Insertion-deletion Systems Research Articles

Regulated insertion-deletion systems

Regulated insertion-deletion systems

Insertion-deletion systems with substitutions I

In this paper, we discuss the addition of substitutions as a further type of operations to (in particular, context-free) insertion-deletion systems, i.e., in addition to insertions and deletions we allow single letter replacements to occur. We investigate the effect of the addition of substitution rules on the context dependency of such systems, thereby also obtaining new characterizations of and even normal forms for context-sensitive (CS) and recursively enumerable (RE) languages and their phrase-structure grammars. More specifically, we prove that for each RE language, there is a system generating this language that only inserts and deletes strings of length two without considering the context of the insertion or deletion site, but which may change symbols (by a substitution operation) by checking a single symbol to the left of the substitution site. When we allow checking left and right single-letter context in substitutions, even context-free insertions and deletions of single letters suffice to reach computational completeness. When allowing context-free insertions only, checking left and right single-letter context in substitutions gives a new characterization of CS. This clearly shows the power of this new type of rules.

On homomorphic images of the Szilard languages of matrix insertion–deletion systems with matrices of size 2

On homomorphic images of the Szilard languages of matrix insertion–deletion systems with matrices of size 2

On the generative capacity of matrix insertion-deletion systems of small sum-norm

A matrix insertion-deletion system (or matrix ins-del system) is described by a set of insertion-deletion rules presented in matrix form, which demands all rules of a matrix to be applied in the given order. These systems were introduced to model very simplistic fragments of sequential programs based on insertion and deletion as elementary operations as can be found in biocomputing. We are investigating such systems with limited resources as formalized in descriptional complexity. A traditional descriptional complexity measure of such a matrix ins-del system is its size s=(k;n,i',i'';m,j',j''), where the parameters from left to right represent the maximal matrix length, maximal insertion string length, maximal length of left contexts in insertion rules, maximal length of right contexts in insertion rules; the last three are deletion counterparts of the previous three parameters. We call the sum n+i'+i''+m+j'+j'' the sum-norm of s. We show that matrix ins-del systems of sum-norm 4 and sizes (3; 1, 0, 0; 1, 2, 0), (3; 1, 0, 0; 1, 0, 2), (2; 1, 2, 0; 1, 0, 0), (2; 1, 0, 2; 1, 0, 0), and (2; 1, 1, 1; 1, 0, 0) describe the recursively enumerable languages. Moreover, matrix ins-del systems of sizes (3; 1, 1, 0; 1, 0, 0), (3; 1, 0, 1; 1, 0, 0), (2; 2, 1, 0; 1, 0, 0) and (2; 2, 0, 1; 1, 0, 0) can describe at least the regular closure of the linear languages. In fact, we show that if a matrix ins-del system of size s can describe the class of linear languages mathrm {LIN}, then without any additional resources, matrix ins-del systems of size s also describe the regular closure of mathrm {LIN}. Finally, we prove that matrix ins-del systems of sizes (2; 1, 1, 0; 1, 1, 0) and (2; 1, 0, 1; 1, 0, 1) can describe at least the regular languages.

Single semi-contextual insertion-deletion systems

Single semi-contextual insertion-deletion systems

Closure properties of bonded sequential insertion-deletion systems

Through the years, formal language theory has evolved through continual interdisciplinary work in theoretical computer science, discrete mathematics and molecular biology. The combination of these areas resulted in the birth of DNA computing. Here, language generating devices that usually considered any set of letters have taken on extra restrictions or modified constructs to simulate the behavior of recombinant DNA. A type of these devices is an insertion-deletion system, where the operations of insertion and deletion of a word have been combined in a single construct. Upon appending integers to both sides of the letters in a word, bonded insertion-deletion systems were introduced to accurately depict chemical bonds in chemical compounds. Previously, it has been shown that bonded sequential insertion-deletion systems could generate up to recursively enumerable languages. However, the closure properties of these systems have yet to be determined. In this paper, it is shown that bonded sequential insertion-deletion systems are closed under union, concatenation, concatenation closure, λ-free concatenation closure, substitution and intersection with regular languages. Hence, the family of languages generated by bonded sequential insertion-deletion systems is shown to be a full abstract family of languages.

Adding Matrix Control: Insertion-Deletion Systems with Substitutions III

Insertion-deletion systems have been introduced as a formalism to model operations that find their counterparts in ideas of bio-computing, more specifically, when using DNA or RNA strings and biological mechanisms that work on these strings. So-called matrix control has been introduced to insertion-deletion systems in order to enable writing short program fragments. We discuss substitutions as a further type of operation, added to matrix insertion-deletion systems. For such systems, we additionally discuss the effect of appearance checking. This way, we obtain new characterizations of the family of context-sensitive and the family of recursively enumerable languages. Not much context is needed for systems with appearance checking to reach computational completeness. This also suggests that bio-computers may run rather traditionally written programs, as our simulations also show how Turing machines, like any other computational device, can be simulated by certain matrix insertion-deletion-substitution systems.

Universal insertion grammars of size two

In this paper, we show that pure insertion grammars of size 2 (i.e., inserting two symbols in a left and right context, each consisting of two symbols) can characterize all recursively enumerable languages. This is achieved by either applying an inverse morphism and a weak coding, or a left (right) quotient with a regular LOC(2) language, or an intersection with a LOC(2) language and a weak coding. The obtained results improve the descriptional complexity of insertion grammars and complete the picture of known results on insertion-deletion systems that are motivated from the DNA computing area.

Computational completeness of simple semi-conditional insertion–deletion systems of degree (2,1)

Insertion–deletion (or ins–del for short) systems are simple models of bio-inspired computing. They are well studied in formal language theory, especially regarding their computational completeness. This concerns the question if all recursively enumerable languages can be generated. This ultimately addresses the question if one can build general-purpose computers rooted in this formalism. The descriptional complexity of an ins–del system is typically measured by its size, a 6-dimensional tuple of non-negative integers $$(e,e',e'';d,d',d'')$$ where e is the maximum length of the insertion string, $$e'$$ (and $$e''$$ ) is the maximum length of the left (and right) context used for insertion; the last three parameters $$d,d',d''$$ are similarly understood for deletion rules. Computational completeness for ins–del systems can even be achieved with rule size (1, 1, 1; 1, 1, 1) but with no rule size strictly smaller than this. This fact has motivated to study ins–del systems in combination with regulation mechanisms. In this context, the six-tuple explained above is called the ID size of a system. Several regulations like graph-control, matrix and semi-conditional have been imposed on ins–del systems. Typically, the computational completeness results are obtained as trade-offs, reducing the ID size, say, to (1, 1, 0; 1, 1, 0) at the expense of increasing other measures of descriptional complexity. In this paper, we study simple semi-conditional ins–del systems, where an ins–del rule can be applied only in the presence or absence of substrings of the derivation string. This brings along two further natural parameters to measure descriptional complexity, namely, the maximum permitting string length p and the maximum forbidden string length f, usually summarized as the degree $$d=(p,f)$$ . We show that simple semi-conditional ins–del systems of degree (2, 1) and with ID sizes $$(1+e,e',e'';1+d,d',d'')$$ are computationally complete for any $$e,e',e'',d,d',d''\in \{0,1\}$$ , with $$e+e'+e''=1$$ and $$d+d'+d''=1$$ . The obtained results complement and improve on the existing results known from the literature. To prove our results, we also show a new normal form for type-0 grammars that appears to be interesting in its own right.

On describing super-linear languages by matrix insertion–deletion systems

Matrix insertion–deletion (matrix ins–del) systems combine the idea of matrix control (as established in regulated rewriting) with that of insertion and deletion operations (as opposed to replacements). There are matrix ins–del systems of small sizes that are known to describe linear languages but not even context-free languages. Our aim is to study the generative power of such matrix ins–del systems. In this regard, we consider six language classes, called super-linear languages between LIN and CFL, namely $${\mathrm {CLIN}}$$ , $${\mathrm {SLIN}}$$ , $${\mathrm {CSLIN}}$$ , $${\mathrm {SCLIN}}$$ , $${\mathrm {CSCLIN}}$$ and $${\mathrm {SCSLIN}}$$ obtained by applying concatenation or/and Kleene closure operation on linear languages. These classes deserve special attention due to the fact that $${\mathrm {LIN}}$$ is closed neither under concatenation nor under Kleene closure. In this paper, we discuss the context-free grammars that generate these language classes and also simulate them by matrix ins–del systems.

On path-controlled insertion–deletion systems

A graph-controlled insertion–deletion system is a regulated extension of an insertion–deletion system. It has several components and each component contains some insertion–deletion rules. These components are the vertices of a directed control graph. A transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule. This also describes the arcs of the control graph. Starting from an axiom in the initial component, strings thus move through the control graph. The language of the system is the set of all terminal strings collected in the final component. In this paper, we investigate a variant of the main question in this area: which combinations of size parameters (the maximum number of components, the maximal length of the insertion string, the maximal length of the left context for insertion, the maximal length of the right context for insertion; plus three similar restrictions with respect to deletion) are sufficient to maintain computational completeness of such restricted systems under the additional restriction that the (undirected) control graph is a path? Notice that these results also bear consequences for the domain of insertion–deletion P systems, improving on a number of previous results from the literature, concerning in particular the number of components (membranes) that are necessary for computational completeness results.

On describing the regular closure of the linear languages with graph-controlled insertion-deletion systems

A graph-controlled insertion-deletion (GCID) system has several components and each component contains some insertion-deletion rules. A transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule. The language of the system is the set of all terminal strings collected in the final component. When resources are very limited (especially, when deletion is demanded to be context-free and insertion to be one-sided only), then GCID systems are not known to describe the class of recursively enumerable languages. Hence, it becomes interesting to explore the descriptional complexity of such GCID systems of small sizes with respect to language classes below RE and even below CF. To this end, we consider so-called closure classes of linear languages defined over the operations concatenation, Kleene star and union. We show that whenever GCID systems (with certain syntactical restrictions) describe all linear languages (LIN) with t components, we can extend this to GCID systems with just one more component to describe, for instance, the concatenation of two languages from the language family that can be described as the Kleene closure of linear languages. With further addition of one more component, we can extend the construction to GCID systems that describe the regular closure of LIN.

Investigations on the power of matrix insertion-deletion systems with small sizes

Matrix insertion-deletion systems combine the idea of matrix control (a control mechanism well established in regulated rewriting) with that of insertion and deletion (as opposed to replacements). Given a matrix insertion-deletion system, the size of such a system is given by a septuple of integers $$(k;n,i',i'';m,j',j'')$$ . The first integer k denotes the maximum number of rules in (length of) any matrix. The next three parameters $$n,i',i''$$ denote the maximal length of the insertion string, the maximal length of the left context, and the maximal length of the right context of insertion rules, respectively. The last three parameters $$m,j',j''$$ are similarly understood for deletion rules. In this paper, we improve on and complement previous computational completeness results for such systems, showing that matrix insertion-deletion systems of size (1) (3; 1, 0, 1; 1, 0, 1), (3; 1, 0, 1; 1, 1, 0), (3; 1, 1, 1; 1, 0, 0) and (3; 1, 0, 0; 1, 1, 1) (2) (2; 1, 0, 1; 2, 0, 0), (2; 2, 0, 0; 1, 0, 1), (2; 1, 1, 1; 1, 1, 0) and (2; 1, 1, 0; 1, 1, 1), are computationally complete. Further, we also discuss linear and metalinear languages and we show how to simulate grammars characterizing them by matrix insertion-deletion systems of size (3; 1, 1, 0; 1, 0, 0), (3; 1, 0, 1; 1, 0, 0), (2; 2, 1, 0; 1, 0, 0) and (2; 2, 0, 1; 1, 0, 0). We also generate non-semilinear languages using matrices of length three with context-free insertion and deletion rules.

Universality and Computational Completeness of Controlled Leftist Insertion-Deletion Systems

In this article, we consider leftist insertion-deletion systems (LIDS), in which all rules have contexts on the same (left) side, and may only insert or delete one symbol at a time. We start by introducing extended rules, in which the contexts may be specified as regular expressions, instead of fix ed words. We prove that in this case the computational completeness is achieved when additional control mechanisms are used (graph control with two states, matrix control with binary matrices and random-context control). We then show how rules with regular contexts can be simulated by conventional rules checking one-symbol (resp. two-symbol) left contexts for insertion and two-symbol (resp. one-symbol) left contexts for deletion. This simulation does not generally hold in the controlled case, however. Hence, we provide a construction simulating an arbitrary 2-tag system using extended rules and which can be rewritten in terms of conventional rules of types above, which implies that the latter systems are universal.

On the computational completeness of graph-controlled insertion–deletion systems with binary sizes

A graph-controlled insertion–deletion (GCID) system is a regulated extension of an insertion–deletion system. Such a system has several components and each component has some insertion–deletion rules. The transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule. The language of the system is the set of all terminal strings collected in the final component. The parameters in the size (k;n,i′,i″;m,j′,j″) of a GCID system denote (from left to right) the maximum number of components, the maximal length of the insertion string, the maximal length of the left context for insertion, the maximal length of the right context for insertion; the last three parameters follow a similar representation with respect to deletion.In this paper, we discuss the computational completeness of the families of GCID systems of size (k;1,i′,i″;1,j′,j″) with k∈{3,5} and for (nearly) all values of i′,i″j′,j″∈{0,1}. All proofs are based on the simulation of type-0 grammars given in Special Geffert Normal Form (SGNF). The novelty in our proof presentation is that the context-free and the non-context-free rules of the given SGNF grammar are simulated by GCID systems of different sizes and finally we combine them by stitching and overlaying to characterize the recursive enumerable languages. This proof presentation greatly simplifies and unifies the proof of such characterization results. We also connect some of the obtained GCID simulations to the domain of insertion–deletion P systems.

Modelling DNA and RNA secondary structures using matrix insertion–deletion systems

Abstract Insertion and deletion are operations that occur commonly in DNA processing and RNA editing. Since biological macromolecules can be viewed as symbols, gene sequences can be represented as strings and structures can be interpreted as languages. This suggests that the bio-molecular structures that occur at different levels can be theoretically studied by formal languages. In the literature, there is no unique grammar formalism that captures various bio-molecular structures. To overcome this deficiency, in this paper, we introduce a simple grammar model called the matrix insertion–deletion system, and using it we model several bio-molecular structures that occur at the intramolecular, intermolecular and RNA secondary levels.

Random Context and Semi-conditional Insertion-deletion Systems

In this article we introduce the operations of insertion and deletion working in random context and semi-conditional modes. We show that conditional application of insertion and deletion rules strictly increases the computational power. In the case of semi-conditional insertion-deletion systems, context-free insertion and deletion rules of one symbol are sufficient to achieve computational completeness. In the random context case, our results expose asymmetry between the computational power of insertion and deletion rules: semi-conditional systems of size 2, 0, 0; 1, 1, 0 with context-free two-symbol insertion rules, and one-symbol deletion rules with one-symbol left context are computationally complete, while systems of size 1, 1, 0; 2, 0, 0 and, more generally, of size 1, 1, 0; p, 1, 1 are not.

Matrix insertion–deletion systems

We investigate in this article the operations of insertion and deletion working in a matrix-controlled manner. We show that this allows to us strictly increase the computational power: in the case of systems that are not computationally complete (with total size equal to 4), the computational completeness can be obtained by introducing the matrix control and using only binary matrices.

ON THE AMBIGUITY OF INSERTION SYSTEMS

Gene insertion and deletion are the operations that occur commonly in DNA processing and RNA editing. Based on these evolutionary transformations, a computing model has been formulated in formal language theory known as insertion-deletion systems. In this paper, we study about the ambiguity issues of insertion systems. First, we define six levels of ambiguity for insertion systems based on the components used in the derivation such as axiom, contexts and strings. Next, we show that there are inherently i-ambiguous insertion languages which are j-unambiguous for the combinations (i, j) ∈ {(5,0), (5,4), (4,3), (4,2), (3,1),(2,1), (1,0), (0,1)}. Finally, we prove an important result that the ambiguity problem of insertion systems is undecidable.

On the study of ambiguity and the trade-off between measures and ambiguity in insertion–deletion languages

Gene insertion and deletion are the operations that occur commonly in DNA processing and RNA editing. Based on these operations, a computing model has been formulated in formal language theory known as insertion–deletion systems. In this paper we study about ambiguity issues of these systems. First, we define six levels of ambiguity for insertion–deletion systems that are based on the components used in the derivation such as axiom, contexts and strings. Next, we show that there are inherently i -ambiguous insertion–deletion languages which are j -unambiguous for the combinations ( i , j ) ∈ { ( 5 , 4 ) , ( 4 , 3 ) , ( 4 , 2 ) , ( 3 , 1 ) , ( 2 , 1 ) , ( 1 , 0 ) , ( 0 , 1 ) } . As an application, we discuss with an example that how some of these ambiguity levels can be interpreted in gene sequences. Further, we prove an important result that the ambiguity problem of insertion–deletion systems is undecidable. Then, we define six new measures for insertion–deletion systems based on used contexts and strings. Finally, we analyze the trade-off between ambiguity levels and measures. We note that there are languages which are inherently i -ambiguous (for i = 5 , 4 , 2 , 0 ) when a measure M is minimal for the languages but they are i -unambiguous otherwise.