From Automata to Multiautomata via Theory of Hypercompositional Structures

Abstract

In this paper, we study two important problems related to quasi-multiautomata: the complicated nature of verification of the GMAC condition for systems of quasi-multiautomata, and the fact that the nature of quasi-multiautomata has deviated from the original nature of automata as seen by the theory of formal languages. For the former problem, we include several new conditions that simplify the procedure. For the latter problem, we close this gap by presenting a construction of quasi-multiautomata, which corresponds to deterministic automata of the theory of formal languages and is based on the operation of concatenation.

Preface to the Special Issue “Algebraic Structures and Graph Theory”

“…Furthermore, they prove that for any positive integer n greater than 1, there exists an infinite quotient hyperfield of characteristic n. A similar result holds for the C-characteristic. The manuscripts [11,12] cover some applications of hypercompositional algebra to automata theory. Massouros et al [11] study the binary state machines with magma of two elements as their environment.…”

“…Massouros et al [11] study the binary state machines with magma of two elements as their environment. Another aspect of automata theory is discussed in [12], where the authors propose several conditions for simplifying the verification of the GMAC condition for systems of quasi-multiautomata. Furthermore, using the concatenation, they construct quasi-multiautomata corresponding to the deterministic automata of the theory of formal languages.…”

State Machines and Hypergroups

“…The free monoid of the words generated by an alphabet Σ can be endowed with the B-hypergroup structure, and so become a join hyperringoid [21,[72][73][74], which is named linguistic hyperringoid [14,21,73,74]. If the B-hypergroup is fortified with a strong identity [31], which is necessary for the theory of formal languages and automata [14,21], then the join hyperring comes into being [72][73][74]. Proof.…”

“…There followed more papers by the same author and Ch. Massouros, e.g., [13][14][15][16][17][18][19][20][21], as well as other researchers such as J. Chvalina [22][23][24][25][26][27][28], L. Chvalinová [22], Š. Hošková-Mayerová [24,25], M. Novák [26][27][28][29][30][31][32], S. Křehlík [26,27,[29][30][31]33], M.M. Zahedi [34], M. Ghorani [34,35], D. Heidari and S. Doostali [36], R.A. Borzooei et al [37].…”

New Aspects in the Theory of Complete Hypergroups

“…Starting in 1934, when F. Marty introduced the hypergroup as a new algebraic structure extending the classical one of the group, and satisfying the same properties, i.e., the associativity and reproductivity, the theory of hypercompositional structures (also called hyperstructure theory) has experienced a rapid growth, succeeding to impose itself as a branch of Abstract Algebra. Currently, this theory offers a strong background for studies in algebraic geometry [1], number theory [2], automata theory [3], graph theory [4], matroids theory [5], and association schemes [6], to list just some of the research fields where hypercompositional structures are deeply involved. These structures are non-empty sets endowed with at least one hyperoperation, i.e., a multivalued operation resulting in a subset of the underlying set, usually denoted as • : H × H −→ P * (H), where (P) * (H) denotes the set of non-empty subsets of H. A non-empty set H equipped with a hyperoperation that satisfies: (i) the associativity: (x • y) • z = x • (y • z) for any x, y, z ∈ H, where (x • y) • z = u∈x•y u • z and x • (y • z) = v∈y•z x • v, and (ii) the reproductivity:…”