In this paper, we give a direct proof of the result of Latteux and Turakainen that the full class of recursively enumerable languages can be obtained from minimal linear languages (which are generated by linear context-free grammars with only one nonterminal symbol) by Dyck reductions (which reduce pairs of parentheses to the empty word).

The Firing Squad Synchronization Problem (FSSP), one of the most well-known problems related to cellular automata, was originally proposed by Myhill in 1957 and became famous through the work of Moore [1]. The first solution to this problem was given by Minsky and McCarthy [2] and a minimal time solution was given by Goto [3]. A significant amount of research has also dealt with variants of this problem. In this paper, from a theoretical interest, we will extend this problem to number patterns on a seven-segment display. Some of these problems can be generalized as the FSSP for some special trees called segment trees. The FSSP for segment trees can be reduced to a FSSP for a one-dimensional array divided evenly by joint cells that we call segment array. We will give algorithms to solve the FSSPs for this segment array and other number patterns, respectively. Moreover, we will clarify the minimal time to solve these problems and show that there exists no such solution.

Watson-Crick automata were introduced as a new computer model and have been intensively investigated regarding their computational power. In this paper, aiming to establish the relations among language families defined by Watson-Crick automata and the family of context-free languages completely, we obtain the following results. (1) F1WK = FSWK = FWK, (2) FWK = AWK, (3) there exists a language which is not context-free but belongs to NWK, and (4) there exists a context-free language which does not belong to AWK.

Let L be any class of languages, L' be a class of languages which is closed under λ-free homomorphisms, and Σ be any alphabet. In this paper, we show that if the following statement (1) holds, then the statement (2) holds. (1) For any language L in L over Σ, there exist an alphabet of k pairs of matching parentheses Xk, Dyck reduction Red over Xk, and a language L1 in L' over ΣXk such that L=Red(L1)Σ*. (2) For any language L in L over Σ, there exist an alphabet Γ including Σ, a homomorphism h : Γ*Σ*, a Dyck language D over Γ, and a language L2 in L' over Γ such that L=h(DL2). We also give an application of this result.

Watson-Crick automata, recently introduced in, are new types of automata in the DNA computing framework, working on tapes which are double stranded sequences of symbols related by a complementarity relation, similar to a DNA molecule. The automata scan separately each of the two strands in a corelated mannar. Some restricted variants of them were also introduced and the relationship between the families of languages recognized by them were investigated in. In this paper, we clarify some relations between the families of languages recognized by the restricted variants of Watson-Crick finite automata and the families in the Chomsky hierarchy.

Let L be any class of languages, L' be one of the classes of context-free, context-sensitive and recursively enumerable languages, and Σ be any alphabet. In this paper, we show that if the following statement (1) holds, then the statement (2) holds. (1) For any language L in L over Σ, there exist an alphabet Γ including Σ, a homomorphism h:Γ*Σ* defined by h(a)=a for aΣ and h(a)=λ (empty word) for aΓ-Σ, a Dyck language D over Γ, and a language L1 in L' over Γ such that L=h(DL1). (2) For any language L in L over Σ, there exist an alphabet of k pairs of matching parentheses Xk, Dyck reduction Red over Xk, and a language L2 in L' over ΣXk such that L=Red(L2)Σ*. We also give an application of this result.

An Insertion-Deletion system, first introduced in [1], is a theoretical computing model in the DNA computing framework based on insertion and deletion operations. When insertion and deletion operations work together, as expected, they are very powerful. In fact, it has been shown that even the very restricted Insertion-Deletion systems can characterize the class of recursively enumerable languages [1]-[4]. In this paper, we investigate the computational power of Insertion-Deletion systems and show that they preserve the computational universality without using contexts.