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There are three important definitions of splicing systems that are well-studied in the literature.

Definition (Head’s [13]) A splicing systemS_H = (A,I,B,C)consists of a finite alphabetA, a

finite setI ⊆A∗ of initial strings, and finite setsBandCof triples (c,x,d)withc, x andd in

A∗. Each such triple in BorCis called a pattern. For each such triple the stringcxd is called a site and the stringxis called a crossing. Patterns inBare called left patterns and patterns in

C are called right patterns. The left patterns are associated with restriction enzymes produc- ing 5’ overhangs and the right patterns are associated with restriction enzymes producing 3’ overhangs. Splicing can happen between two words only if both the corresponding patterns involved in the operation are either left (i.e. both are in B) or right (i.e. both are in C). Given two wordsucxdvandpex f qinL, if(c,x,d)and(e,x,f)are patterns inB(resp.C), the splicing operation generates the stringsucx f qand pexdv.

is a set of splicing rules.

Definition (Paun’s [44]) A splicing systemS_PA= (A,I,R)consists of a finite setI⊆A∗as the initial language and a corresponding splicing scheme(A,R). Each rule r in Ris of the form

r= (u1,u2;u3,u4)(and represented by the stringu1#u2$u3#u4), withui∈A∗, fori=1,2,3,4

and #,$∈/ A. Given two words x=x₁u₁u₂x₂, y=y₁u₃u₄y₂, and the rule r=u₁#u₂$u₃#u₄, the splicing operation produces w =x₁u₁u₄y₂ and w0 = y₁u₃u₂x₂. Formally, we can write

(x,y)`r{w,w0}.

The Paun’s definition of splicing is the most widely used model of splicing system in the area of DNA computing.

Definition (Pixton’s [41]) A splicing system S_PI = (A,I,R) consists of a finite alphabet A, a finite set of strings I ⊆A∗ as initial language, a set of rules R where for all r in R, we have r = (α,α0;β), for α,α0,β ∈A∗. Given two words x=ε α η, y=ε0α0η0 and the rule

r= (α,α0;β), the splicing operation produces w=ε β η0 and w0=ε0β η. Formally, we can

write(ε α η,ε0α0η0)`r {ε β η0,ε0β η}.

Note that in Pixton’s definition, the wordβ is introduced to make the system more generic

than Paun’s system. This way we are not only cutting and pasting at the recognition site but also substituting it with a new word given by the splicing rule.

Every splicing system generates a language by the iterated application of splicing rules to its initial language. Thus, every splicing system is associated with a corresponding splicing language. Formally, letR0(L) =LandR(L) ={w∈A∗| ∃w0,w00∈L,∃r∈R:(w0,w00)`r w}.

For each non-negative integeri, we haveRi+1(L) =Ri(L)∪R(Ri(L)). The languageR∗(L) =

∪{Ri(L):i≥0} is the language generated fromLthrough the iterated application of the rule

setR.

A language is said to be a splicing language if there is a splicing system that can generate it. Formally, a languageLis a splicing language ifL=R∗(I)for some splicing systemS= (A,I,R)

A splicing system S= (A,I,R) is said to be a finite splicing system if both R and I are finite sets. A language generated by such a system is called finite splicing language. For finite splicing systems, it was proved that the family of languages generated by Head’s system is strictly included in the family generated by Paun’s system which is in turn strictly included in the family generated by Pixton’s system [5].

Sometimes, splicing systems that have a splicing scheme with some restrictions are studied.

Definition [15] A splicing schemeRisreflexiveif for every splicing rule(u,u0;v0,v)(Paun’s definition) inR, there is a corresponding rule(u,u0;u,u0)that is inR. A splicing system using a reflexive splicing scheme is called reflexive splicing system and correspondingly, the language generated by such a system is called as areflexive splicing language.

Definition [15] A splicing schemeRissymmetricif for every splicing rule(u,u0;v0,v)(Paun’s definition) inR, there is a corresponding rule (v0,v;u,u0) that is inR. A splicing system with a symmetric splicing scheme is called a symmetric splicing system and correspondingly, the language generated by such a system is called as asymmetric splicing language.

Restricted versions of splicing systems based on the type and size of splicing rules are also defined. These include the following:

Definition [40] A splicing systemS= (A,I,R)(Paun’s version) in which all rules in Rhave the form(a,λ;a,λ)wherea∈Ais calledsimple splicing system.

Definition [11] A splicing systemS= (A,I,R) (Paun’s version) in which I and A are finite and every rule inRhas the form(a,λ;b,λ), wherea,bare inAis called asemi-simple splicing

system.

Definition [13] A null-context splicing system is a splicing system S= (A,I,B,C) (Head’s version) for which each cleavage pattern inBandChas the form(λ,x,λ).

Definition [28] A splicing system (A,I,R) (Paun’s version) in which I and R are finite and every rule in R has the form(u,λ;v,λ), whereu , v are in A+ is called a semi-null splicing

system.

Definition [13] A uniform splicing systemis a null-context splicing system S= (A,I,X,X)

(Head’s version) for which there is a positive integerPsuch thatX =AP.

Definition [1] A splicing systemS= (A,I,R)(Paun’s version) in whichAis finite and every rule in R has the form(u₁,u₂;u₃,u₄), whereu₁,u₂,u₃,u₄ are in A or equal toλ is called an

alphabetic splicing system.

Several relationships among the splicing systems defined above are established: If _A,_B

are two classes of splicing systems, let_A⊆_B(i.e., subset inclusion) mean that_Ais a special case of_B. Then, it was proved that, simple splicing system⊆semi-simple splicing system⊆

semi-null splicing system⊆uniform splicing system⊆null-context splicing system [49, 55].