13th International Conference on Membrane Computing - MTA Sztaki

One-membrane symport P systems with few extra symbols
We denote the family of all recursively enumerable sets of non-negative integers
by NRE. By N j RE we denote the family {{x + j | x ∈ M} | M ∈ NRE},
i.e., the family of all recursively enumerable sets of non-negative integers, such
that j has been added to each element of every set (or, equivalently, {M ∈
NRE | x ≤ j, x ∈ M}, i.e., the family of all recursively enumerable sets of
integers not smaller than j).
2.1 Finite Automata
Definition 1. A finite automaton is a tuple A = (Σ, Q, q 0 , δ, F ), where Σ is an
input alphabet, Q is the set of states, q 0 ∈ Q is the initial state, F ⊆ Q is the
set of final states, and δ : Q × Σ −→ 2 Q is the transition mapping.
The function δ is naturally extended from symbols to strings. The language
accepted by A is the set {w ∈ Σ ∗ | δ(q 0 , w) ∩ F ≠ ∅}.
Throughout the paper we assume the following property holds for finite automata:
there is at least one transition from every non-final state. This does not
restrict the generality, since adding transitions from each non-final dead state to
itself leads to an equivalent automaton satisfying the needed property.
2.2 Counter Automata
In the universality proofs of this paper we will use counter automata with conflicting
counters. We use slightly different semantics of conflicting counters in
Theorems 1, 2, so we introduce them locally.
Definition 2. A non-deterministic counter automaton is a construct M = (Q,
q 0 , q f , P, C), where
– Q is the set of states,
– q 0 ∈ Q is the initial state,
– q f ∈ Q is the final state,
– P is the set of instructions of types (q → q ′ , i+), (q → q ′ , i−) and (q →
q ′ , i = 0), modifying the state and incrementing or decrementing counter i
by one, or verifying the value of the counter is zero,
– C is the set of counters.
A computation of M consists of transitions between the states from Q with
updating/checking the counters. Attempting to decrement a counter with value
zero, or to zero-test a counter with a non-zero value leads to aborting a computation
without producing a result. Without restricting generality, we assume that
for every state, there is at least one instruction from it. Counter automata are
known to be computationally complete: for every recursively enumerable set U
of non-negative integers there exists a counter automaton starting in q 0 from the
empty counters and generating in q f precisely elements of U in the first counter,
all others having zero value.
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