersection utomton (or: product utomton) to e B = Σ, S, S 0,, F, where S = S 1 S 2 {1, 2}, S 0 = S 0 1 S0 2 {1}, F = F 1 S 2 {1} s defined on the next slide 15

16 ( s, t, 1,, s, t, 1 ) iff (s,, s ) 1, (t,, t ) 2, s / F 1 ( s, t, 1,, s, t, 2 ) iff (s,, s ) 1, (t,, t ) 2, s F 1 ( s, t, 2,, s, t, 2 ) iff (s,, s ) 1, (t,, t ) 2, t / F 2 ( s, t, 2,, s, t, 1 ) iff (s,, s ) 1, (t,, t ) 2, t F 2 16

17 Intersection: Exmple s0 s1 t0 t1 B1 B2 s0,t0,1 B1 x B2 s1,t1,2 s0,t0,2 s1,t1,1 17

18 Union nd Complement Union: Juxtpose oth utomt (like in the finite cse) Complement: Complicted! The complement construction, when pplied to Büchi utomton with n sttes, results in n utomton with O(n!) sttes. We will skip it in this course. Detils on Complementtion: see e.g. W. Thoms, Automt on Infinite Ojects, Chpter 4 in Hndook of Theoreticl Computer Science, or I. Wlukiewicz, Lecture notes on Automt nd Logic, Chpter 3, igw/ppers/igw-eefss01.ps 18

19 Emptiness check Oservtion: L(B) iff there is n ccepting stte s F tht is rechle from n initil stte nd rechle from itself (the ltter with non-empty pth). s s s... This condition cn e checked with n lgorithm tht tkes liner time in the size of the Büchi utomton. (We will see how lter.) 19

20 Note on determinism In the cse of finite utomt, we could convert ech non-deterministic utomton into lnguge-equivlent deterministic utomton. This is not the cse with Büchi utomt. In other words, non-deterministic Büchi utomt re strictly more expressive thn deterministic Büchi utomt., Eventully, only will occur. s0 s1 Not expressile y deterministic Büchi utomton! 20

21 Generlised Büchi Automt A vrint of the model re so-clled generlised Büchi utomt. They differ from (norml) Büchi utomt only in the cceptnce condition, which is set of cceptnce sets, i.e. F 2 S In generlised Büchi utomton, run ρ is ccepting iff, for F = {F 1,..., F n } nd ech index 1 i n, we hve tht ρ visits infinitely mny sttes from F i. Generlised Büchi utomt cn e trnslted ck into Büchi utomt. (Tke the n-fold intersection of the utomton with itself, with cceptnce sets F 1 through F n.) 21

Formal Methods in Software Engineering

Formal 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 information

CS 267: Automated Verification. Lecture 8: Automata Theoretic Model Checking. Instructor: Tevfik Bultan

CS 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 information

Software Engineering using Formal Methods

Software 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 information

Automata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ω-automata. Ralf Huuck.

Automata 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 information

Probabilistic Model Checking Michaelmas Term Dr. Dave Parker. Department of Computer Science University of Oxford

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 information

Lecture 08: Feb. 08, 2019

Lecture 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 information

Chapter 2 Finite Automata

Chapter 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 information

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1

Chapter 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 information

Minimal DFA. minimal DFA for L starting from any other

Minimal 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 information

CSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science

CSCI 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 information

Finite-State Automata: Recap

Finite-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 information

Regular expressions, Finite Automata, transition graphs are all the same!!

Regular 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 information

4 Deterministic Büchi Automata

4 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 information

Theory of Computation Regular Languages. (NTU EE) Regular Languages Fall / 38

Theory 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 information

1 Nondeterministic Finite Automata

1 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 information

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 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 information

Anatomy of a Deterministic Finite Automaton. Deterministic Finite Automata. A machine so simple that you can understand it in less than one minute

Anatomy 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 information

Convert the NFA into DFA

Convert 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 information

CMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014

CMPSCI 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 information

Designing finite automata II

Designing 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 information

Finite Automata-cont d

Finite 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 information

CHAPTER 1 Regular Languages. Contents

CHAPTER 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 information

Speech Recognition Lecture 2: Finite Automata and Finite-State Transducers

Speech 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 information

More on automata. Michael George. March 24 April 7, 2014

More 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 information

Design and Analysis of Distributed Interacting Systems

Design and Analysis of Distributed Interacting Systems Design nd Anlysis of Distriuted Intercting Systems Lecture 6 LTL Model Checking Prof. Dr. Joel Greenyer My 16, 2013 Some Book References (1) C. Bier, J.-P. Ktoen: Principles of Model Checking. The MIT

More information

Formal Languages and Automata

Formal 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 information

Introduction to ω-autamata

Introduction to ω-autamata Fridy 25 th Jnury, 2013 Outline From finite word utomt ω-regulr lnguge ω-utomt Nondeterministic Models Deterministic Models Two Lower Bounds Conclusion Discussion Synthesis Preliminry From finite word

More information

Non-deterministic Finite Automata

Non-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 information

Coalgebra, Lecture 15: Equations for Deterministic Automata

Coalgebra, Lecture 15: Equations for Deterministic Automata Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined

More information

Theory of Computation Regular Languages

Theory 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 information

CHAPTER 1 Regular Languages. Contents. definitions, examples, designing, regular operations. Non-deterministic Finite Automata (NFA)

CHAPTER 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 information

Deterministic Finite Automata

Deterministic 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 information

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4

Intermediate 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 information

AUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton

AUTOMATA 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 information

Converting Regular Expressions to Discrete Finite Automata: A Tutorial

Converting 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 information

Homework 3 Solutions

Homework 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 information

NFAs continued, Closure Properties of Regular Languages

NFAs 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 information

Nondeterminism and Nodeterministic Automata

Nondeterminism 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 information

Java II Finite Automata I

Java 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 information

Speech Recognition Lecture 2: Finite Automata and Finite-State Transducers. Mehryar Mohri Courant Institute and Google Research

Speech 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 information

Non-deterministic Finite Automata

Non-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 information

Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages

Grammar. 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 information

Finite Automata. Informatics 2A: Lecture 3. Mary Cryan. 21 September School of Informatics University of Edinburgh

Finite 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 information

Formal Language and Automata Theory (CS21004)

Formal 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 information

Non-Deterministic Finite Automata. Fall 2018 Costas Busch - RPI 1

Non-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 information

NFAs and Regular Expressions. NFA-ε, continued. Recall. Last class: Today: Fun:

NFAs 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 information

Tutorial Automata and formal Languages

Tutorial Automata and formal Languages Tutoril Automt nd forml Lnguges Notes for to the tutoril in the summer term 2017 Sestin Küpper, Christine Mik 8. August 2017 1 Introduction: Nottions nd sic Definitions At the eginning of the tutoril we

More information

Fundamentals of Computer Science

Fundamentals 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 information

CMSC 330: Organization of Programming Languages

CMSC 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 information

On Determinisation of History-Deterministic Automata.

On 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 information

5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9.

5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9. Regulr Expressions, Pumping Lemm, Right Liner Grmmrs Ling 106 Mrch 25, 2002 1 Regulr Expressions A regulr expression descries or genertes lnguge: it is kind of shorthnd for listing the memers of lnguge.

More information

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. Comparing DFAs and NFAs (cont.) Finite Automata 2

Types 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

Good-for-Games Automata versus Deterministic Automata.

Good-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 information

Finite Automata. Informatics 2A: Lecture 3. John Longley. 22 September School of Informatics University of Edinburgh

Finite 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 information

From LTL to Symbolically Represented Deterministic Automata

From LTL to Symbolically Represented Deterministic Automata Motivtion nd Prolem Setting Determinizing Non-Confluent Automt Det. vi Automt Hierrchy From LTL to Symoliclly Represented Deterministic Automt Andres Morgenstern Klus Schneider Sven Lmerti Mnuel Gesell

More information

Non-Deterministic Finite Automata

Non-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 information

NFAs continued, Closure Properties of Regular Languages

NFAs 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 information

CS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata

CS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata CS103B ndout 18 Winter 2007 Ferury 28, 2007 Finite Automt Initil text y Mggie Johnson. Introduction Severl childrens gmes fit the following description: Pieces re set up on plying ord; dice re thrown or

More information

Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene-*

Regular 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 information

CS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University

CS415 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 information

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.

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 information

LTL Translation Improvements in Spot

LTL Translation Improvements in Spot LTL Trnsltion Improvements in Spot Alexndre Duret-Lutz http://www.lrde.epit.fr/~dl/ VECoS'11 16 September 2011 Alexndre Duret-Lutz LTL Trnsltion Improvements 1 / 19 Context High-level

More information

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.

1. 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 information

CS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018

CS 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 information

Closure Properties of Regular Languages

Closure Properties of Regular Languages Closure Properties of Regulr Lnguges Regulr lnguges re closed under mny set opertions. Let L 1 nd L 2 e regulr lnguges. (1) L 1 L 2 (the union) is regulr. (2) L 1 L 2 (the conctention) is regulr. (3) L

More information

Automata, Games, and Verification

Automata, 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 information

12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016

12.1 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 information

Non Deterministic Automata. Linz: Nondeterministic Finite Accepters, page 51

Non 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 information

State Minimization for DFAs

State 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 information

Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)

Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints) C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,

More information

NFA DFA Example 3 CMSC 330: Organization of Programming Languages. Equivalence of DFAs and NFAs. Equivalence of DFAs and NFAs (cont.

NFA 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 information

Assignment 1 Automata, Languages, and Computability. 1 Finite State Automata and Regular Languages

Assignment 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 information

CS 275 Automata and Formal Language Theory

CS 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 information

PART 2. REGULAR LANGUAGES, GRAMMARS AND AUTOMATA

PART 2. REGULAR LANGUAGES, GRAMMARS AND AUTOMATA PART 2. REGULAR LANGUAGES, GRAMMARS AND AUTOMATA RIGHT LINEAR LANGUAGES. Right Liner Grmmr: Rules of the form: A α B, A α A,B V N, α V T + Left Liner Grmmr: Rules of the form: A Bα, A α A,B V N, α V T

More information

Bisimulation. R.J. van Glabbeek

Bisimulation. R.J. van Glabbeek Bisimultion R.J. vn Glbbeek NICTA, Sydney, Austrli. School of Computer Science nd Engineering, The University of New South Wles, Sydney, Austrli. Computer Science Deprtment, Stnford University, CA 94305-9045,

More information

Myhill-Nerode Theorem

Myhill-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 information

CS 275 Automata and Formal Language Theory

CS 275 Automata and Formal Language Theory CS 275 utomt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Prolem (II) Chpter II.5.: Properties of Context Free Grmmrs (14) nton Setzer (Bsed on ook drft y J. V. Tucker nd K. Stephenson)

More information

1.4 Nonregular Languages

1.4 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 information

CDM Automata on Infinite Words

CDM 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 information

80 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 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 information

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 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 information

Lexical Analysis Finite Automate

Lexical 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 information

Topological, Automata-Theoretic and Logical Characterization of Finitary Languages

Topological, Automata-Theoretic and Logical Characterization of Finitary Languages Topologicl, Automt-Theoretic nd Logicl Chrcteriztion of Finitry Lnguges Krishnendu Chterjee nd Nthnël Fijlkow IST Austri (Institute of Science nd Technology Austri) Am Cmpus 1 A-3400 Klosterneuurg Technicl

More information

Lecture 3: Equivalence Relations

Lecture 3: Equivalence Relations Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts

More information

a,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

a,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 information

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018

Finite 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 information

Chapter 1, Part 1. Regular Languages. CSC527, Chapter 1, Part 1 c 2012 Mitsunori Ogihara 1

Chapter 1, Part 1. Regular Languages. CSC527, Chapter 1, Part 1 c 2012 Mitsunori Ogihara 1 Chpter 1, Prt 1 Regulr Lnguges CSC527, Chpter 1, Prt 1 c 2012 Mitsunori Ogihr 1 Finite Automt A finite utomton is system for processing ny finite sequence of symols, where the symols re chosen from finite

More information

Non Deterministic Automata. Formal Languages and Automata - Yonsei CS 1

Non 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 information

12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2014

12.1 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 information

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations. Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

More information

Lecture 09: Myhill-Nerode Theorem

Lecture 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 information

Scanner. Specifying patterns. Specifying patterns. Operations on languages. A scanner must recognize the units of syntax Some parts are easy:

Scanner. 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 information

FABER Formal Languages, Automata and Models of Computation

FABER 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 information

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

Homework 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 information

Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem. Kleene s Theorem 2/16/15

Kleene 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 information

The size of subsequence automaton

The size of subsequence automaton Theoreticl Computer Science 4 (005) 79 84 www.elsevier.com/locte/tcs Note The size of susequence utomton Zdeněk Troníček,, Ayumi Shinohr,c Deprtment of Computer Science nd Engineering, FEE CTU in Prgue,

More information

3 Regular expressions

3 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 information

STRUCTURE OF CONCURRENCY Ryszard Janicki. Department of Computing and Software McMaster University Hamilton, ON, L8S 4K1 Canada

STRUCTURE OF CONCURRENCY Ryszard Janicki. Department of Computing and Software McMaster University Hamilton, ON, L8S 4K1 Canada STRUCTURE OF CONCURRENCY Ryszrd Jnicki Deprtment of Computing nd Softwre McMster University Hmilton, ON, L8S 4K1 Cnd jnicki@mcmster.c 1 Introduction Wht is concurrency? How it cn e modelled? Wht re the

More information

CISC 4090 Theory of Computation

CISC 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 information

11.1 Finite Automata. CS125 Lecture 11 Fall Motivation: TMs without a tape: maybe we can at least fully understand such a simple model?

11.1 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 information