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Finite Automata We present one application of finite automata: non trivial text search algorithm Given a finite set of words find if there are occurences of one of these words in a given text 1 Nondeterministic Finite Automata A nondeterministic finite automaton (NFA) is one for which the next state is not uniquely determined by the current state and the coming symbol Informally, the automaton can choose between different states 1 5kr choc coffee 0 5kr 2 A nondeterministic vending machine 2 Nondeterministic Finite Automata When does nondeterminism appear?? Tossing a coin (probabilistic automata) When there is incomplete information about the state For example, the behaviour of a distributed system might depend on messages from other processes that arrive at unpredictable times 3 Nondeterministic Finite Automata When does a NFA accepts a word?? Intuitively, the automaton accepts w iff there is at least one computation path starting from the start state to an accepting state It is helpful to think that the automaton can guess the succesful computation (if there is one) 0,1 q0 0 q1 1 q2 NFA accepting all words that end in 01 What are all possible computations for the word 1010?? 4 Nondeterministic Finite Automata Another example: automaton accepting only the words such that the second last symbol from the right is 1 0,1 q0 1 q1 0,1 q2 The automaton “guesses” when the word finishes 5 Nondeterministic Finite Automata Σ = {1} 1 1 1 1 1 1 1 1 1 1 NFA accepting all words of length multiple of 3 or 5 The automaton guesses the right direction, and then verifies that |w| is correct! How to define mathematically a non deterministic machine?? 6 NFA and DFA We saw on examples that it is much easier to build a NFA accepting a given language than to build a DFA accepting this language We are going to give an algorithm that produces a DFA from a given NFA accepting the same language This is surprising because a DFA cannot “guess” First we have to define mathematically what is a NFA Both this definition and the algorithm uses in a crucial way the powerset operation if A is a set, we denote by P ow(A) the set of all subsets of A (in particular the empty set ∅ is in P ow(A)) 7 Nondeterministic Finite Automata Definition A nondeterministic finite automaton (NFA) consists of 1. a finite set of states (often denoted Q) 2. a finite set Σ of symbols (alphabet) 3. a transition function that takes as argument a state and a symbol and returns a set of states (often denoted δ); this set can be empty 4. a start state 5. a set of final or accepting states (often denoted F ) We have, as before, q0 ∈ Q F ⊆ Q 8 Nondeterministic Finite Automata The transition function of a NFA is a function δ : Q × Σ → P ow(Q) Each symbol a ∈ Σ defines a binary relation on the set Q a q1 → q2 iff q2 ∈ δ(q1 , a) 9 Nondeterministic Finite Automata 0,1 q0 1 q1 0,1 has for transition table 0 1 →q0 {q0 } {q0 , q1 } q1 {q2 } {q2 } ∗q2 ∅ ∅ δ(q0 , 1) = {q0 , q1 }; we have δ(q0 , 1) ∈ P ow(Q) 10 q2 Extending the Transition Function to Strings We define δ̂(q, x) by induction BASIS δ̂(q, ) = {q} INDUCTION suppose x = ay δ̂(q, ay) = δ̂(p1 , y) ∪ . . . ∪ δ̂(pk , y) where δ(q, a) = {p1 , . . . , pk } S δ̂(q, ay) = p∈δ(q,a) δ̂(p, y) We write q.x ∈ P ow(Q) instead of δ̂(q, x) 11 Extending the Transition Function to Strings A word x is accepted iff q0 .x ∩ F 6= ∅ i.e. there is at least one accepting state in q0 .x δ̂ : Q × Σ∗ → P ow(Q) and each word x defines x a binary relation on Q: q1 → q2 iff q2 ∈ q1 .x L(A) = {x ∈ Σ∗ | q0 .x ∩ F 6= ∅} 12 Extending the Transition Function to Strings x Intuitively: q1 → q2 means that there is one path from q1 to q2 having x for sequence of events x We can define q1 → q2 inductively BASIS: q1 → q2 iff q1 = q2 ay y STEP: q1 → q2 iff there exists q ∈ δ(q1 , a) such that q → q2 x Then we have q1 → q2 iff q2 ∈ q1 .x 13 Representation in functional programming next :: Q -> E -> [Q] run :: Q -> [E] -> [Q] run q [] = [q] run q (a:x) = concat (map (\ p -> run p x) (next q a)) 14 Representation in functional programming We use -- map f [a1,...,an] = [f a1,...,f an] map f [] = [] map f (a:x) = (f a):(map f x) -- concat [x1,...,xn] = x1 ++ ... ++ xn concat [] = [] concat (x:xs) = x ++ concat xs 15 Representation in functional programming It is nicer to take next :: E -> Q -> [Q] we define run :: [E] -> Q -> [Q] run [] q = [q] run (a:x) q = concat (map (run x) (next a q)) 16 Representation in functional programming In the monadic notation (with the list monad) run :: [E] -> Q -> [Q] run [] q = return q run (a:x) q = next a q >>= run x accept :: [E] -> Bool accept x = or (map final (run x q0)) 17 Representation in functional programming List monad: clever notations for programs with list -- return :: a -> [a] return x = [x] -- (>>=) :: [a] -> (a->[b]) -> [b] xs >>= f = concat (map f xs) This is exactly what is needed to define run (a:x) q 18 Representation in functional programming Other notation: do notation run :: [E] -> Q -> [Q] run [] q = return q run (a:x) q = next a q >>= run x is written run :: [E] -> Q -> [Q] run [] q = return q run (a:x) q = do p <- next a q run x p 19 The Subset Construction This corresponds closely to Ken Thompson’s implementation We can now indicate how, given a NFA, to build a DFA that accepts the same language.This DFA may require more states. Intuitive idea of the construction for a NFA N : there are only finitely many subsets of Q, hence only finitely many possible situations 20 Extending the Transition Function to Strings We start from a NFA N = (Q, Σ, δ, q0 , F ) where δ : Q × Σ → P ow(Q) We define δD : P ow(Q) × Σ → P ow(Q) S δD (X, a) = q∈X δ(q, a) If X = {p1 , . . . , pk } then δD (X, a) = δ(p1 , a) ∪ . . . ∪ δ(pk , a) δD (∅, a) = ∅, δD ({q}, a) = δ(q, a) 21 The Subset Construction This function satisfies also δD (X1 ∪ X2 , a) = δD (X1 , a) ∪ δD (X2 , a) S δD (X, a) = p∈X δD ({p}, a) 22 The Subset Construction We build the following DFA QD = P ow(Q) δD : P ow(Q) × Σ → P ow(Q) qD = {q0 } ∈ QD FD = {X ⊆ Q | X ∩ F 6= ∅} 23 Representation in functional programming Given next :: E -> Q -> [Q] we define its parallel version pNext :: E -> [Q] -> [Q] pNext a qs = concat (map (next a) qs) 24 Representation in functional programming With the monadic notation pNext :: E -> [Q] -> [Q] pNext a qs = qs >>= next a pNext :: E -> [Q] -> [Q] pNext a qs = do q <- qs next a q 25 Representation in functional programming We can now define run’ :: [E] -> [Q] -> [Q] run’ [] qs = qs run’ (a:x) qs = run’ x (pNext a qs) 26 Representation in functional programming We state that we have for all x run’ x [q] = run x q run’ [a1,a2] [q] = pNext a2 (pNext a1 [q]) = [q] >>= next a1 >>= next a2 = next a1 q >>= next a2 run [a1,a2] q = next a1 q >>= run [a2] = next a1 q >>= (\ p -> next a2 p >>= return) = next a1 q >>= next a2 27 The Subset Construction Lemma 1: For all word z and all set of states X we have S ˆ δD (X, z) = p∈X δˆD ({p}, z) Lemma 2: For all words x we have q.x = δˆD ({q}, x) Proof: By induction. The inductive case is when x = ay and then q.(ay) = S p.y by definition S ˆ = p∈δ(q,a) δD ({p}, y) by induction = δˆD (δ(q, a), y) by lemma 1 = δˆD (q, ay) by definition p∈δ(q,a) 28 The Subset Construction Lemma: For all words x we have q.x = δˆD ({q}, x) Theorem: The language accepted by the NFA N is the same as the language accepted by the DFA (QD , Σ, δD , qD , FD ) Proof: We have x ∈ L(N ) iff δ̂(q0 , x) ∩ F 6= ∅ iff δ̂(q0 , x) ∈ FD iff δˆD (qD , x) ∈ FD . We use the Lemma to replace δ̂(q0 , x) by δˆD ({q0 }, x) which is the same as δˆD (qD , x) Q.E.D. 29 The Subset Construction It seems that if we start with a NFA that has n states we shall need 2n states for building the corresponding DFA In practice, often a lot of states are not accessible from the start state and we don’t need them 30 The Subset Construction 0,1 q0 1 q1 0,1 q2 We start from A= {q0 } (only one start state) If we get 0, we can only go to the state q0 If we get 1, we can go to q0 or to q1 . We represent this by going to the state B= {q0 , q1 } = δD (A, 1) From B, if we get 0, we can go to q0 or to q2 ; we go to the state C= {q0 , q2 } = δD (B, 0) From B, if we get 1, we can go to q0 or q1 or q2 ; we go to the state D= {q0 , q1 , q2 } = δD (B, 1) etc... 31 The Subset Construction We get the following automaton A= {q0 } B= {q0 , q1 } C= {q0 , q2 } D= {q0 , q1 , q2 } 0 1 →A A B B D C ∗C A B ∗D C D 32 The Subset Construction Same automaton, as a transition system B 1 0 A 0 0 1 D 1 C 1 0 The DFA “remembers” the last two bits seen and accepts if the next-to-last bit is 1 33 The Subset Construction Another example: words ending by 01 0,1 q0 0 q1 1 q2 The new states are A = {q0 } B = {q0 , q1 } C = {q0 , q2 } 0 1 →A B A B B C ∗C B A 34 The Subset Construction The DFA is 1 0 0 A 0 B 1 C 1 A = {q0 } B = {q0 , q1 } C = {q0 , q2 } This DFA has only 3 states (and not 8). It is correct i.e. accepts only the word ending by 01 by construction We had only to prove the general correctness of the subset construction 35 Example: password If we apply the subset construction to the NFA t h e n we get exactly the following DFA q0 6=t q5 t q1 6=h 6=e h q2 e q3 6=n For this NFA, δ is a partial function with a “stop” or “dead” state q5 = ∅ 36 n q4 An Application: Text Search Suppose we are given a set of words, called keywords, and we want to find occurences of any of these words. For such a problem, a useful way to proceed is to design a NFA which recognizes, by entering in an accepting state, that it has seen one of the keywords. The NFA is only a nondeterministic program, but we can run it using lists or transform it to a DFA and get a deterministic (efficient) program Once again, we know that this DFA will be correct by construction This is a good example of a derivation of a program (DFA) from a specification (NFA) 37 An Application: Text Search The following NFA searches for the keyword web and ebay B e C b D w Σ A e E b F a G y H Almost no thinking needed to write this NFA What is a corresponding DFA?? Notice that this has the same number of states as the NFA 38 Representation in functional programming \slideheading{Representation in functional programming} data Q = A | B | C | D | E | F| G | H next next next next next next next next next next next ’w’ ’e’ _ A ’e’ ’b’ ’b’ ’a’ ’y’ _ D _ H _ _ A A = B C E F G = = = = [A,B] = [A,E] [A] = [C] = [D] = [F] = [G] = [H] [D] [H] [] 39 Representation in functional programming run :: String -> Q -> [Q] run [] q = return q run (a:x) q = next a q >>= run x final final final final :: Q -> Bool D = True H = True _ = False accept :: String -> Bool accept x = or (map final (run x A)) 40 An Application: Text Search Even for searching an occurence of one keyword this gives an interesting program This is connected to the Knuth-Morris-Pratt string searching algorithm Better than the naive string searching algorithm 41 A Bad Case for the Subset Construction Theorem: Any DFA recognising the same language as the NFA 0,1 q0 1 q1 0,1 q2 has at least 25 = 32 states! 42 0,1 q3 0,1 q4 0,1 q5 A Bad Case for the Subset Construction Lemma 1: If A is a DFA then q.(xy) = (q.x).y for any q ∈ Q and x, y ∈ Σ∗ We have proved this last time 43 A Bad Case for the Subset Construction We define Ln = {x1y | x ∈ Σ∗ , y ∈ Σn−1 } A = (Q, Σ, δ, q0 , F ) Theorem: If |Q| < 2n then L(A) 6= Ln Lemma 2: If |Q| < 2n there exists x, y ∈ Σ∗ and u, v ∈ Σn−1 with q0 .(x0u) = q0 .(y1v) Proof of the Theorem, given Lemma 2: If L(A) = Ln we have y1v ∈ L(A) and x0u ∈ / L(A), so q0 .(y1v) ∈ F and q0 .(x0u) ∈ /F This contradicts q0 .(x0u) = q0 .(y1v) Q.E.D. 44 A Bad Case for the Subset Construction Proof of the lemma: The map z 7−→ q0 .z, injective because |Q| < 2n = |Σn | Σn → Q is not So we have a1 . . . an 6= b1 . . . bn with q0 .(a1 . . . an ) = q0 .(b1 . . . bn ) We can assume ai = 0, bi = 1. We take x = a1 . . . ai−1 , y = b1 . . . bi−1 and u = ai+1 . . . an 0i−1 , v = bi+1 . . . bn 0i−1 Notice then that (∗) implies, by Lemma 1 q0 .(a1 . . . an 0i−1 ) = q0 .(b1 . . . bn 0i−1 ) Q.E.D. 45 (∗)