Fuzzy tree language recognizability

Fuzzy Sets and Systems 161 (2010) 716–734www.elsevier.com/locate/fss

Fuzzy tree language recognizability

Symeon Bozapalidisa,∗, Olympia Louscou Bozapalidoyb

aDepartment of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, GreecebSection of Mathematics and Informatics, Technical Institute of West Macedonia, Koila, Kozani, Greece

Received 17 March 2008; received in revised form 28 August 2009; accepted 31 August 2009Available online 6 September 2009

Abstract

A fuzzy tree language with membership grades in an arbitrary set is syntactically recognizable (s-recognizable) if its syntacticalgebra is finite. The equality problem for such languages is decidable and their syntactic algebra can be effectively constructedprovided that they are s-recognizable. Linear (but non arbitrary) tree homomorphisms preserve s-recognizability. Tree automatawhose transitions are weighted over the unit interval and whose behavior is computed with respect to a pair made of a t-normdistributive over a t-conorm have the syntactic recognition power and thus their equivalence problem is decidable. However, s-recognizability is more powerful when dealing with non-distributive pairs of such operations.© 2009 Elsevier B.V. All rights reserved.

Keywords: Syntactic recognizability; Fuzzy tree language; Fuzzy tree automata

1. Introduction

Fuzzy automata on words have long history (cf. [14,15,17,4,3]). Automata theory based on residuated lattices areestablished in [18,19]. They are used in fuzzy switching (cf. [11]), modelling and control of fuzzy discrete event systems(cf. [13,20]), Turing machines (cf. [21]) and lexical analysis (cf. [16]). At the level of trees Inagaki and Fukumuraproposed a model of fuzzy automata whose slight modification can be viewed as a tree automaton over an �-continuoussemiring (cf. [2]).

Esik and Liu studied fuzzy tree automata with membership in a (completely) distributive lattice and an equivalencebetween recognizability and equationality of fuzzy tree languages was established (cf. [6]).

Here we deal with tree automata whose behavior is computed with respect to a pair (∇, �) where ∇ ranges overthe classical t-conorms max, ∇L , ∇D and � ranges over the classical t-norms min, �L , �D product, respectively. (∇L ,�L and ∇D , �D stand for the Lucasiewics and drastic t-conorms and t-norms.) We show that tree automata over(max, min), (max, �L ) and (max, �D) have the same recognition power which also coincides with the recognitionpower of deterministic tree automata with a final state distribution in [0, 1].

The syntactic representation of a fuzzy tree language S is effectively constructed by means of right derivatives of Sand a pumping lemma is presented.

In the case that � does not distribute over ∇ then the behavior of a fuzzy tree automaton computed with respectto (∇, �) can be defined collectively but not inductively. By using the non-distributive Lukasiewics pair (∇L , �L )

∗ Corresponding author.E-mail address: [email protected] (S. Bozapalidis).

0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2009.08.008

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S. Bozapalidis, O.L. Bozapalidoy / Fuzzy Sets and Systems 161 (2010) 716 –734 717

we find the rather surprising result that the behaviors of the corresponding automata are included into the previouslydetermined class of fuzzy tree languages.

2. Preliminaries

2.1. Basics on theory of trees

A ranked alphabet is a set � together with a function rank : � → N (N the natural numbers). We write � = ⋃k≥0�k

where �k = { f/ f ∈ �, rank( f ) = k} is the set of k-ranked symbols.Let X be a set whose elements are called variables. The set of trees over � and X is the smallest set T�(X ) inductively

defined by the next two items:

• �0 ∪ X ⊆ T�(X ).• If f ∈ �k(k≥1) and t1, . . . , tk ∈ T�(X ), then the scheme f (t1, . . . , tk) ∈ T�(X ).

We write T� instead of T�(∅). Frequently f (t1, . . . , tk) is depicted asf

� . . . �

t1 tkhence the denomination “tree”.

A familiar way to measure trees is the height: it is the length of the longest branch of a tree. Formally, the functionheight : T�(X ) → N is defined by

• height(a) = 0, for a ∈ �0 ∪ X ;• height( f (t1, . . . , tk)) = 1 + max{height(ti )/1≤i≤k};for all f ∈ �k (k≥1) and t1, . . . , tk ∈ T�(X ).

Trees are interpreted in the real world by algebras. Precisely, a �-algebra is a pair A = (A, �), where A is a set (thecarrier of A) and

� = (� f : Ak → A) f ∈�k , k≥0

is a set of functions called structural operations of A.For instance, T�(X ) can be converted into a �-algebra by defining the structural operation of f ∈ �k to be

(t1, . . . , tk)� f (t1, . . . , tk).Given �-algebras A = (A, �) and B = (B, �), a function h : A → B commuting with the structural operations, i.e.

h(� f (q1, . . . , qk)) = � f (h(q1), . . . , h(qk)), f ∈ �k, q1, . . . , qk ∈ A, k≥0

is called a morphism of �-algebras.For instance, for each �-algebra A = (A, �) a canonical morphism hA : T� → A is obtained by

hA( f (t1, . . . , tk)) = � f (hA(t1), . . . , hA(tk))

for f ∈ �k, t1, . . . , tk ∈ T� (k≥0).Given a �-algebra A = (A, �) any equivalence relation ∼ on A compatible with the structural operations, i.e.

f ∈ �k(k≥1), q1 ∼ q ′1, . . . , qk ∼ q ′

k

implies

� f (q1, . . . , qk) ∼ � f (q ′1, . . . , q ′

k)

is called a congruence on A.Then the quotient set A/ ∼ admits a �-algebra structure by defining

(�∼) f : (A/ ∼)k → A/ ∼, f ∈ �k, k≥0

718 S. Bozapalidis, O.L. Bozapalidoy / Fuzzy Sets and Systems 161 (2010) 716 –734

by the clause

(�∼) f ([q1], . . . , [qk]) = [� f (q1, . . . , qk)],

where [q] stands for the ∼-class of q ∈ A. The pair A/ ∼= (A/ ∼, �∼) is the quotient �-algebra of A by ∼.Congruences on T� can be characterized through an action that we are going to describe. Denote by P� the set of all

trees of T�(x) where the variable x occurs exactly once. P� with multiplication the substitution at x is converted into amonoid: for �, � ∈ P�, �� = �[�/x]. Actually P� is a free monoid, since every element � ∈ P�

� :

f1� . . . | . . . �

t (1)1 f2 t (1)

k1

� . . . | . . . �

t (2)1

... t (2)k2|

f p

� . . . | . . . �t (p)1 x t (p)

kp

is uniquely written as a product of trees of the formf

� . . . | . . . �t1 x tk

The number of such trees appearing in the above factorization of � ∈ P� is denoted by |�|, i.e. |�| = p.The clones of � ∈ P� is the set of all oblique trees of �

Cl(�) = {t (1)1 , . . . , t (1)

k1, t (2)

1 , . . . , t (2)k2

, . . . , t (p)1 , . . . , t (p)

kp}.

P� acts on T� again by substitution at x: if � ∈ P� and t ∈ T�, then �t = �[t/x].It is well known that

Proposition 1 (cf. Bozapalidis [3], Dubois and Prade [4], and Esik and Lin [6]). An equivalence ∼ on T� is a con-gruence iff it is compatible with the P�-action, i.e.

t ∼ t ′ and � ∈ P� implies �t ∼ �t ′.

More generally P� acts on any �-algebra A = (A, �) as follows:

• if � = f (t1, . . . , ti−1, x, ti+1, . . . , tk) and q ∈ A then

� · q = � f (hA(t1), . . . , hA(ti−1), q, hA(ti+1), . . . , hA(tk));• if � = �1�2 (�1, �2 ∈ P�) and q ∈ A then

� · q = �1 · (�2 · q).

Fact. For every � ∈ P�, t ∈ T� it holds

hA(�t) = � · hA(t).

In order to make some effective constructions, we need the following finite set:

Pn = {�/� ∈ P�, |�|≤n, height(t) ≤n for all t ∈ Cl(�)},where n is a fixed positive integer.


A last tree theoretic notion: given ranked alphabets � and �, a tree homomorphism from � to � is just a sequenceof functions

hk : �k → T�(k), k≥0,

where = {1, 2, . . .} is a set of auxiliary variables and k = {1, . . . , k}, k≥1, 0 = ∅. Such a sequence (hk)determines a function h : T� → T� by setting

• h(d) = h0(d), d ∈ �0;• h( f (t1, . . . , tk)) = hk( f )[h(t1)/1, . . . , h(tk)/k], f ∈ �k, t1, . . . , tk ∈ T� where the right hand side notation means

the tree obtained by substituting the tree h(ti ) at all occurrences of i inside hk( f ), 1≤i≤k.

A tree homomorphism h : T� → T� is said to be linear if for each k≥1 and each f ∈ �k , the variables 1, . . . , koccur in hk( f ) at most once.

The simpler linear homomorphisms are of the following three types:

(i) projections, i.e. homomorphisms of the form h : T�∪� → T� with rank(�) = n and h(�) = xk(1≤k≤n), h(�) = �for all � ∈ �;

(ii) relabellings, i.e. homomorphisms of the form h : T�∪� → T�∪ with h(�) = (x11 , . . . , xi p ), i1, . . . , i p distinctelements of {1, . . . , p}, p = rank( );

(iii) elementary homomorphisms, i.e. homomorphisms of the form h : T�∪� → T�∪{ 1, 2} with rank(�) = k +l, rank( 1) = l +1, rank( 2) = k and h(�) = 1(x1, . . . , xm, 2(xm+1, . . . , xm+k), xm+k+1, . . . , xk+l ), h(�) = �for all � ∈ �.

Here is an important result due to Arnold and Dauchet stating that

Theorem 1 (cf. Arnold and Dauchet [1]). Any linear homomorphism can be factorized into projections, relabellingsand elementary homomorphisms.

2.2. Basics from fuzzy set theory

Recall that a t-norm (resp. t-conorm) is an associative, commutative and monotonic operation on the unit interval[0, 1], admitting 1 (resp. 0) as the neutral element.

Throughout this article we use the traditional t-norms

• Lukasiewicz intersection:

x�L y = max(0, x + y − 1), x, y ∈ [0, 1];

• drastic intersection:

x�D y = x if y = 1

= y if x = 1

= 0 else;

• ordinary multiplication:

x�m y = xy;

• min operation

and their associated t-conorms.

• Lukasiewicz union:

x�L y = min(1, x + y);


• drastic union:

x�D y = y if x = 0

= x if y = 0

= 1 else;

• x�a y = x + y − xy;• max operation,

respectively (cf. [5]).

Proposition 2. Let A be a finite subset of [0, 1]. For � = max, min, �L , �L , �D, �D we have

{a1� . . . �an/a1, . . . , an ∈ A, n≥0} = {a1� . . . �an/a1, . . . , an ∈ A, n≤�(A)},where �(A) is a positive integer exclusively depending on A.

Proof. For � = max, min we have nothing to prove. � = �L . Without loss of generality we may assume that 1 /∈ A.We put a = max A. Then

a1�L . . . �Lan≤a�L . . . �La = max(0, na − n + 1).

For n > 1/(1 − a) we have b = na − n + 1 < 0 and so max(0, b) = 0. Therefore a1�L . . . �Lan = 0.� = �D . For a1, . . . , an ∈ A we have either

a1�D . . . �Dan = ai if a j = 1 for j�i

or

a1�D . . . �Dan = 0 if for at least a pair of indices (i, j) it holds ai < 1, a j < 1.

In other words

A�D . . . �D A = {0} if 1 /∈ A

= A if 1 ∈ A.

The cases � = �L , �D are dual to the above ones. �

3. Tree languages with membership in an arbitrary set

Let � be a finite ranked alphabet and E an arbitrary set. Functions S : T� → E are called fuzzy tree languages overE. Denote by Fuzzy(�, E) the set of such objects.

A representation of S : T� → E is a pair (A, �), where A = (A, �) is a �-algebra and � : A → E is a functionsuch that � ◦ hA = S.

T�

E

AS

hA

��

��

(A, �) is said to be finite (resp. surjective) whenever the set A is finite (resp. hA is surjective).The syntactic �-algebra AS = (AS, �S) associated with S : T� → E is constructed as follows. Let ∼S be the

equivalence on T� (named syntactic) defined by

t ∼S t ′ iff S(�t) = S(�t ′) for all � ∈ P�.


In fact ∼S is a congruence (use Proposition 1) and so the quotient AS = T�/ ∼S admits a canonical �-algebra structure,its structural operation corresponding to f ∈ �k(k≥0) being the function (�S) f : Ak

S → AS defined by

(�S) f ([t1], . . . , [tk]) = [ f (t1, . . . , tk)] for all t1, . . . , tk ∈ T�,

where [t] stands for the ∼S-class of t ∈ T�. The syntactic representation of S is then (AS, �S) with �S : AS →E, �S([t]) = S(t). It is well defined and surjective. Moreover

Proposition 3. For any surjective representation (A, �) of S, there is a unique epimorphism h : A → AS makingcommutative the diagram

A

T�

AS

E

��

��

��

��

�

hA hS

h

� �S

where hS is the canonical morphism sending every tree ∈ T� to its class [t] ∈ AS .

Proof. By hypothesis, every element q ∈ A can be written as q = hA(t), for some t ∈ T�. We put h(q) = hS(t). Itholds

h(t) = h(t ′) implies hS(t) = hS(t ′).

Indeed, for all � ∈ P� we have

S(� · t) = (� ◦ hA)(� · t) = �(hA(� · t)) = �(� · hA(t)) = �(� · hA(t ′)) = S(� · t ′),

i.e. t ∼S t ′ and so [t] = [t ′] as wanted. The rest of the proof is left to the reader. �

Remark. For a syntactic theory of ordinary tree languages see [7].

Imitating Eilenberg (cf. [2]), for �-algebras A,B we write A < B whenever there exists a �-algebra M togetherwith a monomorphism � : M → B and an epimorphism h : M → A.

Proposition 4. For every fuzzy tree language S : T� → E , any tree homomorphism h : T� → T� and any functionf : E → E ′, we have

AS◦h < AS, A f ◦S < AS .

In the case h is surjective and f is injective we get the isomorphisms

AS◦h � AS, A f ◦S � AS .

Proof. We only establish the last isomorphism A f ◦S � AS . We have

t ∼ f ◦S t ′ iff ( f ◦ S)(�t) = ( f ◦ S)(�t ′) for all � ∈ P�

iff f (S(�t)) = f (S(�t ′)) for all � ∈ P�f inj.iff S(�t) = S(�t ′) for all � ∈ P�

iff t ∼ t ′.

It turns out that the mapping sending the ∼S-class of t into the ∼ f ◦S-class of t is a bijection. Actually, it is an isomorphismof �-algebras, as the reader can verify. �


Now, assume that an operation � : E × E → E is given. Then the �-product of two fuzzy tree languagesS1, S2 : T� → E is defined pointwise:

(S1�S2)(t) = S1(t)�S2(t), t ∈ T�.

Proposition 5. It holds that

AS1�S2 < AS1 × AS2 .

4. Syntactic recognizability

We say that an equivalence relation ∼ on T�

• is compatible with S : T� → E whenever

t ∼ t ′ implies S(t) = S(t ′);• has finite index whenever it has finitely many

equivalence classes.

The right (resp. left) derivative of S : T� → E at t ∈ T� (resp. at � ∈ P�) is the function St−1 : P� → E (resp.�−1S : T� → E) defined by

(St−1)(�) = S(�t)(resp. (�−1S)(t) = S(�t))

for all � ∈ P� (resp. for all t ∈ T�).

Theorem 2. Next conditions are equivalent for a fuzzy tree language S : T� → E .

(i) S has a finite representation (A, �).(ii) There is a congruence ∼ on T� compatible with S and having finite index.

(iii) ∼S has finite index.(iv) The syntactic representation (AS, �S) is finite.(v) S has finitely many right derivatives

card{St−1/t ∈ T�} < ∞.

(vi) S has finitely many left derivatives

card{�−1S/� ∈ P�} < ∞.

Proof. The logical equivalence (i) ⇔ (iii) comes immediately from Proposition 3 whereas (iii) ⇔ (iv) is obvious.(i) ⇒ (ii). Assume that (A, �) represents S. Then the equivalence on T�

t ∼A t ′ iff hA(t) = hA(t ′)

is a finite index congruence compatible with S because

t ∼A t ′ implies hA(t) = hA(t ′)implies �(hA(t)) = �(hA(t ′))implies S(t) = S(t ′).

(ii) ⇒ (i). Let ∼ be a finite index congruence on T� compatible with S. Then the quotient algebra A = T�/ ∼ isfinite and the function

� : T�/ ∼→ E, �([t]) = S(t)

is well defined. It turns out that (A, �) represents S.


(iii) ⇔ (v). We have

t ∼S t ′ iff S(�t) = S(�t ′) for all � ∈ P�

iff St−1 = St ′−1.

Therefore the mapping [t]�St−1 is a well defined bijection (as usual [t] stands for the ∼S-class of t).(v) ⇒ (vi). Suppose that St−1

1 , . . . , St−1n are the distinct right derivatives of S; this means that for any St−1 we have

St−1 = St−1i for some i = 1, . . . , n, i.e.

S(�t) = S(�ti ) for some i and all � ∈ P�. (1)

We are going to show that the assignment

�−1�(S(�t1), . . . , S(�tn))

is a well defined injection. For this, assume that for �, �′ ∈ P� we have

S(�t j ) = S(�′t j ) for all j = 1, . . . , n. (2)

Then for all t ∈ T� we have

(�−1S)(t) = S(�t)(1)= S(�ti )

(2)= S(�′ti )(1)= S(�′t) = (�′−1S)(t),

i.e. �−1S = �′−1S. Since (v) ⇒ (i), the function S has finite range. We put

k = card{S(t)/t ∈ T�}.

Then the result follows by observing that there are at most kn distinct n-tuples (S(�t1), . . . , S(�tn)).Finally the implication (vi) ⇒ (v) is obtained by dualizing the above arguments. �

Call a fuzzy tree language S : T� → E syntactically recognizable (s-recognizable) whenever it satisfies one (andthus all) of the conditions of the previous theorem. Recs(�, E) denotes the so defined set.

An immediate application of Propositions 4 and 5 yields

Proposition 6. For any tree homomorphism h : T� → T� and any function : E → E ′ we have

S ∈ Recs(�, E) implies S ◦ h ∈ Recs(�, E)

and

S ∈ Recs(�, E) implies ◦ S ∈ Recs(�, E ′).

Moreover, if � is an arbitrary operation on E, then

S1, S2 ∈ Recs(�, E) implies S1�S2 ∈ Recs(�, E).

Example 1. Recall that a tree language L ⊆ T� is recognizable whenever a finite �-algebra A = (A, �) exists, so thatL = h−1

A (P), for some P ⊆ A (cf. [3,4]).Assuming now that E has at least two points e0, e1, the function SL : T� → E defined by

SL (t) = e1 if t ∈ L

= e0 else

is manifestly s-recognizable. In other words L recognizable iff SL s-recognizable.


Now taking into account that

SLt−1 = SLt−1 for all t ∈ T�,

with

Lt−1 = {�/� ∈ P�, �t ∈ L}we refind the next classical characterization result:

L recognizable iff card{Lt−1/t ∈ T�} < ∞.

5. Some decidability results

We start with a lemma.

Lemma 1. Assume that S : T� → E is represented by (A = (A, �), �), with card A = n < ∞. Then

(i) For every t ∈ T� we can effectively determine t ∈ T� with height(t)≤n, so that

S(�t) = S(�t) for all � ∈ P�.

(ii) For all � ∈ P� we can effectively determine � ∈ Pn , so that

S(�t) = S(�t) for all t ∈ T�.

Proof. (i) Assume that

t = �1 . . . �pc with �i ∈ P�, |�i | = 1 for i = 1, . . . , p and c ∈ �0.

If p > n, the elements

hA(�1 . . . �pc), hA(�2 . . . �pc), . . . , hA(�pc), hA(c)

cannot be distinct. Thus we must have

hA(�i . . . �pc) = hA(� j . . . �pc) with i < j.

But then, for all � ∈ P� we have

S(�t) = �(hA(�t))

= �(hA(��1 . . . �i . . . � j . . . �pc))

= �(�1 . . . �i−1hA(�i . . . � j . . . �pc))

= �(��1 . . . �i−1hA(� j . . . �pc))

= �(hA(��1 . . . �i−1� j . . . �pc))

= S(��1 . . . �i−1� j . . . �pc).

The item (i) comes by repeating, if necessary, the above argument.(ii) For each � ∈ P�, we denote by �(�) ∈ P�, the tree obtained by replacing every s ∈ Cl(�) by its associated s

according to item (i).Clearly it holds

S(�t) = S(�(�)t) for all t ∈ T�.

If |�(�)| > n, we proceed as in (i) in order to determine � ∈ Pn with the desired properties. �


Assume now that Si : T� → E is represented by (Ai = (Ai , �i ), �i ) with card Ai = ni < ∞(i = 1, 2). Then

S : T� → E1 × E2, S(t) = (S1(t), S2(t))

is manifestly represented by (A1 × A2, �1 × �2). Applying item (i) of Lemma 1 to S, for each t ∈ T� we gett, height(t)≤n1 + n2, so that S(t) = S(t) i.e. Si (t) = Si (t) (i = 1, 2).

Hence, if S1, S2 coincide on the trees with height≤n1 + n2, then they are equal since

S1(t) = S1(t) = S2(t) = S2(t).

Consequently

Proposition 7. The equality problem is decidable for s-recognizable fuzzy tree languages.

Our next task will be the effective construction of the syntactic algebra AS of an s-recognizable fuzzy languageS : T� → E .

According to Proposition 3 and Theorem 2, AS can be identified (up to isomorphism) with the �-algebra whosecarrier set is the set of all right derivatives of S

AS = {St−1/t ∈ T�}and whose structural operations are given by

(�S) f (St−11 , . . . , St−1

k ) = S f (t1, . . . , tk)−1, f ∈ �k, k≥0.

By virtue of Lemma 1, item (i) we have St−1 = St−1 and thus AS consists of all distinct elements of the finite list

St−1, height(t)≤n.

On the other hand we can decide whether or not two right derivatives St−1 and St ′−1 are equal or not. Indeed, assumethat

S(�t) = S(�t ′) for all � ∈ Pn . (3)

Then for any � ∈ P� we have

S(�t) = S(�t) (by Lemma 1(ii))

= S(�t ′) (by (3))

= S(�t ′) (by Lemma 1(ii)).

Thus, the set AS is effectively determined.Now for f ∈ �k(k≥0) and St−1

i , height(ti )≤n(i = 1, . . . , k) we get

S f (t1, . . . , tk)−1 = St−1, height(t)≤n.

Consequently

(�S) f (St−11 , . . . , St−1

k ) = St−1

and �S is completely computed. We conclude

Theorem 3. For any s-recognizable fuzzy tree language S : T� → E , its syntactic �-algebra AS can be effectivelyconstructed.

We close this section by giving a pumping result.


Proposition 8. Let S : T� → E be an s-recognizable fuzzy language. Then, there exists an integer N > 0 such thatfor any tree t with height(t) > N , we have a factorization

t = ��u, �, � ∈ P�, u ∈ T�, |�| > 0, height(�u)≤N

so that

S(t) = S(��ku) for all k≥0.

Proof. Argue as in Lemma 1(i). �

6. The homomorphic image problem

A fundamental result in Tree Language Theory states that recognizability is preserved by linear (non-arbitrary) treehomomorphisms. In the present section we shall attack this problem in the fuzzy setup.

In order to define the image under the tree homomorphism h : T� → T� of the fuzzy tree language S :T� → E , we need to assume that E is equipped with infinite (countable) �’s. This means that to each family(ai )i∈I of elements of E we can assign an element �i∈I ai of E such that next associativity-commutativity axiomholds:

�i∈I, j∈J

ai j = �i∈I

(�j∈J

ai j

).

Then the image of S under h with respect to � is the fuzzy tree language

h(S) : T� → E, h(S)(u) = �h(t)=u

S(t).

Here, our interest is focused in the case that E = [0, 1], � = min, �L , �D, �a and

�i∈I

ai = infF⊆I

F finite

�i∈F

ai

as well as � = max, �L , �D, �a and

%i∈I

ai = supF⊆I

Ffinite

%i∈F

ai .

Theorem 4. With respect either to min, �L , �D or to max, �L , �D if h : T� → T� is linear, then

S : T� → [0, 1] s-recognizable implies that h(S) : T� → [0, 1] s-recognizable.

Proof. We first treat the case of a projection h : T�∪� → T�, h(�) = xk(1≤k≤rank(�)), h(�) = � for all � ∈ �. Letus point out that for any tree

t =

f� . . . �f1 f p

� . . . � � . . . �...

......

...


of T�, h−1(t) is the set of all trees of the form�

� . . . | . . . �T�∪� : T�∪�

...

|�

� . . . | . . . �T�∪� f T�∪�

� . . . . . . . . . ��

� . . . .| . . . .� � . . . .| . . . .�T�∪� : T�∪� T�∪� : T�∪�

......

| |� �

� . . . | . . . � � . . . | . . . �T�∪� f1 T�∪� T�∪� f p T�∪�� . . . | . . . � � . . . | . . . �

......

......

For t ∈ T� and � ∈ P� it holds that

h−1(�t) = h−1(�)h−1(t).

Next, we are going to calculate the left derivatives of h(S). For � ∈ P� we have

[�−1h(S)](t) = h(S)(�t)

= �h(u)=�t

S(u)

= �h(�)=�h(w)=t

S(�w)

= �h(�)=�

(�

h(w)=tS(�w)

)= �

h(�)=�h(�−1S). �

At this moment we need two lemmas.

Lemma 2. With respect either to min, �L , �D or to max, �L , �D if S : T�∪� → [0, 1] has finite range, then so doesh(S) : T� → [0, 1].

Proof. We have to apply Proposition 2. �

Lemma 3. Let D be an arbitrary set. The D-power of a ∈ [0, 1] (resp. of g : B → [0, 1]) with respect to � is definedby

aD = �d∈D

ad , ad = a for all d ∈ D

(resp. gD(b) = g(b)D for all b ∈ B).If g : B → [0, 1] has finite range then for � = min, �L , �D, max, �L , �D there are finitely many powers of g.

Proof. For � = max, min the result is immediate.


� = �L . Set A = {g(b)/b ∈ B}. By virtue of Proposition 2, for any b ∈ B such that g(b) < 1, we have g(b)n = 0for all n > 1/1 − max(A) the function gn coincides with the characteristic function of the set g−1(1) = {b/g(b) = 1}.

� = �D . Again, according to Proposition 2, for any g(b) < 1, we have g(b)2 = 0 and thus for n≥2, gn = 0.Finally, the cases � = �L , �D can be treated in an analogous way. �

Now, let us return to the proof of our Theorem.Since by hypothesis S is s-recognizable, there will be finitely many distinct left derivatives of it

�−11 S, . . . , �−1

k S.

Thus, the set P�∪� can be partitioned into a finite number of classes C1, . . . , Ck with Ci = {�/�−1S = �−1i S}.

For each � ∈ P�, we set

Ci,� = Ci ∩ h−1(�).

Then, the established identity

�−1h(S) = �h(�)=�

h(�−1S)

can be rewritten as

�−1h(S) = h(�−11 S)C1,�� . . . �h(�−1

k S)Ck,� .

By Lemma 2, h(�−1i S) has finite range and by Lemma 3, h(�−1

i S) has finitely many powers. It follows that

card{�−1h(S)/� ∈ P�} < ∞as wanted.

The case that h is a relabelling is similar to the above.We are going to apply item (i) of Theorem 2. Next, let h : T�∪� → T�∪{ 1, 2} with rank( 1) = k + 1, rank( 2) =

l, rank(�) = k + l and h(�) = 1(x1, . . . , xm, 2(xm+1, . . . , xm+l ), xm+l+1, . . . , xk+l ), h(�) = � for all � ∈ �.Let (A = (A, �), �) be a representation of S : T�∪� → [0, 1]. We introduce the �-algebra A′ = (A′, �′) where

A′ = A ∪ {p 2(q1 . . .qk )/q1, . . . , qk ∈ A} ∪ {p+},

�′f = � f for all f ∈ �, �′

2(q1, . . . , qk) = p 2(q1 . . .qk )

, �′ 1

(p1, . . . , pm, p 2(qm+1 . . .qm+k ), pm+k+1, . . . , pk+l ) = �′

�(p1,

. . . , pm, qm+1, . . . , qm+k, pm+k+1, . . . , pk+l ).In all other cases the value of �′ is equal to p+.An easy calculation shows that (A′, �′) with �′ : A′ → [0, 1] defined by �′(q) = �(q) for all q ∈ A �′(p) = 0 for

all p ∈ A′ − A computes h(S).The result comes by applying Theorem 1. �

Non-linear tree homomorphisms do not preserve in general s-recognizability as confirms next simple example.

Example 2. Take � = {a, c}, � = {�, c}, rank(c) = 0, rank(a) = 1, rank(�) = 2 and let h : T� → T� be definedby h(c) = c, h(a) = �(x, x). Then although S : T� → [0, 1], S(t) = 1

2 for all t ∈ T� is recognizable, its image fails todo so.

7. Fuzzy tree automata

A fuzzy tree automaton over (�, �) is a system

M = (�, Q, �, F),

where � is the input alphabet, Q is the finite set of states, F : Q → [0, 1] is the final distribution and

� = (� f : Qk → Fuzzy(Q)) f ∈�k , k≥0

is the transition family of M.


The function � f is extended into a function

� f : Fuzzy(Q)k → Fuzzy(Q)

by setting

� f (X1, . . . , Xk) = %q1,. . .,qk∈Q

X1(q1)� . . . �Xk(qk)�� f (q1, . . . , qk)

for all X1, . . . , Xk ∈ Fuzzy(Q).The reachability map of M

�M : T� → Fuzzy(Q)

is inductively defined by

�M( f (t1, . . . , tk)) = � f (�M(t1, . . . , tM(tk)))

and the behavior of M is the fuzzy tree language |M| : T� → [0, 1] given by the formula

|M|(t) = %q∈Q

�M(t)(q)�F(q), t ∈ T�.

Fuzzy tree languages obtained as behaviors of such automata are called recognizable and their set is denoted byRec(�, �, �).

Proposition 9. It holds that

Recs(�) ⊆ Rec(�, �, �)

for any distributive pair (�, �).

Proof. Let S ∈ Recs(�) and consider a finite representation (A = (A, �), �) of S. Then the fuzzy tree automaton

M(A) = (�, A, �, �)

with � f : Ak → Fuzzy(A) defined by

� f (q1, . . . , qk)(q) = 1 if � f (q1, . . . , qk) = q

= 0 else,

computes S. �

Example 3. The inclusion in Proposition 9 is proper when dealing with the distributive pair (max, �m). Indeed, thefuzzy tree language S : T� → [0, 1], S(t) = asize(t), 0 < a < 1 which is the behavior of the automaton

M = (�, {q}, �, F)

� f (q, . . . , q)(q) = 1( f ∈ �), F(q) = 1

is not s-recognizable for the reason it has not finite range (remember that si ze(t) denotes the number of symbols in �occurring in t).

The opposite inclusion also holds for some interesting special cases.

Theorem 5. If (�, �) is one of the following pairs

(max, min), (max, �L ), (max, �D) (4)

then

Rec(�, �, �) = Recs(�).


Proof. By virtue of the previous proposition, it suffices to establish only the inclusion

Rec(�, �, �) ⊆ Recs(�).

Let M = (�, Q, �, F) be a fuzzy tree automaton over (�, �). For t = f (t1, . . . , tn) and q ∈ Q, the equality

�M( f (t1, . . . , tn))(q) = %q1,. . .,qn∈Q

�M(t1)(q1)� . . . ��M(tn)(qn)�� f (q1, . . . , qk)(q)

shows that �M(t)(q) can be expressed as a finite �’s of �-products of elements of the finite list

� f (q1, . . . , qn)(q), f ∈ �n, n≥0, q1, . . . , qn, q ∈ Q.

By the choice of (�, �) the set {�M(t)(q)/t ∈ T�, q ∈ Q} is finite and thus so is the set A = {�M(t)/t ∈ T�}. Weconvert A into a �-algebra by defining the structural operation � f : An → A by

� f (�M(t1), . . . , �M(tn)) = �M( f (t1, . . . , tn)).

This formula is consistent for if

�M(t ′1) = �M(t1), . . . , �M(t ′n) = �M(tn)

then

�M( f (t ′1, . . . , t ′n)) = � f (�M(t ′1), . . . , �M(t ′n))

= � f (�M(t1), . . . , �M(tn))

= �M( f (t1, . . . , tn)).

The morphism hA : T� → A is clearly equal to �M, and so (A, �, �), with

� : A → [0, 1], �(X ) = %q∈Q

X (q)�F(q), X ∈ A

represents the behavior |M|, i.e. |M| = � ◦ hA.Consequently |M| is s-recognizable and the proof is completed. �

Corollary 1. The equivalence problem for tree automata over the pairs (4) is decidable.

A new behavior of a tree automaton M = (�, Q, �, F) can be computed, if we use a pair (�, �) consisting of twot-norms, such that � is distributive over �:

x�(y�z) = (x�y)�(x�z), x, y, z ∈ [0, 1].

It is the case of the pairs

(min, min), (min, �L ), (min, �D), (min, �m).

More precisely, the extension of � f is given by

� f (X1, . . . , Xn) = �q1,. . .,qn∈Q

X (q1)� . . . �X (qn)�� f (q1, . . . , qn),

whereas �M is given by the corresponding inductive formula

�M( f (t1, . . . , tk)) = � f (�M(t1), . . . , �M(tk)).

The proof of Theorem 5 remains valid only for the pairs

(min, min), (min, �L ), (min, �D) (5)

which render the set A = {�M(t)/t ∈ T�} finite.


Theorem 6. The behavior of a tree automaton M computed with respect to the pairs of the list (5) is an s-recognizablefuzzy tree language.

The use of (min, �m) provides a non s-recognizable fuzzy language as indicates again Example 3.The example below shows the rather unexpectable fact that there are s-recognizable fuzzy tree languages which

cannot be computed by any tree automaton.

Example 4. Consider the ranked alphabet

� = { f, c, d}, rank( f ) = 2, rank(c) = rank(d) = 0

and the representation (A = (A, �), �) with

A = {q1, q2}, �(q1) = 1, �(q2) = 0, �c = q1, �d = q2,

� f (q1, q1) = q1 = � f (q2, q2), � f (q1, q2) = q2 = � f (q2, q1).

The computed fuzzy tree language S = � ◦ hA : T� → [0, 1] satisfies the equalities

S(c) = 1 = S( f (d, d)), S(d) = 0.

Suppose there exists a tree automaton M = (�, Q, �, F) whose behavior with respect to (min, min) is equal to S.Then from |M|(c) = 1 or minq∈Q(�c(q), F(q)) = 1, we get F(q) = 1, for all q ∈ Q. Thus from |M|(d) = 0 we get�d (q) = 0 for some q ∈ Q. But then for all p ∈ Q we have

�M( f (d, d))(p) = minq,q ′∈Q

(�d (q), �d (q ′), � f (q, q ′)(p)) = 0

and so |M|( f (d, d)) = 0, contradiction.The above machinery also works if instead of (min, min) we use (min, �L ), (min, �D).

Theorem 7. It holds that

Rec(�, �, �) ⊂ Recs(�) (strict inclusion)

with (�, �) running over the list

(max, max), (max, �L ), (max, �D). (6)

Finally we consider tree automata with respect to (�, �), where � is a t-conorm distributive over the t-norm �. It isthe case of the pairs

(min, max), (min, �L ), (min, �D). (7)

Theorem 8. It holds that

Recs(�) = Rec(�, �, �)

where (�, �) ranges over (7).

Proof. We only have to dualize the arguments in Proposition 9 and Theorem 5. �

8. Non distributive pairs

Most of the pairs (�, �) with � = max, �L , �D, �a and � = min, �L , �D, �m are not distributive and the questionis whether we can compute the behavior of a tree automaton with respect to such a pair as well as to examine whatproperties these behaviors have.

Consider a tree automaton M = (�, Q, �, F) and a non-distributive pair (�, �).


τ :

(q1,..., qi−1, p, qi+1,..., qk, f/a, q)

t1 ti−1 x ti+1 tk

Fig. 1.

Weighted runs of trees are needed. In order to construct them, we introduce the transition alphabet C by setting

C0 = {(c/a, q)/c ∈ �0 and �c(q) = a},Ck = {(q1, . . . , qk, f/a, q)/ f ∈ �k(k≥1) and � f (q1, . . . , qk)(q) = a}.

We simultaneously define the tree languages Lq ⊆ TC and the mappings valq : Lq → [0, 1] as follows:

• c = (c/a, q) ∈ Lq and valq (c) = a, c ∈ �0, q ∈ Q.• If f = (q1, . . . , qk, f/a, q) ∈ �k and t1 ∈ Lq1 , . . . , tk ∈ Lqk , then f(t1, . . . , tk) ∈ Lq and valq (f(t1, . . . , tk)) =

a�valq1 (t1)� . . . �valqk (tk).

For all t ∈ T� and q ∈ Q we define

�M(q) = %hq (t)=t

valq (t)

where hq : Lq → T� is the restriction on Lq of the tree homomorphism h : TC → T� given by

• h((c/a, q)) = c, c ∈ �0, q ∈ Q;• h((q1, . . . , qk, f/a, q)) = f (x1, . . . , xk), f ∈ �k and q1, . . . , qk ∈ Q.

In the case that � = max, �L , �D and � = min, �L , �D the set A = {�M(t)/t ∈ T�} is finite but the constructionof Theorem 5 does not work. However

Theorem 9. Under the above made considerations, the behavior |M| is an s-recognizable fuzzy tree language.

Proof. The result follows by showing that |M| has finitely many left derivatives and then applying Theorem 2, item(vi).

For p, q ∈ Q we denote by �(q,p)C the set consisting of all trees in PC of the form shown in Fig. 1, with f ∈ �k(k≥1)

and t j ∈ Lq j ( j�i).We set

P (q,p)C =

⋃p1,...,pl ∈Q

l≥0

�(q,p1)C �(p1,p2)

C . . . �(pl ,q)C

and define val(q,p) : P (q,p)C → [0, 1] as follows:

• if s is as in Fig. 1 then

val(q,p)(s) = a�

(�j�i

valq j (t j )

);

• if p = s1 . . . sl with s j as in Fig. 1, 1≤ j≤l then

val(q,p)(p) = val(q,p1)(s1)� . . . �val(pl ,q)(sl ).

Consider the fuzzy tree languages

G�,q : T� → [0, 1], G�,q (t) = �M(�t)(q)

(� ∈ P�, q ∈ Q). Each u ∈ Lq with hq (u) = �t can be written as

u = spt, sp ∈ P (q,p)C , t ∈ L p, h(q,p)(sp) = �, h p(t) = t


and

valq (u) = val(q,p)(sp)�valp(t).

We have

G�,q (t) = %hq (u)=�t

valq (u) = %q,p∈Q

val(q,p)(sp)�valp(t).

By virtue of Proposition 9

card{val(q,p)(sp)/h(q,p)(s) = �, � ∈ P�, p, q ∈ Q} < ∞and thus

card{G�,q/� ∈ P�, q ∈ Q} < ∞.

Consequently

card{G�/� ∈ P�} < ∞with

G� : T� → Fuzzy(Q), G�(t) = �M(t)

and finally

card{�−1|M|/� ∈ P�} < ∞as wanted. �

Corollary 2. The equivalence problem for tree automata with respect to a non-distributive pair (�, �), is decidable.

9. Conclusion

In the present paper next equivalent conditions have been established for a fuzzy language S : T� → [0, 1]:

• S is recognized by a deterministic finite tree automaton with final state distribution in [0, 1] (called representation ofS).

• The minimal representation AS of S is finite.• The syntactic congruence ∼S has finite index.• S has finitely many right derivatives.• S has finitely many left derivatives.

Then we effectively have constructed AS provided that S is s-recognizable.The above common class of fuzzy tree languages denoted by Recs has nice closure properties, namely it is closed

under arbitrary inverse tree homomorphism, arbitrary negation, Hadamard product but the most important fact is theclosure of Recs under linear (non-arbitrary) homomorphisms. The equality problem in Recs is decidable.

Tree automata whose transitions are weighted over the unit interval [0, 1] and whose behavior is computed withrespect to a pair (�, �) of a t-conorm � and a t-norm � have been investigated. In the case that � is distributive over�, the above automata recognize exactly the fuzzy languages of the class Recs .

Finally, fuzzy tree automata with respect to (�, �) with � non-distributive over � (which is the most frequentcase) have been considered. The behaviors of such automata are no longer defined inductively, but collectively. Therecognition power of these automata is weaker than s-recognizability and thus the equivalence problem is decidablefor them.

Acknowledgment

We thank the referees for their fruitful remarks and suggestions.


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