Research Article Generative Power and Closure Properties of Watson-Crick … · 2019. 7. 30. ·...

Research ArticleGenerative Power and Closure Properties ofWatson-Crick Grammars

Nurul Liyana Mohamad Zulkufli, Sherzod Turaev, Mohd Izzuddin Mohd Tamrin,and Azeddine MessikhKulliyyah of Information and Communication Technology, International Islamic University Malaysia, 53100 Kuala Lumpur, Malaysia

Correspondence should be addressed to Nurul Liyana Mohamad Zulkufli; [email protected]

Received 27 November 2015; Revised 19 February 2016; Accepted 2 June 2016

Academic Editor: Ryotaro Kamimura

Copyright © 2016 Nurul Liyana Mohamad Zulkufli et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

WedefineWKlinear grammars, as an extension ofWKregular grammarswith linear grammar rules, andWKcontext-free grammars,thus investigating their computational power and closure properties.We show thatWK linear grammars can generate some context-sensitive languages. Moreover, we demonstrate that the family of WK regular languages is the proper subset of the family of WKlinear languages, but it is not comparable with the family of linear languages. We also establish that the Watson-Crick regulargrammars are closed under almost all of the main closure operations.

1. Introduction

DNAcomputing appears as a challenge to design new types ofcomputing devices, which differ from classical counterpartsin fundamental way, to solve wide spectrum of computa-tionally intractable problems. DNA (deoxyribonucleic acid)is double-stranded chain of nucleotides, which differ by theirchemical bases that are adenine (A), guanine (G), cytosine (C),and thymine (T), and they are paired as A-T, C-G according totheWatson-Crick complementary as it is illustrated in Figure 1[1]. The massive parallelism, another fundamental feature ofDNAmolecules, allows performing millions of cut and pasteoperations simultaneously on DNA strands until a completeset of new DNA strands performing are generated.These twofeatures give high hope for the use of DNA molecules andDNA based biooperations to develop powerful computingparadigms and devices.

Since a DNA strand can be interpreted as a double strandsequence of symbols, the DNA replication and synthesizeprocesses can be modeled using methods and techniquesof formal language theory. Watson-Crick (WK) automata[2], one of the recent computational models abstractingthe properties of DNA molecules, are finite automata withtwo reading heads, working on complete double-strandedsequences where characters on corresponding positions from

the two strands of the input are related by a complementarityrelation similar to the Watson-Crick complementarity ofDNA nucleotides. The two strands of the input are separatelyread from left to right by heads controlled by a common state.Several variants have been introduced and studied in recentpapers [3–7].

WK regular grammars [8], a grammar counterpart ofWK automata, generate double-stranded strings related bya complementarity relation as in a WK automaton but userules as in a regular grammar. The approach of using formalgrammars in the study of biological and computationalproperties of DNA molecules by formal grammars is a newdirection in the field of DNA computing: we can introducepowerful variants of WK grammars, such as WK linear, WKcontext-free, and WK regulated grammars, and use them inthe investigation of the properties of DNA structures andalso inDNA applications in food authentication, gene diseasedetection, and so forth. In this paper, we introduce WKlinear grammars and study the generative capacity in therelationship of Chomsky grammars.

Further, as a motivation, we show synthesis processes, forinstance, in DNA replication (Figure 2) can be simulated byderivations inWK grammars.The replication of DNA beginsat the origin(s).The double strand then separated by proteins,producing bubble-like shape(s).The synthesis of new strands

Hindawi Publishing CorporationApplied Computational Intelligence and So ComputingVolume 2016, Article ID 9481971, 12 pageshttp://dx.doi.org/10.1155/2016/9481971

2 Applied Computational Intelligence and Soft Computing

N

N N-H

N

O

N-HH

H-N

O

N

H

N

N-H

N

N

N

H

N

O

O

H-N

N

Adenine (A)

Guanine (G)

Thymine (T)

Cytosine (C)

CH3

Figure 1: The structure of DNA strand.

Lagging strand

DNA pol I DNA ligaseDNA pol III

DNA pol III

Primer

Primase

Helicase

Parental DNA

OverviewOrigin of replication

Lagging strand Leading strandLeading strand

Parental DNA

5

3

5

3

5

35

3

5

3

Figure 2: Synthesis process in bacterial DNA replication.

using the parental strands as templates starts from the originsand proceeds in the 5 to 3 direction of both strands [1].

This synthesis process in general can be seen as a stringgeneration. The enzymes responsible for the synthesizing,DNA polymerases, cannot initiate the process by themselvesbut can only add nucleotides to an existing RNA chain. Thischain is called primer which is produced by the enzymeprimase. From the grammar perspective, the primase canbe interpreted as the start symbol 𝑆. After the primer hasbeen connected to the parental strand, one of the synthesizingenzymes, called DNA polymerase III, continues to addnucleotides one by one to the primer and are complementedwith the parental strand.The synthesis finishes with replacingRNA primer with DNA nucleotides using the enzyme DNApolymerase I and joining with DNA ligase. Again, from thegrammar perspective, DNA polymerase I and polymerase IIIact as production rules in the grammar, specially, DNA ligaseresembles to a terminal production (see Figure 3).

The paper is organized as follows. In Section 2, we givesome notions and definitions from the theories of formallanguages and DNA computing needed in the sequel. InSection 3, we define WK grammars and languages generated

AwS

Figure 3: The simulation of a synthesis process with a derivation.

by these grammars. Section 4 is devoted to the study ofthe generative capacity of WK regular and linear grammars.In Section 5, we investigate the closure properties of WKgrammars. Furthermore, we show the application of WKgrammars in the analyses of DNA structures and program-ming language structures in Section 6. As the conclusion, wediscuss open problems and interesting future research topicsrelated to WK grammars in Section 7.

2. Preliminaries

We assume readers are familiar with formal languages theoryand automata. Readers are referred to [3, 9–11].

Applied Computational Intelligence and Soft Computing 3

Throughout this paper we use the following notations.Let ∈ be the belonging relationship of an element to thecorresponding set and ∉ indicates its negation. The symbol⊆ indicates the inclusion while ⊂ notes the proper inclusion.The notations 0, |𝐴|, and 2𝐴 denote the empty set, thecardinality of a set 𝐴, and the power set of 𝐴, respectively.When Σ is an alphabet (a finite set of symbols), the set ofall finite strings is denoted by Σ∗, while Σ+ shows similarmeaningwithout including empty strings (we use 𝜆 for emptystring). The length of a string 𝑎 ∈ Σ∗ is shown by |𝑎|. Alanguage is a subset 𝐿 ⊆ Σ∗.

Next we recall some terms regarding the closure proper-ties of languages. The union of two languages, 𝐿

1∪ 𝐿2, is the

set of strings including the elements contained in both sets of𝐿1and 𝐿

2. The concatenation of two languages is yielded by

lining two strings from both languages which is shown by

𝐿1𝐿2= {𝑎𝑏 | 𝑎 ∈ 𝐿

1, 𝑏 ∈ 𝐿

2} . (1)

The Kleene-star closure is the closure under the Kleene ∗operation, the set of all possible strings in 𝐿 including theempty string. The mirror image of a word 𝑎 = 𝑎

1𝑎2⋅ ⋅ ⋅ 𝑎𝑛is

𝑎𝑅= 𝑎𝑛⋅ ⋅ ⋅ 𝑎2𝑎1. For language 𝐿, its mirror image is

𝐿𝑅= {𝑎𝑅| 𝑎 ∈ 𝐿} . (2)

A substitution is a mapping 𝑠 : Σ∗ → 𝑇∗ where 𝑠(𝜆) = 𝜆and 𝑠(𝑎

1, 𝑎2) = 𝑠(𝑎

1)𝑠(𝑎2) for 𝑎

1, 𝑎2∈ Σ∗. The substitution

for a language 𝐿 ⊆ Σ∗; that is, 𝑠(𝐿) is the union of 𝑠(𝑦), where𝑦 ∈ 𝐿. A substitution 𝑠 is called finite if its length |𝑠(𝑎)| is finitefor each 𝑎 ∈ Σ. Amorphism is a substitution where its lengthis 1.

A Chomsky grammar is defined by 𝐺 = (𝑁, 𝑇, 𝑆, 𝑃)where𝑁 is the set ofnonterminal symbols and𝑇 is the set of terminalsymbols, and 𝑁 ∩ 𝑇 = 0. 𝑆 ∈ 𝑁 is the start symbol while𝑃 ⊆ (𝑁 ∪ 𝑇)

∗𝑁(𝑁 ∪ 𝑇)

∗× (𝑁 ∪ 𝑇)

∗ is the set of productionrules. We write 𝛼 → 𝛽 indicating the rewriting process of thestrings based on the production rules (𝛼, 𝛽) ∈ 𝑃. The term 𝑥directly derives 𝑦 is written as 𝑥 ⇒ 𝑦 when

𝑥 = 𝑎1𝛼𝑎2,

𝑦 = 𝑎1𝛽𝑎2

(3)

for some production rules 𝛼 → 𝛽 ∈ 𝑃. A grammar generatesa language defined by

𝐿 (𝐺) = {𝑤 ∈ 𝑇∗: 𝑆 ⇒

∗𝑤} . (4)

According to the forms of production rules, grammars areclassified as follows. A grammar 𝐺 = (𝑁, 𝑇, 𝑆, 𝑃) is called

(i) context-sensitive if each production has the form𝑢1𝐴𝑢2→ 𝑢1𝑢𝑢2, where 𝐴 ∈ 𝑁, 𝑢

1, 𝑢2, ∈ (𝑁 ∪ 𝑇)

∗,and 𝑢 ∈ (𝑁 ∪ 𝑇)+;

(ii) context-free if each production has the form 𝐴 → 𝑢,where 𝐴 ∈ 𝑁 and 𝑢 ∈ (𝑁 ∪ 𝑇)∗;

(iii) linear if each production has the form 𝐴 → 𝑢1𝐵𝑢2or

𝐴 → 𝑢, where 𝐴, 𝐵 ∈ 𝑁 and 𝑢1, 𝑢2, 𝑢 ∈ 𝑇

∗;

(iv) right-linear if each production has the form 𝐴 → 𝑢𝐵or 𝐴 → 𝑢, where 𝐴, 𝐵 ∈ 𝑁 and 𝑢 ∈ 𝑇∗;

(v) left-linear if each production has the form𝐴 → 𝐵𝑢 or𝐴 → 𝑢, where 𝐴, 𝐵 ∈ 𝑁 and 𝑢 ∈ 𝑇∗.

A right-linear and left-linear grammars are called regular.The families of languages generated by these grammars

are REG, LIN, CF, and CS, respectively. The families ofrecursive enumerable languages are denoted by RE while thefamilies of finite languages are denoted by FIN.Thus the nextrelation holds [11].

Theorem 1 (Chomsky hierarchy). Consider

𝐹𝐼𝑁 ⊂ 𝑅𝐸𝐺 ⊂ 𝐿𝐼𝑁 ⊂ 𝐶𝐹 ⊂ 𝐶𝑆 ⊂ 𝑅𝐸. (5)

We recall the definition of a finite automaton. A finiteautomaton (FA) is a quintuple𝑀 = (𝑄,𝑉, 𝑞

0, 𝐹, 𝛿), where 𝑄

is the set of states, 𝑞0∈ 𝑄 is the initial state, and 𝐹 ⊆ 𝑄 is set

of final states. Meanwhile 𝑉 is an alphabet and 𝛿 : 𝑄 × 𝑉 →2𝑄 is called the transition function. The set (language) of allstrings accepted by 𝑀 is denoted by 𝐿(𝑀). We denote thefamily of languages accepted by finite automata by FA. Then,FA = REG (see [11]).

Next, we cite some basic definitions and results ofWatson-Crick automata.

The key feature of WK automata is the symmetric relationon an alphabet 𝑉; that is, 𝜌 ⊆ 𝑉 × 𝑉. In this paper, forsimplicity, we use the form ⟨𝑢/V⟩ to mention the elements(𝑢, V) in the set of all pairs of strings 𝑉×𝑉 (which we chooseto write as [𝑉/𝑉]), and, instead of𝑉∗×𝑉∗, we write ⟨𝑉∗/𝑉∗⟩.

Watson-Crick domain is the set of well-formed double-stranded strings (molecules)

WK𝜌(𝑉) = [

𝑉

𝑉]

∗

𝜌

= {[𝑎

𝑏] : 𝑎, 𝑏 ∈ 𝑉, (𝑎, 𝑏) ∈ 𝜌} . (6)

WK+𝜌(𝑉) holds the similar meaning without including 𝜆. We

write

[𝑎1

𝑏1

] [𝑎2

𝑏2

] ⋅ ⋅ ⋅ [𝑎𝑛

𝑏𝑛

] ∈WK𝜌

(7)

as [𝑢/V] where the upper strand is 𝑢 = 𝑎1𝑎2⋅ ⋅ ⋅ 𝑎𝑛and the

lower strand is V = 𝑏1𝑏2⋅ ⋅ ⋅ 𝑏𝑛. Note that when the elements in

the upper strand are complemented and have the same lengthwith the lower strand,

⟨𝑢

V⟩ = [

𝑢

V] . (8)

A Watson-Crick finite automaton (WKFA) is 6-tuple

𝑀 = (𝑄,𝑉, 𝑞0, 𝐹, 𝛿, 𝜌) , (9)


where 𝑄, 𝑉, 𝑞0, and 𝐹 are the same as a FA. Meanwhile the

transition function 𝛿 is

𝛿 : 𝑄 × ⟨𝑉∗

𝑉∗⟩ → 2

𝑄, (10)

where 𝛿(𝑞, ⟨𝑢/V⟩) is not an empty set only for finitely manytriples (𝑞, 𝑢, V) ∈ 𝑄×𝑉∗ ×𝑉∗. Similar to FA, we can write therelation in transition function 𝑞

2∈ 𝛿(𝑞1, ⟨𝑢/V⟩) as a rewriting

rule in grammars; that is,

𝑞1⟨𝑢

V⟩ → ⟨

𝑢

V⟩𝑞2. (11)

We describe the reflexive and transitive closure of→ as→∗.The language accepted by a WKFA𝑀 is

𝐿 (𝑀) = {𝑢 : [𝑢

V] ∈WK

𝜌(𝑉) , 𝑞

0[𝑢

V]

→∗[𝑢

V] 𝑞, where 𝑞 ∈ 𝐹} .

(12)

The family of languages accepted is indicated by WKFA. It isshown in [10, 12] that

REG ⊂WKFA ⊂ CS. (13)

3. Definitions

In this section we slightly modified the definition of Watson-Crick regular grammars introduced in [8] in order to extendthe concept to linear grammars and context-free grammars.

Definition 2. A Watson-Crick (WK) grammar 𝐺 = (𝑁, 𝑇, 𝑆,𝑃, 𝜌) is called

(i) regular if each production has the form

𝐴 → ⟨𝑢

V⟩𝐵

or 𝐴 → ⟨𝑢V⟩ ,

(14)

where 𝐴, 𝐵 ∈ 𝑁 and ⟨𝑢/V⟩ ∈ ⟨𝑇∗/𝑇∗⟩;(ii) linear if each production has the form

𝐴 → ⟨𝑢1

V1

⟩𝐵⟨𝑢2

V2

⟩

or 𝐴 → ⟨𝑢V⟩ ,

(15)

where 𝐴, 𝐵 ∈ 𝑁 and ⟨𝑢1/V1⟩, ⟨𝑢2/V2⟩, ⟨𝑢/V⟩ ∈

⟨𝑇∗/𝑇∗⟩;

(iii) context-free if each production has the form

𝐴 → 𝛼, (16)

where 𝐴 ∈ 𝑁 and 𝛼 ∈ (𝑁 ∪ (⟨𝑇∗/𝑇∗⟩))∗.

Definition 3. Let𝐺 = (𝑁, 𝑇, 𝜌, 𝑆, 𝑃) be aWK linear grammar.We say that 𝑥 ∈ (𝑁 ∪ ⟨𝑇∗/𝑇∗⟩)∗ directly derives 𝑦 ∈ (𝑁 ∪⟨𝑇∗/𝑇∗⟩)∗, denoted by 𝑥 ⇒ 𝑦, iff

𝑥 = ⟨𝑢1

V1

⟩𝐴⟨𝑢2

V2

⟩ ,

𝑦 = ⟨𝑢1

V1

⟩⟨𝑢3

V3

⟩𝐵⟨𝑢4

V4

⟩⟨𝑢2

V2

⟩

or 𝑦 = ⟨𝑢1V1

⟩⟨𝑢

V⟩⟨𝑢2

V2

⟩ ,

(17)

where 𝐴, 𝐵 ∈ 𝑁, 𝑢𝑖, V𝑖∈ ⟨𝑇∗/𝑇∗⟩, 𝑖 = 1, 2, 3, 4, and

𝐴 → ⟨𝑢3

V3

⟩𝐵⟨𝑢4

V4

⟩ ,

𝐴 → ⟨𝑢

V⟩ ∈ 𝑃.

(18)

Definition 4. Let 𝐺 = (𝑁, 𝑇, 𝜌, 𝑆, 𝑃) be a WK context-freegrammar. We say that 𝑥 ∈ (𝑁 ∪ ⟨𝑇∗/𝑇∗⟩)∗ directly derives𝑦 ∈ (𝑁 ∪ ⟨𝑇

∗/𝑇∗⟩)∗, denoted by 𝑥 ⇒ 𝑦, if and only if

𝑥 = ⟨𝑢1

V1

⟩𝐴⟨𝑢2

V2

⟩ ,

𝑦 = ⟨𝑢1

V1

⟩𝛼⟨𝑢2

V2

⟩ ,

(19)

where 𝐴, 𝐵 ∈ 𝑁, 𝑢𝑖, V𝑖∈ ⟨𝑇∗/𝑇∗⟩, 𝑖 = 1, 2, 3, 4, and 𝐴 → 𝛼 ∈

𝑃.

Remark 5. We use a common notion “Watson-Crick gram-mars” referring to any type of WK grammars.

Definition 6. The language generated by a WK grammar is aquintuple 𝐺 which is defined as

𝐿 (𝐺) = {𝑢 : [𝑢

V] ∈WK

𝜌(𝑇) , 𝑆 ⇒

∗[𝑢

V]} . (20)

4. Generative Power ofWatson-Crick Grammars

In this section, we establish results regarding the computa-tional power of WK grammars.

4.1. A Normal Form for Watson-Crick Linear Grammars.Next, we define 1-normal form for WK linear grammarsand show that, for every WK linear grammar 𝐺, there is anequivalent WK linear grammar 𝐺 in the normal form; thatis, 𝐿(𝐺) = 𝐿(𝐺).

Definition 7. A linearWK grammar𝐺 = (𝑁, 𝑇, 𝜌, 𝑃, 𝑆) is saidto be in the 1-normal form if each rule in 𝑃 of the form

𝐴 → ⟨𝑢1

V1

⟩𝐵⟨𝑢2

V2

⟩

or 𝐴 → ⟨𝑢1V1

⟩ ,

(21)

where |𝑢𝑖| ≤ 1, |V

𝑖| ≤ 1, 𝑖 = 1, 2, and 𝐴, 𝐵 ∈ 𝑁.


Lemma 8. For every WK linear grammar 𝐺, there exists anequivalent WK linear grammar 𝐺 in the 1-normal form.

Proof. Let 𝐺 = (𝑁, 𝑇, 𝜌, 𝑆, 𝑃) be a WK linear grammar. Let

𝐴 → ⟨

𝑎11⋅ ⋅ ⋅ 𝑎1𝑚1

𝑏11⋅ ⋅ ⋅ 𝑏1𝑛1

⟩𝐵⟨

𝑎2𝑚2

⋅ ⋅ ⋅ 𝑎21

𝑏2𝑛2

⋅ ⋅ ⋅ 𝑏21

⟩ (22)

be a production in 𝑃 where 𝑚1> 1, 𝑚

2> 1, 𝑛

1> 1, or

𝑛2> 1. Without loss of generality, we assume that 𝑚

1≥ 𝑛1

and𝑚2≥ 𝑛2.Then, we define the following sequence of right-

linear and left-linear production rules:

𝐴 → ⟨𝑎11

𝑏11

⟩𝐴𝑟

11,

𝐴𝑟

11→ ⟨

𝑎12

𝑏12

⟩𝐴𝑟

12

.

.

.

𝐴𝑟

1𝑛1−1→ ⟨

𝑎1𝑛1

𝑏1𝑛1

⟩𝐴𝑟

1𝑛1

,

𝐴𝑟

1𝑛1

→ ⟨

𝑎1𝑛1+1

𝜆⟩𝐴𝑟

1𝑛1+1

.

.

.

𝐴𝑟

1𝑚1−1→ ⟨

𝑎1𝑚1

𝜆⟩𝐴𝑟

1𝑚1

,

𝐴𝑟

1𝑚1

→ 𝐴𝑟

21⟨𝑎21

𝑏21

⟩ ,

𝐴𝑟

21→ 𝐴

𝑟

22⟨𝑎22

𝑏22

⟩

.

.

.

𝐴𝑟

1𝑛2−1→ 𝐴

𝑟

2𝑛2

⟨

𝑎2𝑛2

𝑏2𝑛2

⟩,

𝐴𝑟

2𝑛2

→ 𝐴𝑟

2𝑛2+1⟨

𝑎2𝑛2+1

𝜆⟩

.

.

.

𝐴𝑟

2𝑚2−1→ ⟨

𝑎2𝑚2

𝜆⟩ ,

(23)

where 𝐴𝑟1𝑗, 1 ≤ 𝑗 ≤ 𝑚

1, and 𝐴𝑟

2𝑗, 1 ≤ 𝑗 ≤ 𝑚

2− 1, are new

nonterminals.

Let

𝑟 : 𝐴 → ⟨𝑎1𝑎2⋅ ⋅ ⋅ 𝑎𝑚

𝑏1𝑏2⋅ ⋅ ⋅ 𝑏𝑛

⟩ ∈ 𝑃, (24)

where 𝑚 > 1 or 𝑛 > 1. Without loss of generality, we assumethat 𝑚 ≥ 𝑛. Then, we define the following sequence of right-linear production rules:

𝐴 → ⟨𝑎1

𝑏1

⟩𝐴𝑟

1,

𝐴𝑟

1→ ⟨

𝑎2

𝑏2

⟩𝐴𝑟

2,

𝐴𝑟

𝑛−1→ ⟨

𝑎𝑛

𝑏𝑛

⟩𝐴𝑟

𝑛,

𝐴𝑟

𝑛→ ⟨

𝑎𝑛+1

𝜆⟩𝐴𝑟

𝑛+1

.

.

.

𝐴𝑟

𝑚−1→ ⟨

𝑎𝑚

𝜆⟩ ,

(25)

where 𝐴𝑟𝑖, 1 ≤ 𝑖 ≤ 𝑚 − 1, are new nonterminals.

We construct a WK linear grammar 𝐺 = (𝑁 ∪𝑁, 𝑇, 𝜌, 𝑆, 𝑃 ∪ 𝑃

), where 𝑃 consists of productions defined

above for each 𝐴 → ⟨𝑢1/V1⟩𝐵⟨𝑢2/V2⟩ ∈ 𝑃 with |𝑢

1| > 1,

|V1| > 1, |𝑢

2| > 1, or |V

2| > 1 and 𝐴 → ⟨𝑢/V⟩ ∈ 𝑃 with

|𝑢| > 1 or |V| > 1. Then, it is not difficult to see that, in everyderivation, productions in the form of (22) and (24) in 𝐺 canbe replaced by the sequences of productions (23) and (25) in𝐺 and vice versa. Thus, 𝐿(𝐺) = 𝐿(𝐺).

4.2.TheGenerative Power. The following results immediatelyfollow from the definition of WK grammars.

Lemma 9. The following inclusions hold:

𝑊𝐾𝑅𝐸𝐺 ⊆ 𝑊𝐾𝐿𝐼𝑁,

𝐿𝐼𝑁 ⊆ 𝑊𝐾𝐿𝐼𝑁,

𝐶𝐹 ⊆ 𝑊𝐾𝐶𝐹.

(26)

Next, we show that WK grammars can generate non-context-free languages:

{𝑎𝑛𝑐𝑛𝑏𝑛: 𝑛 ≥ 1} ,

{𝑎𝑛𝑏𝑚𝑐𝑛𝑑𝑚: 𝑛, 𝑚 ≥ 1} ,

{𝑤𝑐𝑤 : 𝑤 ∈ {𝑎, 𝑏}∗} .

(27)


Example 10. Let 𝐺1= ({𝑆, 𝐴, 𝐵}, {𝑎, 𝑏, 𝑐}, {(𝑎, 𝑎), (𝑏, 𝑏), (𝑐, 𝑐)},

𝑃, 𝑆) be a WK linear grammar, where 𝑃 consists of the rules

𝑆 → ⟨𝑎

𝜆⟩𝑆⟨

𝑏

𝜆⟩ ,

𝑆 → ⟨𝑎

𝜆⟩𝐴⟨

𝑏

𝜆⟩ ,

𝐴 → ⟨𝑐

𝑎⟩𝐴,

𝐴 → ⟨𝜆

𝑐⟩𝐵⟨

𝜆

𝑏⟩ ,

𝐵 → ⟨𝜆

𝑐⟩𝐵⟨

𝜆

𝑏⟩ ,

𝐵 → ⟨𝜆

𝜆⟩ .

(28)

In general, we have the derivation

𝑆 ⇒∗⟨𝑎𝑛−1

𝜆⟩𝑆⟨

𝑏𝑛−1

𝜆⟩

⇒ ⟨𝑎𝑛

𝜆⟩𝐴⟨

𝑏𝑛

𝜆⟩

⇒∗⟨𝑎𝑛𝑐𝑛

𝑎𝑛⟩𝐴⟨

𝑏𝑛

𝜆⟩

⇒ ⟨𝑎𝑛𝑐𝑛

𝑎𝑛𝑐⟩𝐵⟨

𝑏𝑛

𝑏⟩

⇒∗⟨𝑎𝑛𝑐𝑛

𝑎𝑛𝑐𝑛⟩𝐵⟨

𝑏𝑛

𝑏𝑛⟩

⇒ [𝑎𝑛𝑐𝑛𝑏𝑛

𝑎𝑛𝑐𝑛𝑏𝑛] .

(29)

Thus, 𝐺1generates the language 𝐿(𝐺

1) = {𝑎

𝑛𝑐𝑛𝑏𝑛: 𝑛 ≥ 1} ∈

CS − CF.

Example 11. Let

𝐺2= ({𝑆, 𝐴, 𝐵, 𝐶,𝐷} , {𝑎, 𝑏, 𝑐, 𝑑} ,

{(𝑎, 𝑎) , (𝑏, 𝑏) , (𝑐, 𝑐) , (𝑑, 𝑑)} , 𝑃, 𝑆)

(30)

be a WK regular grammar, and 𝑃 consists of the rules

𝑆 → ⟨𝑎

𝜆⟩𝑆 | ⟨

𝑎

𝜆⟩𝐴,

𝐴 → ⟨𝑏

𝜆⟩𝐴 | ⟨

𝑏

𝜆⟩𝐵,

𝐵 → ⟨𝑐

𝑎⟩𝐵 | ⟨

𝑐

𝑎⟩𝐶,

𝐶 → ⟨𝑑

𝑏⟩𝐶 | ⟨

𝑑

𝑏⟩𝐷,

𝐷 → ⟨𝜆

𝑐⟩𝐷 | ⟨

𝜆

𝑑⟩𝐷 | ⟨

𝜆

𝜆⟩ .

(31)

Then, we have the following derivation for 𝑛,𝑚 ≥ 1:

𝑆 ⇒∗⟨𝑎𝑛−1

𝜆⟩𝑆 ⇒ ⟨

𝑎𝑛

𝜆⟩𝐴

⇒∗⟨𝑎𝑛𝑏𝑚−1

𝜆⟩𝐴 ⇒ ⟨

𝑎𝑛𝑏𝑚

𝜆⟩𝐵

⇒∗⟨𝑎𝑛𝑏𝑚𝑐𝑛−1

𝑎𝑛−1

⟩𝐵 ⇒ ⟨𝑎𝑛𝑏𝑚𝑐𝑛

𝑎𝑛⟩𝐶

⇒∗⟨𝑎𝑛𝑏𝑚𝑐𝑛𝑑𝑚−1

𝑎𝑛𝑏𝑚−1

⟩𝐶 ⇒ ⟨𝑎𝑛𝑏𝑚𝑐𝑛𝑑𝑚

𝑎𝑛𝑏𝑚⟩𝐷

⇒∗⟨𝑎𝑛𝑏𝑚𝑐𝑛𝑑𝑚

𝑎𝑛𝑏𝑚𝑐𝑛⟩𝐷

⇒∗⟨𝑎𝑛𝑏𝑚𝑐𝑛𝑑𝑚

𝑎𝑛𝑏𝑚𝑐𝑛𝑑𝑚⟩𝐷

⇒ [𝑎𝑛𝑏𝑚𝑐𝑛𝑑𝑚

𝑎𝑛𝑏𝑚𝑐𝑛𝑑𝑚] .

(32)

Hence, 𝐿(𝐺2) = {𝑎

𝑛𝑏𝑚𝑐𝑛𝑑𝑚: 𝑛, 𝑚 ≥ 1} ∈ CS − CF.

Example 12. Let 𝐺3= ({𝑆, 𝐴, 𝐵, 𝐶}, {𝑎, 𝑏, 𝑐}, {(𝑎, 𝑎), (𝑏, 𝑏)},

𝑃, 𝑆) be a WK linear grammar with 𝑃 consisting of thefollowing rules:

𝑆 → ⟨𝑎

𝜆⟩𝑆 | ⟨

𝑏

𝜆⟩𝑆 | ⟨

𝑐

𝜆⟩𝐴,

𝐴 → ⟨𝑎

𝑎⟩𝐴 | ⟨

𝑏

𝑏⟩𝐴 | ⟨

𝜆

𝑐⟩𝐵,

𝐵 → ⟨𝜆

𝑎⟩𝐵 | ⟨

𝜆

𝑏⟩𝐵 | ⟨

𝜆

𝜆⟩ .

(33)

By rules 𝑆 → ⟨𝑎/𝜆⟩ and 𝑆 → ⟨𝑏/𝜆⟩, we obtain a sententialform ⟨𝑤/𝜆⟩𝑆 where 𝑤 ∈ {𝑎, 𝑏}∗. The derivation is continuedby only possible rule 𝑆 → ⟨𝑐/𝜆⟩𝐴 and we have ⟨𝑤𝑐/𝜆⟩𝐴.Further, we can only apply rules 𝐴 → ⟨𝑎/𝑎⟩𝐴 and 𝐴 →⟨𝑏/𝑏⟩𝐴. By the symmetric relation 𝜌, the derivation resultsin ⟨𝑤𝑐𝑤/𝑤⟩𝐴. Then, we can only apply 𝐴 → ⟨𝜆/𝑐⟩𝐵continuing with rules 𝐵 → ⟨𝜆/𝑎⟩𝐵 and 𝐵 → ⟨𝜆/𝑏⟩𝐵 andobtain ⟨𝑤𝑐𝑤/𝑤𝑐𝑤⟩𝐵. Finally, by rule 𝐵 → ⟨𝜆/𝜆⟩, we get[𝑤𝑐𝑤/𝑤𝑐𝑤]. Illustratively,

𝑆 ⇒∗⟨𝑤

𝜆⟩𝑆 ⇒ ⟨

𝑤𝑐

𝜆⟩𝐴

⇒∗⟨𝑤𝑐𝑤

𝑤⟩𝐴 ⇒ ⟨

𝑤𝑐𝑤

𝑤𝑐⟩𝐵

⇒∗⟨𝑤𝑐𝑤

𝑤𝑐𝑤⟩𝐵 ⇒ [

𝑤𝑐𝑤

𝑤𝑐𝑤] .

(34)

Thus, 𝐿(𝐺3) = {𝑤𝑐𝑤 : 𝑤 ∈ {𝑎, 𝑏}

∗} ∈ CS − CF.

The following theorem follows from Lemma 9 and Exam-ples 10, 11, and 12.


Theorem 13. The following inclusions hold:

𝐿𝐼𝑁 ⊊ 𝑊𝐾𝐿𝐼𝑁,

𝑊𝐾𝑅𝐸𝐺 − 𝐶𝐹 ̸= 0,

𝑊𝐾𝐿𝐼𝑁 − 𝐶𝐹 ̸= 0.

(35)

The following example shows that some WK linearlanguages cannot be generated by WK regular grammars.

Lemma 14. The following language is not a WK regularlanguage:

𝐿1= {𝑎𝑛𝑏𝑚𝑎𝑛: 2𝑛 ≤ 𝑚 ≤ 3𝑛}

∈ 𝑊𝐾𝐿𝐼𝑁 −𝑊𝐾𝑅𝐸𝐺.

(36)

Proof. The language𝐿1can be generated by the followingWK

linear grammar 𝐺4= ({𝑆, 𝐴, 𝐵}, {𝑎, 𝑏}, {(𝑎, 𝑎), (𝑏, 𝑏)}, 𝑆, 𝑃),

where 𝑃 consists of the rules:

𝑆 → ⟨𝑎

𝜆⟩𝑆⟨

𝑎

𝑎⟩ | ⟨

𝑎

𝜆⟩𝐴⟨

𝑎

𝑎⟩ ,

𝐴 → ⟨𝑏𝑏

𝑎⟩𝐴 | ⟨

𝑏𝑏𝑏

𝑎⟩𝐴 | ⟨

𝜆

𝑏⟩𝐵,

𝐵 → ⟨𝜆

𝑏⟩𝐵 | ⟨

𝜆

𝜆⟩ .

(37)

It is not difficult to see that

𝑆 ⇒∗⟨𝑎𝑛−1

𝜆⟩𝑆⟨

𝑎𝑛−1

𝑎𝑛−1⟩ ⇒ ⟨

𝑎𝑛

𝜆⟩𝐴⟨

𝑎𝑛

𝑎𝑛⟩

⇒∗⟨𝑎𝑛𝑏𝑚

𝑎𝑛⟩𝐴⟨

𝑎𝑛

𝑎𝑛⟩ ⇒ ⟨

𝑎𝑛𝑏𝑚

𝑎𝑛𝑏⟩𝐵⟨

𝑎𝑛

𝑎𝑛⟩

⇒∗⟨𝑎𝑛𝑏𝑚

𝑎𝑛𝑏𝑚⟩𝐵⟨

a𝑛

𝑎𝑛⟩ ⇒ [

𝑎𝑛𝑏𝑚𝑎𝑛

𝑎𝑛𝑏𝑚𝑎𝑛] ,

(38)

where 2𝑛 ≤ 𝑚 ≤ 3𝑛.Next, we show that 𝐿

1∉WKREG.

We suppose, by contradiction, that𝐿1can be generated by

a WK regular grammar 𝐺 = (𝑁, {𝑎, 𝑏}, 𝜌, 𝑆, 𝑃). Without lossof generality, we assume that 𝐺 is in 1-normal form. Then,for each rule 𝑢 → V in 𝑃, we have 𝑢 ∈ 𝑁 and

V ∈ {⟨𝑎

𝜆⟩ ,⟨

𝜆

𝑎⟩ ,⟨

𝑎

𝑎⟩ ,⟨

𝑏

𝜆⟩ ,⟨

𝜆

𝑏⟩ ,⟨

𝑏

𝑏⟩ ,⟨

𝑎

𝑏⟩ ,

⟨𝑏

𝑏⟩} (𝑁 ∪ {𝜆}) .

(39)

Let 𝑤 = 𝑎𝑟𝑏𝑠𝑎𝑟 be a string in 𝐿1such that 𝑟 > |𝑃|. Then, the

double-stranded sequence [𝑎𝑟𝑏𝑠𝑎𝑟/𝑎𝑟𝑏𝑠𝑎𝑟] is generated by thegrammar 𝐺.

Case 1. In any derivation for this string, first 𝑏 can occur inthe upper (or lower) strand if 𝑎𝑟 has already been generated

in the upper (or lower) strand. Thus, we obtain two possiblesuccessful derivations:

𝑆 ⇒∗⟨𝑎𝑟𝑏

𝑎𝑘⟩

or 𝑆 ⇒∗⟨𝑎𝑟𝑏

𝑎𝑟𝑏⟩ ,

(40)

where 𝑘 ≤ 𝑟. In the latter derivation in (40), we cannot controlthe number of occurrences of 𝑏; that is, the derivation maynot be successful. In the former derivation in (40), using thesecond strand, we can generate 𝑏𝑠:


𝑎𝑘⟩⇒

∗⟨𝑎𝑟𝑏𝑠

𝑎𝑟𝑏𝑡⟩ , 𝑡 ≤ 𝑠. (41)

Equation (41) is continued by generating 𝑎’s in the first strandand we can use the second strand to control their number.Consider


𝑎𝑘⟩⇒

∗⟨𝑎𝑟𝑏𝑠

𝑎𝑟𝑏𝑡⟩⇒

∗⟨𝑎𝑟𝑏𝑠𝑎𝑖

𝑎𝑟𝑏𝑠𝑎𝑗⟩, (42)

and 𝑖 is related to 𝑠. Since 2𝑟 ≤ 𝑠 ≤ 3𝑟, generally, 𝑖 is not thesame as 𝑟 for all derivations.

Case 2.We can control the number of 𝑎’s after 𝑏’s by using thesecond strand for 𝑎’s before 𝑏’s. In this case, the number of 𝑏’scannot be related to the number of 𝑎’s:

𝑆 ⇒∗⟨𝑎𝑟𝑏𝑙𝑎𝑟

𝑎𝑟⟩. (43)

In both cases, we cannot control the number of 𝑏’s and thenumber of 𝑎’s after 𝑏’s at the same time using WK regularrules.

Since strings 𝑎𝑛𝑏𝑚𝑎𝑛 are palindrome strings for even𝑚’s,the language {𝑤𝑤𝑅 : 𝑤 ∈ {𝑎, 𝑏}∗} is not in WKREG; that is,we have the following.

Corollary 15. The following holds:

𝐿𝐼𝑁 −𝑊𝐾𝑅𝐸𝐺 ̸= 0. (44)

4.3. Hierarchy of the Families of Watson-Crick Languages.Combining the results above, we obtain the following theo-rem.

Theorem 16. The relations in Figure 4 hold; the dotted linesdenote incomparability of the language families and the arrowsdenote proper inclusions of the lower families into the upperfamilies, while the dotted arrows denote inclusions.

5. Closure Properties

In this section, we establish results regarding the closureproperties of WK grammars. The families of WK languagesare shown to be higher in the hierarchy than their respectiveChomsky language families; thus it is interesting to see how


RE

WKCF

WKLIN

WKREG

REG

CS

CF

LIN

Figure 4: The hierarchy of WK and Chomsky language families.

WKgrammarswork in terms of closure properties as the onesfor the Chomsky languages. Moreover, researching closureproperties ofWKgrammars ensure the safety and correctnessof the results yielded when performing operations on the setsof DNA molecules generated by some WK grammars.

5.1. Watson-Crick Regular Grammars. Let 𝐿1, 𝐿2∈WKREG,

and 𝐺1= (𝑁1, 𝑇, 𝑆1, 𝑃1, 𝜌), 𝐺

2= (𝑁2, 𝑇, 𝑆2, 𝑃2, 𝜌) be Watson-

Crick regular grammars generating languages 𝐿1and 𝐿

2,

respectively; that is, 𝐿1= 𝐿(𝐺

1) and 𝐿

2= 𝐿(𝐺

2). Without

loss of generality, we can assume that𝑁1∩𝑁2̸= 0 and 𝑆

1and

𝑆2do not appear on the right-hand side of any production

rule.

Lemma 17 (union). 𝑊𝐾𝑅𝐸𝐺 is closed under union.

Proof. Define𝐺 = (𝑁, 𝑇, 𝑆, 𝑃, 𝜌) by setting𝑁 = 𝑁1∪𝑁2∪{𝑆}

with 𝑆 ∉ 𝑁1∪ 𝑁2, and

𝑃 = 𝑃1∪ 𝑃2∪ {𝑆 → 𝑆

1} ∪ {𝑆 → 𝑆

2} . (45)

Then it is not difficult to see that 𝐿(𝐺) = 𝐿1∪ 𝐿2.

Lemma 18 (concatenation). 𝑊𝐾𝑅𝐸𝐺 is closed under concate-nation.

Proof. Define 𝐺 = (𝑁, 𝑇, 𝑆, 𝑃, 𝜌), where 𝑁 = 𝑁1∪ 𝑁2and

𝑆 = 𝑆1. We define

𝑃 = 𝑃2∪ (𝑃1− {𝐴 → ⟨

𝑢

V⟩ ∈ 𝑃

1})

∪ {𝐴 → ⟨𝑢

V⟩𝑆2| 𝐴 → ⟨

𝑢

V⟩ ∈ 𝑃

1} .

(46)

Then it is obvious that 𝐿(𝐺) = 𝐿1⋅ 𝐿2.

Lemma 19 (Kleene-star). 𝑊𝐾𝑅𝐸𝐺 is closed under Kleene-star operation.

Proof. Define the WK regular grammar 𝐺 = (𝑁1, 𝑇1, 𝑆, 𝑃1, 𝜌)

with

𝑃1= (𝑃1− 𝑃𝑇) ∪ 𝑃𝑆∪ {𝑆1→ 𝜆} , (47)

where

𝑃𝑇= {𝐴 → ⟨

𝑢

V⟩ ∈ 𝑃 : ⟨

𝑢

V⟩ ∈ ⟨

𝑇∗

𝑇∗⟩} ,

𝑃𝑆= {𝐴 → ⟨

𝑢

V⟩𝑆1| 𝐴 → ⟨

𝑢

V⟩ ∈ 𝑃

𝑇} .

(48)

Then, 𝐿(𝐺) = 𝐿∗1.

Lemma 20 (finite substitution and homomorphism).𝑊𝐾𝑅𝐸𝐺 is closed under finite substitution and homomor-phism.

Proof. We show that the finite substitution (homomorphism)of 𝐿1is also in WKREG. Let 𝑠 : 𝑇∗ → Σ∗ be a finite

substitution (homomorphism).Define𝐺 = (𝑁1, 𝑇1, 𝑆1, 𝑃, 𝜌1),

where 𝐿(𝐺) = 𝑠(𝐿1). Without loss of generality, we assume

that 𝐺1is in the 1-normal form. Then, 𝑃

1can contain

production rules of the forms

𝐴 → ⟨𝑎

𝑏⟩𝐵,

𝐴 → ⟨𝑎

𝑏⟩ ,

𝐴 → ⟨𝑎

𝜆⟩𝐵,

𝐴 → ⟨𝑎

𝜆⟩ ,

𝐴 → ⟨𝜆

𝑏⟩𝐵,

𝐴 → ⟨𝜆

𝑏⟩ ,

𝐴 → ⟨𝜆

𝜆⟩ .

(49)

We define 𝑃 as the set of production rules of the forms

𝐴 → ⟨𝑠 (𝑎)

𝑠 (𝑏)⟩𝐵,

𝐴 → ⟨𝑠 (𝑎)

𝑠 (𝑏)⟩ ,

𝐴 → ⟨𝑠 (𝑎)

𝜆⟩𝐵,

𝐴 → ⟨𝑠 (𝑎)

𝜆⟩ ,

𝐴 → ⟨𝜆

𝑠 (𝑏)⟩𝐵,


𝐴 → ⟨𝜆

𝑠 (𝑏)⟩ ,

𝐴 → ⟨𝜆

𝜆⟩ .

(50)

Since the substitution/homomorphism 𝑠 is finite, 𝑃 is afinite set too; that is, 𝐺 is a WK regular grammar, and 𝐿(𝐺) =𝑠(𝐿1).

Lemma 21 (mirror image). 𝑊𝐾𝑅𝐸𝐺 is closed under mirrorimage.

Proof. We show that 𝐿𝑅1∈ WKREG. Define 𝐺 = (𝑁

1∪

{𝑆}, 𝑇1, 𝑆, 𝑃, 𝜌

1) generating the language 𝐿𝑅

1, where 𝑆 is a new

nonterminal and 𝑃 consists of production rules defined asfollows:

(i) 𝐴 → 𝐵⟨𝑢/V⟩, where 𝐴 → ⟨𝑢/V⟩𝐵 ∈ 𝑃1,

(ii) 𝐴 → ⟨𝑢/V⟩, where 𝐴 → ⟨𝑢/V⟩ ∈ 𝑃1.

The results obtained from the lemmas above are summa-rized in the following theorem.

Theorem 22. The family of Watson-Crick regular languages isclosed under union, concatenation, Kleene-star, finite substitu-tion, homomorphism, and mirror image.

This shows thatWK regular grammars preserve almost allof the closure properties of regular grammars. Other closureproperties ofWK regular grammars are left for future studies.

5.2. Watson-Crick Linear Grammars. Similar to the subsec-tion above, a Watson-Crick linear grammar 𝐺 is constructedfor the purpose of investigating the closure properties ofWKLIN.

Let 𝐿1, 𝐿2∈WKLIN, and

𝐺1= (𝑁1, 𝑇, 𝑆1, 𝑃1, 𝜌) ,

𝐺2= (𝑁2, 𝑇, 𝑆2, 𝑃2, 𝜌)

(51)

be Watson-Crick linear grammars generating 𝐿1and 𝐿

2,


1) and 𝐿

2= 𝐿(𝐺

2). Without

loss of generality, we can assume that𝑁1∪ 𝑁2̸= 0.

Lemma 23 (union). 𝑊𝐾𝐿𝐼𝑁 is closed under union.

Proof. Define 𝐺 = (𝑁, 𝑇, 𝑆, 𝑃, 𝜌), where 𝑁 = 𝑁1∪ 𝑁2∪ {𝑆}

with 𝑆 ∉ 𝑁1∪ 𝑁2and

𝑃 = 𝑃1∪ 𝑃2∪ {𝑆 → 𝑆

1} ∪ {𝑆 → 𝑆

2} . (52)

Then, 𝐿(𝐺) = 𝐿1∪ 𝐿2.

Lemma 24 (homomorphism). 𝑊𝐾𝐿𝐼𝑁 is closed underhomomorphism.

Proof. Define 𝐺 = (𝑁1, 𝑇, 𝑆1, 𝑃, 𝜌), where 𝐿(𝐺) = ℎ(𝐿

1). We

show that the homomorphism of 𝐿1is also in WKLIN. Let

ℎ : 𝑇∗→ Σ∗ be a homomorphism.Without loss of generality,

we assume that 𝐺1is in the 1-normal form. For each rule

𝑟 : 𝐴 → ⟨𝑢1

V1

⟩𝐵⟨𝑢2

V2

⟩ ∈ 𝑃1, (53)

where 𝑢1, V1, 𝑢2, V2∈ 𝑇 ∪ {𝜆}, we construct

ℎ (𝑟) : 𝐴 → ⟨ℎ (𝑢1)

ℎ (V1)⟩𝐵⟨

ℎ (𝑢2)

ℎ (V2)⟩ (54)

in 𝑃.In every successful derivation of 𝐺

1generating [𝑤/𝑤] ∈

[𝑇/𝑇]∗, we replace production rule 𝑟 ∈ 𝑃

1with the

production rule ℎ(𝑟) ∈ 𝑃 and obtain the string [ℎ(𝑤)/ℎ(𝑤)] ∈[Σ/Σ]

∗. Thus, ℎ(𝑤) ∈ ℎ(𝐿1).

Theorem 25. The family of Watson-Crick linear languages isclosed under union and homomorphism.

It is compelling to prove if the concatenation of two WKlinear grammars is still included in WKLIN or not. Thisalso decides whether WK linear grammars are unique fromlinear grammars or not, as linear grammars are not closedunder concatenation and thus not closed under Kleene-staroperation.

5.3. Watson-Crick Context-Free Grammars. Let 𝐿1, 𝐿2, 𝐿 ∈

WKCF, and 𝐺1= (𝑁1, 𝑇, 𝑆1, 𝑃1, 𝜌), 𝐺

2= (𝑁2, 𝑇, 𝑆2, 𝑃2, 𝜌) be

Watson-Crick context-free grammars generating 𝐿1and 𝐿

2,


1) and 𝐿

2= 𝐿(𝐺

2). Without

loss of generality, we can assume that𝑁1∪ 𝑁2̸= 0.

Lemma 26 (union). 𝑊𝐾𝐶𝐹 is closed under union operation.

Proof. Define the WK context-free grammar 𝐺 =(𝑁, 𝑇, 𝑆, 𝑃, 𝜌), where 𝑁 = 𝑁

1∪ 𝑁2∪ {𝑆} with 𝑆 ∉ 𝑁

1∩ 𝑁2,

and

𝑃 = 𝑃1∪ 𝑃2∪ {𝑆 → 𝑆

1} ∪ {𝑆 → 𝑆

2} . (55)

Then it is obvious that 𝐿(𝐺) = 𝐿1∪ 𝐿2.

Lemma 27 (concatenation). 𝑊𝐾𝐶𝐹 is closed under concate-nation.

Proof. Define 𝐺 = (𝑁1∪ 𝑁2∪ {𝑆}, 𝑇

1∪ 𝑇2, 𝑆, 𝑃, 𝜌) where 𝑆 ∉

(𝑁1∪ 𝑁2), and

𝑃 = 𝑃1∪ 𝑃2∪ {𝑆 → 𝑆

1𝑆2} . (56)

Then, 𝐿(𝐺) ∈WKCF.

Lemma 28 (Kleene-star). 𝑊𝐾𝐶𝐹 is closed under Kleene-staroperation.

Proof. Define 𝐺 = (𝑁1∪ {𝑆}, 𝑇

1, 𝑆, 𝑃, 𝜌) ∈WKCF setting

𝑃 = 𝑃1∪ {𝑆 → 𝑆

1𝑆 | 𝜆} . (57)

Then it is not difficult to see that 𝐿(𝐺) ∈WKCF.


Lemma 29 (homomorphism). 𝑊𝐾𝐶𝐹 is closed under homo-morphism.

Proof. Define 𝐺 = (𝑁1, 𝑇, 𝑆1, 𝑃, 𝜌), where 𝐿(𝐺) = ℎ(𝐿

1). We

show that the homomorphism of 𝐿1is also in WKCF. Let 𝑠 :

𝑇∗→ Σ∗ be a homomorphism. 𝑃

1contains production rules

of the form

𝑟 : 𝐴

→ ⟨𝑢1

V1

⟩𝐵1⟨𝑢2

V2

⟩𝐵2⋅ ⋅ ⋅ ⟨

𝑢𝑘

V𝑘

⟩𝐵𝑘⟨𝑢𝑘+1

V𝑘+1

⟩ ,

(58)

where 𝑘 ≥ 0.For each rule 𝑟 ∈ 𝑃, we construct

ℎ (𝑟) : 𝐴 → ⟨ℎ (𝑢1)

ℎ (V1)⟩𝐵1⟨ℎ (𝑢2)

ℎ (V2)⟩𝐵2

⋅ ⋅ ⋅⟨ℎ (𝑢𝑘)

ℎ (V𝑘)⟩𝐵𝑘⟨ℎ (𝑢𝑘+1)

ℎ (V𝑘+1)⟩ ,

(59)

where 𝑘 ≥ 0.In every successful derivation of 𝐺

1generating [𝑤/𝑤] ∈

[𝑇/𝑇]∗, we replace production rule 𝑟 ∈ 𝑃

1with the

production rule ℎ(𝑟) ∈ 𝑃 and obtain the string [ℎ(𝑤)/ℎ(𝑤)] ∈[Σ/Σ]

∗. Thus, ℎ(𝑤) ∈ ℎ(𝐿1).

With the lemmas provided above in this subsection, thenext theorem follows.

Theorem 30. The family of Watson-Crick context-free lan-guages is closed under union, concatenation, Kleene-star, andhomomorphism.

Closure of WKCF under complement and intersectiondepends on the generative capacity ofWKCF.Thenonclosureof context-free (CF) grammars for intersection was shownwith the famous example that the intersection of two CFlanguages results in a string that cannot be generated by a CFgrammars. If one can provide some examples of strings thatcannot be generated byWK context-free grammars, then thenonclosure of WKCF can be proven.

6. Applications of Watson-Crick Grammars

In this section, we consider two examples of the applicationsofWatson-Crick grammars in the analyses of DNA structuresand programming language structures.

6.1. DNA Structure Analysis. Since Watson-Crick grammarsare developed based on the structure and recombinant behav-ior of DNA molecules, they can suitably be implemented inthe study of DNA related problems.

The analysis of DNA strings provides useful information:for instance, the finding of a specific pattern in a DNAstring and the identification of the repeats of a pattern arevery important for detecting mutation. One of the diseasescaused by mutation is Huntington disease, resulting fromtrinucleotide repeat disorders [13–15]. It is discovered that

the number of repeats of the trinucleotide 𝐶𝑇𝐺/𝐶𝐴𝐺 in thepatient’s DNA with Huntington disease is not normal. Therepeats are also useful for finding the origin of replication ofmicroorganisms [16].

In this section, we show thatWatson-Crick grammars canbe used for analyzing the repeats in DNA strings. We use theDNA of a breed of pig, Sus scrofa breed mixed chromosome1, Sscrofa10.2 provided by the The National Center forBiotechnology Information (NCBI) database (NCBI Refer-ence Sequence: NC 010443.4) [17].

Consider a part of the upper strand of the DNA from thebreed stated above with the length of 100 nucleotides:𝑎𝑎𝑡𝑐𝑔𝑎𝑐𝑔𝑐𝑐 𝑎𝑐𝑎𝑐𝑔cag𝑔𝑐 cag𝑡𝑡𝑐𝑐𝑔𝑎𝑔 ctg𝑐𝑎𝑡ctg𝑔 𝑔𝑎𝑐ctg𝑐𝑡𝑐𝑐

𝑎𝑡𝑎𝑡ctg𝑎𝑎𝑔 𝑔𝑐𝑎𝑎𝑡𝑐ctg𝑔 𝑎𝑡𝑐cag𝑔𝑔𝑎𝑡 𝑡𝑔𝑐𝑎𝑐ctg𝑐𝑐 𝑡𝑡𝑐𝑡𝑐𝑡𝑐𝑐𝑡𝑎.(60)

In this example, the pattern 𝑐𝑡𝑔 is being repeated for sixtimes in the direct strand (upper strand) and two times in thereverse strand (lower strand) which can be seen as 𝑐𝑎𝑔 in theupper strand. Focusing on the repeats of 𝑐𝑡𝑔 pattern, a DNAstring containing such a pattern can be expressed as

({𝑎, 𝑡, 𝑐, 𝑔}∗

𝑐𝑡𝑔 {𝑎, 𝑡, 𝑐, 𝑔}∗

)∗

. (61)

We now construct a simple Watson-Crick regular gram-mar 𝐺 to generate the above language.

Let 𝐺 = (𝑁, 𝑇, 𝑆, 𝑃, 𝜌), where 𝑇 = {𝑎, 𝑡, 𝑐, 𝑔}, 𝜌 ={(𝑎, 𝑡), (𝑐, 𝑔)}, and 𝑃 consists of the following productions:

𝑆 → ⟨𝑎

𝑡⟩ 𝑆 | ⟨

𝑡

𝑎⟩ 𝑆 | ⟨

𝑔

𝑐⟩ 𝑆 | ⟨

𝑐

𝑔⟩𝐴,

𝐴 → ⟨𝑐

𝑔⟩𝐴 | ⟨

𝑎

𝑡⟩ 𝑆 | ⟨

𝑔

𝑐⟩ 𝑆 | ⟨

𝑡

𝑎⟩𝐵,

𝐵 → ⟨𝑐

𝑔⟩𝐴 | ⟨

𝑎

𝑡⟩ 𝑆 | ⟨

𝑡

𝑎⟩ 𝑆 | ⟨

𝑔

𝑐⟩𝐶,

𝐶

→ ⟨𝑎

𝑡⟩𝐶 | ⟨

𝑡

𝑎⟩𝐶 | ⟨

𝑔

𝑐⟩𝐶 | ⟨

𝑐

𝑔⟩𝐴 | ⟨

𝜆

𝜆⟩ .

(62)

For instance, a derivation resulting in six repeats of 𝑐𝑡𝑔pattern can be obtained as follows:

𝑆 ⇒∗⟨𝑥

𝑦⟩𝑆 ⇒ ⟨

𝑥𝑐

𝑦𝑔⟩𝐴 ⇒ ⟨

𝑥𝑐𝑡

𝑦𝑔𝑎⟩𝐵

⇒ ⟨𝑥𝑐𝑡𝑔

𝑦𝑔𝑎𝑐⟩𝐶

⇒∗⟨𝑥𝑐𝑡𝑔𝑥

𝑦𝑔𝑎𝑐𝑦⟩𝐴 ⇒ ⟨

𝑥𝑐𝑡𝑔𝑥𝑐

𝑦𝑔𝑎𝑐𝑦𝑔⟩𝐴

⇒ ⟨𝑥𝑐𝑡𝑔𝑥𝑐𝑡

𝑦𝑔𝑎𝑐𝑦𝑔𝑎⟩𝐵

⇒ ⟨𝑥𝑐𝑡𝑔𝑥𝑐𝑡𝑔

𝑦𝑔𝑎𝑐𝑦𝑔𝑎𝑐⟩𝐶⇒

∗⟨𝑥𝑐𝑡𝑔𝑥𝑐𝑡𝑔𝑥

𝑦𝑔𝑎𝑐𝑦𝑔𝑎𝑐𝑦⟩𝐶

⇒∗[𝑥𝑐𝑡𝑔𝑥𝑐𝑡𝑔𝑥𝑐𝑡𝑔𝑥𝑐𝑡𝑔𝑥𝑐𝑡𝑔𝑥𝑐𝑡𝑔𝑥

𝑦𝑔𝑎𝑐𝑦𝑔𝑎𝑐𝑦𝑔𝑎𝑐𝑦g𝑎𝑐𝑦𝑔𝑎𝑐𝑦𝑔𝑎𝑐𝑦] ,

(63)


where 𝑥 ∈ {𝑎, 𝑡, 𝑔}∗ and 𝑦 is their complementarity symbolsbased on 𝜌, respectively.

Though, in this example, the functionality of WK regulargrammars is not used to the fullest, the DNA structures canbe naturally and effectively analyzed with WK grammars.

6.2. Programming Language Structure Analysis. The abil-ity to differentiate between parentheses that are correctlybalanced and those that are unbalanced is an importantpart of recognizing many programming language structures.Balanced parentheses mean that each opening symbol has acorresponding closing symbol and the pairs of parenthesesare properly nested.

Theparsing algorithms of compilers and interpreters haveto check the correctness of balanced parentheses in the blocksof codes including algebraic and arithmetic expressions.

Although context-free grammars are used to developparsers for programming languages, many programminglanguage structures are context-sensitive.Thus, it is of interestto develop parsers based on grammars which are able toanalyze non-context-free language structures.

Further, we show an example of balanced parentheses thatcan be generated by WK regular grammars but cannot begenerated by context-free grammars. One can see that just byincorporating the concept of double-stranded string bondedwith Watson-Crick complementarity, even WK grammarsthat are based of just regular rules can enhance the power ofthe parsing techniques.

To avoid confusion, we denote “(” as the open parenthesisterminal symbol and “)” as the close parenthesis terminalsymbol in bold font.

Example 31. Let 𝐺5= ({𝑆, 𝐴, 𝐵}, {(, )}, {((, ))}, 𝑆, 𝑃

5) be a WK

regular grammar. 𝑃5consists of the rules

𝑆 → ⟨(𝜆⟩𝑆

𝑆 → ⟨(𝜆⟩𝐴,

𝐴 → ⟨)(⟩𝐴

𝐴 → ⟨)(⟩𝐵

𝐵 → ⟨()⟩𝐵

𝐵 → ⟨𝜆

)⟩𝐵

𝐵 → ⟨𝜆

𝜆⟩

𝐵 → 𝐴.

(64)

From this, we obtain the derivation

𝑆 ⇒ ⟨(𝜆⟩𝑆⇒

∗⟨(𝑛

𝜆⟩𝐴

⇒ ⟨(𝑛)(⟩𝐴⇒

∗⟨

(𝑛)𝑛

(𝑛⟩𝐵

⇒ ⟨(𝑛)𝑛

(𝑛)⟩𝐵⇒

∗⟨

(𝑛)𝑛

(𝑛)𝑛⟩𝐵

⇒ ⟨(𝑛)𝑛((𝑛)𝑛⟩𝐵⇒

∗⟨

(𝑛)𝑛(𝑛

(𝑛)𝑛⟩𝐴

⇒ ⟨(𝑛)𝑛(𝑛)(𝑛)𝑛

⟩𝐴⇒∗⟨

(𝑛)𝑛(𝑛)𝑛

(𝑛)𝑛⟩𝐵

⇒∗⋅ ⋅ ⋅ ⇒

∗[((𝑛)𝑛)𝑘

((𝑛)𝑛)𝑘] ,

(65)

where 𝑘 ≥ 1. Hence, the language obtained is

𝐿5= {((𝑛)𝑛)

𝑘

: 𝑛, 𝑘 ≥ 1} ∈WKREG − CF. (66)

7. Conclusion

In this paper, we defined Watson-Crick regular grammars,Watson-Crick linear grammars, and Watson-Crick context-free grammars. Further, we investigated their computationalpower and closure properties. We showed that

(1) WK linear grammars can generate some context-sensitive languages;

(2) the families of linear languages and WK regularlanguages are strictly included in the family of WKlinear grammars;

(3) the families of WK regular languages and linearlanguages are not comparable;

(4) the family of WK linear languages is not comparablewith the family of context-free languages;

(5) WK regular grammars preserves the closure proper-ties similar to the ones of regular languages.

The following problems related to the topic remain open:

(1) Are the family of context-free languages proper subsetof the family of WK linear languages or are they notcomparable?

(2) What is the generative capacity of Watson-Crickcontext-free grammars?

(3) What are the remaining closure properties ofWatson-Crick (regular, linear, and context-free) grammars?These depend on their generative capacity.

Moreover, there are many interesting topics for furtherresearch; for instance, we can define WK context-free andregulated WK context-free grammars and use them in thestudy of DNA properties and in the DNA based applicationssuch as food authentication and disease gene detection.


Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work has been supported through International IslamicUniversity Endowment B research Grant EDW B14-136-1021and Fundamental ResearchGrant Scheme FRGS13-066-0307,Ministry of Education, Malaysia. The first author would liketo thank both organizations for the scholarship through theIIUM fellowship program.

References

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[2] R. Freund, G. Paun, G. Rozernberg, and A. Salomaa, Watson-Crick Finite Automata, vol. 48 of DIMACS Series in DiscreteMathematics and Theoretical Computer Science, 1999.

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