Finite automata. III. Finite automata: language recognizers. Nondeterministic Finite Automata. Nondeterministic Finite Automata with λ-moves

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1 . Finite automata: language recognizers n F can e descried y a laeled directed graph, where the nodes, called states, are laeled with a (unimportant) name edges, called transitions, are laeled with symols from the alphaet/vocaulary one node is designated as the initial or start state zero or more nodes are designated as final states (doule circle, n F recognizes/accepts a string w if there is a path from the initial state to the final state laeled y the letters in w (e.g., for = {,}, consider vs ) [ ] - - [ ] - - [ ] - - [ ] : not accepted [ ] - - [ ] - - [ ] - [ ] [ ] : accepted Finite automata lternate terminology: a symol s can e used to make a move from state s to state s 2 if there is a transition laeled from s to s 2 ; a string is recognized if it causes a sequence of moves from the initial state to end in some final state. The F graph can also e descried y a transition tale, showing the moves that can e made from each state, on all possile input symols:, S25 Fa6.orgida S25 Fa6.orgida 2, Nondeterministic Finite utomata, From any state there are or more transitions on same symol. (So transition tale can have more than one entry this is what distinguishes NF from DF!) e.g. of possile path of moves for the NF aove on : [] --[]--[]--[]--[] : rejected ecause is not final [] --[]--[]--[]-- : << stuck, so rejected [] --[]--[]--[]--[] : accepted, ecause is final n NF recognizes/accepts a string if there exists SOME path to some final state. So is accepted aove y the third path, and the language is Strings over {,}, ending with n F is not a DF if some entry in the tale has 2 or more entries. ( entries can e replaced y a dead state ) S25 Fa6.orgida 3 Nondeterministic Finite utomata with λ-moves, λ λ, λ-move can e taken, if present, without consuming input (= n a concatenation of edge laels along a path, inserting a null string has no effect on the path name, i.e., the string eing accepted) e.g. possile sequences of moves for the NF aove: [] --[]--[]--[]-λ-[] and the lael of this path is λ = λ-moves are useful, among others, for things like optional, and repeating (jump ack to eginning state) We will only consider λ-moves in writing NFs more conveniently. S25 Fa6.orgida 4

2 (λ-moves: an aside for good students) λ, λ,, Someone suggested that one might want to add a λ loop to every state. This will not affect the language accepted, and just complicates the automaton Note that λ-moves make any F e nondeterministic since you must guess when to follow one You will never HVE to use use λ-moves in this class. They just make things easier. Relationship of regular language specifiers. For every Nondeterministic F (without λ moves) there is a deterministic F recognizing the same language. (Though the deterministic F may have many more states). NF recognition is simulated y keeping track of all possile states you could e in after seeing symols so far, y using sets: e.g. recognizing using the NF elow {}- - {} - - {,} - - {,} {} - - {,} {} {} f the set at the end contains a final state, then the string could lead to it, so the string is recognized., S25 Fa6.orgida 5 S25 Fa6.orgida 6 NF 2 DF construction eginning with {StartState}, construct sets of states one can move to on each symol. Then repeat process for each of the new sets appearing (underlined),, NF 2 DF construction eg.2 {} {} {,} {,} {} {,,} {,,} {} {,,} FNL STTES: ny set containing a final state in the original NFS e.g. here it is any set containing : {,,} FNL STTES: any set containing : {,} {,,} Draw diagram for this DF S25 Fa6.orgida 8 S25 Fa6.orgida 7 {} {} {,} {,} {,} {,,} {,,} {,} {,,} {,} {} {,,}

3 Larger example from ook Diagram of corresp. DF (work it out) egin with {s}... Next slide shows resulting DF S25 Fa6.orgida 9 S25 Fa6.orgida Relationship of regular language specifiers.for every Nondeterministic F there is a deterministic F recognizing the same language. (Though the deterministic F may have many more states). NF recognition is simulated y keeping track of all possile states you could e in after seeing symols so far: {}- - {} - - {,} - - {,} {} - - {,} {} {} f the set at the end has a final state, then string could lead there, so string recognized. (side: some NF require exponentially larger DF:, n Relationships (cont d) 2. For every Regular Expression there is an NF descriing the same language. connect start and end state with regexp R [s] R-- >[[s-end]] replace [x] R T > [y] y [x] R > [v] T > [y] replace [x] R T > [y] y [x] R > [y], [x] T > [y] replace [x] R* > [y] y [x] λ > [v] R > [w] λ > [y] plus [v] < λ [w] replace [x] Ø > [y] y [x] [y] //no transition replace [x]! > [y] y [x] λ > [y] leave [x] > [y] as is (3 We do not prove in this course that from DF you can get an RE y a process similar to the aove in reverse) S25 Fa6.orgida S25 Fa6.orgida 2

4 Relationships (cont d) 2. For every Regular Expression there is an NF descriing the same language. Example: algorithm applied to (a ( * c)) connect start and end state with regexp R: [s] R [[s-end]] replace [x] (R T) [y] y [x] R [v] T [y] replace [x] (R T) [y] y [x] R [y], [x] T [y] replace [x] R* [y] y [x] λ [v] R [v] λ [y] replace [x] Ø [y] y [x] [y] //no transition replace [x]! [y] y [x] λ [y] leave as is [x] [y] for in alphaet R* is the only tricky one here. (side: n all ut one ook, it is more complicated: replace [x] R* [y] y [x] λ [v] R [w] λ [y] plus λ moves from [x] to [y] (to recognize λ alone), and from [w] to [v] (to allow repetition) e.g., do /*anyone willing to draw these steps for my notes? */ S25 Fa6.orgida 3 S25 Fa6.orgida 4 States:,2,3,4 Tale of result for(a *c) a c " Final state: 2 nitial state: Why not simply collapse states? onsider (a (* c)) F a F * F c F *c : collapse final state of F * with initial state of F c E G H GH c a c F S25 Fa6.orgida 5 F (a *c) : collapse initial state of F a with initial state of F *c ut this F also accepts a which is not in L( a (* c) ) EGH c S25 Fa6.orgida 6 a F

5 Relationships (cont d) ( 3 We do not prove in this course that from a DF you can get an RE y a process similar to the one in the previous slide ) Languages, RE, (D/N)F Questions one can ask relating to F,RE:. s some specific string accepted y a given F/RE 2. What is the language descried y an F/RE 3. Given a language, find an F/RE descriing it Examples Strings over {,}, starting with (incomplete DF: some states have no transitions on some symol = transition to a dead state) Strings over {,}, ending with (NF,DF) float numers integers, decimal numers, exponent notation (null transitions) Strings over {,} with (no) consecutive s S25 Fa6.orgida 7 S25 Fa6.orgida 8