On a category of deterministic fuzzy automata

Priyanka Pala and S.P. TiwaribaDepartment of Applied Mathematics, IIT(ISM), Dhanbad-826004, India, priyankapal2192@gmail.combDepartment of Applied Mathematics, IIT(ISM), Dhanbad-826004, India, sptiwarimaths@gmail.com

Abstract

The purpose of the present work is to usethe concepts of reachability and observabil-ity maps to provide a solution of a well-known characterization of fuzzy regular lan-guages. In between, we associate two de-terministic fuzzy automata for a given de-terministic fuzzy automaton and show thatthe reachability and observability maps of thegiven deterministic fuzzy automaton turn outto be morphisms in the category of determin-istic fuzzy automata.

Keywords: Deterministic fuzzy automaton;reachability map; observability map; fuzzylanguage.

1 Introduction

Fuzzy automata theory, firstly proposed by Wee [39]has now developed significantly due to its importancein reducing the gap between formal languages andnatural languages (cf., [5, 10–20, 22, 23, 27, 30, 31, 40]).Among these studies, the concept of a deterministicfuzzy automaton (deterministic automaton equippedwith fuzzy set of final states) was introduced andstudied in [5,10,20]. The importance of such automa-ton is justified from the fact that (i) a fuzzy languageis accepted by some fuzzy automaton iff it is acceptedby some deterministic fuzzy automaton (cf., [5, 20]),and (ii) the Nerode automaton of a fuzzy automatonis a deterministic fuzzy automaton equivalent to givenfuzzy automaton (cf., [10]).

One of the most significant branches of the al-gebraic theory of languages and automata isMyhill-Nerode’s theory [28, 29], in which formallanguages and deterministic automata are studiedthrough right congruences and congruences on freemonoid. These right congruences on a free monoid

have been shown to be useful in the constructionof a minimal deterministic automaton recognizinga given language. After the introduction of fuzzyautomata and fuzzy languages, the Myhill-Nerode’stheory has also been used to construct a minimaldeterministic fuzzy automaton for a given fuzzylanguage [11]. Further, in [11], the concept of deriva-tive of fuzzy languages was used for such construction.

In addition to, the usefulness of certain ideas,concepts and tools from category theory in thedevelopment of many aspects of theoretical computerscience is well-known, specifically, there have beenmany research dealing with the categorical approachto automata theory (cf., [2–4, 8, 9, 35, 36]). Moreover,the usefulness of category theory in the study of fuzzyautomata were reported in [1, 21,24–26,32–34,41].

In this paper, we introduce the categories of de-terministic fuzzy automata and fuzzy languages basedon a complete residuated lattice without zero divisors.Next, for a given deterministic fuzzy automaton, weconstruct two new deterministic fuzzy automata. Fur-ther, we show that the reachability and observabilitymaps of the given deterministic fuzzy automaton turnout to be the DFA-morphisms between the givendeterministic fuzzy automaton and the constructeddeterministic fuzzy automata. Finally, the observabil-ity map of one of the associated deterministic fuzzyautomaton led to deduce the well-known relationshipbetween a fuzzy regular language and Myhill-Nerodeequivalence.

2 Preliminaries

In this section, we recall the concepts related to resid-uated lattices and category theory, which are useful inthe subsequent sections. We begin with the followingconcept of a residuated lattice. The details on residu-ated lattices can be found in [6, 37,38].

11th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2019)

Copyright © 2019, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Atlantis Studies in Uncertainty Modelling, volume 1

205

Definition 1 A residuated lattice is an algebra(L,∧,∨,⊗,→, 0, 1) such that

(i) (L,∧,∨, 0, 1) is a lattice with the least element 0and the greatest element 1;

(ii) (L,⊗, 1) is a commutative monoid with unit 1;and

(iii) ⊗ and → form an adjoint pair, i.e., ∀a, b, c ∈L, a⊗ b ≤ c⇔ a ≤ b→ c.

If, in addition (L,∨,∧, 0, 1) is a complete lattice, thenthe residuated lattice is called a complete residuatedlattice. Further, (L,∧,∨,⊗,→, 0, 1) is called com-plete residuated lattice without zero divisors iffor all x 6= 0, y 6= 0 ⇒ x ⊗ y 6= 0. For example, aresiduated lattice (L,∧,∨,⊗,→, 0, 1) with Lukasiewiczt-norm x⊗y = max(x+y−1, 0) is with zero divisors;while the same with Godel t-norm x ⊗ y = min(x, y)is without zero divisors.

The fuzzy sets considered in this paper take themembership values in a complete residuated lattice Lwithout zero divisors. The set of all fuzzy subsets ofX is denoted by LX . For a fixed set X, X∗ denotesthe free monoid generated by X. We shall denote bye, the identity element of X∗.

Now, we recall some concepts associated withfuzzy finite automata, deterministic fuzzy automataand fuzzy languages from [5,10,11,20,33,34].

Definition 2 A fuzzy automaton is a 5-tuple M =(Q,X, δ, σ0, σ1), where Q and X are nonempty sets,called the set of states and the set of inputs, respec-tively, δ : Q × X × Q −→ L is a fuzzy subset ofQ × X × Q, called a fuzzy transition function, andσ0, σ1 : Q −→ L are fuzzy subsets of Q, called fuzzyinitial and final state, respectively.

Remark 1 In [5], it has been shown that the fuzzytransition function δ : Q×X×Q −→ L can be extendedto δ∗ : Q×X∗×Q −→ L such that ∀p, q ∈ Q, ∀u ∈ X∗,and ∀x ∈ X

δ∗(q, e, p) =

{1 if q = p0 if q 6= p, and

δ∗(p, ux, q) = ∨{δ∗(p, u, r)⊗ δ(r, x, q) : r ∈ Q}.

Also, in [5], it has been observed that

δ∗(p, uv, q) = ∨{δ∗(p, u, r)⊗ δ∗(r, v, q) : r ∈ Q},

∀p, q ∈ Q,∀u, v ∈ X∗.

A fuzzy automaton whose set of states is finite is calleda fuzzy finite automaton.

Definition 3 (i) A fuzzy subset of X∗ is called afuzzy language in X.(ii) A fuzzy language accepted by a fuzzy automatonM = (Q,X, δ, σ0, σ1), denoted as fM : X∗ −→ L iffM (u) = ∨{σ0(p) ⊗ δ∗(p, u, q) ⊗ σ1(q) : p, q ∈ Q}, forall u ∈ X∗.(iii) A fuzzy language which is accepted by a fuzzy fi-nite automaton is called a regular language.

Now, we recall the following concept of a deterministicfuzzy automaton.

Definition 4 A deterministic fuzzy automatonis a 5-tuple M = (Q,X, δ, q0, β), where

(i) Q and X are sets called the state-set and theinput-set, respectively.

(ii) δ : Q×X −→ Q is a map called transition map.

(iii) β : Q −→ L is a map called fuzzy set of finalstates.

(iv) q0 ∈ Q is a fixed state called the initial state.

Remark 2 Let M = (Q,X, δ, q0, β) be a deterministicfuzzy automaton. The map δ can be extended to a mapδ∗ : Q×X∗ −→ Q such that

(a) δ∗(q, e) = q,

(b) δ∗(q, wx) = δ(δ∗(q, w), x),∀q ∈ Q,w ∈ X∗ andx ∈ X.

Remark 3 (i) For a fuzzy language f : X∗ −→ L, therelation R = {(u, v) : f(uw) = f(vw), for allw ∈ X∗}is called Myhill-Nerode relation on X∗. Also, X∗/Rturns out to be the state-set of minimal determin-istic fuzzy automaton Nf = (Qf , X, δf , [e]f , βf ),where Qf = X∗/R, δf : Qf × X −→ Qf andβf : Qf −→ L are maps such that δf ([w], x) = [wx]and βf ([w]) = f(w), ∀ [w] ∈ Qf , x ∈ X. Also, Nfaccepts the fuzzy language f .

(ii) For a fuzzy language f : X∗ −→ L andu ∈ X∗, a fuzzy language fu : X∗ −→ L defined byfu(v) = f(uv),∀v ∈ X∗, is called a derivative of fwith respect to u. Also, the set of all such derivativesturn out to be the state-set of a minimal automatonaccepting the fuzzy language f .

The following shows that the deterministic fuzzy au-tomata are equally powerful as fuzzy automata.

Proposition 1 [5] A fuzzy language is accepted bysome fuzzy automaton if and only if it is accepted bysome deterministic fuzzy automaton.

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Definition 5 Let M = (Q,X, δ, q0, β) and M ′ =(Q′, X ′, δ′, q′0, β

′) be deterministic fuzzy automata. Ahomomorphism f : M −→M ′ is a pair (a, b), wherea : X −→ X ′ and b : Q −→ Q′ are maps such that forall q ∈ Q, x ∈ X and for some l ∈ L \ 0,

(i) b(q0) = q′0,

(ii) b(δ(q, x)) = δ′(b(q), a(x)), and

(iii) β(q) −→ β′(b(q)) ≥ l.

Proposition 2 The deterministic fuzzy automataand their homomorphisms form a category undercomponent-wise composition of maps.

We shall denote by DFA the category of deterministicfuzzy automata. By abuse of language, we shall alsodenote the object-class of the category DFA by DFAitself.

Remark 4 (i) Let DFA(X) be the class of DFA-objects having the fixed input set X and whose mor-phisms are all DFA-morphisms of the form (b, idX).Then DFA(X) is also a category. Obviously, it is asubcategory of DFA.(ii) For DFA(X)-objects M = (Q,X, δ, q0, β) andM ′ = (Q′, X, δ′, q′0, β

′), we can describe a morphism(b, idX) simply by the map b. Thus a morphism M −→M ′ in DFA(X) is a map b : Q −→ Q′ such that forall q ∈ Q, x ∈ X and for some l ∈ L \ 0,

(i) b(q0) = q′0,

(ii) b(δ(q, x)) = δ′(b(q), x), and

(iii) β(q) −→ β′(b(q)) ≥ l.

3 Reachability and observability of adeterministic fuzzy automaton

In this section, for a given deterministic fuzzy au-tomaton, we construct two new deterministic fuzzyautomata. Further, we show that the reachabilityand observability maps of the given deterministicfuzzy automaton turn out to be the DFA-morphismsbetween the given deterministic fuzzy automatonand the constructed deterministic fuzzy automata.Interestingly, the observability map of one of theassociated deterministic fuzzy automaton leads usto provide a much simpler proof of the well-knownrelationship among a fuzzy regular language, its setof derivatives and the index of Myhill-Nerode relationgiven in [10].

We begin with the following concept of reacha-bility map of a deterministic fuzzy automaton.

Definition 6 (a) Let M=(Q,X, δ, q0, β)∈ DFA(X).The reachability map r of M is a map r : X∗ −→ Qsuch that

(i) r(e) = q0,

(ii) r(wa) = δ(r(w), a),∀w ∈ X∗ and ∀a ∈ X.

Remark 5 It can be easily seen that

(i) r(w) = δ∗(q0, w),∀w ∈ X∗, and

(ii) r(ww′) = δ∗(r(w), w′),∀w,w′ ∈ X∗.

Reachability of a deterministic fuzzy automaton meansthat all of its states should be reached from the ini-tial state, i.e., (Q,X, δ, q0, β) ∈ DFA(X) is said to bereachable if for each q ∈ Q there exists w ∈ X∗ suchthat δ∗(q0, w) = q. Which in terms of the reachabilitymap can be characterized as the following.

Proposition 3 (Q,X, δ, q0, β)∈ DFA(X) is reach-able iff r is onto.

Definition 7 Given M= (Q,X, δ, q0, β) ∈ DFA(X),the fuzzy language fq accepted by M in state qis a map fq : X∗ −→ L such that fq(w) = β(δ∗(q, w)),∀w ∈ X∗.

The fuzzy language accepted by M in state q0 is calledthe fuzzy language accepted by M .

Remark 6 It can be easily seen that fq0 = β ◦ r.

Now, let M = (Q,X, δ, q0, β) ∈ DFA(X) which ac-cepts fuzzy language f . Also, define a map µ :X∗ × X −→ X∗ such that µ(w, x) = wx,∀w ∈ X∗

and x ∈ X. Then M1 = (X∗, X, µ, e, f) is also a de-terministic fuzzy automaton, which accepts the fuzzylanguage f . Now, we have the following.

Proposition 4 The reachability map r of M =(Q,X, δ, q0, β) ∈ DFA(X) is a DFA(X)-morphismfrom M1 to M .

Proof: To show that the reachability map r : X∗ −→Q is a DFA(X)-morphism from M1 to M , it is enoughto show that for all q ∈ Q,w ∈ X∗, x ∈ X and for somel ∈ L \ 0,

(i) r(e) = q0,

(ii) r(µ(w, x)) = δ((r × idX)(w, x)), and

(iii) f(w)→ β(r(w))) ≥ l.

Now, (i) is trivial. Also, r(µ(w, x)) = r(wx) =δ(r(w), x) = δ((r × idX)(w, x)). Finally, for all w ∈X∗, f(w) → β(r(w)) = f(w) → β(δ∗(q0, w)) =f(w)→ f(w) = 1.

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Proposition 5 Let M = (Q,X, δ, q0, β) and M ′ =(Q′, X ′, δ′, q′0, β

′) be deterministic fuzzy automata withreachability maps r : X∗ −→ Q and r′ : X ′∗ −→ Q′

respectively and DFA-morphism (a, b) : M −→ M ′,then (b ◦ r)(w) = (r′ ◦ a)(w).

Proof: We prove this by induction on the lengthof strings in X∗. Let w ∈ X∗. For |w| = 0,(b ◦ r)(e) = b(r(e)) = b(q0) = q′0 and (r′ ◦ a)(e) =r′(a∗(e)) = r′(e) = q′0. Thus the equality holds for|w| = 0. We now assume that the result is true forall strings of length less than or equal to n, i.e., forall w ∈ X∗ such that |w| ≤ n, (b ◦ r)(w) = (r′ ◦a)(w), i.e., b(r(w)) = r′(a∗(w)). Then (b ◦ r)(wx) =b(r(wx)) = b(δ(r(w)), x) = δ′(b(r(w), a(x))), as (a, b)is a DFA(X)-morphism. Furthermore, (r′◦a∗)(wx) =r′(a∗(wx)) = r′(a∗(w)a(x)) = δ′(r′(a∗(w)), a(x)) =δ′(b(r(w)), a(x)) follows from the fact that the resultis true for all strings of length less than or equal to n.Hence the equality holds for |w| = n+ 1.

Definition 8 (a) Given M = (Q,X, δ, q0, β) ∈DFA(X), the observability map σM of M is a mapσM : Q −→ LX

∗such that σM (q) = fq,∀q ∈ Q.

(b) M is called observable, if σM is one-one.

Remark 7 (i) Note that, σM (q0) = fq0 , wherefq0(w) = (β ◦ r)(w) = β(r(w)) = β(δ∗(q0, w)), ∀w ∈X∗.(ii) The observability of M ∈ DFA(X) tells that dif-ferent fuzzy languages are assigned to distinct states.

Again, let M = (Q,X, δ, q0, β) ∈ DFA(X) whichaccepts a given fuzzy language f . Also, define themaps ν : LX

∗ × X −→ LX∗

such that ν(g, x) = gx,where gx(w) = g(xw), ∀g ∈ LX

∗, w ∈ X∗, x ∈ X

and ρ : LX∗ −→ L such that ρ(g) = g(e). Then

M2 = (LX∗, X, ν, f, ρ) is also a deterministic fuzzy au-

tomaton, which accepts the fuzzy language f . Now,we have the following.

Proposition 6 The observability map σM of M =(Q,X, δ, q0, β) ∈ DFA(X) is a DFA(X)-morphismfrom M to M2.

Proof : To show that the observability map σM :Q −→ LX

∗is a DFA(X)-morphism from M to M2, it

is enough to show that for all q ∈ Q,w ∈ X∗, x ∈ Xand for some l ∈ L \ 0,

(i) σM (q0) = fq0 ,

(ii) ν((σM × idX)(q, x)) = σM (δ(q, x)), and

(iii) ρ(σM (q)) −→ β(q) ≥ l.

(i) is trivial. Now, for all (q, x) ∈ Q × X,ν(σM × idX)(q, x)) = ν(σM (q), x) = ν(fq, x) = (fq)x.

Also, σM (δ)(q, x) = σM (δ(q, x)) = fδ(q,x) and(fq)x(w) = fq(xw) = β(δ∗(q, xw)) = β(δ∗(δ(q, x), w))= fδ(q,x)(w). Thus (ii) holds. Finally, for all q ∈ Q,ρ(σM (q)) −→ β(q) = ρ(fq) −→ β(q) = fq(e) −→β(q) = β(δ∗(q, e)) −→ β(q) = β(q) −→ β(q) = 1.

The observability map of σM1: X∗ −→ LX

∗of de-

terministic fuzzy automaton M1 = (X∗, X, µ, e, f) isdefined as σM1

(u) = fu, where fu(w) = f(µ∗(u,w)) =f(uw),∀w ∈ X∗. Now, we have the following.

Proposition 7 The observability map σM1of M1 is

a DFA(X)-morphism from M1 to M2.

Proof: Similar to that of Proposition 5.

Proposition 8 Let σM1 : X∗ −→ LX∗

be the ob-servability map of deterministic fuzzy finite automatonM1 = (X∗, X, µ, e, f). Then

(i) ker(σM1) is the Myhill-Nerode relation on X∗

(ii) Im(σM1) is the set of all derivatives of fuzzy lan-guage f .

(iii) σM ◦ r = σM1 .

Proof: (i) ker(σM1) = {(u, v) : σM1(u) = σM1(v)} ={(u, v) : fu = fv} = {(u, v) : fu(w) = fv(w),∀w ∈X∗} = {(u, v) : f(uw) = f(vw),∀w ∈ X∗}. This isprecisely the Myhill-Nerode relation on X∗.(ii) Im(σM1

) = {σM1(u) : u ∈ X∗} = {fu : fu(w) =

f(uw), u, w ∈ X∗}, which is precisely the set of allderivatives of fuzzy language f .(iii) For all u ∈ X∗, (σM ◦ r)(u) = σM (r(u)) =σM (δ∗(q0, u)) = fδ∗(q0,u). Also, for all w ∈ X∗,fδ∗(q0,u)(w) = f(δ∗(δ∗(q0, u), w) = f(δ∗(q0, uw)) =f(uw). Thus σM ◦ r = σM1

.

Now, the observability map σM1 leads us to pro-vide a simple proof of the following well-knownequivalence given in [10].

Proposition 9 Let f : X∗ −→ L be a fuzzy language.Then the following are equivalent.

(i) f is regular.

(ii) The set of derivatives of f is finite.

(iii) The Myhill-Nerode equivalence has finite index.

Proof: (i) ⇒ (ii): Let f be a fuzzy regular lan-guage. Then there exists a deterministic finite fuzzyautomaton M = (Q,X, δ, q0, β) which accepts f , andtherefore M1 = (X∗, X, µ, e, f) ∈ DFA(X) accepts f .Now, Im(σM1) = {σM1(u) : u ∈ X∗} = {σM (r(u)) :u ∈ X∗}, which is finite as r(u) ∈ Q for all u ∈ X∗ and

208

Q is finite. Thus Im(σM1), i.e., the set of derivatives

of f is finite.(ii) ⇒ (iii): Follows from the definition of Im(σM1

)and ker(σM1

).(iii) ⇒ (i): Let the Myhill-Nerode equivalence hasfinite index. Then there exists a deterministic finitefuzzy automaton Nf = (Qf , X, δf , [e]f , βf ) whichaccepts f . Hence the fuzzy language f is regular.

In the following, we introduce the concept ofhomomorphism between two fuzzy languages.

4 Functorial relationship betweenDFA and FL

In this section, we study a functorial relationship be-tween the category of deterministic fuzzy automataand category of fuzzy languages. We begin with thefollowing.

Definition 9 A homomorphism from a fuzzy lan-guage f : X∗ −→ L to fuzzy language f ′ : X ′∗ −→ Lis a map a : X −→ X ′ such that

f(w)→ f ′(a∗(w)) ≥ l,

where a∗ is the free extension of the map a definedinductively by a∗(e) = e and a∗(wx) = a∗(w)a(x) forevery w ∈ X∗, x ∈ X and l ∈ L \ 0.

Proposition 10 For a given set X, the class of fuzzylanguages and their morphisms form a category.

Remark 8 We shall denote by FL, the category offuzzy languages.

The following is towards the functorial relationship be-tween the category of deterministic fuzzy automataand category of fuzzy languages.

Proposition 11 Let E : DFA −→ FL be such thatE(M) = f , ∀M ∈ DFA and for all DFA- morphism(a, b) : M −→M ′, E(a, b) = a. Then E is a functor.

Proof: We will only show that a is an FL-morphism.Since (a, b) : M −→ M ′ is a DFA-morphism, for w ∈X∗,

f ′(a∗(w)) = β′(δ′∗(b(q), a(w)))

= β′(b(δ∗(q, w)))

≥ l ⊗ β(δ∗(q, w))

= l ⊗ f(w).

i.e., f(w)→ f ′(a∗(w)) ≥ l. Thus E is a functor.

5 Conclusion

In this paper,

• we have introduced the concept of reachabilitymap, observability map, categories of determinis-tic fuzzy automata and fuzzy language based on acomplete residuated lattice without zero divisors;

• we have shown that the observability map of oneof the associated deterministic fuzzy automatonleads us to deduce the well-known result of a fuzzyregular language;

• further, a functorial relationship is establishedbetween the category of deterministic fuzzy au-tomata and that of fuzzy languages.

A generalized concept of a deterministic fuzzy automa-ton is recently introduced in [7]. In future, we will tryto study the categorical concepts for such automaton.

Acknowledgement

The work of second author is supported by SERB,Ministry of Science & Technology, DST, Govt. of In-dia, under grant No. EMR/2017/001108.

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