DNAQL is a query language for databases implemented as DNA molecules in solution, with well-known DNA wetlab procedures as operators. DNAQL focusses on faithful implementability in DNA, and, as a result, does not include powerful, but difficult to implement, features like nonterminating/recursive hybridization needed for general computation (i.e., Turing-completeness). In this article, we show that DNAQL is relationally complete, i.e., the full relational algebra can be performed by DNAQL programs.

We investigate the expressive power of MATLANG, a formal language for matrix manipulation based on common matrix operations and linear algebra. The language can be extended with the operation inv for inverting a matrix. In MATLANG + inv, we can compute the transitive closure of directed graphs, whereas we show that this is not possible without inversion. Indeed, we show that the basic language can be simulated in the relational algebra with arithmetic operations, grouping, and summation. We also consider an operation eigen for diagonalizing a matrix. It is defined such that for each eigenvalue a set of mutually orthogonal eigenvectors is returned that span the eigenspace of that eigenvalue. We show that inv can be expressed in MATLANG + eigen. We put forward the open question whether there are Boolean queries about matrices, or generic queries about graphs, expressible in MATLANG + eigen but not in MATLANG + inv. Finally, the evaluation problem for MATLANG + eigen is shown to be complete for the complexity class ∃ R.

This paper presents an exposition of the authors' past and present work on the query language DNAQL for querying databases in DNA. In DNA computing, data is represented and stored in DNA molecules. Accordingly, a logical data model is defined that models complexes of DNA molecules in a graph-oriented fashion. Next, a set of formal operations on DNA complexes is defined, much in the spirit of the operations of the relational algebra in the relational data model. These operations model laboratory operations on DNA in solution. Their combination leads to the query language DNAQL; but in order for programs to be well-defined on prescribed types of inputs, a type system is superimposed on the language. Finally a correspondence is shown between well-typed DNAQL programs and programs in a relational-algebra query language.

Grammar inference deals with determining (preferably simple) models/grammars consistent with a set of observations. There is a large body of research on grammar inference within the theory of formal languages. However, there is surprisingly little known on grammar inference for graph grammars. In this article, we take a further step in this direction and work within the framework of node label controlled (NLC) graph grammars. Specifically, given a graph G and a set 𝒮 of disjoint and isomorphic subgraphs of G, we characterize whether or not there is a graph grammar consisting of one production such that G may be derived from G0, the graph obtained from G by ‘contraction’ of each subgraph in 𝒮 to a node labelled by N. This generalizes a previous result that assumes boundary NLC graph grammars, and leads one to consider the more involved ‘non-confluent’ graph grammar rules.

We show that the symmetric-difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case. We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition, we obtain similar characterizations for matroids). In this way, we find that, e.g., the delta-matroids obtained after loop complementation and after pivot on a single element together with the original delta-matroid fulfill the property that two of them have equal “null space” while the third has a larger dimension.

We consider the Schur complement operation for symmetric matrices over GF(2), which we identify with graphs through the adjacency matrix representation. It is known that Schur complementation for such a matrix (i.e., for a graph) can be decomposed in