The destiny of constant structure discrete time closed semantic systems
Constant structure closed semantic systems are the systems each element of which receives its definition through the correspondent unchangeable set of other elements of the system. Discrete time means here that the definitions of the elements change iteratively and simultaneously based on the “neighbor portraits” from the previous iteration. I prove that the iterative redefinition process in such class of systems will quickly degenerate into a series of pairwise isomorphic states and discuss some directions of further research.
Keywords: closed semantic system, graph, isomorphism.
A closed semantic system (CSS) may be thought as a system each element of which is defined through other elements of this system. One of the most natural and important CSSs is language. Every child at the age of 3–5 years is full of questions: “Why?” , “What for?” , “How?” . At that time, the child’s world view is growing and getting as closed as possible: every word claims to be explained in terms of other words.
One of the most simple CSSs is a discrete time system with a permanent structure: 1) the number of the elements that are involved in the definition of each element of such system does not change with time; 2) all the elements are redefined simultaneously, basing on the states of the neighbors taken at the previous simultaneous iteration. In this paper, I study the behavior of such trivial systems, starting from the “zero” state, where all the system’s elements are equal. In contrast with the above mentioned growing CSS of small kids, a CSS with a constant structure resembles an adult’s world view, where the addition of new notions and connections between them ceases as the personal world view approaches the best known contemporary world view of the humanity. The simultaneous iterative changes in discrete time is a reasonable assumption as far as we consider an artificial CSS designed for a deterministic Turing machine equivalent . However, the discreteness can hardly correspond to real-world examples, so it would be challenging to get rid of it in the future.
The development of the described constant-structure discrete time CSS may be modelled as a discrete time coloring process on a connected graph. From now and till almost the end of the paper I will consider the iterative redefinition process on the vertices only, assuming that the edges are plain connectors, which do not have their own colors and do not participate in the definition of the incident vertices. Initially, all the vertices have the same “type” , so at the first iteration the only difference between any 2 vertices is the number of their neighbors (degree). For illustrative purposes, we may assign each degree a specific “color” . At the second iteration the “neighbor portrait” of each vertex becomes more complicated (e.g., “5 neighbors” at the first iteration becomes “2 red and 3 green” at the second). This iterative coloring process (see Fig.1) produces equitable partitions  (perfect colorings, regular partitions and graph divisors), which are well known [4, 5, 6] and were successfully used in e.g. graph isomorphism heuristic Nauty  (the coloring process in the latter is similar to the Algorithm of this paper).
How will this iterative coloring process behave? Will the size of the palette ever increase or it can decrease and then oscillate? Will the process become self-repeating? If yes, then how fast and what will be the cycle size? The next section gives the answers to these questions.
1 Mathematical model
There are some little less common mathematical notations used below, which I would rather state explicitly: a) the number of adjacent vertices for each in simple is equal to the degree of and is referred to as ; b) the image of the mapping is referred to as and c) an arbitrary (possible multi-valued) mapping between and is referred to as .
Let be a simple graph with vertices and edges . Each vertex possesses a color, which changes iteratively depending on the colors of the vertex’s neighbors. Let be the set (palette) of colors and be the coloring function at the -th iteration.
The iterative coloring process may be described as follows:
How will this IICP behave at infinity? Below I prove that, starting from some iteration, all the following colorings will be pairwise isomorphic in the sense of the following definition:
Lemma 1 makes the first step in this direction. It shows that two consequent isomorphic colorings result in the degeneration of the following IICP.
Two colorings and are isomorphic (), if there exists a bijection such that .
If for some , then all the following colorings will be pairwise isomorphic: .
It is enough to prove that . Let us select an arbitrary and take the portrait of from the previous iteration, which corresponds to :
Let us consider the mapping :
and prove that it is a bijection.
1) is defined for all elements of , since it is defined for all .
2) Each portrait from corresponds to at least one vertex at the -th iteration; selecting this in (2), we get at least one preimage for in , which means that is surjective.
3) The single-valuedness of is slightly more complicated to prove; suppose there is a portrait from
that is mapped by into two portraits from
Since , there exists an isomorphism such that
Considering this, we can write
which means that and are equal elementwise and proves the single-valuedness of .
4) The injectiveness of may be proved in the same manner; suppose there are two portraits from
that are mapped by into one portrait from
Returning to (4), we have
which proves that is injective and therefore is a bijection.
Applying to the right-hand side of (1), we get an element of , which can be mapped further to a color from by . The final sought-after mapping is the composition of the three above-mentioned mappings,
which is a bijection since all the composing mappings are bijections. ∎
At this point, we know that two consequent isomorphic colorings make the further coloring process degenerated, but is this situation impossible, probable or inevitable? The answer to this question starts with the analysis of the behavior of the palette size. The number of colors in the palettes evidently belongs to , and intuitively this number either increases or remains constant during the IICP.
The size of the palette never decreases: .
Let us use induction on . The base case for is true since all the vertices have the same color at the first iteration: .
Inductive step: assume the statement is true for all and prove it for . Let be the adjacent vertices of and be the adjacent vertices of in .
Since and is a bijection,
which means that and there exists a bijection , such that
which means that and thus
Now we know that at each iteration the number of colors either increases or remains constant. We also know that the palette size cannot be arbitrary large, it is limited by . Putting together both results, we get that an IICP may contain only a finite number of the steps where the palette size grows. In not more than iterations, an IICP will come to the situation, where the palettes at two consequent iterations have the same size. The consequences will be radical.
Either the size of the palette increases or the two last colorings are isomorphic: .
I will prove that if , then the mapping , defined as , is a bijection. Let’s argue by contradiction: suppose , but is not a bijection. Since is defined for all , it is completely defined on and is a surjection onto ; it is also injective by Lemma 2. The last option for not to be a bijection is multi-valuedness. Suppose that
Let and (see Fig.3). The set is not empty, since otherwise
There are no colors from among the colors from since otherwise (see Fig.3) but, by construction of , , which contradicts the injectiveness of . Thus,
The number of distinct colors in is not less than since is injective and does not map distinct colors into the same one.
By the definition of and assumption (6), we have and
which means that
and contradicts the assumption . ∎
The results of the previous lemmas may be summed up as a theorem.
The coloring process described in the Algorithm converges to an unchanging equitable partition in at most iterations.
Until now, all objects in the CSS were defined through other objects by means of one-type connections. One of the important generalizations of such approach is the introduction of connections of different types. As far as we talk about closed semantic systems, the “type” of a connection should be defined within the system. By analogy to the objects defined through their neighbors, the connections could be defined through the objects, which they relate. In the considered graph model of CSS, it means that the edges, just as the vertices, are subjected to an iteration coloring process based on the portraits of neighbors. During such an expanded coloring process, the color of a vertex is defined by the colors of the adjacent vertices and incident edges, and the color of an edge is defined by the colors of its 2 incident vertices.
This iterative coloring process for vertices and edges may be converted to the vertices-only coloring case by the addition of “virtual” vertices corresponding to the edges (Fig. 4) assuming the colors for the vertices and for the edges are taken from the same palette and assigned in accordance with the same laws.
In the present paper I showed that each CSS with a constant structure and discrete time ceases to change rather quickly. The future research could aim at getting the CSS model closer to the real world. The consideration of more complex unsynchronized dynamical structures will change, to all appearances, the further research methods from pure mathematical to statistical and computational. Below I concern several conceivable directions for the further investigations:
1. The first step towards real complex systems is to get rid of the discreteness of time. In the majority of real complex systems it seems unnatural to change the states of all elements simultaneously. Such desynchronization will mix the palettes, so the corresponding formal coloring process should be considerably different.
2. The second step is the consideration of the systems that start from an arbitrary state, not only the “zero state” of Algorithm, where . It is easy to see that in this case Lemma 2 is not always true (see Fig.5), therefore, Lemma 3 would also need a different proof since the current one relies on Lemma 2. Nevertheless, it seems the proof of convergence can be generalized to the case of arbitrary initial states.
3. The most interesting research direction is the analysis of behavior of the CSSs with (stochastically) varying sets of objects and connections. Such CSSs could presumably be addressed by means of applied statistics and multi-agent modelling. In this respect, the underlying graph model could be replaced with the model of horizontal gene transfer  where each agent is “defined” through the influence of the agents encountered during stochastic motion.
The author thanks Dr. D.V. Khlopin and Dr. A.L. Gavrilyuk for their comments.
-  Bloom P. How Children Learn the Meanings of Words. A Bradford Book. MIT Press, 2002.
-  Papadimitriou C.H. Computational Complexity. Pearson, 1993.
-  Godsil C.D. Algebraic Combinatorics. Chapman and Hall Mathematics Series. Chapman and Hall, 1993.
-  Unger S.H. GIT – a heuristic program for testing pairs of directed line graphs for isomorphism. Commun. ACM, 1964, vol. 7, no. 1, pp. 26–34.
-  Weisfeiler B., Lehman A.A. [A reduction of a graph to a canonical form and an algebra arising during this reduction] (in Russian). Nauchno-Technicheskaya Informatsia [Scientific-technical information], 1968, vol. 2, no. 9, pp. 12–16.
-  Arlazarov V.L., Zuev I.I., Uskov A.V., Faradzhev I.A. An algorithm for the reduction of finite non-oriented graphs to canonical form. USSR Computational Mathematics and Mathematical Physics, 1974, vol. 14, no. 3, pp. 195–201.
-  McKay B.D., Piperno A. Practical graph isomorphism, II. Journal of Symbolic Computation, 2014, no. 60, pp. 94–112.
-  Syvanen M. Horizontal gene transfer: Evidence and possible consequences. Annual Review of Genetics, 1994, vol. 28, no. 1, pp. 237–261.