On Intuitionistic Fuzzy Context-Free Languages

Abstract

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 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 context-free 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 lattice-valued 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 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 context-free 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, 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,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)∣𝑥∈𝑋,(1) where the functions 𝜇_𝐴∶𝑋→[0,1] and 𝜈_𝐴∶𝑋→[0,1] denote the degree of membership (i.e., 𝜇_𝐴(𝑥)) and the degree of nonmembership (𝜈_𝐴(𝑥)) of each element 𝑥∈𝑋 to the set 𝐴, respectively, and the two quantities satisfy the following inequalities: 0≤𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1,∀𝑥∈𝑋.(2) For the sake of simplicity, we use the notation 𝐴=(𝜇_𝐴,𝜈_𝐴) instead of 𝐴={(𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(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∈𝐼𝐴_𝑖=𝑥,_𝑖∈𝐼𝜇_𝐴𝑖(𝑥),_𝑖∈𝐼𝜈_𝐴𝑖(𝑥)∣𝑥∈𝑋_𝑖∈𝐼,𝐴_𝑖=𝑥,_𝑖∈𝐼𝜇_𝐴𝑖(𝑥),_𝑖∈𝐼𝜈_𝐴𝑖(𝑥)∣𝑥∈𝑋,(3) where ⋁ and ⋀ denote supremum and infimum of real numbers in [0,1], respectively.
For two IFSs 𝐴=(𝜇_𝐴,𝜈_𝐴) and 𝐵=(𝜇_𝐵,𝜈_𝐵), we say 𝐴=𝐵 if 𝜇_𝐴=𝜇_𝐵 and 𝜈_𝐴=𝜈_𝐵. In addition, if the IFS 𝐴=(𝜇_𝐴,𝜈_𝐴) in 𝑋 satisfies the condition that, for any 𝑥∈𝑋, 𝜇_𝐴(𝑥)+𝜈_𝐴(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,𝜇_𝐸𝑥,𝑦,𝜈_𝐸𝑥,𝑦∣𝑥∈𝑋,𝑦∈𝑌,(4) where the mappings 𝜇_𝐸∶𝑋×𝑌→[0,1] and 𝜈_𝐸∶𝑋×𝑌→[0,1] satisfy 0≤𝜇_𝐸𝑥,𝑦+𝜈_𝐸𝑥,𝑦≤1,∀𝑥,𝑦∈𝑋×𝑌.(5) An IFBR over 𝑋 is an IFS of 𝑋×𝑋. Let 𝑃=(𝜇_𝑃,𝜈_𝑃) and 𝐸=(𝜇_𝐸,𝜈_𝐸) be IFRs in 𝑋×𝑌 and 𝑌×𝑍, respectively. Define the composition of IFRs, 𝑃∘𝐸=(𝜇_𝑃∘𝐸,𝜈_𝑃∘𝐸) in 𝑋×𝑍, by 𝜇_𝑃∘𝐸(𝑥,𝑧)=_𝑦∈𝑌𝜇_𝑃𝑥,𝑦∧𝜇_𝐸,𝜈𝑦,𝑧_𝑃∘𝐸(𝑥,𝑧)=_𝑦∈𝑌𝜈_𝑃𝑥,𝑦∨𝜈_𝐸𝑦,𝑧,(6) for all (𝑥,𝑧)∈𝑋×𝑍.
Furthermore, if 𝑅 is an IFBR over 𝑋, then its reflexive and transitive closure is 𝑅^∗=⋃^∞_𝑛=0𝑅^𝑛, where 𝑅^𝑛+1=𝑅^𝑛∘𝑅,𝑛≥0, and 𝑅⁰=𝑖𝑑=(𝜇_𝑖𝑑,𝜈_𝑖𝑑), that is, 𝜇_𝑖𝑑()=𝜈𝑢,𝑣1,if𝑢=𝑣0,if𝑢≠𝑣,_𝑖𝑑(𝑢,𝑣)=0,if𝑢=𝑣1,if𝑢≠𝑣,(7) for all (𝑢,𝑣)∈𝑋×𝑋.

Definition 2. Let 𝐴=(𝜇_𝐴,𝜈_𝐴) be an IFS in 𝑋. Then the image set of 𝐴, denoted as Im(𝐴), is given as Im(𝐴)=Im𝜇_𝐴∪Im𝜈_𝐴,(8) where Im(𝜇_𝐴)={𝜇_𝐴(𝑥)∣𝑥∈𝑋} and Im(𝜈_𝐴)={𝜈_𝐴(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∣𝜇_𝐴(𝑥)≥𝜆,𝜈_𝐴(𝑥)≤𝜃.(9) And the support set of 𝐴, denoted as supp(𝐴), is defined by supp(𝐴)=𝑥∈𝑋∣𝜇_𝐴(𝑥)>0,𝜈_𝐴(𝑥)<1.(10) If supp(𝐴) is finite, then 𝐴 is called a finite IFS in 𝑋.

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|. 𝑁_𝑘={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)𝑄is 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;(iv)𝛿=(𝜇_𝛿,𝜈_𝛿) is a finite IFS in 𝑄×(Σ∪{𝜀})×Γ×𝑄×Γ^∗;(v)𝑍₀∈Γ is called the start stack symbol;(vi)𝐼=(𝜇_𝐼,𝜈_𝐼) and 𝐹=(𝜇_𝐹,𝜈_𝐹) are intuitionistic fuzzy subsets in 𝑄, which are called the intuitionistic fuzzy subsets of initial and final states, respectively.

Definition 4. An intuitionistic fuzzy pushdown automaton with empty stack (IFPDA^∅, for short) is a seven tuple 𝒩=(𝑄,Σ,Γ,𝛿,𝐼,𝑍₀,∅), where 𝑄,Σ,Γ,𝛿,𝐼 and 𝑍₀ are the same as those in IFPDA ℳ, and ∅ 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 𝜇_⊢ℳ,=𝜇𝑞,𝜔,𝛽𝑝,𝑢,𝛼_𝛿𝛽𝛽,𝛽𝜇𝑞,𝜀,head,𝑝,𝛼⧵tailif𝑢=𝜔,tail≤𝛼_𝛿(𝜔)𝛽𝛽,(𝜔)𝛽𝜈𝑞,head,head,𝑝,𝛼⧵tailif𝑢=tail,tail≤𝛼0,otherwise,_⊢ℳ,=𝜈𝑞,𝜔,𝛽𝑝,𝑢,𝛼_𝛿𝛽𝛽,𝛽𝜈𝑞,𝜀,head,𝑝,𝛼⧵tailif𝑢=𝜔,tail≤𝛼_𝛿(𝜔)𝛽𝛽,(𝜔)𝛽𝑞,head,head,𝑝,𝛼⧵tailif𝑢=tail,tail≤𝛼0,otherwise(11) for any (𝑞,𝜔,𝛽),(𝑝,𝑢,𝛼)∈𝑄×Σ^∗×Γ^∗. Here, for any nonempty string 𝑢=𝑥₁⋯𝑥_𝑛, 𝑛≥1, head(𝑢)=𝑥₁, tail(𝑢)=𝑥₂⋯𝑥_𝑛, 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)=(𝜇_ℒ(ℳ),𝜈_ℒ(ℳ)), 𝜇_ℒ(ℳ), and 𝜈_ℒ(ℳ) are functions from Σ^∗ to the unit interval [0,1], and𝜇_ℒ(ℳ)(𝜔) = ⋁{𝜇_𝐼(𝑞₀)∧𝜇_⊢∗ℳ((𝑞₀,𝜔,𝑧₀), (𝑝,𝜀,𝑟))∧ 𝜇_𝐹(𝑝)∣𝑞₀,𝑝∈𝑄,𝑟∈Γ^∗}, 𝜈_ℒ(ℳ)(𝜔) = ⋀{𝜈_𝐼(𝑞₀)∨𝜈_⊢^∗_ℳ((𝑞₀,𝜔,𝑧₀), (𝑝,𝜀,𝑟))∨𝜈_𝐹 (𝑝)∣𝑞₀,𝑝∈𝑄,𝑟∈Γ^∗}for any 𝜔∈Σ^∗.

Definition 7. Let 𝒩=(𝑄,Σ,Γ,𝛿,𝐼,𝑍₀,∅) be an IFPDA^∅. Then we call ℒ(𝒩) an intuitionistic fuzzy language accepted by 𝒩 with empty stack, where ℒ(𝒩)=(𝜇_ℒ(𝒩),𝜈_ℒ(𝒩)), 𝜇_ℒ(𝒩) and 𝜈_ℒ(𝒩) are functions from Σ^∗ to the unit interval [0,1], and 𝜇_ℒ(𝒩)(𝜔) = ⋁{𝜇_𝐼(𝑞₀)∧𝜇_⊢∗𝒩((𝑞₀,𝜔,𝑧₀), (𝑝,𝜀,𝜀)) ∣𝑞₀,𝑝∈𝑄}, 𝜈_ℒ(𝒩)(𝜔) = ⋀{𝜈_𝐼(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_∨, is also finite, where 𝑋_∧={𝑥₁∧⋯∧𝑥_𝑘∶𝑘≥1,𝑥₁,…,𝑥_𝑘∈𝑋}∪{1}, and 𝑋_∨={𝑥₁∨⋯∨𝑥_𝑘∶𝑘≥1,𝑥₁,…,𝑥_𝑘∈𝑋}∪{0}.

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=(𝜇_𝑓,𝜈_𝑓) is an IFS in Σ^∗).
It suffices to show that 0≤𝜇_𝑓(𝜔)+𝜈_𝑓(𝜔)≤1, for any 𝜔=𝑢₁⋯𝑢_𝑛, 𝑢_𝑖∈Σ∪{𝜀}, 𝑖=1,…,𝑛.
Clearly, 𝜇_𝑓(𝜔)=⋁{𝜇_𝐼(𝑞₀)∧𝜇_⊢∗ℳ((𝑞₀, 𝜔,𝑧₀),(𝑝,𝜀,𝑟))∧𝜇_𝐹(𝑝)∣ 𝑞₀,𝑝∈𝑄,𝑟∈Γ^∗} = ⋁{𝜇_𝐼(𝑞₀)∧𝜇_⊢((𝑞₀,𝜔,𝑧₀), (𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁))∧𝜇_⊢((𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁),(𝑞₂,𝑢₃⋯𝑢_𝑛, 𝑧₂𝑟₂))∧⋯∧𝜇_⊢((𝑞_𝑛−1, 𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1), (𝑞_𝑛,𝜀,𝑟_𝑛))∧𝜇_𝐹(𝑞_𝑛)∣ (𝑞₀,𝑞₁,…,𝑞_𝑛)∈𝑄^𝑛+1,𝑧₁,…,𝑧_𝑛−1∈Γ, 𝑟₁,…,𝑟_𝑛∈Γ^∗}, and 𝜈_𝑓(𝜔)=⋀{𝜈_𝐼(𝑞₀)∨𝜈_⊢^∗_ℳ((𝑞₀, 𝜔,𝑧₀),(𝑝,𝜀,𝑟))∨𝜈_𝐹(𝑝)∣ 𝑞₀,𝑝∈𝑄,𝑟∈Γ^∗} = ⋀{𝜈_𝐼(𝑞₀)∨𝜈_⊢((𝑞₀,𝜔,𝑧₀), (𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁))∨𝜈_⊢((𝑞₁, 𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁), (𝑞₂,𝑢₃⋯𝑢_𝑛,𝑧₂𝑟₂))∨⋯∨𝜈_⊢((𝑞_𝑛−1, 𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1), (𝑞_𝑛,𝜀,𝑟_𝑛))∨𝜈_𝐹(𝑞_𝑛)∣ (𝑞₀,𝑞₁,…,𝑞_𝑛)∈𝑄^𝑛+1,𝑧₁, …,𝑧_𝑛−1∈Γ, 𝑟₁,…,𝑟_𝑛∈Γ^∗}.
On the one hand, 0≤𝜇_𝑓(𝜔)+𝜈_𝑓(𝜔); on the other hand, there exists a sequence (𝑞₀,𝑞₁,…,𝑞_𝑛)∈𝑄^𝑛+1,𝑧₁, …,𝑧_𝑛−1∈Γ,𝑟₁,…,𝑟_𝑛∈Γ^∗ such that 𝜇_𝑓(𝜔)=𝜇_𝐼(𝑞₀)∧𝜇_⊢((𝑞₀,𝜔,𝑧₀), (𝑞₁,𝑢₂⋯𝑢_𝑛, 𝑧₁𝑟₁))∧𝜇_⊢((𝑞₁,𝑢₂⋯𝑢_𝑛, 𝑧₁𝑟₁), (𝑞₂,𝑢₃⋯𝑢_𝑛, 𝑧₂𝑟₂)) ∧⋯∧𝜇_⊢((𝑞_𝑛−1,𝑢_𝑛, 𝑧_𝑛−1𝑟_𝑛−1), (𝑞_𝑛,𝜀,𝑟_𝑛))∧𝜇_𝐹(𝑞_𝑛). Hence 𝜈_𝑓(𝜔)≤𝜈_𝐼(𝑞₀)∨𝜈_⊢((𝑞₀, 𝜔,𝑧₀), (𝑞₁,𝑢₂⋯𝑢_𝑛, 𝑧₁𝑟₁))∨𝜈_⊢((𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁), (𝑞₂,𝑢₃⋯𝑢_𝑛, 𝑧₂𝑟₂))∨⋯∨𝜈_⊢((𝑞_𝑛−1,𝑢_𝑛, 𝑧_𝑛−1𝑟_𝑛−1), (𝑞_𝑛,𝜀,𝑟_𝑛))∨𝜈_𝐹(𝑞_𝑛).
Therefore, 𝜇_𝑓(𝜔)+𝜈_𝑓(𝜔)≤(𝜇_𝐼(𝑞₀)+ 𝜈_𝐼(𝑞₀))∨(𝜇_⊢((𝑞₀,𝜔,𝑧₀),(𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁))+𝜈_⊢((𝑞₀,𝜔,𝑧₀), (𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁))) ∨(𝜇_⊢((𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁),(𝑞₂,𝑢₃⋯𝑢_𝑛,𝑧₂𝑟₂))+ 𝜈_⊢((𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁), (𝑞₂,𝑢₃⋯𝑢_𝑛,𝑧₂𝑟₂)))∨⋯∨ (𝜇_⊢((𝑞_𝑛−1,𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1), (𝑞_𝑛,𝜀,𝑟_𝑛))+𝜈_⊢((𝑞_𝑛−1,𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1), (𝑞_𝑛,𝜀,𝑟_𝑛)))∨(𝜇_𝐹(𝑞_𝑛)+𝜈_𝐹(𝑞_𝑛))≤1∨1∨⋯∨1=1.
Claim 2 (Im(𝑓) is finite).
In fact, let 𝑋=Im(𝜇_𝐼)∪Im(𝜇_𝛿)∪Im(𝜇_𝐹) and 𝑌=Im(𝜈_𝐼)∪Im(𝜈_𝛿)∪Im(𝜈_𝐹). Then 𝑋_∧={𝑥₁∧⋯∧𝑥_𝑘∣𝑘≥1,𝑥₁, …,𝑥_𝑘∈𝑋}∪{1} and 𝑋_∨={𝑥₁∨⋯∨𝑥_𝑘∣𝑘≥1,𝑥₁,…,𝑥_𝑘∈𝑋}∪{0} are finite sets by Lemma 8. Since 𝛿=(𝜇_𝛿,𝜈_𝛿) is a finite IFS, for any 𝜔=𝑢₁⋯𝑢_𝑛, 𝑢_𝑖∈Σ∪{𝜀}, 𝑖=1,…,𝑛, there exists a natural number 𝑑∈𝑁 such that 𝜇_𝑓(𝜔)=⋁{𝜇_𝐼(𝑞₀)∧𝜇_⊢((𝑞₀,𝜔,𝑧₀), (𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁))∧ 𝜇_⊢((𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁), (𝑞₂,𝑢₃⋯𝑢_𝑛,𝑧₂𝑟₂))∧⋯∧𝜇_⊢((𝑞_𝑛−1,𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1), (𝑞_𝑛,𝜀,𝑟_𝑛))∧𝜇_𝐹(𝑞_𝑛)∣(𝑞₀,𝑞₁,…,𝑞_𝑛)∈𝑄^𝑛+1, 𝑧₁,…,𝑧_𝑛−1∈Γ, 𝑟₁,…,𝑟_𝑛∈Γ^∗} = (𝑎₁₀∧⋯∧𝑎_1,𝑛−1∧𝑎_1𝑛)∨⋯∨ (𝑎_𝑑0∧⋯∧𝑎_{𝑑,𝑛−1}∧𝑎_𝑑𝑛)), where 𝑎_𝑖𝑗∈𝑋, 𝑖=1,…,𝑑; 𝑗=0,…,𝑛. By Lemma 8, (𝑋_∧)_∨ is also finite. Since 𝜇_𝑓(𝜔)∈(𝑋_∧)_∨ for any 𝜔∈Σ^∗, Im(𝜇_𝑓)⊆(𝑋_∧)_∨. Hence Im(𝜇_𝑓) is a finite subset of [0,1].
Similarly, it follows that Im(𝜈_𝑓) is also a finite subset of [0,1].
Therefore, Im(𝑓)=Im(𝜇_𝑓)∪Im(𝜈_𝑓) is finite.
If 𝑓 can be accepted by some IFPDA^∅𝒩=(𝑄,Σ,Γ,𝛿,𝐼,𝑍₀,∅), then, by Definition 7, for any 𝜔=𝑢₁⋯𝑢_𝑛, 𝑢_𝑖∈Σ∪{𝜀}, 𝑖=1,…,𝑛, we have 𝜇_𝑓(𝜔)=⋁{𝜇_𝐼(𝑞₀)∧𝜇_⊢∗𝒩((𝑞₀,𝜔,𝑧₀), (𝑝,𝜀,𝜀))∣𝑞₀,𝑝∈𝑄} = ⋁{𝜇_𝐼(𝑞₀)∧𝜇_⊢((𝑞₀,𝜔,𝑧₀),(𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁))∧𝜇_⊢((𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁), (𝑞₂,𝑢₃⋯𝑢_𝑛,𝑧₂𝑟₂))∧⋯∧𝜇_⊢((𝑞_𝑛−1,𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1), (𝑞_𝑛,𝜀,𝜀))∣(𝑞₀,𝑞₁,…,𝑞_𝑛)∈𝑄^𝑛+1, 𝑧₁,…,𝑧_𝑛−1∈Γ,𝑟₁,…,𝑟_𝑛−1∈Γ^∗}, and 𝜈_𝑓(𝜔)=⋀{𝜈_𝐼(𝑞₀)∨𝜈_⊢^∗_𝒩((𝑞₀,𝜔,𝑧₀), (𝑝,𝜀,𝜀))∣𝑞₀,𝑝∈𝑄}=⋀{𝜈_𝐼(𝑞₀)∨𝜈_⊢((𝑞₀,𝜔,𝑧₀), (𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁))∨𝜈_⊢((𝑞₁, 𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁),(𝑞₂,𝑢₃⋯𝑢_𝑛, 𝑧₂𝑟₂))∨⋯∨𝜈_⊢((𝑞_𝑛−1,𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1), (𝑞_𝑛,𝜀,𝜀))∣(𝑞₀,𝑞₁,…,𝑞_𝑛)∈𝑄^𝑛+1, 𝑧₁,…,𝑧_𝑛−1∈Γ,𝑟₁,…,𝑟_𝑛−1∈Γ^∗}.
In a similar manner, it is concluded that the following must be true.

Proposition 10. If 𝑓 can be accepted by some IFPDA^∅𝒩=(𝑄,Σ,Γ,𝛿,𝐼,𝑍₀,∅), 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)𝐴can be accepted by an IFPDA ℳ=(𝑄,Σ,Γ,𝛿,𝐼,𝑍₀,𝐹);(ii)𝐴can 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 𝜇_𝐼′𝑞=1,if𝑞=𝑞₀0,if𝑞≠𝑞₀,𝜈_𝐼′𝑞=0,if𝑞=𝑞₀1,if𝑞≠𝑞₀.(12)
Define an IFS 𝐹^′ in 𝑄^′ by 𝜇_𝐹′𝑞=0,if𝑞=𝑞₀𝜇_𝐹𝑞,if𝑞≠𝑞₀,𝜈_𝐹′𝑞=1,if𝑞=𝑞₀𝜈_𝐹𝑞,if𝑞≠𝑞₀.(13)
Define an IFS 𝛿^′ in 𝑄^′×(Σ∪{𝜀})×Γ^′×𝑄^′×Γ^′∗ by mappings 𝜇_𝛿′,𝜈_𝛿′∶𝑄^′×(Σ∪{𝜀})×Γ^′×𝑄^′×Γ^′∗→[0,1], 𝜇_𝛿′(𝑞₀,𝜀,𝑋₀,𝑝,𝑍₀)=𝜇_𝐼(𝑝), 𝜈_𝛿′(𝑞₀,𝜀,𝑋₀,𝑝,𝑍₀)=𝜈_𝐼(𝑝), 𝜇_𝛿′(𝑞,𝜏,𝑧,𝑝,𝛾)=𝜇_𝛿(𝑞,𝜏,𝑧,𝑝,𝛾), 𝜈_𝛿′(𝑞,𝜏,𝑧,𝑝,𝛾)=𝜈_𝛿(𝑞,𝜏,𝑧,𝑝,𝛾), where 𝑞,𝑝∈𝑄, 𝜏∈Σ∪{𝜀}, 𝑧∈Γ, 𝛾∈Γ^∗; otherwise, 𝜇_𝛿′(𝑞₀,𝜏,𝑧,𝑝,𝛾)=0 and 𝜈_𝛿′(𝑞₀,𝜏,𝑧,𝑝,𝛾)=1.
Then for any 𝜔=𝑢₁⋯𝑢_𝑛∈Σ^∗, 𝑢_𝑖∈Σ∪{𝜀},𝑖=1,…,𝑛, we have 𝜇_ℒ(ℳ′)(𝜔)=⋁{𝜇_𝐼′(𝑞)∧𝜇_⊢ℳ′((𝑞,𝜔,𝑋₀), (𝑝₀,𝜔,𝑍₀))∧𝜇_⊢ℳ′((𝑝₀,𝜔,𝑍₀), (𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁))∧𝜇_⊢ℳ′((𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁), (𝑞₂,𝑢₃⋯𝑢_𝑛,𝑧₂𝑟₂))∧⋯∧𝜇_⊢ℳ′((𝑞_𝑛−1,𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1),(𝑞_𝑛,𝜀,𝑟_𝑛))∧𝜇_𝐹′(𝑞_𝑛)∣𝑞∈𝑄^′, (𝑝₀,𝑞₁,…,𝑞_𝑛)∈𝑄^′𝑛+1,𝑧₁,…,𝑧_𝑛−1∈Γ^′, 𝑟₁,…,𝑟_𝑛∈Γ^′∗}=⋁{1∧𝜇_𝐼(𝑝₀)∧𝜇_⊢ℳ((𝑝₀,𝜔,𝑍₀),(𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁))∧𝜇_⊢ℳ((𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁),(𝑞₂,𝑢₃⋯𝑢_𝑛,𝑧₂𝑟₂))∧⋯∧𝜇_⊢ℳ((𝑞_𝑛−1,𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1),(𝑞_𝑛,𝜀,𝑟_𝑛))∧𝜇_𝐹(𝑞_𝑛)∣(𝑝₀,𝑞₁,…,𝑞_𝑛)∈𝑄^𝑛+1,𝑧₁,…,𝑧_𝑛−1∈Γ,𝑟₁,…,𝑟_𝑛∈Γ^∗}=𝜇_ℒ(ℳ)(𝜔), and 𝜈_ℒ(ℳ′)(𝜔)=⋀{𝜈_𝐼′(𝑞)∨𝜈_⊢ℳ′((𝑞,𝜔,𝑋₀),(𝑝₀,𝜔,𝑍₀))∨𝜈_⊢ℳ′((𝑝₀,𝜔,𝑍₀), (𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁))∨𝜈_⊢ℳ′((𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁), (𝑞₂,𝑢₃⋯𝑢_𝑛,𝑧₂𝑟₂))∨⋯∨𝜈_⊢ℳ′((𝑞_𝑛−1,𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1), (𝑞_𝑛,𝜀,𝑟_𝑛))∨𝜈_𝐹′(𝑞_𝑛)∣𝑞∈𝑄^′, (𝑝₀,𝑞₁,…,𝑞_𝑛)∈𝑄^′𝑛+1,𝑧₁,…,𝑧_𝑛−1∈Γ^′, 𝑟₁,…,𝑟_𝑛∈Γ^′∗}=⋀{1∨𝜈_𝐼(𝑝₀)∨𝜈_⊢ℳ((𝑝₀, 𝜔,𝑍₀),(𝑞₁,𝑢₂⋯𝑢_𝑛, 𝑧₁𝑟₁))∨𝜈_⊢ℳ((𝑞₁,𝑢₂⋯𝑢_𝑛,𝑧₁𝑟₁), (𝑞₂,𝑢₃⋯𝑢_𝑛,𝑧₂𝑟₂))∨⋯∨𝜈_⊢ℳ((𝑞_𝑛−1, 𝑢_𝑛,𝑧_𝑛−1𝑟_𝑛−1),(𝑞_𝑛,𝜀,𝑟_𝑛))∨𝜈_𝐹(𝑞_𝑛)∣(𝑝₀,𝑞₁, …,𝑞_𝑛)∈𝑄^𝑛+1,𝑧₁,…,𝑧_𝑛−1∈Γ, 𝑟₁,…,𝑟_𝑛∈Γ^∗}=𝜈_ℒ(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 𝜇_𝐼𝑞=1,if𝑞=𝑞₀0,if𝑞≠𝑞₀,𝜈_𝐼𝑞=0,if𝑞=𝑞₀1,if𝑞≠𝑞₀.(14)
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.

Proposition 12. Let 𝐴 be an IFS in a nonempty set Σ^∗. Then the following statements are equivalent:(i)𝐴can be accepted by an IFPD 𝐴^∅ℳ=(𝑄,Σ,Γ,𝛿,𝐼,𝑍₀,∅);(ii)There exists an IFPD 𝐴^∅ℳ^′=(𝑄^′,Σ,Γ^′,𝛿^′,𝑞₀,𝑋₀,∅)recognizing 𝐴, where 𝑞₀∈𝑄^′.

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 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.

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^′,𝑞^′₀,𝑍₀,𝐹^′) as follows:(i)𝑄^′=𝑄×(𝐿₁−{0})×(𝐿₂−{1}), where 𝐿₁=𝑋_∧, 𝐿₂=𝑌_∨, 𝑋=Im(𝜇_𝛿)∪Im(𝜇_𝐹) and 𝑌=Im(𝜈_𝛿)∪Im(𝜈_𝐹);(ii)𝑞^′₀=(𝑞₀,1,0)∈𝑄^′;(iii)𝛿^′=(𝜇_𝛿′,𝜈_𝛿′) is an IFS in 𝑄^′×(Σ∪{𝜀})×Γ×𝑄^′×Γ^∗, where the mappings 𝜇_𝛿′,𝜈_𝛿′∶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^∗, 𝜇_𝛿′((𝑞,𝑎,𝑏),𝜏,𝑋,(𝑞^′,𝑐,𝑑),𝛾)=1, and 𝜈_𝛿′((𝑞,𝑎,𝑏),𝜏,𝑋,(𝑞^′,𝑐,𝑑),𝛾)=0 whenever there exist 𝑎^′ and 𝑏^′ such that 𝜇_𝛿(𝑞,𝜏,𝑋,𝑞^′,𝛾)=𝑎^′>0, 𝜈_𝛿(𝑞,𝜏,𝑋,𝑞^′,𝛾)=𝑏^′<1, 𝑐=𝑎∧𝑎^′ and 𝑑=𝑏∨𝑏^′. Otherwise, 𝜇_𝛿′((𝑞,𝑎,𝑏),𝜏,𝑋,(𝑞^′,𝑐,𝑑),𝛾)=0 and 𝜈_𝛿′((𝑞,𝑎,𝑏),𝜏,𝑋,(𝑞^′,𝑐,𝑑),𝛾)=1;(iv)𝐹^′=(𝜇_𝐹′,𝜈_𝐹′) is an IFS in 𝑄^′. For any (𝑞,𝑎,𝑏)∈𝑄^′, 𝜇_𝐹′𝑞,𝑎,𝑏=𝑎∧𝜇_𝐹𝑞𝜈,if0≤𝑎+𝑏≤10,if𝑎+𝑏>1,_𝐹′𝑞,𝑎,𝑏=𝑏∨𝜈_𝐹𝑞,if0≤𝑎+𝑏≤11,if𝑎+𝑏>1.(15)
Now, it is claimed that, for any 𝜔=𝜏₁⋯𝜏_𝑛∈Σ^∗, 𝜏_𝑖∈Σ∪{𝜀}, 𝑖∈{1,…,𝑛} and for any (𝑞_𝑛,𝑎_𝑛,𝑏_𝑛)∈𝑄^′, 𝑍_𝑛,𝛾_𝑛∈Γ^∗, 𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑍₀),((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑍_𝑛𝛾_𝑛))=1 and 𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔,𝑍₀),((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑍_𝑛𝛾_𝑛))=0 whenever the following condition is satisfied.
(P1) There exist 𝑞₁,…,𝑞_𝑛−1∈𝑄, 𝑍₁,…,𝑍_𝑛−1∈Γ and 𝛾₁,…,𝛾_𝑛−1∈Γ^∗ such that 𝑎_𝑛=𝜇_⊢ℳ((𝑞₀,𝜔,𝑍₀), (𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∧𝜇_⊢ℳ((𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁), (𝑞₂,𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂))∧⋯∧𝜇_⊢ℳ((𝑞_𝑛−1,𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1), (𝑞_𝑛,𝜀,𝑍_𝑛𝛾_𝑛)) and 𝑏_𝑛=𝜈_⊢ℳ((𝑞₀,𝜔,𝑍₀), (𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∨𝜈_⊢ℳ((𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁), (𝑞₂,𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂))∨⋯∨𝜈_⊢ℳ((𝑞_𝑛−1,𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1), (𝑞_𝑛,𝜀,𝑍_𝑛𝛾_𝑛)). Otherwise, 𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑍₀),((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑍_𝑛𝛾_𝑛))=0 and 𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔,𝑍₀),((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑍_𝑛𝛾_𝑛))=1.
It is proved by induction. In fact, if |𝜔|=0, then 𝜔=𝜀, 𝜇_⊢∗ℳ′((𝑞₀,𝜀,𝑍₀),(𝑞₀,𝜀,𝑍₀))=1 and 𝜈_⊢^∗_ℳ((𝑞₀,𝜀,𝑍₀),(𝑞₀,𝜀,𝑍₀))=0. Hence 𝜇_⊢∗ℳ(((𝑞₀,1,0),𝜀,𝑍₀),((𝑞₀,1,0),𝜀,𝑍₀))=1 and 𝜈_⊢^∗_ℳ′(((𝑞₀,1,0),𝜀,𝑍₀),((𝑞₀,1,0),𝜀,𝑍₀))=0.
Suppose the result still holds whenever |𝜔|≤𝑛, 𝑛∈𝑁. If |𝜔|=𝑛+1, 𝜔=𝜏₁⋯𝜏_𝑘𝜏_𝑘+1=𝑥𝜏_𝑘+1 and 𝜏_𝑘+1∈Σ, then |𝑥|=𝑛 and 𝑥=𝜏₁⋯𝜏_𝑘.
Next, for any (𝑞_𝑘+1,𝑎_𝑘+1,𝑏_𝑘+1)∈𝑄^′, 𝑍_𝑘+1,𝛾_𝑘+1∈Γ^∗, whenever (P1) is satisfied; that is, there exists a sequence of states 𝑞₁,…,𝑞_𝑘∈𝑄, 𝑍₁,…,𝑍_𝑘∈Γ, 𝛾₁,…,𝛾_𝑘∈Γ^∗ such that 𝜇_⊢ℳ((𝑞_𝑖,𝜏_𝑖+1⋯𝜏_𝑘+1,𝑍_𝑖𝛾_𝑖), (𝑞_𝑖+1,𝜏_𝑖+2⋯𝜏_𝑘+1, 𝑍_𝑖+1𝛾_𝑖+1)) = 𝑐_𝑖+1>0, 𝜈_⊢ℳ((𝑞_𝑖,𝜏_𝑖+1⋯𝜏_𝑘+1,𝑍_𝑖𝛾_𝑖), (𝑞_𝑖+1,𝜏_𝑖+2⋯𝜏_𝑘+1, 𝑍_𝑖+1𝛾_𝑖+1)) = 𝑑_𝑖+1<1, 𝜇_𝛿(𝑞_𝑘,𝜏_𝑘+1,𝑍_𝑘,𝑞_𝑘+1,𝑍_𝑘+1)=𝑐_𝑘+1>0,  𝜈_𝛿(𝑞_𝑘,𝜏_𝑘+1,𝑍_𝑘,𝑞_𝑘+1,𝑍_𝑘+1)=𝑑_𝑘+1<1, where 𝛾₀=𝜀, 𝑖=0,1,…,𝑘−1.
Let 𝑎_𝑙=𝑐₁∧⋯∧𝑐_𝑙, 𝑏_𝑙=𝑑₁∨⋯∨𝑑_𝑙, 𝑙∈{1,…,𝑘+1}. Then 𝜇_⊢ℳ′(((𝑞_𝑖,𝑎_𝑖,𝑏_𝑖),𝜏_𝑖+1⋯𝜏_𝑘+1,𝑍_𝑖𝛾_𝑖), ((𝑞_𝑖+1,𝑎_𝑖+1,𝑏_𝑖+1), 𝜏_𝑖+2⋯𝜏_𝑘+1,𝑍_𝑖+1𝛾_𝑖+1)) = 1, 𝜈_⊢ℳ′(((𝑞_𝑖,𝑎_𝑖,𝑏_𝑖),𝜏_𝑖+1⋯𝜏_𝑘+1,𝑍_𝑖𝛾_𝑖), ((𝑞_𝑖+1,𝑎_𝑖+1,𝑏_𝑖+1), 𝜏_𝑖+2⋯𝜏_𝑘+1,𝑍_𝑖+1𝛾_𝑖+1))=0, 𝜇_⊢ℳ′(((𝑞_𝑘,𝑎_𝑘,𝑏_𝑘),𝜏_𝑘+1,𝑍_𝑘𝛾_𝑘), ((𝑞_𝑘+1,𝑎_𝑘+1,𝑏_𝑘+1), 𝜀,𝑍_𝑘+1𝛾_𝑘+1))=1, and 𝜈_⊢ℳ′(((𝑞_𝑘,𝑎_𝑘,𝑏_𝑘),𝜏_𝑘+1,𝑍_𝑘𝛾_𝑘), ((𝑞_𝑘+1,𝑎_𝑘+1,𝑏_𝑘+1), 𝜀,𝑍_𝑘+1𝛾_𝑘+1))=0,where 𝑎₀=1, 𝑏₀=0, 𝛾₀=𝜀, 𝑖=0,1,…,𝑘−1.
By assumption, 𝜇_⊢∗ℳ′((𝑞^′₀,𝑥,𝑍₀), ((𝑞_𝑘,𝑎_𝑘,𝑏_𝑘),𝜀,𝑍_𝑘𝛾_𝑘))=1,  𝜈_⊢^∗_ℳ′((𝑞^′₀,𝑥,𝑍₀), ((𝑞_𝑘,𝑎_𝑘,𝑏_𝑘),𝜀,𝑍_𝑘𝛾_𝑘))=0, and so 𝜇_⊢∗ℳ′((𝑞^′₀,𝑥𝜏_𝑘+1,𝑍₀), ((𝑞_𝑘,𝑎_𝑘,𝑏_𝑘),𝜏_𝑘+1,𝑍_𝑘𝛾_𝑘))=1, 𝜈_⊢^∗_ℳ′((𝑞^′₀,𝑥𝜏_𝑘+1,𝑍₀), ((𝑞_𝑘,𝑎_𝑘,𝑏_𝑘), 𝜏_𝑘+1,𝑍_𝑘𝛾_𝑘))=0.
Therefore, 𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑍₀), ((𝑞_𝑘+1,𝑎_𝑘+1,𝑏_𝑘+1), 𝜀,𝑍_𝑘+1𝛾_𝑘+1))=⋁{𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑍₀), ((𝑞_𝑘,𝑎_𝑘,𝑏_𝑘), 𝜏_𝑘+1,𝑍_𝑘𝛾_𝑘))∧𝜇_⊢ℳ′(((𝑞_𝑘,𝑎_𝑘,𝑏_𝑘),𝜏_𝑘+1,𝑍_𝑘𝛾_𝑘), ((𝑞_𝑘+1,𝑎_𝑘+1,𝑏_𝑘+1), 𝜀,𝑍_𝑘+1𝛾_𝑘+1))∣(𝑞_𝑘,𝑎_𝑘,𝑏_𝑘)∈𝑄^′, 𝑍_𝑘∈Γ,𝛾_𝑘∈Γ^∗}=1∧1=1.
𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔,𝑍₀), ((𝑞_𝑘+1,𝑎_𝑘+1,𝑏_𝑘+1), 𝜀,𝑍_𝑘+1𝛾_𝑘+1)) = ⋀{𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔,𝑍₀), ((𝑞_𝑘,𝑎_𝑘,𝑏_𝑘), 𝜏_𝑘+1,𝑍_𝑘𝛾_𝑘))∨𝜈_⊢ℳ′(((𝑞_𝑘,𝑎_𝑘,𝑏_𝑘),𝜏_𝑘+1,𝑍_𝑘𝛾_𝑘), ((𝑞_𝑘+1,𝑎_𝑘+1,𝑏_𝑘+1), 𝜀,𝑍_𝑘+1𝛾_𝑘+1))∣(𝑞_𝑘,𝑎_𝑘,𝑏_𝑘)∈𝑄^′,𝑍_𝑘∈Γ,𝛾_𝑘∈Γ^∗}=0∨0=0.
For any (𝑞_𝑘+1,𝑎_𝑘+1,𝑏_𝑘+1)∈𝑄^′, 𝑍_𝑘+1,𝛾_𝑘+1∈Γ^∗, if (P1) is not satisfied, then it follows that 𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑍₀),((𝑞_𝑘+1,𝑎_𝑘+1,𝑏_𝑘+1),𝜀,𝑍_𝑘+1𝛾_𝑘+1))=0,  𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔,𝑍₀),((𝑞_𝑘+1,𝑎_𝑘+1,𝑏_𝑘+1),𝜀,𝑍_𝑘+1𝛾_𝑘+1))=1.
Hence, for any 𝜔=𝜏₁⋯𝜏_𝑛∈Σ^∗, 𝜏_𝑖∈Σ∪{𝜀}, 𝑖∈{1,…,𝑛}, we have 𝜇_ℒ(ℳ′)(𝜔)=⋁{𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑍₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀, 𝛾))∧𝜇_𝐹′((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛))∣(𝑞_𝑛, 𝑎_𝑛,𝑏_𝑛)∈𝑄^′,𝛾∈Γ^∗} = ⋁{𝑎_𝑛∧𝜇_𝐹(𝑞_𝑛)∣ 𝑞₁,…,𝑞_𝑛−1∈𝑄, 𝑍₁,…,𝑍_𝑛−1∈Γ, 𝛾₁,…,𝛾_𝑛−1∈Γ^∗, 𝑎_𝑛=𝜇_⊢ℳ((𝑞₀,𝜔,𝑍₀), (𝑞₁,𝜏₂⋯𝜏_𝑛, 𝑍₁𝛾₁))∧𝜇_⊢ℳ((𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁), (𝑞₂,𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂))∧⋯∧𝜇_⊢ℳ((𝑞_𝑛−1, 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1),(𝑞_𝑛,𝜀,𝛾)), 𝑏_𝑛=𝜈_⊢ℳ((𝑞₀,𝜔,𝑍₀), (𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∨𝜈_⊢ℳ((𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁),    (𝑞₂,𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂))∨⋯∨𝜈_⊢ℳ((𝑞_𝑛−1,  𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1), (𝑞_𝑛,𝜀,𝛾))}= ⋁{𝜇_⊢ℳ((𝑞₀,𝜔,𝑍₀),(𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∧𝜇_⊢ℳ((𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁),(𝑞₂,𝜏₃⋯𝜏_𝑛, 𝑍₂𝛾₂))∧⋯∧𝜇_⊢ℳ((𝑞_𝑛−1, 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1),(𝑞_𝑛,𝜀,𝛾))∧𝜇_𝐹(𝑞_𝑛)∣ 𝑞₁,…,𝑞_𝑛−1∈𝑄,𝑍₁,…,𝑍_𝑛−1∈Γ, 𝛾₁,…,𝛾_𝑛−1∈Γ^∗}=𝜇_ℒ(ℳ)(𝜔), 𝜈_ℒ(ℳ′)(𝜔)=⋀{𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔,𝑍₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀, 𝛾))∨𝜈_𝐹′((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛))∣(𝑞_𝑛,𝑎_𝑛,𝑏_𝑛)∈𝑄^′, 𝛾∈Γ^∗}=⋀{𝑏_𝑛∨𝜈_𝐹(𝑞_𝑛)∣ 𝑞₁,…,𝑞_𝑛−1∈𝑄, 𝑍₁,…,𝑍_𝑛−1∈Γ, 𝛾₁,…,𝛾_𝑛−1∈Γ^∗, 𝑎_𝑛=𝜇_⊢ℳ((𝑞₀,𝜔,𝑍₀),(𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∧𝜇_⊢ℳ((𝑞₁, 𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁),(𝑞₂,𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂))∧⋯∧𝜇_⊢ℳ((𝑞_𝑛−1, 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1),(𝑞_𝑛,𝜀,𝛾)), 𝑏_𝑛=𝜈_⊢ℳ((𝑞₀,𝜔,𝑍₀), (𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∨𝜈_⊢ℳ((𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁), (𝑞₂,𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂))∨⋯∨𝜈_⊢ℳ((𝑞_𝑛−1, 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1), (𝑞_𝑛,𝜀,𝛾))}= ⋀{𝜈_⊢ℳ((𝑞₀,𝜔,𝑍₀), (𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∨𝜈_⊢ℳ((𝑞₁, 𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁),(𝑞₂,𝜏₃⋯𝜏_𝑛, 𝑍₂𝛾₂))∨⋯∨𝜈_⊢ℳ((𝑞_𝑛−1,𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1), (𝑞_𝑛,𝜀,𝛾))∨𝜈_𝐹(𝑞_𝑛)∣ 𝑞₁,…,𝑞_𝑛−1∈𝑄,𝑍₁,…,𝑍_𝑛−1∈Γ, 𝛾₁,…,𝛾_𝑛−1∈Γ^∗}=𝜈_ℒ(ℳ)(𝜔).
Therefore, ℒ(ℳ^′)=ℒ(ℳ).

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 ℒ(ℳ)=𝜔∈Σ^∗∣𝑞₀,𝜔,𝑍₀≻^∗_ℳ𝑝,𝜀,𝛾,𝑝∈𝐹,𝛾∈Γ^∗.(16)

Definition 15. A collection of classical pushdown automata with final states 𝑆=ℳ_𝑎𝑏∣ℳ_𝑎𝑏=𝑄,Σ,Γ,𝛿,𝑞₀,𝑍₀,𝐹_𝑎𝑏,0≤𝑎+𝑏≤1,𝑎,𝑏∈[0,1](17) is called a cover if the following conditions hold:(i)𝑎₁≤𝑎₂ and 𝑏₂≤𝑏₁ imply 𝐹_𝑎2𝑏2⊆𝐹_𝑎1𝑏1;(ii)𝐹₀₁=𝑄.
For a cover 𝑆, its recognized intuitionistic fuzzy language 𝑓_𝑆=(𝜇_𝑓𝑆,𝜈_𝑓𝑆) in Σ^∗ is given by 𝜇_𝑓𝑆(𝜔)=⋁{𝑎∈[0,1]∣ℳ_𝑎𝑏 accepts 𝜔,ℳ_𝑎𝑏∈𝑆}, 𝜈_𝑓𝑆(𝜔)=⋀{𝑏∈[0,1]∣ℳ_𝑎𝑏 accepts 𝜔,ℳ_𝑎𝑏∈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 𝑆=ℳ_𝑎𝑏∣ℳ_𝑎𝑏=𝑄,Σ,Γ,𝛿^′,𝑞₀,𝑍₀,𝐹_𝑎𝑏,0≤𝑎+𝑏≤1,𝑎,𝑏∈[0,1],(18) where 𝐹_𝑎𝑏={𝑞∈𝑄∣𝜇_𝐹(𝑞)≥𝑎,𝜈_𝐹(𝑞)≤𝑏}; the mapping 𝛿^′∶𝑄×(Σ∪{𝜀})×Γ→2^𝑄×Γ∗ is given by 𝛿^′(𝑞,𝜏,𝑍)={(𝑝,𝛾)∣𝜇_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=1,𝑝∈𝑄,𝛾∈Γ^∗}, for all (𝑞,𝜏,𝑍)∈𝑄×(Σ∪{𝜀})×Γ.
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,for all 𝑎,𝑏∈[0,1] with 𝑎+𝑏≤1, for all 𝜔∈Σ^∗,𝛾∈Γ^∗. 𝜇_𝑓𝑆(𝜔)=⋁{𝑎∈[0,1]∣ℳ_𝑎𝑏 accepts 𝜔,ℳ_𝑎𝑏∈𝑆} =⋁{𝑎∈[0,1]∣𝑞∈𝐹_𝑎𝑏,(𝑞₀,𝜔,𝑍₀)≻^∗_ℳ𝑎𝑏(𝑞,𝜀,𝛾),𝛾∈Γ^∗} =⋁{𝑎∈[0,1]∣𝜇_𝐹(𝑞)≥𝑎,𝜈_𝐹(𝑞)≤𝑏,(𝑞₀,𝜔,𝑍₀)≻^∗_ℳ𝑎𝑏(𝑞,𝜀,𝛾),𝛾∈Γ^∗} =⋁{𝑎∈[0,1]∣𝜇_𝐹(𝑞)≥𝑎,(𝑞₀,𝜔,𝑍₀)≻^∗_ℳ𝑎𝑏(𝑞,𝜀,𝛾),𝑞∈𝑄,𝛾∈Γ^∗} =⋁{𝜇_𝐹(𝑞)∣𝜇_⊢∗ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾))=1,𝑞∈𝑄,𝛾∈Γ^∗}=𝜇_𝑓(𝜔)=𝜇_ℒ(ℳ)(𝜔), 𝜈_𝑓𝑆(𝜔)=⋀{𝑏∈[0,1]∣ℳ_𝑎𝑏 accepts 𝜔,ℳ_𝑎𝑏∈𝑆} =⋀{𝑏∈[0,1]∣𝑞∈𝐹_𝑎𝑏,(𝑞₀,𝜔,𝑍₀)≻^∗_ℳ𝑎𝑏(𝑞,𝜀,𝛾),𝛾∈Γ^∗} =⋀{𝑏∈[0,1]∣𝜇_𝐹(𝑞)≥𝑎,𝜈_𝐹(𝑞)≤𝑏,(𝑞₀,𝜔,𝑍₀)≻^∗_ℳ𝑎𝑏(𝑞,𝜀,𝛾),𝛾∈Γ^∗} =⋀{𝑏∈[0,1]∣𝜈_𝐹(𝑞)≤𝑏,(𝑞₀,𝜔,𝑍₀)≻^∗_ℳ𝑎𝑏(𝑞,𝜀,𝛾),𝑞∈𝑄,𝛾∈Γ^∗} =⋀{𝜈_𝐹(𝑞)∣𝜇_⊢∗ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾))=1,𝑞∈𝑄,𝛾∈Γ^∗}=𝜈_𝑓(𝜔)=𝜈_ℒ(ℳ)(𝜔).
Therefore, 𝑓_𝑆=ℒ(ℳ)=𝑓.
Conversely, suppose 𝑓 can be recognized by a cover 𝑆=ℳ_𝑎𝑏∣ℳ_𝑎𝑏=𝑄,Σ,Γ,𝛿^′,𝑞₀,𝑍₀,𝐹_𝑎𝑏,0≤𝑎+𝑏≤1,𝑎,𝑏∈[0,1].(19) Then we construct an IFSPDA ℳ=(𝑄,Σ,Γ,𝜂,𝑞₀,𝑍₀,𝐹), where 𝜂=(𝜇_𝜂,𝜈_𝜂) is an IFS in 𝑄×(Σ∪{𝜀})×Γ×𝑄×Γ^∗, and the mappings 𝜇_𝜂,𝜈_𝜂∶𝑄×(Σ∪{𝜀})×Γ×𝑄×Γ^∗→{0,1} are defined as 𝜇_𝜂𝑞,𝜏,𝑍,𝑝,𝛾=1,if𝑝,𝛾∈𝛿^′𝜈𝑞,𝜏,𝑍0,otherwise,_𝜂𝑞,𝜏,𝑍,𝑝,𝛾=0,if𝑝,𝛾∈𝛿^′𝑞,𝜏,𝑍1,otherwise,(20) for any (𝑞,𝜏,𝑍,𝑝,𝛾)∈𝑄×(Σ∪{𝜀})×Γ×𝑄×Γ^∗.
𝐹=(𝜇_𝐹,𝜈_𝐹) is an IFS in 𝑄, where 𝜇_𝐹(𝑞)=⋁{𝑎∈[0,1]∣𝑞∈𝐹_𝑎𝑏} and 𝜈_𝐹(𝑞)=⋀{𝑏∈[0,1]𝑞∈𝐹_𝑎𝑏},forall𝑞∈𝑄. Then ℒ(ℳ)=𝑓_𝑆. In fact, for any 𝜔∈Σ^∗, 𝜇_ℒ(ℳ)(𝜔)=⋁{𝜇_𝐹(𝑞)∣𝜇_⊢∗ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾))=1,𝑞∈𝑄,𝛾∈Γ^∗}=⋁{⋁{𝑎∈[0,1]∣𝑞∈𝐹_𝑎𝑏𝜇}∣_⊢∗ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾))=1,𝑞∈𝑄,𝛾∈Γ^∗}=⋁{𝑎∈[0,1]∣𝑞∈𝐹_𝑎𝑏,𝜇_⊢∗ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾))=1,𝑞∈𝑄,𝛾∈Γ^∗} =⋁{𝑎∈[0,1]∣𝑞∈𝐹_𝑎𝑏,(𝑞₀,𝜔,𝑍₀)≻^∗_ℳ𝑎𝑏(𝑞,𝜀,𝛾),𝛾∈Γ^∗} =⋁{𝑎∈[0,1]∣ℳ_𝑎𝑏 accepts 𝜔,ℳ_𝑎𝑏∈𝑆}=𝜇_𝑓𝑆(𝜔), 𝜈_ℒ(ℳ)(𝜔)=⋀{𝜈_𝐹(𝑞)∣𝜇_⊢∗ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾))=1,𝑞∈𝑄,𝛾∈Γ^∗}=⋀{⋀{𝑏∈[0,1]∣𝑞∈𝐹_𝑎𝑏𝜇}∣_⊢∗ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾))=1,𝑞∈𝑄,𝛾∈Γ^∗}=⋀{𝑏∈[0,1]∣𝑞∈𝐹_𝑎𝑏,(𝑞₀,𝜔,𝑍₀)≻^∗_ℳ𝑎𝑏(𝑞,𝜀,𝛾),𝛾∈Γ^∗} =⋀{𝑏∈[0,1]∣ℳ_𝑎𝑏 accepts 𝜔,ℳ_𝑎𝑏∈𝑆}=𝜈_𝑓𝑆(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,…,ℒ_𝑛⊆Σ^∗, and (𝑎_𝑖,𝑏_𝑖)∈(0,1]×[0,1) with 0≤𝑎_𝑖+𝑏_𝑖≤1 for 𝑖=1,…,𝑛 such that 𝐴=𝜇_𝐴,𝜈_𝐴=^𝑛_𝑖=1𝑎_𝑖,𝑏_𝑖⋅𝟏_ℒ𝐢,(⋆) where 𝟏_ℒ𝐢=(𝜇_𝟏ℒ𝐢,𝜈_𝟏ℒ𝐢) represents the intuitionistic characterized function of ℒ_𝑖, 𝑖=1,…,𝑛, that is, 𝜇_𝟏ℒ𝐢(𝜔)=1,if𝜔∈ℒ_𝑖0,if𝜔∉ℒ_𝑖,𝜈_𝟏ℒ𝐢(𝜔)=0,if𝜔∈ℒ_𝑖1,if𝜔∉ℒ_𝑖.(21) And the equation (⋆) means that the following equations hold: 𝜇_𝐴(𝜔)=^𝑛_𝑖=1𝑎_𝑖∧𝜇_𝟏ℒ𝐢(𝜔),𝜈_𝐴(𝜔)=^𝑛_𝑖=1𝑏_𝑖∨𝜈_𝟏ℒ𝐢(𝜔),∀𝜔∈Σ^∗.(22)

Noting that the family of all the intuitionistic fuzzy recognizable step functions over Σ^∗ is denoted by Step^𝐶(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)∈ Ste𝑝^𝐶(Σ).

Proof. By Proposition 14, there is an IFSPDA ℳ=(𝑄,Σ,Γ,𝛿,𝑞₀,𝑍₀,𝐹) equivalent to ℳ^′₁.
Let 𝑅={(𝜇_𝐹(𝑞),𝜈_𝐹(𝑞))∣𝑞∈𝑄}⧵{(0,1)}={(𝑎_𝑖,𝑏_𝑖)∣𝑖∈𝑁_𝑘}, 𝑁_𝑘={1,…,𝑘}. Put 𝐹_𝑖={𝑞∈𝑄∣𝜇_𝐹(𝑞)=𝑎_𝑖,𝜈_𝐹(𝑞)=𝑏_𝑖}, for all 𝑖∈𝑁_𝑘. Then we construct a PDA ℳ_𝑖=(𝑄,Σ,Γ,𝛿^′,𝑞₀,𝑍₀,𝐹_𝑖), where the mapping 𝛿^′∶𝑄×(Σ∪{𝜀})×Γ→2^𝑄×Γ∗ is defined by 𝛿^′(𝑞,𝜏,𝑋)={(𝑝,𝛾)∣𝜇_𝛿(𝑞,𝜏,𝑋,𝑝,𝛾)=1,𝑝∈𝑄,𝛾∈Γ^∗}, for all (𝑞,𝜏,𝑋)∈𝑄×(Σ∪{𝜀})×Γ.
Then ℒ(ℳ_𝑖)=ℒ_𝑖={𝜔∈Σ^∗∣(𝑞₀,𝜔, 𝑍₀)≻^∗_ℳ𝑖(𝑞,𝜀,𝛾), 𝑞∈𝐹_𝑖,𝛾∈Γ^∗}.
Therefore, for any 𝜔∈Σ^∗, we have 𝜇_ℒ(ℳ′1)(𝜔) = 𝜇_ℒ(ℳ)(𝜔) = ⋁{𝜇_𝐹(𝑞)∣𝜇_⊢∗ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾)) = 1,𝑞∈𝑄,𝛾∈Γ^∗}=⋁{𝑎_𝑖∣𝜇_⊢∗ℳ((𝑞₀,𝜔,𝑍₀),   (𝑞,𝜀,𝛾))=1,𝜇_𝐹(𝑞) = 𝑎_𝑖,𝜈_𝐹(𝑞)=𝑏_𝑖,𝑖∈𝑁_𝑘,𝛾∈Γ^∗}=⋁{𝑎_𝑖∣(𝑞₀,𝜔,𝑍₀)≻^∗_ℳ𝑖(𝑞,𝜀,𝛾),𝑞∈𝐹_𝑖,𝑖∈𝑁_𝑘,𝛾∈Γ^∗}=⋁{𝑎_𝑖∣𝜔∈ℒ(ℳ_𝑖),𝑖∈𝑁_𝑘}=⋁_{𝑖∈𝑁𝑘}𝑎_𝑖∧𝜇_{𝟏𝐢)ℒ(ℳ}(𝜔), and 𝜈_ℒ(ℳ′1)(𝜔)=𝜈_ℒ(ℳ)(𝜔)=⋀{𝜈_𝐹(𝑞)∣𝜈_⊢^∗_ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾))=0,𝑞∈𝑄,𝛾∈Γ^∗}=⋀{𝑏_𝑖∣𝜇_⊢∗ℳ((𝑞₀,𝜔,𝑍₀), (𝑞,𝜀,𝛾))=1,𝜇_𝐹(𝑞)=𝑎_𝑖,𝜈_𝐹(𝑞)=𝑏_𝑖,𝑖∈𝑁_𝑘,𝛾∈Γ^∗}=⋀{𝑏_𝑖∣(𝑞₀,𝜔,𝑍₀)≻^∗_ℳ𝑖(𝑞,𝜀,𝛾),𝑞∈𝐹_𝑖,𝑖∈𝑁_𝑘Γ,𝛾∈_∗}=⋀{𝑏_𝑖∣𝜔∈ℒ(ℳ_𝑖),𝑖∈𝑁_𝑘}=⋀_{𝑖∈𝑁𝑘}𝑏_𝑖∨𝜈_{𝟏𝐢)ℒ(ℳ}(𝜔).
So ℒ(ℳ)=∐_{𝑖∈𝑁𝑘}(𝑎_𝑖,𝑏_𝑖)⋅𝟏_ℒ𝐢.

Proposition 18 shows that the set of the languages recognized by all the IFPDAs is a subset of Step^𝐶(Σ). In fact, Step^𝐶(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∈ Step^𝐶(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∈ Step^𝐶(Σ). Then there is a finite natural number 𝑛∈𝑁, recognizable context-free languages ℒ₁,…,ℒ_𝑛⊆Σ^∗, and (𝑎_𝑖,𝑏_𝑖)∈(0,1]×[0,1) with 0≤𝑎_𝑖+𝑏_𝑖≤1 for 𝑖=1,…,𝑛 such that 𝐴=(𝜇_𝐴,𝜈_𝐴)=∐^𝑛_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅𝟏_ℒ𝐢. Let ℒ_𝑖 be recognized by a PDA ℳ_𝑖=(𝑄_𝑖,Σ,Γ_𝑖,𝛿_𝑖,𝑞_0𝑖,𝑍_0𝑖,𝐹_𝑖), and 𝑄_𝑖∩𝑄_𝑗=∅ whenever 𝑖≠𝑗, 𝑖,𝑗∈𝑁_𝑛.
Next, construct an IFSPDA ℳ=(𝑄,Σ,Γ,𝛿,𝑞₀,𝑍₀,𝐹) as follows: 𝑄=⋃^𝑛_𝑖=1𝑄_𝑖∪{𝑞₀}, Γ=⋃^𝑛_𝑖=1Γ_𝑖∪{𝑍₀}, where 𝑞₀∉⋃^𝑛_𝑖=1𝑄_𝑖, 𝑍₀∉⋃^𝑛_𝑖=1Γ_𝑖. 𝛿=(𝜇_𝛿,𝜈_𝛿) is an IFS over 𝑄×(Σ∪{𝜀})×Γ×𝑄×Γ^∗, where the mappings 𝜇_𝛿,𝜈_𝛿∶𝑄×(Σ∪{𝜀})×Γ×𝑄×Γ^∗→{0,1} are defined by 𝜇_𝛿(𝑞₀,𝜀,𝑍₀,𝑝,𝛾)=1 and 𝜈_𝛿(𝑞₀,𝜀,𝑍₀,𝑝,𝛾)=0 if (𝑝,𝛾)=(𝑞_0𝑖,𝑍_0𝑖), 𝑖∈𝑁_𝑛; 𝜇_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=𝜇_𝛿𝑖(𝑞,𝜏,𝑍,𝑝,𝛾) and 𝜈_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=𝜈_𝛿𝑖(𝑞,𝜏,𝑍,𝑝,𝛾) if 𝑝,𝑞∈𝑄_𝑖, 𝑍∈Γ_𝑖,𝛾∈Γ^∗_𝑖, 𝜏∈Σ∪{𝜀}, 𝑖∈𝑁_𝑛. Otherwise, 𝜇_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=0 and 𝜈_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=1.
𝐹=(𝜇_𝐹,𝜈_𝐹) is an IFS in 𝑄, where 𝜇_𝐹𝑞₀=𝑎_𝑖∈[0,1]∣𝑞_0𝑖∈𝐹_𝑖,𝑖∈𝑁_𝑛,𝜈_𝐹𝑞₀=𝑏_𝑖∈[0,1]∣𝑞_0𝑖∈𝐹_𝑖,𝑖∈𝑁_𝑛,𝜇_𝐹𝑞=𝑎_𝑖,if𝑞∈𝐹_𝑖0,if𝑞∉𝐹_𝑖∪𝑞₀,𝜈_𝐹𝑞=𝑏_𝑖,if𝑞∈𝐹_𝑖1,if𝑞∉𝐹_𝑖∪𝑞₀.(23)
Therefore, for any 𝜔∈Σ^∗, we have 𝜇_ℒ(ℳ)(𝜔)=⋁{𝜇_⊢∗ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾))∧𝜇_𝐹 (𝑞)∣𝑞∈𝑄,𝛾∈Γ^∗},  𝜈_ℒ(ℳ)(𝜔)=⋀{𝜈_⊢^∗_ℳ((𝑞₀,𝜔,𝑍₀),(𝑞,𝜀,𝛾))∨𝜈_𝐹 (𝑞)∣𝑞∈𝑄,𝛾∈Γ^∗}.
If 𝜔=𝜀, then 𝜇_ℒ(ℳ)(𝜀)=(𝜇_⊢∗ℳ((𝑞₀,𝜀,𝑍₀),(𝑞₀,𝜀,𝑍₀))∧𝜇_𝐹(𝑞₀))∨(⋁{𝜇_⊢∗ℳ((𝑞₀,𝜀,𝑍₀),(𝑞_0𝑖,𝜀,𝑍_0𝑖))∧𝜇_𝐹(𝑞_0𝑖)∣𝑞_0𝑖∈𝑄_𝑖,𝑍_0𝑖∈Γ_𝑖,𝑖∈𝑁_𝑛})=𝜇_𝐹(𝑞₀)∨(⋁{𝜇_𝐹(𝑞_0𝑖)∣𝑞_0𝑖∈𝑄_𝑖,𝑍_0𝑖∈Γ_𝑖,𝑖∈𝑁_𝑛})=⋁{𝑎_𝑖∈[0,1]∣𝑞_0𝑖∈𝐹_𝑖,𝑖∈𝑁_𝑛}=⋁^𝑛_𝑖=1𝑎_𝑖∧𝜇_𝟏ℒ𝐢(𝜀); 𝜈_ℒ(ℳ)(𝜀)=(𝜈_⊢^∗_ℳ((𝑞₀,𝜀,𝑍₀),(𝑞₀,𝜀,𝑍₀))∨𝜈_𝐹(𝑞₀))∧(⋀{𝜈_⊢^∗_ℳ((𝑞₀,𝜀,𝑍₀), (𝑞_0𝑖,𝜀,𝑍_0𝑖))∨𝜈_𝐹(𝑞_0𝑖)∣𝑞_0𝑖∈𝑄_𝑖, 𝑍_0𝑖∈Γ_𝑖,𝑖∈𝑁_𝑛})=𝜈_𝐹(𝑞₀)∧(⋀{𝜈_𝐹(𝑞_0𝑖)∣𝑞_0𝑖∈𝑄_𝑖, 𝑍_0𝑖∈Γ_𝑖,𝑖∈𝑁_𝑛})=⋀{𝑏_𝑖∈[0,1]∣𝑞_0𝑖∈𝐹_𝑖, 𝑖∈𝑁_𝑛}=⋀^𝑛_𝑖=1𝑏_𝑖∨𝜈_𝟏ℒ𝐢(𝜀). If 𝜔∈Σ^∗⧵{𝜀}, then 𝜇_ℒ(ℳ)(𝜔)=⋁{𝜇_⊢∗ℳ((𝑞_0𝑖,𝜔,𝑍_0𝑖), (𝑞,𝜀,𝛾))∧𝜇_𝐹(𝑞)∣𝑞_0𝑖∈𝑄_𝑖, 𝛾∈Γ^∗,𝑞∈𝑄,𝑖∈𝑁_𝑛}=⋁{𝜇_𝐹(𝑞)∣𝑞_0𝑖∈𝑄_𝑖, 𝛾∈Γ^∗_𝑖,𝑞∈𝑄,𝑖∈𝑁_𝑛,𝜇_⊢∗ℳ((𝑞_0𝑖,𝜔,𝑍_0𝑖),(𝑞,𝜀,𝛾))=1} = ⋁{𝑎_𝑖∣𝑞∈𝐹_𝑖,𝑖∈𝑁_𝑛,𝜇_⊢∗ℳ((𝑞_0𝑖,𝜔,𝑍_0𝑖),(𝑞,𝜀,𝛾)) = 1,𝛾∈Γ^∗_𝑖}=⋁{𝑎_𝑖∣𝜔∈ℒ_𝑖,𝑖∈𝑁_𝑛}=⋁^𝑛_𝑖=1𝑎_𝑖∧𝜇_𝟏ℒ𝐢(𝜔), 𝜈_ℒ(ℳ)(𝜔)=⋀{𝜈_⊢^∗_ℳ((𝑞_0𝑖,𝜔,𝑍_0𝑖), (𝑞,𝜀,𝛾))∨𝜈_𝐹(𝑞)|𝑞_0𝑖∈𝑄_𝑖,𝛾∈Γ^∗,𝑞∈𝑄,𝑖∈𝑁_𝑛} = ⋀{𝜈_𝐹(𝑞)∣𝑞_0𝑖∈𝑄_𝑖,𝛾∈Γ^∗_𝑖,𝑞∈𝑄,𝑖∈𝑁_𝑛, 𝜇_⊢∗ℳ((𝑞_0𝑖,𝜔,𝑍_0𝑖),(𝑞,𝜀,𝛾))=1} = ⋀{𝑏_𝑖∣𝑞∈𝐹_𝑖,𝑖∈𝑁_𝑛,𝜇_⊢∗ℳ((𝑞_0𝑖,𝜔,𝑍_0𝑖),(𝑞,𝜀,𝛾)) = 1,𝛾∈Γ^∗_𝑖}=⋀{𝑏_𝑖∣𝜔∈ℒ_𝑖,𝑖∈𝑁_𝑛} = ⋀^𝑛_𝑖=1𝑎_𝑖∨𝜈_𝟏ℒ𝐢(𝜔).
(iii) implies (ii): obviously.
(ii) implies (i): it is concluded by Proposition 18.

Next, we will discuss the characterization of IFPDA^∅.

Proposition 20. Let ℳ=(𝑄,Σ,Γ,𝛿,𝑞₀,𝑍₀,∅) be an IFPDA^∅. Then there is a special IFPDA^∅ ℳ^′=(𝑄^′,Σ,Γ^′,𝛿^′,𝑞^′₀,𝑋₀,∅) equivalent to ℳ.

Proof. Given ℳ, we construct an IFPDA^∅ ℳ^′=(𝑄^′,Σ,Γ^′,𝛿^′,𝑞^′₀,𝑋₀,∅) as follows:(1)𝑄^′=𝑄₂×(𝐿₁⧵{0})×(𝐿₂⧵{1}), where 𝑄₂=𝑄∪{𝑝₀}, Γ^′=Γ∪{𝑋₀}, 𝐿₁=𝑋_∧, 𝐿₂=𝑌_∨, 𝑋=Im(𝜇_𝛿), 𝑌=Im(𝜈_𝛿), and 𝑞^′₀=(𝑝₀,1,0);(2)𝛿^′=(𝜇_𝛿′,𝜈_𝛿′) is an IFS in 𝑄^′×(Σ∪{𝜀})×Γ^′×𝑄^′×Γ^′∗, where the mappings 𝜇_𝛿′,𝜈_𝛿′∶𝑄^′×(Σ∪{𝜀})×Γ^′×𝑄^′×Γ^′∗→[0,1] are defined by(i)𝜇_𝛿′((𝑝₀,1,0),𝜀,𝑋₀,(𝑞₀,1,0),𝑍₀𝑋₀)=1, 𝜈_𝛿′((𝑝₀,1,0),𝜀,𝑋₀,(𝑞₀,1,0),𝑍₀𝑋₀)=0;(ii)𝜇_𝛿′((𝑞,𝑎,𝑏),𝜏,𝑋,(𝑞^′,𝑎∧𝑎^′,𝑏∨𝑏^′),𝛾)=1 and 𝜈_𝛿′((𝑞,𝑎,𝑏),𝜏,𝑋,(𝑞^′,𝑎∧𝑎^′,𝑏∨𝑏^′),𝛾)=0 whenever 𝜇_𝛿(𝑞,𝜏,𝑋,𝑞^′,𝛾)=𝑎^′>0 and 𝜈_𝛿(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 𝜇_𝛿′((𝑞,𝑎,𝑏), 𝜀,𝑋₀, (𝑞,𝑎,𝑏),𝜀)=𝑎 and 𝜈_𝛿′((𝑞,𝑎,𝑏), 𝜀,𝑋₀, (𝑞,𝑎,𝑏),𝜀)=𝑏. If 𝑎+𝑏>1, then 𝜇_𝛿′((𝑞,𝑎,𝑏),𝜀,𝑋₀,(𝑞,𝑎,𝑏),𝜀)=0 and 𝜈_𝛿′((𝑞,𝑎,𝑏),𝜀,𝑋₀,(𝑞,𝑎,𝑏),𝜀)=1;(iv)In other cases, 𝜇_𝛿′((𝑞,𝑎,𝑏),𝜏,𝑋, (𝑝,𝑐,𝑑),𝛾)=0 and 𝜈_𝛿′((𝑞,𝑎,𝑏),𝜏, 𝑋,(𝑝,𝑐,𝑑),𝛾)=1.
Obviously, 𝛿^′=(𝜇_𝛿′,𝜈_𝛿′) is a finite IFS in 𝑄^′×(Σ∪{𝜀})×Γ^′×𝑄^′×Γ^′∗.
Next, we show ℒ(ℳ^′)=ℒ(ℳ).
Firstly let us show that, for any 𝜔=𝜏₁⋯𝜏_𝑛∈Σ^∗, 𝜏_𝑖∈Σ∪{𝜀}, 𝑖∈{1,…,𝑛}, 𝑞_𝑛∈𝑄, 𝜇_⊢∗ℳ′𝑞^′₀,𝜔,𝑋₀,𝑞_𝑛,𝑎_𝑛,𝑏_𝑛,𝜀,𝑋₀=()𝜈1,ifP2issatisfied0,otherwise,_⊢∗ℳ′𝑞^′₀,𝜔,𝑋₀,𝑞_𝑛,𝑎_𝑛,𝑏_𝑛,𝜀,𝑋₀=0,if(P2)issatisfied1,otherwise,(24) where the condition (P2) is the following:(P2) there exist 𝑞₁,…,𝑞_𝑛−1∈𝑄, 𝑍₁,…,𝑍_𝑛−1∈Γ, 𝛾₁,…,𝛾_𝑛−1∈Γ^∗, s.t. 𝑎_𝑛=𝜇_⊢ℳ((𝑞₀,𝜔,𝑍₀), (𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∧𝜇_⊢ℳ((𝑞₁, 𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁), (𝑞₂,𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂))∧⋯∧𝜇_⊢ℳ((𝑞_𝑛−1, 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1), (𝑞_𝑛,𝜀,𝜀))>0 and 𝑏_𝑛=𝜈_⊢ℳ((𝑞₀,𝜔,𝑍₀), (𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∨𝜈_⊢ℳ((𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁), (𝑞₂,𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂))∨⋯∨𝜈_⊢ℳ((𝑞_𝑛−1, 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1), (𝑞_𝑛,𝜀,𝜀))<1.
In fact, if (P2) is satisfied, then let 𝜇_⊢ℳ((𝑞_𝑖,𝜏_𝑖+1⋯𝜏_𝑛,𝑍_𝑖𝛾_𝑖),(𝑞_𝑖+1,𝜏_𝑖+2⋯𝜏_𝑛,𝑍_𝑖+1𝛾_𝑖+1))=𝑐_𝑖, 𝜈_⊢ℳ((𝑞_𝑖,𝜏_𝑖+1⋯𝜏_𝑛,𝑍_𝑖𝛾_𝑖),(𝑞_𝑖+1,𝜏_𝑖+2⋯𝜏_𝑛,𝑍_𝑖+1𝛾_𝑖+1))=𝑑_𝑖, 𝑖=0,1,…,𝑛−2, where 𝛾₀=𝜀. We have  𝜇_⊢ℳ′((𝑞₀,1,0), 𝜔,𝑍₀𝑋₀), ((𝑞₁,𝑐₀,𝑑₀), 𝜏₂⋯𝜏_𝑛, 𝑍₁𝛾₁𝑋₀)) = 1,  𝜈_⊢ℳ′((𝑞₀,1,0), 𝜔,𝑍₀𝑋₀), ((𝑞₁,𝑐₀,𝑑₀), 𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁𝑋₀)) = 0,  𝜇_⊢ℳ′(((𝑞₁,𝑐₀,𝑑₀), 𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁𝑋₀), ((𝑞₂,𝑐₀∧𝑐₁,𝑑₀∨𝑑₁), 𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂𝑋₀)) = 1,  𝜈_⊢ℳ′(((𝑞₁,𝑐₀,𝑑₀),𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁𝑋₀), ((𝑞₂,𝑐₀∧𝑐₁,𝑑₀∨𝑑₁), 𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂𝑋₀)) = 0, ⋮ 𝜇_⊢ℳ′(((𝑞_𝑛−1, 𝑐₀∧𝑐₁∧⋯∧𝑐_𝑛−2, 𝑑₀∨𝑑₁∨⋯∨𝑑_𝑛−2), 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1𝑋₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛), 𝜀,𝑋₀))=1, 𝜈_⊢ℳ′(((𝑞_𝑛−1,𝑐₀∧𝑐₁∧⋯∧𝑐_𝑛−2, 𝑑₀∨𝑑₁∨⋯∨𝑑_𝑛−2), 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1𝑋₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑋₀))=0.
Hence 𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑋₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛), 𝜀,𝑋₀)≥𝜇_⊢ℳ′((𝑞₀,1,0), 𝜔,𝑍₀𝑋₀), ((𝑞₁,𝑐₀,𝑑₀), 𝜏₂⋯𝜏_𝑛, 𝑍₁𝛾₁𝑋₀))∧𝜇_⊢ℳ′(((𝑞₁, 𝑐₀,𝑑₀), 𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁𝑋₀), ((𝑞₂,𝑐₀∧𝑐₁,𝑑₀∨𝑑₁), 𝜏₃⋯𝜏_𝑛, 𝑍₂𝛾₂𝑋₀))∧⋯∧𝜇_⊢ℳ′(((𝑞_𝑛−1, 𝑐₀∧𝑐₁∧⋯∧𝑐_𝑛−2, 𝑑₀∨𝑑₁∨⋯∨𝑑_𝑛−2), 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1𝑋₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑋₀))=1, 𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔,𝑋₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑋₀))≤𝜈_⊢ℳ′((𝑞₀,1,0), 𝜔,𝑍₀𝑋₀),((𝑞₁,𝑐₀,𝑑₀), 𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁𝑋₀))∨𝜈_⊢ℳ′(((𝑞₁,𝑐₀, 𝑑₀),𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁𝑋₀), ((𝑞₂,𝑐₀∧𝑐₁,𝑑₀∨𝑑₁), 𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂𝑋₀))∨⋯∨𝜈_⊢ℳ′(((𝑞_𝑛−1, 𝑐₀∧𝑐₁∧⋯∧𝑐_𝑛−2, 𝑑₀∨𝑑₁∨⋯∨𝑑_𝑛−2), 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1𝑋₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑋₀)) = 0.
Since 𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑋₀), ((𝑞_𝑛,𝑎_𝑛, 𝑏_𝑛), 𝜀, 𝑋₀))≤1 and 𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔, 𝑋₀), ((𝑞_𝑛,𝑎_𝑛, 𝑏_𝑛),𝜀,𝑋₀))≥0, 𝜇_⊢∗ℳ′((𝑞^′₀, 𝜔,𝑋₀), ((𝑞_𝑛,𝑎_𝑛, 𝑏_𝑛), 𝜀,𝑋₀))=1 and 𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔, 𝑋₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛), 𝜀,𝑋₀))=0.
If (P2) is not satisfied, then we assume 𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑋₀),((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛), 𝜀,𝑋₀))>0.
So there at least exist 𝑞₁,…,𝑞_𝑛−1∈𝑄, 𝑍₁,…,𝑍_𝑛−1∈Γ, 𝛾₁,…,𝛾_𝑛−1∈Γ^∗, s.t. 𝜇_⊢ℳ′((𝑞₀,1,0),𝜔,𝑍₀𝑋₀), ((𝑞₁,𝑐₀,𝑑₀), 𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁𝑋₀))∧𝜇_⊢ℳ′(((𝑞₁, 𝑐₀,𝑑₀), 𝜏₂⋯𝜏_𝑛, 𝑍₁𝛾₁𝑋₀), ((𝑞₂,𝑐₀∧𝑐₁, 𝑑₀∨𝑑₁), 𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂𝑋₀))∧⋯∧𝜇_⊢ℳ′(((𝑞_𝑛−1, 𝑐₀∧𝑐₁∧⋯∧𝑐_𝑛−2, 𝑑₀∨𝑑₁∨⋯∨𝑑_𝑛−2),𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1𝑋₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑋₀))>0.
Hence 𝑎_𝑛=𝜇_⊢ℳ((𝑞₀,𝜔,𝑍₀), (𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∧𝜇_⊢ℳ((𝑞₁, 𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁), (𝑞₂,𝜏₃⋯𝜏_𝑛, 𝑍₂𝛾₂))∧⋯∧𝜇_⊢ℳ((𝑞_𝑛−1, 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1), (𝑞_𝑛,𝜀,𝜀))>0 and 𝑏_𝑛=𝜈_⊢ℳ((𝑞₀,𝜔,𝑍₀), (𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁))∨𝜈_⊢ℳ((𝑞₁, 𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁), (𝑞₂,𝜏₃⋯𝜏_𝑛, 𝑍₂𝛾₂))∨⋯∨𝜈_⊢ℳ((𝑞_𝑛−1, 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1), (𝑞_𝑛,𝜀,𝜀))<1.
It contradicts with the assumption. So 𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑋₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑋₀)=0 if (P2) is not satisfied. In a similar way, it is easily concluded that 𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔,𝑋₀),((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝑋₀)=1 if (P2) is not satisfied.
Secondly, for any 𝜔=𝜏₁⋯𝜏_𝑛∈Σ^∗, 𝜏_𝑖∈Σ∪{𝜀}, 𝑖∈{1,…,𝑛}, 𝜇_ℒ(ℳ′)(𝜔) = ⋁{𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑋₀), ((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛), 𝜀,𝑋₀))∧𝜇_⊢ℳ′((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛), 𝜀,𝑋₀),((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛),𝜀,𝜀)) ∣(𝑞_𝑛,𝑎_𝑛,𝑏_𝑛)∈𝑄^′} = ⋁{𝜇_⊢ℳ((𝑞₀,𝜔,𝑍₀),(𝑞₁,𝜏₂⋯𝜏_𝑛, 𝑍₁𝛾₁))∧𝜇_⊢ℳ((𝑞₁,𝜏₂⋯𝜏_𝑛,𝑍₁𝛾₁), (𝑞₂,𝜏₃⋯𝜏_𝑛,𝑍₂𝛾₂))∧⋯∧𝜇_⊢ℳ((𝑞_𝑛−1, 𝜏_𝑛,𝑍_𝑛−1𝛾_𝑛−1),(𝑞_𝑛,𝜀,𝜀))∣𝑞₁,…,𝑞_𝑛∈𝑄,𝑍₁,…,𝑍_𝑛−1∈Γ, 𝛾₁,…,𝛾_𝑛−1∈Γ^∗} = 𝜇_ℒ(ℳ)(𝜔).
Similarly, 𝜈_ℒ(ℳ′)(𝜔)=𝜈_ℒ(ℳ)(𝜔).
Hence ℒ(ℳ^′)=ℒ(ℳ).

Remark 21. Proposition 20 presents an equivalence of an IFPDA^∅. 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 IFPDA^∅ is an intuitionistic fuzzy recognizable step function.

Proposition 22. Let ℳ=(𝑄,Σ,Γ,𝛿,𝑞₀,𝑍₀,∅) be an IFPD A^∅. Then the language recognized by ℳ is an intuitionistic fuzzy recognizable step function over Σ^∗, that is, ℒ(ℳ)∈ Step^𝐶(Σ).

Proof. Let ℳ=(𝑄,Σ,Γ,𝛿,𝑞₀,𝑍₀,∅) be an IFPDA^∅. Then there is a special IFPDA^∅ ℳ^′=(𝑄^′,Σ,Γ^′,𝛿^′,𝑞^′₀,𝑋₀,∅) constructed by Proposition 20, which is equivalent to ℳ. For any (𝑎,𝑏)∈(𝐿₁⧵{0})×(𝐿₂⧵{1}) with 0<𝑎+𝑏≤1, construct a classical PDA with empty stack ℳ_𝑎𝑏=(𝑄^′,Σ,Γ^′,𝛿^′′_𝑎𝑏,𝑞^′₀,𝑋₀,∅), where 𝑄^′,Σ,Γ^′,𝑞^′₀, and 𝑋₀ are the same as those in ℳ^′, and the function 𝛿^′′_𝑎𝑏∶𝑄^′×Σ×Γ^′→2^{𝑄′×Γ′∗} is defined by(i)𝛿^′′_𝑎𝑏((𝑝₀,1,0),𝜀,𝑋₀)={((𝑞₀,1,0),𝑍₀𝑋₀)};(ii)𝛿^′′_𝑎𝑏((𝑞,𝑐,𝑑),𝜏,𝑋)={((𝑞^′,𝑐∧𝑐₁,𝑑∨𝑑₁),𝛾)∣𝑐₁=𝜇_𝛿(𝑞,𝜏,𝑋,𝑞^′,𝛾)>0,𝑑₁=𝜈_𝛿(𝑞,𝜏,𝑋,𝑞^′, 𝛾)<1,𝑞^′∈𝑄,𝛾∈Γ^∗};(iii)𝛿^′′_𝑎𝑏((𝑞,𝑎,𝑏),𝜀,𝑋₀)={((𝑞,𝑎,𝑏),𝜀)};(iv)𝛿^′′_𝑎𝑏((𝑞,𝑐,𝑑),𝜏,𝑋)=∅ in other cases.
Then for any 𝜔∈Σ^∗, we have 𝜇_ℒ(ℳ)(𝜔) = 𝜇_ℒ(ℳ′)(𝜔) = ⋁{𝜇_⊢∗ℳ′((𝑞^′₀,𝜔, 𝑋₀), ((𝑞_𝑛,𝑎_𝑛, 𝑏_𝑛), 𝜀,𝑋₀))∧𝜇_⊢ℳ′((𝑞_𝑛, 𝑎_𝑛,𝑏_𝑛), 𝜀,𝑋₀),((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛), 𝜀,𝜀))∣ (𝑞_𝑛,𝑎_𝑛,𝑏_𝑛)∈𝑄^′}= ⋁{𝑎_𝑛∣(𝑞_𝑛,𝑎_𝑛,𝑏_𝑛)∈𝑄^′, 0<𝑎_𝑛+𝑏_𝑛≤1, 𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑋₀),((𝑞_𝑛,𝑎_𝑛,𝑏_𝑛), 𝜀,𝑋₀))=1} = ⋁{𝑎_𝑛∣(𝑞^′₀,𝜔,𝑋₀)≻^∗_ℳ𝑎𝑏((𝑞_𝑛, 𝑎_𝑛,𝑏_𝑛),𝜀,𝜀), 𝑞_𝑛∈𝑄} = ⋁_{𝑎∈𝐿1⧵{0}}𝑎∧𝜇_{1)ℒ(ℳ𝐚𝐛}(𝜔), and 𝜈_ℒ(ℳ)(𝜔)=𝜈_ℒ(ℳ′)(𝜔) = ⋀{𝜈_⊢^∗_ℳ′((𝑞^′₀,𝜔,𝑋₀), ((𝑞,𝑎,𝑏), 𝜀,𝑋₀))∨𝜈_⊢ℳ′((𝑞, 𝑎,𝑏), 𝜀,𝑋₀),((𝑞,𝑎,𝑏), 𝜀,𝜀))∣(𝑞,𝑎, 𝑏)∈𝑄^′} = ⋀{𝑏∣(𝑞,𝑎,𝑏)∈𝑄^′, 0<𝑎 + 𝑏≤1,𝜇_⊢∗ℳ′((𝑞^′₀,𝜔,𝑋₀), ((𝑞,𝑎,𝑏),𝜀,𝑋₀)) = 1}=⋀{𝑏∣ (𝑞^′₀,𝜔,𝑋₀)≻^∗_ℳ𝑎𝑏((𝑞, 𝑎,𝑏),𝜀,𝑋₀), (𝑞,𝑎,𝑏)∈𝑄^′} = ⋀_{𝑏∈𝐿2⧵{1}}𝑏∨𝜈_{1)ℒ(ℳ𝐚𝐛}(𝜔).
Therefore ℒ(ℳ)∈ Step^𝐶(Σ).

Theorem 23. Let 𝐴 be an IFS over Σ^∗. Then the following statements are equivalent:(1)𝐴∈ 𝑆𝑡𝑒𝑝^𝐶(Σ);(2)𝐴 is accepted by a certain 𝐼𝐹𝑃𝐷𝐴^∅ ℳ.

Proof. (i) implies (ii). Suppose 𝐴∈ Step^𝐶(Σ). Then there are a finite natural number 𝑛∈𝑁, recognizable context-free languages ℒ₁,…,ℒ_𝑛⊆Σ^∗, and (𝑎_𝑖,𝑏_𝑖)∈(0,1]×[0,1) with 0≤𝑎_𝑖+𝑏_𝑖≤1 for 𝑖=1,…,𝑛 such that 𝐴=(𝜇_𝐴,𝜈_𝐴)=∐^𝑛_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅𝟏_ℒ𝐢. Let ℒ_𝑖 be recognized by PDA ℳ_𝑖=(𝑄_𝑖,Σ,Γ_𝑖,𝛿_𝑖,𝑞_0𝑖,𝑍_0𝑖,∅), and 𝑄_𝑖∩𝑄_𝑗=∅ whenever 𝑖≠𝑗, 𝑖,𝑗∈𝑁_𝑛.
Next, construct an IFPDA^∅ ℳ=(𝑄,Σ,Γ,𝛿,𝐼,𝑍₀,∅) as follows: 𝑄=⋃^𝑛_𝑖=1𝑄_𝑖, Γ=⋃^𝑛_𝑖=1Γ_𝑖∪{𝑍₀}, and 𝐼=(𝜇_𝐼,𝜈_𝐼) are an IFS over 𝑄, defined by 𝜇_𝐼(𝑞_0𝑖)=𝑎_𝑖, 𝜈_𝐼(𝑞_0𝑖)=𝑏_𝑖, 𝑖∈𝑁_𝑛; otherwise 𝜇_𝐼(𝑞)=0 and 𝜈_𝐼(𝑞)=1. 𝛿=(𝜇_𝛿,𝜈_𝛿) is an IFS over 𝑄×(Σ∪{𝜀})×Γ×𝑄×Γ^∗, where the mappings 𝜇_𝛿,𝜈_𝛿∶𝑄×(Σ∪{𝜀})×Γ×𝑄×Γ^∗→{0,1} are defined by 𝜇_𝛿(𝑞_0𝑖,𝜀,𝑍₀,𝑞_0𝑖,𝑍_0𝑖)=1 and 𝜈_𝛿(𝑞_0𝑖,𝜀,𝑍₀,𝑞_0𝑖,𝑍_0𝑖)=0, 𝑖∈𝑁_𝑛; 𝜇_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=1 and 𝜈_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=0 if 𝑝,𝑞∈𝑄_𝑖, 𝑍∈Γ_𝑖,𝛾∈Γ^∗_𝑖, 𝜏∈Σ, 𝑖∈𝑁_𝑛 and (𝑝,𝛾)∈𝛿(𝑞,𝜏,𝑍). Otherwise, 𝜇_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=0 and 𝜈_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=1.
Then 𝐴=ℒ(ℳ)=∐^𝑛_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅𝟏_ℒ𝐢.
(ii) implies (i). It is concluded by Proposition 22.

One can see that IFPDAs and IFPDAs^∅ are equivalent to a type of intuitionistic fuzzy recognizable step functions over a set, respectively, by Theorems 19 and 23. Therefore, IFPDAs and IFPDAs^∅ 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 IFPDA^∅ ℳ^′ equivalent to ℳ. For any IFPDA^∅ 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 grammars is introduced in the section. Then the relationship between intuitionistic fuzzy context-free grammars, IFPDAs, IFPDAs^∅, 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=∅;(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)^∗, 𝜇_𝜌(𝑥→𝑦)>0,𝜈_𝜌(𝑥→𝑦)<1},where 𝜌=(𝜇_𝜌,𝜈_𝜌) is an IFS over (𝑁∪𝑇)^∗×(𝑁∪𝑇)^∗, 𝜇_𝜌(𝑥,𝑦), and 𝜈_𝜌(𝑥,𝑦) mean the membership degree and the nonmembership degree that 𝑥 will be replaced by 𝑦, respectively, denoted by 𝜇_𝜌(𝑥,𝑦)=𝜇_𝜌(𝑥→𝑦), 𝜈_𝜌(𝑥,𝑦)=𝜈_𝜌(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 𝜇_𝜌(𝛼𝑥𝛽⇒𝛼𝑦𝛽)=𝜇_𝜌(𝑥→𝑦), 𝜈_𝜌(𝛼𝑥𝛽⇒𝛼𝑦𝛽)=𝜈_𝜌(𝑥→𝑦).

If 𝛼₁,…,𝛼_𝑚 are strings in (𝑁∪𝑇)^∗ and 𝛼₁→𝛼₂,…,𝛼_𝑚−1→𝛼_𝑚∈𝑃, then 𝛼₁ is said to derive 𝛼_𝑚 in 𝐺, or, equivalently, 𝛼_𝑚 is derivable from 𝛼₁ in 𝐺. This is expressed by 𝛼₁⇒^∗_𝐺𝛼_𝑚 or simply 𝛼₁⇒^∗𝛼_𝑚. The expression 𝛼₁→𝛼₂→⋯→𝛼_𝑚 is referred to as a derivation chain from 𝛼₁ to 𝛼_𝑚.

An intuitionistic fuzzy grammar 𝐺 generates an intuitionistic fuzzy language ℒ(𝐺)=(𝜇_𝐺,𝜈_𝐺) in the following manner. For any 𝜃=𝜔_𝑛∈𝑇^∗, 𝑛≥1, 𝜇_𝐺(𝜃)=⋁{𝜇_𝐼(𝜔₀)∧𝜇_𝜌(𝜔₀⇒𝜔₁)∧⋯∧𝜇_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝜔₀∈𝑁,𝜔₁,…,𝜔_𝑛−1∈(𝑁∪𝑇)^∗}, and 𝜈_𝐺(𝜃)=⋀{𝜈_𝐼(𝜔₀)∨𝜈_𝜌(𝜔₀⇒𝜔₁)∨⋯∨𝜈_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝜔₀∈𝑁,𝜔₁,…,𝜔_𝑛−1∈(𝑁∪𝑇)^∗}.

𝜇_𝐺(𝜃) and 𝜈_𝐺(𝜃) express the membership and nonmembership degree of 𝜃 in the language generated by grammar 𝐺, respectively. Obviously, ℒ(𝐺)=(𝜇_𝐺,𝜈_𝐺) is well defined. In fact, for any 𝜃=𝜔_𝑛∈𝑇^∗, 𝑛≥1, there is the strongest derivation from 𝜔₀ to 𝜔_𝑛, that is, 𝜔₀⇒𝜔^′₁⇒⋯⇒𝜔^′_𝑛−1⇒𝜔_𝑛, such that 𝜇_𝐺(𝜃)=𝜇_𝐼(𝜔₀)∧𝜇_𝜌(𝜔₀⇒𝜔^′₁)∧⋯∧𝜇_𝜌(𝜔^′_𝑛−1⇒𝜔_𝑛). So 𝜈_𝐺(𝜃)≤𝜈_𝐼(𝜔₀)∨𝜈_𝜌(𝜔₀⇒𝜔^′₁)∨⋯∨𝜈_𝜌(𝜔^′_𝑛−1⇒𝜔_𝑛), and 𝜇_𝐺(𝜃)+𝜈_𝐺(𝜃)≤𝜇_𝐺(𝜃)+𝜈_𝐼(𝜔₀)∨𝜈_𝜌(𝜔₀⇒𝜔^′₁)∨⋯∨𝜈_𝜌(𝜔^′_𝑛−1⇒𝜔_𝑛)=(𝜇_𝐺(𝜃)+𝜈_𝐼(𝜔₀))∨(𝜇_𝐺(𝜃)+𝜈_𝜌(𝜔₀⇒𝜔^′₁))∨⋯∨(𝜇_𝐺(𝜃)+𝜈_𝜌(𝜔^′_𝑛−1⇒𝜔_𝑛))≤(𝜇_𝐼(𝜔₀)+𝜈_𝐼(𝜔₀))∨(𝜇_𝜌(𝜔₀⇒𝜔^′₁)+𝜈_𝜌(𝜔₀⇒𝜔^′₁))∨⋯∨(𝜇_𝜌(𝜔^′_𝑛−1⇒𝜔_𝑛)+𝜈_𝜌(𝜔^′_𝑛−1⇒𝜔_𝑛))≤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(𝜇_𝐼)∪Im(𝜈_𝐼)={0,1} and supp(𝐼)={𝑆}, then 𝐺 is also written as 𝐺=(𝑁,𝑇,𝑃,𝑆).

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), 𝜇_𝜌(𝑆→𝑞)=𝜇_𝐼(𝑞),𝜈_𝜌(𝑆→𝑞)=𝜈_𝐼(𝑞)}.
Next we show that ℒ(𝐺^′)=ℒ(𝐺). In fact, 𝐺^′=(𝑁^′,𝑇^′,𝑃^′,𝐼^′), where 𝐼^′ is an IFS over 𝑁^′, 𝜇_𝐼′(𝑆)=1, 𝜈_𝐼′(𝑆)=0; 𝜇_𝐼′(𝑞)=0 and 𝜈_𝐼′(𝑞)=1 when 𝑞∈𝑁.
For any 𝜃=𝜔_𝑛∈𝑇^∗, 𝑛≥1, 𝜇_𝐺′(𝜃)=⋁{𝜇_𝐼′(𝜔₀)∧𝜇_𝜌(𝜔₀⇒𝜔₁)∧⋯∧𝜇_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝜔₀∈𝑁^′,𝜔₁,…,𝜔_𝑛−1∈(𝑁^′∪𝑇^′)^∗} =⋁{𝜇_𝜌(𝑆⇒𝜔₁)∧𝜇_𝜌(𝜔₁⇒𝜔₂)∧⋯∧𝜇_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝜔₁,…,𝜔_𝑛−1∈(𝑁^′∪𝑇^′)^∗} =⋁{𝜇_𝜌(𝑆⇒𝑞)∧𝜇_𝜌(𝑞⇒𝜔₂)∧⋯∧𝜇_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝑞∈𝑁,𝜔₂,…,𝜔_𝑛−1∈(𝑁∪𝑇)^∗} =⋁{𝜇_𝐼(𝑞)∧𝜇_𝜌(𝑞⇒𝜔₂)∧⋯∧𝜇_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝑞∈𝑁,𝜔₂,…,𝜔_𝑛−1∈(𝑁∪𝑇)^∗}=𝜇_𝐺(𝜃) and 𝜈_𝐺′(𝜃)=⋀{𝜈_𝐼′(𝜔₀)∨𝜈_𝜌(𝜔₀⇒𝜔₁)∨⋯∨𝜈_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝜔₀∈𝑁^′,𝜔₁,…,𝜔_𝑛−1∈(𝑁^′∪𝑇^′)^∗} =⋀{𝜈_𝜌(𝑆⇒𝜔₁)∨𝜈_𝜌(𝜔₁⇒𝜔₂)∨⋯∨𝜈_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝜔₁,…,𝜔_𝑛−1∈(𝑁^′∪𝑇^′)^∗} =⋀{𝜈_𝜌(𝑆⇒𝑞)∨𝜈_𝜌(𝑞⇒𝜔₂)∨⋯∨𝜈_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝑞∈𝑁,𝜔₂,…,𝜔_𝑛−1∈(𝑁∪𝑇)^∗} =⋀{𝜈_𝐼(𝑞)∨𝜈_𝜌(𝑞⇒𝜔₂)∨⋯∨𝜈_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝑞∈𝑁,𝜔₂,…,𝜔_𝑛−1∈(𝑁∪𝑇)^∗}=𝜈_𝐺(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 simple-typed intuitionistic fuzzy context-free grammar (IFSCFG) if 𝑃=𝐴⟶𝑥∣𝐴∈𝑁,𝑥∈(𝑁∪𝑇)^∗,𝜇_𝜌(𝐴⟶𝑥)=1.(25)

Proposition 28. Let 𝐺=(𝑁,𝑇,𝑃,𝐼) be an IFCFG and ℒ(𝐺)=(𝜇_𝐺,𝜈_𝐺) 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(𝜇_𝜌) and Im(𝜈_𝜌) are finite. For any 𝜃=𝜔_𝑛∈𝑇^∗, 𝑛≥1, there exist natural numbers 𝑑,𝑙 such that 𝜇_𝐺(𝜃)=⋁{𝜇_𝐼(𝑆)∧𝜇_𝜌(𝑆⇒𝜔₁)∧𝜇_𝜌(𝜔₁⇒𝜔₂)∧⋯∧𝜇_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝑆∈𝑁,𝜔₁,…,𝜔_𝑛−1∈(𝑁∪𝑇)^∗}=(𝑎₁₀∧⋯∧𝑎_1𝑛−1∧𝑎_1𝑛)∨⋯∨(𝑎_𝑑0∧⋯∧𝑎_𝑑𝑛−1∧𝑎_𝑑𝑛) and 𝜈_𝐺(𝜃)=⋀{𝜈_𝐼(𝑆)∨𝜈_𝜌(𝑆⇒𝜔₁)∨𝜈_𝜌(𝜔₁⇒𝜔₂)∨⋯∨𝜈_𝜌(𝜔_𝑛−1⇒𝜔_𝑛)∣𝑆∈𝑁,𝜔₁,…,𝜔_𝑛−1∈(𝑁∪𝑇)^∗}=(𝑏₁₀∨⋯∨𝑏_1𝑛−1∨𝑏_1𝑛)∧⋯∧(𝑏_𝑙0∨⋯∨𝑏_𝑙𝑛−1∨𝑏_𝑙𝑛), where 𝑎_𝑖𝑘∈Im(𝜇_𝐼)∪Im(𝜇_𝜌), 𝑏_𝑗𝑘∈Im(𝜈_𝐼)∪Im(𝜈_𝜌), 𝑖∈{1,…,𝑑}, 𝑗∈{1,…,𝑙}, 𝑘∈{0,1,…,𝑛}. Let 𝑋=Im(𝜇_𝐼)∪Im(𝜇_𝜌) and 𝑌=Im(𝜈_𝐼)∪Im(𝜈_𝜌). Then 𝑋 and 𝑌 are finite subsets of the interval [0,1]. (𝑋_∧)_∨ and (𝑌_∨)_∧ are also finite by Lemma 8. Since 𝜇_𝐺(𝜃)∈(𝑋_∧)_∨ and 𝜈_𝐺(𝜃)∈(𝑌_∨)_∧ for any 𝜃∈𝑇^∗, we have Im(𝜇_𝐺)⊆(𝑋_∧)_∨ and Im(𝜈_𝐺)⊆(𝑌_∨)_∧. Hence Im(ℒ(𝐺))=Im(𝜇_𝐺)∪Im(𝜈_𝐺) is finite.

Proposition 29. Let 𝐺=(𝑁,𝑇,𝑃,𝑆) be an IFCFG. Then ℒ(𝐺)∈ Step^𝐶(𝑇).

Proof. Let 𝑋=Im(𝜇_𝜌) and 𝑌=Im(𝜈_𝜌). Then 𝑋 and 𝑌 have finite elements because 𝑃 is a finite collection of productions over 𝑇∪𝑁. Suppose 𝐿₁=𝑋_∧ and 𝐿₂=𝑌_∨. Then 𝐿₁ and 𝐿₂ are finite by Lemma 8. For any (𝑎,𝑏)∈(𝐿₁⧵{0})×(𝐿₂⧵{1}), 0≤𝑎+𝑏≤1, we construct a classical context-free grammar 𝐺_𝑎𝑏=(𝑁^′,𝑇,𝑃^′_𝑎𝑏,𝑆^′) as follows:
𝑁^′=𝑁×(𝐿₁⧵{0})×(𝐿₂⧵{1}), 𝑆^′=(𝑆,1,0)∈𝑁^′, 𝑃^′_𝑎𝑏 consists of the form:(1)(𝐴,𝑎₁,𝑏₁)→𝐷₁⋯𝐷_𝑘 whenever 𝜇_𝜌(𝐴→𝜏₁⋯𝜏_𝑘)>0 and 𝜈_𝜌(𝐴→𝜏₁⋯𝜏_𝑘)<1, where 𝐷_𝑖=𝜏_𝑖,𝑎₂,𝑏₂,if𝜏_𝑖𝜏∈𝑁_𝑖,if𝜏_𝑖∈𝑇(26)𝑖=1,…,𝑘; 𝑎₂=𝑎₁∧𝜇_𝜌(𝐴→𝜏₁⋯𝜏_𝑘) and 𝑏₂=𝑏₁∨𝜈_𝜌(𝐴→𝜏₁⋯𝜏_𝑘);(2)(𝐴,𝑎₁,𝑏₁)→𝑥 whenever 𝑎≤𝑎₁∧𝜇_𝜌(𝐴→𝑥) and 𝑏≥𝑏₁∨𝜈_𝜌(𝐴→𝑥), for all 𝐴∈𝑁,𝑥∈(𝑁∪𝑇)^∗.
Then ℒ(𝐺_𝑎𝑏)={𝜔∈𝑇^∗∣𝑆^′⇒^∗_𝐺𝑎𝑏𝜔} ={𝜔∈𝑇^∗∣𝑎≤𝜇_𝜌(𝑆⇒𝑢₁)∧𝜇_𝜌(𝑢₁⇒𝑢₂)∧⋯∧𝜇_𝜌(𝑢_𝑛−1⇒𝜔),𝑏≥𝜈_𝜌(𝑆⇒𝑢₁)∨𝜈_𝜌(𝑢₁⇒𝑢₂)∨⋯∨𝜈_𝜌(𝑢_𝑛−1⇒𝜔),𝑢₁,…,𝑢_𝑛−1∈(𝑁∪𝑇)^∗}.
Next it suffices to show that ℒ(𝐺)=∐(𝑎,𝑏)⋅𝟏_{ℒ(𝐺𝑎𝑏)}, that is, 𝜇_𝐺(𝜔)=⋁_{𝑎∈𝐿1⧵{0}}𝑎∧𝜇_{1)ℒ(𝐺𝑎𝑏}(𝜔) and 𝜈_𝐺(𝜔)=⋀_{𝑏∈𝐿2⧵{1}}𝑏∨𝜈_{1)ℒ(𝐺𝑎𝑏}(𝜔), for all 𝜔∈𝑇^∗.
Suppose 𝜇_𝐺(𝜔)=𝑎_𝑘>0. Then there exist 𝑢^′₁,…,𝑢^′_𝑛−1∈(𝑁∪𝑇)^∗ such that 𝑎_𝑘=𝜇_𝜌(𝑆⇒𝑢^′₁)∧𝜇_𝜌(𝑢^′₁⇒𝑢^′₂)∧⋯∧𝜇_𝜌(𝑢^′_𝑛−1⇒𝜔). Put 𝑐=𝜈_𝜌(𝑆⇒𝑢^′₁)∨𝜈_𝜌(𝑢^′₁⇒𝑢^′₂)∨⋯∨𝜈_𝜌(𝑢^′_𝑛−1⇒𝜔). Then 𝑐<1 and 𝜔∈ℒ(𝐺_𝑘𝑐). Therefore, 𝜇_𝐺(𝜔)≤⋁_{𝑎∈𝐿1⧵{0}}𝑎∧𝜇_{1)ℒ(𝐺𝑎𝑏}(𝜔) and 𝜈_𝐺(𝜔)≥⋀_{𝑏∈𝐿2⧵{1}}𝑏∨𝜈_{1)ℒ(𝐺𝑎𝑏}(𝜔) for any 𝜔∈𝑇^∗. In addition, ⋁_{𝑎∈𝐿1⧵{0}}𝑎∧𝜇_{1)ℒ(𝐺𝑎𝑏}(𝜔)=⋁{𝑎∈𝐿₁⧵{0}∣𝜔∈ℒ(𝐺_𝑎𝑏)} ≤⋁{𝜇_𝜌(𝑆⇒𝑢₁)∧𝜇_𝜌(𝑢₁⇒𝑢₂)∧⋯∧𝜇_𝜌(𝑢_𝑛−1⇒𝜔)∣𝜔∈ℒ(𝐺_𝑎𝑏),𝑢₁,…,𝑢_𝑛−1∈(𝑁∪𝑇)^∗} ≤⋁{𝜇_𝜌(𝑆⇒𝑢₁)∧𝜇_𝜌(𝑢₁⇒𝑢₂)∧⋯∧𝜇_𝜌(𝑢_𝑛−1⇒𝜔)∣𝑢₁,…,𝑢_𝑛−1∈(𝑁∪𝑇)^∗}=𝜇_𝐺(𝜔), 𝜈_𝐺(𝜔)=⋀{𝜈_𝜌(𝑆⇒𝑢₁)∨𝜈_𝜌(𝑢₁⇒𝑢₂)∨⋯∨𝜈_𝜌(𝑢_𝑛−1⇒𝜔)∣𝑢₁,…,𝑢_𝑛−1∈(𝑁∪𝑇)^∗} ≤⋀{𝑏∈𝐿₂⧵{1}∣𝑏≥𝜈_𝜌(𝑆⇒𝑢₁)∨𝜈_𝜌(𝑢₁⇒𝑢₂)∨⋯∨𝜈_𝜌(𝑢_𝑛−1⇒𝜔),𝑢₁,…,𝑢_𝑛−1∈(𝑁∪𝑇)^∗} ≤⋀{𝑏∈𝐿₂⧵{1}∣𝑏≥𝜈_𝜌(𝑆⇒𝑢₁)∨𝜈_𝜌(𝑢₁⇒𝑢₂)∨⋯∨𝜈_𝜌(𝑢_𝑛−1⇒𝜔),𝑎≤𝜇_𝜌(𝑆⇒𝑢₁)∧𝜇_𝜌(𝑢₁⇒𝑢₂)∧⋯∧𝜇_𝜌(𝑢_𝑛−1⇒𝜔),𝑢₁,…,𝑢_𝑛−1∈(𝑁∪𝑇)^∗} =⋀{𝑏∈𝐿₂⧵{1}∣𝜔∈ℒ(𝐺_𝑎𝑏)}=⋀_{𝑏∈𝐿2⧵{1}}𝑏∨𝜈_{1)ℒ(𝐺𝑎𝑏}(𝜔).
It is concluded that 𝜇_𝐺(𝜔)=⋁_{𝑎∈𝐿1⧵{0}}𝑎∧𝜇_{1)ℒ(𝐺𝑎𝑏}(𝜔) and 𝜈_𝐺(𝜔)=⋀_{𝑏∈𝐿2⧵{1}}𝑏∨𝜈_{1)ℒ(𝐺𝑎𝑏}(𝜔), for all 𝜔∈𝑇^∗.

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∈ Step^𝐶(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=∐^𝑘_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅1_ℒ𝑖, where 𝑎_𝑖,𝑏_𝑖∈[0,1], 𝑎_𝑖+𝑏_𝑖≤1, 𝑖∈𝑁_𝑘, and ℒ₁,…,ℒ_𝑘⊂𝑇^∗ are classical context-free languages. Suppose ℒ_𝑖 is generated by a context-free grammar 𝐺_𝑖=(𝑁_𝑖,𝑇,𝑃_𝑖,𝑆_0𝑖) and 𝑁_𝑖∩𝑁_𝑗=∅ whenever 𝑖≠𝑗. Then we construct an IFSCFG 𝐺=(𝑁,𝑇,𝑃,𝐼) as follows: 𝑁=⋃^𝑘_𝑖=1𝑁_𝑖, 𝑃=⋃^𝑘_𝑖=1𝑃_𝑖, 𝐼=(𝜇_𝐼,𝜈_𝐼) is an IFS over 𝑁, where the mappings 𝜇_𝐼,𝜈_𝐼∶𝑁→[0,1] are defined by 𝜇_𝐼(𝑆_0𝑖)=𝑎_𝑖, 𝜈_𝐼(𝑆_0𝑖)=𝑏_𝑖; 𝜇_𝐼(𝑞)=0 and 𝜈_𝐼(𝑞)=1 whenever 𝑞∈𝑁⧵{𝑆_0𝑖∣𝑖=1,…,𝑘}.
Next, we show 𝐴=ℒ(𝐺)=∐^𝑘_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅1_ℒ𝑖. In fact, for any 𝜔=𝑢_𝑛∈𝑇^∗, 𝜇_𝐺(𝜔)=⋁{𝜇_𝐼(𝑆)∧𝜇_𝜌(𝑆⇒𝑢₁)∧𝜇_𝜌(𝑢₁⇒𝑢₂)∧⋯∧𝜇_𝜌(𝑢_𝑛−1⇒𝜔)∣𝑆∈𝑁,𝑢₁,…,𝑢_𝑛−1∈(𝑁∪𝑇)^∗} =⋁{𝜇_𝐼(𝑆_0𝑖)∣𝑆_0𝑖⇒^∗_𝐺𝑖𝜔,𝑆_0𝑖∈𝑁,𝑖∈𝑁_𝑘} =⋁{𝑎_𝑖∣𝜔∈ℒ_𝑖,𝑖∈𝑁_𝑘}=⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_1ℒ𝑖(𝜔) and 𝜈_𝐺(𝜔)=⋀{𝜈_𝐼(𝑆)∨𝜈_𝜌(𝑆⇒𝑢₁)∨𝜈_𝜌(𝑢₁⇒𝑢₂)∨⋯∨𝜈_𝜌(𝑢_𝑛−1⇒𝜔)∣𝑆∈𝑁,𝑢₁,…,𝑢_𝑛−1∈(𝑁∪𝑇)^∗} =⋀{𝜈_𝐼(𝑆_0𝑖)∣𝑆_0𝑖⇒^∗_𝐺𝑖𝜔,𝑆_0𝑖∈𝑁,𝑖∈𝑁_𝑘} =⋀{𝑏_𝑖∣𝜔∈ℒ_𝑖,𝑖∈𝑁_𝑘}=⋀^𝑘_𝑖=1𝑏_𝑖∨𝜈_1ℒ𝑖(𝜔).
Hence ℒ(𝐺)=(𝜇_𝐺,𝜈_𝐺)=∐^𝑘_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅1_ℒ𝑖=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∈ Step^𝐶(Σ);(2)there is an IFPDA ℳ such that 𝐴=ℒ(ℳ);(3)there is an IFPDA^∅ 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).

Proof. (1) implies (5). Let 𝐴∈ Step^𝐶(Σ). Then suppose 𝐴=∐^𝑘_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅1_ℒ𝑖, where 𝑎_𝑖,𝑏_𝑖∈[0,1], 𝑎_𝑖+𝑏_𝑖≤1, 𝑖∈𝑁_𝑘, and ℒ₁,…,ℒ_𝑘⊂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∣𝜇_𝜌(𝑆→𝜀)=𝜇_𝐴(𝜀),𝜈_𝜌(𝑆→𝜀)=𝜇_𝐴(𝜀)}. Clearly, ℒ(𝐺)=𝐴.
If supp(𝐴)⧵{𝜀}≠∅, then there is a Chomsky normal form grammar 𝐺_𝑖=(𝑁_𝑖,Σ,𝑃_𝑖,𝑆_0𝑖) such that ℒ(𝐺_𝑖)=ℒ_𝑖⧵{𝜀}, for any ℒ_𝑖 with ℒ_𝑖⧵{𝜀}≠∅, 𝑖=1,…,𝑘. Let 𝑁_𝑖∩𝑁_𝑗=∅ whenever 𝑖≠𝑗. Then we construct an IFSCFG 𝐺=(𝑁,Σ,𝑃,𝐼) according to the method constructed by Theorem 30, where 𝑁=⋃^𝑘_𝑖=1𝑁_𝑖, 𝑃=⋃^𝑘_𝑖=1𝑃_𝑖, 𝐼=(𝜇_𝐼,𝜈_𝐼) is an IFS over 𝑁, and the mappings 𝜇_𝐼,𝜈_𝐼∶𝑁→[0,1] are defined by 𝜇_𝐼(𝑆_0𝑖)=𝑎_𝑖, 𝜈_𝐼(𝑆_0𝑖)=𝑏_𝑖; 𝜇_𝐼(𝑞)=0 and 𝜈_𝐼(𝑞)=1 whenever 𝑞∈𝑁⧵{𝑆_0𝑖∣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 𝜇_𝜌(𝑆→𝜀)=𝜇_𝐴(𝜀),𝜈_𝜌(𝑆→𝜀)=𝜇_𝐴(𝜀) whenever 𝜀∈ supp(𝐴);(𝐸2) 𝑆→𝐵𝐶 with 𝜇_𝜌(𝑆→𝐵𝐶)=𝜇_𝐼(𝑆_0𝑖) and 𝜈_𝜌(𝑆→𝐵𝐶)=𝜈_𝐼(𝑆_0𝑖) whenever 𝑆_0𝑖∈ supp(𝐼) and 𝑆_0𝑖→𝐵𝐶∈𝑃_𝑖, 𝑖=1,…,𝑘;(𝐸3) 𝑆→𝛼 with 𝜇_𝜌(𝑆→𝛼)=⋁{𝜇_𝐼(𝑆_0𝑖)∣𝑆_0𝑖∈ supp(𝐼), 𝑆_0𝑖→𝛼∈𝑃_𝑖,𝑖=1,…,𝑘} and 𝜈_𝜌(𝑆→𝛼)=⋀{𝜈_𝐼(𝑆_0𝑖)∣𝑆_0𝑖∈ supp(𝐼), 𝑆_0𝑖→𝛼∈𝑃_𝑖,𝑖=1,…,𝑘} whenever 𝑆_0𝑖∈ supp(𝐼) and 𝑆_0𝑖→𝛼∈𝑃_𝑖,𝑖=1,…,𝑘.
Then we have ℒ(𝐺^′)=(𝜇_𝐺′,𝜈_𝐺′)=𝐴. In fact, for any 𝜔∈Σ^∗, if 𝜔=𝜀, then 𝜇_𝐺′(𝜀)=𝜇_𝜌(𝑆→𝜀)=𝜇_𝐴(𝜀) and 𝜈_𝐺′(𝜀)=𝜈_𝜌(𝑆→𝜀)=𝜈_𝐴(𝜀); if 𝜔≠𝜀, then 𝜇_𝐺′(𝜔)=⋁{𝜇_𝜌(𝑆⇒_𝐺′𝑢₁)∧𝜇_𝜌(𝑢₁⇒_𝐺′𝑢₂)∧⋯∧𝜇_𝜌(𝑢_𝑛−1⇒_𝐺′𝜔)∣𝑢₁,𝑢₂,…,𝑢_𝑛−1∈(𝑁^′∪Σ)^∗}=⋁{𝜇_𝐼(𝑆_0𝑖𝜇)∧_𝜌(𝑆_0𝑖⇒_𝐺𝑖𝑢₁)∧𝜇_𝜌(𝑢₁⇒_𝐺𝑖𝑢₂)∧⋯∧𝜇_𝜌(𝑢_𝑛−1⇒_𝐺𝑖𝜔)∣𝑢₁,𝑢₂,…,𝑢_𝑛−1∈(𝑁∪Σ)^∗,𝑖∈𝑁_𝑘}=⋁{𝑎_𝑖∣𝜔∈ℒ(𝐺_𝑖),𝑖∈𝑁_𝑘}=⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_1ℒ𝑖(𝜔) and 𝜈_𝐺′(𝜔)=⋀{𝜈_𝜌(𝑆⇒_𝐺′𝑢₁)∨𝜈_𝜌(𝑢₁⇒_𝐺′𝑢₂)∨⋯∨𝜈_𝜌(𝑢_𝑛−1⇒_𝐺′𝜔)∣𝑢₁,𝑢₂,…,𝑢_𝑛−1∈(𝑁^′∪Σ)^∗}=⋀{𝜈_𝐼(𝑆_0𝑖𝜈)∨_𝜌(𝑆_0𝑖⇒_𝐺𝑖𝑢₁)∨𝜈_𝜌(𝑢₁⇒_𝐺𝑖𝑢₂)∨⋯∨𝜈_𝜌(𝑢_𝑛−1⇒_𝐺𝑖𝜔)∣𝑢₁,𝑢₂,…,𝑢_𝑛−1∈(𝑁∪Σ)^∗,𝑖∈𝑁_𝑘}=⋀{𝑎_𝑖∣𝜔∈ℒ(𝐺_𝑖),𝑖∈𝑁_𝑘}=⋀^𝑘_𝑖=1𝑎_𝑖∨𝜇_1ℒ𝑖(𝜔).
(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=(𝜇_𝐴,𝜈_𝐴) and 𝐵=(𝜇_𝐵,𝜈_𝐵) 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(i)𝐴∪𝐵=(𝜇_𝐴∪𝐵,𝜈_𝐴∪𝐵), 𝜇_𝐴∪𝐵(𝜔)=𝜇_𝐴(𝜔)∨𝜇_𝐵(𝜔),𝜈_𝐴∪𝐵(𝜔)=𝜈_𝐴(𝜔)∧𝜈_𝐵(𝜔);(ii)(𝜆,𝜃)𝐴=(𝜆∧𝜇_𝐴,𝜃∨𝜈_𝐴), (𝜆∧𝜇_𝐴)(𝜔)=𝜆∧𝜇_𝐴(𝜔), (𝜃∨𝜈_𝐴)(𝜔)=𝜃∨𝜈_𝐴(𝜔);(iii)𝐴⁻¹=(𝜇_𝐴−1,𝜈_𝐴⁻¹), 𝜇_𝐴−1(𝜔)=𝜇_𝐴(𝜔⁻¹), 𝜈_𝐴⁻¹(𝜔)=𝜈_𝐴(𝜔⁻¹);(iv)𝐴𝐵=(𝜇_𝐴𝐵,𝜈_𝐴𝐵), 𝜇_𝐴𝐵(𝜔)=⋁{𝜇_𝐴(𝜔₁)∧𝜇_𝐵(𝜔₂)∣𝜔₁𝜔₂=𝜔}, 𝜈_𝐴𝐵(𝜔)=⋀{𝜈_𝐴(𝜔₁)∨𝜈_𝐵(𝜔₂)∣𝜔₁𝜔₂=𝜔};(v)𝐴^∗=(𝜇_𝐴∗,𝜈_𝐴^∗), 𝜇_𝐴∗(𝜔)=⋁{𝜇_𝐴(𝜔₁)∧⋯∧𝜇_𝐴(𝜔_𝑘)∶𝑘≥1,𝜔=𝜔₁⋯𝜔_𝑘}, 𝜈_𝐴^∗(𝜔)=⋀{𝜈_𝐴(𝜔₁)∨⋯∨𝜈_𝐴(𝜔_𝑘)∶𝑘≥1,𝜔=𝜔₁⋯𝜔_𝑘} for any 𝜔∈Σ^∗, where 𝜔⁻¹ represents the reversal of 𝜔, that is, if 𝜔=𝜔₁⋯𝜔_𝑘, then 𝜔⁻¹=𝜔_𝑘⋯𝜔₁, for all 𝜔∈Σ^∗.

Theorem 32. (1) The family Step^𝐶(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^∗∈ Step^𝐶(Σ), for any 𝐴,𝐵∈ Step^𝐶(Σ), 𝜆,𝜃∈[0,1], 0≤𝜆+𝜃≤1.
(2) Let ℎ∶Σ^∗₁→Σ^∗₂ be a homomorphism. If 𝐴∈ Step^𝐶(Σ₂), then ℎ⁻¹(𝐴)=𝐴∘ℎ∈ Step^𝐶(Σ₁). (3) Let ℎ∶Σ^∗₁→Σ^∗₂ be a homomorphism. If ℎ satisfies, for 𝜏∈Σ, ℎ(𝜏)≠𝜀, and 𝑔=(𝜇_𝑔,𝜈_𝑔)∈ Step^𝐶(Σ₁), then ℎ(𝑔)=(𝜇_ℎ(𝑔),𝜈_ℎ(𝑔))∈ Step^𝐶(Σ₂), where 𝜇_ℎ(𝑔)(𝜔)=⋁{𝜇_𝑔(𝛼)∣ℎ(𝛼)=𝜔,𝛼∈Σ^∗₁}, 𝜈_ℎ(𝑔)(𝜔)=⋀{𝜈_𝑔(𝛼)∣ℎ(𝛼)=𝜔,𝛼∈Σ^∗₁} for any 𝜔∈Σ^∗₂.

Proof. (1) Let 𝐴,𝐵∈ Step^𝐶(Σ). By Definition 17, we can assume 𝐴=(𝜇_𝐴,𝜈_𝐴)=∐^𝑘_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅1_ℒ𝑖,𝐵=(𝜇_𝐵,𝜈_𝐵)=∐^𝑛_𝑗=1(𝑐_𝑗,𝑑_𝑗)⋅1_ℳ𝑗, where all ℒ_𝑖 and ℳ_𝑗 are classical context-free languages, 0≤𝑎_𝑖+𝑏_𝑖≤1,0≤𝑐_𝑗+𝑑_𝑗≤1,𝑎_𝑖,𝑏_𝑖,𝑐_𝑗,𝑑_𝑗∈[0,1],𝑖∈𝑁_𝑘, and 𝑗∈𝑁_𝑛. With respect to the union, we have 𝐴∪𝐵∈ Step^𝐶(Σ). That is, 𝐴∪𝐵=(𝜇_𝐴∪𝐵,𝜈_𝐴∪𝐵)=(𝜇_𝐴∨𝜇_𝐵,𝜈_𝐴∧𝜈_𝐵), 𝜇_𝐴∪𝐵(𝜔)=𝜇_𝐴(𝜔)∨𝜇_𝐵(𝜔)=(⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_1ℒ𝑖(𝜔))∨(⋁^𝑛_𝑗=1𝑐_𝑗∧𝜇_1ℳ𝑗(𝜔)), 𝜈_𝐴∪𝐵(𝜔)=𝜈_𝐴(𝜔)∧𝜈_𝐵(𝜔)=(⋀^𝑘_𝑖=1𝑏_𝑖∨𝜈_1ℒ𝑖(𝜔))∧(⋀^𝑛_𝑗=1𝑑_𝑗∨𝜈_1ℳ𝑗(𝜔)), for all 𝜔∈Σ^∗.
With respect to the scalar product, for each (𝜆,𝜃)∈[0,1]×[0,1],0≤𝜆+𝜃≤1, we have (𝜆,𝜃)𝐴=(𝜆∧𝜇_𝐴,𝜃∨𝜈_𝐴), (𝜆∧𝜇_𝐴)(𝜔)=𝜆∧𝜇_𝐴(𝜔)=𝜆∧(⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_1ℒ𝑖(𝜔))=⋁^𝑘_𝑖=1((𝜆∧𝑎_𝑖)∧𝜇_1ℒ𝑖(𝜔)), (𝜃∨𝜈_𝐴)(𝜔)=𝜃∨𝜈_𝐴(𝜔)=𝜃∨(⋀^𝑘_𝑖=1𝑏_𝑖∨𝜈_1ℒ𝑖(𝜔))=⋀^𝑘_𝑖=1(𝜃∨𝑏_𝑖)∨𝜈_1ℒ𝑖(𝜔),forall𝜔∈Σ^∗. By Definition 17, (𝜆,𝜃)𝐴∈ Step^𝐶(Σ). For the reversal operation, ℒ_𝑖⁻¹={𝜔⁻¹∈Σ^∗∣𝜔∈ℒ_𝑖}, and ℒ_𝑖⁻¹ is a context-free language because ℒ_𝑖 is a context-free language, 𝑖∈𝑁_𝑘. For any 𝜔∈Σ^∗,𝜇_𝐴−1(𝜔)=𝜇_𝐴(𝜔⁻¹)=⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_1ℒ𝑖(𝜔⁻¹)=⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_1ℒ𝑖−1(𝜔),𝜈_𝐴⁻¹(𝜔)=𝜈_𝐴(𝜔⁻¹)=⋀^𝑘_𝑖=1𝑏_𝑖∨𝜈_1ℒ𝑖(𝜔⁻¹)=⋀^𝑘_𝑖=1𝑏_𝑖∨𝜈_1ℒ𝑖−1(𝜔), that is 𝐴⁻¹∈ Step^𝐶(Σ). For the operation of concatenation, since 𝜇_𝐴𝐵(𝜔)=⋁^𝑘_𝑖=1⋁^𝑛_𝑗=1(𝑎_𝑖∧𝑐_𝑗)∧𝜇_{1ℒ𝑖ℳ𝑗}(𝜔),𝜈_𝐴𝐵(𝜔)=⋀^𝑘_𝑖=1⋀^𝑛_𝑗=1(𝑏_𝑖∨𝑑_𝑗)∨𝜈_{1ℒ𝑖ℳ𝑗}(𝜔), for all 𝜔∈Σ^∗, where the elements of the family set {ℒ_𝑖ℳ_𝑗∣𝑖∈𝑁_𝑘,𝑗∈𝑁_𝑛} are also context-free languages since ℒ_𝑖 and ℳ_𝑗 are context-free languages. Hence 𝐴𝐵∈ Step^𝐶(Σ).
For the Kleene closure, 𝐴^∗=(𝜇_𝐴∗,𝜈_𝐴^∗) is defined by 𝜇_𝐴∗(𝜔)=⋁{𝜇_𝐴(𝜔₁)∧⋯∧𝜇_𝐴(𝜔_𝑘)∶𝑘≥1,𝜔=𝜔₁⋯𝜔_𝑘}, 𝜈_𝐴^∗(𝜔)=⋀{𝜈_𝐴(𝜔₁)∨⋯∨𝜈_𝐴(𝜔_𝑘)∶𝑘≥1,𝜔=𝜔₁⋯𝜔_𝑘} for any 𝜔∈Σ^∗. Since 𝐴∈ Step^𝐶(Σ), we assume that the IFSPDA ℳ=(𝑄,Σ,Γ,𝛿,𝑞₀,𝑍₀,𝐹) accepts 𝐴 by Theorem 19. Let 𝑅={(𝜇_𝐹(𝑞),𝜈_𝐹(𝑞))∣𝑞∈𝑄}⧵{(0,1)}={(𝑎_𝑖,𝑏_𝑖)∣𝑖∈𝑁_𝑘}. Then 𝐴=∐^𝑘_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅1_ℒ𝑖, where ℒ_𝑖 is accepted by a PDA ℳ_𝑖=(𝑄,Σ,Γ,𝛿^′,𝑞₀,𝑍₀,𝐹_𝑖), the mapping 𝛿^′∶𝑄×(Σ∪{𝜀})×Γ→2^𝑄×Γ∗ is defined by 𝛿^′(𝑞,𝜏,𝑋)={(𝑝,𝛾)∣𝜇_𝛿(𝑞,𝜏,𝑋,𝑝,𝛾)=1,𝑝∈𝑄,𝛾∈Γ^∗}, for all (𝑞,𝜏,𝑋)∈𝑄×(Σ∪{𝜀})×Γ, and 𝐹_𝑖={𝑞∈𝑄∣𝜇_𝐹(𝑞)=𝑎_𝑖,𝜈_𝐹(𝑞)=𝑏_𝑖}, for all 𝑖∈𝑁_𝑘.
For any nonempty subset 𝐽 of the set {1,2,…,𝑘}, we can assume that 𝐽={𝑖₁,…,𝑖_𝑠}. Let 𝑟_𝐽=𝑎_𝑖1∧⋯∧𝑎_𝑖𝑠,𝑡_𝐽=𝑏_𝑖1∨⋯∨𝑏_𝑖𝑠, and ℒ(𝐽)=⋃_{𝑝1⋯𝑝𝑠}ℒ⁺_𝑝1ℒ⁺_𝑝2ℒ^∗_𝑝1ℒ⁺_𝑝3(ℒ_𝑝1∪ℒ_𝑝2)^∗⋯ℒ⁺_𝑝𝑠−1 (ℒ_𝑝1∪⋯∪ℒ_𝑝𝑠−2)^∗ℒ⁺_𝑝𝑠(ℒ_𝑝1∪⋯∪ℒ_𝑝𝑠)^∗, where 𝑝₁⋯𝑝_𝑠 is a permutation of {𝑖₁,…,𝑖_𝑠}, and ℒ(𝐽) takes unions under all permutations of {𝑖₁,…,𝑖_𝑠}. Hence ℒ(𝐽) is a context-free language. It is easily verified that 𝐴^∗=(∐_{∅≠𝐽⊆𝑁𝑘}(𝑟_𝐽,𝑡_𝐽)⋅1_ℒ(𝐽))∪((𝜇_𝐴(𝜀),𝜈_𝐴(𝜀))⋅1_{𝜀}). Therefore, 𝐴^∗∈ Step^𝐶(Σ).
(2) If 𝐴∈ Step^𝐶(Σ₂), then 𝐴=∐^𝑘_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅1_ℒ𝑖, where 𝑎_𝑖,𝑏_𝑖∈[0,1], 𝑎_𝑖+𝑏_𝑖≤1, 𝑖∈𝑁_𝑘, and ℒ₁,…,ℒ_𝑘⊂Σ^∗₂ are classical context-free languages. Since ℎ∶Σ^∗₁→Σ^∗₂ is a homomorphism, 𝜇_ℎ⁻¹_(𝐴)(𝜔)=𝜇_𝐴∘ℎ(𝜔)=𝜇_𝐴(ℎ(𝜔))=⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_1ℒ𝑖(ℎ(𝜔))=⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_{1ℎ𝑖)−1(ℒ}(𝜔), and 𝜈_ℎ⁻¹_(𝐴)(𝜔)=𝜈_𝐴∘ℎ(𝜔)=𝜈_𝐴(ℎ(𝜔))=⋀^𝑘_𝑖=1𝑏_𝑖∨𝜇_1ℒ𝑖(ℎ(𝜔))=⋀^𝑘_𝑖=1𝑏_𝑖∨𝜈_{1ℎ𝑖)−1(ℒ}(𝜔), where 𝜔∈Σ^∗₁,ℎ⁻¹(ℒ_𝑖)={𝜔₁∈Σ^∗₁∣ℎ(𝜔₁)∈ℒ_𝑖},𝑖∈𝑁_𝑘, and the elements of the set {ℎ⁻¹(ℒ_𝑖)∣𝑖∈𝑁_𝑘} are classical context-free languages. Hence ℎ⁻¹(𝐴)=𝐴∘ℎ=∐^𝑘_𝑖=1(𝑎_𝑖,𝑏_𝑖)⋅1_ℎ⁻¹_(ℒ𝑖). And so ℎ⁻¹(𝐴)∈ Step^𝐶(Σ₁). (3) If ℎ(𝜏)≠𝜀, for all 𝜏∈Σ, and 𝑔=(𝜇_𝑔,𝜈_𝑔)∈ Step^𝐶(Σ₁), then 𝜇_𝑔(𝜔)=⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_1ℒ𝑖(𝜔),𝜈_𝑔(𝜔)=⋀^𝑘_𝑖=1𝑏_𝑖∨𝜇_1ℒ𝑖(𝜔), for any 𝜔∈Σ₁, where 𝑎_𝑖,𝑏_𝑖∈[0,1], 𝑎_𝑖+𝑏_𝑖≤1, 𝑖∈𝑁_𝑘, ℒ₁,…,ℒ_𝑘⊂Σ^∗₁ are classical context-free languages. Since ℎ∶Σ^∗₁→Σ^∗₂ is a homomorphism, ℎ(ℒ_𝑖)={ℎ(𝜔)∈Σ^∗₂∣𝜔∈ℒ_𝑖} is also a classical context-free language, 𝑖∈𝑁_𝑘. For any 𝑥∈Σ^∗₂, 𝜇_ℎ(𝑔)(𝑥)=⋁{𝜇_𝑔(𝛼)∣ℎ(𝛼)=𝑥}=⋁{⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_1ℒ𝑖(𝛼)∣ℎ(𝛼)=𝑥}=⋁^𝑘_𝑖=1𝑎_𝑖∧𝜇_{1𝑖)ℎ(ℒ}(𝑥), 𝜈_ℎ(𝑔)(𝑥)=⋀{𝜈_𝑔(𝛼)∣ℎ(𝛼)=𝑥}=⋀{⋀^𝑘_𝑖=1𝑏_𝑖∨𝜈_1ℒ𝑖(𝛼)∣ℎ(𝛼)=𝑥}=⋀^𝑘_𝑖=1𝑏_𝑖∨𝜈_{1𝑖)ℎ(ℒ}(𝑥). Hence ℎ(𝑔)=(𝜇_ℎ(𝑔),𝜈_ℎ(𝑔))∈ Step^𝐶(Σ₂).

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=(𝜇_𝐴,𝜈_𝐴) 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=𝑢₁𝑣₁𝑤₁𝑥₁𝑦₁, |𝑣𝑤𝑥|≤𝑛, |𝑣₁𝑤₁𝑥₁|≤𝑛, |𝑣𝑥|≥1, |𝑣₁𝑥₁|≥1, and 𝜇_𝐴(𝑢𝑣^𝑖𝑤𝑥^𝑖𝑦)≥𝜇_𝐴(𝑢𝑣𝑤𝑥𝑦),𝜈_𝐴(𝑢₁𝑣^𝑖₁𝑤₁𝑥^𝑖₁𝑦₁)≤𝜈_𝐴(𝑢₁𝑣₁𝑤₁𝑥₁𝑦₁), for all  𝑖≥0.

Proof. Let 𝐴=(𝜇_𝐴,𝜈_𝐴) 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)⋅𝟏_{ℒ(𝐺𝑎𝑏)}, where (𝑎,𝑏)∈(𝑋_∧⧵{0})×(𝑌_∨⧵{1}), 𝑋=Im(𝜇_𝜌), 𝑌=Im(𝜈_𝜌), 0≤𝑎+𝑏≤1 and the classical context-free grammar 𝐺_𝑎𝑏=(𝑁^′,𝑇,𝑃^′_𝑎𝑏,𝑆^′) is shown in the proof process of of Proposition 29. Let 𝐺 have 𝑚 variables. That means, |𝑁|=𝑚. Choose 𝑛=2^𝑚. Next, suppose |𝑧|≥𝑛, 𝜇_𝐺(𝑧)=𝑎₀>0 and 𝜈_𝐺(𝑧)=𝑏₀<1. Then there exist 𝑏_𝑘∈𝑌_∨⧵{1} and 𝑎_𝑙∈𝑋_∧⧵{0} with 0<𝑎₀+𝑏_𝑘≤1 and 0<𝑎_𝑙+𝑏₀≤1 such that 𝑧∈ℒ(𝐺_{𝑎0𝑏𝑘})∩ℒ(𝐺_{𝑎𝑙𝑏0}). 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 𝑢𝑣^𝑖𝑤𝑥^𝑖𝑦∈ℒ(𝐺_{𝑎0𝑏𝑘}), for all 𝑖≥0. Then 𝜇_𝐴(𝑢𝑣^𝑖𝑤𝑥^𝑖𝑦)≥𝜇_𝐴(𝑢𝑣𝑤𝑥𝑦) for any 𝑖≥0 since 𝜇_𝐴(𝑢𝑣^𝑖𝑤𝑥^𝑖𝑦)=⋁_{𝑎∈𝑋∧⧵{0}}𝑎∧1_{ℒ(𝐺𝑎𝑏)}(𝑢𝑣^𝑖𝑤𝑥^𝑖𝑦)≥𝑎₀.
Similarly, there are 𝑢₁,𝑣₁,𝑤₁,𝑥₁,𝑦₁∈Σ^∗ satisfying 𝑧=𝑢₁𝑣₁𝑤₁𝑥₁𝑦₁, |𝑣₁𝑤₁𝑥₁|≤𝑛 and |𝑣₁𝑥₁|≥1 such that 𝑢₁𝑣^𝑖₁𝑤₁𝑥^𝑖₁𝑦₁∈ℒ(𝐺_{𝑎𝑙𝑏0}), for all 𝑖≥0. Then 𝜈_𝐴(𝑢₁𝑣^𝑖₁𝑤₁𝑥^𝑖₁𝑦₁)≤𝜈_𝐴(𝑢₁𝑣₁𝑤₁𝑥₁𝑦₁) for any 𝑖≥0 since 𝜈_𝐴(𝑢₁𝑣^𝑖₁𝑤₁𝑥^𝑖₁𝑦₁)=⋀_{𝑏∈𝑌∨⧵{1}}𝑏∨1_{ℒ(𝐺𝑎𝑏)}(𝑢₁𝑣^𝑖₁𝑤₁𝑥^𝑖₁𝑦₁)≤𝑏₀.

Next, let us look at an example to negate an intuitionistic fuzzy language to be an IFCFL.

Example 34. Let 𝐴=(𝜇_𝐴,𝜈_𝐴) be an IFS over 𝑇^∗. The mappings 𝜇_𝐴,𝜈_𝐴∶𝑇^∗→[0,1] are defined by 𝜇_𝐴(𝑧)=0.5,if𝑧=𝑎^𝑖𝑏^𝑗𝑐^𝑘,𝜈𝑖<𝑗<𝑘0,otherwise,_𝐴(𝑧)=0.3,if𝑧=𝑎^𝑖𝑏^𝑗𝑐^𝑘𝑖<𝑗<𝑘,1,otherwise,(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=𝑎^𝑛𝑏^𝑛+1𝑐^𝑛+2. Hence, 𝜇_𝐴(𝑧)=𝜇_ℒ(𝐺)(𝑧)=0.5 and 𝜈_𝐴(𝑧)=𝜈_ℒ(𝐺)(𝑧)=0.3. Let 𝑧=𝑢𝑣𝑤𝑥𝑦, where |𝑣𝑤𝑥|≤𝑛 and |𝑣𝑥|≥1. If 𝑣𝑤𝑥 does not have 𝑐’s, then 𝑢𝑣³𝑤𝑥³𝑦 has at least 𝑛+2 𝑎’s or 𝑏’s; if 𝑣𝑤𝑥 has at least a 𝑐, then it has not an 𝑎 since |𝑣𝑤𝑥|≤𝑛. And so 𝑢𝑤𝑦 has 𝑛 𝑎^′s, but no more than 2𝑛+2 𝑏’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 𝜇_𝐴(𝑢𝑤𝑦)=0 and 𝜈_𝐴(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.

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=(𝜇_𝛿,𝜈_𝛿) in 𝑄×(Σ∪{𝜀})×Γ×𝑄×Γ^∗ is defined by 𝜇_𝛿(𝑞₀,0,𝑍₀,𝑞₀,0𝑍₀)=0.7, 𝜈_𝛿(𝑞₀,0,𝑍₀,𝑞₀,0𝑍₀)=0.2, 𝜇_𝛿(𝑞₀,1,𝑍₀,𝑞₀,1𝑍₀)=0.6, 𝜈_𝛿(𝑞₀,1,𝑍₀,𝑞₀,1𝑍₀)=0.3, 𝜇_𝛿(𝑞₀,0,0,𝑞₀,00)=0.3, 𝜈_𝛿(𝑞₀,0,0,𝑞₀,00)=0.6, 𝜇_𝛿(𝑞₀,0,1,𝑞₀,01)=0.3, 𝜈_𝛿(𝑞₀,0,1,𝑞₀,01)=0.5, 𝜇_𝛿(𝑞₀,1,0,𝑞₀,10)=0.5, 𝜈_𝛿(𝑞₀,1,0,𝑞₀,10)=0.4, 𝜇_𝛿(𝑞₁,0,0,𝑞₁,𝜀)=0.6, 𝜈_𝛿(𝑞₁,0,0,𝑞₁,𝜀)=0.3, 𝜇_𝛿(𝑞₁,1,1,𝑞₁,𝜀)=0.5, 𝜈_𝛿(𝑞₁,1,1,𝑞₁,𝜀)=0.35, 𝜇_𝛿(𝑞₀,1,𝑍₀,𝑞₁,𝑍₀)=1, 𝜈_𝛿(𝑞₀,1,𝑍₀,𝑞₁,𝑍₀)=0, 𝜇_𝛿(𝑞₀,1,0,𝑞₁,0)=1, 𝜈_𝛿(𝑞₀,1,0,𝑞₁,0)=0, 𝜇_𝛿(𝑞₀,1,1,𝑞₁,1)=1, 𝜈_𝛿(𝑞₀,1,1,𝑞₁,1)=0, 𝜇_𝛿(𝑞₁,𝜀,𝑍₀,𝑞₂,𝑍₀)=1, 𝜈_𝛿(𝑞₁,𝜀,𝑍₀,𝑞₂,𝑍₀)=0.
Otherwise 𝜇_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=0 and 𝜈_𝛿(𝑞,𝜏,𝑍,𝑝,𝛾)=1 for (𝑞,𝜏,𝑍,𝑝,𝛾)∈𝑄×(Σ∪{𝜀})×Γ×𝑄×Γ^∗.
The IFSs 𝐼=(𝜇_𝐼,𝜈_𝐼) and 𝐹=(𝜇_𝐹,𝜈_𝐹) in 𝑄 are defined by 𝜇_𝐼(𝑞₀)=1, 𝜈_𝐼(𝑞₀)=0, 𝜇_𝐼(𝑞₁)=𝜇_𝐼(𝑞₂)=0, 𝜈_𝐼(𝑞₁)=𝜇_𝐼(𝑞₂)=1, 𝜇_𝐹(𝑞₂)=1, 𝜈_𝐹(𝑞₂)=0, 𝜇_𝐹(𝑞₀)=𝜇_𝐹(𝑞₁)=0 and 𝜈_𝐹(𝑞₀)=𝜇_𝐹(𝑞₁)=1.
And set 𝜂=𝜇_𝛿, 𝜎₀=𝜇_𝐼, and 𝜎₁=𝜇_𝐹.
By computing with the strings, 010, 111, 01110, 10101, 0011100, and 1011101∈Σ^∗, we have 𝜇_ℒ(ℳ)(010)=𝑓_𝒩(010)=0.6, 𝜈_ℒ(ℳ)(010)=0.3, 𝜇_ℒ(ℳ)(111)=𝑓_𝒩(111)=0.5, 𝜈_ℒ(ℳ)(111)=0.35, 𝜇_ℒ(ℳ)(01110)=𝑓_𝒩(01110)=0.5, 𝜈_ℒ(ℳ)(01110)=0.4, 𝜇_ℒ(ℳ)(10101)=𝑓_𝒩(10101)=0.3, 𝜈_ℒ(ℳ)(10101)=0.5, 𝜇_ℒ(ℳ)(0011100)=𝑓_𝒩(0011100)=0.3, 𝜈_ℒ(ℳ)(0011100)=0.6, 𝜇_ℒ(ℳ)(1011101)=𝑓_𝒩(1011101)=0.3, 𝜈_ℒ(ℳ)(1011101)=0.5.
This implies that 111 is better than 01110 because the degree of nonmembership of 𝜈_ℒ(ℳ)(111) is smaller than the 𝜈_ℒ(ℳ)(01110)'s although the degrees of membership of the fuzzy context-free languages 𝑓_𝒩(01110) and 𝑓_𝒩(111) are equal. Comparing the above five strings, 010 is the best and 0011100 is the worst.

6. Conclusions

Taking intuitionistic fuzzy sets as the structures of truth values, we have investigated intuitionistic fuzzy context-free languages and established pumping lemma for the underlying 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 IFPDAs^∅ 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).