Cut, Paste and Filter

We study the combined and separate effects of three parts of finite multi-testtube cut and paste DNA computing. First, we reformulate the ideas of Csuhaj-Varjú, Kari, and Pǎun [1], Freund [3], and Priese, Rogojine and Margenstern [10] about multi-test-tube splicing DNA computing in terms of cutting and pasting as in Pixton’s work. Pixton shows [8, 9] that with finite cutting and pasting only regular sets can be obtained from a finite set of initial molecules. The others listed above show that using filtering between a finite number of test-tubes, each with finite splicing, any recursively enumerable set can be obtained from finite initial contents [1, 3, 10]. We confirm their result for cutting and pasting. Second, we show that when only finite pasting and filtering between tubes with finite initial contents are allowed then the result must be context free and that any context free language can be so obtained. Finally, we consider several forms of filtering and several ways of combining filtering with cutting, pasting or splicing and show that all give equivalent results. 1. Basic Definitions An alphabet A is a non-empty finite set. A∗ is the free monoid generated by A. Its elements are called words or strings (on A), and the identity of A∗, the empty word, is written 1. A subset of A∗ is a language (on A). If L is a language on A, let the set of factors or sub-words of L be FacL = {w ∈ A∗ | uwv ∈ L for some u, v ∈ A∗}. If L is a language on A and M is a language on B then the result of concatenating them is LM = {w ∈ (A ∪B)∗ | w = w1w2 for some w1 ∈ L,w2 ∈M}, a language on A ∪B. If Σ is another finite non-empty set, of end-markers, then an element of ΣA∗Σ is called an end-marked string (on A with end-markers in Σ) and a subset of ΣA∗Σ is called an end-marked language. If M is an end-marked language on A with end-markers in Σ then StripM = {w ∈ A∗ | αwβ ∈M forsomeα, β ∈ Σ}, the stripped version of M , is the language obtained by removing the end-markers from the words of M . As a convention, we will use Roman letters for elements of A and A∗ and Greek letters for elements of Σ. For multi-testtube computing we will need tuples of end-marked languages. Let L = (L1, L2, . . . , Ln) and M = (M1,M2, . . . ,Mn) be n-tuples of end-marked languages. We say L ⊂ M provided Li ⊂Mi for 1 ≤ i ≤ n. Further we define L ∪M component-wise by (L ∪M)i = Li ∪Mi. One way to generate languages is using phrase-structure grammars. The most general grammar we will use here is the type-0 grammar G = (N, T,R, S). Here N is a finite set of non-terminal symbols, T is a finite set of terminal symbols, R ⊂ (N∪T )∗N+(N∪T )∗×(N∪T )∗ is a finite set of production rules or productions, and S ∈ N is the start symbol. It is required that N and T be disjoint. If (u, v) ∈ R we write u → v. The relation → extends to a relation on (N ∪ T )∗ as follows: for x, y ∈ (N ∪ T )∗ and u→ v in R write xuy → xvy. The relation k → is the k-fold iteration of →. The relation ∗ → is the reflexive, transitive closure Date: July 7,2000.