Miklos Kresz- Soliton automata with constant external edges

8/3/2019 Miklos Kresz- Soliton automata with constant external edges

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Soliton automata with constant external edges

Miklos Kresz

Department of Computer Science, Gyula Juhasz Faculty of EducationUniversity of Szeged, Szeged, Hungary

[email protected]

Abstract. Soliton automata are the mathematical models of certainpossible molecular switching devices. Both from theoretical and practicalpoint of view, it is a central question to describe soliton automata withconstant external edges. Extending a result of Dassow and Jurgensen,we characterize soliton automata in this special case.

1 Introduction

Molecular computing is an emerging field in current theoretical and application-oriented research ([25]) as well. The idea of molecular memories goes back toFeynmans pioneer paper (cf. [14]), in which he proposes building small machinesand then using those machines to build still smaller machines and so on, downto the molecular level. During the past decades several promising concepts havebeen worked out for unconventional computing; among these nonlinear mediathat exhibit self-localized mobile patterns in their evolution are potential can-didates for the role of universal dynamical computers.The computational modelcalled soliton cellular automata (see eg. [23], [24] and [26]) uses soliton inter-

actions in the design of collision-based logic gates. The word soliton (solitarywave) is applied to certain types of waves traveling relatively large distance withlittle energy loss. For a survey of unconventional architectures on the principlesof molecular computing, see [1].

Other alternatives of molecular computers are based on the design of con-ventional digital circuits on the molecular level ([10]). The idea of this approachis: if we build up the electronic elements chemically from the molecular level, itwould be possible to make circuits thousands of times smaller. These molecularcircuits would use chemical molecules as electronic switches and be intercon-nected by some sort of ultra-fine conducting wires. One interesting possibility ofthese conductors was proposed by Carter ([9]) and is about using single strandsof the electrically conductive plastic polyacetylene. In this case, soliton wavesare induced by electrons travelling along polyacetylene in little packets. Hence,molecular scale electronic devices constructed from molecular switches and poly-acetylene chains are called soliton circuits.

The practical research in soliton circuits (see e.g. [15] and [16]) has evokedthe need to develop an applied mathematical arsenal in order to obtain a de-tailed understanding of the behavior of these circuits. The mathematical modelof soliton circuits called soliton automata was introduced in [11], but it was not


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until [4] that matching theory ([22]) was recognized as the fundamental theoret-ical background for the study of this model. The underlying object of a soliton

automaton is the topological model of the corresponding molecule chain calledsoliton graph. In this model, a soliton graph comes with a perfect internal match-ing, i.e. a matching that covers all the vertices with degree at least two. Thesevertices called internal model carbon atoms, whereas vertices with degreeone called external represent a suitable chemical interface with the outsideworld.

Soliton graphs and automata have been systematically studied on the groundsof matching theory (see e.g. [3], [4], [6] and [8]). Perhaps the most significantcontribution along this line is [6], where soliton graphs are decomposed intoelementary components, and these components are grouped into pairwise disjointfamilies based on how they can be reached by alternating paths starting fromexternal vertices. This decomposition is carried over to soliton automata in [5],using quasi-direct and 0-products of their component automata. The abovestructural results have significant algorithmic consequences for the verificationand simulation of soliton circuits, as it was outlined in [19]. Moreover, the basicbuilding elements of soliton circuits were characterized by the description ofcomplete systems of soliton automata (cf. [18])

It seems to be a fundamental question to determine the computational powerof soliton automata. Several special cases have been described with respect totheir transition monoids. (see e.g. [11], [12], [13]). In this paper we generalize theresults of [12] where deterministic soliton automata with a single external vertexwere characterized. First we extend this result by considering nondeterministicautomata. It will turn out that the transition monoids of soliton automata witha single external vertex are special null monoids. Then, as a generalization, wedescribe the class of soliton automata with constant external edges by products

of automata characterized in the first part of the paper.Though studying soliton automata with constant external edges is interesting

in itself, the main reason for considering this special case is that these automataplay a central role in the decomposition of soliton automata (cf. [5]). With thehelp of this decomposition, the descriptional complexity of soliton automata canbe significantly reduced ([20]).

2 Perfect internal machings in graphs

As it was mentioned in the introduction, the fundamental technique for the anal-ysis of soliton automata is based on the concept of perfect internal matchings.Therefore, in this section we review some of the basic notions concerning graphsand matchings. Our notation and terminology will be compatible with that of[22], except that point and line will be replaced by the more conventionalterminology vertex and edge, respectively.

By a graph we shall mean a finite undirected graph in the most general sense,i.e. with multiple edges and loops allowed. For a graph G, V(G) and E(G) willdenote the set of vertices and the set of edges of G, respectively. The concept


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of walk, cycle and path can be defined in a usual way. If all edges in a walk aredistinct, the walk is called a trail. The subtrail of a trail between vertices vi

and vj is denoted by [vi, vj ], while the notation 1

will be used to representthe reverse of .

If the vertex set of a graph G can be partitioned into two disjoint non-emptysets, V(G) = A B, such that all edges of G connect a vertex of A to a vertexof B, we call G bipartite and refer to A B as the bipartitionof G.

A vertex v V(G) is called external if its degree d(v) is one, internal ifd(v) 2, and isolated otherwise. The sets of external and internal vertices of Gwill be denoted by Ext(G) and Int(G) respectively. External edges are those ofE(G) that are incident with at least one external vertex, and internal edges arethose connecting two internal vertices. Graph G is called open if it has at leastone external vertex, otherwise G is called closed. For the demonstration of theabove terminology see Figure 1, where the external vertices are u and v, while

the external edges are e and f.A matching M of graph G is a subset ofE(G) such that no vertex ofG occurs

more than once as an endpoint of some edge in M. Again, it is understood bythis definition that loops are not allowed to participate in M. The endpointsof the edges contained in M are said to be covered by M. A perfect internalmatching is a matching that covers all of the internal vertices.

An edge e E(G) is allowed (mandatory) if e is contained in some (respec-tively, all) perfect internal matching(s) of G. Forbidden edges are those that arenot allowed. We shall also use the term constant edge to identify an edge that iseither forbidden or mandatory.

Now consider the graph G in Figure 1 again. It is easy to see that the set{e, h1, h2} determines a perfect internal matching in G and g being its unique

forbidden edge.By the usual definition, a subgraph G ofG is just a collection of vertices and

edges of G. Since in our treatment we are particular about external vertices, wedo not want to allow that new external vertices (i.e. ones that are not present inG) emerge in G. Therefore, whenever this happens, so that vertex v Int(G)becomes external in G, we shall augment G with a loop edge around v. Thisaugmentation will be understood automatically in all subgraphs of G. Finally,for a subgraph G and matching M of G, M(G) will denote the restriction of Mto G.

Let G be an open graph and M be a perfect internal matching of G, fixedfor the rest of this section. An edge e E(G) is said to be M-positive (M-negative) if e M (respectively, e M). An M-alternating path (cycle) in G isa path (respectively, even-length cycle) stepping on M-positive and M-negativeedges in an alternating fashion. An M-alternating loop is an odd-length cyclehaving the same alternating pattern of edges, except that exactly one vertexhas two negative edges incident with it. Let us agree that, if the matching Mis understood or irrelevant in a particular context, then it will not be explicitlyindicated in these terms. An external alternating path is one that has an externalendpoint. If both endpoints of the path are external, then it is called a crossing.


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Fig. 1. Example graph with perfect internal matchings.

An alternating path is positive if it is such at its internal endpoints, meaning thatthe edges incident with those endpoints are positive. In Figure 1, = u,e,w,f,vis an alternating crossing and = z1, l1, z4, h2, z3, l2, z2, h1, z1 is an alternatingcycle with respect to the perfect internal matching M = {f, l1, l2}.

An internal vertex v ofG is called accessible from external vertex w in M (orsimply v is M-accessible from w), if there exists a positive external M-alternatingpath connecting w and v. Furthermore, an alternating cycle is said to be M-accessible from w if some of its vertices is accessible from w in M. Generally itis not true that if a vertex is accessible from an external vertex w in a perfectinternal matching, then it is accessible from w in all perfect internal match-ings. Nevertheless, as it was proved in [6], the accessibility without specifyingthe external vertex is invariant with respect to perfect internal matchings. It is

therefore meaningful to say that vertex v is accessible in G without specifyingthe perfect internal matching M and the external vertex w.

An M-alternating unit is either a crossing or an alternating cycle with respectto M. Switching on an alternating unit amounts to changing the sign of eachedge along the unit. It is easy to see that the operation of switching on anM-alternating unit creates a new perfect internal matching S(M, ) for G.Moreover, as it was proved in [4], every perfect internal matching M of G canbe transformed into any other perfect internal matching M by switching ona group of pairwise disjoint alternating units. A set of pairwise disjoint M-alternating units will be called an M-alternating network, and the alternatingnetwork determined by the symmetric difference of perfect internal matchingsM and M will be denoted by N(M, M). We will also refer to N(M, M) asthe mediator alternating network between M and M. The following importantobservation on alternating units was proved in [4].

Proposition 1. ([5]) An edge e of a graph G having a perfect internal matchingis not constant if and only if there exists an alternating unit passing through ein every perfect internal matching of G.


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3 Soliton graphs and automata

In this section, following [11], we introduce the concept of soliton automata asthe mathematical model of switching at the molecular level by so-called solitonvalves. Towards this goal, we first define the topological model of the underlyingstructure, which is a graph representing a molecule chain in which solitons travelalong. In this simple model, vertices correspond to the atoms or certain groupsof atoms, whereas the edges represent chemical bonds or chains of bonds. It isassumed that the molecules consist of carbon and hydrogen atoms only, and thatamong the neighbors of each carbon atom there exists a unique one to whichthe atom is connected by a double bond. The above property is captured byperfect internal matchings, where the edges contained in the given matchingcorresponding to the double bonds. Therefore, a soliton graph is defined as anopen graph having a perfect internal matching. Since perfect internal matchingsrepresent the states of the corresponding molecule chain, it is justified to referthem as states of the given soliton graph.

A soliton graph G models the underlying molecular structure as follows: Eachinternal vertex v represents a C atom or a C-H group depending on whether d(v)is 3 or 2, respectively. A single (double) edge (v, w) in a given state representsa (CH)-chain with alternating double and single bonds which connects the Catoms of v and w and which begins and ends with a single (repectively, double)bond. As the length of such chains does not affect the logic of the model wedraw them as length 1 chains; physico-chemical reasons may require differentlengths for actual realizations. Finally, external vertices represent the connectionto surrounding structures. Figure 2 shows an example of a soliton graph and apossible chemical interpretation.

Fig. 2. A soliton graph with one of its interpretations

It is clear that in the above model the degree of each internal vertex is atmost 3. This restriction was applied in the original definition ([11], but omit-ting the above condition makes it possible to use more general techniques andconstructions for soliton automata. Nevertheless, it can be proved (cf. [17]) thatsoliton graphs and automata in this generalized sense (soliton graphs without


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restrictions on the degree of the internal vertices) are equivalent to the originalconcept. Therefore, in the rest of the paper, by soliton graph, we mean the ones

without degree restrictions.For the study of the logical aspects of soliton switching we need to give a

graph theoretic formalization of the state transitions induced by soliton waves.Ignoring the physico-chemical details, the effect of a soliton wave propagatingalong a polycetylene chain is to exchange all single and double bonds. This logicalaspect is captured by the concept of soliton walk. Intuitively, a soliton walk is abacktrack-free walk which starts and ends at an external vertex, and alternateson matching covered and uncovered edges. However, the status of the traversededges are exchanged dynamically step by step while making the walk.

Before the mathematical definition we give an informal description of solitonwalks through the example in Figure 3, which illustrates the effects both in thegraph model of a given molecule chain.

M

M

Fig. 3. A soliton walk

The initial state of the system is represented by perfect internal matching M.The walk = v1, e1, v4, e4, v5, e5, v6, e6, v4, e4, e2, v2 corresponding to the givensoliton wave results in the sequence in the figure. In each of them the positionof the soliton is indicated by an arrow. We note that, though during the walk, astep does not necessarily results in a perfect internal matching, but by the timethe walk is finished, a new state M of G is reached.

The walk starts at vertex v1

and after traversing edge e1

, the double bond isexchanged for single one; thus the status of the corresponding edge is exchanged.Then, the status of the traversed edge is exchanged dynamically. During thewalk, if the soliton is about to continue its way on an uncovered edge like inthe second step at vertex v4 , then it might have several alternatives for the nextstep among the adjacent edges (e.g. both e4 and e6 could be chosen). However, ifa situation of two adjacent covered edges occurs like in the third step at vertex


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v5 , then the walk must continue on the appropriate covered edge. (Remember,that a soliton walk is backtrack-free.)

The concept of soliton walks can be formalized in soliton graphs in the follow-ing way. The collection of external alternating walks in G with respect to somestate M, and the concept of switching on such walks are defined recursively asfollows.

(i) The walk = v0ev1, where e = (v0, v1) with v0 being external, is an externalM-alternating walk, and switching on results in the set S(M, ) = M {e}.(The operation is symmetric difference of sets.)

(ii) If = v0e1 . . . envn is an external M-alternating walk ending at an internalvertex vn, and en+1 = (vn, vn+1) is such that en+1 S(M, ) iff en S(M, ),then = en+1vn+1 is an external M-alternating walk and

S(M, ) = S(M, ){en+1}.

It is required, however, that en+1 = en, unless en S(M, ) is a loop.

It is clear by the above definition that S(M, ) is a perfect internal matching iffthe endpoint vn of is external, too. In this case we say that is a soliton walk.

Example. Consider the graph G of Fig.1 again, and let M = {e, h1, h2}.Then = uewgz1h1z2l2z3h2z4l1z1gwfv is a possible soliton walk from u to vwith respect to M. Switching on then results in S(M, ) = {f, l1, l2}.

Graph G gives rise to a soliton automaton AG = (SG, X X, ), the setSG of states of which consists of the perfect internal matchings of G. The inputalphabet X X for AG is the set of all (ordered) pairs of external vertices in G i.e. X = Ext(G) , and the transition function is defined by (M, (v, w)) ={S(M, )| is an M-alternating soliton walk from v to w }. Nevertheless, if no

soliton walk exists from v to w in M, then (M, (v, w)) = {M}. Finally, as usual,we extend the transition function for any word y (X X) including theempty word by (M, ) = {M} and (M,ya) = ((M, y), a) with a X X.

Example. Consider the graph G in Fig.1 again. This graph is a soliton graphhaving states: seh = {e, h1, h2}, s

el = {e, l1, l2}, s

fh = {f, h1, h2}, and s

fl =

{f, l1, l2}. The transitions of AG are the following:

(seh, (u, v)) = (sel , (u, v)) = {s

fh, s

fl },

(sfh, (v, u)) = (sfl , (v, u)) = {s

eh, s

el },

(seh, (v, u)) = {sfh}, (s

el , (v, u)) = {s

fl },

(sfh, (u, v)) = {seh}, (s

fl , (u, v)) = {s

el },

(seh, (u, u)) = {sel }, (s

el , (u, u)) = {s

eh},

(sf

h

, (v, v)) = {sf

l

}, (sf

l

, (v, v)) = {sf

h

}.

As an example, the transition seh sfl on input (u, v) is induced by the soliton

walk:uewgz1h1z2l2z3h2z4l1z1gwfv.

One of the central goals of this paper is to describe constant soliton automata,i.e. soliton automata associated with a graph having constant external edges only


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(constant soliton graphs), in terms of products of smaller automata. Each of theseautomata will be either a soliton automaton itself or constructed from a soliton

automaton in a certain way. Therefore the input alphabet of each automaton isa Cartesian power X2 over a set X. For automata with alphabet of this type westrengthen the concept of isomorphism.

Definition 1. Let X be an alphabet and for i = 1, 2, let Ai = (Si, X X, i)be an automaton. We say that A1 and A2 are strongly isomorphic if there existsa bijection : S1 S2 which satisfies the equation

{(s) | s 1(s, (x, x))} = 2((s), (x, x

))

for every s S1 and every x, x X.

Recall from [11] that an edge e of G is impervious if there is no external

alternating walk passing through e in any state of G. It is clear that imperviousedges have no effect on the operations of soliton automata. Thus, without loss ofgenerality, we can restrict our investigation to soliton graphs without imperviousedges. Therefore, throughout the paper, G will denote a soliton graph withoutimpervious edges.

Moreover, note that, without loss of generality, we can assume that all con-stant external edges of a soliton graph G are mandatory. Indeed, attaching anextra mandatory edge to each forbidden external edge of G results in a graphG for which AG and AG are strongly isomorphic. We shall use this assumptionthroughout the paper without any further reference.

In [5] the transitions of soliton automata have been characterized in termsof alternating trails and networks. First we quote the result describing the tran-sitions between distinct states.

Definition 2. Let M be a state of G and v, w Ext(G). An M-transitionnetwork from v to w is a nonempty M-alternating network such that allelements of , except one crossing from v to w if v = w, are alternating cyclesaccessible from v in M.

Theorem 1. ([5]) Let M, M be distinct states of soliton automaton AG =((SG, (X X), ). Then for any pair of external vertices (v, w) X X,M (M, (v, w)) holds iff N(M, M) is an M-transition network from v tow.

For the characterization of self-transitions, transitions from a state to itself,we need the following concepts.

A soliton trail is an external alternating walk, stepping on positive andnegative edges in such a way that is either a path, or it returns to itself only inthe last step, traversing a negative edge. The trail is a c-trail (l-trail) if it doesreturn to itself, closing up an even-length (respectively, odd-length) cycle. Thatis, = 1 + 2, where 1 is a path and 2 is a cycle. These two components of are called the handle and cycle, in notation, h and c. See Fig. 4 for illustrating


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a)c-trail b)l-trail

Fig. 4. Soliton trails.

the above concepts. In this figure, as well as in the further ones throughout thepaper, double lines indicate edges that belong to the given matching.

An M-alternating double soliton c-trail from external vertex v is a pair of

M-alternating soliton c-trails = (1, 2) from v such that E(1h) E(2c) = ,E(2h) E(

1c) = , and either

1c =

2c or V(

1c) V(

2c) = . The maximal

common external subpath denoted by h of 1h and of2h is called the handle

of . The internal endpoint of h is called the branching vertex of . Figure 5presents simple examples for the above definition.

Fig. 5. Example for double soliton c-trails.

Theorem 2. [5] For any state M of soliton automaton AG = ((SG, (X X), )and for any external vertex v X ofG, M (M, (v, v)) iff one of the followingconditions holds:

(i) G does not contain an M-alternating soliton c-trail from v.(ii) G contains an M-alternating soliton l-trail from v.

(iii) G contains an M-alternating double soliton c-trail from v.

The following important observation on soliton l-trails has been proved in[19].


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Proposition 2. ([19]) LetM be a state of soliton graph G andv Ext(G) suchthat each edge of G is traversed by an external M-alternating trail starting from

v. Then G contains a soliton l-trail from v iff G is non-bipartite.

Making use of the above result, we can refine Theorem 2. To this end wewill use the following notation: For any soliton graph G, state M of G andv Ext(G), let G[M, v] denote the graph determined by the edges traversedby an M-alternating trail starting from v. Since for any maximal external M-alternating trail , M() is clearly a perfect internal matching in the graphdetermined by , G[M, v] is also a soliton graph with Ext(G[M, v]) = Ext(G) V(G[M, v]) and M(G[M,v]) S(G[M, v]).

Theorem 3. For any state M of soliton automatonAG = ((SG, (XX), ) and for any external vertex v X of G, M (M, (v, v)) iff one of the followingconditions holds:

(a) G[M, v] is a non-bipartite graph.

(b) G[M, v] is a bipartite graph containing an M(G[M,v])-alternating doublesoliton c-trail from v.

(c) G[M, v] is a bipartite graph not containing anM(G[M,v])-alternating cycle.

Proof. Immediate by Theorem 2 and Proposition 2.

4 Nondeterministic soliton automata with a single

external vertex

As a first step towards the characterization of nondeterministic soliton automata

with a single external vertex, the transition between distinct states is describedbelow as a straightforward consequence of Theorem 1.

Theorem 4. If G is a soliton graph with a single external vertex v, then M2 (M1, (v, v)) holds for any distinct states M1, M2 of AG = (SG, X X, ).

Proof. Immediate by Theorem 1 and by the observation that the mediator al-ternating network between M1 and M2 consists of alternating cycles accessiblefrom v in M1.

For the analysis of self-transitions, according to Theorem 2, we need to inves-tigate soliton trails. In reaching the above goal the following concept will play acentral role.

Definition 3. States M1 and M2 of soliton graph G are called compatible, ifM1 and M2 cover the same external vertices.

Proposition 3. Let M and M be compatible states of soliton graph G and let be an M-alternating crossing between external vertices v and w. Then thereexists an M-alternating crossing connecting v and w.


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Proof. Let 1, . . . , k (k 0) be the alternating cycles constituting the mediatoralternating network between M and M; and construct the graph G = +

1 + . . . + k. Then it is clear that Ext(G

) = {v, w}, M

(G) S(G

), and theexternal edges incident with v and w are non-constant in G. Therefore, makinguse of Proposition 1, we easily obtain that v and w are connected by an M(G)-alternating crossing, as required.

Proposition 4. LetM and M be compatible states of a soliton graph G. Thenfor any external vertex v Ext(G), G[M, v] = G[M, v] holds.

Proof. By symmetry, it is enough to prove the claim in one direction. To thisend let e be an edge of G traversed by an M-alternating trail starting from v.If e is external, then the statement follows directly from Proposition 3. Supposenow that e is internal. In that case one endpoint of e, let it be denoted by w, isM-accessible from v either by or by the appropriate prefix of . Now let extend

G by a new external edge (w, u) such that u V(G). By the above observationthere is an M-alternating crossing between v and u in G + (w, u). Now applyingProposition 3, we obtain that v and u is connected by an M-alternating crossing. Since M and M are compatible, we conclude that [v, w] is a positive M-alternating alternating path in G. Therefore either or + e will provide anM-alternating trail starting from v and traversing e; as required.

Corollary 1. Let M1 and M2 be compatible states of soliton graph G and v Ext(G). ThenG contains an M1-alternating soliton l-trail from v iff it containsan M2-alternating soliton l-trail from v.

Proof. Immediate by Propositions 2 and 4.

Proposition 5. Let G be a bipartite soliton graph, M be a state of G and v Ext(G). ThenG contains anM-alternating double solitonc-trail fromv iff thereexists anM-alternating double solitonc-trail from v for all states M compatiblewith M.

Proof. Let = (1, 2) be an M-alternating double soliton c-trail from v withbranching vertex w, be an M-alternating cycle, and M = S(M, ). Since anymediator alternating network between compatible states consists of alternatingcycles only, if we prove that an M-alternating double soliton c-trail from v alsoexists, then we are ready by a straightforward induction argument on ||. Forthis goal, consider first the case of V() V(h) = .

If V() V() = , then our statement is trivial. Otherwise, for k = 1, 2, letk

denote the suffix ofk from w to its internal endpoint, and if V(k) V() =, then let k denote the prefix of

k from v to the first vertex common with. (See an example in Figure 6.) Then an M-alternating double soliton c-trail = (1, 2) can be constructed in the following way: If V() V(k) = for k = 1, 2, then let 1h =

1 ,

2h =

2 , and

1c =

2c = . Otherwise, i.e.

V() V(k) = and V() V(3k) = for some k {1, 2}, let

kh =

k ,

kc = , and 3k = 3k.


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Fig. 6. The case V() V(h) = in the proof of Proposition 5

By the preceding paragraph, we can assume for the rest of the proof that

V() V(h) = . In this case, starting from v, let u denote the first vertex atwhich h overlaps with , and let u be the vertex of V() V() such thatthe positive M-alternating path = [u, u] is maximal as a subpath in .(Observe that h[v, u] is negative at the u end.) Moreover, let

be the negativeM-alternating subpath of connecting u and u, i.e. E() = E() \ E(). SeeFigure 7 for an example.

Fig. 7. The case V() V(h) = in the proof of Proposition 5

From now on, assume that is an M-alternating double soliton c-trail fromv, such that the subpath constructed above is maximal. Then the followingholds.

Claim A is edge-disjoint from .


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In order to prove the above claim, let us assume by contradiction, that startingfrom u, the next vertex u of having the property that u V() and u

is different from u.(See Figure 7 again) It is clear that

[u

, u

] is a negativealternating path, so that u and u belong to distinct bipartition class of G.Furthermore, we may suppose without loss of generality that u V(1). Weknow by the choice of that 1[v, u] is positive at its u end, but it is also easyto observe that if u V(k) (k {1, 2}), then k[v, u] is negative at its u

end. Indeed, if k[v, u] terminated in a positive edge at u, then both u andu would be accessible from v in the bipartite graph G + (u, u). However, u

and u belong to distinct bipartition class, which is a contradiction.

For k = 1, 2, now let vk denote the internal endpoint of kh, and let v be the

vertex adjacent to u by positive edge in M. (See Figure 7 again) We will showthat an M-alternating double soliton c-trail = (1, 2) starting from v can beconstructed such that the positive M-alternating path [u, v] is a subpath of

1

. For this, we distinguish four cases.Case A/1: u V(kh) (k {1, 2}), and either u

V(1h) or 1c =

2c .

Note that, in this case, if 1c = 2c , then k = 2. Now it is easy to check that

the trails defined below constitute an M-alternating double soliton c-trail suchthat [u, v] is a positive M-alternating subpath of 1, as required.

1h = 1[v, u] + [u, u] + kh[u

, vk]2h =

kh

1c = 2c =

kc .

Case A/2: u V(kc ) (k {1, 2}) such that either u V(1h), or

1c =

2c with

k = 2.

Then let1h

= 1[v, u] + [u, u],2h =

kh,

and 1c = 2c =

kc .

Again, considering all possible alternatives of this case, we obtain that =(1, 2) is a double soliton c-trail with the required properties.

Case A/3: u, u V(1c ).

Now a suitable = (1, 2) is defined as follows.1h =

1h,

2h = 2h,

1c = [v1, u

] + [u, u] + 1[u, v1].and

2c = 1c , if

1c =

2c

2c = 2c , if 1c = 2c .Case A/4: u V(1c ), u

V(2h), and 1c =

2c .

Now the construction of = (1, 2) below is represented in Figure 8.1h =

1h,

2h = 2h[v, u

],1c =

2c =

[v1, u] + [u, u] + 2[u, v1].


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Fig. 8. Case A/4 in the proof of Proposition 5.

Now considering all the possible combinations of the locations of u and u,below we summarize that Cases A/1 A/4 are indeed sufficient to cover allalternatives.(Remember that u V(1).)

(i) Ifu V(kh) (k {1, 2}) and u V(1h), then consider Case A/1.

(ii) If u V(kh) (k {1, 2}) and u V(1c ), then consider Case A/4

(1c = 2c ) and Case A/1 (

1c =

2c ).

(iii) If u V(kc ) (k {1, 2}) and u V(1h), then consider Case A/2.

(iv) If u

V(k

c ) (k {1, 2}) and u

V(1

c ), then consider Case A/3(k = 1) and Case A/2 (k = 2).

Therefore we obtained in all possible cases that [u, v] is a positive M-alternating subpath of 1. However, the length of [u, v] is greater than thatof , since [u, v] is constructed as + [u, u] + (u, v). The above factcontradicts the choice of , thus Claim A is proved.

By the above claim, we can suppose for the rest of the proof that is edge-disjoint from . As earlier, assume that u V(1) and for i {1, 2}, let videnote the internal endpoint of ih. We will construct an M

-alternating doublesoliton c-trail = (1, 2) in the subgraph determined by and . For this, wemust deal with three cases and several subcases.

Case 1: v1 = v2.In this case 1c = 2c , for which we use the notation c. Now starting from

v, let y denote the last vertex of 1h such that y is incident with an edge inE(2h)\E(

1h), and let x denote the last vertex of

1h preceding y with x V(

2h).

(See Figure 9.) Below we give the construction of = (1, 2), for which, basedon the location of u, we distinguish four subcases.

Subcase 1a: u V(1h[v, x]).


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Fig. 9. Case 1 in the proof of Proposition 5

Then1h = h[v, u] +

+ 1h[u, v1],

2h = h[v, u] + + 1h[u

, x] + 2h[x, v2]1c =

2c = c.

Subcase 1b: u V(1h[x, y]).Then

1h = h[v, u] + + 1h[u

, v1],

2

h = h[v, u] +

+ (

1

h)

1

[u

, x] +

2

h[x, v2]1c = 2c = c.

Subcase 1c: u V(1h[y, v1]).Then

1h = h[v, u] + + 1h[u

, v1],2h = h[v, u] +

+ (1h)1[u, y]

1c = c,2c =

1h[x, y] + (

2h)1[y, x].

Subcase 1d: u V(1c ).

In this case, let denote the negative M-alternating subpath from u to v1in c determined by the edges not contained in

.

Then

1h = h[v, u] + + ()1[u, y],2h = h[v, u] +

+ + (1h)1[v1, y]

1c = 2c =

1h[x, y] + (

2h)1[y, x].

Now it is easy to check that = (1, 2) is indeed a suitable M-alternatingdouble soliton c-trail in all of the above cases.


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Case 2: v1 = v2 and 1c =

2c . In this case consider the bipartite graph G

determined by and , and apply the method of [7] contracting the redexes

of G

. Recall from [7] that a redex r consists of two adjacent edges e = (u, z)and f = (z, w) such that u = w are both internal and d(z) = 2. Contractingr in G means creating a new graph Gr from G

by deleting z and merging uand w into one vertex. Now applying the above method for all redexes of G inan iterative way, we obtain a graph G1 having no redexes. Then let M1 (M

1)

denote the restriction ofM (respectively, M) to the edges ofG1. It is easy to seethat is contracted to an M1-alternating double soliton c-trail = (

1, 2) suchthat 1h and

2h have the same internal endpoint. Therefore, according to Case

1, an M1-alternating double soliton c-trail also exists, which obviously becomesan M-alternating double soliton c-trail after unfolding all the redexes.

Case 3: 1c = 2c .

To handle this case, we need to further break it down into two subcases.

(Remember that u

V(1

.)Subcase 3a: u V(1h).

Consider the subgraph Gh determined by 1h and 2h. Then v1 and v2 are

obviously accessible in Gh, consequently there exists an M(Gh)

-alternating path

1 (2) from v to v1 (respectively, v2). Therefore the external alternating trails1 = 1 +

1c and

2 = 2 + 2c form an M

-alternating double soliton c-trailfrom v.

Subcase 3b: u V(1c ).Now starting from v, let x denote the last vertex of 1h which is also on

2h, and let denote the negative M-alternating subpath from u to v1 in

1c

determined by the edges not contained in . Then a suitable M-alternatingdouble soliton c-trail = (1, 2) is defined as follows.

1h = h[v, u] + + ()1[u, x] + 2h[x, v2],2h = h[v, u] +

+ + (1h)1[v1, x] +

2h[x, v2]

1c = 2c =

2c .

In summary, Cases 1 3 cover all the possible alternatives and we obtained inall cases that an M-alternating double soliton c-trail can be constructed, asrequired. Therefore the proof is complete.

By the above result it is meaningful to say for a bipartite soliton graph G witha single external vertex v that G contains a double soliton c-trail withoutspecifying any state.

Now we are ready to prove our main result concerning self-transitions.

Theorem 5. LetM, M

be compatible states of soliton automatonAG = (SG, XX, ) and v X be an external vertex of G. Then M (M, (v, v)) iff M (M, (v, v)).

Proof. Let Gv denote the subgraph G[M, v] = G[M, v] (see Proposition 4) and

apply Theorem 3 for Gv. It is clear that if condition (c) holds with respectto M or M, then M = M, and there is nothing to prove. The statement is


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also obvious if Gv is non-bipartite, while in other case the theorem follows fromProposition 5.

Making use of the above results, we obtain a characterization of soliton au-tomata with a single external vertex.

Definition 4. Let A = (S,X,) be an automaton such that its alphabet is asingleton, i.e. X = {x}. We say that A is a full (semi-full) automaton if for eachs S, (s, x) = S (respectively, (s, x) = S\ {s} with |S| > 1). Moreover, a fullautomaton with a single state is called trivial.

Theorem 6. LetG be a soliton graph with a single external vertex v. Then AGis either a full or a semi-full automaton. Moreover, AG is semi-full iff G is abipartite graph without double soliton c-trails.

Proof. Since any maximal alternating trail in G is necessarily either an l-trail ora c-trail, G is either non-bipartite or it contains alternating cycle with respect toany of its states. Therefore the argument is straightforward by using Theorems2 and 4 with Propositions 2 and 5.

Having characterized the structure of soliton graphs and automata with asingle external vertex, we can describe the transition monoids of these automata.Since it was proved in [12] that the transition monoid in deterministic case isisomorphic with the symmetric group of order 2 (or order 1, in the trivial case ofan automaton having a single state), we will assume that the considered solitonautomata are not deterministic.

Nevertheless, before stating the closing result of this section, we briefly reviewthe necessary concepts from the theory of automata and semigroups. For any

automaton A = (S,X,) and w X, we can define the relation w inducedby w on S, i.e. (s, s) w iff s

(s, w) for s, s S. Then the set T(A) ofrelations w on Swhich are induced by some w X

, with the usual compositionof relations is a monoid T(A), the transition monoid of A.

A monoid (A, ) with |A| 2 is called a null monoidif there exists an elementa A such that x y = a for all x, y A \ {e}, where e denotes the identityelement. In that case a is called the zero element. It is clear that a null monoidwith two elements A consists of the identity element and the zero element isuniquely determined up to isomorphism, which is called the trivial null monoid.

Now we are ready to state the final result of this section.

Theorem 7. Let G be a soliton graph with a single external vertex v such that

AG is not deterministic. Then T(AG) is a null monoid with at most 3 elements.Moreover, T(AG) is nontrivial iff G is a bipartite graph without double solitonc-trails.

Proof. The identity element of T(AG) is the relation induced by the emptyword. If G is nonbipartite or bipartite with a double soliton c-trail, then basedon Theorem 6, the relation induced by (v, v) is S2G, which clearly serves as


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a zero element. Based on the above fact, T(AG) has no additional elements,consequently T(AG) is trival.

Consider now the case of G being bipartite without double soliton c-trails.Then applying Theorem 6 again, we obtain that the relation induced by (v, v)is v = {(q, q)|q = q SG}. However, since AG is not deterministic, it is easyto see that SG 3, and for n 2, (v, v)

n induces S2G. Therefore, we obtainedthat T(AG) is indeed a null-monoid with 3 elements.

5 Elementary decomposition of soliton graphs

Having characterized soliton automata with a single external vertex, our goalis to describe the class of constant soliton automata in terms of products offull automata. For this end, we will make use of the structure theory of solitongraphs and automata worked out in [5] and [6]. In this section we provide a brief

summary of the most significant results obtained in [6]. Again, let us fix a solitongraph G for the forthcoming discussion.

Recall from [4] and [22] that a graph is elementary if its allowed edges form aconnected subgraph containing all the external vertices. In general, the allowededges of G determine a number of connected components as subgraphs in G.The full subgraphs of G induced by these components are called the elementarycomponents of G. An elementary component is called external if it containsexternal vertices, and internal if this is not the case. A mandatory elementarycomponent is a single mandatory edge e E(G), which might have a loop aroundone or both of its endpoints. It is easy to see that each external elementarycomponent of a constant soliton graph is mandatory.

The concept ofcanonical equivalence was originally introduced for elementary

graphs only (cf. [4], [22]). In [6], it was proved that the following extension ofthe concept results in an equivalence on Int(G) for any soliton graph G.

Let v, w Int(G) be internal vertices. Then u v if u and v belong to thesame elementary component and an extra edge e connecting u and v becomesforbidden in G + e. The classes determined by are called canonical classes andthe corresponding canonical partition of Int(G) is denoted by P(G).

The structure of elementary components in a soliton graph G has been anal-ysed in [6]. To summarize the main results of this analysis, we first need toreview some of the key concepts introduced in that paper. An internal elemen-tary component C is one-way if all external alternating paths (with respect toany perfect internal matching M) enter C in vertices belonging to the samecanonical class of C. This unique class, as well as the vertices belonging to thisclass, are called principal. Furthermore, every external elementary componentis considered a priori one-way (with no principal canonical class, of course). Anelementary component is two-way if it is not one-way.

Example The graph of Fig. 10. has five elementary components, among whichD and E are mandatory external, while C1, C2 and C3 are internal. ComponentC3 is one-way with the canonical class {u, v} being principal, while C1 and C2are two-way.


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DE

Fig. 10. Elementary components in a soliton graph

Let C be an elementary component ofG, and M be a state. An M-alternatingC-ear is a negative M-alternating path or loop having its two endpoints, butno other vertices, in C. The endpoints of the ear will necessarily be in the samecanonical class of C.

We say that elementary component C is two-way accessible from componentC with respect to any perfect internal matching M, in notation CC, if C iscovered by a C-ear. As it was shown in [6], the two-way accessible relationshipis matching invariant. A family of elementary components in G is a block ofthe partition induced by the smallest equivalence relation containing . A familyF is called external if it contains an external elementary component, otherwiseF is internal. It was proved in [6] that any family contains a unique one-wayelementary component called the root of the family.

Example Our example graph in Fig.10. has three families: F1 = {E, C1, C2}, F2 ={D}, F3 = {C3}. Families F1 and F2 are external, whereas F3 is internal. Theroots of the families are E, D and C3.

For two distinct families F1 and F2, F2 is said to follow F1, in notationF1 F2, if there exists an edge in G connecting any non-principal vertex in F1with a principal vertex belonging to the root of F2. The reflexive and transitiveclosure of is denoted by

. As a main result of [6] it was shown that the

relation

is a partial order among the families, by which the external familiesare maximal elements.

6 Constant soliton automata

In this section, synthesizing the results on the characterization of soliton au-tomata with a single external vertex and the structure theory of soliton graphs,we will describe the class of constant soliton automata in terms of products offull automata. As a first step, for constant soliton automata we establish theconnection between soliton transitions and the structure theory presented inSection 5. Let us fix a constant soliton graph G for the forthcoming discussion.


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Definition 5. For any external vertex v of G, let Gv denote the full subgraphof G induced by the vertices belonging to elementary components of families F

for which Fv

F, where Fv denotes the external family containing v.

Proposition 6. For any external vertex v and state M of G, graph Gv is iden-tical with G[M, v].

Proof. Adapting Lemma 3.8 in [5], we obtain that an edge e is viable by anexternal alternating trail starting from v iff e is incident with a non-principalvertex of an elementary component belonging to Gv.

Definition 6. Let v be an external vertex of G and let {C1, . . . , C n} (n > 0)be the set of the elementary components of Gv with C1 being its mandatoryexternal elementary component. Graph Gv is a component-chain graph if it canbe decomposed in the chain-form Gv = C1 + (w1, v2) + C2 + (w2, v3) + . . . +(wn1, vn)+Cn such that for each i [n1], (wi, vi+1) E(G) with wi V(Ci)

and vi+1 V(Ci+1). Furthermore, Gv is called rigid if it does not contain adouble soliton c-trail.

Now we are ready to characterize the self-transitions with respect to Gv.

Proposition 7. Let v be an external vertex of G, M be a state of G and let denote the transition function of AG. ThenM (M, (v, v)) iffGv is a bipartiterigid component-chain graph.

Proof. The If part is straightforward by Theorem 3 and Proposition 6. Inorder to prove the Only if part, observe first that Gv must be rigid bipartiteby Theorem 3 and Proposition 6 again. In that case each family in Gv consistsof a single elementary component, in other case the given family would span

a nonbipartite subraph (see [5]). Therefore if Gv was not a component chain-graph, then either two families are connected by two edges or there exists afamily F such that two distinct families F1 and F2 are originated from F by .However, since any external alternating path reaching the families in the orderdetermined by ([6]), in both cases two distinct external alternating paths canbe constructed from v to the referred vertices of the branching family. Becauseof the same reason both paths can be extended to appropriate soliton c-trailshaving cycles below the branching family. Therefore we would obtain a doublesoliton c-trail in both cases, which is a contradiction.

With the help of the above concepts, we can define the appropriate automataproduct by which constant soliton automata will be characterized.

Definition 7. Consider the automata At = (St, Xt, t) (t [m], m N) andlet X be an alphabet. Suppose that there exists a partial order among X andthe automata A1, . . . Am such that the set of maximal elements is equal to X.Furthermore, let Y be a subset of X such that for each y Y, the elementsAj with Aj y constitute a chain. Then the X2-chain product of A1, . . . Amwith respect to Y and feedback function is the automaton A = (S, X X, )defined in the following way.


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(a) S = S1 . . . Sm(b) The function = (1, . . . m) is given in such a way that for each i [m],

i : X X Xi {} is a mapping subject to the following conditions: If x and y are distinct elements of X, then i(x, y) = . If i(x, x) = for an x X, then Ai x.

(c) For every x, y X and s = (s1, . . . , sm) S,

(s, (x, y)) =

1(s1, 1(x, y)) . . . m(sm, m(x, y)) \ {s}, if x = y Y1(s1, 1(x, y)) . . . m(sm, m(x, y)), otherwise

We will show that any constant soliton automaton can be decomposed intofull automata by an X2-chain product. In order to reach the above goal, we needthe following simple observation on essentially elementary soliton automata, bywhich we mean automata associated with a graph consisting of an elementarycomponent D and that of an external edge attached to D.

Proposition 8. Any essentially elementary soliton automaton AG is either afull or a semi-full automaton.

Proof. Since any essentially elementary soliton automaton has a single externalvertex, the claim is immediate by Theorem 6.

For the product construction, we will need the following technical notion.

Definition 8. Let A = (S, X X, ) be an automaton. The self-transition ex-tensionofA is the automaton Ae = (S, XX, e), where for any state s SofA

and for any input (x, x) XX, e(s, (x, x)) =

(s, (x, x)), if x = x

(s, (x, x)) {s}, otherwise

Now we can give the necessary decomposition.

Proposition 9. LetC1, . . . , C k be the internal elementary components of G andfor j [k], let Aj denote the full automaton with |SCj | number of states. ThenAG is strongly isomorphic with an X2-chain product A of A1, . . . , Ak, whereX = Ext(G).

Proof. During the proof we will use the notations of Definition 7 with the sameparameters. Furthermore, it is clear by part (c) of Definition 7 that we only needto prove that AeG and A

e are strongly isomorphic.Let Y denote the set of external vertices v for which Gv is a rigid bipartite

component-chain graph. Moreover, let be determined by relation

. Moreprecisely, for i, j [k], if Fi and Fj denote the families containing Ci and Cj ,

respectively, then Aj Ai iff Fj Fi and Aj v iff Fv Fj for v X andv belonging to Fv.

Now for each 1 j k, let Cj be a soliton graph obtained from Cj byattaching a new external edge to some vertex of Cj . Consider the self-transitionextension AeC

j= (SCj , {x, x},

ej ) of automaton ACj = (SCj , {x, x}, j) defined

by ej (M, (x, x)) = j(M, (x, x)) {M} for any state M SCj .


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By Proposition 8, Ae(Cj) is strongly isomorphic with the full-automaton Aj .Capitalizing on the above fact, because of technical reasons, when we refer to the

transition function (or the states) of Aj, we will mean the ones of Ae

(Cj). This

assumption will be used throughout the proof without any further reference.

Let G and denote the transition function of AG and that of the productautomaton A. Moreover, let y1, y2 Ext(G) and M SG be arbitrary. Sincethe mapping (M) = (M(C1), . . . , M (Ck)) is clearly a bijection between SG andSC1 . . . SCk , we only have to prove that

{(M) | M eG(M, (y1, y2))} == e((M), (y1, y2))

Then based on part (c) of Definition 7, it is clear that the following holds forthe right side of the above basic equality.

e((M), (y1, y2)) == e1(M(C1), z1) . . .

ek(M(Ck), zk) (1)

Now, in order to study the transition functions and the left side of the basicequality, we distinguish between two cases.

Case 1: No soliton walk exists from y1 to y2 with respect to M.

It is easy to see that the above situation is equivalent to the condition ofy1 = y2.

By Definition 7, we obtain that for each i [k], zi = . Consequently,ei (MCi , zi) = {MCi} (i [k]) holds, which results in the following.

{(M) | M eG(M, (y1, y2))} = {(M)} == e1(MC1 , z1) . . .

ek(MCk , zk) (2)

Now the basic equation is obtained by combining (1) and (2).

Case 2: There is a soliton walk from y1 to y2 with respect to M. It is clearlyequivalent with the condition of y1 = y2

Now, for any M-alternating network , let S(M, ) denote the state obtainedfrom M by switching on the alternating units constituting , and let E()denote the set of edges contained in some alternating unit of . Furthermore,let T(M, y1, y2) denote the set of M-transition networks from y1 to y2, and letT(M, y1, y2) = T(M, y1, y2) .

Finally, for any elementary component C, let TC(M, y1, y2) denote the set ofM-alternating networks for which T(M, y1, y2) and E() E(C).

Then, making use of Theorem 1, we obtain the following for the left side of(1).

{(M) | M = S(M, ), T(M, y1, y2)} =

= {S(MC1 , 1) | 1 TC1(M, y1, y2)} . . .

. . . {S(MCk , k) | k TCk

(M, y1, y2)} (3)Now comparing (2) and (3), we conclude that the proof becomes complete, if weshow that for any i [k],

{S(MCi , i)|i TCi

(M, y1, y2)} = ei (MCi , zi) (4)


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For the above goal, let i [k] be arbitrary. Then applying Definition 7, we obtainthat zi = (xi, xi) iff Ci belongs to Gy1 . Now equation (4) easily follows, and the

proof is complete.

Analyzing the construction of the product in the above proposition, it is clearthat the mandatory internal elementary components have role only in the self-transitions. More exactly, if a loop is present in a mandatory internal elementarycomponent C, then for each vertex v with Gv containing C, there exists a self-transition from v in all states. Therefore, if for each external vertex v identifiedby the above way, we add a loop around the internal vertex of the mandatoryexternal elementary component containing v, we obtain a graph associated witha strongly isomorphic automaton. Repeating the above process for all loopsbelonging to a mandatory internal elementary component, we obtain a graph G

with AG and AG being strongly isomorphic.Now applying the decomposition described in the proof of Proposition 9

for AG and removing the trivial component automata (having a single state)from the product, the resulted product automaton A will be evidently stronglyisomorphic with the original product automaton, if AG is nontrivial (havingat least two states). We will refer to this automaton as the reduced productautomaton of AG. Based on the above argument, we obtained the followingresult.

Proposition 10. Let AG is nontrivial, and let ArG denote its reduced product

automaton. Then AG and ArG are strongly isomorphic.

It is easy to see that the above claim does not hold for trivial soliton au-tomata. Indeed, in that case all the internal elementary components are manda-tory, thus the reduction procedure results in an empty automaton. Since trivial

constant soliton automata are well-characterized, we restrict our analysis to non-trivial automata in the rest of the paper.

Corollary 2. Any nontrivial constant soliton automaton is strongly isomorphicwith an X2-chain product of nontrivial full automata, where X = Ext(G)

Proof. Immediate by Propositions 9 and 10.

In order to obtain a complete characterization of constant automata, now weshow that the reverse direction is also true.

Proposition 11. Let X be an arbitrary alphabet, and let A1, . . . , Ak be non-trivial full automata. Then any X2-chain product of A1, . . . , Ak is strongly iso-

morphic with a nontrivial constant soliton automata.

Proof. For each 1 i k, let Gi be a closed elementary graph consisting ofas many parallel edges as the number of states of Ai. It is clear that each edgeuniquely determines a state, Now construct a soliton graph G in such a way thatone vertex of Gi is distinguished as principal vertex and each elementary graphwill form an internal family. The external families are determined by mandatory


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edges corresponding to the elements ofX. Now the graph is built in such a waythat the edges connecting distinct elementary components must correspond to

the relation in such a way that the relation

in the resulted graph reflectsthis order. The elementary components below the vertices of Y are supposed tobe connected by a single edge in such a way that a chain is built. Finally, a loopis added to each external mandatory component corresponding to an element ofX\ Y.

It is easy to see that the family structure of the resulted constant solitongraph G will reflect the partial order and applying Proposition 9 for AG, weobtain a strongly isomorphic product automaton. The proof is now complete.

Combining Corollary 2 and Proposition 11, as a final result, we obtain thecharacterization of nontrivial constant soliton automata.

Theorem 8. The class of nontrivial constant soliton automata and the class ofautomata obtained by X2-chain products of nontrivial full automata coincide upto strong isomorphism.

7 Conclusion

We have provided a complete characterization of soliton automata with constantexternal edges. We have described the underlying graph structure of the specialcase of single external vertex in terms of soliton trails. With the help of theseresults we proved that any nondeterministic soliton automaton in this specialcase is either a full or a semi-full automaton, the latter holds only if the underly-ing graph is bipartite and it does not contain a double soliton c-trail. Then, weconcluded that the transition monoid of such a (not deterministic) automatonis a null monoid with 2 or 3 elements, depending on whether the given automa-ton is full or semi-full. Finally, generalizing the concept, we introduced constantsoliton automata and characterized their class by products of full automata.

Acknowledgements. The author wishes to thank the anonymous referees fortheir constructive comments on a preliminary version of this paper ([21]).

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