Multilevel System as Multigraph

Multilevel System as Multigraph∗† Jos´e de Jes´ us Cruz Guzm´an and Zbigniew Oziewicz‡ Universidad Nacional Aut´onoma de M´exico Facultad de Estudios Superiores Cuautitl´an Apartado Postal 25, C.P. 54714 Cuautitl´an Izcalli, Estado de M´exico [email protected] [email protected], [email protected]

Waldemar Korczy´ nski University of Arts and Science, ulica Wesola 52 PL - 25353 Kielce, Poland [email protected], [email protected]

March 6, 2003 Lecture Notes in Computer Science Abstract Graph based models of hierarchical systems are usually seen as ‘graph equipped with some refinements’, understood as the (homo)morphisms or (bi)simulations. In such a model it is not possible to consider phenomena happened on different levels of the system. We propose a new formalism of directed multi-graph allowing to see a hierarchical system similar as a formula of second order logic, i.e. to consider all levels parallel ‘at the same time’. The concurrence in hierarchical system is modeled in terms of directed multi-graph. ∗

Supported by el Consejo Nacional de Ciencia y Tecnolog´ıa (CONACyT) de M´exico, proyecto # U41214-F, and by UNAM, DGAPA, Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnologica, proyecto # IN 105402. † March 6, 2003. Lecture Notes in Computer Science (Springer-Verlag) Volume 2658 (2003) pp. 832–840. International Conference on Computational Science ICCS-2003, Sankt Petersburg June 2-4, 2003. ‡ Zbigniew Oziewicz is a member of Sistema Nacional de Investigadores, M´exico, No de expediente 15337.

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Contents 1 Introduction

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2 Presentations of Graph

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3 Multi-graph

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4 Multi-category

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5 Refining and Concurrence

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6 Historical Remark

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1

Introduction

An element of the hierarchical system is said to be a module. Hierarchical system is usually described by refinement of some of its elements namely modules, elements with an extra structure. This way of thinking about hierarchical system as a disjoint union of its modules leads to consideration of any part of the system on another abstraction level which make impossible to consider at the same time, parallel, properties of parts of system being of different levels of abstraction (that could be comparable by means of the hierarchical order) [Korczy´ nski 2000]. We propose a way of seeing hierarchical system, which allow to consider the elements of all levels of abstractions in exactly the same way. The ideology can be seen as a generalization of the notion of the directed graph (≡ one-graph as diagram of arrows). Instead of considering two levels of abstraction, vertices and edges, states and transitions, nodes and links, and many scale-free networks like chemical molecules and chemical reactions, scientific publications and citations, phone numbers and phone calls, actors and films, people and sex relations, et cetera, we can treat a hierarchical system as a sequence of nested three or more families of elements, hyper-links, called in these context cells, connected by two operations (of a generalized type, an orientation) having the same properties as the well known operations of the source and the target in a directed one-graph. The structure thus obtained, diagram of oriented surfaces, diagram of oriented volumes and beyond, is

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called directed (or oriented) multi-graph or more precisely a directed n-graph for n ∈ N, or for n ∈ Z. Our N-graded, or rather N-filtered, N-nested, n-graph G with a sequence of type maps Gi+1 −→ Gi × Gi for i ∈ N must not be confused with a hypergraph where two directed graph maps (orientation) type ≡ {source,target} are generalized to generally disordered assignment E −→ 2V . The origin of the concept of the higher-dimensional graph must be seen in the enriched category theory (life without of elements). Jean B´enabou in 1967 introduced bi-category, Kelly and Street in 1974 2-category, and Kelly in 1982 enriched category theory. Then, starting in 1995, many authors, Baez, Gordon, Power, Street, Leinster, and many other, developed tri-category, and (weak) higher dimensional category, weak n-category, et cetera. The last Chapter XII on structures in categories of the Mac Lane’s monograph [1998] is the best elementary survey on higher-dimensional categories. The concept of the directed multi-graph (n-graph) was introduced by Burroni [1981] and developed among other in the following publications [Oziewicz & V´azquez 1999, Marcinek & Oziewicz 2001, 2003; Obtulowicz 2001, Oziewicz 2003]. Our principal aim is to describe the applications of multi-graph to multi-system theory. The second aim of this paper is to describe the construction of multi-graph based on some presentations of graph. The paper is written in the language of the category theory, however we do not assume that the reader is familiar with the category theory, therefore the paper is in essence self-container. We explain all the notions at the very beginning level. For introduction to category theory we refer to [Mac Lane 1998, Hasse and Michler 1966, Semadeni and Wiweger 1978]. The paper is organized as follows. In Section 1 the principal aims of this work are presented. Section 2 recall some well known definitions of graphs. In Section 3 some interpretations on n-graphs as models of hierarchical systems are proposed. Section 4 describe an extension of n-graph into n-category. In Section 5 category structure is presented for refining concurrence of processes in multi level systems. Section 6 contains historical remarks.

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Presentations of Graph

The directed graphs underlie the theory of categories. A category is a directed graph with additional structure. Pre-category and diagram scheme or just a scheme are just other new names for the old and more familiar directed 3

graph. In what follows a directed graph is identified with a one-graph. A directed graph can be seen as the parallel arrows V ⇔ E, or as the source and the target maps s, t : E −→ V. A one-graph is a pre-sheaf with the two-object domain of the following form source )

V i

E target

Figure 1. A directed one-graph. Instead one can use a type map type

V × V ←−−− E. An element of the set V is called vertex of the graph and accordingly is usually illustrated geometrically as zero-dimensional point. The element of E is called geometrically arrow or edge of the graph and is illustrated as onedimensional directed segment of some line or curve. In the theory of automata there is alternative terminology: a state or 0-transition ∈ V and 1-transition ∈ E [Obtulowicz 2001]. For this reason the more adequate terminology is taken from biology, 0-cell ∈ V and 1-cell ∈ E [Burroni 1981], because the biological ‘cells’ terminology, contrary to geometrical ‘point, line, surface, volume’, allows free flexibility to treat any cell as point or arrow or surface, etc, as would be more convenient in many applications. For example the bi-parti Petri nets will be included automatically in our formalism. In the sequel a (scheme of a) graph on Figure 1 is said to be a 1-graph. In some situations it is suitable to consider a globular graph, i.e. a graph consisting of arrows only. The following one sorted presentation of a 1-graph, or a globular graph, is originated from Hasse & Michler [1966]. 2.1 Definition (One-sorted graph). Let s, t be the two-letter alphabet of a monoid M ≡ {s, t}∗ , w ∈ {s, t}∗ be the N-homogeneous word in this alphabet and |w| ∈ N be a length of w. The unit of this monoid, 1 ∈ M, has the length |1| = 0 ∈ N. A graph (a 1-graph) is presented by globular relations ∀ |w| = 1,

s · w = w,

& t · w = w.

Analogously, for n ∈ N, an n-graph is presented by globular relations ∀ |w| = n,

s · w = w, 4

& t · w = w.

Note that #(words of the fixed length n) = |alphabet|length = 2n . Therefore one-sorted n-graph is presented on two generators {s, t} and given by means of 2n+1 globular relations. A two-sorted 1-graph as a pre-sheaf functor ∈ setScheme , is almost equivalent to one-sorted 1-graph as M -module X over the above monoid, M −→ End X ' X X , id, s, t : X −→ X. Let G ≡ {E ⇔ V } be a graph. Set the disjoint union E ∪ V. One can extend the domain of s and t from E to E ∪ V by s|V = id and t|V = id . Then s, t : (E ∪ V ) −→ (E ∪ V ) is a one sorted 1-graph. We need to check the four globular relations, s2 = s, st = t, t2 = t, ts = s. 2.2 Definition (Reflexive graph). A one-sorted n-graph presented as {s, t|globular relations} is said to be a reflexive graph (or a nested graph) if an alphabet of the monoid is extended by one letter, generators ≡ {s, t, i}, and the globular relations are extended by one-sided inverse relations si = 1 & ti = 1. For example: a graph E −→ V ×V, or {V ⇔ E, s, t}, is said to be reflexive (or nested) if there is an injective (bi)section i : V −→ E, s ◦ i = t ◦ i = idV , i.e. i is one-sided inverse. In the present paper, a graph means reflexive graph, we use the reflexive graphs only. Another presentation of graph defines them as indexed families of sets in the complete analogy to bi-functor ‘hom-sets’ in the category theory. 2.3 Definition (Graph as a family of sets). A ∞-graph (or n-graph) G ≡ {. . . ⇔ Gi ⇔ Gi+1 ⇔}, is a family of maps, where to shorten notation, and in the widespread convention used in the category theory, we use the same symbol Gi for two different meanings, ∀ i ∈ N,

G

Gi × Gi −−−i→ 2Gi+1 .

The subset Gi (x, y) ⊂ Gi+1 for x, y ∈ Gi , is similar to the hom-set in a category where for example in cat(F, G) symbol cat has double meaning, a category with objects F and G, and also a bifunctor objects×objects −→ set. For example here cat can be a ‘category of categories’, or a functor category with a set cat(F, G) of natural transformations (morphisms of functors). Analogously we have other examples with the same convention: category of sets and power set of maps set(A, B) from a set A to a set B, category of k-modules and k-module k − mod(A, B) of k-linear morphisms from a 5

Figure 2: A 1-graph pictured traditionally. module A to module B, a category of algebras (e.g. k-algebras) and a set of algebra maps alg(A, B) from an algebra A to an algebra B. For the present paper would be most important the category of graphs (including n-graphs and ∞-graphs) with a set of graph morphisms graph(G, H) from a graph G to a graph H. Therefore in our Definition 2.3 Gi (x, y) necessarily imply that x, y ∈ Gi and that Gi (x, y) is a set of processes from i-cell x to i-cell y. For example a 1-graph G ≡ {V ⇔ E} is a ‘hom’-map V : V × V −→ 2E . The elements of V are called 0-cells or the vertices and those of the disjoint union 1-cells or the arrows of G, G arrows(G) = V (u, v) (u,v)∈V ×V

For ∀ u, v ∈ V the elements of the set V (u, v) are called 1-cells or arrows from u to v and for any α ∈ V (u, v) ⊂ E the vertex (0-cell) u is called the source and v the target of α. If the sets of the family {(V (x, y)x,y∈V } are pairwise disjoint then we use the set-theoretical union instead of the disjoint direct sum of sets. 2.4 Example. Consider the graph with two vertices V ≡ {u, v}. In Figure 2 the four indexed sets V (x, y) ⊂ E are illustrated. We have V (u, u) = {δ}, V (u, v) = {α, β, γ}, V (v, u) = {ε, ϕ}, V (v, v) = ∅. The set of arrows is the disjoint sum G G(x, y) = G(u, u) + G(u, v) + G(v, u) + G(v, v) = (x,y)∈V ×V

{δ} + {α, β, γ} + {ε, ϕ} + ∅ = {α, β, γ, δ, ε, ϕ}. The above definition allow to see a graph as a pre-sheaf. The pre-sheaf functor category is closed under the formations of product, homomorphic image and taking subalgebras. Let G be a k-graph G0 ⇔ FG1 ⇔ . . . ⇔ Gk , with a partition of G0 . Let H be l-graph, such that G0 = (x,y)∈Hl ×Hl Hl (u, v), ∀ x, y ∈ Hl , Hl (x, y) ⊂ G0 . Then the composed graph, H ◦ G, is (k + l + 1)-graph. If instead Hl (x, y) ∈ G0 , then H is said to be G-graph [Wolff 1974]. We note that G-graph H as introduced by Wolff is not 2-graph. 6

Graphs are tools for describing various phenomena in two levels of abstraction (vertices and edges, nodes and links), including many scale-free networks. Let us list some of them: • The arrows of a graph are processes and vertices are situations being their initializations (births) and deads. • Processes: chemical molecules and chemical reactions. • Schemes of connections, schemes of electrical or telephone networks, flow-charts (flow diagrams in computer sciences). • In the theory of automata: states and transitions. • (Dependence) relations in sets of institutions or persons. • Scientific publications and citations, phone numbers and phone calls, actors and films, people and sex relations, etc. • Schemes of dynamical systems. The vertices G0 of a graph G ≡ {G0 ⇔ G1 } are resources, and the arrows G1 are interpreted as activities (events) consuming these resources. Our aim is to treat a hierarchical system as a sequence of nested families of elements, as the hyper-links, called in these context cells.

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Multi-graph

Graph (1-graph) is a good tool for description dynamical systems with processes running on one level only. The systems, one meets in many branches of everyday life, do not have this property. Typical examples are systems of management where one has to consider the control on the fundamental (lowest) level, the control of processes running on this fundamental level, the control of control processes and so on. 3.1 Example (Medical). An example in medicine: process running in a biological cell, processes running in an organ, in a group of organs and at the level of the whole organism. A biological cell c ∈ G1 in an organ O : the

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situation sc ∈ G0 is its birth and tc ∈ G0 its death. The process α ∈ G2 in the organ O is an arrow of a graph having situations being 1-cells, α

G1 3 c1 −−−→ c2 ∈ G1 . In a new family of 1-graphs {G1 ⇔ G2 }, 1-cells from G1 are vertices for G1 ⇔ G2 , and at the same time are arrows for of a 1-graph G0 ⇔ G1 , illustrating processes in the biological cells, G0

⇔

G1

⇔

G2 .

Continuing this reasoning we obtain a sequence of directed 1-graphs Gi ⇔ Gi+1 , 0 ≤ i ≤ n − 1, called a directed n-graph, describing different levels of an organism. The above considerations are formalized as follows [Burroni 1981, Lawvere 1989, Marcinek & Oziewicz 2001, Obtulowicz 2001, Oziewicz 2003]. 3.2 Definition (n-graph). A directed n-graph G is a sequence of families of i-cells {Gi }, 0 ≤ i ≤ n, such that Gi ⇔ Gi+1 is a directed 1-graph with the source map si : Gi+1 −→ Gi and the target map ti : Gi+1 −→ Gi . 3.3 Definition (Reflexive n-graph). A directed n-graph is said to be reflexive (or nested) if there are sections {ii : Gi −→ Gi+1 }, satisfying the conditions of one-sided invertibility si ◦ ii = idGi = ti ◦ ii .

G0

s0

s1

s2

 i0 Y

 i1 Y

 i2 Y

t0

G1

G2

t1

···

t2

3.4 Definition (Morphisms of n-graphs). Let G and H are n-graphs. A morphism f ∈ graph(G, H) of an n-graph G into an n-graph H is a sequence f ≡ {fi : Gi −→ Hi }, commuting with {s, t, i}, fi ◦ s = s ◦ fi+1 , such that for any i ∈ N, (fi , fi+1 ) is a morphism of 1-graphs, ←−

Gi ←−−− Gi+1     fi+1 y fi y ←−

Hi ←−−− Hi+1 8

Let us note a difference between the approach used in the above example and the standard modular modeling of multilevel system. Seeing a system as an n-graph we can consider it on all levels at the same time [Burroni 1981, Marcinek & Oziewicz, Obtulowicz 2001]. This point of view is similar to that known from the higher order logic when one has to consider variables being individuals, sets of individuals, sets of sets of individuals, etc. Seeing a system as an ‘object consisting of some other objects’, called in this context modules, object of modules, we are always at the same level. Refining of a model doesn’t lead to a new situation; we are still at the same level because one hasn’t to consider any relation between objects of different levels. The last is characteristic for n-graphs and higher order logic. Let us consider an analogous construction for graphs seeing as families of sets. Having a (possibly reflexive) 1-graph G ≡ (G0 ⇔ G1 ), one can consider the family Γ(G) ≡ {G1 ⇔ G2 }, of all 1-graphs with vertices being arrows in G, i.e. graphs of the form G2 ≡ {G1 (α, β), (α, β) ∈ G1 × G1 }. In another words, G1 (α, β) ⊂ G2 is a new set of arrows. This construction is not unique. 3.5 Remark (Normal graph). For a reflexive n-graph G we have Gi ⊂ Gi+1 , vertices(G) ⊆ arrows(G). In this case let G be filtered nested family, G ≡ Gn ⊇ G(n−1) ⊇ G(n−2) . . . G0 , and one can consider not restricted map G×G −→ 2G which do not need in general respect N-grading nor N-filtration of G. Particularly we do not exclude the case when for some arrow α and vertex u of a reflexive nested graph G the set G(α, u) is not empty for different grades, grade α 6= grade u. A reflexive n-graph G in which G(α, β) = ∅ if grade α 6= grade β, will be called normal. 3.6 Example. Figure 3 is showing an example of the not normal 2-graph G0 ⇔ G1 ⇔ G2 . In this case G0 ⇔ G1 is the nested 1-graph pictured in Figure 2. We are adding one set of genuine not trivial 2-cells G1 (α, ε) ∈ G2 , and 3 sets of not homogeneous cells violating grading G(u, δ), G(u, α) and type G(ϕ, γ). For example G(u, δ) −− −→ G0 × G1 . Therefore the considered 2graph is not normal.

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Figure 3: An example of not normal 2-graph.

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Multi-category

A directed reflexive (and normal) n-graph is a carrier for an n-category by adding associative multi-compositions, the puzzle operations, as a partial multivalued grafting for each family of cells separately [Marcinek & Oziewicz 2001, 2003; Oziewicz 2003], ∀ i ∈ N,

grafting

Gi × Gi −−−−→ 2Gi ,

being the set theoretical union of total multivalued grafting ∀ u, v, w ∈ Gi ,

grafting

Gi (u, v) × Gi (v, w) −−−−→ 2Gi (u,w) .

Such set of composition is assumed each to be associative and having two sided identities, i Gi 3 u −−−→ idu ∈ Gi+1 , such that for any (i + 1)-cell α ∈ Gi (u, v) ⊂ Gi+1 , and for any grafting (among i possibilities! which we de not wish to be ordered) it holds idv ◦ α = α ◦ idu = α. In other words, we make each family of cells Gi (with exception perhaps of G0 ) a multi-multiplicative graph with bi-unital associative and pair-wise commuting graftings (multiplications) (related to Godement rule of ‘middle four exchange’), i.e. a multi-category. 4.1 Definition (n-category (Marcinek & Oziewicz 2001). An nested (reflexive) n-graph G ≡ (G0 ⇔ G1 ⇔ . . .) such that every family of i-cells Gi is equipped with a set of i unital associative and pair-wise commuting binary operations, that is for G2 we need to have the two commuting comopositions, horizonthal and vertical, is said to be an n-category. An n-graph G ≡ (G0 ⇔ G1 ⇔ . . .) is said to be semi-group-oidal, monoidal, algebraical, et cetera, if G0 is a semi-group, monoid, an algebra, et cetra.

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Refining and Concurrence

Refining is a method for system describing and analysis. An advantage of this way of treating complicated composed systems is that we are always on 10

Figure 4: Different levels concurrence the same level of description. We use in any moment the first order logic. Sometimes this approach to systems is not suitable. Typical example of such situations can be found while considering concurrence of processes running in different levels of a hierarchical system. This kind of concurrence can not be described by the modular hierarchical modeling approach. One of the examples of such systems can be the biological system mentioned early. Other example can be found in hierarchical management system. Let us consider such a system consisting of two levels G0 ⇔ G1 ⇔ G2 . The first one, let us call it production ∈ G1 , can be seen as a really production system in a factory, in an office or in economical organization. The second level ∈ G2 manage the processes in the first one. So the way of working of such a system can be seen as follows. Firstly, an object, let us call it m, performs a planning action A. As a result one obtains a project, say p ∈ G2 , which will be realized by an object w ∈ G1 on the level 1. Now, one can consider a concurrent activities A0 , and the next another project, say p0 being the result of the activity A0 , can be realized concurrently with another designing activity A00 , and so on. The generalization of this concurrence to multilevel systems is evident. This kind of concurrence is different from the concurrence from Petri nets where one considers concurrence in the set of elements of the same level [Baumgarten 1990, Leszak and Eggert 1988; van Sinderen, Ferreira, Vissers and Katoen 1995]. The category structure (given by the multiplication of arrows)allows for considering a hierarchy of sequential systems. In this case the only concurrence is the concurrence described above. But one can also consider another kind of concurrence - the concurrence of the same level of the system. This concurrence can be modeled by introducing in the categories corresponding to the levels of the system a new operations expressing the parallel composition of the elements (arrows and vertices) of the system. It can be done in the same way as it has be done by Korczy´ nski [2001]. This leads to the notion of (partial) monoidal n-category.

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6

Historical Remark

Some notions used in the paper are not very popular and frequently used. Let us recall the origin of them. The one sorted presentation of graphs has been introduced by Maria Hasse & Michler in 1966. It belongs to the language of the French school of category theory originated by Schuetzenberger and Ehresmann [Ehresmann 1950]. Such an approach allows to consider categories as a kind of ”partial semigroups” and leads to interesting applications of(semi)group theory seeing as a (meta)language for such branches of mathematics as for example topology. Reflexive graphs have been considered firstly by Burroni [1981], Gray and Lavwere [1989]. The idea was similar to that of introducing one-sorted graphs; one wants to consider categories as a kind of of monoids. The embedding of vertices into arrows shows how an ”identity” can be introduced into the classical graphs. This identity assigns to every vertex of a graph a ”loop” being an arrow having its beginning and end - point in these vertex. Burroni [1981] introduced n-graphs as a tool for describing algebraic structures in ‘pure mathematics’, more precisely in category theory seeing as a mathematical language. The papers exploring n-graphs and very interesting applications of n-graphs to some problems in computer sciences have been written by Obtulowicz [2001], and by Marcinek & Oziewicz [2001, 2003].

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