13th International Conference on Membrane Computing - MTA Sztaki

Observer/interpreter P systems
3 Observation / Interpretation
Based on the motivation exposed in the Introduction, one can imagine a computational
device Φ = (Π, M) (called Observer/Interpreter P system) composed
by a pair of systems: a P system Π (called the core system) and a finite state
machine with output M which is able to detect in any configuration a change
in the multiset of a region of the core system and which can perform a certain
operation based on the observation.
Without any loss of generality and for the simplicity of exposition, we may
assume that the core system Π is a P system with symbol objects and multiset
rewriting rules and which has only one membrane, that is Π = (O, C, µ =
[ ] 1 , R 1 , w 1 , i 0 = 1) having the components defined as the P system model presented
in Section 2. Because Π has only one membrane we can define a configuration
of Π as a multiset w ∈ O ∗ . The initial configuration is C 0 = w 1 .
Given two configurations C 1 and C 2 of Π, we say that C 2 is obtained from C 1
in one transition step (denoted by C 1 ⊢ C 2 ) by applying the rules from R 1 in a
nondeterministic maximal parallel manner and with the competition on objects.
The reflexive and transitive closure of ⊢ is denoted by ⊢ ∗ . The system continues
performing parallel steps until there remain no applicable rules; then the system
halts (the underlying computation is a halting one). The number of objects
from O contained in the output region i 0 = 1 is the result of the underlying
computation of Π.
Given a multiset M : O → IN then M(a), a ∈ O, represents the multiplicity
of a in M. For an ordered pair (M 1 , M 2 ) of multisets M 1 , M 2 : O → IN, we
denote by a ↑ the case when M 1 (a) < M 2 (a) (which indicates the increasing of
the number of objects a from M 1 to M 2 ), by a ↓ the case when M 1 (a) > M 2 (a)
(which indicates the decreasing of the number of objects a from M 1 to M 2 ), and
finally by a− the case when M 1 (a) = M 2 (a). A (partial) observation of the pair
(M 1 , M 2 ) is a subset of {a ↑| a ∈ O, M 1 (a) < M 2 (a)} ∪ {a ↓| a ∈ O, M 1 (a) >
M 2 (a)} ∪ {a− | a ∈ O, M 1 (a) = M 2 (a)}. Considering that O = {a 1 , . . . , a k },
then the set of all possible observations is denoted by
O = {{x 1 , . . . , x k } | (∃) {y 1 , . . . , y k } ⊆ O and
x i ∈ {y i ↑, y i ↓, y i −}, for 1 ≤ i ≤ k} .
The Observer/Interpreter P system represents a finite state machine with
output
M = (Q, O, ∆, δ, λ, q 0 , F, r)
where Q is a finite set of states, O is the set of all possible observations, q 0 ∈ Q
is the initial state, F ⊆ Q is the set of final states, and r is a data register able
to store an integer (r is initially set to 0). The transition function δ : Q × O → Q
defines the functioning of the machine: if M is in a state q ∈ Q and given an
observation o ∈ O, then M moves to the state δ(q, o). The interpretation function
λ : Q × O → {inc, dec, skip} describes the output actions performed by M:
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