Hybrid Systems Modeling, Analysis and Control
|
|
- Brent Hampton
- 5 years ago
- Views:
Transcription
1 Hyrid Systems Modeling, Anlysis nd Control Rdu Grosu Vienn University of Tehnology Leture 5
2 Finite Automt s Liner Systems Oservility, Rehility nd More
3 Miniml DFA re Not Miniml NFA (Arnold, Diky nd Nivt s Exmple) x 2 x 4 x 2 L = (* + *)
4 Miniml NFA: How re they Relted? (Arnold, Diky nd Nivt s Exmple) x 2 x 4 x 5 x 2 x 4 x 5 L = No homomorphism of either utomton onto the other.
5 Miniml NFA: How re they Relted? (Arnold, Diky nd Nivt s Exmple) x 2 x 4 x 8 x 5 x 6 x 7 x 8 Crrez s solution: Tke oth in terminl NFA. Is this the est one n do? No! One n use use liner (similrity) trnsformtions.
6 Convergene of Theories Hyrid Systems Computtion nd Control: - onvergene etween ontrol nd utomt theory. Hyrid Automt: n outome of this onvergene - modeling formlism for systems exhiiting oth disrete nd ontinuous ehvior, - suessfully used to model nd nlyze emedded nd iologil systems.
7 Lk of Common Foundtion for HA v V E x& = Ax+ Bu v = Cx v V R Mode dynmis: - Liner system (LS) voltge(mv) v V U Stimulted s / di = t v V / R s v t < V U = Mode swithing: - Finite utomton (FA) Different tehniques: - LS redution - FA minimiztion time(ms) LS & FA tught seprtely: No ommon foundtion!
8 Min Conjeture of this Tlk Finite utomt n e onveniently regrded s time invrint liner systems over semimodules: - liner systems tehniques generlize to utomt Exmples of suh tehniques inlude: - liner trnsformtions of utomt, - minimiztion nd determiniztion of utomt s oservility nd rehility redutions - Z-trnsform of utomt to ompute ssoited regulr expression through Gussin elimintion.
9 Finite Automt s Liner Systems Consider finite utomton M = (X, Σδ,, S,F) with: - finite set of sttes X, finite input lphet - trnsition reltion δ X Σ X, - strting nd finl sets of sttes S, F X Σ,
10 Finite Automt s Liner Systems Consider finite utomton M = (X, Σδ,,S,F) with: - finite set of sttes X, finite input lphet Σ, - trnsition reltion δ X Σ X, - strting nd finl sets of sttes S,F X For eh input letter Σ: - represent δ() X X s oolen mtrix A(), - write A = A() whe re (x) = Σ { if x = } otherwise
11 Finite Automt s Liner Systems Now define the liner system L M = [S,A,C]: x(n+) = x(n)a, x y(n) = S(ε)ε = x(n)c, C = F(ε)ε x nd y re row vetors
12 Finite Automt s Liner Systems Now define the liner system L M = [S,A,C]: x(n+) = x(n)a, x = S(ε)ε y(n) = x(n)c, C = F(ε)ε Exmple: onsider following utomton: L x x 2 A() = A() =, x (ε)' =, C(ε) =
13 Polynomils nd their Opertions A, C, x(n) nd y(n) re polynomils with: - powers: strings in Σ * (the input strings) - oeffiient s: mtries nd vetors over B
14 Polynomils nd their Opertions A, C, x(n) nd y(n) re polynomils - powers: strings in Σ (the input strings) - oeffiients: mtries nd vetors over B Addition nd multiplition: 2 (A() + A()) = * with: done over polynomils A()A() + A()A() + A()A() + A()A( ) = ˆ A() + A() + A() + A()
15 Boolen Semimodules B is douly idempotent, ommuttive semiring: - (B,+,) is ommuttive idempotent monoid (or), - (B,,) is ommuttive idempotent monoid (nd), - multiplition distriutes over ddition, - is n nnihiltor: =
16 Boolen Semimodules B is douly idempotent, ommuttive semiring: - (B,+,) is ommuttive idempotent monoid (or), - (B,,) is ommuttive idempotent monoid (nd), - multiplition distriutes over ddition, - is n nnihiltor: = n B is semimodule over slrs in B: - r(x+y) = rx + ry, (r+s)x = rx + sx, (rs)x = - x = x, x = r(sx),
17 Boolen Semimodules B is douly idempotent, ommuttive semiring: - (B,+,) is ommuttive idempotent monoid (or), - (B,,) is ommuttive idempotent monoid (nd), - multiplition distriutes over ddition, - is n nnihiltor: = n B is semimodule over slrs in B: - r(x+y) = rx + ry, (r+s)x = rx + sx, (rs)x = r(sx), - x = x, x = Note: No dditive nd multiplitive inver ses!
18 Divergene of Clssi/Disrete Mth Cnonil prtil order in semirings: + iff!. + = iff!. =
19 Divergene of Clssi/Disrete Mth Cnonil prtil order in semirings: + iff!. + = iff!. = Exmple of nonil PO for Nturl numers: + 5 iff! = 5
20 Divergene of Clssi/Disrete Mth Cnonil prtil order in semirings: + iff!. + = iff!. = Exmple: Cnonil PO for Nturl numers: + 5 iff! = 5 Exmple: Cnonil PO for Integer numers: iff!(-6). 5 + (-6) = Semiring: Either inverses or prtil order!
21 Oservility Let L = [S,A,C] e n n-stte utomton. It's output: [y() y()... y(n-)] = x [C AC... A n- C] = x O () L is oservle if x is uniquely determined y ().
22 Oservility Let L = [S,A,C] e n n-stte utomton. It's output: [ O t n- t [y() y()... y(n-)] = x C AC... A C] = x () L is oservle if x is uniquely determined y ( ). Exmple: the oservility mtrix O of L is: O = n AC 3 ε x x2 L x x 2
23 Liner Dependene Initil vetor x selets sum of rows from O. Hene: - if L is deterministi nd therefore hs single initil stte, x is uniquely determined if ll rows O in O re distint i
24 Liner Dependene Initil vetor x selets sum of rows from O. Hene: - if L is deterministi nd therefore hs single initil stte, x is uniquely determined if ll rows O in O re distint i - if L is nondeterministi nd hs severl initil sttes, x is not uniquely determined if t here re oolen nd: I,J [..n]. I J = O = i i I i J i O i i i, (2) i
25 Liner Dependene Initil vetor x selets sum of rows from O. Hene: - if L is deterministi nd therefore hs single initil stte, x is uniquely determined if ll rows O in O re distint i - if L is nondeterministi nd hs severl initil sttes, x is not uniquely determined if there re oolen nd: Line I,J [..n]. I J = O = O r dependene: i i I i J - Def (2) generlizes liner dependene in vetor spes i i i i, (2) (2) for finite I,J nd ny vetor set. i
26 Liner Dependene Initil vetor x selets sum of rows from O. Hene: - if L is deterministi nd therefore hs single initil stte, x is uniquely determined if ll rows O in O re distint i - if L is nondeterministi nd hs severl initil sttes, x is not uniquely determined if there re oolen nd: Line I,J [..n]. I J = O = O r dependene: i i I i J - Def (2) generlizes liner dependene in vetor - Liner independene is onsequently: I,J [..n]. I J = spn(o ) i I i i i, (2) (2) for finite I,J nd ny vetor set. spes spn(o J) = {} i
27 Bsis in Boolen Semimodule An ordered set of vetors Y is sis for X if: () Y is independent, () spn(y) = X
28 Bsis in Boolen Semimodule An ordered set of vetors Y is sis for X if: () Y is independent, () spn(y) = X n Theorem (Bsis) If X B hs sis Y then Y is unique.
29 Bsis in Boolen Semimodule An ordered set of vetors Y is sis for X if: () Y is independent, () spn(y) = X Theorem (Bsis) If X n B hs sis Y then Y is unique. O = n AC x ε 2 3 L x x 2 [x x2 x3 ]: row sis
30 Bsis in Boolen Semimodule An ordered set of vetors Y is sis for X if: () Y is independent, () spn(y) = X n Theorem (Bsis) If X B hs sis Y then Y is uniqu e. O = n AC ε x x2 3 L x x 2 [x x x ]: [C( ε) AC() AC()]: 2 3 row sis, olumn sis.
31 Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε)
32 Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5
33 Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5
34 Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5 Define liner trnsf x = x T: T
35 Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5 Define liner trnsf x = x T: T x (n + ) = x (n + )T = x (n)at = x (n)t - AT = x (n)a x (ε) = x (ε)t C(ε) = T - C(ε)
36 Oservility Redution y Rows L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5 Define liner trnsf x = x T: T A () A() x (ε) C(ε) A(x) = [A(x)T] T x (ε) = x (ε)t C(ε) = [C(ε)] T
37 Oservility Redution y Columns L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5 Define liner trnsf x = x T: T A () A() x (ε) C(ε) A(x) = [A(x)T] T x (ε) = x (ε)t C(ε) = [C(ε)] T
38 Mixed Oservility Redution L 2 x 2 x 4 x 5 A() A() x ( ε) C( ε) O ε x x2 x3 x 4 5 Define liner trnsf x = x T: T A () A() x (ε) C(ε) A(x) = [A(x)T] T x (ε) = x (ε)t C(ε) = [C(ε)] T
39 Originl nd Redued Automt L 2 x 2 L 2, x, 3 x 2 x 4 x 5 NFA L 22 y olumns DFA L 2 y rows x, x 2 NFA L 23 mixed x,, 3 x 2
40 Originl nd Redued Automt L 2 x 2 L 2 x 4 x 5 NFA L 22 y olumns,, x 2 DFA L 2 y rows x, x 2 NFA L 23 mixed x,, 3 x 2 Let x = x T in L 2 where t Then L 22 = [A 2, x 2,, C 2 ] ' 't ' L 23 = [A 2, x 2,, C 2 ] T = T ' =
41 Row Bsis ut No Column Bsis 7 L 3 x x 2 x 5 x 6 O ε x x2 x3 x4 x5 x6 x 7 x 4
42 Row Bsis ut No Column Bsis L 3 x x 5 x 2 x 6 O ε x 2 x 4 x 5 x 6 x 7 x 7 x 4
43 Row Bsis ut No Column Bsis L 3 x x 2 x 5 x 6 x 7 x 4 O ε x 2 x 4 x 5 x 6 x 7 O ε x x2 x3 x4 x5 x6 7
44 Row Bsis ut No Column Bsis L 3 x x 2 x 5 x 6 x 7 x 4 O ε x 2 x 4 x 5 x 6 x 7 O ε x x2 x3 x4 x5 x6 7
45 Oservilty Redution Theorem (Cover): Finding (possily mixed) sis T for O L is equivlent to finding miniml over for O L. - either s its set sis over or s its Krnugh over. Theorem (Complexity): Determining over T for O L is NP-omplete (set sis prolem omplexity). Theorem (Rnk): The row (= olumn) rnk of O L is the size of the set over T (size of Krnugh over).
46 Rehility: Dul of Oservility Let L = [S,A,C] e n n-stte utomton. It's output: [y() y()... y(n-)] t = C t [x A t x... (A t ) n- x ] = C t R t (3) where x is now olumn vetor. L is rehle if C is uniquely determined y (3).
47 Rehility: Dul of Oservility Let L = [S,A,C] e n n-stte utomton. It's output: ) R t t t t n- t t [y() y()... y(n-)] = C [x A x... (A x ] = C (3) L is rehle if C is uniquely determined y ( 3. ) Exmple: the rehility mtrix of L is: t R = 2 3 t (A ) x x x n x ε L x x 2 Row sis [x x x [ ε ] = ( ) A () A t t 2 3 x x x ()] ol sis.
48 Oservilty, Rehility nd More DFA Minimiztion: Is prtiulr se of oservility redution (single initil stte requires distintness only) NFA Determiniztion: Is prtiulr se of rehility trnsformtion (tke ll distint olumns s sis ) Miniml utomt: Are relted y liner mps (ut not y grph isomorphisms!). Better definition of minimlity Other tehniques: Are esily formlized in this setting: Pumping lemm, NFA to RE, Z-trnsforms, et.
49 Arnold, Diky & Nivt s Exmple Revisited (Oservility Redution) A x 2 x 4 x 5 A x 23 x 24 4 x 5 Define liner trnsf t t x = x T: A(x) = [A(x)T] T t t T = x ( ε) = x ( ε)t C( ε) = [ C( ε)] T
50 Arnold, Diky & Nivt s Exmple Revisited (Rehility Redution) A x 2 x 4 x 5 A x 23 x 24 4 x 5 Define liner trnsf x = x T: T = A(x) = [A(x)T] T x (ε) = x (ε)t C(ε) = [C(ε)] T
51
The Cayley-Hamilton Theorem For Finite Automata. Radu Grosu SUNY at Stony Brook
The Cyley-Hmilton Theorem For Finite Automt Rdu Grosu SUNY t Stony Brook How did I get interested in this topic? Convergence of Theories Hyrid Systems Computtion nd Control: - convergence etween control
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 informationState Complexity of Union and Intersection of Binary Suffix-Free Languages
Stte Complexity of Union nd Intersetion of Binry Suffix-Free Lnguges Glin Jirásková nd Pvol Olejár Slovk Ademy of Sienes nd Šfárik University, Košie 0000 1111 0000 1111 Glin Jirásková nd Pvol Olejár Binry
More informationTechnische Universität München Winter term 2009/10 I7 Prof. J. Esparza / J. Křetínský / M. Luttenberger 11. Februar Solution
Tehnishe Universität Münhen Winter term 29/ I7 Prof. J. Esprz / J. Křetínský / M. Luttenerger. Ferur 2 Solution Automt nd Forml Lnguges Homework 2 Due 5..29. Exerise 2. Let A e the following finite utomton:
More informationPrefix-Free Regular-Expression Matching
Prefix-Free Regulr-Expression Mthing Yo-Su Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST Prefix-Free Regulr-Expression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings
More informationNondeterministic Finite Automata
Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)
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 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 informationCS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6
CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized
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 informationSpeech Recognition Lecture 2: Finite Automata and Finite-State Transducers. Mehryar Mohri Courant Institute and Google Research
Speech Recognition Lecture 2: Finite Automt nd Finite-Stte Trnsducers Mehryr Mohri Cournt Institute nd Google Reserch mohri@cims.nyu.com Preliminries Finite lphet Σ, empty string. Set of ll strings over
More 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 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 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 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 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 informationNON-DETERMINISTIC FSA
Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is
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 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 informationCONTROLLABILITY and observability are the central
1 Complexity of Infiml Oservle Superlnguges Tomáš Msopust Astrt The infiml prefix-losed, ontrollle nd oservle superlnguge plys n essentil role in the reltionship etween ontrollility, oservility nd o-oservility
More informationJava II Finite Automata I
Jv II Finite Automt I Bernd Kiefer Bernd.Kiefer@dfki.de Deutsches Forschungszentrum für künstliche Intelligenz Finite Automt I p.1/13 Processing Regulr Expressions We lredy lerned out Jv s regulr expression
More 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 informationFinite Automata as Time-Inv Linear Systems Observability, Reachability and More
Finite Automt s Time-Inv Liner Systems Oservility, Rechility nd More Rdu Grosu Deprtment of Computer Science, Stony Brook University Stony Brook, NY 11794-4400, USA Astrct. We show tht regrding finite
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 informationFirst Midterm Examination
Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does
More informationAutomatic Synthesis of New Behaviors from a Library of Available Behaviors
Automti Synthesis of New Behviors from Lirry of Aville Behviors Giuseppe De Giomo Università di Rom L Spienz, Rom, Itly degiomo@dis.unirom1.it Sestin Srdin RMIT University, Melourne, Austrli ssrdin@s.rmit.edu.u
More informationCS 573 Automata Theory and Formal Languages
Non-determinism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Non-deterministi Finite Automton with -moves (NFA) An NFA is tuple
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 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 informationModel Reduction of Finite State Machines by Contraction
Model Reduction of Finite Stte Mchines y Contrction Alessndro Giu Dip. di Ingegneri Elettric ed Elettronic, Università di Cgliri, Pizz d Armi, 09123 Cgliri, Itly Phone: +39-070-675-5892 Fx: +39-070-675-5900
More informationCoalgebra, 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 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 informationDescriptional Complexity of Non-Unary Self-Verifying Symmetric Difference Automata
Desriptionl Complexity of Non-Unry Self-Verifying Symmetri Differene Automt Lurette Mris 1,2 nd Lynette vn Zijl 1 1 Deprtment of Computer Siene, Stellenosh University, South Afri 2 Merk Institute, CSIR,
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 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 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 information= state, a = reading and q j
4 Finite Automt CHAPTER 2 Finite Automt (FA) (i) Derterministi Finite Automt (DFA) A DFA, M Q, q,, F, Where, Q = set of sttes (finite) q Q = the strt/initil stte = input lphet (finite) (use only those
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 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 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 informationCS375: Logic and Theory of Computing
CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tle of Contents: Week 1: Preliminries (set lger, reltions, functions) (red Chpters 1-4) Weeks
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 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 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 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 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 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 informationAbstraction of Nondeterministic Automata Rong Su
Astrtion of Nondeterministi Automt Rong Su My 6, 2010 TU/e Mehnil Engineering, Systems Engineering Group 1 Outline Motivtion Automton Astrtion Relevnt Properties Conlusions My 6, 2010 TU/e Mehnil Engineering,
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 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 informationCompiler Design. Fall Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz
University of Southern Cliforni Computer Science Deprtment Compiler Design Fll Lexicl Anlysis Smple Exercises nd Solutions Prof. Pedro C. Diniz USC / Informtion Sciences Institute 4676 Admirlty Wy, Suite
More informationTalen en Automaten Test 1, Mon 7 th Dec, h45 17h30
Tlen en Automten Test 1, Mon 7 th Dec, 2015 15h45 17h30 This test consists of four exercises over 5 pges. Explin your pproch, nd write your nswer to ech exercise on seprte pge. You cn score mximum of 100
More informationKENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS CLASS - XII MATHEMATICS (Relations and Functions & Binary Operations)
KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS 6-7 CLASS - XII MATHEMATICS (Reltions nd Funtions & Binry Opertions) For Slow Lerners: - A Reltion is sid to e Reflexive if.. every A
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 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 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 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 informationTutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.
Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix
More informationCS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University
CS415 Compilers Lexicl Anlysis nd These slides re sed on slides copyrighted y Keith Cooper, Ken Kennedy & Lind Torczon t Rice University First Progrmming Project Instruction Scheduling Project hs een posted
More informationFinite Automata. 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 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 informationHyers-Ulam stability of Pielou logistic difference equation
vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo
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 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 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 informationAssignment 1 Automata, Languages, and Computability. 1 Finite State Automata and Regular Languages
Deprtment of Computer Science, Austrlin Ntionl University COMP2600 Forml Methods for Softwre Engineering Semester 2, 206 Assignment Automt, Lnguges, nd Computility Smple Solutions Finite Stte Automt nd
More informationRunning an NFA & the subset algorithm (NFA->DFA) CS 350 Fall 2018 gilray.org/classes/fall2018/cs350/
Running n NFA & the suset lgorithm (NFA->DFA) CS 350 Fll 2018 gilry.org/lsses/fll2018/s350/ 1 NFAs operte y simultneously exploring ll pths nd epting if ny pth termintes t n ept stte.!2 Try n exmple: L
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 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 informationThe University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS
The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their
More informationPetri Nets. Rebecca Albrecht. Seminar: Automata Theory Chair of Software Engeneering
Petri Nets Ree Alreht Seminr: Automt Theory Chir of Softwre Engeneering Overview 1. Motivtion: Why not just using finite utomt for everything? Wht re Petri Nets nd when do we use them? 2. Introdution:
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 informationCSCI565 - Compiler Design
CSCI565 - Compiler Deign Spring 6 Due Dte: Fe. 5, 6 t : PM in Cl Prolem [ point]: Regulr Expreion nd Finite Automt Develop regulr expreion (RE) tht detet the longet tring over the lphet {-} with the following
More informationTOPIC: LINEAR ALGEBRA MATRICES
Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A -... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED
More informationBehavior Composition in the Presence of Failure
Behvior Composition in the Presene of Filure Sestin Srdin RMIT University, Melourne, Austrli Fio Ptrizi & Giuseppe De Giomo Spienz Univ. Rom, Itly KR 08, Sept. 2008, Sydney Austrli Introdution There re
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 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 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 informationRevision Sheet. (a) Give a regular expression for each of the following languages:
Theoreticl Computer Science (Bridging Course) Dr. G. D. Tipldi F. Bonirdi Winter Semester 2014/2015 Revision Sheet University of Freiurg Deprtment of Computer Science Question 1 (Finite Automt, 8 + 6 points)
More informationMore general families of infinite graphs
More generl fmilies of infinite grphs Antoine Meyer Forml Methods Updte 2006 IIT Guwhti Prefix-recognizle grphs Theorem Let G e grph, the following sttements re equivlent: G is defined y reltions of the
More informationFormal languages, automata, and theory of computation
Mälrdlen University TEN1 DVA337 2015 School of Innovtion, Design nd Engineering Forml lnguges, utomt, nd theory of computtion Thursdy, Novemer 5, 14:10-18:30 Techer: Dniel Hedin, phone 021-107052 The exm
More informationLecture Notes No. 10
2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite
More informationA tutorial on sequential functions
A tutoril on sequentil functions Jen-Éric Pin LIAFA, CNRS nd University Pris 7 30 Jnury 2006, CWI, Amsterdm Outline (1) Sequentil functions (2) A chrcteriztion of sequentil trnsducers (3) Miniml sequentil
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 informationDFA Minimization and Applications
DFA Minimiztion nd Applictions Mondy, Octoer 15, 2007 Reding: toughton 3.12 C235 Lnguges nd Automt Deprtment of Computer cience Wellesley College Gols for ody o Answer ny P3 questions you might hve. o
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 informationChapter 3. Vector Spaces. 3.1 Images and Image Arithmetic
Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner
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 informationFirst Midterm Examination
24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet
More 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 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 informationIn-depth introduction to main models, concepts of theory of computation:
CMPSCI601: Introduction Lecture 1 In-depth introduction to min models, concepts of theory of computtion: Computility: wht cn e computed in principle Logic: how cn we express our requirements Complexity:
More informationTest Generation from Timed Input Output Automata
Chpter 8 Test Genertion from Timed Input Output Automt The purpose of this hpter is to introdue tehniques for the genertion of test dt from models of softwre sed on vrints of timed utomt. The tests generted
More informationReference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.
I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the
More informationNFA and regex. the Boolean algebra of languages. non-deterministic machines. regular expressions
NFA nd regex l the Boolen lger of lnguges non-deterministi mhines regulr expressions Informtis The intersetion of two regulr lnguges is regulr Run oth mhines in prllel? Build one mhine tht simultes two
More informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
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 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 informationLogic Synthesis and Verification
Logi Synthesis nd Verifition SOPs nd Inompletely Speified Funtions Jie-Hong Rolnd Jing 江介宏 Deprtment of Eletril Engineering Ntionl Tiwn University Fll 2010 Reding: Logi Synthesis in Nutshell Setion 2 most
More information