Abstract
Hypergraphic automata are automata with state sets and input symbol sets being hypergraphs which are invariant under actions of transition and output functions. Universally attracting objects of a category of hypergraphic automata are automata Atm(H1,H2). Here,H1is a state hypergraph,H2is classified as an output symbol hypergraph, andS= EndH1× Hom(H1,H2) is an input symbol semigroup. Such automata are called universal hypergraphic automata. The input symbol semigroupSof such an automaton Atm(H1,H2) is an algebra of mappings for such an automaton. Semigroup properties are interconnected with properties of the algebraic structure of the automaton. Thus, we can study universal hypergraphic automata with the help of their input symbol semigroups. In this paper, we investigated a representation problem of universal hypergraphic automata in their input symbol semigroup. The main result of the current study describes a universal hypergraphic automaton as a multiple-set algebraic structure canonically constructed from autonomous input automaton symbols. Such a structure is one of the major tools for proving relatively elementary definability of considered universal hypergraphic automata in a class of semigroups in order to analyze interrelation of elementary characteristics of universal hypergraphic automata and their input symbol semigroups. The main result of the paper is the solution of this problem for universal hypergraphic automata for effective hypergraphs withp-definable edges. It is an important class of automata because such an algebraic structure variety includes automata with state sets and output symbol sets represented by projective or affine planes, along with automata with state sets and output symbol sets divided into equivalence classes. The article is published in the authors' wording.
Highlights
Automata theory is among major computer science branches studying data conversion devices
The main focus of our research is universal hypergraphic automata. Their subautomata cover all homomorphic images of hypergraphic automata (Theorem 1). Such universal automaton for any hypergraphs H1 and H2 is the automaton Atm(H1, H2) = (H1, S, H2, δ◦, λ◦), where S is the input symbol semigroup consisting of all pairs s = (φ, ψ) of endomorphisms φ of the hypergraph H1 and homomorphisms ψ from the hypergraph H1 to the hypergraph H2, δ◦(x, s) = φ(x) is the transition function and λ◦(x, s) = ψ(x) is the output function (where x is a vertex of H1 and s = (φ, ψ) is an element of S)
An automaton A = (X, S, Y, δ, λ) is a hypergraphic automaton if state set X and output symbol set Y are such hypergraphs H1 = (X, LX) and H2 = (Y, LY ) respectively that for every fixed input symbol s ∈ S the transformation δs : X −→ X is an endomorphism of H1 and the mapping λs : X −→ Y is a homomorphism from H1 to H2
Summary
Automata theory is among major computer science branches studying data conversion devices. Their subautomata cover all homomorphic images of hypergraphic automata (Theorem 1) Such universal automaton for any hypergraphs H1 and H2 is the automaton Atm(H1, H2) = (H1, S, H2, δ◦, λ◦), where S is the input symbol semigroup consisting of all pairs s = (φ, ψ) of endomorphisms φ of the hypergraph H1 and homomorphisms ψ from the hypergraph H1 to the hypergraph H2, δ◦(x, s) = φ(x) is the transition function and λ◦(x, s) = ψ(x) is the output function (where x is a vertex of H1 and s = (φ, ψ) is an element of S). The main result of our current study is Theorem 2 It shows the important property of input symbol semigroup of universal hypergraphic automaton which allows to construct an isomorphic copy of the original automaton using input symbol semigroup. The authors would like to thank the reviewer for his constructive comments on the paper
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
