By Cristian S. Calude

Professor Jozef Gruska is a well-known laptop scientist for his many and huge effects. He used to be the daddy of theoretical laptop technology study in Czechoslovakia and one of the first Slovak programmers within the early Sixties. Jozef Gruska brought the descriptional complexity of grammars, automata, and languages, and is likely one of the pioneers of parallel (systolic) automata. His different major study pursuits contain parallel platforms and automata, in addition to quantum info processing, transmission, and cryptography. he's co-founder of 4 usual sequence of meetings in informatics and in quantum details processing and the Founding Chair (1989-96) of the IFIP expert workforce on Foundations of laptop Science.

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**Example text**

An application of rule r1 = a → a bin2 c2in2 causes a copy of object b and two copies of object c to be sent into the inner membrane 2. The object a is still present after the application, since it appears in the right-hand part of the rule. Rule r2 = a2 → c3in2 , instead, can be applied to a pair of objects a, and results in sending three copies of the object c into membrane 2. The initial state, as depicted, contains two copies of object a in membrane 1, and no objects in membrane 2. At the ﬁrst step, either rule r1 or r2 is applied.

Definition 6 (Stable MBSTA). A MBSTA M = (Λ, Q, F, f, g) is stable iﬀ for each multiset μ ∈ M(Λ) and n1 , n2 ∈ N with |μ| ≤ 2n1 ≤ 2n2 , it holds that μ is accepted by M at level n1 ⇐⇒ μ is accepted by M at level n2 . Proposition 2. Let K = (Λ, Q, F, f, g) be a stable BSTA. Then the MBSTA M = (Λ, Q, F, f, g) having the same structure is stable. Proof. Assume any multiset μ ∈ M(Λ) and any pair of naturals n1 , n2 ∈ N such that |μ| ≤ 2n1 ≤ 2n2 . We prove that if μ is accepted by M at level n1 then μ is accepted by M at level n2 , the converse case is analogous.

The length of w is denoted |w|. The ith element of w is denoted wi . We denote with Λ+ the set Λ∗ \ { }. As usual, a language over Λ is a subset L ⊆ Λ∗ . We denote M(Λ) the set of all the multisets with elements in Λ. The union of Systolic Automata and P Systems 19 multisets is denoted by ⊕, \ denotes both the diﬀerence between sets and the diﬀerence between multisets, and ∅ denotes both the empty set and the empty multiset. Moreover, we denote with P(I) the powerset of I, that is the set of all subsets of the set I.