Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 825249, 16 pages
http://dx.doi.org/10.1155/2013/825249
Research Article
On Intuitionistic Fuzzy Context-Free Languages
Jianhua Jin,1,2 Qingguo Li,1 and Chunquan Li2
1
2
College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
College of Sciences, Southwest Petroleum University, Chengdu 610500, China
Correspondence should be addressed to Qingguo Li; liqingguoli@yahoo.com.cn
Received 19 October 2012; Revised 15 December 2012; Accepted 15 December 2012
Academic Editor: Hak-Keung Lam
Copyright © 2013 Jianhua Jin et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Taking intuitionistic fuzzy sets as the structures of truth values, we propose the notions of intuitionistic fuzzy context-free
grammars (IFCFGs, for short) and pushdown automata with final states (IFPDAs). Then we investigate algebraic characterization
of intuitionistic fuzzy recognizable languages including decomposition form and representation theorem. By introducing the
generalized subset construction method, we show that IFPDAs are equivalent to their simple form, called intuitionistic fuzzy simple
pushdown automata (IF-SPDAs), and then prove that intuitionistic fuzzy recognizable step functions are the same as those accepted
by IFPDAs. It follows that intuitionistic fuzzy pushdown automata with empty stack and IFPDAs are equivalent by classical automata
theory. Additionally, we introduce the concepts of Chomsky normal form grammar (IFCNF) and Greibach normal form grammar
(IFGNF) based on intuitionistic fuzzy sets. The results of our study indicate that intuitionistic fuzzy context-free languages generated
by IFCFGs are equivalent to those generated by IFGNFs and IFCNFs, respectively, and they are also equivalent to intuitionistic fuzzy
recognizable step functions. Then some operations on the family of intuitionistic fuzzy context-free languages are discussed. Finally,
pumping lemma for intuitionistic fuzzy context-free languages is investigated.
1. Introduction
Intuitionistic fuzzy set (IFS) introduced by Atanassov [1–
3], which emerges from the simultaneous consideration of
the degrees of membership and nonmembership with a
degree of hesitancy, has been found to be highly useful in
dealing with problems with vagueness and uncertainty. The
notion of vague set, proposed by Gau and Buehrer [4], is
another generalization of fuzzy sets. However, Burillo and
Bustince [5] showed that it is an equivalence of the IFS and
studied intuitionistic fuzzy relations. Recently, IFS theory
has supported a wealth of important applications in many
fields such as fuzzy multiple attribute decision making, fuzzy
pattern recognition, medical diagnosis, fuzzy control, and
fuzzy optimization [6–10].
In classical theoretical computer science, it is well known
that formal languages are very useful in the description of
natural languages and programming languages. But they are
not powerful in the processing of human languages. For this,
Lee and Zadeh [11] introduced the notion of fuzzy languages
and gave some characterizations, where fuzzy languages took
values in the unit interval [0, 1]. Malik and Mordeson [12–
14] studied algebraic properties of fuzzy languages. They
stated that fuzzy regular languages can be characterized by
fuzzy finite automata, fuzzy regular expressions, and fuzzy
regular grammars. Meanwhile, as one of the generators of
fuzzy languages, fuzzy automata have been used to solve
meaningful issues such as the model of computing with
words [15], clinical monitoring [16], neural networks [17], and
pattern recognition [18]. Also, fuzzy grammars, automata,
and languages tend to the improvement of lexical analysis and
simulating fuzzy discrete event dynamical systems and hybrid
systems [14, 19].
As is well known, quantum logic was proved by Birkhoff
and Von Neumann as a logic of quantum mechanics and
2
is currently understood as a logic with truth values taken
from an orthomodular lattice. To study quantum computation, Ying [20, 21] first proposed automata theory based
on quantum logic where quantum automata are defined to
be orthomodular lattice-valued generalization of classical
automata. More systematic exposition of this theory appeared
in [22, 23]. Moore and Crutchfield [24] defined quantum
version of pushdown automata and regular and contextfree grammars. He showed that the corresponding languages
generated by quantum grammars and recognized by quantum
automata have satisfactory properties in analogy to their
classical counterparts. A basic framework of grammar theory
on quantum logic was established by Cheng and Wang [25].
They proved that the set of lattice-valued quantum regular
languages generated by lattice-valued quantum regular grammars coincides with that of lattice-valued quantum languages
recognized by lattice-valued quantum automata. Then some
algebraic properties of automata based on quantum logic
were discussed by Qiu [26, 27]. To enhance the processing
ability of fuzzy automata, the membership grades were
extended to many general algebraic structures. For example,
by combining the ideas in [20–23] and the idea in Ying’s
another work on topology based on residuated lattice-valued
logic [28], Qiu has primarily established automata theory
based on complete residuated lattice-valued logic [29–31].
And Li and Pedrycz [32] studied automata theory with membership values in lattice-ordered monoids. They showed that
lattice-valued finite automata have more power to recognize
fuzzy languages than that of classical fuzzy finite automata.
Recently, Li [33] studied automata theory with membership
values in lattices, introduced the technique of extended
subset construction to prove the equivalence between latticevalued finite automata and lattice-valued deterministic finite
automata, and then presented a minimization algorithm of
lattice-valued deterministic finite automata. On the basis of
breadth-first and depth-first ways, Jin and Li [34] established
a fundamental framework of fuzzy grammars based on
lattices, which provided a necessary tool for the analysis of
fuzzy automata.
Fuzzy context-free languages, more powerful than fuzzy
regular languages, have also been studied and can be characterized by fuzzy pushdown automata with two distinct
ways and fuzzy context-free grammars, respectively [14, 35].
As a continuation of the work in [29–31], a fundamental
framework of fuzzy pushdown automata theory based on
complete residuated lattice-valued logic has been established
in recent years by Xing et al. [36], and the work generalizes
the previous fuzzy automata theory systematically studied by
Mordeson and Malik to some extent. The pumping lemma
for fuzzy context-free grammar theory in this setting was also
investigated by Xing and Qiu [37].
Using the notions of IFSs and fuzzy finite automata,
Jun [38, 39] presented the concept of intuitionistic fuzzy
finite state machines as a generalization of fuzzy finite state
machines, and Zhang and Li [40] discussed intuitionistic
fuzzy recognizers, intuitionistic fuzzy finite automata, and
intuitionistic fuzzy language. They showed that the languages
recognized by intuitionistic fuzzy recognizers are regular, and
Journal of Applied Mathematics
the intuitionistic fuzzy languages recognized by the intuitionistic fuzzy finite automata and the intuitionistic fuzzy languages recognized by deterministic intuitionistic fuzzy finite
automata are equivalent. Recently Chen et al. [41] utilized the
intuitionistic fuzzy automata to deal with consumers’ advertising involvement when considering the expression of an IFS
characterized by a pair of membership degree and nonmembership degree is similar to human thinking logic with pros
and cons. Due to pushdown automata being another kind of
important computational models [15] and also motivated by
the importance of grammars, languages and models theory
[14], it stands to reason that we ought consider the notions of
intuitionistic fuzzy pushdown automata, intuitionistic fuzzy
context-free grammars, and fuzzy context-free languages
because our discussion in this paper will provide a fundamental framework for studying intuitionistic fuzzy set theory
on fuzzy pushdown automata and generators as well. How to
characterize intuitionistic fuzzy context-free languages and
its pumping lemma in this setting becomes open problems;
however, there is no research on the algebraic characterization of intuitionistic fuzzy context-free languages. We
will try to solve the problems in this paper. Additionally,
some examples are given to illustrate the significance of the
results. In particular, Example 35 presented in this paper
will show that intuitionistic fuzzy pushdown automata have
more power than fuzzy pushdown automata when comparing
two distinct strings although the degrees of membership of
these strings recognized by the underlying fuzzy pushdown
automata are equal. Investigating intuitionistic fuzzy contextfree languages will reduce the gap between the precision of
formal languages and the imprecision of human languages.
The remaining parts of the paper are arranged as follows. Section 2 describes some basic concepts of IFSs. Section 3 gives the definitions of intuitionistic fuzzy pushdown
automata with two distinct ways and their languages. It
is investigated that, for any intuitionistic fuzzy pushdown
automaton with final states (IFPDA, for short), there is
a cover, which consists of a collection of classical pushdown automata, equivalent to the IFPDA. By introducing
intuitionistic fuzzy recognizable step functions, it is shown
that intuitionistic fuzzy pushdown automata with final states
and empty stack are intuitionistic fuzzy recognizable step
functions, respectively, and conversely any intuitionistic
fuzzy recognizable step function can be recognized by an
intuitionistic fuzzy pushdown automaton with final states or
empty stack. It follows that intuitionistic fuzzy pushdown
automata with final states and empty stack are equivalent.
Section 4 studies intuitionistic fuzzy context-free grammars
(IFCFGs) as a type of generator of intuitionistic fuzzy
context-free languages (IFCFLs). The notions of intuitionistic
fuzzy Chomsky normal form (IFCNF) and Greibach normal form (IFGNF) are proposed. The results of our study
indicate that IFCFLs generated by IFCFGs are equivalent to
those generated by IFGNFs and IFCNFs, respectively, and
they are also equivalent to intuitionistic fuzzy recognizable
step functions. The algebraic properties of IFCFLs are also
discussed. Section 5 establishes pumping lemma for IFCFLs.
Some examples are then given to illustrate the application
of pumping lemma and the significance of IFCFLs. Finally,
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conclusions and directions for future work are presented in
Section 6.
2. Basic Concepts
Definition 1 (see [40]). An intuitionistic fuzzy set 𝐴 in a nonempty set 𝑋 is an object having the form:
𝐴 = {(𝑥, 𝜇𝐴 (𝑥) , 𝜈𝐴 (𝑥)) | 𝑥 ∈ 𝑋} ,
(1)
where the functions 𝜇𝐴 : 𝑋 → [0, 1] and 𝜈𝐴 : 𝑋 → [0, 1]
denote the degree of membership (i.e., 𝜇𝐴 (𝑥)) and the degree
of nonmembership (𝜈𝐴(𝑥)) of each element 𝑥 ∈ 𝑋 to the set
𝐴, respectively, and the two quantities satisfy the following
inequalities:
0 ≤ 𝜇𝐴 (𝑥) + 𝜈𝐴 (𝑥) ≤ 1,
∀𝑥 ∈ 𝑋.
(2)
For the sake of simplicity, we use the notation 𝐴 = (𝜇𝐴 , 𝜈𝐴 )
instead of 𝐴 = {(𝑥, 𝜇𝐴 (𝑥), 𝜈𝐴(𝑥)) | 𝑥 ∈ 𝑋}. An intuitionistic
fuzzy set will be abbreviated as an IFS.
Let {𝐴 𝑖 | 𝑖 ∈ 𝐼} be a family of IFSs in 𝑋. Then the infimum
and supremum operations of IFSs are defined as follows:
𝑖∈𝐼
𝑖∈𝐼
(3)
𝑖∈𝐼
0, if 𝑢 = 𝑣
𝜈𝑖𝑑 (𝑢, 𝑣) = {
1, if 𝑢 ≠ 𝑣,
(7)
for all (𝑢, 𝑣) ∈ 𝑋 × 𝑋.
Definition 2. Let 𝐴 = (𝜇𝐴 , 𝜈𝐴 ) be an IFS in 𝑋. Then the image
set of 𝐴, denoted as Im(𝐴), is given as
Im (𝐴) = Im (𝜇𝐴 ) ∪ Im (𝜈𝐴 ) ,
(8)
where Im(𝜇𝐴 ) = {𝜇𝐴 (𝑥) | 𝑥 ∈ 𝑋} and Im(𝜈𝐴 ) = {𝜈𝐴(𝑥) | 𝑥 ∈
𝑋}.
For any 𝜆, 𝜃 ∈ [0, 1], 𝜆 + 𝜃 ≤ 1, the (𝜆, 𝜃)-cut set of 𝐴 is
defined as
𝐴 (𝜆,𝜃) = {𝑥 ∈ 𝑋 | 𝜇𝐴 (𝑥) ≥ 𝜆, 𝜈𝐴 (𝑥) ≤ 𝜃} .
(9)
supp (𝐴) = {𝑥 ∈ 𝑋 | 𝜇𝐴 (𝑥) > 0, 𝜈𝐴 (𝑥) < 1} .
(10)
𝑖∈𝐼
where ⋁ and ⋀ denote supremum and infimum of real numbers in [0, 1], respectively.
For two IFSs 𝐴 = (𝜇𝐴 , 𝜈𝐴 ) and 𝐵 = (𝜇𝐵 , 𝜈𝐵 ), we say 𝐴 = 𝐵
if 𝜇𝐴 = 𝜇𝐵 and 𝜈𝐴 = 𝜈𝐵 . In addition, if the IFS 𝐴 = (𝜇𝐴 , 𝜈𝐴) in
𝑋 satisfies the condition that, for any 𝑥 ∈ 𝑋, 𝜇𝐴 (𝑥) + 𝜈𝐴(𝑥) =
1, then 𝐴 reduces to a fuzzy set in 𝑋. The difference between
intuitionistic fuzzy sets and fuzzy sets is whether the sum
of the degrees of membership and nonmembership of an
element to a set equals one.
An IFR in 𝑋 × 𝑌 is an intuitionistic fuzzy subset of 𝑋 × 𝑌;
that is, it is an expression 𝐸 given by
𝐸 = {((𝑥, 𝑦) , 𝜇𝐸 (𝑥, 𝑦) , 𝜈𝐸 (𝑥, 𝑦)) | 𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌} ,
(4)
where the mappings 𝜇𝐸 : 𝑋 × 𝑌 → [0, 1] and 𝜈𝐸 : 𝑋 × 𝑌 →
[0, 1] satisfy
0 ≤ 𝜇𝐸 (𝑥, 𝑦) + 𝜈𝐸 (𝑥, 𝑦) ≤ 1,
∀ (𝑥, 𝑦) ∈ 𝑋 × 𝑌.
(5)
An IFBR over 𝑋 is an IFS of 𝑋 × 𝑋. Let 𝑃 = (𝜇𝑃 , 𝜈𝑃 ) and
𝐸 = (𝜇𝐸 , 𝜈𝐸 ) be IFRs in 𝑋 × 𝑌 and 𝑌 × 𝑍, respectively. Define
the composition of IFRs, 𝑃 ∘ 𝐸 = (𝜇𝑃∘𝐸 , 𝜈𝑃∘𝐸 ) in 𝑋 × 𝑍, by
𝑦∈𝑌
𝜈𝑃∘𝐸 (𝑥, 𝑧) = ⋀ (𝜈𝑃 (𝑥, 𝑦) ∨ 𝜈𝐸 (𝑦, 𝑧)) ,
𝑦∈𝑌
3. Intuitionistic Fuzzy Pushdown Automata
It is well known that any language accepted by a pushdown
automaton with final states can be accepted by a certain
pushdown automaton with empty stack, and vice versa. As
a natural generalization of pushdown automata, we give the
notions of intuitionistic fuzzy pushdown automata with final
states and empty stack, respectively, and then do research
in the algebraic characterization of their intuitionistic fuzzy
recognizable languages including decomposition form and
representation theorem. Note that Σ∗ is the free monoid
generated from the set Σ with the operator of concatenation,
where the empty string 𝜀 is identified with the identity of
Σ. And the length of the string 𝜔 ∈ Σ∗ is denoted by |𝜔|.
𝑁𝑘 = {1, . . . , 𝑘}.
Definition 3. An intuitionistic fuzzy pushdown automaton
with final states (IFPDA, for short) is a seven tuple M =
(𝑄, Σ, Γ, 𝛿, 𝐼, 𝑍0 , 𝐹), where
(i) 𝑄 is a finite nonempty set of states;
(ii) Σ is a finite nonempty set of input symbols;
(iii) Γ is a finite nonempty set of stack symbols;
𝜇𝑃∘𝐸 (𝑥, 𝑧) = ⋁ (𝜇𝑃 (𝑥, 𝑦) ∧ 𝜇𝐸 (𝑦, 𝑧)) ,
for all (𝑥, 𝑧) ∈ 𝑋 × 𝑍.
if 𝑢 = 𝑣
if 𝑢 ≠ 𝑣,
If supp(𝐴) is finite, then 𝐴 is called a finite IFS in 𝑋.
⋃𝐴 𝑖 = {(𝑥, ⋁𝜇𝐴 𝑖 (𝑥) , ⋀𝜈𝐴 𝑖 (𝑥)) | 𝑥 ∈ 𝑋} ,
𝑖∈𝐼
1,
𝜇𝑖𝑑 (𝑢, 𝑣) = {
0,
And the support set of 𝐴, denoted as supp(𝐴), is defined by
⋂𝐴 𝑖 = {(𝑥, ⋀𝜇𝐴 𝑖 (𝑥) , ⋁𝜈𝐴 𝑖 (𝑥)) | 𝑥 ∈ 𝑋} ,
𝑖∈𝐼
Furthermore, if 𝑅 is an IFBR over 𝑋, then its reflexive and
𝑛
𝑛+1
= 𝑅𝑛 ∘ 𝑅, 𝑛 ≥
transitive closure is 𝑅∗ = ⋃∞
𝑛=0 𝑅 , where 𝑅
0, and 𝑅0 = 𝑖𝑑 = (𝜇𝑖𝑑 , 𝜈𝑖𝑑 ), that is,
(6)
(iv) 𝛿 = (𝜇𝛿 , 𝜈𝛿 ) is a finite IFS in 𝑄 × (Σ ∪ {𝜀}) × Γ × 𝑄 × Γ∗ ;
(v) 𝑍0 ∈ Γ is called the start stack symbol;
(vi) 𝐼 = (𝜇𝐼 , 𝜈𝐼 ) and 𝐹 = (𝜇𝐹 , 𝜈𝐹 ) are intuitionistic fuzzy
subsets in 𝑄, which are called the intuitionistic fuzzy
subsets of initial and final states, respectively.
4
Journal of Applied Mathematics
Definition 4. An intuitionistic fuzzy pushdown automaton
with empty stack (IFPDA0 , for short) is a seven tuple N =
(𝑄, Σ, Γ, 𝛿, 𝐼, 𝑍0 , 0), where 𝑄, Σ, Γ, 𝛿, 𝐼 and 𝑍0 are the same as
those in IFPDA M, and 0 represents an empty set.
Definition 5. Let M = (𝑄, Σ, Γ, 𝛿, 𝐼, 𝑍0 , 𝐹) be an IFPDA.
Define an IFBR on 𝑄 × Σ∗ × Γ∗ , ⊢M = (𝜇⊢M , 𝜈⊢M ) in the form
of
𝜈⊢M ((𝑞, 𝜔, 𝛽) , (𝑝, 𝑢, 𝛼))
Proposition 9. If 𝑓 can be accepted by some IFPDA M =
(𝑄, Σ, Γ, 𝛿, 𝐼, 𝑍0 , 𝐹), then 𝑓 is an IFS in Σ∗ , and the image set
of 𝑓 is finite.
Proof. We have the following.
Claim 1 (𝑓 = (𝜇𝑓 , 𝜈𝑓 ) is an IFS in Σ∗ ).
𝜇⊢M ((𝑞, 𝜔, 𝛽) , (𝑝, 𝑢, 𝛼))
𝜇𝛿 (𝑞, 𝜀, head (𝛽) , 𝑝, 𝛼 \ tail (𝛽)) ,
{
{
{
if 𝑢 = 𝜔, tail (𝛽) ≤ 𝛼
{
{
= {𝜇𝛿 (𝑞, head (𝜔) , head (𝛽) , 𝑝, 𝛼 \ tail (𝛽)) ,
{
{
if 𝑢 = tail (𝜔) , tail (𝛽) ≤ 𝛼
{
{
0,
otherwise,
{
is also finite, where 𝑋∧ = {𝑥1 ∧ ⋅ ⋅ ⋅ ∧ 𝑥𝑘 : 𝑘 ≥ 1, 𝑥1 , . . . , 𝑥𝑘 ∈
𝑋} ∪ {1}, and 𝑋∨ = {𝑥1 ∨ ⋅ ⋅ ⋅ ∨ 𝑥𝑘 : 𝑘 ≥ 1, 𝑥1 , . . . , 𝑥𝑘 ∈ 𝑋} ∪ {0}.
(11)
𝜈𝛿 (𝑞, 𝜀, head (𝛽) , 𝑝, 𝛼 \ tail (𝛽)) ,
{
{
{
if 𝑢 = 𝜔, tail (𝛽) ≤ 𝛼
{
{
= { 𝜈𝛿 (𝑞, head (𝜔) , head (𝛽) , 𝑝, 𝛼 \ tail (𝛽)) ,
{
{
if 𝑢 = tail (𝜔) , tail (𝛽) ≤ 𝛼
{
{
0,
otherwise
{
for any (𝑞, 𝜔, 𝛽), (𝑝, 𝑢, 𝛼) ∈ 𝑄 × Σ∗ × Γ∗ . Here, for any nonempty string 𝑢 = 𝑥1 ⋅ ⋅ ⋅ 𝑥𝑛 , 𝑛 ≥ 1, head(𝑢) = 𝑥1 , tail(𝑢) =
𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 , and tail(𝑢) ≤ 𝑢. ⊢∗M is the reflexive and transitive
closure of ⊢M .
If no confusion, we denote ⊢ and ⊢∗ instead of ⊢M and
∗
⊢M , respectively.
Definition 6. Let M = (𝑄, Σ, Γ, 𝛿, 𝐼, 𝑍0 , 𝐹) be an IFPDA. Then
we call L(M) an intuitionistic fuzzy language accepted by M
with final states, where L(M) = (𝜇L(M) , 𝜈L(M) ), 𝜇L(M) , and
𝜈L(M) are functions from Σ∗ to the unit interval [0, 1], and
𝜇L(M) (𝜔)= ⋁{𝜇𝐼 (𝑞0 ) ∧ 𝜇⊢∗M ((𝑞0 , 𝜔, 𝑧0 ), (𝑝, 𝜀, 𝑟)) ∧
𝜇𝐹 (𝑝) | 𝑞0 , 𝑝 ∈ 𝑄, 𝑟 ∈ Γ∗ },
𝜈L(M) (𝜔) = ⋀{𝜈𝐼 (𝑞0 ) ∨ 𝜈⊢∗M ((𝑞0 , 𝜔, 𝑧0 ), (𝑝, 𝜀, 𝑟)) ∨
𝜈𝐹 (𝑝) | 𝑞0 , 𝑝 ∈ 𝑄, 𝑟 ∈ Γ∗ }
for any 𝜔 ∈ Σ∗ .
Definition 7. Let N = (𝑄, Σ, Γ, 𝛿, 𝐼, 𝑍0 , 0) be an IFPDA0 .
Then we call L(N) an intuitionistic fuzzy language accepted
by N with empty stack, where L(N) = (𝜇L(N) , 𝜈L(N) ),
𝜇L(N) and 𝜈L(N) are functions from Σ∗ to the unit interval
[0, 1], and
𝜇L(N) (𝜔) = ⋁{𝜇𝐼 (𝑞0 ) ∧ 𝜇⊢∗N ((𝑞0 , 𝜔, 𝑧0 ), (𝑝, 𝜀, 𝜀)) |
𝑞0 ,𝑝 ∈ 𝑄},
𝜈L(N) (𝜔) = ⋀{𝜈𝐼 (𝑞0 ) ∨ 𝑣⊢∗N ((𝑞0 , 𝜔, 𝑧0 ), (𝑝, 𝜀, 𝜀))
𝑞0 , 𝑝 ∈ 𝑄}
|
for any 𝜔 ∈ Σ∗ .
Lemma 8 (see [33]). Let 𝑙 be a lattice and 𝑋 a finite subset of
𝑙. Then the ∧-semilattice of 𝑙 generated by 𝑋, written as 𝑋∧ , is
finite, and the ∨-semilattice of 𝑙 generated by 𝑋, denoted as 𝑋∨ ,
It suffices to show that 0 ≤ 𝜇𝑓 (𝜔) + 𝜈𝑓 (𝜔) ≤ 1, for any
𝜔 = 𝑢1 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑢𝑖 ∈ Σ ∪ {𝜀}, 𝑖 = 1, . . . , 𝑛.
Clearly, 𝜇𝑓 (𝜔) = ⋁{𝜇𝐼 (𝑞0 ) ∧ 𝜇⊢∗M ((𝑞0 , 𝜔, 𝑧0 ), (𝑝, 𝜀, 𝑟)) ∧
𝜇𝐹 (𝑝) | 𝑞0 , 𝑝 ∈ 𝑄, 𝑟 ∈ Γ∗ } = ⋁{𝜇𝐼 (𝑞0 ) ∧ 𝜇⊢ ((𝑞0 , 𝜔, 𝑧0 ),
(𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 ,𝑧1 𝑟1 )) ∧ 𝜇⊢ ((𝑞1 ,𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ), (𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 ,
𝑧2 𝑟2 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢ ((𝑞𝑛−1 , 𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑟𝑛 )) ∧ 𝜇𝐹 (𝑞𝑛 ) |
(𝑞0 , 𝑞1 , . . . , 𝑞𝑛 ) ∈ 𝑄𝑛+1 , 𝑧1 , . . . , 𝑧𝑛−1 ∈ Γ, 𝑟1 , . . ., 𝑟𝑛 ∈ Γ∗ },
and 𝜈𝑓 (𝜔) = ⋀{𝜈𝐼 (𝑞0 ) ∨ 𝜈⊢∗M ((𝑞0 , 𝜔, 𝑧0 ), (𝑝, 𝜀, 𝑟)) ∨ 𝜈𝐹 (𝑝) |
𝑞0 , 𝑝 ∈ 𝑄, 𝑟 ∈ Γ∗ } = ⋀{𝜈𝐼 (𝑞0 ) ∨ 𝜈⊢ ((𝑞0 , 𝜔, 𝑧0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅
𝑢𝑛 , 𝑧1 𝑟1 )) ∨ 𝜈⊢ ((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ), (𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 )) ∨ ⋅ ⋅ ⋅ ∨
𝜈⊢ ((𝑞𝑛−1 , 𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑟𝑛 )) ∨ 𝜈𝐹 (𝑞𝑛 ) | (𝑞0 , 𝑞1 , . . . , 𝑞𝑛 ) ∈
𝑄𝑛+1 , 𝑧1 , . . . , 𝑧𝑛−1 ∈ Γ, 𝑟1 , . . . , 𝑟𝑛 ∈ Γ∗ }.
On the one hand, 0 ≤ 𝜇𝑓 (𝜔) + 𝜈𝑓 (𝜔); on the other
hand, there exists a sequence (𝑞0 , 𝑞1 , . . . , 𝑞𝑛 ) ∈ 𝑄𝑛+1 , 𝑧1 ,
. . . , 𝑧𝑛−1 ∈ Γ, 𝑟1 , . . . , 𝑟𝑛 ∈ Γ∗ such that 𝜇𝑓 (𝜔) = 𝜇𝐼 (𝑞0 ) ∧
𝜇⊢ ((𝑞0 , 𝜔, 𝑧0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 )) ∧ 𝜇⊢ ((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ),
(𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢ ((𝑞𝑛−1 , 𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑟𝑛 )) ∧
𝜇𝐹 (𝑞𝑛 ). Hence 𝜈𝑓 (𝜔) ≤ 𝜈𝐼 (𝑞0 ) ∨ 𝜈⊢ ((𝑞0 , 𝜔, 𝑧0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 ,
𝑧1 𝑟1 )) ∨ 𝜈⊢ ((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ), (𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 )) ∨ ⋅ ⋅ ⋅ ∨
𝜈⊢ ((𝑞𝑛−1 , 𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑟𝑛 )) ∨ 𝜈𝐹 (𝑞𝑛 ).
Therefore, 𝜇𝑓 (𝜔) + 𝜈𝑓 (𝜔) ≤ (𝜇𝐼 (𝑞0 ) + 𝜈𝐼 (𝑞0 )) ∨ (𝜇⊢ ((𝑞0 ,
𝜔, 𝑧0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 )) + 𝜈⊢ ((𝑞0 , 𝜔, 𝑧0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 ,
𝑧1 𝑟1 ))) ∨ (𝜇⊢ ((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ), (𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 )) + 𝜈⊢ ((𝑞1 ,
𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ), (𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 ))) ∨ ⋅ ⋅ ⋅ ∨ (𝜇⊢ ((𝑞𝑛−1 , 𝑢𝑛 ,
𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑟𝑛 )) + 𝜈⊢ ((𝑞𝑛−1 , 𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑟𝑛 ))) ∨
(𝜇𝐹 (𝑞𝑛 ) + 𝜈𝐹 (𝑞𝑛 )) ≤ 1 ∨ 1 ∨ ⋅ ⋅ ⋅ ∨ 1 = 1.
Claim 2 (Im(𝑓) is finite).
In fact, let 𝑋 = Im(𝜇𝐼 ) ∪ Im(𝜇𝛿 ) ∪ Im(𝜇𝐹 ) and 𝑌 =
Im(𝜈𝐼 ) ∪ Im(𝜈𝛿 )∪Im(𝜈𝐹 ). Then 𝑋∧ = {𝑥1 ∧⋅ ⋅ ⋅∧𝑥𝑘 | 𝑘 ≥ 1, 𝑥1 ,
. . . , 𝑥𝑘 ∈ 𝑋} ∪ {1} and 𝑋∨ = {𝑥1 ∨ ⋅ ⋅ ⋅ ∨ 𝑥𝑘 | 𝑘 ≥ 1, 𝑥1 , . . . , 𝑥𝑘 ∈
𝑋} ∪ {0} are finite sets by Lemma 8. Since 𝛿 = (𝜇𝛿 , 𝜈𝛿 ) is a
finite IFS, for any 𝜔 = 𝑢1 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑢𝑖 ∈ Σ ∪ {𝜀}, 𝑖 = 1, . . . , 𝑛, there
exists a natural number 𝑑 ∈ 𝑁 such that 𝜇𝑓 (𝜔) = ⋁{𝜇𝐼 (𝑞0 ) ∧
𝜇⊢ ((𝑞0 , 𝜔, 𝑧0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 )) ∧ 𝜇⊢ ((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ),
(𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 ))∧⋅ ⋅ ⋅∧ 𝜇⊢ ((𝑞𝑛−1 , 𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑟𝑛 ))∧
𝜇𝐹 (𝑞𝑛 ) | (𝑞0 , 𝑞1 , . . . , 𝑞𝑛 ) ∈ 𝑄𝑛+1 , 𝑧1 , . . . , 𝑧𝑛−1 ∈ Γ, 𝑟1 , . . . , 𝑟𝑛 ∈
Γ∗ } = (𝑎10 ∧⋅ ⋅ ⋅∧𝑎1,𝑛−1 ∧𝑎1𝑛 )∨⋅ ⋅ ⋅∨(𝑎𝑑0 ∧⋅ ⋅ ⋅∧𝑎𝑑,𝑛−1 ∧𝑎𝑑𝑛 )), where
𝑎𝑖𝑗 ∈ 𝑋, 𝑖 = 1, . . . , 𝑑; 𝑗 = 0, . . . , 𝑛. By Lemma 8, (𝑋∧ )∨ is also
finite. Since 𝜇𝑓 (𝜔) ∈ (𝑋∧ )∨ for any 𝜔 ∈ Σ∗ , Im(𝜇𝑓 ) ⊆ (𝑋∧ )∨ .
Hence Im(𝜇𝑓 ) is a finite subset of [0, 1].
Similarly, it follows that Im(𝜈𝑓 ) is also a finite subset of
[0, 1].
Therefore, Im(𝑓) = Im(𝜇𝑓 ) ∪ Im(𝜈𝑓 ) is finite.
If 𝑓 can be accepted by some IFPDA0 N = (𝑄, Σ, Γ, 𝛿,
𝐼, 𝑍0 , 0), then, by Definition 7, for any 𝜔 = 𝑢1 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑢𝑖 ∈
Σ ∪ {𝜀}, 𝑖 = 1, . . . , 𝑛, we have 𝜇𝑓 (𝜔) = ⋁{𝜇𝐼 (𝑞0 )∧
Journal of Applied Mathematics
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𝜇⊢∗N ((𝑞0 , 𝜔, 𝑧0 ), (𝑝, 𝜀, 𝜀)) | 𝑞0 , 𝑝 ∈ 𝑄} = ⋁{𝜇𝐼 (𝑞0 ) ∧
𝜇⊢ ((𝑞0 , 𝜔, 𝑧0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 )) ∧ 𝜇⊢ ((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ),
(𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢ ((𝑞𝑛−1 , 𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝜀)) |
(𝑞0 , 𝑞1 , . . . , 𝑞𝑛 ) ∈ 𝑄𝑛+1 , 𝑧1 , . . . , 𝑧𝑛−1 ∈ Γ, 𝑟1 , . . . , 𝑟𝑛−1 ∈
Γ∗ }, and 𝜈𝑓 (𝜔) = ⋀ {𝜈𝐼 (𝑞0 ) ∨ 𝜈⊢∗N ((𝑞0 , 𝜔, 𝑧0 ), (𝑝, 𝜀, 𝜀)) | 𝑞0 ,
𝑝 ∈ 𝑄} = ⋀ {𝜈𝐼 (𝑞0 ) ∨ 𝜈⊢ ((𝑞0 , 𝜔, 𝑧0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 )) ∨
𝜈⊢ ((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ), (𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 )) ∨ ⋅ ⋅ ⋅ ∨ 𝜈⊢ ((𝑞𝑛−1 ,
𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝜀)) | (𝑞0 , 𝑞1 , . . . , 𝑞𝑛 ) ∈ 𝑄𝑛+1 , 𝑧1 , . . . , 𝑧𝑛−1 ∈
Γ, 𝑟1 , . . ., 𝑟𝑛−1 ∈ Γ∗ }.
In a similar manner, it is concluded that the following
must be true.
Proposition 10. If 𝑓 can be accepted by some IFPDA0 N =
(𝑄, Σ, Γ, 𝛿, 𝐼, 𝑍0 , 0), then 𝑓 is an IFS in Σ∗ , and the image set
of 𝑓 is finite.
Specially, the IFPDA M = (𝑄, Σ, Γ, 𝛿, 𝐼, 𝑍0 , 𝐹) will be
abbreviated as M󸀠 = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 𝐹), whenever Im(𝐼) ⊆
{0, 1} and supp(𝐼) = {𝑞0 }. Moreover, if Im(𝐼) ∪ Im(𝐹) ∪
Im(𝛿) ⊆ {0, 1} and supp(𝐼) has only one element, then the
IFPDA is a classical PDA.
For two IFPDAs M1 and M2 , we say that they are equivalent if they accept the same intuitionistic fuzzy language.
Proposition 11. Let 𝐴 be an IFS in a nonempty set Σ∗ . Then
the following statements are equivalent:
(i) 𝐴 can be accepted by an IFPDA M = (𝑄, Σ, Γ, 𝛿, 𝐼,
𝑍0 , 𝐹);
(ii) 𝐴 can be accepted by a certain IFPDA M󸀠 = (𝑄󸀠 , Σ,
Γ󸀠 , 𝛿󸀠 , 𝑞0 , 𝑋0 , 𝐹󸀠 ), where 𝑞0 ∈ 𝑄󸀠 .
󸀠
󸀠
Proof. (i) implies (ii). Construct an IFPDA M = (𝑄 , Σ,
Γ󸀠 , 𝛿󸀠 , 𝐼󸀠 , 𝑋0 , 𝐹󸀠 ) as follows: 𝑄󸀠 = 𝑄 ∪ {𝑞0 }, Γ󸀠 = Γ ∪ {𝑋0 },
where 𝑞0 ∉ 𝑄, 𝑋0 ∉ Γ. Define an IFS 𝐼󸀠 in 𝑄󸀠 by
1,
𝜇𝐼󸀠 (𝑞) = {
0,
if 𝑞 = 𝑞0
if 𝑞 ≠ 𝑞0 ,
0, if 𝑞 = 𝑞0
𝜈𝐼󸀠 (𝑞) = {
1, if 𝑞 ≠ 𝑞0 .
󸀠
(12)
1,
if 𝑞 = 𝑞0
𝜈𝐹󸀠 (𝑞) = {
𝜈𝐹 (𝑞) , if 𝑞 ≠ 𝑞0 .
if 𝑞 = 𝑞0
if 𝑞 ≠ 𝑞0 ,
(14)
0, if 𝑞 = 𝑞0
𝜈𝐼 (𝑞) = {
1, if 𝑞 ≠ 𝑞0 .
It follows that the IFPDA M = (𝑄󸀠 , Σ, Γ󸀠 , 𝛿󸀠 , 𝐼, 𝑋0 , 𝐹󸀠 )
accepts 𝐴.
Similarly, it is easily concluded that the following must be
true.
(i) 𝐴 can be accepted by an IFPDA0 M
Γ, 𝛿, 𝐼, 𝑍0 , 0);
=
(𝑄, Σ,
(ii) There exists an IFPDA0 M󸀠 = (𝑄󸀠 , Σ, Γ󸀠 , 𝛿󸀠 , 𝑞0 , 𝑋0 , 0)
recognizing 𝐴, where 𝑞0 ∈ 𝑄󸀠 .
Define an IFS 𝐹 in 𝑄 by
if 𝑞 = 𝑞0
if 𝑞 ≠ 𝑞0 ,
1,
𝜇𝐼 (𝑞) = {
0,
Proposition 12. Let 𝐴 be an IFS in a nonempty set Σ∗ . Then
the following statements are equivalent:
󸀠
0,
𝜇𝐹󸀠 (𝑞) = {
𝜇𝐹 (𝑞) ,
(𝑝0 , 𝜔, 𝑍0 )) ∧ 𝜇⊢M󸀠 ((𝑝0 , 𝜔, 𝑍0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 )) ∧ 𝜇⊢M󸀠
((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ), (𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢M󸀠 ((𝑞𝑛−1 ,
𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑟𝑛 )) ∧ 𝜇𝐹󸀠 (𝑞𝑛 ) | 𝑞 ∈ 𝑄󸀠 , (𝑝0 , 𝑞1 , . . . , 𝑞𝑛 ) ∈
𝑄󸀠𝑛+1 , 𝑧1 , . . . , 𝑧𝑛−1 ∈ Γ󸀠 , 𝑟1 , . . . , 𝑟𝑛 ∈ Γ󸀠∗ } = ⋁{1 ∧ 𝜇𝐼 (𝑝0 )∧
𝜇⊢M ((𝑝0 , 𝜔, 𝑍0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 )) ∧ 𝜇⊢M ((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 ,
𝑧1 𝑟1 ), (𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢M ((𝑞𝑛−1 , 𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ),
(𝑞𝑛 , 𝜀, 𝑟𝑛 )) ∧ 𝜇𝐹 (𝑞𝑛 ) | (𝑝0 , 𝑞1 , . . . , 𝑞𝑛 ) ∈ 𝑄𝑛+1 , 𝑧1 , . . ., 𝑧𝑛−1 ∈
Γ, 𝑟1 , . . . , 𝑟𝑛 ∈ Γ∗ } = 𝜇L(M) (𝜔), and 𝜈L(M󸀠 ) (𝜔) =
⋀{𝜈𝐼󸀠 (𝑞) ∨ 𝜈⊢M󸀠 ((𝑞, 𝜔, 𝑋0 ), (𝑝0 , 𝜔, 𝑍0 )) ∨ 𝜈⊢M󸀠 ((𝑝0 , 𝜔, 𝑍0 ),
(𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 )) ∨ 𝜈⊢M󸀠 ((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ), (𝑞2 , 𝑢3 ⋅ ⋅ ⋅
𝑢𝑛 , 𝑧2 𝑟2 )) ∨ ⋅ ⋅ ⋅ ∨ 𝜈⊢M󸀠 ((𝑞𝑛−1 , 𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑟𝑛 )) ∨ 𝜈𝐹󸀠 (𝑞𝑛 ) |
𝑞 ∈ 𝑄󸀠 , (𝑝0 , 𝑞1 , . . . , 𝑞𝑛 ) ∈ 𝑄󸀠𝑛+1 , 𝑧1 , . . . , 𝑧𝑛−1 ∈ Γ󸀠 , 𝑟1 , . . .,
𝑟𝑛 ∈ Γ󸀠∗ } = ⋀{1 ∨ 𝜈𝐼 (𝑝0 ) ∨ 𝜈⊢M ((𝑝0 , 𝜔, 𝑍0 ), (𝑞1 , 𝑢2 ⋅ ⋅ ⋅
𝑢𝑛 , 𝑧1 𝑟1 )) ∨ 𝜈⊢M ((𝑞1 , 𝑢2 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧1 𝑟1 ), (𝑞2 , 𝑢3 ⋅ ⋅ ⋅ 𝑢𝑛 , 𝑧2 𝑟2 )) ∨
⋅ ⋅ ⋅ ∨ 𝜈⊢M ((𝑞𝑛−1 , 𝑢𝑛 , 𝑧𝑛−1 𝑟𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑟𝑛 )) ∨ 𝜈𝐹 (𝑞𝑛 ) | (𝑝0 , 𝑞1 , . . .,
𝑞𝑛 ) ∈ 𝑄𝑛+1 , 𝑧1 , . . . , 𝑧𝑛−1 ∈ Γ, 𝑟1 , . . ., 𝑟𝑛 ∈ Γ∗ } = 𝜈L(M) (𝜔).
Therefore, L(M󸀠 ) = L(M).
From the construction, it is clearly that M󸀠 can be
denoted as M󸀠 = (𝑄󸀠 , Σ, Γ󸀠 , 𝛿󸀠 , 𝑞0 , 𝑋0 , 𝐹󸀠 ).
(ii) implies (i). Suppose the IFS 𝐴 is accepted by the
IFPDA M󸀠 = (𝑄󸀠 , Σ, Γ󸀠 , 𝛿󸀠 , 𝑞0 , 𝑋0 , 𝐹󸀠 ). Then we construct an
IFS 𝐼 in 𝑄󸀠 by
(13)
Define an IFS 𝛿󸀠 in 𝑄󸀠 × (Σ ∪ {𝜀}) × Γ󸀠 × 𝑄󸀠 × Γ󸀠∗ by
mappings 𝜇𝛿󸀠 , 𝜈𝛿󸀠 : 𝑄󸀠 × (Σ ∪ {𝜀}) × Γ󸀠 × 𝑄󸀠 × Γ󸀠∗ → [0, 1],
𝜇𝛿󸀠 (𝑞0 , 𝜀, 𝑋0 , 𝑝, 𝑍0 ) = 𝜇𝐼 (𝑝), 𝜈𝛿󸀠 (𝑞0 , 𝜀, 𝑋0 , 𝑝, 𝑍0 ) = 𝜈𝐼 (𝑝),
𝜇𝛿󸀠 (𝑞, 𝜏, 𝑧, 𝑝, 𝛾) = 𝜇𝛿 (𝑞, 𝜏, 𝑧, 𝑝, 𝛾), 𝜈𝛿󸀠 (𝑞, 𝜏, 𝑧, 𝑝, 𝛾) = 𝜈𝛿 (𝑞, 𝜏,
𝑧, 𝑝, 𝛾), where 𝑞, 𝑝 ∈ 𝑄, 𝜏 ∈ Σ ∪ {𝜀}, 𝑧 ∈ Γ, 𝛾 ∈ Γ∗ ; otherwise,
𝜇𝛿󸀠 (𝑞0 , 𝜏, 𝑧, 𝑝, 𝛾) = 0 and 𝜈𝛿󸀠 (𝑞0 , 𝜏, 𝑧, 𝑝, 𝛾) = 1.
Then for any 𝜔 = 𝑢1 ⋅ ⋅ ⋅ 𝑢𝑛 ∈ Σ∗ , 𝑢𝑖 ∈ Σ ∪ {𝜀}, 𝑖 =
1, . . . , 𝑛, we have 𝜇L(M󸀠 ) (𝜔) = ⋁{𝜇𝐼󸀠 (𝑞) ∧ 𝜇⊢M󸀠 ((𝑞, 𝜔, 𝑋0 ),
There is especially a simple type of intuitionistic fuzzy
pushdown automata, which is called intuitionistic fuzzy
simple pushdown automata. The definition is given as follows.
Definition 13. An IFPDA M = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 𝐹) is called
an intuitionistic fuzzy simple pushdown automaton
(IFSPDA) if the image set of 𝛿 is contained in the set
{0, 1}.
Next any IFPDA is proven to be an equivalence of a certain IFSPDA by utilizing the generalized subset construction
method. Noting that an IFS requires that the sum of the
degrees of membership and nonmembership of an element
to a set is no more than the natural number 1. So the proof
technique is to some extent different from the technique of
6
Journal of Applied Mathematics
𝜇⊢M ((𝑞𝑖 , 𝜏𝑖+1 ⋅ ⋅ ⋅ 𝜏𝑘+1 , 𝑍𝑖 𝛾𝑖 ), (𝑞𝑖+1 , 𝜏𝑖+2 ⋅ ⋅ ⋅ 𝜏𝑘+1 ,
𝑍𝑖+1 𝛾𝑖+1 )) = 𝑐𝑖+1 > 0,
extended subset construction introduced by Li in [33], and it
is not an easy task to conduct reasoning in the realm of the
modified techniques.
𝜈⊢M ((𝑞𝑖 , 𝜏𝑖+1 ⋅ ⋅ ⋅ 𝜏𝑘+1 , 𝑍𝑖 𝛾𝑖 ), (𝑞𝑖+1 , 𝜏𝑖+2 ⋅ ⋅ ⋅ 𝜏𝑘+1 ,
𝑍𝑖+1 𝛾𝑖+1 )) = 𝑑𝑖+1 < 1,
Proposition 14. Let M be an IFPDA. Then there exists an
IFSPDA M󸀠 such that L(M󸀠 ) = L(M).
Proof. Let M = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 𝐹) be an IFPDA. Then we
construct an IFSPDA M󸀠 = (𝑄󸀠 , Σ, Γ, 𝛿󸀠 , 𝑞0󸀠 , 𝑍0 , 𝐹󸀠 ) as follows:
(i) 𝑄󸀠 = 𝑄 × (𝐿 1 − {0}) × (𝐿 2 − {1}), where 𝐿 1 = 𝑋∧ , 𝐿 2 =
𝑌∨ , 𝑋 = Im(𝜇𝛿 ) ∪ Im(𝜇𝐹 ) and 𝑌 = Im(𝜈𝛿 ) ∪ Im(𝜈𝐹 );
(ii) 𝑞0󸀠 = (𝑞0 , 1, 0) ∈ 𝑄󸀠 ;
(iii) 𝛿󸀠 = (𝜇𝛿󸀠 , 𝜈𝛿󸀠 ) is an IFS in 𝑄󸀠 × (Σ ∪ {𝜀}) × Γ ×
𝑄󸀠 × Γ∗ , where the mappings 𝜇𝛿󸀠 , 𝜈𝛿󸀠 : 𝑄󸀠 × (Σ ∪
{𝜀}) × Γ × 𝑄󸀠 × Γ∗ → {0, 1} are given as follows.
For any (𝑞, 𝑎, 𝑏), (𝑞󸀠 , 𝑐, 𝑑) ∈ 𝑄󸀠 , 𝜏 ∈ Σ ∪ {𝜀}, 𝑋 ∈
Γ and 𝛾 ∈ Γ∗ , 𝜇𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜏, 𝑋, (𝑞󸀠 , 𝑐, 𝑑), 𝛾) = 1,
and 𝜈𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜏, 𝑋, (𝑞󸀠 , 𝑐, 𝑑), 𝛾) = 0 whenever there
exist 𝑎󸀠 and 𝑏󸀠 such that 𝜇𝛿 (𝑞, 𝜏, 𝑋, 𝑞󸀠 , 𝛾) = 𝑎󸀠 > 0,
𝜈𝛿 (𝑞, 𝜏, 𝑋, 𝑞󸀠 , 𝛾) = 𝑏󸀠 < 1, 𝑐 = 𝑎 ∧ 𝑎󸀠 and 𝑑 =
𝑏 ∨ 𝑏󸀠 . Otherwise, 𝜇𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜏, 𝑋, (𝑞󸀠 , 𝑐, 𝑑), 𝛾) = 0
and 𝜈𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜏, 𝑋, (𝑞󸀠 , 𝑐, 𝑑), 𝛾) = 1;
(iv) 𝐹󸀠 = (𝜇𝐹󸀠 , 𝜈𝐹󸀠 ) is an IFS in 𝑄󸀠 . For any (𝑞, 𝑎, 𝑏) ∈ 𝑄󸀠 ,
𝑎 ∧ 𝜇𝐹 (𝑞) , if 0 ≤ 𝑎 + 𝑏 ≤ 1
𝜇𝐹󸀠 ((𝑞, 𝑎, 𝑏)) = {
0,
if 𝑎 + 𝑏 > 1,
𝑏 ∨ 𝜈𝐹 (𝑞) , if 0 ≤ 𝑎 + 𝑏 ≤ 1
𝜈𝐹󸀠 ((𝑞, 𝑎, 𝑏)) = {
1,
if 𝑎 + 𝑏 > 1.
(15)
Now, it is claimed that, for any 𝜔 = 𝜏1 ⋅ ⋅ ⋅ 𝜏𝑛 ∈ Σ∗ ,
𝜏𝑖 ∈ Σ ∪ {𝜀}, 𝑖 ∈ {1, . . . , 𝑛} and for any (𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ) ∈
𝑄󸀠 , 𝑍𝑛 , 𝛾𝑛 ∈ Γ∗ , 𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑍𝑛 𝛾𝑛 )) = 1
M
and 𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑍𝑛 𝛾𝑛 )) = 0 whenever the
M
following condition is satisfied.
(P1) There exist 𝑞1 , . . . , 𝑞𝑛−1 ∈ 𝑄, 𝑍1 , . . . , 𝑍𝑛−1 ∈ Γ and 𝛾1 ,
. . . , 𝛾𝑛−1 ∈ Γ∗ such that 𝑎𝑛 = 𝜇⊢M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞1 , 𝜏2 ⋅ ⋅ ⋅
𝜏𝑛 , 𝑍1 𝛾1 )) ∧ 𝜇⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∧
⋅ ⋅ ⋅ ∧ 𝜇⊢M ((𝑞𝑛−1 , 𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑍𝑛 𝛾𝑛 )) and 𝑏𝑛 =
𝜈⊢M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 )) ∨ 𝜈⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∨ ⋅ ⋅ ⋅ ∨ 𝜈⊢M ((𝑞𝑛−1 , 𝜏𝑛 ,
𝑍𝑛−1 𝛾𝑛−1 ), (𝑞𝑛 , 𝜀, 𝑍𝑛 𝛾𝑛 )). Otherwise, 𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑛 ,
M
𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑍𝑛 𝛾𝑛 )) = 0 and 𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀,
M
𝑍𝑛 𝛾𝑛 )) = 1.
It is proved by induction. In fact, if |𝜔| = 0, then 𝜔 =
𝜀, 𝜇⊢∗ 󸀠 ((𝑞0 , 𝜀, 𝑍0 ), (𝑞0 , 𝜀, 𝑍0 )) = 1 and 𝜈⊢∗M ((𝑞0 , 𝜀, 𝑍0 ), (𝑞0 , 𝜀,
M
𝑍0 )) = 0. Hence 𝜇⊢∗M (((𝑞0 , 1, 0), 𝜀, 𝑍0 ), ((𝑞0 , 1, 0), 𝜀, 𝑍0 )) = 1
and 𝜈⊢∗ 󸀠 (((𝑞0 , 1, 0), 𝜀, 𝑍0 ), ((𝑞0 , 1, 0), 𝜀, 𝑍0 )) = 0.
M
Suppose the result still holds whenever |𝜔| ≤ 𝑛, 𝑛 ∈ 𝑁.
If |𝜔| = 𝑛 + 1, 𝜔 = 𝜏1 ⋅ ⋅ ⋅ 𝜏𝑘 𝜏𝑘+1 = 𝑥𝜏𝑘+1 and 𝜏𝑘+1 ∈ Σ, then
|𝑥| = 𝑛 and 𝑥 = 𝜏1 ⋅ ⋅ ⋅ 𝜏𝑘 .
Next, for any (𝑞𝑘+1 , 𝑎𝑘+1 , 𝑏𝑘+1 ) ∈ 𝑄󸀠 , 𝑍𝑘+1 , 𝛾𝑘+1 ∈ Γ∗ ,
whenever (P1) is satisfied; that is, there exists a sequence of
states 𝑞1 , . . . , 𝑞𝑘 ∈ 𝑄, 𝑍1 , . . . , 𝑍𝑘 ∈ Γ, 𝛾1 , . . . , 𝛾𝑘 ∈ Γ∗ such that
𝜇𝛿 (𝑞𝑘 , 𝜏𝑘+1 , 𝑍𝑘 , 𝑞𝑘+1 , 𝑍𝑘+1 ) = 𝑐𝑘+1 > 0,
𝜈𝛿 (𝑞𝑘 , 𝜏𝑘+1 , 𝑍𝑘 , 𝑞𝑘+1 , 𝑍𝑘+1 ) = 𝑑𝑘+1 < 1,
where 𝛾0 = 𝜀, 𝑖 = 0, 1, . . . , 𝑘 − 1.
Let 𝑎𝑙 = 𝑐1 ∧ ⋅ ⋅ ⋅ ∧ 𝑐𝑙 , 𝑏𝑙 = 𝑑1 ∨ ⋅ ⋅ ⋅ ∨ 𝑑𝑙 , 𝑙 ∈ {1, . . . , 𝑘 + 1}.
Then
𝜇⊢M󸀠 (((𝑞𝑖 , 𝑎𝑖 , 𝑏𝑖 ), 𝜏𝑖+1 ⋅ ⋅ ⋅ 𝜏𝑘+1 , 𝑍𝑖 𝛾𝑖 ), ((𝑞𝑖+1 , 𝑎𝑖+1 , 𝑏𝑖+1 ),
𝜏𝑖+2 ⋅ ⋅ ⋅ 𝜏𝑘+1 , 𝑍𝑖+1 𝛾𝑖+1 )) = 1,
𝜈⊢M󸀠 (((𝑞𝑖 , 𝑎𝑖 , 𝑏𝑖 ), 𝜏𝑖+1 ⋅ ⋅ ⋅ 𝜏𝑘+1 , 𝑍𝑖 𝛾𝑖 ),
𝜏𝑖+2 ⋅ ⋅ ⋅ 𝜏𝑘+1 , 𝑍𝑖+1 𝛾𝑖+1 )) = 0,
((𝑞𝑖+1 , 𝑎𝑖+1 , 𝑏𝑖+1 ),
𝜇⊢M󸀠 (((𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ), 𝜏𝑘+1 , 𝑍𝑘 𝛾𝑘 ), ((𝑞𝑘+1 , 𝑎𝑘+1 , 𝑏𝑘+1 ),
𝜀, 𝑍𝑘+1 𝛾𝑘+1 )) = 1, and
𝜈⊢M󸀠 (((𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ), 𝜏𝑘+1 , 𝑍𝑘 𝛾𝑘 ), ((𝑞𝑘+1 , 𝑎𝑘+1 , 𝑏𝑘+1 ),
𝜀, 𝑍𝑘+1 𝛾𝑘+1 )) = 0,
where 𝑎0 = 1, 𝑏0 = 0, 𝛾0 = 𝜀, 𝑖 = 0, 1, . . . , 𝑘 − 1.
By assumption, 𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝑥, 𝑍0 ), ((𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ), 𝜀, 𝑍𝑘 𝛾𝑘 )) =
M
1, 𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝑥, 𝑍0 ), ((𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ), 𝜀, 𝑍𝑘 𝛾𝑘 )) = 0, and so 𝜇⊢∗ 󸀠 ((𝑞0󸀠 ,
M
M
𝑥𝜏𝑘+1 , 𝑍0 ), ((𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ), 𝜏𝑘+1 , 𝑍𝑘 𝛾𝑘 )) = 1, 𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝑥𝜏𝑘+1 , 𝑍0 ),
M
((𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ), 𝜏𝑘+1 , 𝑍𝑘 𝛾𝑘 )) = 0.
((𝑞𝑘+1 , 𝑎𝑘+1 , 𝑏𝑘+1 ),
𝜀,
Therefore,
𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ),
M
𝑍𝑘+1 𝛾𝑘+1 )) = ⋁{𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ), 𝜏𝑘+1 , 𝑍𝑘 𝛾𝑘 )) ∧
M
𝜇⊢M󸀠 (((𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ), 𝜏𝑘+1 , 𝑍𝑘 𝛾𝑘 ), ((𝑞𝑘+1 , 𝑎𝑘+1 , 𝑏𝑘+1 ),
𝜀,
𝑍𝑘+1 𝛾𝑘+1 )) | (𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ) ∈ 𝑄󸀠 , 𝑍𝑘 ∈ Γ, 𝛾𝑘 ∈ Γ∗ } = 1 ∧ 1 = 1.
𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑘+1 , 𝑎𝑘+1 , 𝑏𝑘+1 ), 𝜀, 𝑍𝑘+1 𝛾𝑘+1 )) =
M
⋀ {𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ), 𝜏𝑘+1 , 𝑍𝑘 𝛾𝑘 )) ∨ 𝜈⊢M󸀠 (((𝑞𝑘 , 𝑎𝑘 ,
M
𝑏𝑘 ), 𝜏𝑘+1 , 𝑍𝑘 𝛾𝑘 ), ((𝑞𝑘+1 , 𝑎𝑘+1 , 𝑏𝑘+1 ), 𝜀, 𝑍𝑘+1 𝛾𝑘+1 )) | (𝑞𝑘 , 𝑎𝑘 , 𝑏𝑘 ) ∈
𝑄󸀠 , 𝑍𝑘 ∈ Γ, 𝛾𝑘 ∈ Γ∗ } = 0 ∨ 0 = 0.
For any (𝑞𝑘+1 , 𝑎𝑘+1 , 𝑏𝑘+1 ) ∈ 𝑄󸀠 , 𝑍𝑘+1 , 𝛾𝑘+1 ∈ Γ∗ , if (P1) is
not satisfied, then it follows that
𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑘+1 , 𝑎𝑘+1 , 𝑏𝑘+1 ), 𝜀, 𝑍𝑘+1 𝛾𝑘+1 )) = 0,
M
𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑘+1 , 𝑎𝑘+1 , 𝑏𝑘+1 ), 𝜀, 𝑍𝑘+1 𝛾𝑘+1 )) = 1.
M
Hence, for any 𝜔 = 𝜏1 ⋅ ⋅ ⋅ 𝜏𝑛 ∈ Σ∗ , 𝜏𝑖 ∈ Σ ∪ {𝜀}, 𝑖 ∈
{1, . . . , 𝑛}, we have 𝜇L(M󸀠 ) (𝜔) = ⋁{𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑛 , 𝑎𝑛 ,
M
𝑏𝑛 ), 𝜀, 𝛾)) ∧ 𝜇𝐹󸀠 ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 )) | (𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ) ∈ 𝑄󸀠 , 𝛾 ∈ Γ∗ }
= ⋁{𝑎𝑛 ∧ 𝜇𝐹 (𝑞𝑛 ) | 𝑞1 , . . . , 𝑞𝑛−1 ∈ 𝑄, 𝑍1 , . . . , 𝑍𝑛−1 ∈ Γ,
𝛾1 , . . . , 𝛾𝑛−1 ∈ Γ∗ , 𝑎𝑛 = 𝜇⊢M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝑍1 𝛾1 )) ∧ 𝜇⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∧ ⋅ ⋅ ⋅ ∧
𝜇⊢M ((𝑞𝑛−1 , 𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 ),(𝑞𝑛 , 𝜀, 𝛾)), 𝑏𝑛 = 𝜈⊢M ((𝑞0 , 𝜔, 𝑍0 ),
(𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 )) ∨ 𝜈⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝑍2 𝛾2 )) ∨ ⋅ ⋅ ⋅ ∨ 𝜈⊢M ((𝑞𝑛−1 , 𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 ), (𝑞𝑛 , 𝜀, 𝛾))} =
⋁{𝜇⊢M ((𝑞0 , 𝜔, 𝑍0 ),(𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 )) ∧ 𝜇⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝑍1 𝛾1 )(𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢M ((𝑞𝑛−1 , 𝜏𝑛 ,
𝑍𝑛−1 𝛾𝑛−1 ), (𝑞𝑛 , 𝜀, 𝛾)) ∧ 𝜇𝐹 (𝑞𝑛 ) | 𝑞1 , . . . , 𝑞𝑛−1 ∈ 𝑄, 𝑍1 , . . .,
𝑍𝑛−1 ∈ Γ, 𝛾1 , . . . , 𝛾𝑛−1 ∈ Γ∗ } = 𝜇L(M) (𝜔), 𝜈L(M󸀠 ) (𝜔) =
⋀{𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑍0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝛾)) ∨ 𝜈𝐹󸀠 ((𝑞𝑛 , 𝑎𝑛 ,
M
Journal of Applied Mathematics
7
𝑏𝑛 )) | (𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ) ∈ 𝑄󸀠 , 𝛾 ∈ Γ∗ } = ⋀{𝑏𝑛 ∨ 𝜈𝐹 (𝑞𝑛 ) | 𝑞1 ,
. . . , 𝑞𝑛−1 ∈ 𝑄, 𝑍1 , . . . , 𝑍𝑛−1 ∈ Γ, 𝛾1 , . . . , 𝛾𝑛−1 ∈ Γ∗ ,
𝑎𝑛 = 𝜇⊢M ((𝑞0 , 𝜔, 𝑍0 ),(𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 )) ∧ 𝜇⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢M ((𝑞𝑛−1 , 𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 ),
(𝑞𝑛 , 𝜀, 𝛾)), 𝑏𝑛 = 𝜈⊢M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 )) ∨ 𝜈⊢M ((𝑞1 ,
𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∨ ⋅ ⋅ ⋅ ∨ 𝜈⊢M ((𝑞𝑛−1 , 𝜏𝑛 ,
𝑍𝑛−1 𝛾𝑛−1 ), (𝑞𝑛 , 𝜀, 𝛾))} = ⋀{𝜈⊢M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝑍1 𝛾1 )) ∨ 𝜈⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∨ ⋅ ⋅ ⋅
∨𝜈⊢M ((𝑞𝑛−1 , 𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 ), (𝑞𝑛 , 𝜀, 𝛾)) ∨ 𝜈𝐹 (𝑞𝑛 ) | 𝑞1 , . . . , 𝑞𝑛−1 ∈
𝑄, 𝑍1 , . . . , 𝑍𝑛−1 ∈ Γ, 𝛾1 , . . .,𝛾𝑛−1 ∈ Γ∗ } = 𝜈L(M) (𝜔).
Therefore, L(M󸀠 ) = L(M).
where 𝐹𝑎𝑏 = {𝑞 ∈ 𝑄 | 𝜇𝐹 (𝑞) ≥ 𝑎, 𝜈𝐹 (𝑞) ≤ 𝑏}; the mapping
∗
𝛿󸀠 : 𝑄 × (Σ ∪ {𝜀}) × Γ → 2𝑄×Γ is given by
Clearly, an IFPDA is a generalization of a classical pushdown automaton (PDA). Next, it will be shown that any
IFPDA can be characterized by a collection of pushdown
automata. To describe the behavior of a pushdown automaton
M = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 𝐹), we need to introduce the concept
of instantaneous description. An instantaneous description is
a three-tuple (𝑞, 𝜔, 𝛾) ∈ 𝑄 × Σ∗ × Γ∗ , which means that the
automaton is in the state 𝑞 and has unexpended input 𝜔 and
stack contents 𝛾. An instantaneous description represents the
configuration of a pushdown automaton at a given instant. To
introduce the transition in a pushdown automaton in terms of
instantaneous descriptions, we define ≻M as a binary relation
on 𝑄 × Σ∗ × Γ∗ . We say (𝑞, 𝑎𝜔, 𝑍𝛾) ≻M (𝑝, 𝜔, 𝜎𝛾) if 𝛿(𝑞, 𝑎, 𝑍)
contains (𝑝, 𝜎), where 𝑝, 𝑞 ∈ 𝑄, 𝑎 ∈ Σ ∪ {𝜀}, 𝜔 ∈ Σ∗ , 𝑍 ∈ Γ,
and 𝛾, 𝜎 ∈ Γ∗ . Furthermore, we define ≻∗M as the reflexive
and transitive closure of ≻M . Then the language accepted by
M with final states is defined as
for all 𝑎, 𝑏 ∈ [0, 1] with 𝑎 + 𝑏 ≤ 1, for all 𝜔 ∈ Σ∗ , 𝛾 ∈ Γ∗ .
𝜇𝑓𝑆 (𝜔) = ⋁{𝑎 ∈ [0, 1] | M𝑎𝑏 accepts 𝜔, M𝑎𝑏 ∈ 𝑆} = ⋁{𝑎 ∈
[0, 1] | 𝑞 ∈ 𝐹𝑎𝑏 , (𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑎𝑏 (𝑞, 𝜀, 𝛾), 𝛾 ∈ Γ∗ } = ⋁{𝑎 ∈
[0, 1] | 𝜇𝐹 (𝑞) ≥ 𝑎, 𝜈𝐹 (𝑞) ≤ 𝑏, (𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑎𝑏 (𝑞, 𝜀, 𝛾), 𝛾 ∈
Γ∗ } = ⋁{𝑎 ∈ [0, 1] | 𝜇𝐹 (𝑞) ≥ 𝑎, (𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑎𝑏 (𝑞, 𝜀, 𝛾), 𝑞 ∈
𝑄, 𝛾 ∈ Γ∗ } = ⋁{𝜇𝐹 (𝑞) | 𝜇⊢∗M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1, 𝑞 ∈
𝑄, 𝛾 ∈ Γ∗ } = 𝜇𝑓 (𝜔) = 𝜇L(M) (𝜔), 𝜈𝑓𝑆 (𝜔) = ⋀{𝑏 ∈ [0, 1] |
M𝑎𝑏 accepts 𝜔, M𝑎𝑏 ∈ 𝑆} = ⋀{𝑏 ∈ [0, 1] | 𝑞 ∈ 𝐹𝑎𝑏 ,
(𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑎𝑏 (𝑞, 𝜀, 𝛾), 𝛾 ∈ Γ∗ } = ⋀{𝑏 ∈ [0, 1] | 𝜇𝐹 (𝑞) ≥
𝑎, 𝜈𝐹 (𝑞) ≤ 𝑏, (𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑎𝑏 (𝑞, 𝜀, 𝛾), 𝛾 ∈ Γ∗ } = ⋀{𝑏 ∈
[0, 1] | 𝜈𝐹 (𝑞) ≤ 𝑏, (𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑎𝑏 (𝑞, 𝜀, 𝛾), 𝑞 ∈ 𝑄, 𝛾 ∈ Γ∗ } =
⋀{𝜈𝐹 (𝑞) | 𝜇⊢∗M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1, 𝑞 ∈ 𝑄, 𝛾 ∈ Γ∗ } =
𝜈𝑓 (𝜔) = 𝜈L(M) (𝜔).
Therefore, 𝑓𝑆 = L(M) = 𝑓.
Conversely, suppose 𝑓 can be recognized by a cover
L (M) = {𝜔 ∈ Σ∗ | (𝑞0 , 𝜔, 𝑍0 ) ≻∗M (𝑝, 𝜀, 𝛾) , 𝑝 ∈ 𝐹, 𝛾 ∈ Γ∗ } .
(16)
𝑆 = {M𝑎𝑏 | M𝑎𝑏 = (𝑄, Σ, Γ, 𝛿󸀠 , 𝑞0 , 𝑍0 , 𝐹𝑎𝑏 ) ,
Definition 15. A collection of classical pushdown automata
with final states
𝑆 = {M𝑎𝑏 | M𝑎𝑏 = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 𝐹𝑎𝑏 ) ,
0 ≤ 𝑎 + 𝑏 ≤ 1, 𝑎, 𝑏 ∈ [0, 1] }
(17)
is called a cover if the following conditions hold:
For a cover 𝑆, its recognized intuitionistic fuzzy language
𝑓𝑆 = (𝜇𝑓𝑆 , 𝜈𝑓𝑆 ) in Σ∗ is given by
𝜇𝑓𝑆 (𝜔) = ⋁{𝑎 ∈ [0, 1] | M𝑎𝑏 accepts 𝜔, M𝑎𝑏 ∈ 𝑆},
𝜈𝑓𝑆 (𝜔) = ⋀{𝑏 ∈ [0, 1] | M𝑎𝑏 accepts 𝜔, M𝑎𝑏 ∈ 𝑆}, for
all 𝜔 ∈ Σ∗ .
Theorem 16. Let 𝑓 be an IFS in Σ∗ . Then 𝑓 can be accepted
by an IFPDA if and only if 𝑓 can be recognized by a cover 𝑆.
Proof. If 𝑓 can be accepted by an IFPDA, then there exists an
IFSPDA M = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 𝐹) such that M accepts 𝑓 by
Proposition 14. Next we construct a cover
0 ≤ 𝑎 + 𝑏 ≤ 1, 𝑎, 𝑏 ∈ [0, 1] } ,
Clearly, the cover 𝑆 is well defined.
Next, we will show that 𝑓 can be recognized by the cover
𝑆. In fact, we have
(𝑞0 , 𝜔, 𝑍0 )≻∗M𝑎𝑏 (𝑞, 𝜀, 𝛾) if and only if 𝜇⊢∗M ((𝑞0 ,
𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1,
0 ≤ 𝑎 + 𝑏 ≤ 1, 𝑎, 𝑏 ∈ [0, 1] } .
(18)
(19)
Then we construct an IFSPDA M = (𝑄, Σ, Γ, 𝜂, 𝑞0 , 𝑍0 , 𝐹),
where 𝜂 = (𝜇𝜂 , 𝜈𝜂 ) is an IFS in 𝑄 × (Σ ∪ {𝜀}) × Γ × 𝑄 × Γ∗ ,
and the mappings 𝜇𝜂 , 𝜈𝜂 : 𝑄 × (Σ ∪ {𝜀}) × Γ × 𝑄 × Γ∗ → {0, 1}
are defined as
1,
𝜇𝜂 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = {
0,
(i) 𝑎1 ≤ 𝑎2 and 𝑏2 ≤ 𝑏1 imply 𝐹𝑎2 𝑏2 ⊆ 𝐹𝑎1 𝑏1 ;
(ii) 𝐹01 = 𝑄.
𝑆 = {M𝑎𝑏 | M𝑎𝑏 = (𝑄, Σ, Γ, 𝛿󸀠 , 𝑞0 , 𝑍0 , 𝐹𝑎𝑏 ) ,
𝛿󸀠 (𝑞, 𝜏, 𝑍) = {(𝑝, 𝛾) | 𝜇𝛿 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = 1, 𝑝 ∈ 𝑄, 𝛾 ∈
Γ∗ }, for all (𝑞, 𝜏, 𝑍) ∈ 𝑄 × (Σ ∪ {𝜀}) × Γ.
if (𝑝, 𝛾) ∈ 𝛿󸀠 (𝑞, 𝜏, 𝑍)
otherwise,
0, if (𝑝, 𝛾) ∈ 𝛿󸀠 (𝑞, 𝜏, 𝑍)
𝜈𝜂 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = {
1, otherwise,
(20)
for any (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) ∈ 𝑄 × (Σ ∪ {𝜀}) × Γ × 𝑄 × Γ∗ .
𝐹 = (𝜇𝐹 , 𝜈𝐹 ) is an IFS in 𝑄, where 𝜇𝐹 (𝑞) = ⋁{𝑎 ∈ [0,
1] | 𝑞 ∈ 𝐹𝑎𝑏 } and 𝜈𝐹 (𝑞) = ⋀{𝑏 ∈ [0, 1]𝑞 ∈ 𝐹𝑎𝑏 }, for all
𝑞 ∈ 𝑄. Then L(M) = 𝑓𝑆 . In fact, for any 𝜔 ∈ Σ∗ ,
𝜇L(M) (𝜔) = ⋁{𝜇𝐹 (𝑞) | 𝜇⊢∗M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1, 𝑞 ∈
𝑄, 𝛾 ∈ Γ∗ } = ⋁{⋁{𝑎 ∈ [0, 1] | 𝑞 ∈ 𝐹𝑎𝑏 } | 𝜇⊢∗M ((𝑞0 , 𝜔,
𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1, 𝑞 ∈ 𝑄, 𝛾 ∈ Γ∗ } = ⋁{𝑎 ∈ [0, 1] |
𝑞 ∈ 𝐹𝑎𝑏 , 𝜇⊢∗M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1, 𝑞 ∈ 𝑄, 𝛾 ∈ Γ∗ } =
⋁{𝑎 ∈ [0, 1] | 𝑞 ∈ 𝐹𝑎𝑏 , (𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑎𝑏 (𝑞, 𝜀, 𝛾), 𝛾 ∈ Γ∗ } =
⋁{𝑎 ∈ [0, 1] | M𝑎𝑏 accepts 𝜔, M𝑎𝑏 ∈ 𝑆} = 𝜇𝑓𝑆 (𝜔),
𝜈L(M) (𝜔) = ⋀{𝜈𝐹 (𝑞) | 𝜇⊢∗M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1, 𝑞 ∈ 𝑄,
𝛾 ∈ Γ∗ } = ⋀{⋀{𝑏 ∈ [0, 1] | 𝑞 ∈ 𝐹𝑎𝑏 } | 𝜇⊢∗M ((𝑞0 , 𝜔,
𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1, 𝑞 ∈ 𝑄, 𝛾 ∈ Γ∗ } = ⋀{𝑏 ∈ [0, 1] | 𝑞 ∈ 𝐹𝑎𝑏 ,
8
Journal of Applied Mathematics
(𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑎𝑏 (𝑞, 𝜀, 𝛾), 𝛾 ∈ Γ∗ } = ⋀{𝑏 ∈ [0, 1] | M𝑎𝑏
accepts 𝜔, M𝑎𝑏 ∈ 𝑆} = 𝜈𝑓𝑆 (𝜔).
Therefore, the IFSPDA M accepts 𝑓.
Theorem 16 shows that every IFPDA is equivalent to a
certain cover; however, the cover may have infinite classical
pushdown automata elements. Is there a finite cover who
is equivalent to the IFPDA? To solve the problem, we
introduce the notion of an intuitionistic fuzzy recognizable
step function as follows.
Definition 17. An IFS 𝐴 over Σ∗ is called an intuitionistic
fuzzy recognizable step function if there are a finite natural number 𝑛 ∈ 𝑁, recognizable context-free languages
L1 , . . . , L𝑛 ⊆ Σ∗ , and (𝑎𝑖 , 𝑏𝑖 ) ∈ (0, 1]×[0, 1) with 0 ≤ 𝑎𝑖 +𝑏𝑖 ≤
1 for 𝑖 = 1, . . . , 𝑛 such that
𝑛
𝐴 = (𝜇𝐴 , 𝜈𝐴) = ∐ (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1Li ,
(⋆)
𝑖=1
where 1Li = (𝜇1L , 𝜈1L ) represents the intuitionistic characi
i
terized function of L𝑖 , 𝑖 = 1, . . . , 𝑛, that is,
1,
𝜇1L (𝜔) = {
i
0,
0,
𝜈1L (𝜔) = {
i
1,
if 𝜔 ∈ L𝑖
if 𝜔 ∉ L𝑖 ,
(21)
if 𝜔 ∈ L𝑖
if 𝜔 ∉ L𝑖 .
And the equation (⋆) means that the following equations
hold:
𝑛
𝜇𝐴 (𝜔) =⋁ 𝑎𝑖 ∧ 𝜇1L (𝜔) ,
𝑖=1
i
𝑛
𝜈𝐴 (𝜔) =⋀ 𝑏𝑖 ∨ 𝜈1L (𝜔) ,
𝑖=1
i
(22)
∗
∀𝜔 ∈ Σ .
Noting that the family of all the intuitionistic fuzzy
recognizable step functions over Σ∗ is denoted by Step𝐶(Σ).
Proposition 18. Let M󸀠1 = (𝑄1 , Σ, Γ1 , 𝛿1 , 𝑞01 , 𝑍01 , 𝐹1 ) be an
IFPDA. Then the language recognized by M󸀠1 is an intuitionistic
fuzzy recognizable step function over Σ∗ , that is, L(M󸀠1 ) ∈
Step𝐶(Σ).
Proof. By Proposition 14, there is an IFSPDA M = (𝑄, Σ, Γ,
𝛿, 𝑞0 , 𝑍0 , 𝐹) equivalent to M󸀠1 .
Let 𝑅 = {(𝜇𝐹 (𝑞), 𝜈𝐹 (𝑞)) | 𝑞 ∈ 𝑄} \ {(0, 1)} = {(𝑎𝑖 , 𝑏𝑖 ) |
𝑖 ∈ 𝑁𝑘 }, 𝑁𝑘 = {1, . . . , 𝑘}. Put 𝐹𝑖 = {𝑞 ∈ 𝑄 | 𝜇𝐹 (𝑞) =
𝑎𝑖 , 𝜈𝐹 (𝑞) = 𝑏𝑖 }, for all 𝑖 ∈ 𝑁𝑘 . Then we construct a PDA
M𝑖 = (𝑄, Σ, Γ, 𝛿󸀠 , 𝑞0 , 𝑍0 , 𝐹𝑖 ), where the mapping 𝛿󸀠 : 𝑄 × (Σ ∪
∗
{𝜀}) × Γ → 2𝑄×Γ is defined by
𝛿󸀠 (𝑞, 𝜏, 𝑋) = {(𝑝, 𝛾) | 𝜇𝛿 (𝑞, 𝜏, 𝑋, 𝑝, 𝛾) = 1, 𝑝 ∈ 𝑄, 𝛾 ∈
Γ∗ }, for all (𝑞, 𝜏, 𝑋) ∈ 𝑄 × (Σ ∪ {𝜀}) × Γ.
Then L(M𝑖 ) = L𝑖 = {𝜔 ∈ Σ∗ | (𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑖 (𝑞, 𝜀, 𝛾),
𝑞 ∈ 𝐹𝑖 , 𝛾 ∈ Γ∗ }.
Therefore, for any 𝜔 ∈ Σ∗ , we have 𝜇L(M󸀠1 ) (𝜔) = 𝜇L(M) (𝜔)
= ⋁{𝜇𝐹 (𝑞) | 𝜇⊢∗M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1, 𝑞 ∈ 𝑄, 𝛾 ∈ Γ∗ } =
⋁{𝑎𝑖 | 𝜇⊢∗M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1, 𝜇𝐹 (𝑞) = 𝑎𝑖 , 𝜈𝐹 (𝑞) = 𝑏𝑖 ,
𝑖 ∈ 𝑁𝑘 , 𝛾 ∈ Γ∗ } = ⋁{𝑎𝑖 | (𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑖 (𝑞, 𝜀, 𝛾), 𝑞 ∈ 𝐹𝑖 ,
𝑖 ∈ 𝑁𝑘 , 𝛾 ∈ Γ∗ } = ⋁{𝑎𝑖 | 𝜔 ∈ L(M𝑖 ), 𝑖 ∈ 𝑁𝑘 } = ⋁𝑖∈𝑁𝑘 𝑎𝑖 ∧
𝜇1L(M ) (𝜔), and 𝜈L(M󸀠1 ) (𝜔) = 𝜈L(M) (𝜔) = ⋀{𝜈𝐹 (𝑞) | 𝜈⊢∗M ((𝑞0 ,
i
𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) = 0, 𝑞 ∈ 𝑄, 𝛾 ∈ Γ∗ } = ⋀{𝑏𝑖 |
𝜇⊢∗M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) = 1, 𝜇𝐹 (𝑞) = 𝑎𝑖 , 𝜈𝐹 (𝑞) = 𝑏𝑖 , 𝑖 ∈
𝑁𝑘 , 𝛾 ∈ Γ∗ } = ⋀{𝑏𝑖 | (𝑞0 , 𝜔, 𝑍0 ) ≻∗M𝑖 (𝑞, 𝜀, 𝛾), 𝑞 ∈ 𝐹𝑖 , 𝑖 ∈ 𝑁𝑘 ,
𝛾 ∈ Γ∗ } = ⋀{𝑏𝑖 | 𝜔 ∈ L(M𝑖 ), 𝑖 ∈ 𝑁𝑘 } = ⋀𝑖∈𝑁𝑘 𝑏𝑖 ∨ 𝜈1L(M ) (𝜔).
i
So L(M) = ∐𝑖∈𝑁𝑘 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1Li .
Proposition 18 shows that the set of the languages recognized by all the IFPDAs is a subset of Step𝐶(Σ). In fact,
Step𝐶(Σ) is also a subset of the set of the languages recognized
by all the IFPDAs. We will prove the decomposition form in
the following.
Theorem 19. Let 𝐴 be an IFS over Σ∗ . Then the following
statements are equivalent:
(i) 𝐴 ∈ Step𝐶(Σ);
(ii) there is an IFPDA M such that 𝐴 = L(M);
(iii) there is an IFSPDA M such that 𝐴 = L(M).
Proof. (i) implies (iii). Suppose 𝐴 ∈ Step𝐶(Σ). Then there is
a finite natural number 𝑛 ∈ 𝑁, recognizable context-free
languages L1 , . . . , L𝑛 ⊆ Σ∗ , and (𝑎𝑖 , 𝑏𝑖 ) ∈ (0, 1] × [0, 1)
with 0 ≤ 𝑎𝑖 + 𝑏𝑖 ≤ 1 for 𝑖 = 1, . . . , 𝑛 such that 𝐴 =
(𝜇𝐴 , 𝜈𝐴) = ∐𝑛𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1Li . Let L𝑖 be recognized by a PDA
M𝑖 = (𝑄𝑖 , Σ, Γ𝑖 , 𝛿𝑖 , 𝑞0𝑖 , 𝑍0𝑖 , 𝐹𝑖 ), and 𝑄𝑖 ∩𝑄𝑗 = 0 whenever 𝑖 ≠ 𝑗,
𝑖, 𝑗 ∈ 𝑁𝑛 .
Next, construct an IFSPDA M = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 𝐹)
as follows: 𝑄 = ⋃𝑛𝑖=1 𝑄𝑖 ∪ {𝑞0 }, Γ = ⋃𝑛𝑖=1 Γ𝑖 ∪ {𝑍0 }, where
𝑞0 ∉ ⋃𝑛𝑖=1 𝑄𝑖 , 𝑍0 ∉ ⋃𝑛𝑖=1 Γ𝑖 . 𝛿 = (𝜇𝛿 , 𝜈𝛿 ) is an IFS over
𝑄 × (Σ ∪ {𝜀}) × Γ × 𝑄 × Γ∗ , where the mappings 𝜇𝛿 , 𝜈𝛿 :
𝑄 × (Σ ∪ {𝜀}) × Γ × 𝑄 × Γ∗ → {0, 1} are defined by 𝜇𝛿 (𝑞0 ,
𝜀, 𝑍0 , 𝑝, 𝛾) = 1 and 𝜈𝛿 (𝑞0 , 𝜀, 𝑍0 , 𝑝, 𝛾) = 0 if (𝑝, 𝛾) = (𝑞0𝑖 ,
𝑍0𝑖 ), 𝑖 ∈ 𝑁𝑛 ; 𝜇𝛿 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = 𝜇𝛿𝑖 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) and 𝜈𝛿 (𝑞,
𝜏, 𝑍, 𝑝, 𝛾) = 𝜈𝛿𝑖 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) if 𝑝, 𝑞 ∈ 𝑄𝑖 , 𝑍 ∈ Γ𝑖 , 𝛾 ∈ Γ𝑖∗ ,
𝜏 ∈ Σ ∪ {𝜀}, 𝑖 ∈ 𝑁𝑛 . Otherwise, 𝜇𝛿 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = 0 and
𝜈𝛿 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = 1.
𝐹 = (𝜇𝐹 , 𝜈𝐹 ) is an IFS in 𝑄, where
𝜇𝐹 (𝑞0 ) = ⋁ {𝑎𝑖 ∈ [0, 1] | 𝑞0𝑖 ∈ 𝐹𝑖 , 𝑖 ∈ 𝑁𝑛 } ,
𝜈𝐹 (𝑞0 ) = ⋀ {𝑏𝑖 ∈ [0, 1] | 𝑞0𝑖 ∈ 𝐹𝑖 , 𝑖 ∈ 𝑁𝑛 } ,
𝜇𝐹 (𝑞) = {
𝑎𝑖 ,
0,
if 𝑞 ∈ 𝐹𝑖
if 𝑞 ∉ 𝐹𝑖 ∪ {𝑞0 } ,
(23)
𝑏 , if 𝑞 ∈ 𝐹𝑖
𝜈𝐹 (𝑞) = { 𝑖
1, if 𝑞 ∉ 𝐹𝑖 ∪ {𝑞0 } .
Therefore, for any 𝜔 ∈ Σ∗ , we have
𝜇L(M) (𝜔) = ⋁{𝜇⊢∗M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) ∧ 𝜇𝐹 (𝑞) | 𝑞 ∈
𝑄, 𝛾 ∈ Γ∗ },
𝜈L(M) (𝜔) = ⋀{𝜈⊢∗M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞, 𝜀, 𝛾)) ∨ 𝜈𝐹 (𝑞) | 𝑞 ∈
𝑄, 𝛾 ∈ Γ∗ }.
Journal of Applied Mathematics
9
If 𝜔 = 𝜀, then 𝜇L(M) (𝜀) = (𝜇⊢∗M ((𝑞0 , 𝜀, 𝑍0 ), (𝑞0 , 𝜀, 𝑍0 )) ∧
𝜇𝐹 (𝑞0 )) ∨ (⋁{𝜇⊢∗M ((𝑞0 , 𝜀, 𝑍0 ), (𝑞0𝑖 , 𝜀, 𝑍0𝑖 )) ∧ 𝜇𝐹 (𝑞0𝑖 ) | 𝑞0𝑖 ∈
𝑄𝑖 , 𝑍0𝑖 ∈ Γ𝑖 , 𝑖 ∈ 𝑁𝑛 }) = 𝜇𝐹 (𝑞0 ) ∨ (⋁{𝜇𝐹 (𝑞0𝑖 ) | 𝑞0𝑖 ∈ 𝑄𝑖 , 𝑍0𝑖 ∈
Γ𝑖 , 𝑖 ∈ 𝑁𝑛 }) = ⋁{𝑎𝑖 ∈ [0, 1] | 𝑞0𝑖 ∈ 𝐹𝑖 , 𝑖 ∈ 𝑁𝑛 } =
⋁𝑛𝑖=1 𝑎𝑖 ∧ 𝜇1L (𝜀); 𝜈L(M) (𝜀) = (𝜈⊢∗M ((𝑞0 , 𝜀, 𝑍0 ), (𝑞0 , 𝜀, 𝑍0 )) ∨
i
𝜈𝐹 (𝑞0 )) ∧ (⋀{𝜈⊢∗M ((𝑞0 , 𝜀, 𝑍0 ), (𝑞0𝑖 , 𝜀, 𝑍0𝑖 )) ∨ 𝜈𝐹 (𝑞0𝑖 ) | 𝑞0𝑖 ∈ 𝑄𝑖 ,
𝑍0𝑖 ∈ Γ𝑖 , 𝑖 ∈ 𝑁𝑛 }) = 𝜈𝐹 (𝑞0 )∧(⋀{𝜈𝐹 (𝑞0𝑖 ) | 𝑞0𝑖 ∈ 𝑄𝑖 , 𝑍0𝑖 ∈ Γ𝑖 , 𝑖 ∈
𝑁𝑛 }) = ⋀{𝑏𝑖 ∈ [0, 1] | 𝑞0𝑖 ∈ 𝐹𝑖 , 𝑖 ∈ 𝑁𝑛 } = ⋀𝑛𝑖=1 𝑏𝑖 ∨ 𝜈1L (𝜀). If
i
𝜔 ∈ Σ∗ \ {𝜀}, then 𝜇L(M) (𝜔) = ⋁{𝜇⊢∗M ((𝑞0𝑖 , 𝜔, 𝑍0𝑖 ), (𝑞, 𝜀, 𝛾)) ∧
𝜇𝐹 (𝑞) | 𝑞0𝑖 ∈ 𝑄𝑖 , 𝛾 ∈ Γ∗ , 𝑞 ∈ 𝑄, 𝑖 ∈ 𝑁𝑛 } = ⋁{𝜇𝐹 (𝑞) | 𝑞0𝑖 ∈ 𝑄𝑖 ,
𝛾 ∈ Γ𝑖∗ , 𝑞 ∈ 𝑄, 𝑖 ∈ 𝑁𝑛 , 𝜇⊢∗M ((𝑞0𝑖 , 𝜔, 𝑍0𝑖 ),(𝑞, 𝜀, 𝛾)) = 1} =
⋁{𝑎𝑖 | 𝑞 ∈ 𝐹𝑖 , 𝑖 ∈ 𝑁𝑛 , 𝜇⊢∗M ((𝑞0𝑖 , 𝜔, 𝑍0𝑖 ), (𝑞, 𝜀, 𝛾)) = 1, 𝛾 ∈ Γ𝑖∗ } =
⋁{𝑎𝑖 | 𝜔 ∈ L𝑖 , 𝑖 ∈ 𝑁𝑛 } = ⋁𝑛𝑖=1 𝑎𝑖 ∧ 𝜇1L (𝜔), 𝜈L(M) (𝜔) =
i
⋀{𝜈⊢∗M ((𝑞0𝑖 , 𝜔, 𝑍0𝑖 ), (𝑞, 𝜀, 𝛾)) ∨ 𝜈𝐹 (𝑞)|𝑞0𝑖 ∈ 𝑄𝑖 , 𝛾 ∈ Γ∗ , 𝑞 ∈
𝑄, 𝑖 ∈ 𝑁𝑛 } = ⋀{𝜈𝐹 (𝑞) | 𝑞0𝑖 ∈ 𝑄𝑖 , 𝛾 ∈ Γ𝑖∗ , 𝑞 ∈ 𝑄, 𝑖 ∈
𝑁𝑛 , 𝜇⊢∗M ((𝑞0𝑖 , 𝜔, 𝑍0𝑖 ),(𝑞, 𝜀, 𝛾)) = 1} = ⋀{𝑏𝑖 | 𝑞 ∈ 𝐹𝑖 , 𝑖 ∈
𝑁𝑛 , 𝜇⊢∗M ((𝑞0𝑖 , 𝜔, 𝑍0𝑖 ), (𝑞, 𝜀, 𝛾)) = 1, 𝛾 ∈ Γ𝑖∗ } = ⋀{𝑏𝑖 | 𝜔 ∈
L𝑖 , 𝑖 ∈ 𝑁𝑛 } = ⋀𝑛𝑖=1 𝑎𝑖 ∨ 𝜈1L (𝜔).
i
(iii) implies (ii): obviously.
(ii) implies (i): it is concluded by Proposition 18.
Next, we will discuss the characterization of IFPDA0 .
Proposition 20. Let M = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 0) be an IFPDA0 .
Then there is a special IFPDA0 M󸀠 = (𝑄󸀠 , Σ, Γ󸀠 , 𝛿󸀠 , 𝑞0󸀠 , 𝑋0 , 0)
equivalent to M.
Proof. Given M, we construct an IFPDA0 M󸀠
(𝑄󸀠 , Σ, Γ󸀠 , 𝛿󸀠 , 𝑞0󸀠 , 𝑋0 , 0) as follows:
Firstly let us show that, for any 𝜔 = 𝜏1 ⋅ ⋅ ⋅ 𝜏𝑛 ∈ Σ∗ , 𝜏𝑖 ∈
Σ ∪ {𝜀}, 𝑖 ∈ {1, . . . , 𝑛}, 𝑞𝑛 ∈ 𝑄,
𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ) , ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ) , 𝜀, 𝑋0 ))
M
1,
={
0,
𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ) , ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ) , 𝜀, 𝑋0 ))
󸀠
󸀠
0,
={
1,
(P2) there exist 𝑞1 , . . . , 𝑞𝑛−1 ∈ 𝑄, 𝑍1 , . . . , 𝑍𝑛−1 ∈ Γ, 𝛾1 ,
. . . , 𝛾𝑛−1 ∈ Γ∗ , s.t. 𝑎𝑛 = 𝜇⊢M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝑍1 𝛾1 ))∧𝜇⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 ))∧
⋅ ⋅ ⋅ ∧ 𝜇⊢M ((𝑞𝑛−1 , 𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 ), (𝑞𝑛 , 𝜀, 𝜀)) > 0
and 𝑏𝑛 = 𝜈⊢M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 )) ∨
𝜈⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∨ ⋅ ⋅ ⋅ ∨
𝜈⊢M ((𝑞𝑛−1 , 𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 ), (𝑞𝑛 , 𝜀, 𝜀)) < 1.
In fact, if (P2) is satisfied, then let 𝜇⊢M ((𝑞𝑖 , 𝜏𝑖+1 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝑍𝑖 𝛾𝑖 ), (𝑞𝑖+1 , 𝜏𝑖+2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍𝑖+1 𝛾𝑖+1 )) = 𝑐𝑖 , 𝜈⊢M ((𝑞𝑖 , 𝜏𝑖+1 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍𝑖 𝛾𝑖 ),
(𝑞𝑖+1 , 𝜏𝑖+2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍𝑖+1 𝛾𝑖+1 )) = 𝑑𝑖 , 𝑖 = 0, 1, . . . , 𝑛 − 2, where
𝛾0 = 𝜀. We have
Obviously, 𝛿󸀠 = (𝜇𝛿󸀠 , 𝜈𝛿󸀠 ) is a finite IFS in 𝑄󸀠 × (Σ ∪ {𝜀}) ×
Γ × 𝑄󸀠 × Γ󸀠∗ .
Next, we show L(M󸀠 ) = L(M).
𝜔, 𝑍0 𝑋0 ),
((𝑞1 , 𝑐0 , 𝑑0 ),
𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝜈⊢M󸀠 ((𝑞0 , 1, 0),
𝑍1 𝛾1 𝑋0 )) = 0,
𝜔, 𝑍0 𝑋0 ),
((𝑞1 , 𝑐0 , 𝑑0 ),
𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝜈⊢M󸀠 (((𝑞1 , 𝑐0 , 𝑑0 ), 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 𝑋0 ), ((𝑞2 , 𝑐0 ∧ 𝑐1 , 𝑑0 ∨
𝑑1 ), 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 𝑋0 )) = 0,
..
.
(2) 𝛿 = (𝜇𝛿󸀠 , 𝜈𝛿󸀠 ) is an IFS in 𝑄 × (Σ ∪ {𝜀}) × Γ × 𝑄 × Γ ,
where the mappings 𝜇𝛿󸀠 , 𝜈𝛿󸀠 : 𝑄󸀠 × (Σ ∪ {𝜀}) × Γ󸀠 × 𝑄󸀠 ×
Γ󸀠∗ → [0, 1] are defined by
󸀠
𝜇⊢M󸀠 ((𝑞0 , 1, 0),
𝑍1 𝛾1 𝑋0 )) = 1,
𝜇⊢M󸀠 (((𝑞1 , 𝑐0 , 𝑑0 ), 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 𝑋0 ), ((𝑞2 , 𝑐0 ∧ 𝑐1 , 𝑑0 ∨
𝑑1 ), 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 𝑋0 )) = 1,
󸀠∗
(i) 𝜇𝛿󸀠 ((𝑝0 , 1, 0), 𝜀, 𝑋0 , (𝑞0 , 1, 0), 𝑍0 𝑋0 )
=
1,
𝜈𝛿󸀠 ((𝑝0 , 1, 0), 𝜀, 𝑋0 , (𝑞0 , 1, 0), 𝑍0 𝑋0 ) = 0;
(ii) 𝜇𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜏, 𝑋, (𝑞󸀠 , 𝑎 ∧ 𝑎󸀠 , 𝑏 ∨ 𝑏󸀠 ), 𝛾) = 1
and 𝜈𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜏, 𝑋, (𝑞󸀠 , 𝑎 ∧ 𝑎󸀠 , 𝑏 ∨ 𝑏󸀠 ), 𝛾) =
0 whenever 𝜇𝛿 (𝑞, 𝜏, 𝑋, 𝑞󸀠 , 𝛾) = 𝑎󸀠 > 0 and
𝜈𝛿 (𝑞, 𝜏, 𝑋, 𝑞󸀠 , 𝛾) = 𝑏󸀠 < 1, for all (𝑞, 𝑎, 𝑏) ∈ 𝑄󸀠 ,
𝜏 ∈ Σ ∪ {𝜀}, 𝑋 ∈ Γ, 𝛾 ∈ Γ∗ ;
(iii) if 0 ≤ 𝑎 + 𝑏 ≤ 1, then 𝜇𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜀, 𝑋0 , (𝑞, 𝑎,
𝑏), 𝜀) = 𝑎 and 𝜈𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜀, 𝑋0 , (𝑞, 𝑎, 𝑏), 𝜀) = 𝑏.
If 𝑎 + 𝑏 > 1, then 𝜇𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜀, 𝑋0 , (𝑞, 𝑎, 𝑏), 𝜀) =
0 and 𝜈𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜀, 𝑋0 , (𝑞, 𝑎, 𝑏), 𝜀) = 1;
(iv) In other cases, 𝜇𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜏, 𝑋, (𝑝, 𝑐, 𝑑), 𝛾) = 0
and 𝜈𝛿󸀠 ((𝑞, 𝑎, 𝑏), 𝜏, 𝑋, (𝑝, 𝑐, 𝑑), 𝛾) = 1.
if (P2) is satisfied
otherwise,
where the condition (P2) is the following:
=
󸀠
(24)
M
(1) 𝑄󸀠 = 𝑄2 × (𝐿 1 \ {0}) × (𝐿 2 \ {1}), where 𝑄2 = 𝑄 ∪ {𝑝0 },
Γ󸀠 = Γ ∪ {𝑋0 }, 𝐿 1 = 𝑋∧ , 𝐿 2 = 𝑌∨ , 𝑋 = Im(𝜇𝛿 ), 𝑌 =
Im(𝜈𝛿 ), and 𝑞0󸀠 = (𝑝0 , 1, 0);
󸀠
if (P2) is satisfied
otherwise,
𝜇⊢M󸀠 (((𝑞𝑛−1 , 𝑐0 ∧ 𝑐1 ∧ ⋅ ⋅ ⋅ ∧ 𝑐𝑛−2 , 𝑑0 ∨ 𝑑1 ∨ ⋅ ⋅ ⋅ ∨ 𝑑𝑛−2 ),
𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) = 1,
𝜈⊢M󸀠 (((𝑞𝑛−1 , 𝑐0 ∧ 𝑐1 ∧ ⋅ ⋅ ⋅ ∧ 𝑐𝑛−2 , 𝑑0 ∨ 𝑑1 ∨ ⋅ ⋅ ⋅ ∨ 𝑑𝑛−2 ),
𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) = 0.
Hence 𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 ) ≥ 𝜇⊢M󸀠 ((𝑞0 ,
M
1, 0), 𝜔, 𝑍0 𝑋0 ), ((𝑞1 , 𝑐0 , 𝑑0 ), 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 𝑋0 )) ∧ 𝜇⊢M󸀠 (((𝑞1 ,
𝑐0 , 𝑑0 ), 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 𝑋0 ), ((𝑞2 , 𝑐0 ∧ 𝑐1 , 𝑑0 ∨ 𝑑1 ), 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 ,
𝑍2 𝛾2 𝑋0 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢M󸀠 (((𝑞𝑛−1 , 𝑐0 ∧ 𝑐1 ∧ ⋅ ⋅ ⋅ ∧ 𝑐𝑛−2 , 𝑑0 ∨
𝑑1 ∨ ⋅ ⋅ ⋅ ∨ 𝑑𝑛−2 ), 𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) = 1,
𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) ≤ 𝜈⊢M󸀠 ((𝑞0 , 1, 0), 𝜔,
M
𝑍0 𝑋0 ), ((𝑞1 , 𝑐0 , 𝑑0 ), 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 𝑋0 )) ∨ 𝜈⊢M󸀠 (((𝑞1 , 𝑐0 , 𝑑0 ), 𝜏2
⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 𝑋0 ), ((𝑞2 , 𝑐0 ∧ 𝑐1 , 𝑑0 ∨ 𝑑1 ), 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 𝑋0 )) ∨
⋅ ⋅ ⋅ ∨ 𝜈⊢M󸀠 (((𝑞𝑛−1 , 𝑐0 ∧ 𝑐1 ∧ ⋅ ⋅ ⋅ ∧ 𝑐𝑛−2 , 𝑑0 ∨ 𝑑1 ∨ ⋅ ⋅ ⋅ ∨ 𝑑𝑛−2 ),
𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) = 0.
Since 𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) ≤ 1 and
M
𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) ≥ 0, 𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ),
M
M
10
Journal of Applied Mathematics
((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) = 1 and 𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ),
M
𝜀, 𝑋0 )) = 0.
If (P2) is not satisfied, then we assume 𝜇⊢∗ 󸀠 ((𝑞0󸀠 ,
M
𝜔, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) > 0.
So there at least exist 𝑞1 , . . . , 𝑞𝑛−1 ∈ 𝑄, 𝑍1 , . . . , 𝑍𝑛−1 ∈ Γ,
𝛾1 , . . . , 𝛾𝑛−1 ∈ Γ∗ , s.t. 𝜇⊢M󸀠 ((𝑞0 , 1, 0), 𝜔, 𝑍0 𝑋0 ), ((𝑞1 , 𝑐0 , 𝑑0 ),
𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 𝑋0 )) ∧ 𝜇⊢M󸀠 (((𝑞1 , 𝑐0 , 𝑑0 ), 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 𝑋0 ),
((𝑞2 , 𝑐0 ∧ 𝑐1 , 𝑑0 ∨ 𝑑1 ), 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 𝑋0 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢M󸀠 (((𝑞𝑛−1 ,
𝑐0 ∧ 𝑐1 ∧ ⋅ ⋅ ⋅ ∧ 𝑐𝑛−2 , 𝑑0 ∨ 𝑑1 ∨ ⋅ ⋅ ⋅ ∨ 𝑑𝑛−2 ),𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 𝑋0 ),
((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) > 0.
Hence 𝑎𝑛 = 𝜇⊢M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 )) ∧
𝜇⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢M ((𝑞𝑛−1 ,
𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 ), (𝑞𝑛 , 𝜀, 𝜀)) > 0 and 𝑏𝑛 = 𝜈⊢M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞1 , 𝜏2
⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 )) ∨ 𝜈⊢M ((𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∨
⋅ ⋅ ⋅ ∨ 𝜈⊢M ((𝑞𝑛−1 , 𝜏𝑛 , 𝑍𝑛−1 𝛾𝑛−1 ), (𝑞𝑛 , 𝜀, 𝜀)) < 1.
It contradicts with the assumption. So 𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ),
M
((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 ) = 0 if (P2) is not satisfied. In a similar
way, it is easily concluded that 𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 ,
M
𝑏𝑛 ), 𝜀, 𝑋0 ) = 1 if (P2) is not satisfied.
Secondly, for any 𝜔 = 𝜏1 ⋅ ⋅ ⋅ 𝜏𝑛 ∈ Σ∗ , 𝜏𝑖 ∈ Σ ∪ {𝜀},
𝑖 ∈ {1, . . . , 𝑛}, 𝜇L(M󸀠 ) (𝜔) = ⋁{𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀,
M
𝑋0 )) ∧ 𝜇⊢M󸀠 ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝜀)) | (𝑞𝑛 , 𝑎𝑛 ,
𝑏𝑛 ) ∈ 𝑄󸀠 } = ⋁{𝜇⊢M ((𝑞0 , 𝜔, 𝑍0 ), (𝑞1 , 𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 )) ∧ 𝜇⊢M ((𝑞1 ,
𝜏2 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍1 𝛾1 ), (𝑞2 , 𝜏3 ⋅ ⋅ ⋅ 𝜏𝑛 , 𝑍2 𝛾2 )) ∧ ⋅ ⋅ ⋅ ∧ 𝜇⊢M ((𝑞𝑛−1 , 𝜏𝑛 ,
𝑍𝑛−1 𝛾𝑛−1 ),(𝑞𝑛 , 𝜀, 𝜀)) | 𝑞1 , . . . , 𝑞𝑛 ∈ 𝑄, 𝑍1 , . . . , 𝑍𝑛−1 ∈ Γ, 𝛾1 ,
. . . , 𝛾𝑛−1 ∈ Γ∗ } = 𝜇L(M) (𝜔).
Similarly, 𝜈L(M󸀠 ) (𝜔) = 𝜈L(M) (𝜔).
Hence L(M󸀠 ) = L(M).
Remark 21. Proposition 20 presents an equivalence of an
IFPDA0 . In particular, due to the underlying truth-valued
domain being an IFS, the proof technique used in Proposition 20 is to some extent different from the technique of
extended subset construction in [33]. Moreover, Proposition 20 plays an important role in proving the fact that any
language recognized by an IFPDA0 is an intuitionistic fuzzy
recognizable step function.
Proposition 22. Let M = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 0) be an IFPDA0 .
Then the language recognized by M is an intuitionistic fuzzy
recognizable step function over Σ∗ , that is, L(M) ∈ Step𝐶(Σ).
Proof. Let M = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 0) be an IFPDA0 . Then
there is a special IFPDA0 M󸀠 = (𝑄󸀠 , Σ, Γ󸀠 , 𝛿󸀠 , 𝑞0󸀠 , 𝑋0 , 0) constructed by Proposition 20, which is equivalent to M. For any
(𝑎, 𝑏) ∈ (𝐿 1 \ {0}) × (𝐿 2 \ {1}) with 0 < 𝑎 + 𝑏 ≤ 1, construct a classical PDA with empty stack M𝑎𝑏 = (𝑄󸀠 , Σ, Γ󸀠 ,
󸀠󸀠
, 𝑞0󸀠 , 𝑋0 , 0), where 𝑄󸀠 , Σ, Γ󸀠 , 𝑞0󸀠 , and 𝑋0 are the same as
𝛿𝑎𝑏
󸀠
󸀠∗
󸀠󸀠
: 𝑄󸀠 × Σ × Γ󸀠 → 2𝑄 ×Γ is
those in M󸀠 , and the function 𝛿𝑎𝑏
defined by
󸀠󸀠
((𝑝0 , 1, 0), 𝜀, 𝑋0 ) = {((𝑞0 , 1, 0), 𝑍0 𝑋0 )};
(i) 𝛿𝑎𝑏
󸀠󸀠
((𝑞, 𝑐, 𝑑), 𝜏, 𝑋) = {((𝑞󸀠 , 𝑐 ∧ 𝑐1 , 𝑑 ∨ 𝑑1 ), 𝛾) | 𝑐1 =
(ii) 𝛿𝑎𝑏
𝜇𝛿 (𝑞, 𝜏, 𝑋, 𝑞󸀠 , 𝛾) > 0, 𝑑1 = 𝜈𝛿 (𝑞, 𝜏, 𝑋, 𝑞󸀠 , 𝛾) < 1, 𝑞󸀠 ∈
𝑄, 𝛾 ∈ Γ∗ };
󸀠󸀠
(iii) 𝛿𝑎𝑏
((𝑞, 𝑎, 𝑏), 𝜀, 𝑋0 ) = {((𝑞, 𝑎, 𝑏), 𝜀)};
󸀠󸀠
((𝑞, 𝑐, 𝑑), 𝜏, 𝑋) = 0 in other cases.
(iv) 𝛿𝑎𝑏
Then for any 𝜔 ∈ Σ∗ , we have 𝜇L(M) (𝜔) = 𝜇L(M󸀠 ) (𝜔)
= ⋁{𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝑋0 )) ∧ 𝜇⊢M󸀠 ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ),
M
𝜀, 𝑋0 ), ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝜀)) | (𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ) ∈ 𝑄󸀠 } = ⋁{𝑎𝑛 |
(𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ) ∈ 𝑄󸀠 , 0 < 𝑎𝑛 + 𝑏𝑛 ≤ 1, 𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ),((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ),
M
𝜀, 𝑋0 )) = 1} = ⋁{𝑎𝑛 | (𝑞0󸀠 , 𝜔, 𝑋0 ) ≻∗M𝑎𝑏 ((𝑞𝑛 , 𝑎𝑛 , 𝑏𝑛 ), 𝜀, 𝜀), 𝑞𝑛 ∈
𝑄} = ⋁𝑎∈𝐿 1 \{0} 𝑎 ∧ 𝜇1L(M ) (𝜔), and 𝜈L(M) (𝜔) = 𝜈L(M󸀠 ) (𝜔)
ab
= ⋀{𝜈⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ), ((𝑞, 𝑎, 𝑏), 𝜀, 𝑋0 )) ∨ 𝜈⊢M󸀠 ((𝑞, 𝑎, 𝑏), 𝜀,
M
𝑋0 ), ((𝑞, 𝑎, 𝑏), 𝜀, 𝜀)) | (𝑞, 𝑎, 𝑏) ∈ 𝑄󸀠 } = ⋀{𝑏 | (𝑞, 𝑎, 𝑏) ∈
𝑄󸀠 , 0 < 𝑎 + 𝑏 ≤ 1, 𝜇⊢∗ 󸀠 ((𝑞0󸀠 , 𝜔, 𝑋0 ), ((𝑞, 𝑎, 𝑏), 𝜀, 𝑋0 )) =
M
1} = ⋀{𝑏 | (𝑞0󸀠 , 𝜔, 𝑋0 ) ≻∗M𝑎𝑏 ((𝑞, 𝑎, 𝑏), 𝜀, 𝑋0 ), (𝑞, 𝑎, 𝑏) ∈ 𝑄󸀠 }
= ⋀𝑏∈𝐿 2 \{1} 𝑏 ∨ 𝜈1L(M ) (𝜔).
ab
Therefore L(M) ∈ Step𝐶(Σ).
Theorem 23. Let 𝐴 be an IFS over Σ∗ . Then the following
statements are equivalent:
(1) 𝐴 ∈ Step𝐶(Σ);
(2) 𝐴 is accepted by a certain IFPDA0 M.
Proof. (i) implies (ii). Suppose 𝐴 ∈ Step𝐶(Σ). Then there are
a finite natural number 𝑛 ∈ 𝑁, recognizable context-free
languages L1 , . . . , L𝑛 ⊆ Σ∗ , and (𝑎𝑖 , 𝑏𝑖 ) ∈ (0, 1] × [0, 1)
with 0 ≤ 𝑎𝑖 + 𝑏𝑖 ≤ 1 for 𝑖 = 1, . . . , 𝑛 such that 𝐴 =
(𝜇𝐴 , 𝜈𝐴) = ∐𝑛𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1Li . Let L𝑖 be recognized by PDA
M𝑖 = (𝑄𝑖 , Σ, Γ𝑖 , 𝛿𝑖 , 𝑞0𝑖 , 𝑍0𝑖 , 0), and 𝑄𝑖 ∩ 𝑄𝑗 = 0 whenever 𝑖 ≠ 𝑗,
𝑖, 𝑗 ∈ 𝑁𝑛 .
Next, construct an IFPDA0 M = (𝑄, Σ, Γ, 𝛿, 𝐼, 𝑍0 , 0) as
follows: 𝑄 = ⋃𝑛𝑖=1 𝑄𝑖 , Γ = ⋃𝑛𝑖=1 Γ𝑖 ∪ {𝑍0 }, and 𝐼 = (𝜇𝐼 , 𝜈𝐼 )
are an IFS over 𝑄, defined by 𝜇𝐼 (𝑞0𝑖 ) = 𝑎𝑖 , 𝜈𝐼 (𝑞0𝑖 ) = 𝑏𝑖 ,
𝑖 ∈ 𝑁𝑛 ; otherwise 𝜇𝐼 (𝑞) = 0 and 𝜈𝐼 (𝑞) = 1. 𝛿 = (𝜇𝛿 , 𝜈𝛿 ) is
an IFS over 𝑄 × (Σ ∪ {𝜀}) × Γ × 𝑄 × Γ∗ , where the mappings
𝜇𝛿 , 𝜈𝛿 : 𝑄 × (Σ ∪ {𝜀}) × Γ × 𝑄 × Γ∗ → {0, 1} are defined by
𝜇𝛿 (𝑞0𝑖 , 𝜀, 𝑍0 , 𝑞0𝑖 , 𝑍0𝑖 ) = 1 and 𝜈𝛿 (𝑞0𝑖 , 𝜀, 𝑍0 , 𝑞0𝑖 , 𝑍0𝑖 ) = 0, 𝑖 ∈
𝑁𝑛 ; 𝜇𝛿 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = 1 and 𝜈𝛿 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = 0 if 𝑝, 𝑞 ∈ 𝑄𝑖 ,
𝑍 ∈ Γ𝑖 , 𝛾 ∈ Γ𝑖∗ , 𝜏 ∈ Σ, 𝑖 ∈ 𝑁𝑛 and (𝑝, 𝛾) ∈ 𝛿(𝑞, 𝜏, 𝑍).
Otherwise, 𝜇𝛿 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = 0 and 𝜈𝛿 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = 1.
Then 𝐴 = L(M) = ∐𝑛𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1Li .
(ii) implies (i). It is concluded by Proposition 22.
One can see that IFPDAs and IFPDAs0 are equivalent to
a type of intuitionistic fuzzy recognizable step functions over
a set, respectively, by Theorems 19 and 23. Therefore, IFPDAs
and IFPDAs0 are equivalent in the sense that they accept or
recognize the same classes of intuitionistic fuzzy languages.
That is to say, the following statement is true.
Corollary 24. For any IFPDA M, there is an
IFPDA0 M󸀠 equivalent to M. For any IFPDA0 M󸀠 , there
is an IFPDA M equivalent to M󸀠 .
4. Intuitionistic Fuzzy Context-Free Grammars
As a type of generator of intuitionistic fuzzy context-free
languages, the notion of intuitionistic fuzzy context-free
Journal of Applied Mathematics
grammars is introduced in the section. Then the relationship
between intuitionistic fuzzy context-free grammars, IFPDAs,
IFPDAs0 , and intuitionistic fuzzy recognizable step functions
is discussed. The algebraic properties of intuitionistic fuzzy
context-free languages are investigated finally.
Definition 25. An intuitionistic fuzzy grammar is a system
𝐺 = (𝑁, 𝑇, 𝑃, 𝐼), where
(i) 𝑁 is a finite nonempty alphabet of variables;
(ii) 𝑇 is a finite nonempty alphabet of terminals and 𝑇 ∩
𝑁 = 0;
(iii) 𝐼 is an intuitionistic fuzzy set over 𝑁;
(iv) 𝑃 is a finite collection of productions over 𝑇 ∪ 𝑁,
and 𝑃 = {𝑥 → 𝑦 | 𝑥 ∈ (𝑁 ∪ 𝑇)∗ 𝑁(𝑁 ∪ 𝑇)∗ ,
𝑦 ∈ (𝑁 ∪ 𝑇)∗ , 𝜇𝜌 (𝑥 → 𝑦) > 0, 𝜈𝜌 (𝑥 → 𝑦) < 1},
where 𝜌 = (𝜇𝜌 , 𝜈𝜌 ) is an IFS over (𝑁 ∪ 𝑇)∗ × (𝑁 ∪ 𝑇)∗ ,
𝜇𝜌 (𝑥, 𝑦), and 𝜈𝜌 (𝑥, 𝑦) mean the membership degree and
the nonmembership degree that 𝑥 will be replaced by 𝑦,
respectively, denoted by 𝜇𝜌 (𝑥, 𝑦) = 𝜇𝜌 (𝑥 → 𝑦), 𝜈𝜌 (𝑥, 𝑦) =
𝜈𝜌 (𝑥 → 𝑦).
For 𝛼, 𝛽 ∈ (𝑁 ∪ 𝑇)∗ , if 𝑥 → 𝑦 ∈ 𝑃, then 𝛼𝑦𝛽 is said to
be directly derivable from 𝛼𝑥𝛽, denoted by 𝛼𝑥𝛽 ⇒ 𝛼𝑦𝛽, and
define 𝜇𝜌 (𝛼𝑥𝛽 ⇒ 𝛼𝑦𝛽) = 𝜇𝜌 (𝑥 → 𝑦), 𝜈𝜌 (𝛼𝑥𝛽 ⇒ 𝛼𝑦𝛽) =
𝜈𝜌 (𝑥 → 𝑦).
If 𝛼1 , . . . , 𝛼𝑚 are strings in (𝑁 ∪ 𝑇)∗ and 𝛼1 → 𝛼2 ,
. . . , 𝛼𝑚−1 → 𝛼𝑚 ∈ 𝑃, then 𝛼1 is said to derive 𝛼𝑚 in 𝐺, or,
equivalently, 𝛼𝑚 is derivable from 𝛼1 in 𝐺. This is expressed
by 𝛼1 ⇒∗𝐺 𝛼𝑚 or simply 𝛼1 ⇒∗ 𝛼𝑚 . The expression 𝛼1 → 𝛼2 →
⋅ ⋅ ⋅ → 𝛼𝑚 is referred to as a derivation chain from 𝛼1 to 𝛼𝑚 .
An intuitionistic fuzzy grammar 𝐺 generates an intuitionistic fuzzy language L(𝐺) = (𝜇𝐺, 𝜈𝐺) in the following
manner. For any 𝜃 = 𝜔𝑛 ∈ 𝑇∗ , 𝑛 ≥ 1, 𝜇𝐺(𝜃) = ⋁{𝜇𝐼 (𝜔0 ) ∧
𝜇𝜌 (𝜔0 ⇒ 𝜔1 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ) | 𝜔0 ∈ 𝑁, 𝜔1 , . . . , 𝜔𝑛−1 ∈
(𝑁 ∪ 𝑇)∗ }, and 𝜈𝐺(𝜃) = ⋀{𝜈𝐼 (𝜔0 ) ∨ 𝜈𝜌 (𝜔0 ⇒ 𝜔1 ) ∨ ⋅ ⋅ ⋅ ∨
𝜈𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ) | 𝜔0 ∈ 𝑁, 𝜔1 , . . . , 𝜔𝑛−1 ∈ (𝑁 ∪ 𝑇)∗ }.
𝜇𝐺(𝜃) and 𝜈𝐺(𝜃) express the membership and nonmembership degree of 𝜃 in the language generated by grammar
𝐺, respectively. Obviously, L(𝐺) = (𝜇𝐺, 𝜈𝐺) is well defined.
In fact, for any 𝜃 = 𝜔𝑛 ∈ 𝑇∗ , 𝑛 ≥ 1, there is the strongest
󸀠
⇒
derivation from 𝜔0 to 𝜔𝑛 , that is, 𝜔0 ⇒ 𝜔1󸀠 ⇒ ⋅ ⋅ ⋅ ⇒ 𝜔𝑛−1
󸀠
󸀠
⇒
𝜔𝑛 , such that 𝜇𝐺(𝜃) = 𝜇𝐼 (𝜔0 ) ∧ 𝜇𝜌 (𝜔0 ⇒ 𝜔1 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝜔𝑛−1
󸀠
󸀠
𝜔𝑛 ). So 𝜈𝐺(𝜃) ≤ 𝜈𝐼 (𝜔0 ) ∨ 𝜈𝜌 (𝜔0 ⇒ 𝜔1 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ),
and 𝜇𝐺(𝜃) + 𝜈𝐺(𝜃) ≤ 𝜇𝐺(𝜃) + 𝜈𝐼 (𝜔0 ) ∨ 𝜈𝜌 (𝜔0 ⇒ 𝜔1󸀠 ) ∨ ⋅ ⋅ ⋅ ∨
󸀠
⇒ 𝜔𝑛 ) = (𝜇𝐺(𝜃) + 𝜈𝐼 (𝜔0 )) ∨ (𝜇𝐺(𝜃) + 𝜈𝜌 (𝜔0 ⇒ 𝜔1󸀠 )) ∨
𝜈𝜌 (𝜔𝑛−1
󸀠
⋅ ⋅ ⋅ ∨ (𝜇𝐺(𝜃) + 𝜈𝜌 (𝜔𝑛−1
⇒ 𝜔𝑛 )) ≤ (𝜇𝐼 (𝜔0 ) + 𝜈𝐼 (𝜔0 )) ∨ (𝜇𝜌 (𝜔0 ⇒
󸀠
󸀠
󸀠
󸀠
𝜔1 ) + 𝜈𝜌 (𝜔0 ⇒ 𝜔1 )) ∨ ⋅ ⋅ ⋅ ∨ (𝜇𝜌 (𝜔𝑛−1
⇒ 𝜔𝑛 ) + 𝜈𝜌 (𝜔𝑛−1
⇒
𝜔𝑛 )) ≤ 1.
For any intuitionistic fuzzy grammars 𝐺1 and 𝐺2 , if
L(𝐺1 ) = L(𝐺2 ) in the sense of equality of intuitionistic fuzzy
sets, then the grammars 𝐺1 and 𝐺2 are said to be equivalent.
For any intuitionistic fuzzy grammar 𝐺 = (𝑁, 𝑇, 𝑃, 𝐼), if
Im(𝐼) = Im(𝜇𝐼 ) ∪ Im(𝜈𝐼 ) = {0, 1} and supp(𝐼) = {𝑆}, then 𝐺
is also written as 𝐺 = (𝑁, 𝑇, 𝑃, 𝑆).
11
Proposition 26. Let 𝐴 be an IFS over 𝑇∗ . Then the following
statements are equivalent:
(i) 𝐴 is generated by a certain intuitionistic fuzzy grammar
𝐺 = (𝑁, 𝑇, 𝑃, 𝐼);
(ii) 𝐴 is generated by a certain intuitionistic fuzzy grammar
𝐺 = (𝑁󸀠 , 𝑇󸀠 , 𝑃󸀠 , 𝑆).
Proof. (i) implies (ii). Let 𝐴 be generated by an intuitionistic
fuzzy grammar 𝐺 = (𝑁, 𝑇, 𝑃, 𝐼). Then we construct an
intuitionistic fuzzy grammar 𝐺󸀠 = (𝑁󸀠 , 𝑇󸀠 , 𝑃󸀠 , 𝑆) as follows:
𝑁󸀠 = 𝑁 ∪ {𝑆}, 𝑆 ∉ 𝑁; 𝑇󸀠 = 𝑇, 𝑃󸀠 = 𝑃 ∪ 𝑃1 , where 𝑃1 = {𝑆 →
𝑞 | 𝑞 ∈ supp(𝐼), 𝜇𝜌 (𝑆 → 𝑞) = 𝜇𝐼 (𝑞), 𝜈𝜌 (𝑆 → 𝑞) = 𝜈𝐼 (𝑞)}.
Next we show that L(𝐺󸀠 ) = L(𝐺). In fact, 𝐺󸀠 =
󸀠
(𝑁 , 𝑇󸀠 , 𝑃󸀠 , 𝐼󸀠 ), where 𝐼󸀠 is an IFS over 𝑁󸀠 , 𝜇𝐼󸀠 (𝑆) = 1, 𝜈𝐼󸀠 (𝑆) =
0; 𝜇𝐼󸀠 (𝑞) = 0 and 𝜈𝐼󸀠 (𝑞) = 1 when 𝑞 ∈ 𝑁.
For any 𝜃 = 𝜔𝑛 ∈ 𝑇∗ , 𝑛 ≥ 1, 𝜇𝐺󸀠 (𝜃) = ⋁{𝜇𝐼󸀠 (𝜔0 )∧𝜇𝜌 (𝜔0 ⇒
𝜔1 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ) | 𝜔0 ∈ 𝑁󸀠 , 𝜔1 , . . . , 𝜔𝑛−1 ∈ (𝑁󸀠 ∪
𝑇󸀠 )∗ } = ⋁{𝜇𝜌 (𝑆 ⇒ 𝜔1 ) ∧ 𝜇𝜌 (𝜔1 ⇒ 𝜔2 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝜔𝑛−1 ⇒
𝜔𝑛 ) | 𝜔1 , . . . , 𝜔𝑛−1 ∈ (𝑁󸀠 ∪ 𝑇󸀠 )∗ } = ⋁{𝜇𝜌 (𝑆 ⇒ 𝑞) ∧ 𝜇𝜌 (𝑞 ⇒
𝜔2 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ) | 𝑞 ∈ 𝑁, 𝜔2 , . . . , 𝜔𝑛−1 ∈ (𝑁 ∪
𝑇)∗ } = ⋁{𝜇𝐼 (𝑞) ∧ 𝜇𝜌 (𝑞 ⇒ 𝜔2 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ) | 𝑞 ∈
𝑁, 𝜔2 , . . . , 𝜔𝑛−1 ∈ (𝑁∪𝑇)∗ } = 𝜇𝐺(𝜃) and 𝜈𝐺󸀠 (𝜃) = ⋀{𝜈𝐼󸀠 (𝜔0 )∨
𝜈𝜌 (𝜔0 ⇒ 𝜔1 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ) | 𝜔0 ∈ 𝑁󸀠 , 𝜔1 , . . . , 𝜔𝑛−1 ∈
(𝑁󸀠 ∪ 𝑇󸀠 )∗ } = ⋀{𝜈𝜌 (𝑆 ⇒ 𝜔1 ) ∨ 𝜈𝜌 (𝜔1 ⇒ 𝜔2 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝜔𝑛−1 ⇒
𝜔𝑛 ) | 𝜔1 , . . . , 𝜔𝑛−1 ∈ (𝑁󸀠 ∪ 𝑇󸀠 )∗ } = ⋀{𝜈𝜌 (𝑆 ⇒ 𝑞) ∨ 𝜈𝜌 (𝑞 ⇒
𝜔2 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ) | 𝑞 ∈ 𝑁, 𝜔2 , . . . , 𝜔𝑛−1 ∈ (𝑁 ∪ 𝑇)∗ } =
⋀{𝜈𝐼 (𝑞) ∨ 𝜈𝜌 (𝑞 ⇒ 𝜔2 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ) | 𝑞 ∈ 𝑁,
𝜔2 , . . . , 𝜔𝑛−1 ∈ (𝑁 ∪ 𝑇)∗ } = 𝜈𝐺(𝜃).
Hence L(𝐺󸀠 ) = L(𝐺).
(ii) implies (i), obviously.
Definition 27. (1) An intuitionistic fuzzy grammar 𝐺 = (𝑁,
𝑇, 𝑃, 𝐼) is called context-free (IFCFG, for short) if it has only
productions of the form 𝐴 → 𝜔 ∈ 𝑃 with 𝐴 ∈ 𝑁 and 𝜔 ∈
(𝑁 ∪ 𝑇)∗ . And the language L(𝐺), generated by the IFCFG
𝐺, is said to be an intuitionistic fuzzy context-free language
(IFCFL).
(2) An IFCFG 𝐺 = (𝑁, 𝑇, 𝑃, 𝑆) is called an intuitionistic
fuzzy Chomsky normal form (IFCNF) if it has only productions of the form 𝐴 → 𝐵𝐶 ∈ 𝑃 or 𝐴 → 𝑎 ∈ 𝑃 or 𝑆 → 𝜀,
where 𝐴, 𝐵, 𝐶 ∈ 𝑁, 𝐵 ≠ 𝑆, 𝐶 ≠ 𝑆 and 𝑎 ∈ 𝑇. (3) An IFCFG
𝐺 = (𝑁, 𝑇, 𝑃, 𝑆) is called an intuitionistic fuzzy Greibach
normal form (IFGNF) if all the productions are of the form
𝐴 → 𝑎𝑥 ∈ 𝑃 or 𝑆 → 𝜀, where 𝐴 ∈ 𝑁, 𝑎 ∈ 𝑇, and 𝑥 ∈
(𝑁 \ {𝑆})∗ . (4) An IFCFG 𝐺 = (𝑁, 𝑇, 𝑃, 𝐼) is called a simpletyped intuitionistic fuzzy context-free grammar (IFSCFG) if
𝑃 = {𝐴 󳨀→ 𝑥 | 𝐴 ∈ 𝑁, 𝑥 ∈ (𝑁 ∪ 𝑇)∗ , 𝜇𝜌 (𝐴 󳨀→ 𝑥) = 1} .
(25)
Proposition 28. Let 𝐺 = (𝑁, 𝑇, 𝑃, 𝐼) be an IFCFG and
L(𝐺) = (𝜇𝐺, 𝜈𝐺) be the intuitionistic fuzzy context-free
language generated by 𝐺. Then the image set of L(𝐺) is a finite
subset of the unit interval [0, 1].
Proof. Let 𝐺 = (𝑁, 𝑇, 𝑃, 𝐼) be an IFCFG. Then Im(𝜇𝜌 ) and
Im(𝜈𝜌 ) are finite. For any 𝜃 = 𝜔𝑛 ∈ 𝑇∗ , 𝑛 ≥ 1, there exist
12
Journal of Applied Mathematics
natural numbers 𝑑, 𝑙 such that 𝜇𝐺(𝜃) = ⋁{𝜇𝐼 (𝑆) ∧ 𝜇𝜌 (𝑆 ⇒
𝜔1 ) ∧ 𝜇𝜌 (𝜔1 ⇒ 𝜔2 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ) | 𝑆 ∈ 𝑁,
𝜔1 , . . . , 𝜔𝑛−1 ∈ (𝑁 ∪ 𝑇)∗ } = (𝑎10 ∧ ⋅ ⋅ ⋅∧ 𝑎1𝑛−1 ∧ 𝑎1𝑛 ) ∨ ⋅ ⋅ ⋅ ∨ (𝑎𝑑0 ∧
⋅ ⋅ ⋅∧𝑎𝑑𝑛−1 ∧𝑎𝑑𝑛 ) and 𝜈𝐺(𝜃) = ⋀{𝜈𝐼 (𝑆)∨𝜈𝜌 (𝑆 ⇒ 𝜔1 )∨𝜈𝜌 (𝜔1 ⇒
𝜔2 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝜔𝑛−1 ⇒ 𝜔𝑛 ) | 𝑆 ∈ 𝑁, 𝜔1 , . . . , 𝜔𝑛−1 ∈ (𝑁 ∪ 𝑇)∗ } =
(𝑏10 ∨ ⋅ ⋅ ⋅ ∨ 𝑏1𝑛−1 ∨ 𝑏1𝑛 ) ∧ ⋅ ⋅ ⋅ ∧ (𝑏𝑙0 ∨ ⋅ ⋅ ⋅ ∨ 𝑏𝑙𝑛−1 ∨ 𝑏𝑙𝑛 ), where
𝑎𝑖𝑘 ∈ Im(𝜇𝐼 ) ∪ Im(𝜇𝜌 ), 𝑏𝑗𝑘 ∈ Im(𝜈𝐼 ) ∪ Im(𝜈𝜌 ), 𝑖 ∈ {1, . . . , 𝑑},
𝑗 ∈ {1, . . . , 𝑙}, 𝑘 ∈ {0, 1, . . . , 𝑛}. Let 𝑋 = Im(𝜇𝐼 ) ∪ Im(𝜇𝜌 ) and
𝑌 = Im(𝜈𝐼 ) ∪ Im(𝜈𝜌 ). Then 𝑋 and 𝑌 are finite subsets of the
interval [0, 1]. (𝑋∧ )∨ and (𝑌∨ )∧ are also finite by Lemma 8.
Since 𝜇𝐺(𝜃) ∈ (𝑋∧ )∨ and 𝜈𝐺(𝜃) ∈ (𝑌∨ )∧ for any 𝜃 ∈ 𝑇∗ ,
we have Im(𝜇𝐺) ⊆ (𝑋∧ )∨ and Im(𝜈𝐺) ⊆ (𝑌∨ )∧ . Hence
Im(L(𝐺)) = Im(𝜇𝐺) ∪ Im(𝜈𝐺) is finite.
Proposition 29. Let 𝐺 = (𝑁, 𝑇, 𝑃, 𝑆) be an IFCFG. Then
L(𝐺) ∈ Step𝐶(𝑇).
Proof. Let 𝑋 = Im(𝜇𝜌 ) and 𝑌 = Im(𝜈𝜌 ). Then 𝑋 and 𝑌 have
finite elements because 𝑃 is a finite collection of productions
over 𝑇 ∪ 𝑁. Suppose 𝐿 1 = 𝑋∧ and 𝐿 2 = 𝑌∨ . Then 𝐿 1 and 𝐿 2
are finite by Lemma 8. For any (𝑎, 𝑏) ∈ (𝐿 1 \ {0}) × (𝐿 2 \ {1}),
0 ≤ 𝑎 + 𝑏 ≤ 1, we construct a classical context-free grammar
󸀠
, 𝑆󸀠 ) as follows:
𝐺𝑎𝑏 = (𝑁󸀠 , 𝑇, 𝑃𝑎𝑏
󸀠
󸀠
𝑁 = 𝑁 × (𝐿 1 \ {0}) × (𝐿 2 \ {1}), 𝑆󸀠 = (𝑆, 1, 0) ∈ 𝑁󸀠 , 𝑃𝑎𝑏
consists of the form:
→
𝐷1 ⋅ ⋅ ⋅ 𝐷𝑘 whenever 𝜇𝜌 (𝐴
(1) (𝐴, 𝑎1 , 𝑏1 )
𝜏1 ⋅ ⋅ ⋅ 𝜏𝑘 ) > 0 and 𝜈𝜌 (𝐴 → 𝜏1 ⋅ ⋅ ⋅ 𝜏𝑘 ) < 1, where
𝐷𝑖 = {
(𝜏𝑖 , 𝑎2 , 𝑏2 ) ,
𝜏𝑖 ,
if 𝜏𝑖 ∈ 𝑁
if 𝜏𝑖 ∈ 𝑇
→
(26)
𝑖 = 1, . . . , 𝑘; 𝑎2 = 𝑎1 ∧ 𝜇𝜌 (𝐴 → 𝜏1 ⋅ ⋅ ⋅ 𝜏𝑘 ) and 𝑏2 =
𝑏1 ∨ 𝜈𝜌 (𝐴 → 𝜏1 ⋅ ⋅ ⋅ 𝜏𝑘 );
(2) (𝐴, 𝑎1 , 𝑏1 ) → 𝑥 whenever 𝑎 ≤ 𝑎1 ∧ 𝜇𝜌 (𝐴 → 𝑥) and
𝑏 ≥ 𝑏1 ∨ 𝜈𝜌 (𝐴 → 𝑥), for all 𝐴 ∈ 𝑁, 𝑥 ∈ (𝑁 ∪ 𝑇)∗ .
Then L(𝐺𝑎𝑏 ) = {𝜔 ∈ 𝑇∗ | 𝑆󸀠 ⇒∗𝐺𝑎𝑏 𝜔} = {𝜔 ∈ 𝑇∗ | 𝑎 ≤
𝜇𝜌 (𝑆 ⇒ 𝑢1 ) ∧ 𝜇𝜌 (𝑢1 ⇒ 𝑢2 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝑢𝑛−1 ⇒ 𝜔), 𝑏 ≥ 𝜈𝜌 (𝑆 ⇒
𝑢1 ) ∨ 𝜈𝜌 (𝑢1 ⇒ 𝑢2 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝑢𝑛−1 ⇒ 𝜔), 𝑢1 , . . . , 𝑢𝑛−1 ∈ (𝑁 ∪
𝑇)∗ }.
Next it suffices to show that L(𝐺) = ∐(𝑎, 𝑏) ⋅ 1L(𝐺𝑎𝑏 ) , that
is, 𝜇𝐺(𝜔) = ⋁𝑎∈𝐿 1 \{0} 𝑎 ∧ 𝜇1L(𝐺 ) (𝜔) and 𝜈𝐺(𝜔) = ⋀𝑏∈𝐿 2 \{1} 𝑏 ∨
𝑎𝑏
𝜈1L(𝐺 ) (𝜔), for all 𝜔 ∈ 𝑇∗ .
𝑎𝑏
󸀠
Suppose 𝜇𝐺(𝜔) = 𝑎𝑘 > 0. Then there exist 𝑢1󸀠 , . . . , 𝑢𝑛−1
∈
∗
󸀠
󸀠
(𝑁 ∪ 𝑇) such that 𝑎𝑘 = 𝜇𝜌 (𝑆 ⇒ 𝑢1 ) ∧ 𝜇𝜌 (𝑢1 ⇒ 𝑢2󸀠 ) ∧
󸀠
⇒ 𝜔). Put 𝑐 = 𝜈𝜌 (𝑆 ⇒ 𝑢1󸀠 ) ∨ 𝜈𝜌 (𝑢1󸀠 ⇒ 𝑢2󸀠 ) ∨
⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝑢𝑛−1
󸀠
⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝑢𝑛−1
⇒ 𝜔). Then 𝑐 < 1 and 𝜔 ∈ L(𝐺𝑘𝑐 ).
Therefore, 𝜇𝐺(𝜔) ≤ ⋁𝑎∈𝐿 1 \{0} 𝑎 ∧ 𝜇1L(𝐺 ) (𝜔) and 𝜈𝐺(𝜔) ≥
𝑎𝑏
⋀𝑏∈𝐿 2 \{1} 𝑏 ∨ 𝜈1L(𝐺 ) (𝜔) for any 𝜔 ∈ 𝑇∗ . In addition,
𝑎𝑏
⋁𝑎∈𝐿 1 \{0} 𝑎 ∧ 𝜇1L(𝐺 ) (𝜔) = ⋁{𝑎 ∈ 𝐿 1 \ {0} | 𝜔 ∈ L(𝐺𝑎𝑏 )} ≤
𝑎𝑏
⋁{𝜇𝜌 (𝑆 ⇒ 𝑢1 ) ∧ 𝜇𝜌 (𝑢1 ⇒ 𝑢2 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝑢𝑛−1 ⇒ 𝜔) | 𝜔 ∈
L(𝐺𝑎𝑏 ), 𝑢1 , . . . , 𝑢𝑛−1 ∈ (𝑁∪𝑇)∗ } ≤ ⋁{𝜇𝜌 (𝑆 ⇒ 𝑢1 )∧𝜇𝜌 (𝑢1 ⇒
𝑢2 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝑢𝑛−1 ⇒ 𝜔) | 𝑢1 , . . . , 𝑢𝑛−1 ∈ (𝑁 ∪ 𝑇)∗ } = 𝜇𝐺(𝜔),
𝜈𝐺(𝜔) = ⋀{𝜈𝜌 (𝑆 ⇒ 𝑢1 ) ∨ 𝜈𝜌 (𝑢1 ⇒ 𝑢2 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝑢𝑛−1 ⇒
𝜔) | 𝑢1 , . . . , 𝑢𝑛−1 ∈ (𝑁 ∪ 𝑇)∗ } ≤ ⋀{𝑏 ∈ 𝐿 2 \ {1} | 𝑏 ≥
𝜈𝜌 (𝑆 ⇒ 𝑢1 )∨𝜈𝜌 (𝑢1 ⇒ 𝑢2 )∨⋅ ⋅ ⋅∨𝜈𝜌 (𝑢𝑛−1 ⇒ 𝜔), 𝑢1 , . . . , 𝑢𝑛−1 ∈
(𝑁 ∪ 𝑇)∗ } ≤ ⋀{𝑏 ∈ 𝐿 2 \ {1} | 𝑏 ≥ 𝜈𝜌 (𝑆 ⇒ 𝑢1 ) ∨ 𝜈𝜌 (𝑢1 ⇒
𝑢2 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝑢𝑛−1 ⇒ 𝜔), 𝑎 ≤ 𝜇𝜌 (𝑆 ⇒ 𝑢1 ) ∧ 𝜇𝜌 (𝑢1 ⇒
𝑢2 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝑢𝑛−1 ⇒ 𝜔), 𝑢1 , . . . , 𝑢𝑛−1 ∈ (𝑁 ∪ 𝑇)∗ } = ⋀{𝑏 ∈
𝐿 2 \ {1} | 𝜔 ∈ L(𝐺𝑎𝑏 )} = ⋀𝑏∈𝐿 2 \{1} 𝑏 ∨ 𝜈1L(𝐺 ) (𝜔).
𝑎𝑏
It is concluded that 𝜇𝐺(𝜔) = ⋁𝑎∈𝐿 1 \{0} 𝑎 ∧ 𝜇1L(𝐺 ) (𝜔) and
𝑎𝑏
𝜈𝐺(𝜔) = ⋀𝑏∈𝐿 2 \{1} 𝑏 ∨ 𝜈1L(𝐺 ) (𝜔), for all 𝜔 ∈ 𝑇∗ .
𝑎𝑏
Proposition 29 states that any language recognized by an
IFCFG is an intuitionistic fuzzy recognizable step function,
where the proof technique is constructive. Next we will
show that the set consisting of all the intuitionistic fuzzy
recognizable step functions coincides with that of languages
recognized by IFCFGs.
Theorem 30. Let 𝐴 be an IFS over 𝑇∗ . Then the following
statements are equivalent:
(1) 𝐴 ∈ Step𝐶(𝑇);
(2) There is an IFCFG 𝐺 such that 𝐴 = L(𝐺);
(3) There is an IFSCFG 𝐺 such that 𝐴 = L(𝐺).
Proof. (1) implies (3). Let 𝐴 = ∐𝑘𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1L𝑖 , where 𝑎𝑖 , 𝑏𝑖 ∈
[0, 1], 𝑎𝑖 + 𝑏𝑖 ≤ 1, 𝑖 ∈ 𝑁𝑘 , and L1 , . . . , L𝑘 ⊂ 𝑇∗ are
classical context-free languages. Suppose L𝑖 is generated by
a context-free grammar 𝐺𝑖 = (𝑁𝑖 , 𝑇, 𝑃𝑖 , 𝑆0𝑖 ) and 𝑁𝑖 ∩ 𝑁𝑗 = 0
whenever 𝑖 ≠ 𝑗. Then we construct an IFSCFG 𝐺 = (𝑁, 𝑇, 𝑃, 𝐼)
as follows: 𝑁 = ⋃𝑘𝑖=1 𝑁𝑖 , 𝑃 = ⋃𝑘𝑖=1 𝑃𝑖 , 𝐼 = (𝜇𝐼 , 𝜈𝐼 ) is an IFS
over 𝑁, where the mappings 𝜇𝐼 , 𝜈𝐼 : 𝑁 → [0, 1] are defined
by 𝜇𝐼 (𝑆0𝑖 ) = 𝑎𝑖 , 𝜈𝐼 (𝑆0𝑖 ) = 𝑏𝑖 ; 𝜇𝐼 (𝑞) = 0 and 𝜈𝐼 (𝑞) = 1 whenever
𝑞 ∈ 𝑁 \ {𝑆0𝑖 | 𝑖 = 1, . . . , 𝑘}.
Next, we show 𝐴 = L(𝐺) = ∐𝑘𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1L𝑖 . In fact, for
any 𝜔 = 𝑢𝑛 ∈ 𝑇∗ , 𝜇𝐺(𝜔) = ⋁{𝜇𝐼 (𝑆) ∧ 𝜇𝜌 (𝑆 ⇒ 𝑢1 ) ∧ 𝜇𝜌 (𝑢1 ⇒
𝑢2 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝑢𝑛−1 ⇒ 𝜔) | 𝑆 ∈ 𝑁, 𝑢1 , . . . , 𝑢𝑛−1 ∈ (𝑁 ∪ 𝑇)∗ } =
⋁{𝜇𝐼 (𝑆0𝑖 ) | 𝑆0𝑖 ⇒∗𝐺𝑖 𝜔, 𝑆0𝑖 ∈ 𝑁, 𝑖 ∈ 𝑁𝑘 } = ⋁{𝑎𝑖 | 𝜔 ∈ L𝑖 , 𝑖 ∈
𝑁𝑘 } = ⋁𝑘𝑖=1 𝑎𝑖 ∧ 𝜇1L (𝜔) and 𝜈𝐺(𝜔) = ⋀{𝜈𝐼 (𝑆) ∨ 𝜈𝜌 (𝑆 ⇒ 𝑢1 ) ∨
𝑖
𝜈𝜌 (𝑢1 ⇒ 𝑢2 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝑢𝑛−1 ⇒ 𝜔) | 𝑆 ∈ 𝑁, 𝑢1 , . . . , 𝑢𝑛−1 ∈
(𝑁 ∪ 𝑇)∗ } = ⋀{𝜈𝐼 (𝑆0𝑖 ) | 𝑆0𝑖 ⇒∗𝐺𝑖 𝜔, 𝑆0𝑖 ∈ 𝑁, 𝑖 ∈ 𝑁𝑘 } = ⋀{𝑏𝑖 |
𝜔 ∈ L𝑖 , 𝑖 ∈ 𝑁𝑘 } = ⋀𝑘𝑖=1 𝑏𝑖 ∨ 𝜈1L (𝜔).
𝑖
Hence L(𝐺) = (𝜇𝐺, 𝜈𝐺) = ∐𝑘𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1L𝑖 = 𝐴.
(3) implies (2). It is true obviously since an IFSCFG is a
special IFCFG by Definition 27.
(2) implies (1). It is straightforward by Propositions 26
and 29.
Theorem 31. Let 𝐴 be an IFS over Σ∗ . Then the following
statements are equivalent:
(1) 𝐴 ∈ Step𝐶(Σ);
(2) there is an IFPDA M such that 𝐴 = L(M);
(3) there is an IFPDA0 M such that 𝐴 = L(M);
(4) there is an IFCFG 𝐺 such that 𝐴 = L(𝐺);
(5) there is an IFCNF 𝐺 such that 𝐴 = L(𝐺);
(6) there is an IFGNF 𝐺 such that 𝐴 = L(𝐺).
Journal of Applied Mathematics
Proof. (1) implies (5). Let 𝐴 ∈ Step𝐶(Σ). Then suppose 𝐴 =
∐𝑘𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1L𝑖 , where 𝑎𝑖 , 𝑏𝑖 ∈ [0, 1], 𝑎𝑖 + 𝑏𝑖 ≤ 1, 𝑖 ∈ 𝑁𝑘 , and
L1 , . . . , L𝑘 ⊂ Σ∗ are classical context-free languages.
If supp(𝐴) = {𝜀}, then we construct an IFCNF 𝐺 = (𝑁,
Σ, 𝑃, 𝑆) as follows: 𝑁 = {𝑆}, 𝑃 = {𝑆 → 𝜀 | 𝜇𝜌 (𝑆 → 𝜀) =
𝜇𝐴 (𝜀), 𝜈𝜌 (𝑆 → 𝜀) = 𝜇𝐴 (𝜀)}. Clearly, L(𝐺) = 𝐴.
If supp(𝐴) \ {𝜀} ≠ 0, then there is a Chomsky normal form
grammar 𝐺𝑖 = (𝑁𝑖 , Σ, 𝑃𝑖 , 𝑆0𝑖 ) such that L(𝐺𝑖 ) = L𝑖 \ {𝜀},
for any L𝑖 with L𝑖 \ {𝜀} ≠ 0, 𝑖 = 1, . . . , 𝑘. Let 𝑁𝑖 ∩ 𝑁𝑗 = 0
whenever 𝑖 ≠ 𝑗. Then we construct an IFSCFG 𝐺 = (𝑁, Σ, 𝑃, 𝐼)
according to the method constructed by Theorem 30, where
𝑁 = ⋃𝑘𝑖=1 𝑁𝑖 , 𝑃 = ⋃𝑘𝑖=1 𝑃𝑖 , 𝐼 = (𝜇𝐼 , 𝜈𝐼 ) is an IFS over 𝑁, and
the mappings 𝜇𝐼 , 𝜈𝐼 : 𝑁 → [0, 1] are defined by 𝜇𝐼 (𝑆0𝑖 ) = 𝑎𝑖 ,
𝜈𝐼 (𝑆0𝑖 ) = 𝑏𝑖 ; 𝜇𝐼 (𝑞) = 0 and 𝜈𝐼 (𝑞) = 1 whenever 𝑞 ∈ 𝑁 \ {𝑆0𝑖 |
𝑖 = 1, . . . , 𝑘}. Next, we construct an IFCNF 𝐺󸀠 = (𝑁󸀠 , Σ, 𝑃󸀠 , 𝑆)
as follows:
(i) 𝑁󸀠 = 𝑁 ∪ {𝑆}, 𝑆 ∉ 𝑁;
(ii) 𝑃󸀠 = 𝑃 ∪ 𝑃󸀠󸀠 , where 𝑃󸀠󸀠 has the productions in the
form of
(𝐸1) 𝑆 → 𝜀 with 𝜇𝜌 (𝑆 → 𝜀) = 𝜇𝐴 (𝜀), 𝜈𝜌 (𝑆 → 𝜀) =
𝜇𝐴 (𝜀) whenever 𝜀 ∈ supp(𝐴);
(𝐸2) 𝑆 → 𝐵𝐶 with 𝜇𝜌 (𝑆 → 𝐵𝐶) = 𝜇𝐼 (𝑆0𝑖 ) and
𝜈𝜌 (𝑆 → 𝐵𝐶) = 𝜈𝐼 (𝑆0𝑖 ) whenever 𝑆0𝑖 ∈ supp(𝐼)
and 𝑆0𝑖 → 𝐵𝐶 ∈ 𝑃𝑖 , 𝑖 = 1, . . . , 𝑘;
(𝐸3) 𝑆 → 𝛼 with 𝜇𝜌 (𝑆 → 𝛼) = ⋁{𝜇𝐼 (𝑆0𝑖 ) | 𝑆0𝑖 ∈
supp(𝐼), 𝑆0𝑖 → 𝛼 ∈ 𝑃𝑖 , 𝑖 = 1, . . . , 𝑘} and 𝜈𝜌 (𝑆 →
𝛼) = ⋀{𝜈𝐼 (𝑆0𝑖 ) | 𝑆0𝑖 ∈ supp(𝐼), 𝑆0𝑖 → 𝛼 ∈ 𝑃𝑖 , 𝑖 =
1, . . . , 𝑘} whenever 𝑆0𝑖 ∈ supp(𝐼) and 𝑆0𝑖 → 𝛼 ∈
𝑃𝑖 , 𝑖 = 1, . . . , 𝑘.
Then we have L(𝐺󸀠 ) = (𝜇𝐺󸀠 , 𝜈𝐺󸀠 ) = 𝐴. In fact, for any 𝜔 ∈
Σ , if 𝜔 = 𝜀, then 𝜇𝐺󸀠 (𝜀) = 𝜇𝜌 (𝑆 → 𝜀) = 𝜇𝐴 (𝜀) and 𝜈𝐺󸀠 (𝜀) =
𝜈𝜌 (𝑆 → 𝜀) = 𝜈𝐴(𝜀); if 𝜔 ≠ 𝜀, then 𝜇𝐺󸀠 (𝜔) = ⋁{𝜇𝜌 (𝑆⇒𝐺󸀠 𝑢1 ) ∧
𝜇𝜌 (𝑢1 ⇒𝐺󸀠 𝑢2 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝜌 (𝑢𝑛−1 ⇒𝐺󸀠 𝜔) | 𝑢1 , 𝑢2 , . . . , 𝑢𝑛−1 ∈ (𝑁󸀠 ∪
Σ)∗ } = ⋁{𝜇𝐼 (𝑆0𝑖 ) ∧ 𝜇𝜌 (𝑆0𝑖 ⇒𝐺𝑖 𝑢1 ) ∧ 𝜇𝜌 (𝑢1 ⇒𝐺𝑖 𝑢2 ) ∧ ⋅ ⋅ ⋅ ∧
𝜇𝜌 (𝑢𝑛−1 ⇒𝐺𝑖 𝜔) | 𝑢1 , 𝑢2 , . . . , 𝑢𝑛−1 ∈ (𝑁 ∪ Σ)∗ , 𝑖 ∈ 𝑁𝑘 } =
∗
⋁{𝑎𝑖 | 𝜔 ∈ L(𝐺𝑖 ), 𝑖 ∈ 𝑁𝑘 } = ⋁𝑘𝑖=1 𝑎𝑖 ∧ 𝜇1L (𝜔) and 𝜈𝐺󸀠 (𝜔) =
𝑖
⋀{𝜈𝜌 (𝑆⇒𝐺󸀠 𝑢1 ) ∨ 𝜈𝜌 (𝑢1 ⇒𝐺󸀠 𝑢2 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝑢𝑛−1 ⇒𝐺󸀠 𝜔) |
𝑢1 , 𝑢2 , . . . , 𝑢𝑛−1 ∈ (𝑁󸀠 ∪ Σ)∗ } = ⋀{𝜈𝐼 (𝑆0𝑖 ) ∨ 𝜈𝜌 (𝑆0𝑖 ⇒𝐺𝑖 𝑢1 ) ∨
𝜈𝜌 (𝑢1 ⇒𝐺𝑖 𝑢2 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝜌 (𝑢𝑛−1 ⇒𝐺𝑖 𝜔) | 𝑢1 , 𝑢2 , . . . , 𝑢𝑛−1 ∈ (𝑁 ∪
Σ)∗ , 𝑖 ∈ 𝑁𝑘 } = ⋀{𝑎𝑖 | 𝜔 ∈ L(𝐺𝑖 ), 𝑖 ∈ 𝑁𝑘 } = ⋀𝑘𝑖=1 𝑎𝑖 ∨ 𝜇1L (𝜔).
𝑖
(1) implies (6), similarly. The proof is omitted.
(5) implies (4), (6) implies (4), obviously, since IFCNF
and IFGNF are special IFCFGs respectively.
(1), (2), (3), and (4) are equivalent mutually by Theorems
19, 23, and 30.
Theorem 31 states that IFCFLs, the set of intuitionistic
fuzzy languages recognized by IFPDA and the set of intuitionistic fuzzy recognizable step functions, coincide with
each other. Next we discuss some operations on the family
of IFCFLs.
Let 𝐴 = (𝜇𝐴 , 𝜈𝐴) and 𝐵 = (𝜇𝐵 , 𝜈𝐵 ) be IFSs over Σ∗ ,
𝜆, 𝜃 ∈ [0, 1], and 0 ≤ 𝜆 + 𝜃 ≤ 1. Then the operations of union,
scalar product, reversal, concatenation and Kleene closure are
defined, respectively, by
13
(i) 𝐴 ∪ 𝐵 = (𝜇𝐴∪𝐵 , 𝜈𝐴∪𝐵 ), 𝜇𝐴∪𝐵 (𝜔) = 𝜇𝐴 (𝜔) ∨ 𝜇𝐵 (𝜔),
𝜈𝐴∪𝐵 (𝜔) = 𝜈𝐴 (𝜔) ∧ 𝜈𝐵 (𝜔);
(ii) (𝜆, 𝜃)𝐴 = (𝜆 ∧ 𝜇𝐴 , 𝜃 ∨ 𝜈𝐴), (𝜆 ∧ 𝜇𝐴 )(𝜔) = 𝜆 ∧ 𝜇𝐴 (𝜔),
(𝜃 ∨ 𝜈𝐴 )(𝜔) = 𝜃 ∨ 𝜈𝐴(𝜔);
(iii) 𝐴−1 = (𝜇𝐴−1 , 𝜈𝐴−1 ), 𝜇𝐴−1 (𝜔) = 𝜇𝐴 (𝜔−1 ), 𝜈𝐴−1 (𝜔) =
𝜈𝐴 (𝜔−1 );
(iv) 𝐴𝐵 = (𝜇𝐴𝐵 , 𝜈𝐴𝐵 ), 𝜇𝐴𝐵 (𝜔) = ⋁{𝜇𝐴 (𝜔1 ) ∧ 𝜇𝐵 (𝜔2 ) |
𝜔1 𝜔2 = 𝜔}, 𝜈𝐴𝐵 (𝜔) = ⋀{𝜈𝐴 (𝜔1 ) ∨ 𝜈𝐵 (𝜔2 ) | 𝜔1 𝜔2 = 𝜔};
(v) 𝐴∗ = (𝜇𝐴∗ , 𝜈𝐴∗ ), 𝜇𝐴∗ (𝜔) = ⋁{𝜇𝐴 (𝜔1 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝐴 (𝜔𝑘 ) :
𝑘 ≥ 1, 𝜔 = 𝜔1 ⋅ ⋅ ⋅ 𝜔𝑘 }, 𝜈𝐴∗ (𝜔) = ⋀{𝜈𝐴 (𝜔1 ) ∨ ⋅ ⋅ ⋅ ∨
𝜈𝐴 (𝜔𝑘 ) : 𝑘 ≥ 1, 𝜔 = 𝜔1 ⋅ ⋅ ⋅ 𝜔𝑘 } for any 𝜔 ∈ Σ∗ ,
where 𝜔−1 represents the reversal of 𝜔, that is, if 𝜔 =
𝜔1 ⋅ ⋅ ⋅ 𝜔𝑘 , then 𝜔−1 = 𝜔𝑘 ⋅ ⋅ ⋅ 𝜔1 , for all 𝜔 ∈ Σ∗ .
Theorem 32. (1) The family Step𝐶(Σ) is closed under the
operations of union, scalar product, reversal, concatenation,
and Kleene closure. That is, 𝐴 ∪ 𝐵, (𝜆, 𝜃)𝐴, 𝐴−1 , 𝐴𝐵, 𝐴∗ ∈
Step𝐶(Σ), for any 𝐴, 𝐵 ∈ Step𝐶(Σ), 𝜆, 𝜃 ∈ [0, 1], 0 ≤ 𝜆 + 𝜃 ≤ 1.
(2) Let ℎ : Σ∗1 → Σ∗2 be a homomorphism. If 𝐴 ∈
Step𝐶(Σ2 ), then ℎ−1 (𝐴) = 𝐴 ∘ ℎ ∈ Step𝐶(Σ1 ). (3) Let ℎ : Σ∗1 →
Σ∗2 be a homomorphism. If ℎ satisfies, for 𝜏 ∈ Σ, ℎ(𝜏) ≠ 𝜀,
and 𝑔 = (𝜇𝑔 , 𝜈𝑔 ) ∈ Step𝐶(Σ1 ), then ℎ(𝑔) = (𝜇ℎ(𝑔) , 𝜈ℎ(𝑔) ) ∈
Step𝐶(Σ2 ), where 𝜇ℎ(𝑔) (𝜔) = ⋁{𝜇𝑔 (𝛼) | ℎ(𝛼) = 𝜔, 𝛼 ∈ Σ∗1 },
𝜈ℎ(𝑔) (𝜔) = ⋀{𝜈𝑔 (𝛼) | ℎ(𝛼) = 𝜔, 𝛼 ∈ Σ∗1 } for any 𝜔 ∈ Σ∗2 .
Proof. (1) Let 𝐴, 𝐵 ∈ Step𝐶(Σ). By Definition 17, we can
assume 𝐴 = (𝜇𝐴 , 𝜈𝐴 ) = ∐𝑘𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1L𝑖 , 𝐵 = (𝜇𝐵 , 𝜈𝐵 ) =
∐𝑛𝑗=1 (𝑐𝑗 , 𝑑𝑗 ) ⋅ 1M𝑗 , where all L𝑖 and M𝑗 are classical contextfree languages, 0 ≤ 𝑎𝑖 + 𝑏𝑖 ≤ 1, 0 ≤ 𝑐𝑗 + 𝑑𝑗 ≤ 1, 𝑎𝑖 , 𝑏𝑖 , 𝑐𝑗 , 𝑑𝑗 ∈
[0, 1], 𝑖 ∈ 𝑁𝑘 , and 𝑗 ∈ 𝑁𝑛 . With respect to the union, we
have 𝐴 ∪ 𝐵 ∈ Step𝐶(Σ). That is, 𝐴 ∪ 𝐵 = (𝜇𝐴∪𝐵 , 𝜈𝐴∪𝐵 ) =
(𝜇𝐴 ∨ 𝜇𝐵 , 𝜈𝐴 ∧ 𝜈𝐵 ), 𝜇𝐴∪𝐵 (𝜔) = 𝜇𝐴 (𝜔) ∨ 𝜇𝐵 (𝜔) = (⋁𝑘𝑖=1 𝑎𝑖 ∧
𝜇1L (𝜔)) ∨ (⋁𝑛𝑗=1 𝑐𝑗 ∧ 𝜇1M (𝜔)), 𝜈𝐴∪𝐵 (𝜔) = 𝜈𝐴(𝜔) ∧ 𝜈𝐵 (𝜔) =
𝑖
𝑗
(⋀𝑘𝑖=1 𝑏𝑖 ∨ 𝜈1L (𝜔)) ∧ (⋀𝑛𝑗=1 𝑑𝑗 ∨ 𝜈1M (𝜔)), for all 𝜔 ∈ Σ∗ .
𝑖
𝑗
With respect to the scalar product, for each (𝜆, 𝜃) ∈
[0, 1]×[0, 1], 0 ≤ 𝜆+𝜃 ≤ 1, we have (𝜆, 𝜃)𝐴 = (𝜆∧𝜇𝐴, 𝜃∨𝜈𝐴 ),
(𝜆 ∧ 𝜇𝐴)(𝜔) = 𝜆 ∧ 𝜇𝐴 (𝜔) = 𝜆 ∧ (⋁𝑘𝑖=1 𝑎𝑖 ∧ 𝜇1L (𝜔)) =
𝑖
⋁𝑘𝑖=1 ((𝜆 ∧ 𝑎𝑖 ) ∧ 𝜇1L (𝜔)), (𝜃 ∨ 𝜈𝐴)(𝜔) = 𝜃 ∨ 𝜈𝐴(𝜔) = 𝜃 ∨
𝑖
(⋀𝑘𝑖=1 𝑏𝑖 ∨ 𝜈1L (𝜔)) = ⋀𝑘𝑖=1 (𝜃 ∨ 𝑏𝑖 ) ∨ 𝜈1L (𝜔), for all 𝜔 ∈ Σ∗ .
𝑖
𝑖
By Definition 17, (𝜆, 𝜃)𝐴 ∈ Step𝐶(Σ). For the reversal
−1
operation, L−1
∈ Σ∗ | 𝜔 ∈ L𝑖 }, and L−1
𝑖 = {𝜔
𝑖 is a context-free language because L𝑖 is a context-free language,
𝑖 ∈ 𝑁𝑘 . For any 𝜔 ∈ Σ∗ , 𝜇𝐴−1 (𝜔) = 𝜇𝐴 (𝜔−1 ) = ⋁𝑘𝑖=1 𝑎𝑖 ∧
𝜇1L (𝜔−1 ) = ⋁𝑘𝑖=1 𝑎𝑖 ∧ 𝜇1L−1 (𝜔),𝜈𝐴−1 (𝜔) = 𝜈𝐴(𝜔−1 ) = ⋀𝑘𝑖=1 𝑏𝑖 ∨
𝑖
𝑖
𝜈1L (𝜔−1 ) = ⋀𝑘𝑖=1 𝑏𝑖 ∨ 𝜈1L−1 (𝜔), that is 𝐴−1 ∈ Step𝐶(Σ). For the
𝑖
𝑖
operation of concatenation, since 𝜇𝐴𝐵 (𝜔) = ⋁𝑘𝑖=1 ⋁𝑛𝑗=1 (𝑎𝑖 ∧
𝑐𝑗 ) ∧ 𝜇1L M (𝜔), 𝜈𝐴𝐵 (𝜔) = ⋀𝑘𝑖=1 ⋀𝑛𝑗=1 (𝑏𝑖 ∨ 𝑑𝑗 ) ∨ 𝜈1L M (𝜔), for
𝑖
𝑗
𝑖
𝑗
all 𝜔 ∈ Σ∗ , where the elements of the family set {L𝑖 M𝑗 | 𝑖 ∈
𝑁𝑘 , 𝑗 ∈ 𝑁𝑛 } are also context-free languages since L𝑖 and M𝑗
are context-free languages. Hence 𝐴𝐵 ∈ Step𝐶(Σ).
14
Journal of Applied Mathematics
For the Kleene closure, 𝐴∗ = (𝜇𝐴∗ , 𝜈𝐴∗ ) is defined by
𝜇𝐴∗ (𝜔) = ⋁{𝜇𝐴 (𝜔1 ) ∧ ⋅ ⋅ ⋅ ∧ 𝜇𝐴 (𝜔𝑘 ) : 𝑘 ≥ 1, 𝜔 = 𝜔1 ⋅ ⋅ ⋅ 𝜔𝑘 },
𝜈𝐴∗ (𝜔) = ⋀{𝜈𝐴 (𝜔1 ) ∨ ⋅ ⋅ ⋅ ∨ 𝜈𝐴(𝜔𝑘 ) : 𝑘 ≥ 1, 𝜔 = 𝜔1 ⋅ ⋅ ⋅ 𝜔𝑘 } for
any 𝜔 ∈ Σ∗ . Since 𝐴 ∈ Step𝐶(Σ), we assume that the IFSPDA
M = (𝑄, Σ, Γ, 𝛿, 𝑞0 , 𝑍0 , 𝐹) accepts 𝐴 by Theorem 19. Let 𝑅 =
{(𝜇𝐹 (𝑞), 𝜈𝐹 (𝑞)) | 𝑞 ∈ 𝑄} \ {(0, 1)} = {(𝑎𝑖 , 𝑏𝑖 ) | 𝑖 ∈ 𝑁𝑘 }. Then
𝐴 = ∐𝑘𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1L𝑖 , where L𝑖 is accepted by a PDA M𝑖 =
(𝑄, Σ, Γ, 𝛿󸀠 , 𝑞0 , 𝑍0 , 𝐹𝑖 ), the mapping 𝛿󸀠 : 𝑄 × (Σ ∪ {𝜀}) × Γ →
∗
2𝑄×Γ is defined by 𝛿󸀠 (𝑞, 𝜏, 𝑋) = {(𝑝, 𝛾) | 𝜇𝛿 (𝑞, 𝜏, 𝑋, 𝑝, 𝛾) =
1, 𝑝 ∈ 𝑄, 𝛾 ∈ Γ∗ }, for all (𝑞, 𝜏, 𝑋) ∈ 𝑄 × (Σ ∪ {𝜀}) × Γ, and
𝐹𝑖 = {𝑞 ∈ 𝑄 | 𝜇𝐹 (𝑞) = 𝑎𝑖 , 𝜈𝐹 (𝑞) = 𝑏𝑖 }, for all 𝑖 ∈ 𝑁𝑘 .
For any nonempty subset 𝐽 of the set {1, 2, . . . , 𝑘}, we can
assume that 𝐽 = {𝑖1 , . . . , 𝑖𝑠 }. Let 𝑟𝐽 = 𝑎𝑖1 ∧ ⋅ ⋅ ⋅ ∧ 𝑎𝑖𝑠 , 𝑡𝐽 =
𝑏𝑖1 ∨ ⋅ ⋅ ⋅ ∨ 𝑏𝑖𝑠 , and L(𝐽) = ⋃𝑝1 ⋅⋅⋅𝑝𝑠 L+𝑝1 L+𝑝2 L∗𝑝1 L+𝑝3 (L𝑝1 ∪
L𝑝2 )∗ ⋅ ⋅ ⋅ L+𝑝𝑠−1 (L𝑝1 ∪ ⋅ ⋅ ⋅ ∪ L𝑝𝑠−2 )∗ L+𝑝𝑠 (L𝑝1 ∪ ⋅ ⋅ ⋅ ∪ L𝑝𝑠 )∗ ,
where 𝑝1 ⋅ ⋅ ⋅ 𝑝𝑠 is a permutation of {𝑖1 , . . . , 𝑖𝑠 }, and L(𝐽) takes
unions under all permutations of {𝑖1 , . . . , 𝑖𝑠 }. Hence L(𝐽)
is a context-free language. It is easily verified that 𝐴∗ =
(∐0 ≠ 𝐽⊆𝑁𝑘 (𝑟𝐽 , 𝑡𝐽 )⋅1L(𝐽) )∪((𝜇𝐴 (𝜀), 𝜈𝐴(𝜀))⋅1{𝜀} ). Therefore, 𝐴∗ ∈
Step𝐶(Σ).
(2) If 𝐴 ∈ Step𝐶(Σ2 ), then 𝐴 = ∐𝑘𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1L𝑖 , where 𝑎𝑖 ,
𝑏𝑖 ∈ [0, 1], 𝑎𝑖 + 𝑏𝑖 ≤ 1, 𝑖 ∈ 𝑁𝑘 , and L1 , . . . , L𝑘 ⊂ Σ∗2 are
classical context-free languages. Since ℎ : Σ∗1 → Σ∗2 is a
homomorphism, 𝜇ℎ−1 (𝐴) (𝜔) = 𝜇𝐴∘ℎ (𝜔) = 𝜇𝐴 (ℎ(𝜔)) = ⋁𝑘𝑖=1 𝑎𝑖 ∧
𝜇1L (ℎ(𝜔)) = ⋁𝑘𝑖=1 𝑎𝑖 ∧ 𝜇1ℎ−1 (L ) (𝜔), and 𝜈ℎ−1 (𝐴) (𝜔) = 𝜈𝐴∘ℎ (𝜔) =
𝑖
𝑖
𝜈𝐴(ℎ(𝜔)) = ⋀𝑘𝑖=1 𝑏𝑖 ∨ 𝜇1L (ℎ(𝜔)) = ⋀𝑘𝑖=1 𝑏𝑖 ∨ 𝜈1ℎ−1 (L ) (𝜔), where
𝑖
𝑖
𝜔 ∈ Σ∗1 , ℎ−1 (L𝑖 ) = {𝜔1 ∈ Σ∗1 | ℎ(𝜔1 ) ∈ L𝑖 }, 𝑖 ∈ 𝑁𝑘 , and the
elements of the set {ℎ−1 (L𝑖 ) | 𝑖 ∈ 𝑁𝑘 } are classical contextfree languages. Hence ℎ−1 (𝐴) = 𝐴 ∘ ℎ = ∐𝑘𝑖=1 (𝑎𝑖 , 𝑏𝑖 ) ⋅ 1ℎ−1 (L𝑖 ) .
And so ℎ−1 (𝐴) ∈ Step𝐶(Σ1 ). (3) If ℎ(𝜏) ≠ 𝜀, for all 𝜏 ∈ Σ, and
𝑔 = (𝜇𝑔 , 𝜈𝑔 ) ∈ Step𝐶(Σ1 ), then 𝜇𝑔 (𝜔) = ⋁𝑘𝑖=1 𝑎𝑖 ∧ 𝜇1L (𝜔),
𝑖
𝜈𝑔 (𝜔) = ⋀𝑘𝑖=1 𝑏𝑖 ∨ 𝜇1L (𝜔), for any 𝜔 ∈ Σ1 , where 𝑎𝑖 , 𝑏𝑖 ∈ [0, 1],
𝑖
𝑎𝑖 + 𝑏𝑖 ≤ 1, 𝑖 ∈ 𝑁𝑘 , L1 , . . . , L𝑘 ⊂ Σ∗1 are classical context-free
languages. Since ℎ : Σ∗1 → Σ∗2 is a homomorphism, ℎ(L𝑖 ) =
{ℎ(𝜔) ∈ Σ∗2 | 𝜔 ∈ L𝑖 } is also a classical context-free language,
𝑖 ∈ 𝑁𝑘 . For any 𝑥 ∈ Σ∗2 , 𝜇ℎ(𝑔) (𝑥) = ⋁{𝜇𝑔 (𝛼) | ℎ(𝛼) = 𝑥} =
⋁{⋁𝑘𝑖=1 𝑎𝑖 ∧𝜇1L (𝛼) | ℎ(𝛼) = 𝑥} = ⋁𝑘𝑖=1 𝑎𝑖 ∧𝜇1ℎ(L ) (𝑥), 𝜈ℎ(𝑔) (𝑥) =
𝑖
𝑖
⋀{𝜈𝑔 (𝛼) | ℎ(𝛼) = 𝑥} = ⋀{⋀𝑘𝑖=1 𝑏𝑖 ∨ 𝜈1L (𝛼) | ℎ(𝛼) = 𝑥} =
⋀𝑘𝑖=1 𝑏𝑖 ∨ 𝜈1ℎ(L ) (𝑥). Hence ℎ(𝑔)
𝐶
Step (Σ2 ).
𝑖
=
𝑖
(𝜇ℎ(𝑔) , 𝜈ℎ(𝑔) )
∈
5. Pumping Lemma for IFCFLs
In this section, we mainly discuss the pumping lemma for
IFCFLs, which will become a powerful tool for proving a
certain intuitionistic fuzzy language noncontext-free.
Theorem 33. Let 𝐴 = (𝜇𝐴 , 𝜈𝐴) be an IFCFL over Σ∗ . Then
there exists a finite natural number 𝑛 such that for any 𝑧 ∈ Σ∗
with 𝑛 ≤ |𝑧|, there have 𝑢, 𝑣, 𝑤, 𝑥, 𝑦, 𝑢1 , 𝑣1 , 𝑤1 , 𝑥1 , 𝑦1 ∈ Σ∗
such that 𝑧 = 𝑢𝑣𝑤𝑥𝑦 = 𝑢1 𝑣1 𝑤1 𝑥1 𝑦1 , |𝑣𝑤𝑥| ≤ 𝑛,
|𝑣1 𝑤1 𝑥1 | ≤ 𝑛, |𝑣𝑥| ≥ 1, |𝑣1 𝑥1 | ≥ 1, and 𝜇𝐴 (𝑢𝑣𝑖 𝑤𝑥𝑖 𝑦) ≥
𝜇𝐴 (𝑢𝑣𝑤𝑥𝑦),𝜈𝐴 (𝑢1 𝑣1𝑖 𝑤1 𝑥1𝑖 𝑦1 ) ≤ 𝜈𝐴(𝑢1 𝑣1 𝑤1 𝑥1 𝑦1 ), for all 𝑖 ≥ 0.
Proof. Let 𝐴 = (𝜇𝐴 , 𝜈𝐴) be an IFCFL over Σ∗ . Then there
is an IFCNF 𝐺 = (𝑁, 𝑇, 𝑃, 𝑆) who accepts 𝐴. According to
Proposition 29, L(𝐺) = ∐(𝑎, 𝑏) ⋅ 1L(𝐺𝑎𝑏 ) , where (𝑎, 𝑏) ∈
(𝑋∧ \ {0}) × (𝑌∨ \ {1}), 𝑋 = Im(𝜇𝜌 ), 𝑌 = Im(𝜈𝜌 ), 0 ≤ 𝑎 + 𝑏 ≤ 1
󸀠
and the classical context-free grammar 𝐺𝑎𝑏 = (𝑁󸀠 , 𝑇, 𝑃𝑎𝑏
, 𝑆󸀠 )
is shown in the proof process of of Proposition 29. Let 𝐺 have
𝑚 variables. That means, |𝑁| = 𝑚. Choose 𝑛 = 2𝑚 . Next,
suppose |𝑧| ≥ 𝑛, 𝜇𝐺(𝑧) = 𝑎0 > 0 and 𝜈𝐺(𝑧) = 𝑏0 < 1. Then
there exist 𝑏𝑘 ∈ 𝑌∨ \ {1} and 𝑎𝑙 ∈ 𝑋∧ \ {0} with 0 < 𝑎0 + 𝑏𝑘 ≤ 1
and 0 < 𝑎𝑙 + 𝑏0 ≤ 1 such that 𝑧 ∈ L(𝐺𝑎0 𝑏𝑘 ) ∩ L(𝐺𝑎𝑙 𝑏0 ).
By pumping lemma for context-free languages, there are
𝑢, 𝑣, 𝑤, 𝑥, 𝑦 ∈ Σ∗ satisfying 𝑧 = 𝑢𝑣𝑤𝑥𝑦, |𝑣𝑤𝑥| ≤ 𝑛 and
|𝑣𝑥| ≥ 1 such that 𝑢𝑣𝑖 𝑤𝑥𝑖 𝑦 ∈ L(𝐺𝑎0 𝑏𝑘 ), for all 𝑖 ≥ 0. Then
𝜇𝐴 (𝑢𝑣𝑖 𝑤𝑥𝑖 𝑦) ≥ 𝜇𝐴 (𝑢𝑣𝑤𝑥𝑦) for any 𝑖 ≥ 0 since 𝜇𝐴 (𝑢𝑣𝑖 𝑤𝑥𝑖 𝑦) =
⋁𝑎∈𝑋∧ \{0} 𝑎 ∧ 1L(𝐺𝑎𝑏 ) (𝑢𝑣𝑖 𝑤𝑥𝑖 𝑦) ≥ 𝑎0 .
Similarly, there are 𝑢1 , 𝑣1 , 𝑤1 , 𝑥1 , 𝑦1 ∈ Σ∗ satisfying 𝑧 =
𝑢1 𝑣1 𝑤1 𝑥1 𝑦1 , |𝑣1 𝑤1 𝑥1 | ≤ 𝑛 and |𝑣1 𝑥1 | ≥ 1 such that
𝑢1 𝑣1𝑖 𝑤1 𝑥1𝑖 𝑦1 ∈ L(𝐺𝑎𝑙 𝑏0 ), for all 𝑖 ≥ 0. Then 𝜈𝐴(𝑢1 𝑣1𝑖 𝑤1 𝑥1𝑖 𝑦1 ) ≤
𝜈𝐴 (𝑢1 𝑣1 𝑤1 𝑥1 𝑦1 ) for any 𝑖 ≥ 0 since 𝜈𝐴 (𝑢1 𝑣1𝑖 𝑤1 𝑥1𝑖 𝑦1 ) =
⋀𝑏∈𝑌∨ \{1} 𝑏 ∨ 1L(𝐺𝑎𝑏 ) (𝑢1 𝑣1𝑖 𝑤1 𝑥1𝑖 𝑦1 ) ≤ 𝑏0 .
Next, let us look at an example to negate an intuitionistic
fuzzy language to be an IFCFL.
Example 34. Let 𝐴 = (𝜇𝐴 , 𝜈𝐴 ) be an IFS over 𝑇∗ . The
mappings 𝜇𝐴 , 𝜈𝐴 : 𝑇∗ → [0, 1] are defined by
0.5,
0,
if 𝑧 = 𝑎𝑖 𝑏𝑗 𝑐𝑘 (𝑖 < 𝑗 < 𝑘) ,
otherwise,
0.3,
𝜈𝐴 (𝑧) = {
1,
if 𝑧 = 𝑎𝑖 𝑏𝑗 𝑐𝑘 (𝑖 < 𝑗 < 𝑘) ,
otherwise,
𝜇𝐴 (𝑧) = {
(27)
where 𝑖, 𝑗, and 𝑘 are natural numbers.
Suppose 𝐴 is an IFCFL. Then there exists a certain IFCNF
𝐺 such that L(𝐺) = 𝐴. For constant 𝑛, put 𝑧 = 𝑎𝑛 𝑏𝑛+1 𝑐𝑛+2 .
Hence, 𝜇𝐴 (𝑧) = 𝜇L(𝐺) (𝑧) = 0.5 and 𝜈𝐴(𝑧) = 𝜈L(𝐺) (𝑧) = 0.3.
Let 𝑧 = 𝑢𝑣𝑤𝑥𝑦, where |𝑣𝑤𝑥| ≤ 𝑛 and |𝑣𝑥| ≥ 1. If 𝑣𝑤𝑥
does not have 𝑐’s, then 𝑢𝑣3 𝑤𝑥3 𝑦 has at least 𝑛 + 2𝑎’s or 𝑏’s;
if 𝑣𝑤𝑥 has at least a 𝑐, then it has not an 𝑎 since |𝑣𝑤𝑥| ≤ 𝑛.
And so 𝑢𝑤𝑦 has 𝑛𝑎󸀠 s, but no more than 2𝑛 + 2𝑏’s and 𝑐’s in
total, that is, |𝑢𝑤𝑦| ≤ 𝑛 + 2𝑛 + 2. Therefore, it is impossible
that 𝑢𝑤𝑦 has more 𝑏’s than 𝑎’s and also has more 𝑐’s than 𝑏’s.
By calculation, we have 𝜇𝐴 (𝑢𝑤𝑦) = 0 and 𝜈𝐴 (𝑢𝑤𝑦) = 1. No
matter how 𝑧 is broken into 𝑢𝑣𝑤𝑥𝑦, we have a contradiction
with Theorem 33. Therefore, 𝐴 is not an IFCFL.
The following example will show that intuitionistic fuzzy
pushdown automata have more power than fuzzy pushdown
automata when comparing two distinct strings although the
degrees of membership of these strings recognized by the
underlying fuzzy pushdown automata are equal.
Journal of Applied Mathematics
15
Example 35. Let Σ = {0, 1}. Then 𝐿 = {𝜔1𝜔−1 | 𝜔 ∈ Σ∗ } ⊆ Σ∗
is clearly a context-free language but not a regular language by
classical automata theory, where 𝜔−1 represents the reversal of
the string 𝜔. Given an IFPDA M = (𝑄, Σ, Γ, 𝛿, 𝐼, 𝑍0 , 𝐹) and
a fuzzy pushdown automaton N = (𝑄, Σ, Γ, 𝜂, 𝜎0 , 𝑍0 , 𝜎1 ). Put
𝑄 = {𝑞0 , 𝑞1 , 𝑞2 }, Γ = {𝑍0 , 0, 1}, an IFS 𝛿 = (𝜇𝛿 , 𝜈𝛿 ) in 𝑄 × (Σ ∪
{𝜀}) × Γ × 𝑄 × Γ∗ is defined by
𝜇𝛿 (𝑞0 , 0, 𝑍0 , 𝑞0 , 0𝑍0 ) = 0.7, 𝜈𝛿 (𝑞0 , 0, 𝑍0 , 𝑞0 , 0𝑍0 ) =
0.2,
𝜇𝛿 (𝑞0 , 1, 𝑍0 , 𝑞0 , 1𝑍0 ) = 0.6, 𝜈𝛿 (𝑞0 , 1, 𝑍0 , 𝑞0 , 1𝑍0 ) =
0.3,
𝜇𝛿 (𝑞0 , 0, 0, 𝑞0 , 00) = 0.3, 𝜈𝛿 (𝑞0 , 0, 0, 𝑞0 , 00) = 0.6,
𝜇𝛿 (𝑞0 , 0, 1, 𝑞0 , 01) = 0.3, 𝜈𝛿 (𝑞0 , 0, 1, 𝑞0 , 01) = 0.5,
𝜇𝛿 (𝑞0 , 1, 0, 𝑞0 , 10) = 0.5, 𝜈𝛿 (𝑞0 , 1, 0, 𝑞0 , 10) = 0.4,
𝜇𝛿 (𝑞1 , 0, 0, 𝑞1 , 𝜀) = 0.6, 𝜈𝛿 (𝑞1 , 0, 0, 𝑞1 , 𝜀) = 0.3,
𝜇𝛿 (𝑞1 , 1, 1, 𝑞1 , 𝜀) = 0.5, 𝜈𝛿 (𝑞1 , 1, 1, 𝑞1 , 𝜀) = 0.35,
𝜇𝛿 (𝑞0 , 1, 𝑍0 , 𝑞1 , 𝑍0 ) = 1, 𝜈𝛿 (𝑞0 , 1, 𝑍0 , 𝑞1 , 𝑍0 ) = 0,
𝜇𝛿 (𝑞0 , 1, 0, 𝑞1 , 0) = 1, 𝜈𝛿 (𝑞0 , 1, 0, 𝑞1 , 0) = 0,
𝜇𝛿 (𝑞0 , 1, 1, 𝑞1 , 1) = 1, 𝜈𝛿 (𝑞0 , 1, 1, 𝑞1 , 1) = 0,
𝜇𝛿 (𝑞1 , 𝜀, 𝑍0 , 𝑞2 , 𝑍0 ) = 1, 𝜈𝛿 (𝑞1 , 𝜀, 𝑍0 , 𝑞2 , 𝑍0 ) = 0.
Otherwise 𝜇𝛿 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = 0 and 𝜈𝛿 (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) = 1
for (𝑞, 𝜏, 𝑍, 𝑝, 𝛾) ∈ 𝑄 × (Σ ∪ {𝜀}) × Γ × 𝑄 × Γ∗ .
The IFSs 𝐼 = (𝜇𝐼 , 𝜈𝐼 ) and 𝐹 = (𝜇𝐹 , 𝜈𝐹 ) in 𝑄 are defined
by 𝜇𝐼 (𝑞0 ) = 1, 𝜈𝐼 (𝑞0 ) = 0, 𝜇𝐼 (𝑞1 ) = 𝜇𝐼 (𝑞2 ) = 0, 𝜈𝐼 (𝑞1 ) =
𝜇𝐼 (𝑞2 ) = 1, 𝜇𝐹 (𝑞2 ) = 1, 𝜈𝐹 (𝑞2 ) = 0, 𝜇𝐹 (𝑞0 ) = 𝜇𝐹 (𝑞1 ) = 0 and
𝜈𝐹 (𝑞0 ) = 𝜇𝐹 (𝑞1 ) = 1.
And set 𝜂 = 𝜇𝛿 , 𝜎0 = 𝜇𝐼 , and 𝜎1 = 𝜇𝐹 .
By computing with the strings, 010, 111, 01110, 10101,
0011100, and 1011101∈ Σ∗ , we have
𝜇L(M) (010) = 𝑓N (010) = 0.6, 𝜈L(M) (010) = 0.3,
𝜇L(M) (111) = 𝑓N (111) = 0.5, 𝜈L(M) (111) = 0.35,
𝜇L(M) (01110) = 𝑓N (01110) = 0.5, 𝜈L(M) (01110) =
0.4,
𝜇L(M) (10101) = 𝑓N (10101) = 0.3, 𝜈L(M) (10101) =
0.5,
=
𝜇L(M) (0011100)
𝜈L(M) (0011100) = 0.6,
𝑓N (0011100)
=
0.3,
𝜇L(M) (1011101)
=
𝜈L(M) (1011101) = 0.5.
𝑓N (1011101)
=
0.3,
This implies that 111 is better than 01110 because the
degree of nonmembership of 𝜈L(M) (111) is smaller than the
𝜈L(M) (01110)’s although the degrees of membership of the
fuzzy context-free languages 𝑓N (01110) and 𝑓N (111) are
equal. Comparing the above five strings, 010 is the best and
0011100 is the worst.
languages. Firstly, the notions of intuitionistic fuzzy pushdown automata (IFPDAs) and their recognizable languages
are introduced and discussed in detail. Using the generalized
subset construction method, we show that IFPDAs are equivalent to IFSPDAs and then prove that intuitionistic fuzzy
step functions are the same as those accepted by IFPDAs.
Furthermore, we have presented algebraic characterization of
intuitionistic fuzzy recognizable languages including decomposition form and representation theorem. It follows that
the languages accepted by IFPDAs are equivalent to those
accepted by IFPDAs0 by classical automata theory. Secondly,
we have introduced the notions of IFCFGs, IFCNFs, and
IFGNFs. It is shown that they are equivalent in the sense
that they generate the same classes of intuitionistic fuzzy
context-free languages (IFCFLs). In particular, IFCFGs are
proven to be an equivalence of IFPDAs as well. Then some
operations on the family of IFCFLs are discussed. Finally
pumping lemma for IFCFLs has been established. Thus,
together with [38–40], we have more systematically established intuitionistic fuzzy automata theory as a generalization
of fuzzy automata theory.
As mentioned in Section 1, IFS and fuzzy automata theory
have supported a wealth of important applications in many
fields. The next step is to consider the potential application
of IFPDAs and IFCFLs such as in model checking and
clinical monitoring. Additionally, many related researches in
theories, such as IFPDAs based on the composition of t-norm
and t-conorm and the minimal algorithm of IFPDAs, will be
studied in the future.
Acknowledgments
The authors would like to thank the editor and the anonymous reviewers for their valuable comments that helped
to improve the quality of this paper. This work is supported by National Science Foundation of China (Grant no.
11071061), 973 Program (2011CB311808), and Natural Science
Foundation of Southwest Petroleum University (Grant no.
2012XJZ030).
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