Arithmetical Metabolic P Systems - Foundations on Natural and Artificial Computation - page 291

k
4

N 1 N 2 N 3

s

k

s s

t M 1 M 2 M 3

h

0

t

1 1

1

A

R 1 R 2 R 3 R 4

Fig. 5. A 3-digit abacus with carry

yielded by the carry rules, each trying to consume 10 units from its right region
to produce 1 carry unit on its left region.
The computation, including the carry, may take up to k = 4 steps counted by
the countdown network as follows. When s becomes 1, the reaction r 10 : k

t
occurs leaving k =3and t =1.Ateachnextstep k is decreased, while the rules
consuming t are blocked because they want to globally consume t + s = t +1 units of
t .When k becomes 1 the rule r 13 : s

→

→ h unblocks producing the Halt substance

h and decreasing the value of s to 0. Now r 11 : t k can refill k making the counter

network re-usable. The overall eciency, as in digital computers, has an important
limitation in the carry mechanism unless some additional optimizations are imple-
mented. Notice that the reaction arrows can transform matter either in the same
region or also communicate substances across membranes. The MP grammar in
standard boundary notation is given by:

→

R Φ

ϕ 1 = u

r 1 :[ N 1 u → [ R 2 u

r 2 :[ M 1 u → [ R 2 u 2 = u

r 3 :[ N 2 u → [ R 3 u 3 = u

r 4 :[ M 2 u → [ R 3 u 5 = u

r 5 :[ N 3 u → [ R 4 u 4 = u

In = {N 1 ,N 2 ,N 3 ,M 1 ,M 2 ,M 2 }⊆L
Out = {R 1 ,R 2 ,R 3 ,r 4 }⊆L
Start

r 6 :[ M 3 u → [ R 4 u 6 = u

= s ∈ S

r 7 :[ R 4 10 u → [ R 3 uϕ 7 =1

Halt = h ∈ S

r 8 :[ R 3 10 u → [ R 2 uϕ 8 =1

r 9 :[ R 2 10 u → [ R 1 uϕ 9 =1

r 10 :[ A k → [ A t 10 = s

r 11 :[ A t → [ A k 11 = t

r 12 :[ A t → [ A k 12 = s

r 13 :[ A s → [ A h 13 = k

With the initial configuration
C =[ A 4 K [7 u ] N 1 [2 u ] N 2 [8 u ] N 3 [3 u ] M 1 [2 u ] M 2 [2 u ] M 3 [] R 1 [] R 2 [] R 3 [] R 4 ].