Bond-Free Languages: Formalizations, Maximality and Construction Methods

Abstract

Abstract. The problem of negative design of DNA languages is addressed, that is, properties and construction methods of large sets of words that prevent undesired bonds when used in DNA computations. We recall a few existing formalizations of the problem and then define the property of sim-bond-freedom, where sim is a similarity relation between words. We show that this property is decidable for context-free languages and polynomial-time decidable for regular languages. The maximality of this property also turns out to be decidable for regular languages and polynomial-time decidable for an important case of the Hamming similarity. Then we consider various construction methods for Hamming bond-free languages, including the recently introduced method of templates, and obtain a complete structural characterization of all maximal Hamming bond-free languages. This result is applicable to the θ-k-code property introduced by Jonoska and Mahalingam.

Bond-Free Languages: Formalizations, Maximality and Construction Methods

“…P3 and P5: Important Note. Proofs of the results obtained in this paper can be found in the full version [17].…”

“…This implies that, if S 1 = S then S 1 ⊗ = S ⊗ . Moreover we note that, given S, one can construct in linear time a DFA accepting S ⊗ [17].…”

“…An advantage of the method of Theorem 6 is that we can construct (τ, H d,k )-bond-free languages with a large ratio d/k. Another advantage of some codes K defined in the previous theorem is that one can encode and decode information in linear time [17].…”

“…It is evident that the relation H d,k defined in Subsection 1.4 is an example of a similarity relation with parameters (0, k). It can be shown that also Lev d,k is a similarity relation, with parameters (d, d + k) -see [17]. Theorem 1.…”

“…More specifically, consider the language L 2 accepted by the 2-block DFA (Σ, {1, 2, 3, 4}, 1, {2, 3, 4} The language L 2 is a proper superset of K + 1 and is a (τ, H 0,2 )-bond-free subset of (Σ 2 ) + . In fact it can be shown that L 2 is maximal with this property [17].…”