By Cristian S. Calude

Professor Jozef Gruska is a widely known computing device scientist for his many and vast effects. He was once the daddy of theoretical desktop technology study in Czechoslovakia and one of the first Slovak programmers within the early Nineteen Sixties. Jozef Gruska brought the descriptional complexity of grammars, automata, and languages, and is among the pioneers of parallel (systolic) automata. His different major learn pursuits contain parallel structures and automata, in addition to quantum info processing, transmission, and cryptography. he's co-founder of 4 typical sequence of meetings in informatics and in quantum details processing and the Founding Chair (1989-96) of the IFIP expert crew on Foundations of laptop Science.

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**Extra info for Computing with New Resources: Essays Dedicated to Jozef Gruska on the Occasion of His 80th Birthday**

**Sample text**

Wn one at a time. The amplitude distribution after processing the i-th symbol is denoted as si , with si (y) = x∈Q si−1 (x) · δ (wi , x, y) 14 K. Balodis for every y ∈ Q. After the n-th symbol, the end marker $ is processed in the same way, obtaining the final amplitude distribution sn+1 . , if x∈F sn+1 (x) p λ, then the word w is said to be accepted; otherwise, it is rejected. Here we show an optimal ultrametric automaton for counting. Theorem 9. For each n and each prime p there exists a 1Up FA that recognizes Cn with 2 states.

In that case τa is considered as a mapping of Q into Q, that is as a transformation of Q. The set of transformations of Q induced by τ is a monoid with the multiplication deﬁned as above, the transition monoid T (A) of A. The transformation τλ is the identity element of T (A). Inputs u and v of A are said to be equivalent if and only if τu = τv . As the transition monoid T (A) is a submonoid of the monoid of all mappings of Q into Q, the full transformation monoid TQ of Q, the order (cardinality) of T (A) is at most |Q||Q| .

Let R = (Λ, Q, F, f, g) be a RMBSTA, and let ΠR = (Λ ∪ C ∪ {T}, ∅, μ1 , R1 ) be the corresponding acceptor P system. Then, for any multiset μ ∈ M(Λ) the following implications hold: 1. μ is accepted by R at level n =⇒ μ can accepted by ΠR in no more than n + 2 steps; 28 R. Barbuti et al. x→y if y = g(x), x ∈ Λ xy → z if z = f (x, y); x ∈ Qo , y, z ∈ Qo ∪ Qo x →z if z = f (x, ); x, z ∈ Qo ∪ Qo x →T if x ∈ F ∩ (Qo ∪ Qo ) x →F if x ∈ (Qo ∪ Qo ) \ F →T if ∈F →F if ∈ /F → x→♦ if x ∈ Qo ∪ Qo xT → ♦ if x ∈ Qo ∪ Qo xF → ♦ if x ∈ Qo ∪ Qo ♦→♦ Fig.