Parallel Fuzzy Regular Expression and its ...

The Computer Journal Advance Access published December 17, 2015 c The British Computer Society 2015. All rights reserved.  For Permissions, please email: [email protected] doi:10.1093/comjnl/bxv104

Parallel Fuzzy Regular Expression and its Conversion to Epsilon-Free Fuzzy Automaton Sunita Garhwal1∗ and Ram Jiwari2

∗Corresponding author: [email protected]

Inspired by the applications of fuzzy automata and parallel regular expressions, we propose a new mathematical concept of parallel fuzzy regular expressions. In this paper, we investigate the equivalence between parallel fuzzy regular expressions and parallel fuzzy finite automata. An algorithm is proposed for carrying out the conversion of parallel fuzzy regular expressions to ε-free non-deterministic fuzzy automata. Furthermore, an analysis of the number of states obtained in the fuzzy automaton will be carried out using the proposed algorithm. Finally, the proposed algorithm is illustrated by a numerical example. Keywords: parallel fuzzy regular expression; parallel fuzzy finite automata; non-deterministic fuzzy automata; shuffle operator Received 23 August 2015; revised 10 October 2015 Handling editor: Mariangiola Dezani-Ciancaglini

1.

INTRODUCTION

Fuzzy regular expressions and their features are used to specify patterns for various applications such as learning systems, clinical monitoring, databases, descriptions of programming and natural languages [1–7]. The concept of fuzzy languages was introduced by Santos [8]. Fuzzy automata are a generalization of the classical automata, closing the gap between classical automata theory and natural languages [7]. The inclusion of shuffle operators in the regular expression is known as a parallel regular expression. It appears in various fields of computer science such as concurrency of processes, process algebra, multipoint communication, plan recognition, XML schema language Relax NG and modeling and verification of systems [9–14]. Motivated by the applications of fuzzy automata and parallel regular expressions, we propose a new mathematical model that involves fuzziness in paral´ c [15] developed an lel finite automata. Stamenkovi´c and Ciri´ approach for the conversion of fuzzy regular expressions to fuzzy automata using the concept of position automata [16, 17]. We extend their approach for the conversion of parallel fuzzy regular expressions to ε-free non-deterministic fuzzy automata.

Prior work. Various aspects of fuzzy automata and fuzzy regular expressions have been studied in formal language theory, and they reduce the gap between natural and programming languages. Since 1960, the membership values in fuzzy automata and languages have been considered from the Gödel structure [7]. In recent years, the membership values in fuzzy automata have derived from certain algebraic structures such as the lattice-ordered monoid [15, 18–22] and the complete residuated lattice [23, 24]. Li and Pedrycz [21] considered the integral lattice-ordered monoid and proved that a fuzzy language can be represented by a fuzzy regular expression if, and only if, a fuzzy automaton can represent it. Li et al. [22] introduced the concept of ε-move in fuzzy ´ c [15] treated the automata. Recently, Stamenkovi´c and Ciri´ scalar appearing in the fuzzy regular expression as the extended alphabet. Applying the concept of position automata, they have converted fuzzy regular expressions into fuzzy automata. Kuske [25] proposed a similar approach for the weighted regular expression over semiring. Mendivil and Garitago´ c [15] approach and tia [18] extended the Stamenkovi´c and Ciri´ removed the ε-transition from the fuzzy automata.

Section C: Computational Intelligence, Machine Learning and Data Analytics The Computer Journal, 2015

Downloaded from http://comjnl.oxfordjournals.org/ at Indian Institute of Technology Roorkee on March 21, 2016

1 Thapar University, Patiala 147004, India 2 IIT Roorkee, Uttarakhand 247667, India

2

S. Garhwal and R. Jiwari

(i) We introduce the concept of parallel fuzzy regular expressions and parallel fuzzy finite automata in Section 3. (ii) In Section 4, we propose an algorithm for the conversion of parallel fuzzy regular expressions to ε-free fuzzy automata. (iii) In Section 5, the complete procedure is illustrated by a numerical example. Finally, we conclude with the future perspectives.

2.

PRELIMINARIES

Let  denote an alphabet that consists of symbols/letters. ε represents the empty string.  ∗ [36] represents the free monoid generated by concatenation on , including the empty string. The length of a word w is denoted by |w|. A language over  is a subset of  ∗ . In this paper, we use integral -monoid £ = (L, ∧, ∨, ⊗, 0, 1) for defining the fuzzy sets. A fuzzy language FL over the alphabet  is a fuzzy subset of  ∗ , i.e. FL :  ∗ → L. Definition 2.1. A regular expression [36] can be recursively defined as follows: (i) ε, φ, a ∈  are regular expressions. (ii) If r1 and r2 are regular expressions, then r1∗ , r1 r2 , r1 + r2 are also regular expressions. All regular expressions can be obtained by applying above rules finite many times. Definition 2.2. Let X be a non-empty set. A fuzzy set A [6] is defined as ordered pair {(x, μA (x))} for ∃x ∈ X . μA (x) is interpreted as grade of membership and it is defined by membership function μA : X → [0, 1]. Definition 2.3. A fuzzy regular expression [21, 22] fre can be recursively defined as follows: (i) ε, φ, a ∈  are fuzzy regular expressions. (ii) λr1 ∈ fre , λ ∈ L and r1 ∈ fre . (iii) If r1 and r2 are fuzzy regular expressions, then r∗ , r1 + r2 , r1 r2 are also fuzzy regular expressions. All fuzzy regular expressions can be obtained by applying above rules finite many times. Definition 2.4. Shuffle operator [36, 37] is represented by &. It can be inductively defined as follows: (i) x & ε = ε & x = {x} (ii) a.x & b.y = {a.x1 |x1 ∈ x & b.y} ∪ {b.y1 |y1 ∈ a.x & y} Definition 2.5. A parallel regular expression [28] can be recursively defined as follows: (i) ε, φ, a ∈  are regular expressions. (ii) If r1 and r2 are regular expressions, then r1∗ , r1 r2 , r1 + r2 , r1 & r2 are also parallel regular expressions. All parallel regular expressions can be obtained by applying above rules finite many times. Definition 2.6. A parallel fuzzy regular expression fpre can be recursively defined as follows: (i) ε, φ, a ∈  are parallel fuzzy regular expressions. (ii) λr1 ∈ fpre , λ ∈ L and r1 ∈ fpre

Section C: Computational Intelligence, Machine Learning and Data Analytics The Computer Journal, 2015

Downloaded from http://comjnl.oxfordjournals.org/ at Indian Institute of Technology Roorkee on March 21, 2016

To the best of the author’s knowledge, there is almost no prior work on parallel fuzzy regular expressions. Parallel regular expressions were introduced by Garg [26] in the late 1990s. Subsequently, the transformation of parallel regular expressions to finite automata was established by various researchers (see [9, 10, 27–31] and the references therein). However, the concept of parallel fuzzy regular expressions and parallel fuzzy finite automata has thus far remained unexplored. In this paper, we convert parallel fuzzy regular expressions into ε-free fuzzy automata. First, we convert the fuzzy regular expression into the extended alphabet  ∪ Y , where represents the alphabet and Y is the set representing symbols associated with different representing scalars appearing as the membership values in the parallel fuzzy regular expression r. Secondly, the concept of modular decomposition will be used for converting the parallel fuzzy regular expressions to ε-fuzzy automata. ε-Fuzzy automata are converted into ε-free non-deterministic fuzzy automata by using the concept proposed by Mohri [32, 33] for the ε-removal in weighted automata. A similar methodology was used by Mendivil and Garitagotia [18] for the construction of ε-free fuzzy automata from fuzzy regular expressions. It should be noted that the concept of the extended alphabet was used by Kuske [25] in the context of the weighted regular expression. It was also applied by Stamenkovi´c and ´ c [15] and Mendivil and Garitagotia [18] in the context of Ciri´ the construction of fuzzy automata from fuzzy regular expressions. The concept of modular decomposition was proposed by Yamamoto [34, 35] for the conversion of semi-extended regular expressions to non-deterministic finite automata (NFA). The concept of position automata was proposed independently by Glushkov [16] and McNaughton and Yamada [17]. Position automata are used to convert a regular expression into an ε-free non-deterministic finite automaton. Positions are assigned to each letter of the regular expression; then using the concept of first, last and follow, regular expression is converted into an ε-free non-deterministic finite automaton. There exists a variety of shuffle operators, such as weak synchronized shuffling, strong synchronized shuffling, synchronous composition and asynchronous interleaving. In this paper, a shuffle operator refers to the asynchronous interleaving. The paper is organized as follows. Basic definitions and notations are introduced in Section 2. We claim the following contributions in this paper:

Parallel Fuzzy Regular Expression and its Conversion to Epsilon-Free Fuzzy Automaton

(iv) σ is a fuzzy subset of Q (starting fuzzy state), (v) τ is the fuzzy set of final states, (vi) δ represents the state transition function, and it is a fuzzy subset of Q × ( ∪ λ) → 2Q , (vii) γ represents the node transition function, and it is a N fuzzy subset of 2N × ( ∪ λ) → 22

(iii) If r1 and r2 are parallel fuzzy regular expressions, then r1∗ , r1 r2 , r1 + r2 , r1 & r2 are also parallel fuzzy regular expressions. All parallel fuzzy regular expressions can be obtained by applying above rules finite many times. Definition 2.7. A finite automaton [36] is a quintuple (Q, , σ , τ , δ), where Q is the set of states,  represents an alphabet, σ is the starting state and σ ∈ Q, τ ⊆ Q is the set of final states, δ is a transition function and δ ⊆ Q × ( ∪ ε) × Q.

Definition 2.8. A fuzzy automaton [18] is a quintuple (Q, , σ , τ , δ), where (i) (ii) (iii) (iv) (v)

Q is the set of states,  represents an alphabet, σ is a fuzzy subset of Q (starting fuzzy state), τ is the fuzzy subset of Q (final states), δ is a fuzzy subset of Q × ( ∪ ε) × Q (fuzzy transition relation).

Definition 2.11. For regular expression r, assign a subscript to each letter from left to right which will refer its position. The position set First, Last and Follow are computed inductively as follows [38]: First(φ) = First(ε) = Last(φ) = Last(ε) = φ First(ai ) = Last(ai ) = {i} First(r1 + r2 ) = First(r1 ) + First(r2 )

Last(r1 + r2 ) = Last(r1 ) + Last(r2 )

First(r1∗ ) = First(r1 )

Last(r1∗ ) = Last(r1 )

If ε ∈ r1

If ε ∈ r2

First(r1 r2 ) = First(r1 ) ∪ First(r2 ) Else

Last(r1 r2 ) = Last(r1 ) ∪ Last(r2 ) Else

First(r1 r2 ) = First(r1 ) EndIf

Last(r1 r2 ) = Last(r2 ) EndIf

Follow(φ) = Follow(ε) = Follow(a) = φ Follow(r1 + r2 ) = Follow(r1 ) ∪ Follow(r2 ) Follow(r1 r2 ) = Follow(r1 ) ∪ Follow(r2 ) ∪ Last(r1 ) × First(r2 )

Let w = a1 a2 · · · an

Follow(r1∗ ) = Follow(r1 ) ∪ Last(r1 ) × First(r1 )

FL(M )w = ∨σ (q0 ) ⊗ δa1 (q0 , q1 ) ⊗ · · · ⊗ δan (qn−1 , qn ) ⊗ τ (qn ) Let w = ε

FL(M )w = ∨ σ (q) ⊗ τ (q) q∈Q

Definition 2.9. A parallel finite automaton [28] is septuple (Q, N, , σ , τ , δ, γ ), where Q is the finite set of states and Q ⊆ 2N , N is a finite set of nodes,  represents an alphabet, σ is the starting state and σ ∈ Q, τ ⊆ Q is the set of final states, δ represents the state transition function defined by Q × ( ∪ λ) → 2Q , (vii) γ represents the node transition function defined by N 2N × ( ∪ λ) → 22 . (i) (ii) (iii) (iv) (v) (vi)

A state in a parallel finite automaton represents a set of nodes. Additional symbol λ in δ and γ represents the path divergence and convergence [9]. Definition 2.10. A parallel fuzzy finite automaton is septuple (Q, N, , σ , τ , δ, γ ), where (i) Q is the finite set of states and Q ⊆ 2N , (ii) N is a finite set of nodes, (iii)  represents an alphabet,

3.

PARALLEL FINITE AUTOMATA

Estrade et al. [9] investigated the equivalence between parallel finite automata and shuffle operation on regular languages. They designed a software tool, FLAT [39], for the conversion of parallel regular expressions to finite automata. In parallel finite automata, a state is represented by a set of active states. They have used λ to denote the divergence and convergence from a node. Parallel fuzzy finite automata are similar to parallel finite automata, except that the membership value is associated with each transition. This membership value is chosen from a lattice-ordered monoid. Example 3.1. Given parallel fuzzy regular expression r = (0.4yx & 0.8y)0.2x. Let μ = 0.4, ν = 0.8, κ = 0.2, Y = {μ, ν, κ}, Extended alphabet = {μ, ν, κ, x, y}, r1 = (μyx & νy)κx, r1 = (μ1 y2 x3 & ν4 y5 )κ6 x7 . Using position automata: follow(1) = {2}, follow(6) = {7}.

follow(2) = {3},

follow(4) = {5} and

λ represent divergence and convergence at q0 to (q1 , q5 ) and (q4 , q7 ) to q8 . For each transition (qi , a, qj ) ∧ a ∈ Y convert it into (qi , ε|a, qj ).

Section C: Computational Intelligence, Machine Learning and Data Analytics The Computer Journal, 2015

Downloaded from http://comjnl.oxfordjournals.org/ at Indian Institute of Technology Roorkee on March 21, 2016

(i) (ii) (iii) (iv) (v)

3

4

S. Garhwal and R. Jiwari 4.

Source state

Input

Output

Target state

[q0 ] [q1 , q5 ] [q1 , q5 ] [q2 , q5 ] [q2 , q5 ] [q1 , q6 ] [q1 , q6 ] [q1 , q7 ] [q2 , q6 ] [q2 , q6 ] [q3 , q5 ] [q3 , q5 ] [q3 , q6 ] [q3 , q6 ] [q2 , q7 ] [q4 , q5 ] [q4 , q6 ] [q3 , q7 ] [q4 , q7 ] [q8 ] [q9 ]

λ ε ε y ε ε y ε y y x ε x y y ε y x λ ε x

– 0.4 0.8 1 0.8 0.4 1 0.4 1 1 1 0.8 1 1 1 0.8 1 1 – 0.2 1

[q1 , q5 ] [q2 , q5 ] [q1 , q6 ] [q3 , q5 ] [q2 , q6 ] [q2 , q6 ] [q1 , q7 ] [q2 , q7 ] [q3 , q6 ] [q2 , q7 ] [q4 , q5 ] [q3 , q6 ] [q4 , q6 ] [q3 , q7 ] [q3 , q7 ] [q4 , q6 ] [q4 , q7 ] [q4 , q7 ] [q8 ] [q9 ] [qf ]

Given a parallel fuzzy regular expression r. Let Y be the set representing the symbols associated with different scalars appearing as the membership values in the parallel fuzzy regular expression r. The parallel fuzzy regular expression r is converted into r1 , where scalars are replaced by their symbol from Y . This step resembles the approaches used by various researchers (Refer to [15, 18, 25] for more details). Convert r1 into r1 by assigning a position to each symbol of r1 with respect to its appearance. Convert r1 into a parse tree such that  φ, ε, a ∈ ∪Y appear as a leaf node. Operators (union, concatenation, Kleene closure and shuffle) appear at an internal node, and in-order traversal of the parse tree is equal to r1 . Now, we will partition the parse tree into a sub-tree using modular decomposition. The modular decomposition technique was proposed by Yamamoto [34, 35] for the conversion of semi-extended regular expressions to non-deterministic finite automata and further used by Kumar and Verma [40] for the improvement of the similar conversion. In modular decomposition [34], we will partition the parse tree into sub-trees such that:

FIGURE 1. Depicts the equivalent parallel fuzzy finite automata for r = (0.4yx & 0.8y)0.2x.

  For each transition (qi , a, qj ) ∧ a ∈ ∪ε convert it into (qi , a|1, qj ). Each state is represented by a set of active nodes. Table 1 depicts different states of the parallel fuzzy finite automata shown in Fig. 1. Consider £ be the product structure. Membership of string w = yxyx can be determined using: FL(M )w = ∨σ (q0 ) ⊗ δa1 (q0 , q1 ) ⊗ · · · ⊗ δan (qn−1 , qn ) ⊗ τ (qn ) = ∨((1 ⊗ 0.4 ⊗ 1 ⊗ 1 ⊗ 0.8 ⊗ 1 ⊗ 0.2 ⊗ 1 ⊗ 1), (1 ⊗ 0.8 ⊗ 1 ⊗ 0.4 ⊗ 1 ⊗ 1 ⊗ 0.2 ⊗ 1 ⊗ 1), . . .) = 0.064

CONVERSION OF PARALLEL FUZZY REGULAR EXPRESSIONS TO FUZZY AUTOMATA

(1) The child of a shuffle operator or root of the parse tree will now be the root of the sub-tree. (2) The leaf node of a subtree can be any symbol from the extended alphabet or shuffle operator. (3) Each sub-tree does not contain a shuffle operator as the internal node. We compute the first, last and follow of each sub-tree using the well-known technique known as position automata [16, 17]. The root of the parse tree appears in a subtree known as the root sub-tree. Nodes of a sub-tree represent a regular expression. The left and right sub-trees of the shuffle operator are regarded as a divergence. Once the last position of the left and right sub-trees occurs, they will converge. Divergence and convergence in a module are represented by an edge λ(i, j), where the ith and jth modules are children of the current module. We convert a non-deterministic automaton into an ε-fuzzy automaton using Mendivil and Garitagoitia’s approach (Refer to [18] for more details). The algorithm FPRE_FA is proposed for the conversion of parallel fuzzy regular expressions to ε-fuzzy automata. Algorithm 1. FPRE_FA(r, M ) Input: Parallel Fuzzy Regular Expression r. Output: ε-Free fuzzy automata M such that FL(M ) = FL(r). 1.  Define a bijective function F : L − {0} → Y such that ∩Y = φ. 2. Convert r into r1 by replacing each scalar a by F(a). 3. Convert r1 into r1 by assigning positions to the symbol of r1 . 4. Construct parse tree Pt for r1 such that

Section C: Computational Intelligence, Machine Learning and Data Analytics The Computer Journal, 2015

Downloaded from http://comjnl.oxfordjournals.org/ at Indian Institute of Technology Roorkee on March 21, 2016

TABLE 1. State transition function for parallel fuzzy finite automaton (Q, N, , q0 /1, qf /1, δ, γ ).

Parallel Fuzzy Regular Expression and its Conversion to Epsilon-Free Fuzzy Automaton (a) a ∈  ∪ Y , φ, ε appear as a leaf node. (b) Operators ·, |, ∗ and & appear as internal nodes. (c) Inorder traversal of Pt is equal to r1 . 5. Partition the parse tree Pt using modular decomposition. 6. For each subtree of Pt

5

The ε-fuzzy automata obtained using the FPRE_FA algorithm is converted into ε-free fuzzy automata using the ε-removal approach used by Mohri [32] for the weighted automata. All states having only incoming ε-transitions are removed, which causes the number of states will be reduced.

Compute First, Last and Follow.

Convert transition (qi , a, qk ) to (qi , ε|F −1 (a), qk ). Else Convert transition (qi , a, qk ) to (qi , a|1, qk ). End If End For Theorem 4.1. Let r be a parallel fuzzy regular expression over integral -monoid. The number of states obtained in the ε-fuzzy automata using the FPRE_FA algorithm is equal to   I∈S m∈I (|m| + 1) + 2|s| in the worst case. Proof. Let M represents a set of modules obtained by dividing r into the parse tree using the modular decomposition technique. Let |m| be the total number of symbol appear in m ∈ M from the extended alphabet. We have applied the position automata in each module, and the number of states is equal to |m| + 1 for each module. Let S be a set representing all independent subset of M [34, 35]. The dependent modules are joined using the shuffle operator, and we have constructed fuzzy automata by taking the Cartesian products of the states appearing in the dependent modules. Two additional states are added for each shuffle operator for representing divergence and convergence transitions in the parent module. For s shuffle operators in r, 2∗ s states are added for the convergence and divergence. Hence,  the number ofstates in the dependent modules is equal the number of states to I∈S,m∈I (|m| + 1) + 2. Therefore,    ∗ in ε-fuzzy automata is equal to I∈S m∈I (|m| + 1) + 2 s.

5.

NUMERICAL EXAMPLE

In this section, we give an example to illustrate the conversion of parallel fuzzy regular expressions to ε-free non-deterministic fuzzy finite automata. Example 5.1. Given parallel fuzzy regular expression r = (0.1x∗ )((0.3yx & 0.8y) + 0.2y). Consider £ be the product structure. Y = {μ, ν, o, η}, r1 = (μx∗ )((νyx & oy) + ηy) and r1 = (μ1 x∗2 )((ν3 y4 x5 & o6 y7 ) + η8 y9 ). Figure 2 illustrates the parse tree for r1 . R1 and R2 modules are children of R0 . NFA for module R0 is represented in Fig. 3. NFA for sub-tree R1 and R2 are shown in Figs. 4 and 5, respectively. Green color state depicts the final state. Figure 6 represent ε-NFA for parallel fuzzy regular expression r. Divergence and Convergence represented by λ12 are replaced by ε and the Cartesian product of NFA for R1 and R2 gives us ε-NFA as shown in Fig. 6. ε-NFA of Fig. 6 can be converted into ε-fuzzy automata as shown in Fig. 7 using the following rules: For each transition (qi , a, qj ) ∧ a ∈ Y convert it into (qi , ε|a, qj ).

x

FIGURE 2. Parse tree for fuzzy parallel regular expression r1 = (μx∗ )((νyx & oy) + ηy).

Section C: Computational Intelligence, Machine Learning and Data Analytics The Computer Journal, 2015

Downloaded from http://comjnl.oxfordjournals.org/ at Indian Institute of Technology Roorkee on March 21, 2016

End For 7. Divergence and Convergence in a module are represented by an edge λ(i, j) where ith and jth module are children of the current module. 8. Let Ai (Q1 ,  ∪ Y , σ1 , τ1 , δ1 ) and Aj (Q2 ,  ∪ Y , σ2 /1, τ2 , δ2 ) be two ε-free fuzzy automata representing the left (ith) and right (jth) sub-tree of a shuffle operator. Define AS (Q1 × Q2 ,  ∪ Y , (σ1 , σ2 ), (τ1 , τ2 ), δ) such ql ) such that (qi , a, that δ((qi , qj ), a) = (qk , qj ) ∪ (qi , qk ) ∈ δ1 ∧ (qj , a, ql ) ∈ δ2 ∧ a ∈ ∪Y ∪ ε . Replace σ (i, j) with respective AS after adding ε-transition entering and leaving AS . 9. For each transition (qi , a, qk )/* Conversion of NFA to ε-Fuzzy Automata */ If (a ∈ Y )

6

S. Garhwal and R. Jiwari For each transition (qi , a, qj ) ∧ a ∈ into (qi , a|1, qj ).



 ∪ε convert it

Mohri [32] proposed the generic epsilon-removal for the weighted automata. Fuzzy automata are a special case of weighted automata. Using the ε-removal rules proposed by

Mohri [32], we obtain the ε-free fuzzy automata as shown in Fig. 8.

6.

CONCLUSIONS

In this paper, we have introduced the concept of parallel fuzzy regular expressions and parallel fuzzy finite automata. Further, an algorithm is designed for the conversion of parallel fuzzy

FIGURE 3. NFA (Q, {Y ∪ {x, y}}, {q0 |1}, {q9 |1, q36 |1}, δ) for module R0 , λ12 represent divergence to module R1 and R2 .

FIGURE 5. NFA (Q, {o, y}, {qs |1}, {q7 |1}, δ) for module R2 .

FIGURE 6. ε-NFA (Q, {Y ∪ {x, y}}, {q0 |1}, {q9 |1, q7 |1}, δ) for fuzzy parallel regular expression r = (0.1x∗ )((0.3yx & 0.8y) + 0.2y).

Section C: Computational Intelligence, Machine Learning and Data Analytics The Computer Journal, 2015

Downloaded from http://comjnl.oxfordjournals.org/ at Indian Institute of Technology Roorkee on March 21, 2016

FIGURE 4. NFA (Q, {ν, y, x}, {qs |1}, {q5 |1}, δ) for module R1 .

Parallel Fuzzy Regular Expression and its Conversion to Epsilon-Free Fuzzy Automaton

FIGURE 8. ε-Free fuzzy automata (Q, {Y ∪ {x, y}}, {q0 |1}, {q9 |1, qf |1}, δ) for r = (0.1x∗ )((0.3yx & 0.8y) + 0.2y).

Section C: Computational Intelligence, Machine Learning and Data Analytics The Computer Journal, 2015

Downloaded from http://comjnl.oxfordjournals.org/ at Indian Institute of Technology Roorkee on March 21, 2016

FIGURE 7. ε-Fuzzy automata (Q, {Y ∪ {x, y}}, {q0 |1}, {q9 |1, q36 |1}, δ) for r = (0.1x∗ )((0.3yx & 0.8y) + 0.2y).

7

8

S. Garhwal and R. Jiwari

regular expressions to ε-free non-deterministic fuzzy automata. The proposed approach uses the concept of modular decomposition, position automata, extended alphabets, and the generic epsilon-removal algorithm. Parallel fuzzy regular expressions will provide new technical support in information retrieval and have a wide range of applications in various fields such as the concurrency of processes, process algebra and the verification of systems. In the future, we will try to reduce the number of states fuzzy automata obtained from parallel fuzzy regular expressions.

[1] Wee, W.G. and Fu, K.S. (1969) A formulation of fuzzy automata and its application as a model of learning systems. IEEE Trans. Syst. Man Cybernet., 5, 215–223. [2] Ying, M.S. (2002) A formal model of computing with words. IEEE Trans. Fuzzy Syst., 10, 640–652. [3] Qiu, D. and Wang, H. (2005) A probabilistic model of computing with words. J. Comput. Syst. Sci., 70, 176–200. [4] Lin, F. and Ying, H. (2002) Modeling and control of fuzzy discrete event systems. IEEE Trans. Syst. Man Cybernet. B: Cybernet., 32, 408–415. [5] Qiu, D. (2005) Supervisory control of fuzzy discrete event systems: a formal approach. IEEE Trans. Syst. Man Cybernet. B: Cybernet., 35, 72–88. [6] Zadeh, L.A. (1971) Fuzzy Languages and their Relation to Human and Machine Intelligence. Proc. Int. Conf. Man and Computer, Bordeaux, France, pp. 130–165. [7] Mordeson, J.N. and Malik, D.S. (2002) Fuzzy Automata and Languages: Theory and Applications. Chapman & Hall/ CRC, Boca Raton, FL. [8] Santos, E.S. (1968) Maximin automata. Inform. Control, 13, 363–377. [9] Estrade, B.D., Perkins, A.L. and Harris, J.M. (2006) Explicitly Parallel Regular Expressions. Proc. 1st Int. Multi-symposiums on Computer and Computational Sciences (IMSCCS06), IEEE, Hangzhou, China, pp. 402–409. [10] Estrade, B.D. (2005) An investigation of equivalent serialized forms of parallel finite automata, Master’s Thesis, The University of Southern Mississippi, USA. [11] Nivat, M. (1981) Behaviours of synchronized systems of processes. Report L.I.T.P. no. 81-64, 7, Université de Paris. [12] Baeten, J.C.M. and Weijland, W.P. (1990) Process Algebra. Cambridge tracts in Theoretical Computer Science 18. Cambridge University Press, Cambridge. [13] Clark, J. and Murata, M. (2001) RELAX NG specifications. http://relaxng.org/spec-20011203.html (accessed December 3, 2001). [14] Iwama, K. (1983) Unique Decomposability of Shuffled Strings: A Formal Treatment of Asynchronous Time-multiplexed Communication. Proc. 15th Annual ACM Symp. Theory of Computing, Boston, MA, USA, April 25–27, pp. 374–381. ´ c, M. (2012) Construction of fuzzy [15] Stamenkovi´c, A. and Ciri´ automata from fuzzy regular expressions. Fuzzy Sets Syst., 199, 1–27.

Section C: Computational Intelligence, Machine Learning and Data Analytics The Computer Journal, 2015

Downloaded from http://comjnl.oxfordjournals.org/ at Indian Institute of Technology Roorkee on March 21, 2016

REFERENCES

[16] Glushkov, V.M. (1961) The abstract theory of automata. Russ. Math. Surv., 16, 1–53. [17] McNaughton, R. and Yamada, H. (1960) Regular expressions and state graphs for automata. IEEE Trans. Electron. Comput., 9, 39–47. [18] Mendivil, J.R.G.D. and Garitagoitia, J.R. (2015) A comment on “Construction of fuzzy automata from fuzzy regular expressions”. Fuzzy Sets Syst., 262, 102–110. [19] Kumar, R. and Kumar, A. (2014) Metamorphosis of fuzzy regular expressions to fuzzy automata using the follow automata. preprint arXiv:1411.2865. [20] Li, Y. (2011) Finite automata theory with membership values in lattices. Inform. Sci., 181, 1003–1017. [21] Li, Y. and Pedrycz, W. (2005) Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids. Fuzzy Sets Syst., 156, 68–92. [22] Li, Z., Li, P. and Li, Y.M. (2006) The relationships among several types of fuzzy automata. Inform. Sci., 176, 2208–2226. ´ c, M., Stamenkovi´c, A., Ignjatovi´c, J. and Petkovi´c, T. (2010) [23] Ciri´ Fuzzy relation equations and reduction of fuzzy automata. J. Comput. Syst. Sci., 76, 609–633. ´ c, M. and Bogdanovi´c, S. (2008) Determiniza[24] Ignjatovi´c, J., Ciri´ tion of fuzzy automata with membership values in complete residuated lattices. Inform. Sci., 178, 164–180. [25] Kuske, D. (2008) Schützenberger’s theorem on formal power series follows from Kleene’s theorems. Theor. Comput. Sci., 401, 243–248. [26] Garg, V.K. (1989) Modeling of Distributed Systems by Concurrent Regular Expressions. Proc. 2nd Int. Conf. Formal Description Techniques for Distributed Systems and Communication Protocols, Vancouver, BC, Canada, December 5–8, pp. 313–327. [27] Kumar, A. and Verma, A.K. (2013) A novel algorithm for the conversion of shuffle regular expressions into non-deterministic finite automata. Maejo Int. J. Sci. Technol., 7, 396–407. [28] Stotts, P.D. and Pugh, W. (1994) Parallel finite automata for modeling concurrent software systems. J. Syst. Softw., 27, 27–43. [29] Kumar, A. and Verma, A.K. (2014) A novel algorithm for the conversion of parallel regular expressions to non-deterministic finite automata. Appl. Math. Inf. Sci., 8, 95–105. [30] Gelade, W. (2010) Succinctness of regular expressions with interleaving, intersection and counting. Theor. Comput. Sci., 411, 2987–2998. [31] Gelade, W., Martens, W. and Neven, F. (2006) Optimizing Schema Languages for XML: Numerical Constraints and Interleaving. 11th Int. Conf., Barcelona, Spain, January 10–12, Lecture Notes in Computer Science 4353, pp. 269–283. Springer, Berlin, Heidelberg. [32] Mohri, M. (2002) Generic Epsilon-removal and input Epsilonnormalization algorithms for weighted transducers. Int. J. Found. Comput. Sci., 13, 129–143. [33] Allauzen, C. and Mohri, M. (2006) A Unified Construction of the Glushkov, Follow, and Antimirov Automata. Proc. 31st Int. Symp., MFCS 2006, Stará Lesná, Slovakia, August 28– September 1, Lecture Notes in Computer Science 4162, pp. 110–121. Springer, Berlin, Heidelberg.

Parallel Fuzzy Regular Expression and its Conversion to Epsilon-Free Fuzzy Automaton

Theory and Applications (LATA 2015), Nice, France, March 2–6, Lecture Notes in Computer Science 8977, pp. 275–286. Springer, Berlin, Heidelberg. [38] Gruber, H. and Holzer, M. (2014) From Finite Automata to Regular Expressions and Back-A Summary on Descriptional Complexity. Proc. 14th Int. Conf. Automata and Formal Languages, AFL 2014, Szeged, Hungary, May 27–29, Electronic Proceedings in Theoretical Computer Science 151, pp. 25–48. [39] http://search.cpan.org/∼estrabd/FLAT-0.9.1/lib/FLAT.pm (accessed December 6, 2007). [40] Kumar, A. and Verma, A.K. (2015) An improved algorithm for the metamorphosis of semi-extended regular expressions to deterministic finite automata. Comput. J., 58, 448–456.

Section C: Computational Intelligence, Machine Learning and Data Analytics The Computer Journal, 2015

Downloaded from http://comjnl.oxfordjournals.org/ at Indian Institute of Technology Roorkee on March 21, 2016

[34] Yamamoto, H. (2001) A New Recognition Algorithm for Extended Regular Expressions. Proc. 12th Int. Symp., ISAAC 2001 Christchurch, New Zealand, December 19–21, Lecture Notes in Computer Science 2223, pp. 257–267. Springer, Berlin, Heidelberg. [35] Yamamoto, H. (2003) A New Translation from Semi-extended Regular Expressions into NFA and Its Application to an Approximate Matching Problem. Proc. 14th Int. Symp., ISAAC’03, Kyoto, Japan, December 15–17, Lecture Notes in Computer Science 2906, pp. 158–167. Springer, Berlin, Heidelberg. [36] Hopcroft, J.E., Motwani, R. and Ullman, J.D. (2000) Introduction to Automata Theory, Languages and Computation. Pearson Education, Singapore. [37] Sulzmann, M. and Thiemann, P. (2015) Derivatives for Regular Shuffle Expressions. Proc. 9th Int. Conf. Language and Automata

9