"Going back to our roots": second generation biocomputing

Researchers in the field of biocomputing have, for many years, successfully "harvested and exploited" the natural world for inspiration in developing systems that are robust, adaptable and capable of generating novel and even "creative" solutions to human-defined problems. However, in this position paper we argue that the time has now come for a reassessment of how we exploit biology to generate new computational systems. Previous solutions (the "first generation" of biocomputing techniques), whilst reasonably effective, are crude analogues of actual biological systems. We believe that a new, inherently inter-disciplinary approach is needed for the development of the emerging "second generation" of bio-inspired methods. This new modus operandi will require much closer interaction between the engineering and life sciences communities, as well as a bidirectional flow of concepts, applications and expertise. We support our argument by examining, in this new light, three existing areas of biocomputing (genetic programming, artificial immune systems and evolvable hardware), as well as an emerging area (natural genetic engineering) which may provide useful pointers as to the way forward.Comment: Submitted to the International Journal of Unconventional Computin

Nullity Invariance for Pivot and the Interlace Polynomial

We show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given (square) matrix is described by the symmetric difference operator (for sets). We consider its consequences for graphs, and in particular generalize the recursive relation of the interlace polynomial and simplify its proof.Comment: small revision of Section 8 w.r.t. v2, 14 pages, 6 figure

Maximal Pivots on Graphs with an Application to Gene Assembly

We consider principal pivot transform (pivot) on graphs. We define a natural variant of this operation, called dual pivot, and show that both the kernel and the set of maximally applicable pivots of a graph are invariant under this operation. The result is motivated by and applicable to the theory of gene assembly in ciliates.Comment: modest revision (including different latex style) w.r.t. v2, 16 pages, 5 figure

Pivots, Determinants, and Perfect Matchings of Graphs

We give a characterization of the effect of sequences of pivot operations on a graph by relating it to determinants of adjacency matrices. This allows us to deduce that two sequences of pivot operations are equivalent iff they contain the same set S of vertices (modulo two). Moreover, given a set of vertices S, we characterize whether or not such a sequence using precisely the vertices of S exists. We also relate pivots to perfect matchings to obtain a graph-theoretical characterization. Finally, we consider graphs with self-loops to carry over the results to sequences containing both pivots and local complementation operations.Comment: 16 page

The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems

We study the interplay between principal pivot transform (pivot) and loop complementation for graphs. This is done by generalizing loop complementation (in addition to pivot) to set systems. We show that the operations together, when restricted to single vertices, form the permutation group S_3. This leads, e.g., to a normal form for sequences of pivots and loop complementation on graphs. The results have consequences for the operations of local complementation and edge complementation on simple graphs: an alternative proof of a classic result involving local and edge complementation is obtained, and the effect of sequences of local complementations on simple graphs is characterized.Comment: 21 pages, 7 figures, significant additions w.r.t. v3 are Thm 7 and Remark 2