Hybrid Control and Switched Systems. Lecture #2 How to describe a hybrid system? Formal models for hybrid system
|
|
- Jayson Grant Peters
- 5 years ago
- Views:
Transcription
1 Hyrid Control nd Switched Systems Lecture #2 How to descrie hyrid system? Forml models for hyrid system João P. Hespnh University of Cliforni t Snt Brr Summry. Forml models for hyrid systems: Finite utomt Differentil equtions Hyrid utomt Open hyrid utomton 2. Nondeterministic vs. stochstic systems Non-deterministic hyrid utomt Stochstic hyrid utomt
2 Exmple #5: Multiple-tnk system pump pump-on inflow λ constnt outflow μ y gol prevent the tnk from emptying or filling up δ dely etween commnd is sent to pump nd the time it is executed gurd condition pump off τ δ? wit to off y y min? τ ú 0 stte reset τ ú 0 wit to on τ δ? pump on y y mx? How to formlly descrie this hyrid system? utomt M Deterministic finite utomton Q ú {q, q 2,, q n } finite set of sttes Σ ú {,, c, } finite set of input symols (lphet) Φ : Q Σ Q trnsition function Exmple: q Q 2 2 locking stte s Σ / Φ(q,s) 2 Grph representtion: 2 one node per stte (except for locking stte ) one directed edge (rrow) from q to Φ(q, s) with lel s for ech pir (q, s) for which Φ(q, s) 2
3 2 Deterministic finite utomton Nottion: Given set A string finite sequence of symols empty string A * set of ll strings of symols in set A e.g., A = {, } s = A * s[] = (rd element) s = 8 (length of string) Definition: Given initil stte q Q set of finl sttes F Q M ccepts string s Σ * with length n ú s if there exists sequence of sttes q Q * with length q = n+ (execution) such tht. q[] = q (strts t initil stte) 2. q[i+] = Φ(q[i], s[i] ), i {,2,,n} (follows rrows with correct lel). q[n+] F (ends in set of finl sttes) Definition: lnguge ccepted y utomton M L(M) ú { set of ll strings ccepted y M } There is no concept of time the whole string is ccepted instntneously 2 Deterministic finite utomton Definition: Given initil stte q Q set of finl sttes F Q M ccepts string s Σ * with length n ú s if there exists sequence of sttes q Q * with length q = n+ (execution) such tht. q[] = q (strts t initil stte) 2. q[i+] = Φ(q[i], s[i] ), i {,2,,n} (follows rrows with correct lel). q[n+] = F (ends in set of finl sttes) Definition: lnguge ccepted y utomton M L(M) ú { set of ll strings ccepted y M } Exmple: q ú F ú {} L(M)= {,,,,,, } = ( ()* ()* )* Questions in forml lnguge theory: Is there finite utomton tht ccepts given lnguge? Do two given utomt ccept the sme lnguge? Wht is the smllest utomton tht ccepts given lnguge? etc.
4 ordinry differentil eqution with input Σ Differentil eqution R m stte spce input spce f : R m vector field Definition: Given n input signl u : [0, ) R m A signl x : [0, ) is solution to Σ (in the sense of Crtheodory) if. x is piecewise differentile 2. If x is solution then t ny time t for which the derivtive exists ordinry differentil eqution without input Σ Differentil eqution (no inputs) f : stte spce vector field Definition: A signl x : [0, ) is solution to Σ (in the sense of Crtheodory) if. x is piecewise differentile 2. If x is solution then t ny time t for which the derivtive exists 4
5 Hyrid Automton (Exmple #2: Thermostt) x men temperture x 7? room off mode on mode heter x 77? Q set of discrete sttes continuous stte-spce f : Q vector field ϕ : Q Q discrete trnsition Exmple: Q ú { off, on } n ú note closed inequlities ssocited with jump nd open inequlities with flow Hyrid Automton (Exmple #2: Thermostt) x men temperture x 7? room off mode on mode heter x 77? Q set of discrete sttes continuous stte-spce f : Q vector field ϕ : Q Q discrete trnsition ϕ (q,x) = q 2? mode q 2 mode q mode q ϕ (q,x) = q? resets? 5
6 Hyrid Automton Q set of discrete sttes continuous stte-spce f : Q vector field ϕ : Q Q discrete trnsition ρ : Q reset mp x ú ρ (q,x ) mode q 2 ϕ (q,x ) = q 2? mode q mode q ϕ( q,x ) = q? x ú ρ (q,x ) Q f : Q Φ : Q Q Hyrid Automton set of discrete sttes continuous stte-spce vector field discrete trnsition (& reset mp) x ú Φ 2 (q,x ) mode q 2 Φ (q,x ) = q 2? mode q mode q Φ (q,x ) = q? x ú Φ 2 (q,x ) Compct representtion of hyrid utomton 6
7 Exmple #5: Multiple-tnk system pump pump-on inflow λ constnt outflow μ y gol prevent the tnk from emptying or filling up δ dely etween commnd is sent to pump nd the time it is executed pump off τ δ? wit to off y y min? τ ú 0 τ ú 0 wit to on τ δ? pump on y y mx? Exmple #5: Multiple-tnk system τ δ? pump off wit to off y y min? τ ú 0 τ ú 0 wit to on τ δ? pump on y y mx? Q ú { off, won, on, woff } R 2 continuous stte-spce 7
8 Solution to hyrid utomton x ú Φ 2 (q,x ) Φ (q,x ) = q 2? mode q 2 mode q Definition: A solution to the hyrid utomton is pir of right-continuous signls x : [0, ) q : [0, ) Q such tht. x is piecewise differentile & q is piecewise constnt 2. on ny intervl (t,t 2 ) on which q is constnt nd x continuous continuous evolution. discrete trnsitions Hyrid Automton (Exmple #2: Thermostt) x men temperture x 7? room off mode on mode heter x 77? 77 7 x x = 77, q = on q = off no trnsition would occur if the jump rnch hd strict inequlity x > 77 q = on off off off on on on note closed inequlities ssocited with jumps nd open inequlities with flows 8
9 Exmple #7: Server system with congestion control incoming rte r q mx q Additive increse/multiplictive decrese congestion control (AIMD): while q < q mx increse r linerly when q reches q mx instntneously multiply r y γ (0,) queue dynmics q q mx? server B rte of service (ndwidth) congestion controller q(t) r ú γ r t incoming rte r Open Automton (Exmple #7: Server system with congestion control) queue dynmics q q mx? q mx congestion controller r ú γ r q server vrile r B rte of service (ndwidth) queue dynmics event q q mx congestion controller 9
10 q mx incoming rte r server B rte of service (ndwidth) Open Automton (Exmple #7: Server system with congestion control) q queue dynmics vrile r congestion controller q q mx? queue-full event queue-full queue-full? r ú γ r synchronized trnsitions (ll gurds must hold for trnsition to occur) events re nothing more thn symolic lels for trnsitions incoming rte r q mx server B rte of service (ndwidth) Open Automton (Exmple #7: Server system with congestion control) q queue dynmics vrile r congestion controller q q mx? queue-full event queue-full r > queue-full r ú γ r synchronized trnsitions (ll gurds must hold for trnsition to occur) events re nothing more thn symolic lels for trnsitions 0
11 Open Automton (Exmple #5: Multiple-tnk system) pump pump-on inflow λ constnt outflow μ y gol prevent the tnk from emptying or filling up δ dely etween commnd is sent to pump nd the time it is executed pump off τ δ? wit to off y y min? τ ú 0 τ ú 0 wit to on τ δ? pump on y y mx? Open Automton (Exmple #5: Multiple-tnk system) pump pump-on inflow λ constnt outflow μ y gol prevent the tnk from emptying or filling up δ dely etween commnd is sent to pump nd the time it is executed sk y y min? pump off event chnge idle chnge? sk? chnge y y mx? chnge? pump on event sk τ ú 0 dely τ δ? sk
12 utomt M Deterministic finite utomton Q ú {q, q 2,, q n } finite set of sttes Σ ú {,, c, } finite set of input symols (lphet) Φ : Q Σ Q trnsition function Exmple: q Q 2 2 locking stte s Σ / Φ(q,s) 2 Grph representtion: 2 one node per stte (except for locking stte ) one directed edge (rrow) from q to Φ(q, s) with lel s for ech pir (q, s) for which Φ(q, s) utomt M Nondeterministic finite utomton Q ú {q, q 2,, q n } finite set of sttes Σ ú {,, c, } finite set of input symols (lphet) Φ : Q Σ 2 Q trnsition set-vlued function Exmple: q Q 2 2 locking stte s Σ / Φ(q,s) {2} { } { } {,} {} { } { } Grph representtion: 2 Nottion: Given set A, 2 A power-set of A, i.e., the set of ll susets of A e.g., A = {,2} 2 A = {, {}, {2}, {,2} } When A hs n < elements then 2 A hs 2 n elements 2
13 2 Nondeterministic finite utomton Exmple: q ú F ú {} L(M)= {,,,,,, } = ( ()* ()* )* Definition: Given initil stte q Q set of finl sttes F Q M ccepts string s Σ * with length n ú s if there exists sequence of sttes q Q * with length q = n+ (execution) such tht. q[] = q (strts t initil stte) 2. q[i+] Φ(q[i], s[i] ), i {,2,,n} (follows rrows with correct lel). q[n+] F (ends in set of finl sttes) Definition: lnguge ccepted y utomton M L(M) ú { set of ll strings ccepted y M } Determiniztion From forml lnguge theory: For every nondeterministic finite utomton there is deterministic one tht ccepts the sme lnguge (ut generlly the deterministic one needs more sttes) nondeterministic utomton M 2 deterministic utomton N 2 or 2 or from only ccepts nd goes to 2 from 2 only ccepts nd cn go to either or from or only ccepts nd goes to 2 or resp. from (or 2) cn ccepts nd go to 2 from ( or) 2 cn ccept nd go to or Sme lnguge: L(M) = L(N) = ( ()* ()* )* M provides more compct representtion
14 η η 2 η η 4 Exmple #: Trnsmission throttle u [-,] position θ g {,2,,4} ger ω velocity η k efficiency of the k th ger ω velocity [Hedlund, Rntzer 999] Exmple #: Semi-utomtic trnsmission v(t) { up, down, keep } drivers input (discrete) v = up or ω ω 2? v = up or ω ω? v = up or ω ω 4? g = g = 2 g = g = 4 v = down or ω ϖ? v = down or ω ϖ 2? v = down or ω ϖ? ω 2 g = ω g = 2 ϖ g = ϖ 2 ω 4 g = 4 ϖ 4
15 Nondeterministic Hyrid Automton (Exmple #: Semi-utomtic trnsmission) Suppose we wnt to consider ll possile driver inputs: ω ϖ? ω ϖ 2? ω ϖ? g = g = 2 g = g = 4 ω ω ϖ ω ω ϖ 2 ω ω 4 2 ω ϖ invrince condition (must hold to remin in discrete stte) g = ω ω 2? ω ω? ω ω 4? ϖ ϖ 2 ω 2 ω g = 2 g = ϖ gurd condition (does not force jump, simply llows it) ω 4 g = 4 Nondeterministic Hyrid Automton Q set of discrete sttes continuous stte-spce f : Q vector field ϕ : Q 2 Q set-vlued discrete trnsition ρ : Q 2 Rn set-vlued reset mp Δ Q domin or invrint set x ρ(q,x ) mode q 2 (q 2,x) Δ q 2 ϕ(q,x )? mode q mode q (q,x) Δ (q,x) Δ q ϕ(q,x )? x ρ(q,x ) 5
16 Nondeterministic Hyrid Automton Q set of discrete sttes continuous stte-spce f : Q vector field Φ: Q 2 Q Rn set-vlued discrete trnsition (& reset & domin)? ú (q 2, x) Φ(q,x ) mode q 2 mode q mode q (q 2,x) Φ(q 2,x ) (q,x) Φ(q,x ) (q,x) Φ(q,x ) (q, x) Φ(q,x ) Compct representtion of nondeterministic hyrid utomton gurd condition (does not force jump, simply llows it) Nondeterministic Hyrid Automton (Exmple #: Semi-utomtic trnsmission) ω ϖ? g = g = 2 g = ϖ ω 2 g = 2 invrince condition (must hold to remin in discrete stte) ω ω 2 ϖ ω ω ω ω 2? Q ú {, 2 } R 2 continuous stte-spce 6
17 Solution to nondeterministic hyrid utomton (q 2, x) Φ(q,x ) mode q 2 mode q (q 2,x) Φ(q 2,x ) (q,x) Φ(q,x ) Definition: A solution to the hyrid utomton is pir of right-continuous signls x : [0, ) q : [0, ) Q such tht. x is piecewise differentile & q is piecewise constnt 2. on ny intervl (t,t 2 ) on which q is constnt nd x continuous continuous evolution. discrete trnsition & resets & domin Stochstic finite utomton: controlled Mrkov chin controlled Mrkov chin M 2 20% 80% Q ú {q, q 2,, q n } finite set of sttes Σ ú {,, c, } finite set of input symols Φ : Q Q Σ [0,] trnsition proility function Φ(q, q 2, s ) proility of trnsitioning to stte q 2, when in stte q nd symol s is selected By convention, typiclly edges drwn without proilities correspond to trnsitions tht occur with proility self-loops my e omitted 7
18 Stochstic finite utomton: controlled Mrkov chin 00% 00% 2 00% 20% 00% 80% By convention, typiclly edges drwn without proilities correspond to trnsitions tht occur with proility self loops my e omitted 00% Q ú {, 2, } Σ ú {, } Φ(q, q 2, s ) proility of trnsitioning to stte q 2, when in stte q nd symol s is selected Stochstic Hyrid Automton Q set of discrete sttes continuous stte-spce f : Q vector field ϕ : Q Q [0, ] discrete trnsition proility ρ : Q Q reset mp (deterministic) x ú ρ (q, q 2, x ) ϕ (q, q 2, x ) mode q 2 (Poisson-like model) mode q mode q ϕ (q, q, x ) x ú ρ (q, q, x ) 8
19 Stochstic Hyrid Automton Q set of discrete sttes continuous stte-spce f : Q vector field Φ : Q Q [0, ] discrete trnsition proility & reset ϕ (q, q 2, x, x) (Poisson-like model) mode q 2 mode q mode q ϕ (q, q, x, x) More s specil topic lter Next clss. Trjectories of hyrid systems: Solution to hyrid system Execution of hyrid system 2. Degenercies Finite escpe time Chttering Zeno trjectories Non-continuous dependency on initil conditions 9
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationChapter 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 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. 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 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 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 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 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 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 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 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 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 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 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 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 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 informationSome Theory of Computation Exercises Week 1
Some Theory of Computtion Exercises Week 1 Section 1 Deterministic Finite Automt Question 1.3 d d d d u q 1 q 2 q 3 q 4 q 5 d u u u u Question 1.4 Prt c - {w w hs even s nd one or two s} First we sk whether
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 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 informationHybrid Control and Switched Systems. Lecture #1 Hybrid systems are everywhere: Examples
Hybrid Control and Switched Systems Lecture #1 Hybrid systems are everywhere: Examples João P. Hespanha University of California at Santa Barbara Summary Examples of hybrid systems 1. Bouncing ball 2.
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 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 informationCS103B 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 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 informationA Symbolic Approach to Control via Approximate Bisimulations
A Symolic Approch to Control vi Approximte Bisimultions Antoine Girrd Lortoire Jen Kuntzmnn, Université Joseph Fourier Grenole, Frnce Interntionl Symposium on Innovtive Mthemticl Modelling Tokyo, Jpn,
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 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 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 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 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. 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 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 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 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 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 informationContext-Free Grammars and Languages
Context-Free Grmmrs nd Lnguges (Bsed on Hopcroft, Motwni nd Ullmn (2007) & Cohen (1997)) Introduction Consider n exmple sentence: A smll ct ets the fish English grmmr hs rules for constructing sentences;
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 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 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 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 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 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 informationNormal Forms for Context-free Grammars
Norml Forms for Context-free Grmmrs 1 Linz 6th, Section 6.2 wo Importnt Norml Forms, pges 171--178 2 Chomsky Norml Form All productions hve form: A BC nd A vrile vrile terminl 3 Exmples: S AS S AS S S
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 informationCS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS
CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)
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 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 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 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 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 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 information12.1 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 informationNFAs 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 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 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 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 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 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 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 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 informationReinforcement learning II
CS 1675 Introduction to Mchine Lerning Lecture 26 Reinforcement lerning II Milos Huskrecht milos@cs.pitt.edu 5329 Sennott Squre Reinforcement lerning Bsics: Input x Lerner Output Reinforcement r Critic
More information12.1 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 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 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 informationCSCI 340: Computational Models. Transition Graphs. Department of Computer Science
CSCI 340: Computtionl Models Trnsition Grphs Chpter 6 Deprtment of Computer Science Relxing Restrints on Inputs We cn uild n FA tht ccepts only the word! 5 sttes ecuse n FA cn only process one letter t
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 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 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 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 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 informationCS 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 informationa b b a pop push read unread
A Finite Automton A Pushdown Automton 0000 000 red unred b b pop red unred push 2 An Exmple A Pushdown Automton Recll tht 0 n n not regulr. cn push symbols onto the stck cn pop them (red them bck) lter
More informationSWEN 224 Formal Foundations of Programming WITH ANSWERS
T E W H A R E W Ā N A N G A O T E Ū P O K O O T E I K A A M Ā U I VUW V I C T O R I A UNIVERSITY OF WELLINGTON Time Allowed: 3 Hours EXAMINATIONS 2011 END-OF-YEAR SWEN 224 Forml Foundtions of Progrmming
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 informationPART 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( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
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 informationSynchronizing Automata with Random Inputs
Synchronizing Automt with Rndom Inputs Vldimir Gusev Url Federl University, Ekterinurg, Russi 9 August, 14 Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 1 / 13 Introduction Synchronizing
More informationThoery of Automata CS402
Thoery of Automt C402 Theory of Automt Tle of contents: Lecture N0. 1... 4 ummry... 4 Wht does utomt men?... 4 Introduction to lnguges... 4 Alphets... 4 trings... 4 Defining Lnguges... 5 Lecture N0. 2...
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 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 informationGNFA GNFA GNFA GNFA GNFA
DFA RE NFA DFA -NFA REX GNFA Definition GNFA A generlize noneterministic finite utomton (GNFA) is grph whose eges re lele y regulr expressions, with unique strt stte with in-egree, n unique finl stte with
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 informationCS 330 Formal Methods and Models Dana Richards, George Mason University, Spring 2016 Quiz Solutions
CS 330 Forml Methods nd Models Dn Richrds, George Mson University, Spring 2016 Quiz Solutions Quiz 1, Propositionl Logic Dte: Ferury 9 1. (4pts) ((p q) (q r)) (p r), prove tutology using truth tles. p
More information