Graph multiset transformation: a new framework for massively parallel computation inspired by DNA computing

Abstract

In this paper, graph multiset transformation is introduced and studied as a novel type of parallel graph transformation. The basic idea is that graph transformation rules may be applied to all or at least some members of a multiset of graphs simultaneously providing a computational step with the possibility of massive parallelism in this way. As a consequence, graph problems in the class NP can be solved by a single computation of polynomial length for each input graph.

Appendix

This appendix recalls the notions and notations of multisets used in the paper.

• 1. Let X be a set. Then a multiset (over X) is a mapping ${M:X \rightarrow \mathbb{N}}$, where M(x) is the multiplicity of x in M.

• 2. The carrier of M contains all elements of X with positive multiplicity, i.e.

  $$ car(M)=\{x\in X\;\vert\;M(x)>0\}. $$

• 3. A multiset is finite if its carrier is a finite set.

• 4. Let M and M′ be multisets. Then M′ is a sub-multiset of M, denoted by $M^{\prime}\leq M$, if $M^{\prime}(x)\leq M(x)$ for all x $\in$ X.

• 5. Let M and M′ be multisets. Then the sum (difference) of M and M′ is the multiset defined by

  $$ (M\pm M^{\prime})(x)=M(x)\pm M^{\prime}(x) \;\text{for\;all}\;x\in X. $$

  Here + and −  are the usual sum and difference of non-negative integers with $m-n=0$ if $m\,\leq\,n$ in particular.

• 6. Using the sum of multisets, the multiplication of multisets with non-negative numbers can be defined inductively for all multisets M by

  • (i) $0 \cdot M = {\bf 0}$ and

  • (ii) $(k+1) \cdot M = k \cdot M + M$ for all ${k \in {\mathbb{N}}}$

    where the multiset 0 is the multiset with the constant multiplicity 0, i.e. 0(x) = 0 for all $x \in X$.

• 7. Each sequence $w\in X^{*}$ induces a multiset [w] by counting the number of occurrences of each x in w, i.e., for all x, $y \in X$ and $w\in X^{*}$,

  • $[\lambda]$(x) = 0

  • $[yw](x)= {if}\ x=y \, then \; [w](x)+1 \, else \, [w](x).$

• 8. Let M be a finite multiset. Then the set of all sequences w with [w] = M is denoted by Perm(M). An element of Perm(M) is called a sequential representation of M. Note that Perm(M) contains all permutations of w if $[w]=M$.

• 9. The set of multisets over X as well as the set of finite multisets over X give rise to a commutative monoid with the multiset 0 as null and the sum as inner composition. Moreover, the set of finite sultisets over X is generated by the singletons [x] for all x  $\in$ X so that the finite multisets are characterized as the free commutative monoid over X.