Finite set of states t ypically 2 Alphab et of input symb ols t ypically 3 One state is the startinitial state t ypically 4 Zero or more nalac epting states the set is ypically 5 tr ansition function t ypically This function ak es a stat ID: 27504 Download Pdf

Finite set of states t ypically 2 Alphab et of input symb ols t ypically 3 One state is the startinitial state t ypically 4 Zero or more nalac epting states the set is ypically 5 tr ansition function t ypically This function ak es a stat

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ormal De niti on of Finite Automaton 1. Finite set of states ,t ypically 2. Alphab et of input symb ols ,t ypically . 3. One state is the start/initial state, t ypically 4. Zero or more nal/ac epting states; the set is ypically 5. tr ansition function ,t ypically . This function: ak es a state and input sym bol as argumen ts. Returns a state. One \rule" of ould b e written q; a )= , where and are states, and is an input sym b ol. In tuitiv ely: if the F A is in state , and input is receiv ed, then the F Agoesto state (note: OK). AF A is represen ted as the v e-tuple: Q; ; ;q

;F ). Example: Clamping Logic ema y think of an accepting state as represen ting a \1" output and nonaccepting states as represen ting \0" out. A \clamping" circuit w aits for a 1 input, and forev er after mak es a 1 output. Ho ev er, to a oid clamping on spurious noise, w e'll design a F A that aits for t o 1's in a ro w, and \clamps" only then. In general, w ema y think of a state as represen ting a summary of the history of what has b een seen on the input so far. The states w e need are: 1. State , the start state, sa ys that the most recen t input (if there w as one) w as not a 1, and w

eha e nev er seen t o 1's in a ro w. 2. State sa ys w eha e nev er seen 11, but the previous input w as 1. 3. State is the only accepting state; it sa ys that w eha e at some time seen 11. Th us, =( ;q ;q ; ;q ), where is giv en b y:

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By marking the start state with and accepting states with , the tr ansition table that de nes also sp eci es the en tire F A. Con en tions It helps if w e can a oid men tioning the t yp e of ev ery name b y follo wing some rules: Input sym b ols are , etc., or digits. Strings of input sym b ols are u;v;:::;z States are , etc. ransition Diagram AF

A can b e represen ted b y a graph; no des = states; arc from to is lab eled b y the set of input sym b ols suc h that q; a )= No arc if no suc Start state indicated b yw ord \start" and an arro w. Accepting states get double circles. Example or the clamping F A: Start 0,1 Extension of to P aths In tuitiv ely ,a F ac epts a string if there is a path in the transition diagram that: 1. Begins at the start state, 2. Ends at an accepting states, and 3. Has sequence of lab els ;a ;:::;a ormally ,w e extend transition function to q; w ), where can b e an y string of input sym b ols:

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Basis: q; )= (i.e., on no input, the F do esn't go an ywhere. Induction: q; wa )= q; w ;a , where is a string, and a single sym b ol (i.e., see where the F Agoeson , then lo ok for the transition on the last sym b ol from that state). Imp ortan t fact with a straigh tforw ard, inductiv e pro of: really represen ts paths. That is, if , and ;a )= +1 for all =0 ;:::;n 1, then ;w )= Acceptance of Strings AF =( Q; ; ;q ;F ) accepts string if ;w )isin Language of a F accepts the language )= ;w )is in Aside: T yp e Errors A ma jor source of confusion when dealing with automata (or

mathematics in general) is making \t yp e errors." Example: Don't confuse ,a F A, i.e., a program, with ), whic hisoft yp e \set of strings." Example: the start state is of t yp e \state," but the accepting states is of t yp e \set of states." ric kier example: Is a sym b ol or a string of length 1? Answ er: it dep ends on the con text, e.g., is it used in q; a ), where it is a sym b ol, or q; a ), where it is a string? Nondetermini st ic Finite Automata Allo w (deterministic) F Atoha eac hoice of 0 or more next states for eac h state-input pair. Imp ortan t to ol for designing string pro

cessors, e.g., grep , lexical analyzers. But \imaginary " in the sense that it has to b e implemen ted deterministically Example In this somewhat con triv ed example, w e shall design an NF A to accept strings o er alphab et suc h that the last sym b ol app ears

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previously , without an yin terv ening higher sym b ol, e.g., 11, 21112, 312123. ric k: use start state to mean \I guess I ha en't seen the sym b ol that matc hes the ending sym bol y et. Three other states represen t a guess that the matc hing sym b ol has b een seen, and remem b ers what that sym b ol is.

1,2 1,2,3 Start ormal NF =( Q; ; ;q ;F ), where all is as DF A, but: q; a )isa set of states, rather than a single state. Extension to Basis: q; )= Induction: Let: q; w )= ;p ;:::;p ;a )= for =1 :::;k Then q; wa )= [[ Language of an NF An NF A accepts if any path from the start state to an accepting state is lab eled .F ormally: )= ;w ;g Subset Construction or ev ery NF A there is an quivalent (accepts the same language) DF A. But the DF A can ha e exp onen tially man states.

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Let ; ;q ;F )beanNF A. The equiv alen tDF A constructed b y the subset construction is =( ; ;F

), where: 1. =2 ;. i.e., is the set of all subsets of 2. is the set of sets in suc h that ;q ;:::;q ;a )= ;a ;a [[ ;a ). Key theorem (induction on , pro of in b o ok): ;w )= ;w ). Consequence: )= ). Example: Subset Construction F rom Previous NF An imp ortan t practical tric k, used in lexical analyzers and other text-pro cessors is to ignore the (often man y) states that are not accessible from the start state (i.e., no path leads there). or the NF A example ab o e, of the 32 p ossible subsets, only 15 are accessible. Computing transitions \on demand" giv es the follo wing pq pr ps pq pq t

pr ps pq t pq t pr ps pr pq r pr t ps pr t pq r pr t ps ps pq s pr s pst pst pq s pr s pst pr s pqrs pr st pst pr st pqrs pr st pst pq s pq st pr s pst pq st pq st pr s pst pq r pqrt pr t ps pqrt pqrt pr t ps pqrs pqrst prst pst pqrst pqrst prst pst

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