A proposal to a generalised splicing with a self assembly approach

Theory of splicing is an abstract model of the recombinant behaviour of DNAs. In a splicing system, two strings to be spliced are taken from the same set and the splicing rule is from another set. Here we propose a generalised splicing (GS) model with three components, two strings from two languages and a splicing rule from third component. We propose a generalised self assembly (GSA) of strings. Two strings $u_1xv_1$ and $u_2xv_2$ self assemble over $x$ and generate $u_1xv_2$ and $u_2xv_1$. We study the relationship between GS and GSA. We study some classes of generalised splicing languages with the help of generalised self assembly.

💡 Research Summary

The paper introduces two novel operations—Generalised Splicing (GS) and Generalised Self‑Assembly (GSA)—that extend the classical DNA‑inspired splicing model. In the traditional H‑system, two strings drawn from the same language are cut and recombined according to a rule of the form α # β $ α′ # β′; the rule set may be infinite, and the computational power of the resulting splicing language depends on the classes of the input language and the rule language.

The authors propose a three‑component framework: two input languages L₁ and L₂, and a rule language L₃. Definition 1 formalises GS(L₁, L₂, L₃) as the set of all pairs (z₁, z₂) that can be obtained by applying a rule r∈L₃ to a pair (x, y) with x∈L₁ and y∈L₂. When L₁ = L₂, this collapses to the ordinary H‑system.

Next, they define GSA as a string‑level operation: given two strings u₁ x v₁ and u₂ x v₂ that share a non‑empty substring x, the strings self‑assemble to produce u₁ x v₂ and u₂ x v₁. This is exactly the effect of a splicing rule x # $ x #, establishing an equivalence between GS and GSA (Theorem 1). Consequently, GSA can be viewed as a concrete implementation of the abstract GS operation.

The core of the paper investigates whether the GSA operation preserves various language families.

1. Finite languages: Since each language contains finitely many words, only finitely many common substrings exist, and thus only finitely many new words can be generated. Theorem 2 shows that GSA(FIN, FIN) = FIN.

2. Regular languages: For two regular grammars G₁ and G₂, the authors construct a new grammar G that contains all original productions plus, for each common terminal a, cross‑productions A → a B′ and A′ → a B. Theorem 3 proves L(G) = GS A(L(G₁), L(G₂)). An equivalent construction is given for deterministic finite automata (DFA): the two automata are merged, a fresh ε‑transition connects their start states, and for every common label a the transition edges are “cross‑wired”. Theorem 4 shows that the language of the merged automaton equals the GSA of the two original languages. Hence regular languages are closed under GSA (Theorem 5).

3. Linear and Context‑Free languages: By converting CFGs to Greibach normal form, the same cross‑production technique can be applied. The resulting grammar generates exactly the GSA of the two original CFLs, demonstrating closure of the linear and context‑free families under GSA.

Throughout, the rule language L₃ is restricted to V⁺ ∪ {(w₁,w₂) | w₁∈L₁, w₂∈L₂}, where V is the set of symbols common to L₁ and L₂. The V⁺ part yields rules of the form w # $ w #, while the pair (w₁,w₂) encodes the “cut after the whole word” operation, ensuring that the parent strings themselves are also present in the generated set. This restriction keeps the model tractable while still capturing the essential combinatorial power of DNA recombination.

By establishing the equivalence GS = GSA and proving closure results for several important language classes, the paper provides a unified theoretical framework that can model more general DNA‑like recombination events, including those where the overlapping segment occurs in the interior of the strings rather than only at ends. This opens avenues for further research on computational properties of generalized splicing systems and their potential applications in synthetic biology and formal language theory.