Natural systems may have all a hierarchy of components: individuals, groups of individuals cooperating to realize some task, groups of groups, ...; for instance in a organism: atoms, molecules, cells, tissues,.... Such a hierarchical is modeled by the notion of a hierarchical system. 

An Hierarchical System (HS) is an evolutive system (ES) in which each of the state-categories is a hierarchical category in the following sense: the objects of the category are partitioned into a sequence of complexity levels 0, 1,..., m, so that each object N of level n+1 is the colimit of at least one pattern of linked objets Ni of level n.

An objet N plays a double rôle ('Janus'): it is 'complex' with respect to its components Ni of the lower level, but 'simple' when it is considered itself as one of the components of a higher level object. Thus a cell of a tissue is simple if it is compared to the tissue, but complex if its own decomposition into organites and macromolecules is taken into account.

The hierarchy of a HS is of a structural nature, not necessarily involving a command of the higher levels on the lower ones. Indeed, the levels can be entangled, with both links from lower levels to higher levels and conversely. Thus an object can receive information from objects both of a lower or higher level, and conversely send messages to any other level, as long as the energetic constraints are satisfied. The number of levels can increase in time; for instance, in a neural system, learning helps to memorize more and more complex behaviors. 

An enterprise has such a hierarchical structure: The objects of level 0 represent the employees. Higher level objects represent departments, from small production units specialized in a particular function, to the highest management levels. The links between the members of a department correspond to exchanges of information so that they may cooperate. Higher levels can send commands to lower levels, but they also depend on these lower levels; for instance the construction of a machine is interrupted in case of a long enough strike of a lower level unit producing some components necessary for its construction. 

 

 

Ramifications of an object

In a HS, an object N of level n+1 as at least one decomposition P of level n. It has also a more intricate internal organization of level n-1. Indeed, as each object Ni of P is itself the colimit of a pattern Pi of level n-1, the object N is the 2-iterated colimit of the 2-ramification (P,(Pi)) arriving at the level n-1. This ramification can be thought of as a two steps internal organization of N.

More generally, N has several steps internal organizations, inductively defined, formed for each k £ n by k-ramifications arriving at the level n+1-k. At each step, they take into account both the 'vertical' links from the objects of a pattern to the colimit of the pattern, and the 'horizontal' links formed by the distinguished links of the patterns.

A k-ramification of N allows to re-construct N from the level n+1-k in k steps. However, in some cases, N may also be re-constructed from this level in a unique step, skipping the intermediate levels, so that the level of an object does not determine its 'real' complexity.

 

 

Complexity order of an object

The (complexity) order of the object N of level n+1 is the smallest p £ n such that there exists a pattern of linked objects of level p admitting N as its colimit. And N is said to be q-reducible for each q less or equal to its order.

By definition, any object of level n+1 is n-reducible. When is it p-reducible, for some p<n? To answer, we distinguish several kinds of links in a HS.

A link between two objects of level n+1 is n-simple if it binds a cluster between two patterns of level less or equal to n. It is n-complex if it is the composition of n-simple links binding non-adjacent clusters, without being itself n-simple. Moreover, there might also exist links which are nor n-simple nor n-complex, and which represent constraints unrelated to the lower levels.

Then the following result is proved (Ehresmann & Vanbremeersch, 1996): An object N of level n+1 is (n-1)-reducible if it admits a decomposition into a pattern of objects Ni of level n in which the distinguished links are (n-1)-simple links binding adjacent clusters between the Ni. Otherwise, N might not be (n-1)-reducible. This result extends to lower levels. 

 

 

  Based HS

The preceding result shows the limits of the strict reductionnist program which assumes that any object is reducible, in one step, to the lowest level, i.e., is of order 0. However in most natural HS, there is a kind of reduction to lower levels, but in several steps and with the emergence at each level of new properties reflecting wholistic properties of the preceding level. These systems are modeled by the based HS.

A HS is k-based if, for each n greater than k, its links of level n+1 are n-simple or n-complex, i.e., no external constraints are added in the passage from the level k up. In such a HS, not only each object is stepwise reconstructed from the level k up through a ramification (that is true in any HS), but the links can also be stepwise reconstructed since they either bind clusters of the next lower level, or are composites of such. However as soon as there exist complex links, these reconstructions depend not only on the 'local' properties of the components of level k of the object or link (as required by the reductionnist program), but also on the global structure of each successive level.

Indeed, let us prove it first for a complex link gg' of level k+1, from N to N", which is the composite of a (Pi,P')-simple link g from N to N' and of a (Q',Pj)-simple link g' from N' to N". The properties of this link are deduced from the local properties of the two clusters of level k that g and g' bind, and from the fact that the patterns P' and Q' have the same colimit N'. This last condition means that the two patterns impose the same constraints on each object of level k or less, and so it takes into account the whole level k. Thus the constraints imposed on N and N" by the complex link gg' cannot be reduced to local constraints imposed on their components in Pi and Pj, but emerge from the global structure of level k.

Now let A be a level k+2 object. It can be reconstructed from level k as a 2-iterated colimit of a ramification (R,(Pi)); but, if some of the links of R are complex, it follows that they also impose on A properties emerging from the global structure of level k, and not reflected from the sole local properties of the lower components of A.

Thus, for k-based HS, we can speak of an 'emergentist reductionnism' (precising the notion introduced by Mario Bunge). In particular, for 0-based HS the non-linear language of the system will be entirely decoded given the primitive terms, i.e., the 'atoms' of level 0 and their links, and the 'syntax' which indicates how they are binded together to progressively construct the higher levels objects and links in several steps, with, at each level, emergence of new properties depending on the global structure of the preceding level.