CC BY
12
On Definability of Universal Graphic Automata by Their Input Symbol Semigroups
V. A. Molchanov, R. A. Farakhutdinov
Vladimir A. Molchanov, https://orcid.org/0000-0001-6509-3090, Saratov State University, 83 Ast-rakhanskaya St., Saratov 410012, Russia, molchanovva@mail.ru
Renat A. Farakhutdinov, https://orcid.org/0000-0002-2877-8557, Saratov State University, 83 Ast-rakhanskaya St., Saratov 410012, Russia, renatfara@mail.ru
Universal graphic automaton Atm(G, G') is the universally attracting object in the category of automata, for which the set of states is equipped with the structure of a graph G and the set of output symbols is equipped with the structure of a graph G' preserved by transition and output functions of the automata. The input symbol semigroup of the automaton is S(G, G') = End G x Hom(G, G'). It can be considered as a derivative algebraic system of the mathematical object Atm(G, G') which contains useful information about the initial automaton. It is common knowledge that properties of the semigroup are interconnected with properties of the algebraic structure of the automaton. Hence, we can study universal graphic automata by researching their input symbol semigroups. For these semigroups it is interesting to study the problem of definability of universal graphic automata by their input symbol semigroups — under which conditions are the input symbol semigroups of universal graphic automata isomorphic. This is the subject we investigate in the present paper. The main result of our study states that the input symbol semigroups of universal graphic automata over reflexive graphs determine the initial automata up to isomorphism and duality of graphs if the state graphs of the automata contain an edge that does not belong to any cycle.
Keywords: generalized Galois theory, automaton, graph, semigroup, isomorphism.
Received: 28.02.2019 / Accepted: 24.03.2019 / Published: 02.03.2020
This is an open access article distributed under the terms of Creative Commons Attribution License (CC-BY 4.0)
DOI: https://doi.org/10.18500/1816-9791-2020-20-1-42-50
INTRODUCTION
One of the main trends of the modern algebra is the generalized Galois theory, the basis of which was laid in the research of E. Galois and which is devoted to the study of mathematical objects by means of some derivative algebraic systems connected with the objects in a special way. Various algebraic systems, topological spaces, formal languages and many others were considered as initial mathematical objects, and automorphism groups and endomorphism semigroups of algebraic systems, homeomorphism groups and semigroups of continuous transformations of topological spaces, syntactic monoids of formal languages and many others were considered as derivative algebraic systems. Such studies for automorphism groups of algebraic systems, endomorphism semigroups of graphs, endomorphism rings of modules, and other derivative algebraic systems were very successfully carried out by B. I. Plotkin [1], A. G. Pinus [2,3], L. M. Gluskin [4,5], Yu. M. Vazhenin [6,7], A. V. Mikhalev [8] and many other algebraists. S. Ulam in his well-known book [9] has mentioned the problem of characterization of mathematical objects by their endomorphisms and automorphisms.
The generalized Galois theory also includes studies of structured automata, in which the systems of states and output symbols are objects of some category K (see, for example, review in [10]). In this case an automaton is considered as an initial mathematical object and the semigroup of input symbols is considered as a derivative algebraic system. For example, L. M. Gluskin, Yu. M. Vazhenin and other authors (see, review in [11]) investigated the endomorphism semigroups of graphs that can be considered as graphic semi-automata (i.e. automata without output symbols, for which the state systems are objects of the graph category Gr). In particular, Yu. M. Vazhenin in his studies [7] considered graphic semi-automata for graphs that contain an edge which does not belong to any cycle. The main result of the paper states that such graphic semi-automata are completely determined (up to isomorphism and duality of graphs) by their semigroups of input symbols.
In this paper we generalize this result for graphic automata, in which the systems of states and output symbols are objects of the graph category Gr. For any graphs G, G', in the category of all graphic automata with the state graph G and the output symbol graph G' there exists the universally attracting object Atm(G, G') which is called a universal graphic automata over the graphs G, G'. The main result of the paper — Theorem 1 — states that for a reflexive graph G that contains an edge which does not belong to any cycle and a reflexive graph G' the automaton Atm(G, G') is completely determined (up to isomorphism and duality of graphs) by its input symbol semigroup.
The main result of this paper was announced in [12].
1. PRELIMINARIES
As a preliminary, we introduce some basic terminology from the theory of semigroups, the theory of automata and the theory of graphs that will be used and state the necessary notation. The reader interested in further details may consult, for instance, A. H. Clifford and G. B. Preston [13], B. I. Plotkin [10], A. M. Bogomolov and V. N. Sa-lii [14], F. Harary [15].
Let X, Y be arbitrary sets and let p c X x Y be a binary relation. We put domp = {x G X : (3y G Y)(x,y) G p}, p-1 = {(y,x) G Y x X : (x, y) G p}. By a mapping of a set X to a set Y, denoted p : X ^ Y we mean a single-valued binary relation p c X x Y such that dom p = X. For an element x e X, the image of x under p is denoted by p(x). For a subset A c X, let us denote p(A) = {p(x) | x e A}.
A mapping p : X ^ {x} denoted cx is called a constant mapping of a set X to an element x. A mapping of a set X to itself is called a transformation of X. The identity transformation of a set X is denoted by AX. A one-to-one mapping p : X ^ Y satisfying p(X) = Y is called a bijection of X onto Y. In this case the inverse mapping p-1 : Y ^ X is defined by the formula p-1 (y) = x ^ p(x) = y.
For mappings p : X ^ Y, 0 : Y ^ Z, the composition p0 : X ^ Z is defined by the formula (p0)(x) = 0(p(x)) for x e X. Denote the Cartesian product p x p = p2, where P2(xi, x2) = (p(xi), (p(x2)) for xi, x2 G X.
Following [14], by a graph we mean an algebraic structure G = (X,p), where X is a non-empty set and p is a binary relation p c X x X that is called an adjunct relation. The elements of X and p are called vertices and edges, respectively. The edge (x, x) G p is called a loop. Vertices x,y G X are called adjacent if (x,y) G p or (y,x) G p. If e = (x,y) is an edge then e is said to join the vertex x with the vertex y. Edges e1 = (x1 ,y1 ),e2 = (x2,y2) are called adjacent if y1 = x2. A sequence of adjacent edges
(x0, xi), (#i ,x2), ..., (xn-2 ,xn-i), (xn-i ,xn) is called a path from the vertex x0 to the vertex xn. In case of equality x0 = xn, the path is called a cycle.
A graph G = (X, p) is called reflexive if its adjunct relation p is a reflexive binary relation, i.e. (x,x) G p for any x G X. Further, only reflexive graphs are considered. For a graph G = (X, p) the graph G = (X, p-1) is called the dual graph of G. Let G = (X, p), G' = (X', p') be graphs. A mapping ^ : X ^ X' is called a homomorphism of G to G', denoted ^ : G ^ G', if the following condition holds:
(Vx, y G X)((x, y) G p ^ (p(x), p(y)) G p').
We denote by Hom(G, G') the set of all homomorphisms of G to G'. Obviously, for any vertex x e X', satisfying (x,x) e p', the constant mapping cx : X ^ {x} is a homomorphism of G to G'.
A homomorphism ^ : G ^ G' is called an isomorphism of G onto G' if ^ is a bijection of X onto X' and the following condition holds:
(Vx, y G X)((x, y) G p ^ (p(x), p(y)) G p').
An isomorphism of G onto G' is called an anti-isomorphism of G onto G'. Graphs G, G' are called isomorphic (respectively, anti-isomorphic) if there is an isomorphism (respectively, an anti-isomorphism) of G onto G'.
A homomorphism of a graph G = (X, p) to itself is called an endomorphism of G. We denote by End G the semigroup of all endomorphisms of G under the composition. Obviously, the identity transformation AX of the vertex set X is an endomorphism of G. Moreover, for any x G X, the constant mapping cx : X ^ {x} is an endomorphism of G.
For graphs G, G', S(G, G') let us denote the semigroup End GxHom(G,G') equipped with the following binary operation [10]
(») • ) = (wi^i^
where (<p, , ) G End G x Hom(G, G').
We denote by Z(G,G') the set of all right zeros of the semigroup S(G,G') and by U(G,G') the set of all left identities of the semigroup S(G,G'). These sets are respectively defined in the semigroup S(G, G') by the formulas of the semigroup theory $(x) = (Vy)(yx = x) and ^(x) = (Vy)(xy = y). By analogy with the lemmas 2.12.3 [16], it is easy to prove the following results.
Lemma 1. For any reflexive graphs G = (X, p), G' = (X',p'), the semigroup S(G, G') satisfies the following conditions:
1) an element s G S(G,G') is a right zero of the semigroup S(G, G') if and only if there exist a G X, b G X' such that s = (ca, cb) for the constant mappings ca : X ^ {a}, cb : X ^ {b};
2) an element s G S(G, G') is a left identity of the semigroup S(G, G') if and only if s = (AX, for some ^ G Hom(G, G').
Lemma 2. Let G = (X,p), G' = (X',p') be reflexive graphs. Then the formula of the semigroup theory
E(x, y) = $(x) A $(y) A (Ve)(^(e) ^ xe = ye)
defines the binary relation £(G)G') on the semigroup S(G,G'), such that the following statements hold:
1) £(g,g') is an equivalence on the set Z(G, G') such that for any elements S,S1 G Z(G, G'), s = S1 (£(G,G')) if and only if s = (ca, Cx), S1 = (ca, cy), for some a G X and x, y G X';
2) for any right zero s = (ca ,cb) of the semigroup S(G, G') the equivalence class £(G,G') (s) is {(ca,cx) | x G X'}.
Following [10], by an automaton we mean a system A = (X, S, X',5, A) consisting of a set of states X, a semigroup of input symbols S, a set of output symbols X', a transition function 5 : X x S ^ X and an output function A : X x S ^ X' such that
5(x, st) = 5(5(x, s), t) and A(x, st) = A(5(x, s), t)
for any x G X and s, t G S.
For every s G S, we define the mappings 5s : X ^ X, As : X ^ X' by the formulas
5s(x) = 5(x,s), As(x) = A(x, s),
where x G X.
An automaton A = (X, S, X', 5, A) is said to be graphic if its sets X and X' are equipped with structures of graphs G = (X, p) and G' = (X',p') such that for every s G S the transition function 5s is an endomorphism of G and the output function As is a homomorphism of G to G' respectively. In this case we denote the automaton by
A = (G, S, G',5, A).
By a homomorphism of a graphic automaton A = (G, S, G', 5, A) into a graphic automaton A1 = (G1 ,S1,G/1,51, A1) we mean an ordered triplet y = (f, n,g), consisting of homomorphisms f : G ^ G1, n : S ^ S1, g : G' ^ G1, such that, for any x G X, s,t G S, the following conditions hold
n(s ■ t) = n(s) ■ n(t), f (5(x, s)) = 51 (f(x), n(s)), g(A(x, s)) = A1 (f (x), n(s)).
A homomorphism 7 = (f, n, g) of A to A1 is called an isomorphism of the automaton A onto the automaton A1 if f : G ^ G1, n : S ^ S1, g : G' ^ G1 are isomorphisms.
An important example of a graphic automaton is given by the following algebraic system
Atm(G, G') = (G, S(G, G'), G', 5°, A0),
where G = (X,p), G' = (X',p') are graphs and for every x G X, (p,0) G S(G, G') the equalities 50(x, (p,0)) = p(x), A°(x, (p,0)) = 0(x) hold. It is easy to verify that the automaton Atm(G, G') satisfies the following universal property: for any graphic automaton A = (G, S, G',5, A) there exists a homomorphism n : S ^ S(G,G') such that Y = (AX, n, AX') is a homomorphism of A to Atm(G,G'). For this reason the automaton Atm(G,G') is called [10] a universal graphic automaton over the graphs G, G'.
2. MAIN RESULT
The main result of this paper gives us the following solution of the problem of definability of universal graphic automata by their input symbol semigroups.
Theorem 1. Let G, G', G1, G1 be reflexive graphs, such that the graph G contains an edge which does not belong to any cycle, and Atm(G,G'), Atm(G1 ,G1) be the universal graphic automata over the graphs G, G' and G1, G1 respectively. Then the following conditions are equivalent:
1) the graphs G, G' are isomorphic to the graphs G1, G1 or to the dual graphs G1, G1 respectively;
2) the semigroups of input symbols S(G, G'), S(Gb Gi) of the automata Atm(G, G'), Atm(G1,G/1) are isomorphic;
3) the automaton Atm(G, G') is isomorphic to the automaton Atm(G1, G/) or to the automaton Atm(G1, G/) over the dual graphs G1, G/.
Proof. Obviously, 3) implies 1), 2). On the other hand, any isomorphisms f : G ^ G1, g : G' ^ G1 determine a mapping n : S(G, G') ^ S(G1, G1) by the formula
n) = (f2(p), (f x g)),
where (p,^) G S(G, G'). Indeed, by the definition, the image f2(p) = f-1 pf is an endomorphism of G1 and the image (f x g) = f-Vg is a homomorphism of G1 to G1. It is easy to verify that n is an isomorphism of the semigroup S(G, G') onto the semigroup S(G1, G1) such that the ordered triplet 7 = (f, n,g) is an isomorphism of the automaton Atm(G, G') onto the automaton Atm(G1,G1).
Similarly, any isomorphisms f, g of the graphs G, G' onto the dual graphs G1 , G'1 determine an isomorphism n : S(G, G') ^ S(G1,G1) by the formula n(p,^) = (f2(p), (f x g)(^)) (here (p>) G S(G, G')) such that the ordered triplet y = (f, n,g) is an isomorphism of the automaton Atm(G, G') onto the automaton Atm(G1,G1). Hence, 1) implies 3).
Let us now verify that 2) implies 1). Let n be an isomorphism of the semigroup S(G, G') onto the semigroup S(G1, G1). It is well-known that any isomorphism holds the true value of formulas of the semigroup theory. In particular, the isomorphism n preserves the true value of the formulas Ф(х), Ф(ж) and £(x,y). It follows that n maps the set Z(G, G') of right zeros of the semigroup S(G, G') onto the set Z(G1, G1) of right zeros of the semigroup S(G1, G1) and the set U(G, G') of left identities of the semigroup S(G, G') onto the set U(G1, G1) of left identities of the semigroup S(G1, G1). Moreover, by lemma 2 the Cartesian product n2 maps the equivalence 6 = £{g,g') on the semigroup S(G, G') onto the equivalence 61 = £(Gl)G'1) on the semigroup S(G1, G1). In view of lemma 1, for any a G X, a' G X', there exist a1 G X1, a1 G X1 such that n(ca, ca') = (cai, cai). Hence, n maps the equivalence class 6(ca, ca') onto the equivalence class 61 (cai, cai). It follows that the formulas f (a) = a1 and ga(a') = a1 determine the mappings f : X ^ X1 and ga : X' ^ X1 (a G X) such that the equality holds
n(ca,ca') = (Cf(a), Cga(a')). (1)
It is easy to verify that f is a bijection of X onto X1 and for every a G X, ga is a bijection of X' onto X1.
Consider (p,^) G S(G, G') and for arbitrary a G X we denote
p(a) = b, ^(a) = d. (2)
By definition of S(G, G') for any y G X' the equalities hold
(ca,cy) • (p,^) = (cap,ca^) = (c^(a), c^(a)).
Then (2) is equivalent to the equality (ca,cy) • (p,^) = (cb, cd).
By definition of the isomorphism n
n(ca, cy) • n(p, 0) = n(cb, cd).
(3)
Let us denote n(p,0) = (p',0'). By (1),
n(ca,cy) = (cf(a) ,cga(y)), n(cb,cd) = (cf(b),cgb(d)).
Then (3) implies
(cf(a) ,cga(y)) • (p',0') = (cf (b) ,cgb(d)).
On the other hand
(4)
Then (4) implies that fp' = pf, p' = f 1pf = f2 (p) and 0' = 0^, where the mapping 0^ : X1 ^ X1 is defined by the formula
It follows that the mapping n1 = f2 is a bijection of the set End G onto the set End G1. Moreover, it is obvious that f2 preserves the composition. Hence, for any p,p1 G End G the equality n1 (pp1) = n1 (p)n1 (p1) holds. Then the mapping n1 is an isomorphism of the semigroup End G onto the semigroup End G1. By the condition of the theorem G is a reflexive graph with an edge (u0,v0) G p which does not belong to any cycle. It follows from [7] that the mapping f is an isomorphism of the graph G onto the graph G1 or onto the dual graphs G1.
Suppose that f is an isomorphism of G onto G1. Hence (f(u0),f(v0)) G p1 is an edge of the graph G1, which does not belong to any cycle. Next we show that, for any point a G X the mapping ga is an isomorphism of the graph G' onto the graph G1. It is easy to verify that for vertices x0,y0 G X' the condition (x0,y0) G p' is equivalent to the condition 02(u0,v0) = (x0,y0) for some homomorphism 0 G Hom(G, G'). Indeed, if 02(u0,v0) = (x0,y0) for some homomorphism 0 : G ^ G' then (x0,y0) G p' by the definition of graph homomorphism. On the other hand, let (x0,y0) G p'. Define a mapping 0 : X ^ X' by the following formula:
(5)
y0, if there exists a path from the vertex v0 to the vertex u, x0, otherwise.
MareMamna
We show that ^ is a homomorphism of the graph G to the graph G'. Let (s,t) G p. If there exists a path from the vertex vo to the vertex s, then by the condition (s,t) G p there exists a path from the vertex vo to the vertex t. Then by our definition of ^ the conditions ^(s) = ^(t) = y0 hold. Since the graph G is reflexive, the conditions (^(s),^(t)) = (y0,y0) G p' hold. If there is no a path from the vertex vo to the vertex s, then by our definition of ^ the condition ^(s) = x0 holds. Since ^(t) G {x0,y0} and the graph G is reflexive, the condition (^(s),^(t)) G p' holds.
Thus, in any case the condition (s,t) G p implies (^(s),^(t)) G p'. Hence ^ G Hom(G, G').
Since (f(uo),f(vo)) G pi is an edge of the graph Gi, which does not belong to any cycle, it is similar for vertices x0,y0 G Xi that the condition (x0,y0) G pi is equivalent to the condition (f(uo),f(vo)) = (x0,y0) for some homomorphism ^ G Hom(Gi, Gi).
Suppose that for vertices x0,y0 G X' the condition (x0,y0) G p' holds. It is equivalent to the condition (uo, vo) = (x0,y0) for some ^ G Hom(G,G'). Hence (caG S(G,G') and, as it was proved before, n((ca,^)) = (c/(a)) for the homomorphism G Hom(Gi,Gi). By our definition of the mapping for any x G X the following equalities hold:
^ (f (x)) = gca (x) (^ (x)) = ga(^(x)).
In particular, for uo, vo G X the following equalities hold:
(f (uo)) = ga(^(uo)) = ga(xo), (f (Vo)) = ga(^(Vo)) = ga(y0).
It follows that the homomorphism ^Ca G Hom(Gi, Gi) maps the ordered pair (f(uo),f(vo)) G pi to the ordered pair (ga(xo),ga(yo)) and the condition (ga(xo) , ga(yo)) G pi holds.
Let us assume the converse case — for some vertices xo,yo G Xi the condition (xo, yo) G pi holds. It is equivalent to the condition ^2(f(uo),f(vo)) = (xo,yo) for some homomorphism G Hom(Gi,Gi). Hence, (c/(a)) G S(Gi,Gi) and by our definition of the isomorphism n : S(G,G') ^ S(Gi, Gi) there exists G S(G,G') such that
= (c/(a),^i). From the previously proven part follows that c/(a) = f2and
= ^. Since c/(a) = f-i^f the condition ^ = fc/(a)f-i = ca holds. Hence
x'o = ^i (f (uo)) = (f (uo)) = ga(uo)), ^(uo) = g- K),
yo = ^i(f (Vo)) = ^Ca (f (Vo)) = ga(Vo)), ^(Vo) = ga-i(yo)
and the homomorphism ^ G Hom(G,G') maps the ordered pair (uo,vo) G p to the ordered pair (g-i(x'o),g-i(yo)). Thus, (g-i(x/o),g-i(yo)) G p' and ga is an isomorphism of the graph G' onto the graph Gi.
Similarly, if f is an isomorphism of the graph G onto the dual graphs G1, then all mappings ga are isomorphisms of the graph G' onto the dual graphs Gi, since in this case the condition (uo,vo) G p is equivalent to the condition (f(vo),f(uo)) G pi and hence (xo,yo) G p' is equivalent to (ga(yo),ga(xo)) G pi. Therefore, 2) implies 1). □
CONCLUSION
The results obtained shows that universal graphic automata over the graphs from a wide class of graphs are completely determined (up to isomorphism and the duality of
graphs) by their input symbol semigroups. Consequently, on the one hand, the characteristics of the input symbol semigroups of universal graphic automata must be defined by the characteristics of the automata, and on the other hand, the characteristics of these semigroups must characterize the automata to same extent. This approach allows us to study the properties of such automata by studying properties of their input symbols semigroups. References
1. Plotkin B. I. Groups of Automorphisms of Algebraic Systems. Groningen, The Netherlands, WoLters-Noordhoff Publ., 1972. 502 p.
2. Pinus A. G. On the elementary equivalence of derived structures of free lattices. Russian Math. (Iz. VUZ), 2002, vol. 46, no. 5, pp. 42-45.
3. Pinus A. G. Elementary Equivalence of Derived Structures of Free Semigroups. Algebra and Logic, 2004, vol. 43, no. 6, pp. 408-417. DOI: https://doi.org/10.1023/B:ALL0.0000048829.60182.48
4. Gluskin L. M. Semigroups and rings of endomorphisms of linear spaces. Izv. Akad. Nauk SSSR. Ser. Mat., 1959, vol. 23, iss. 6, pp. 841-870 (in Russian).
5. Gluskin L. M. Semi-groups of isotone transformations. Uspekhi Mat. Nauk, 1961, vol. 16, iss. 5 (101), pp. 157-162 (in Russian).
6. Vazhenin Yu. M. Ordered sets and their inf-endomorphisms. Math. Notes, 1970, vol. 7, iss. 3, pp. 204-208. DOI: https://doi.org/10.1007/BF01093116
7. Vazhenin Yu. M. The elementary definability and elementary characterizability of classes of reflexive graphs. Izv. Vyssh. Uchebn. Zaved. Mat., 1972, no. 7, pp. 3-11 (in Russian).
8. Markov V. T., Mikhalev A. V., Skornyakov L. A., Tuganbaev A. A. Rings of endomor-phisms of modules and lattices of submodules. J. Soviet Math., 1985, vol. 31, iss. 3, pp. 3005-3051. DOI: https://doi.org/10.1007/BF02106808
9. Ulam S. M. A Collection of Mathematical Problems. NewYork, Interscience, 1960. 150 p.
10. Plotkin B. I., Greenglaz L. Ja., Gvaramija A. A. Algebraic structures in automata and databases theory. River Edge, NJ, World Scientific Publ. Co., 1992. 296 p.
11. Molchanov V. A. Semigroups of mappings on graphs. Semigroup Forum, 1983, vol. 27, pp. 155-199. DOI: https://doi.org/10.1007/BF02572738
12. Molchanov V. A., Farakhutdinov R. A. On universal graphic automata. In: Komp'iuternye nauki i informatsionnye tekhnologii: materialy mezhdunarodnoy nauchnoy konferentsii [Computer Science and Information Technologies: Materials of the Int. Sci. Conf.]. Saratov, Izdatel'skii tsentr "Nauka", 2018, pp. 276-279 (in Russian).
13. Clifford A. H., G. B. Preston. The algebraic theory of semigroups. Providence, RI, Amer. Math. Soc., 1964. 224 p.
14. Bogomolov A. M., Salii V. N. Algebraicheskie osnovy teorii diskretnykh sistem [Algebraic foundations of the theory of discrete systems]. Moscow, Nauka, 1997. 368 p. (in Russian).
15. Harary F. Graph Theory. Reading, MA, Addison-Wesley, 1969. 274 p.
16. Molchanov V. A. A universal planar automaton is determined by its semigroup of input symbols. Semigroup Forum, 2011, vol. 82, iss. 1, pp. 1-9. DOI: https://doi.org/10.1007/ s00233-010-9256-8
Cite this article as:
Molchanov V. A., Farakhutdinov R. A. On Definability of Universal Graphic Automata by Their Input Symbol Semigroups. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2020, vol. 20, iss. 1, pp. 42-50. DOI: https://doi.org/10.18500/1816-9791-2020-20-1-42-50
УДК 519.713.2
Об определяемости универсальных графических автоматов своими полугруппами входных сигналов
В. А. Молчанов, Р. А. Фарахутдинов
Молчанов Владимир Александрович, доктор физико-математических наук, профессор кафедры теоретических основ компьютерной безопасности и криптографии, Саратовский национальный исследовательский государственный университет имени Н. Г. Чернышевского, Россия, 410012, г. Саратов, ул. Астраханская, д. 83, molchanovva@mail.ru
Фарахутдинов Ренат Абуханович, студент кафедры теоретических основ компьютерной безопасности и криптографии, Саратовский национальный исследовательский государственный университет имени Н. Г. Чернышевского, Россия, 410012, г. Саратов, ул. Астраханская, д. 83, renatfara@mail.ru
Универсальный графический автомат Atm(G, G') — это универсально притягивающий объект в категории автоматов, у которых множество состояний наделено структурой графа G и множество выходных сигналов — структурой графа G', сохраняющимися функциями переходов и выходов автоматов. Полугруппа входных сигналов такого автомата имеет вид S(G, G') = End G х Hom(G, G'). Она может рассматриваться как производная алгебраическая система математического объекта Atm(G,G'), которая содержит полезную информацию об исходном объекте. Хорошо известно, что свойства такой полугруппы взаимосвязаны со свойствами алгебраической структуры автомата. Следовательно, универсальные графические автоматы можно изучать путем исследования их полугрупп входных сигналов. Для таких полугрупп представляет интерес проблема определяемости универсальных графических автоматов своими полугруппами входных сигналов: при каких условиях полугруппы входных сигналов универсальных графических автоматов будут изоморфны. В данной работе мы исследовали эту проблему. Основной результат нашей работы состоит в том, что универсальные графические автоматы над рефлексивными графами определяются полугруппами своих входных сигналов с точностью до изоморфизма и двойственности графов, если в графе состояний автомата найдется дуга, не входящая ни в один орцикл.
Ключевые слова: обобщенная теория Галуа, автомат, граф, полугруппа, изоморфизм. Поступила в редакцию: 28.02.2019 / Принята: 24.03.2019 / Опубликована: 02.03.2020 Статья опубликована на условиях лицензии Creative Commons Attribution License (CC-BY4.0)
Образец для цитирования:
Molchanov V. A., Farakhutdinov R. A. On Definability of Universal Graphic Automata by Their Input Symbol Semigroups [Молчанов В. А., Фарахутдинов Р. А. Об определяемости универсальных графических автоматов своими полугруппами входных сигналов] // Изв. Сарат. ун-та. Нов. сер. Сер. Математика. Механика. Информатика. 2020. Т. 20, вып. 1. С. 42-50. DOI: https://doi.org/10.18500/1816-9791-2020-20-1-42-50
