1.2 Classical finite-state automata theory

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April 23, 2004

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Basic formal concepts

1.1 Formal Background

1.1.1 Monoid

In this section we present a brief introduction in the monoid theory.

Definition 1.1.1 AmonoidM(semigroup with unit element) is a triplehM,◦,ei where

• M is a non-empty set, the set of monoid elements;

• ◦:M×M→M is the monoid operation (we will use infix notation);

• e∈M is the monoid unit element, and the following conditions hold:

• ∀a, b, c∈M(a◦(b◦c) = (a◦b)◦c) (associativity),

• ∀a∈M(a◦e=e◦a=a).

Later we often useM to denote the monoidM=hM,◦,ei.

LetM=hM,◦,eibe a monoid. ForT1, T2⊆M, we define:

T1◦T2:={t1◦t2|t1∈T1, t2∈T2}.

It is easy to show that for any monoid M = hM,◦,ei the triple Mc = 2^M,◦,{e}

is again a monoid. Note the we use the same symbol for the basic monoid operation and for the “lifted” version that acts on subsets.

Definition 1.1.2 LetM=hM,◦,eiis a monoid and T ⊆M, then for every n∈N, we defineTⁿ inductively:

1. T⁰={e}

2. T^k+1=T^k◦T andT⁺=S∞

k=1T^k,T^∗=S∞

k=0T^k. We callT^∗theiterationorKleene-Star ofT.

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4 CHAPTER 1. BASIC FORMAL CONCEPTS Definition 1.1.3 A subsetT ⊆M of a monoidMis asubmonoidofMif

e∈T andT²⊆T.

Proposition 1.1.4 For any subset T ⊆ M of a monoid M the set T^∗ is the least submonoid of Mcontaining T.

Definition 1.1.5 LetM1 =hM1,◦,e1iand M2=hM2,•,e2ibe monoids. A function ϕ:M1→M2 is amonoid homomorphismiff the following conditions hold:

• ϕ(e1) =e2

• ∀a, b∈M1(ϕ(a◦b) =ϕ(a)•ϕ(b))

LetT ⊆M1, then withϕ(T)⊆M2 we will denote the set{ϕ(m)|m∈T}.

Proposition 1.1.6 The composition of two homomorphisms is again homomorphism.

Proposition 1.1.7 Letϕ:M1→M2 be a monoid homomorphism.

1. LetT_α⊆M1, for everyα∈A=⇒ϕ(S

α∈AT_α) =S

α∈Aϕ(Tα) 2. LetT1, T2⊆M1=⇒ϕ(T1◦T2) =ϕ(T1)•ϕ(T2)

3. LetT ∈M1=⇒ϕ(T^∗) =ϕ(T)^∗

Proof. The proofs of Points 1, 2 are straightforward. The proof of Point 3 is based on a simple induction, using Points 1, 2.

Using Point 2 of the above proposition it is easy shown that ϕ induces a monoid homomorphismϕ: 2^M¹→2^M².

Definition 1.1.8 Let M1 = hM1,◦,e1i and M2 = hM2,•,e2i be monoids.

Then the triple M1× M2=hM1×M2,◦ × •,he1,e2iiis a monoid called the monoid cartesian product. Here◦ × •denotes the function ◦ × •:M1×M2→ M1×M2 defined ashu1, v1i ◦ × • hu2, v2i=hu1◦u2, v1•v2i.

Example 1.1.9 Some examples of monoids are listed below:

1. Let a set X be given. The set of all relations r ⊆X×X with relation composition as monoid operation and the identity function as unit element is a monoidRel(X).

2. The set of all (partial) functions f : X → X is a submonoid of Rel(X) written (pFun(X))Fun(X).

3. The set of all bijection ofX defines a submonoid ofRX, which is a group called the permutation group ofX.

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Analphabet Σ is a set of symbols.

Definition 1.1.10 A wordwover an alphabet Σ is an-tuple w=ha1, . . . , a_ni, where (n≥0) anda_i∈Σ fori= 1, . . . , n.

The integernis called thelengthofw. Later we use|w|to denote the length of w. ifn= 0 we have the only 0-tupleεcalled theempty word. Theconcatenation of two wordsu=ha1, . . . , aniandw=hb1, . . . , bmiover Σ is the word over Σ

u·v=ha1, . . . , a_n, b1, . . . , b_mi.

Clearly|u·v|=|u|+|v|and|ε|= 0. Later we may write w=a1. . . an for w=ha1, . . . , ani. Recall that words represent finite strings.

Definition 1.1.11 The set of words over an alphabet Σ with concatenation as monoid operation and the empty word ε as unit element is called the free monoidΣ^∗=hΣ^∗,·, εiover Σ.

Clearly, the only submonoid of Σ^∗ containing Σ is Σ^∗, and for each element of Σ^∗ there exists a unique presentation as a finite concatenation of elements of Σ.

Proposition 1.1.12 Let M = hM,◦,ei be a monoid, let Σ be an alphabet and ϕ : Σ → M be a function. Then the natural extension φ of ϕ over Σ^∗, inductively defined as

1. φ(ε) =e

2. φ(σ·a) =φ(σ)◦ϕ(a), whereσ ∈Σ^∗, a∈Σ,

is the only homomorphism between the monoidsΣ^∗andMwhich is an extension ofϕ.

Definition 1.1.13 Lets∈Σ^∗. Then v∈Σ^∗ is aninfix ofsifs=u·v·wfor u, w∈Σ^∗. Ifu=εthenv is aprefix ofs. Ifw=εthenv is a sufixofs.

Definition 1.1.14 Thereverse functionρ: Σ^∗ →Σ^∗ is defined as ρ(ε) :=ε, ∀a∈Σ :ρ(a) :=a, ∀u, v∈Σ^∗:ρ(u·v) :=ρ(v)·ρ(u).

A subset L⊆Σ^∗ is called a languageover Σ. Later we shall consider finite alphabets only.

1.2 Classical finite-state automata theory

1.2.1 Finite-state automata and automaton languages

Definition 1.2.1 A finite state automaton (FSA) is a tuple of the formA= hΣ, Q, I, F,∆i where Σ is the input alphabet,Q is the set of states, I ⊆Q is the set of initial states,F ⊆Q is the set of final states, and ∆⊆Q×Σ^ε×Q is the transition relation. The transitionhq, a, si ∈∆ beginsatq,endsatpand has thelabela. Here Σ^ε:= Σ∪ {ε}.

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6 CHAPTER 1. BASIC FORMAL CONCEPTS If we have ∆⊆Q×Σ×Q, then the FSAAis calledε-free.

Definition 1.2.2 Apathc inAis a finite seqence ofk >0 transitions:

c=hq0, a1, q1i hq1, a2, q2i. . .hq_k−1, ak, qki, where hq_i−1, ai, qii ∈∆ for i= 1. . . k.

The integerk is called thelength of c. The wordw=a1·. . .·ak is called the labelofc. We may denote the pathc as

c=q0→^a¹ q1. . .→^a^kqk.

Thenull pathofq∈Qis 0q beginning and ending inqwith labelε.

Definition 1.2.3 Thegeneralized transition relation∆^∗is defined as the small- est subset ofQ×Σ^∗×Qwith the following closure properties:

• for allq∈Qwe havehq, ε, qi ∈∆^∗.

• For all q1, q2, q3 ∈ Q and w ∈ Σ^∗, a ∈ Σ^ε: if hq1, w, q2i ∈ ∆^∗ and hq2, a, q3i ∈∆, then alsohq1, w·a, q3i ∈∆^∗.

Clearly, the set ∆^∗ contains exactly the triples of the begining, label and ending of the paths in A. For q ∈ Qwe write L_A(q) := {w ∈ Σ^∗| ∃q1 ∈ F : hq, w, q1i ∈∆^∗}for the language of all words, which are labels of paths leading fromqto a final state. The languageacceptedbyAisL(A) :=L_A(I). GivenA as above, the set of active statesfor input w∈Σ^∗ is{q∈Q| hq0, w, qi ∈ ∆^∗}.

The FSAA1 andA2are equivalent if L(A1) =L(A2).

1.2.2 -closure of finite-state automata

Definition 1.2.4 Theε-closureCε:Q→2^Q is defined as Cε(q) ={q⁰ ∈Q| hq, ε, q⁰i ∈∆^∗}.

Proposition 1.2.5 For any FSAA=hΣ, Q, I, F,∆ithere exists an equivalent ε-free FSA.

Proof. Let us consider the FSAA⁰=hΣ, Q, I, F∪ {q∈I|Cε(q)∩F 6=∅},∆⁰i, where

∆⁰={hq1, a, q2i |q1, q2∈Q, a∈Σ,∃q⁰∈Cε(q1) :hq⁰, a, q2i ∈∆}.

1.2.3 Closure properties of finite-state automata and carte- sian product construction

Proposition 1.2.6 1. For A_∅=hΣ,∅,∅,∅,∅iwe haveL(A_∅) =∅.

2. ForAε=hΣ,{q},{q},{q},∅iwe haveL(Aε) ={ε}.

3. Leta∈Σ. For the FSAAa=hΣ,{q0, q1},{q0},{q1},{hq0, a, q1i}iwe have L(Aa) ={a}.

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1. For the FSAA=hΣ, Q1∪Q2, I1∪I2, F1∪F2,∆1∪∆2iwe haveL(A) = L(A1)∪L(A2).

2. For the FSAA=hΣ, Q1∪Q2, I1, F2,∆1∪∆2∪ {hq1, ε, q2i |q1∈F1&q2∈I2}i we have L(A) =L(A1)·L(A2).

3. Letq0 be a new state. For the FSA

A=hΣ, Q1∪ {q0},{q0}, F1∪q0,∆1∪ {hq0, ε, q1i |q1∈I1} ∪ {hq2, ε, q0i |q2∈F1}i we have L(A) =L(A1)^∗.

4. For the FSAA=hΣ, Q1, F1, I1,∆⁰i, where∆⁰={hq2, a, q1i | hq1, a, q2i ∈

∆1}, we haveL(A) =ρ(L(A1)).

5. IfA1andA2areε-free FSA then for the FSAA=hΣ, Q1×Q2, I1×I2, F1×F2,∆⁰i, where∆⁰:={hhq1, q2i, a,hr1, r2ii | hq1, a, r1i ∈∆1 & hq2, a, r2i ∈∆2}we

have L(A) =L(A1)∩L(A2).

1.2.4 Deteminization of finite-state automata

A finite state automatonAisdeterministiciff the transition relation is a function δ:Q×Σ→Qand|I|= 1. LetA=hΣ, Q, q0, F, δibe a deterministic FSA, let δ^∗:Q×Σ^∗→Qdenote the generalized transition function, which is defined as in the nondeterministic case.

The following theorem gives us an effective procedure for determinization of FSA.

Theorem 1.2.8 For any FSAA=hΣ, Q, I, F,∆ithere exists an deterministic FSAAD such thatL(A) =L(AD).

Proof. Let AD :=

Σ,2^Q, I, F_D, δ

, where F_D :={S ⊆2^Q|C_ε(S)∩F 6=∅}

andδ(S, a) :={q∈Q| ∃q1∈S :hq1, a, qi ∈∆^∗}forS⊆Q, a∈Σ.

Clearly, the functionδ in the above proof is a total function.

Proposition 1.2.9 For any deterministic FSA A=hΣ, Q, q0, F, δiwhere δ is a total function the following holds: for the FSA A⁰ = hΣ, Q, q0, Q\F, δi we haveL(A⁰) = Σ^∗\L(A).

1.2.5 Regular languages and regular expressions

A language over Σ is called anautomaton languageiff it is recognized by some FSA. The class of automaton languages contains all finite languages and is closed under union, concatenation, iteration, intersection, reversal and complementa- tion.

Definition 1.2.10 Let Σ be a finite alphabet. We define a regular language over Σ by induction:

1. ∅is a regular language;

2. {ε}is a regular language;

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8 CHAPTER 1. BASIC FORMAL CONCEPTS 3. ifσ∈Σ, then{σ}is a regular language;

4. ifL1, L2⊆Σ^∗ are regular languages then

• L1∪L2 is a regular language (union),

• L1·L2 is a regular language (concatenation),

• L^∗₁is a regular language (Kleene closure).

In a similar way we define regular expression.

Definition 1.2.11 Let a finite alphabet Σ be given. Aregular expressionover Σ is a word over Σ∪ {(,),∗,+,·,∅}, defined by induction:

1. ∅is a regular expression;

2. εis a regular expression;

3. ifσ∈Σ, thenσ is a regular expression;

4. ifE1 andE2 are regular expressions over Σ, then

• (E1+E2) is a regular expression,

• (E1·E2) is a regular expression,

• (E₁^∗) is a regular expression.

Each regular expressions naturally corresponds to a regular language and vice versa by the obvious correspondence between the definitions.

The following theorem presents a deeper result for the correspondence between automaton languages and regular languages.

1.2.6 Equivalence between regular languages and automa- ton languages

Theorem 1.2.12 (Kleene) A language is regular if and only if it is an automaton language.

Proof. (=⇒) This direction follows directly from the closure properties given in Propositions 1.2.6,1.2.7.

(⇐=) LetA=hΣ, Q, q1, F, δiis a deterministic FSA. Let Q={q1, . . . , qn}, for 0≤k≤nlet andQ^k:={q1, . . . , qk}. We define

R_ij^k ={v∈Σ^∗|v=a1. . . am&δ^∗(qi, v) =qj&∀l∈ {1, . . . , m−1}:δ^∗(qi, a1. . . al)∈Q^k}.

Note that the final condition∀l∈ {1, . . . , m−1}:δ^∗(qi, a1. . . a_l)∈ {q1, . . . , q_k} is vacuous in the situation wherem= 1. Clearly

R⁰_ij =

{a∈Σ|δ(qi, a) =qj}, ifi6=j {ε} ∪ {a∈Σ|δ(qi, a) =qj}, ifi=j and

R^k_ij =R^k−1_ij ∪(R^k−1_ik ·(R^k−1_kk )^∗·R^k−1_kj )

From the above presentation it follows by induction that for any i, j, k ∈ {1, . . . , n}the languagesR^k_ij are regular. We have that

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L(A) = [

{j|qj∈F}

Rⁿ_1j

Hence L(A) is a regular language.

It is well-known that for any regular languageLthere exists a deterministic FSAA_Lsuch thatL(A) =LandA_Lis minimal (w.r.t. number of states) among all deterministic FSA acceptingL. AL is unique up to renaming of states.

1.2.7 Minimal finite-state automata and the Myhill-Nerod equivalence relation

Let a finite alphabet Σ be given.

Definition 1.2.13 An equivalence relationR⊆Σ^∗×Σ^∗is calledright invariant if

∀u, v∈Σ^∗:u R v⇒(∀w∈Σ^∗:u·w R v·w).

Proposition 1.2.14 LetA=hΣ, Q, q0, F, δi is a deterministic FSA. Then the relation

R_A={hu, vi ∈Σ^∗×Σ^∗|δ^∗(q0, u) =δ^∗(q0, v)}

is a right invariant equivalence relation and|{[s]RA|s∈Σ^∗}| ≤ |Q|.

Proposition 1.2.15 LetL⊆Σ^∗ is a language overΣ. Then the relation RL={hu, vi ∈Σ^∗×Σ^∗| ∀w∈Σ^∗:u·w∈L↔v·w∈L}

is a right invariant equivalence relation.

Proposition 1.2.16 LetL⊆Σ^∗ is a language over Σ, such that the index of R_L is finite. Then for the deterministic FSA

AL=hΣ,{[s]RL|s∈Σ^∗},[ε]RL,{[s]RL|s∈L},{h[u]RL, a,[u·a]RLi |u∈Σ^∗, a∈Σ}i we have thatL(AL) =L.

Proposition 1.2.17 LetA=hΣ, Q, q0, F, δi is a deterministic FSA. Then we have that

RA⊆R_L(A).

Theorem 1.2.18 For any deterministic FSA there exists a unique (up to state renaming) equivalent deterministic FSA, which is minimal with respect to the number of states.