C2.1 Regular Grammars
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1 Theory of Computer Science March 6, 26 C2. Regular Languages: Finite Automata Theory of Computer Science C2. Regular Languages: Finite Automata Malte Helmert University of Basel March 6, 26 C2. Regular Grammars C2.2 DFAs C2.3 NFAs C2.4 Summary M. Helmert (Univ. Basel) Theorie March 6, 26 / 29 M. Helmert (Univ. Basel) Theorie March 6, 26 2 / 29 C2. Regular Languages: Finite Automata Regular Grammars C2. Regular Languages: Finite Automata Regular Grammars Repetition: Regular Grammars C2. Regular Grammars Definition (Regular Grammars) A regular grammar is a 4-tuple Σ, V, P, S with Σ finite alphabet of terminals 2 V finite set of variables (with V Σ = ) 3 P (V (Σ ΣV )) { S, ε } finite set of rules 4 if S ε P, there is no X V, y Σ with X ys P 5 S V start variable. Rule X ε is only allowed if X = S and S never occurs in the right-hand side of a rule. How restrictive is this? M. Helmert (Univ. Basel) Theorie March 6, 26 3 / 29 M. Helmert (Univ. Basel) Theorie March 6, 26 4 / 29
2 C2. Regular Languages: Finite Automata Regular Grammars Epsilon Rules C2. Regular Languages: Finite Automata Regular Grammars Epsilon Rules For every grammar G with rules P V (Σ ΣV {ε}) there is a regular grammar G with L(G) = L(G ). Proof. Let G = Σ, V, P, S be a grammar s.t. P V (Σ ΣV {ε}). Let V ε = {A V A ε P}. Let P be the rule set that is created from P by removing all rules of the form A ε (A S). Additionally, for every rule of the form B xa with A V ε, B V, x Σ we add a rule B x to P.... M. Helmert (Univ. Basel) Theorie March 6, 26 5 / 29 For every grammar G with rules P V (Σ ΣV {ε}) there is a regular grammar G with L(G) = L(G ). Proof (continued). Then L(G) = L( Σ, V, P, S ) and P contains no rule A ε with A S. If S ε P, we are done. Otherwise, let S be a new variable and construct P from P by replacing rules X as where X V, a Σ with X as, 2 for every rule S ax where X V, a Σ adding the rule S ax, and 3 for every rule S a where a Σ adding the rule S a. Then L(G) = L( Σ, V {S }, P, S ). M. Helmert (Univ. Basel) Theorie March 6, 26 6 / 29 Finite Automata: C2.2 DFAs q q q 2 When reading the input the automaton visits the states q, q, q, q, q, q 2. M. Helmert (Univ. Basel) Theorie March 6, 26 7 / 29 M. Helmert (Univ. Basel) Theorie March 6, 26 8 / 29
3 Finite Automata: Terminology and Notation Deterministic Finite Automaton: Definition q q q 2 Definition (Deterministic Finite Automata) A deterministic finite automaton (DFA) is a 5-tuple M = Q, Σ, δ, q, E where Q is the finite set of states states Q = {q, q, q 2 } input alphabet Σ = {, } transition function δ start state q end states {q 2 } δ(q, ) = q δ(q, ) = q δ(q, ) = q 2 δ(q, ) = q δ(q 2, ) = q 2 δ(q 2, ) = q δ q q q q q 2 q q 2 q 2 q table form of δ Σ is the input alphabet (with Q Σ = ) δ : Q Σ Q is the transition function q Q is the start state E Q is the set of end states German: deterministischer endlicher Automat, Zustände, Eingabealphabet, Überführungs-/Übergangsfunktion, Startzustand, Endzustände M. Helmert (Univ. Basel) Theorie March 6, 26 9 / 29 M. Helmert (Univ. Basel) Theorie March 6, 26 / 29 DFA: Recognized Words DFA: Accepted Language Definition (Words Recognized by a DFA) DFA M = Q, Σ, δ, q, E recognizes the word w = a... a n if there is a sequence of states q,..., q n Q with q = q, 2 δ(q i, a i) = q i for all i {,..., n} and 3 q n E. German: DFA erkennt das Wort q q q 2 recognizes: does not recognize: ε Definition (Language Accepted by a DFA) Let M be a deterministic finite automaton. The language accepted by M is defined as L(M) = {w Σ w is recognized by M}. q q q 2 The DFA accepts the language {w {, } w ends with }. M. Helmert (Univ. Basel) Theorie March 6, 26 / 29 M. Helmert (Univ. Basel) Theorie March 6, 26 2 / 29
4 Languages Accepted by DFAs are Regular Languages Accepted by DFAs are Regular Every language accepted by a DFA is regular (type 3). Proof. Let M = Q, Σ, δ, q, E be a DFA. We define a regular grammar G with L(G) = L(M). Define G = Σ, Q, P, q where P contains a rule q aq for every δ(q, a) = q, and a rule q ε for every q E. (We can eliminate forbidden epsilon rules as described at the start of the chapter.)... Every language accepted by a DFA is regular (type 3). Proof (continued). For every w = a a 2... a n Σ : w L(M) iff there is a sequence of states q, q,..., q n with iff q = q, q n E and δ(q i, a i) = q i for all i {,..., n} iff there is a sequence of variables q, q,..., q n with iff q is start variable and we have q a q a a 2 q 2 iff a a 2... a n q n a a 2... a n. iff w L(G) : blackboard M. Helmert (Univ. Basel) Theorie March 6, 26 3 / 29 M. Helmert (Univ. Basel) Theorie March 6, 26 4 / 29 Question Is the inverse true as well: for every regular language, is there a DFA that accepts it? That is, are the languages accepted by DFAs exactly the regular languages? C2.3 NFAs Yes! We will prove this later (via a detour). Picture courtesy of imagerymajestic / FreeDigitalPhotos.net M. Helmert (Univ. Basel) Theorie March 6, 26 5 / 29 M. Helmert (Univ. Basel) Theorie March 6, 26 6 / 29
5 Nondeterministic Finite Automata Nondeterministic Finite Automata: Why are DFAs called deterministic automata? What are nondeterministic automata, then?, q q q 2 differences to DFAs: multiple start states possible transition function δ can lead to zero or more successor states for the same a Σ automaton recognizes a word if there is at least one accepting sequence of states Picture courtesy of stockimages / FreeDigitalPhotos.net M. Helmert (Univ. Basel) Theorie March 6, 26 7 / 29 M. Helmert (Univ. Basel) Theorie March 6, 26 8 / 29 Nondeterministic Finite Automaton: Definition NFA: Recognized Words Definition (Nondeterministic Finite Automata) A nondeterministic finite automaton (NFA) is a 5-tuple M = Q, Σ, δ, S, E where Q is the finite set of states Σ is the input alphabet (with Q Σ = ) δ : Q Σ P(Q) is the transition function (mapping to the power set of Q) S Q is the set of start states E Q is the set of end states German: nichtdeterministischer endlicher Automat DFAs are (essentially) a special case of NFAs. Definition (Words Recognized by an NFA) NFA M = Q, Σ, δ, S, E recognizes the word w = a... a n if there is a sequence of states q,..., q n Q with q S, 2 q i δ(q i, a i) for all i {,..., n} and 3 q n E., q q q 2 recognizes: does not recognize: ε M. Helmert (Univ. Basel) Theorie March 6, 26 9 / 29 M. Helmert (Univ. Basel) Theorie March 6, 26 2 / 29
6 NFA: Accepted Language NFAs are No More Powerful than DFAs Definition (Language Accepted by an NFA) Let M = Q, Σ, δ, S, E be a nondeterministic finite automaton. The language accepted by M is defined as L(M) = {w Σ w is recognized by M}., q q q 2 The NFA accepts the language {w {, } w = or {w {, } w ends with }. (Rabin, Scott) Every language accepted by an NFA is also accepted by a DFA. Proof. For every NFA M = Q, Σ, δ, S, E we can construct a DFA M = Q, Σ, δ, q, E with L(M) = L(M ). Here M is defined as follows: Q := P(Q) (the power set of Q) q := S E := {Q Q Q E } For all Q Q : δ (Q, a) := q Q δ(q, a)... M. Helmert (Univ. Basel) Theorie March 6, 26 2 / 29 M. Helmert (Univ. Basel) Theorie March 6, / 29 NFAs are No More Powerful than DFAs NFAs are More Compact than DFAs (Rabin, Scott) Every language accepted by an NFA is also accepted by a DFA. Proof (continued). For every w = a a 2... a n Σ : w L(M) iff there is a sequence of states q, q,..., q n with iff q S, q n E and q i δ(q i, a i ) for all i {,..., n} iff there is a sequence of subsets Q, Q,..., Q n with iff Q = q, Q n E and δ (Q i, a i ) = Q i for all i {,..., n} iff w L(M ) : blackboard For k consider the language L k = {w {, } w k and the k-th last symbol of w is }. The language L k can be accepted by an NFA with k + states:, q,,, q q 2... q k There is no DFA with less than 2 k states that accepts L k (without proof). NFAs can often represent languages more compactly than DFAs. M. Helmert (Univ. Basel) Theorie March 6, / 29 M. Helmert (Univ. Basel) Theorie March 6, / 29
7 Regular Grammars are No More Powerful than NFAs For every regular grammar G there is an NFA M with L(G) = L(M). Proof. Let G = Σ, V, P, S be a regular grammar. Define NFA M = Q, Σ, δ, S, E with Q = V {X }, X V S = {S} { {S, X } if S ε P E = {X } if S ε P B δ(a, a) if A ab P X δ(a, a) if A a P M. Helmert (Univ. Basel) Theorie March 6, / 29 Regular Grammars are No More Powerful than NFAs For every regular grammar G there is an NFA M with L(G) = L(M). Proof (continued). For every w = a a 2... a n Σ with n : w L(G) iff there is a sequence on variables A, A 2,..., A n with iff a A a a 2 A 2 a a 2... a n A n a a 2... a n. iff there is a sequence of variables A, A 2,..., A n with iff A δ(s, a ), A 2 δ(a, a 2 ),..., X δ(a n, a n ). iff w L(M). Case w = ε is also covered because S E iff S ε P. M. Helmert (Univ. Basel) Theorie March 6, / 29 Finite Automata and Regular Languages C2. Regular Languages: Finite Automata Summary regular grammar C2.4 Summary DFA NFA In particular, this implies: Corollary L regular L is accepted by a DFA. L regular L is accepted by an NFA. M. Helmert (Univ. Basel) Theorie March 6, / 29 M. Helmert (Univ. Basel) Theorie March 6, / 29
8 C2. Regular Languages: Finite Automata Summary Summary We now know three formalisms that all describe exactly the regular languages: regular grammars, DFAs and NFAs We will get to know a fourth formalism in the next chapter. DFAs are automata where every state transition is uniquely determined. NFAs recognize a word if there is at least one accepting sequence of states. M. Helmert (Univ. Basel) Theorie March 6, / 29
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