13th International Conference on Membrane Computing - MTA Sztaki

A.E. Porreca, A. Leporati, G. Mauri, C. Zandron
These rules are replicated for all 0 ≤ i < n, 0 ≤ w < s(n).
This completes the description of the family of P systems Π = {Π x : x ∈ Σ ⋆ }
simulating M. Each P system Π x only requires O(log |x|) membranes and objects
besides the input objects (and these are not modied nor created during the
computation).
In order to prove Theorem 1 we still need to show that the family Π is indeed
(DLT, DLT)-uniform. Here we provide a proof sketch for this result.
Consider the mapping x ↦→ w x , encoding each input string of M as a multiset
over the alphabet of Π n (with n = |x|): each symbol of x has to be subscripted
with an index of l(n) bits representing its position in x. The corresponding
encoding predicate is
encoding(x, i, a j ) ⇐⇒ j = i ∧ x i = a.
It is easy to check in DLOGTIME if the predicate holds for each (x, i, a j ). First,
we copy the portions of the input representing i and a j (of length O(log n)) on
auxiliary work tapes and we check if the third argument is indeed of the form
a j for some a ∈ Σ by simulating a nite state automaton. By scanning i and
j we can ensure that i = j. Then, we extract the i-th symbol of x by copying
i on the address tape of the machine, and we check if that symbol is a. Since
symbol-by-symbol comparisons require linear time with respect to the length of
the strings, the evaluation of encoding can be carried out in logarithmic time.
The alphabet of Π n can be represented by using O(l(n)) bits, where the
hidden constants also depend on the size of the alphabet Σ of M. For simplicity,
we can use kl(n) for some appropriate k as an upper bound, and set
alphabet(1 n , m) ⇐⇒ m = kl(n).
This predicate can be checked in DLOGTIME, as multiplication by a constant
can be implemented by repeated additions. The reasoning for the predicate
labels is similar.
The membrane structure of Π n (see Fig. 1 for an example with n = 5) is
described as follows:
inside(1 n , h 1 , h 2 ) ⇐⇒ (h 1 = i l(n)−1 ∧ h 2 = h) ∨
(h 1 = i j ∧ h 2 = i j+1 ∧ 0 ≤ j < l(n) − 1) ∨
(h 1 = w j ∧ h 2 = h ∧ 0 ≤ j < s(n)) ∨
(h 1 = a i ∧ h 2 = h ∧ a ∈ Σ) ∨
(h 1 = a w ∧ h 2 = h ∧ a ∈ Σ)
that is, by a disjunction of a constant number of conjuncts, each one consisting
of a constant number of terms whose truth can be veried in DLOGTIME by
executing comparisons or simple computations on numbers of O(log n) bits. The
input membrane is identied by
input(1 m , h) ⇐⇒ h = i 0 .
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