13th International Conference on Membrane Computing - MTA Sztaki

Fast distributed DFS solutions for edge-disjoint paths in digraphs
elementary symbols and i, j, X are free variables (assuming that these are not
listed among elementary symbols): b(2) = b 2 , c(i) = c i , d(i, j) = d i,j , e(a 2 b 3 ),
f(j, c 5 ) = f j (c 5 ), f(j, Xc) = f j (Xc).
Besides modelling complex data structures, such as lists, stacks, trees and
dictionaries, or emulating procedure calls, complex symbols are useful for representing
and processing any number of cell IDs with a fixed vocabulary. Thus,
complex symbols allow the design of fixed-size P system algorithms, i.e. algorithms
having a fixed number of rules, which does not depend on the number of
cells in the underlying P systems.
Here we assume that each cell σ i is “blessed” with a unique complex cell
ID symbol, ι(i), typically abbreviated as ι i , which is exclusively used as an
immutable promoter.
Generic rules: To process complex symbols, we use high-level generic rules,
which are instantiated using free variable matching [1]. A generic rule is identified
by an extended version of the classical rewriting mode, in fact, a combined
instantiation.rewriting mode, where (1) the instantiation mode is one of {min,
max, dyn} and (2) the rewriting mode in one of {min, max}.
The instantiation mode indicates how many instance rules are conceptually
generated: (a) the mode min indicates that the generic rules is nondeterministically
instantiated only once, if possible; (b) the mode max indicates that the
generic rule is instantiated as many times as possible, without superfluous instances
(i.e. without duplicates or instances which are not applicable); (c) the
newly proposed mode dyn indicates a dynamic instantiation mode, which will
be described later. The rewriting mode indicates how each instantiated rule is
applied (as in the classical framework).
As an example, consider a system where cell σ 7 contains multiset f 2 f 3 2 v, and
the generic rule ρ α , where α ∈ {min.min, min.max, max.min, max.max} and i and
j are free variables:
(ρ α ) S 20 f j → α S 20 (b i )↕ j | v ι i
1. ρ min.min nondeterministically generates one of the following rule instances:
S 20 f 2 → min S 20 (b 7 )↕ 2
S 20 f 3 → min S 20 (b 7 )↕ 3
2. ρ min.max nondeterministically generates one of the following rule instances:
S 20 f 2 → max S 20 (b 7 )↕ 2
S 20 f 3 → max S 20 (b 7 )↕ 3
3. ρ max.min generates both following rule instances:
S 20 f 2 → min S 20 (b 7 )↕ 2
S 20 f 3 → min S 20 (b 7 )↕ 3
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