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. . Theory of Relations (2) Sequential Machines and Finite Automata Course of Mathematics Pusan National University... . . Yoshhiro Mizoguchi Institute of Mathematics for Industry Kyushu University, JAPAN ym@imi.kyushu-u.ac.jp November 3-4, 2011 Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 1 / 34
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Table of Contents ... 1 Sequential Machine Preliminary Reachable and Observable Mimimal Realization ... 2 Finite Automata Introduction to Theory of Automata The Myhill-Nerode theorem Minimal Realization ... 3 Applications of Relational Calculus to Theory of Automata Nondeterministic Finite Automaton Coproduct and Product Automaton Reverse, Concatenate, Closure Examples ... 4 References Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 2 / 34
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Sequential Circuit (1) 1 2 0 1 0 2 0 0 1 0 0 1 1 0 dc S c Q 0 1 0 0 0 1 0 1 0 0 1 0 0 0 dc c R One of the elementary units of sequential circuits, the RS Flip-ﬂop circuitproduces an output signal sequence according to the sequence of inputsignals. Output signals are 1(on) or 0(o f f ) so we write the outputs asY = {0, 1}. There are two input signals S(et) and R(eset). We consider thepair (SR) of S and R, and we denote 0 = (00), 1 = (01) and 2 = (10). Sowe can consider the inputs as X = {0, 1, 2}.The model of a sequential circuit consists of the state set Q = {a, b}, thestate transition function δ : Q × X → Q, and the output functionβ : Q → Y. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 3 / 34
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Sequential Circuit (2)The table below is the value of δ and β. The ﬁgure is the state transitiondiagram of the RS Flip-ﬂop circuit. q x δ(q, x) a 0 a a 1 a q β(q) 0or1 0or2 ‡ ‡ a 2 b a 0 a/0 2E b/1 b 0 b b 1 i1 b 1 a b 2 bThe labels of a vertex consists of a state and an output symbol. An edgemeans a state transition and the label on an edge corresponds to the inputsymbol. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 4 / 34
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Sequential Circuit (3) An sequential circuit can be considered as a function from an input word to an output word. That is we can consider an sequential circuit as a function f : X∗ → Y ∗ where X∗ is the set of words over X including an empty string ε. . Problem ... Construct an algebraic model (Sequential Machine) of sequential circuits using a state set, a state transition function and an output function. What kind of function from input words to output words is realizable by a ﬁnite state sequential machine? How to construct an efﬁcient sequential machine with small number of. states... . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 5 / 34
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Epi-Mono FactorizationA function F : X → Y from a set X to a set Y can be represented by acomposition F(x) = m(e(x)) (x ∈ X) of an injective function m : Z → Yand an surjective function e : X → Z . The set Z is uniquely determined byf up to isomorphism. That is Z F(X) = {F(x) ∈ Y | x ∈ X} andZ X/ ∼ = {[x] | x ∈ X}, where the equivalent relation ∼ on X is deﬁnedby [ x ∼ x′ ⇔ F(x) = F(x′ )] and [x] = {x′ ∈ X | x ∼ x′ } is a set ofequivalence class including x.(Note)A function e : X → Z is surjective if there exists an element x ∈ X satisfyinge(x) = z for any element z ∈ Z . A function m : Z → Y is injective if m(z1 ) m(z2 )for any two elements z1 , z2 ∈ Z satisfying z1 z2 . We denote an injection by anarrow with tail , and a surjective by an arrow with head . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 6 / 34
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Sequential Machine.Deﬁnition (Sequential Machine) ... A sequential mmachine is a sextuple M = (X, Q, δ, q0 , Y, β) where X is the set of inputs, Q is the set of states, δ:Q×X →Q is the transition function, q0 ∈ Q is the initial state, Y is the set of outputs, and. β:Q→Y is the output map... . .(Note) This sequential machine (SM) is called Moore style SM. The Mealy styleSM is deﬁned by an alternate output map λ : Q × X → Y insted of β. These twomodel are equivalent. Mealy style does not have an output for the initial state, butthe rest of relations between inputs and outputs are mutually transformable.We sometime deﬁne SM by pentad without an initial state. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 7 / 34
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Run map and Response map.Deﬁnition (Run map) ... Let δ : Q × X → Q be a transition function. We deﬁne its run map to be the unique map δ∗ : Q × X∗ → Q deﬁned inductively by δ∗ (q, ε) = q, and δ∗ (q, xw) = δ∗ (δ(q, x), w) ( q ∈ Q, x ∈ X, w ∈ X∗ ). . .. . . Let f : X∗ → Y be a function. We also deﬁne the function f∗ : X∗ → Y ∗ deﬁned inductively by f∗ (ε) = f (ε), and f∗ (wx) = f∗ (w) f (wx) ( x ∈ X, w ∈ X∗ ). .Deﬁnition (Response map) ... Let M = (X, Q, δ, q0 , Y, β) be a sequential machine. We deﬁne its response map to be the map ( f M )∗ : X∗ → Y ∗ where f M : X∗ → Y is deﬁned by f M (w) = β(δ∗ (q0 , w)) (w ∈ X∗ ). . .. . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 8 / 34
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Realization (1) . Deﬁnition ... Let t : X ∗ → Y ∗ be a function. If there exists a sequential machine M such that t = ( f M )∗ , then M is called a realization of t . . .. . . We call t : X∗ → Y ∗ is realizable if there exisits a function f : X∗ → Y such that t = f∗ . . Proposition ... A function t : X ∗ → Y ∗ is realizable if and only if for any w ∈ X ∗ , x ∈ X there exists y ∈ Y such that t(wx) = t(w)y. . .. . . The condition is equivalent that the value t(wx) is depending on only w and is not depending on x. We call a function t : X∗ → Y ∗ satisfying the condition as a sequential function. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 9 / 34
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Realization (2).Proposition ... Let t : X∗ → Y ∗ be a sequential function. Then there exist a sequential machine M which is a realization of t . . .. . . Let f : X∗ → Y be a function and t = f∗ . We introduce two kinds of sequential machines which is a realization of t . M I = (X, X∗ , δ I , ε, Y, f ) δ I (w, x) = wx (w ∈ X∗ , x ∈ X). ∗ MT = (X, Y X , δT , f, Y, βT ) ∗ Y X is the set of all maps from X∗ to Y , that is { f | f : X∗ → Y}, δT ( f, x) : X∗ → Y is deﬁned by δT ( f, x)(w) = f (xw) ( x ∈ X, w ∈ X∗ ), ∗ and βT ( f ) = f (ε) ( f ∈ Y X ).We can verify easily f = f MI = f MT . We note that both M I and MT is nota ﬁnite sequential machine. That is the state set is not a ﬁnite set. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 10 / 34
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Finite Realization . Problem ... What kind of sequential function f : X ∗ → Y which have a ﬁnite state realization? . .. . . The state sets of MT is inﬁnite but the most of states are unreachable from the initial state f . If the states reachable from f is ﬁnite, then we can construct a ﬁnite sequential machine from MT . That is the condition to ∗ have a ﬁnite representation is that the set Z = {δ∗ ( f, w) ∈ Y X | w ∈ X∗ } is T ﬁnite. ∗ We deﬁne a function F : X∗ → Y X by F(w) = δ∗ ( f, w) (w ∈ X∗ ). F is T divided to the composition of a surjection and an injection. Since Z = F(X∗ ), we have Z = X∗ / ∼ by the equivalence relation ∼ deﬁned by [w ∼ w′ ⇔ δ∗ ( f, w) = δ∗ ( f, w′ )]. We note δ∗ ( f, w)(z) = δ∗ ( f, w′ )(z) is T T T T f (wz) = f (w′ z) for any z ∈ X∗ . If the number of the equivalence class is ﬁnite, then there exists a ﬁnite realization. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 11 / 34
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Reachable and ObservableLet M = (X, Q, δ, q0 , Y, β) be a sequential machine and its response map ∗f M : X∗ → Y . We deﬁne F : X∗ → Y X by F(w) = δ∗ ( f M , w). T ∗F is divided into a composition of fe : X∗ → Q and f m : Q → Y X suchthat F(w) = f m( fe (w)),where fe (w) = δ∗ (q0 , w), and f m(q)(w) = β(δ∗ (q, w)) (w ∈ X∗ , q ∈ Q). If fe is a surjection then we call M as reachable. If f m is an injection then we call M as obserbable or reduced. We note that a reachable and obserbable sequential machine is minimal representation of f M . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 12 / 34
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Minimal RealizationLet M = (X, Q, δ, q0 , Y, β) be a sequential machine. and F = f m ◦ fe . ∗Assume fe : X∗ → Q is a surjection. If f m : Q → Y X is not an injection,then we can construct a minimal realization using the epi-monofactorization of f m.The equivalence relation [ q ∼ q′ ⇔ f m(q) = f m(q′ )] on Q is that f m(q)(w) = f m(q′ )(w) for any w ∈ X∗ . That is β(δ∗ (q, w)) = β(δ∗ (q′ , w)). IfQ is a ﬁnite set and |Q| = n, then it is sufﬁcient to check the conditionβ(δ∗ (q, w)) = β(δ∗ (q′ , w)) for ﬁnite number of words w with |w| n. So wecan check q ∼ q′ for any state q and q′ in ﬁnite steps, and we canconstruct a minimal state sequential machine. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 13 / 34
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Introduction to Theory of AutomataThe ﬁrst published study on automata [6] is the Automata studies in 1956edited by C. E. Shanon who is famous as an originator of the informationtheory and J. McCarthy who is a famous researcher of the ﬁelds artiﬁcialintelligence.At that time, they formalized an abstract model of a sequentil circuit andinvestigated relationship between inputs and outputs analyzing a statetransition functions.Once the notion of accept states is introduced, a machine is considered asan acceptor and investigations of recognized language are started. This isthe origin of the theory of language and automata.The ﬁrst paper [4] about ﬁnite state automata is ’ﬁnite Automata and TheirDecision Problems’ by M. O. Rabin and D. Scott pubshed in 1959. Theywere awarded an ACM Turing aword in 1976 for this research. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 14 / 34
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Alphabet, Word, ConcatenationAn alphabet is a ﬁnite, nonempty set. The elements of an plphabet arefeferred to as letters, or symbols. A word over an alphabet is a ﬁnitestring consisting of zero or more letters of the alphabet, in which the sameletter may occur several times. The string consisting of zero letters iscalled the empty word, written ε. The length of a word w, denoted by |w|,is the number of letters in w. Again by deﬁnition, |ε| = 0.Let Σ be an alphabet. The set of all words over an alphabet Σ is denotedby Σ∗ . The sets Σ∗ is inﬁnite for any Σ. Algebraically speaking, Σ∗ is thefree monoid with the identity ε generated by Σ.For words w1 and w2 , the juxtaposition w1 w2 is called the concatenationof w1 and w2 . The empty word is an identity with respect to concatenation,εw = wε = w holds for all word w. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 15 / 34
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Kleene ClosureSubsets of Σ∗ are referred to as formal languages, or brieﬂy, languagesover Σ.The concatenation (or product) of two languages L1 and L2 is deﬁned byL1 L2 = {w1 w2 | w1 ∈ L1 , w2 ∈ L2 }.The (Kleene) closure L∗ of a language L is deﬁned to be the union of allpowers of L, that is L∗ = ∪∞ L n where L0 = {ε}, and L n = L n−1 · L n=0( n ≥ 1).The closure of all words Σ is Σ∗ and there is no confusions deﬁned as aset of all words over Σ. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 16 / 34
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Deterministic Automata (1) . Deﬁnition ... Deterministic Automata A determinisiic ﬁnite automaton is a pentad M = (Σ, Q, δ, q0 , F) where Σ is the alphabet, Q is the ﬁnite set of states, δ:Q×Σ→Q is the transition function, q0 ∈ Q is the initial state, and. F⊂Q is the set of accept states... . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 17 / 34
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Deterministic Automata (2) s1 s2 · · · s n inputs T E move head to right after a state transition Controler( q)Finite Automaton MA ﬁnite automaton is illustrated as above ﬁgure. The input letters are oninput tape. The ﬁrst state of controller is the initial state q0 . If the state is qand input letter is s then the state is changed to δ(q, s). After changing thestate the head is moved to right. Repeating these procedures until the endof an input word. If the head reached to the end of an input word, thencheck the state is accept state or not. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 18 / 34
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Recognized Language Let δ : Q × Σ → Q be a state transition function. The function δ∗ : Q × Σ∗ → Q is uniquely determined by δ∗ (q, ε) = q, and δ∗ (q, wa) = δ(δ∗ (q, w), a) (w ∈ Σ∗ , a ∈ Σ). . Deﬁnition ... The recognized language L(M) ⊂ Σ∗ is deﬁned by L(M) = {w ∈ Σ∗ | δ∗ (q0 , w) ∈ F}.L(M) is referred to as the language accepted by M.... . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 19 / 34
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Example . Example ... Let M = (Q, Σ, δ, q0 , F) be a ﬁnite automaton where Q = {q0 , q1 , q2 }, F = {q1 }, q0 ∈ Q. The state transition function δ : Q × Σ → Q is deﬁned as follows δ(q0 , a) = q1 , δ(q0 , b) = q2 , δ(q1 , a) = q2 , δ(q1 , b) = q0 , δ(q2 , a) = q2 , δ(q2 , b) = q2 .For example, the word w = aba is acceptable δ∗ (q0 , aba)= δ∗ (δ(q0 , a), ba) = δ∗ (q1 , ba) = δ∗ (δ(q1 , b), a) = δ∗ (q0 , a) = δ∗ (δ(q0 , a), ε)= ∗. δ (q1 , ε) = q1 ∈ F... . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 20 / 34
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State Transition DiagramA ﬁnite automaton M is denoted by a following ﬁgure called state transitiondiagram. Vertices are states and the initial state has an arrow withoutlabel. According to the input letters follow arrows with same label withinput letter. The vertex corresponding to an accept state has double circleand if the following the input letters ended at the vertices with double circlethen the input word is accepted. a, b c q0 b q2 i A b q E a 1 a Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 21 / 34
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The Myhill-Nerode theorem (1) . Deﬁnition ... An equivalence relation ∼ on Σ∗ is said to be right invariant if ”w1 ∼ w2 ⇒ w1 z ∼ w2 z (∀z ∈ Σ∗ )” for any w1 , w2 ∈ Σ∗ . An equivalence relation ∼ is ﬁnite index if the number of equivalence classes is ﬁnite. That is {[x] | x ∈ Σ∗ } is ﬁnite set where. [x] = {x′ | x ∼ x′ }... . .Let L Σ∗ . We deﬁne a relation w1 ∼ L w2 on Σ∗ by w1 z ∈ L ⇔ w2 z ∈ L (∀z ∈ Σ∗ ).The relation ∼ L is a right invariant equivalent relation. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 22 / 34
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The Myhil-Nerode theorem (2).Theorem (Nerode) ... Let L Σ∗ be a language on Σ. Then, the following three conditions (1), (2) and (3) are equivalent. (1) The set L is acpted by some ﬁnite automaton. (2) L is the union of some of the equivalent classes of a right invariant equivalence relation of ﬁnite index. (3) The right invariant equivalence relation ∼ L induced by L is of ﬁnite. index... . .Let L be the language L in (3) and Q = {[w] | w ∈ Σ ∗ }, δ([w], a) = [wa],q0 = [ε], F = {[w] | w ∈ L}. Then M = (Q, Σ, δ, q0 , F) is a ﬁnite automatonand L = L(M). Further, if L = L(M′ ) for some ﬁnite automataM′ = (Q′ , Σ, δ′ , q′ , F′ ) then |Q| ≤ |Q′ |. That is M is a minimal state 0automaton with L = L(M). Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 23 / 34
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Minimal RealizationLet M = (X, Q, X, δ, q0 , Y, β) be a sequential machine and Y = {0, 1}.There is an one-to-one coresspondence between an output mapβ : Q → {0, 1} and a subset F = {q ∈ Q | β(q) = 1} of Q. That is a ﬁniteautomata is considered as a sequential machine with Y = {0, 1}.A ﬁnite sequential machine is exactly a ﬁnite automaton. Further, afunction f : X∗ → Y is corresponde to a subset of X∗ that is a recognizedlanguage.In the previous section, we construct a minimal realization of f using anequivalence relation w ∼ w′ deﬁned by f (wz) = f (wz′ ) (∀z ∈ X∗ ). This isthe equivalence relation ∼ L induced by L. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 24 / 34
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Example.Example (Minimalize) ... The automton M′ in right ﬁgure is the minimalized automaton of M in left ﬁgure. We note p0 = {q0 , q2 }, p1 = {q1 , q3 }, and p2 = {q4 , q5 }. . .. . .
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b j $ q1 WB $ X a a b B a Ep2 a M M′ Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 25 / 34
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Nondeterministic Finite AutomatonLet Σ be an alphabet, I one point set {∗}..Deﬁnition (Kawahara 1988[3]) ... A nondeterministic ﬁnite automaton (NFA) M = (Q, τ, δ a (a ∈ Σ), β) is a quadruple where Q is a ﬁnite set of states, τ:I⇁Q is the inital state relation, δa : Q ⇁ Q is the set of state-transition relations, and. β:Q⇁I is the ﬁnal state relation... . .For an input string w = σ1 σ2 · · · σ n ∈ Σ∗ (σi ∈ Σ, 1 ≤ i ≤ n, 0 n), the iterativestate-transition relation δw : Q ⇁ Q is deﬁned by δw = δσ1 δσ2 · · · δσn . . Deﬁnition ... The language accepted by a NFA M is deﬁned by. L(M) = {w ∈ Σ∗ | τδw β♯ = id I }... . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 26 / 34
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Coproduct Automaton . Deﬁnition ... Let M = (Q, τ, δ a (a ∈ Σ), β) and M ′ = (Q′ , τ′ , δ′ (a ∈ Σ), β′ ) be NFAs. The a coproduct automaton of M and M′ is deﬁned by M + M′ = (Q + Q′ , τ, δ a (a ∈ Σ), β) where τ = τ ⊥ τ′ , δ a = δ a + δ′ , and ˆ ˆ ˆ ˆ ˆ a β=β⊥β ˆ ′. . .. . . . Proposition ... Let M + M′ be the coproduct automaton of M and M′ . Then (a) δw = δw + δ′ for w ∈ Σ∗ . ˆ w (b) τδw β♯ = τδw β♯ ⊔ τ′ δ′ (β′ )♯ . ˆˆ ˆ w ′ ′. (c) L(M + M ) = L(M) ∪ L(M )... . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 27 / 34
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Product Automatan . Deﬁnition ... Let M = (Q, τ, δ a (a ∈ Σ), β) and M ′ = (Q′ , τ′ , δ′ (a ∈ Σ), β′ ) be NFAs. The a product automaton of M and M ′ is deﬁned by M × M′ = (Q × Q′ , τ, δ a (a ∈ Σ), β) where τ = τ⊤τ′ , δ a = δ a × δ′ , and ˆ ˆ ˆ ˆ ˆ a β = β⊤β ˆ ′. . .. . . . Proposition ... Let M + M′ be the coproduct automaton of M and M′ . Then (a) δw = δw × δ′ for w ∈ Σ∗ . ˆ w (b) τδw β♯ = τδw β♯ ⊓ τ′ δ′ (β′ )♯ . ˆˆ ˆ w ′ ′. (c) L(M × M ) = L(M) ∩ L(M )... . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 28 / 34
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Reverse Automaton . Deﬁnition ... Let M = (Q, τ, δ a (a ∈ Σ), β) be a NFA. The reverse automaton of M is ♯ deﬁned by M R = (Q, τ R , δ R (a ∈ Σ), β R ) where τ R = β, δ R = δ a and a a . = τ. βR .. . . .Proposition ... Let M R be the reverse automaton of M. Then (a) δw = (δwR )♯ where w R is a reverse string of w ∈ Σ∗ . R (b) τ R δw (β R )♯ = (τδwR β♯ )♯ . R. (c) L(M ) = L(M) where L(M) = {w | w ∈ L(M)}. R R R R.. . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 29 / 34
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Concatenate Automatan . Deﬁnition ... Let M = (Q, τ, δ a (a ∈ Σ), β) and M ′ = (Q′ , τ′ , δ′ (a ∈ Σ), β′ ) be NFAs. The a concatenate automaton of M and M ′ is deﬁned by M · M′ = (Q + Q′ , τ, δ a (a ∈ Σ), β) where γ = β♯ τ′ , τ = τ(i ⊔ γ j), ˆ ˆ ˆ ˆ ˆ a = i♯ δ a i ⊔ i♯ γδ′ j ⊔ j♯ δ′ j, and β = β′ (γ♯ i ⊔ j). δ ˆ . a a .. . . The function δw : Q ⇁ Q′ (w ∈ Σ∗ ) is uniquely determined by δε = 0QQ′ o o and δwa = (δw γ ⊔ δw )δ′ . o o w . Proposition ... Let M · M′ be the concatenate automaton of M and M′ . Then (a) δw = i♯ δw i ⊔ i♯ δw j ⊔ j♯ δ′ j. ˆ o w (b) δw γ ⊔ δw ⊔ γδ′ = ⊔w=uv δu γδ′ . o w v (c) τδw (β)♯ = ∪w=uv τδu γδ′ (β′ )♯ . ˆˆ ˆ v ′ ′ . (d) L(M · M ) = L(M) · L(M ). .. . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 30 / 34
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Closure Automaton . Deﬁnition ... Let M = (Q, τ, δ a (a ∈ Σ), β) be a NFA. The coosure automaton of M + is deﬁned by M+ = (Q, τ, δ+ (a ∈ Σ), β) where γ = β♯ τ, and δ+ = (idQ ⊔ γ)δ a . . a a .. . . . Proposition ... Let M+ be the closure automaton of M. Then (a) δ+ = ⊔w=u1 ···u k ,u k w ε,k0 δu1 γδu2 γ · · · γδu k for w ε. (b) τδ+ β♯ = ⊔w=u1 ···u k ,k0 τδu1 γδu2 γ · · · γδu k β♯ w for w ε.. (c) L(M+ ) = L(M)+ ... . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 31 / 34
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Examples . Proposition ... Let Mϕ = (I, τ, δ a (a ∈ Σ), β) be a NFS where β = 0 II . Then L(Mϕ ) = ϕ. Let Mε = (I, τ, δ a (a ∈ Σ), β)) be a NFS where τ = id I , δ a = 0 II ( a ∈ Σ), and β = id I . Then L(Mε ) = {ε}. Let σ ∈ Σ and Mσ = (I + I, δ a (a ∈ Σ), β) where τ = i, δσ = i♯ j, . δ a = 0 I+I,I+I ( a σ), and β = j. Then L(Mσ ) = {σ}. .. . . The Kleene closure automaton of M is deﬁned by M ∗ = Mε + M+ . Then we have T(M∗ ) = L(Mε + M+ ) = T(Mε ) ∪ T(M+ ) = {ε} ∪ T(M)+ = T(M)∗ . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 32 / 34
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Regular Language . Deﬁnition ... A language L ⊂ Σ∗ is a regular if there exists a NFA M such that . = L(M). L .. . . .Proposition ... Let L and L′ be regular languages. Then L = ϕ, L = {ε} and L = {σ} (σ ∈ Σ) are regular languages. Σ∗ − L, L R , L+ and L∗ are regular languages.. L ∩ L′ , L ∪ L′ and L · L′ are regular languages... . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 33 / 34
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References[1] M.A.Arbib and E.G.Manes: Machines in a cagegory, an expository introduction, SIAM Review, 16(1974), 163-192.[2] J.E.Hopcroft and J.D.Ullman: Formal Languages and Their Relation to Automata (2nd. Ed.), Addison-Wesley(2001).[3] Y. Kawahara, Applications of relational calculus to computer mathematics, Bulletin of Informatics and Cybernetics, 23(1988), 67-78.[4] M.O.Rabin and D.Scott: Finite Automata and Their Decision Problems, IBM Journal, 3(1959), 114-125.[5] A. Salomaa: Computation and Automata, Cambridge University Press(1985).[6] C.E.Shannon and J.Mac Carthy (eds.): Automata Studies, Princeton University Press(1956). Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata November 3-4, 2011 34 / 34
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