Analysis of the fundamentals of Subsumption

Subsumption is generically identified with a scheme for the optimal combinations of two or more orders delivered by two agents of a wider society. Subsumption has been introduced in robotics by Brooks in a number of pioneering papers. Given a couple of orders elicited by two agents after their respective analysis of the same environmental situation. Here we concentrate on orders that can be represented as the superposition of a basic set of elementary ones. Some of such elementary orders are submitted to constraints due to the physical nature of the system. Orders are not an implicit representation of an underlying classification. So consider a couple of such orders delivered by agents A₁ and A₂.

Classical Brook’s subsumption scheme simply introduces an a priori ordering on the class of possible global orders. Then the order with the higher priority is executed.

We briefly discuss in the following the general problem of subsumption, as a scheme of combining two orders represented by strings of elementary instructions. Outside the classical scheme classification-action, the combination of elementary orders to obtain the optimal final global action is not translated into an out-out problem.

Let us represent by the subsumption operation between two global orders. The compatibility between elementary actions can be translated into a compatibility between all the elementary orders that correspond directly to elementary actions. The above discussion shows that there can exist elementary orders which do not correspond to any elementary action. To simplify the discussion, we assume that they can be combined with any set of elementary actions without further constraints. We assume further that any global order is reduced to its essential components; ancillary, not strictly necessary elementary orders have been removed to avoid unnecessary complications.

The most general subsumption scheme (those composed by the largest possible number of elementary orders coming from the two global orders) is evidently given by:

(1)

The above equation is evidently meaningless from the point of view of amount of experience transmitted by such a “mechanical” subsumption scheme. The system has no experience about the validity of the resulting combination of elementary orders. So one of the two orders must at least be adopted. This implies some hierarchy between the global orders. This observation is coherent with one of the basic features of the subsumption scheme proposed by Brooks [1]. Rather than adopting without further analysis the existence of an a priori hierarchy of the global orders, we discuss such a point making use of a general utility function X. We assume that:

(2)

The first vector is taken “as it is” and all the elementary orders of the second one that a compatible with every element of the first are added. So the following relations holds:

(3)

in general.

We then introduce the utility function, and assume that:

(4)

The above equation simply implies that, at least in general, the activation of elementary orders has some relevance for the utility of the system.

It useful introducing the following quantity:

(5)

As, in general such a quantity will be different from zero, its value defines, at least in principle, a local hierarchy between whatever couple of global orders contemporarily elicited by different agents.

The evaluation of DX can be applied in form of a chain rule:

(6)

The computational task increases very rapidly with the number of contemporarily elicited global orders. This results partially justify the choice of an a priori hierarchy of global orders, as that adopted by Brooks in his pioneering works [1]. It is interesting to underline two points implicitly associated with the above observations:

1) the system could have just a partial knowledge about local ordering

2) the system could have a non-homogeneous knowledge of the local hierarchy in the global order space

Both these hypotheses look quite interesting as they would represent some form of “meta-competence” , and will be the object of further research.

For the moment we prefer to adopt some simplifying assumptions.

The modified subsumption scheme we suggest, gives as final result the following final global action:

(7)

where C represents a compatibility relation. Equation (7) states that the agent A₂ elicit all the orders that are compatible with all the elements of O₁.

Interestingly (10) can be easily interpreted as a strategy to minimise the consequences of the possible errors of A₂.

If we represent by S the hypothetical state of the external environment, which is only partially accessible, this situation can be captured making use of the see-functions See_i(S), which correspond to the information extracted from the environmental state S by the monitoring modules of the i-thagent.

One can make different hypotheses on the relation between the different see-functions. One can assume that the A model below holds:

Model

We describe as Model A that where the last term of the RHS of the above equation is zero, model B correspond to the case where this term is not null.

Here the mutual information I(X,Y) is defined as:

We can write as well:

In model A the suggested subsumption scheme in the orders space can be applied without any change. However it’s hard to detect any advantage for A₂ to receive information about See₁(S), as it refers to a sub-domain of S completely uncorrelated with the sub-domain from which S₂(S) extracts its content. Information fluxes between agents can be a useful support for knowledge implementation as far as (A2) is false or if an higher order agent is introduced, which can take their inputs as information to be further evaluated. Such a hierarchical structure of agents would contemporarily imply an incoherent implementation of the subsumption process. In facts, one can easily see that:

does not necessarily identifies the best subsumption scheme for the four involved orders. So at each hierarchical level, the agents need to re-compute from the beginning the best ordering of global orders elicited at the lowest level.

If subsumption scheme of orders is accepted as the basic “reasoning tool” of the system under investigation, a hierarchically ordering of agents is not only inefficient, but produces incoherent results as well. This is an important consequence of avoiding the assumption that, due to the relevance of logic in our formal reasoning, the system must necessarily implement a logically describable decision model. But, as it would be totally “illogical” to assume that the operators applied by the system in its orders space should be logical, when such a space does not need to be such. On the other hand, if we reconsider for a moment the utility function X, the only “available experience for the system is the dependence of X from the orders. The adoption of other softer form of logical reasoning (modal logic, fuzzy logic) would result in an arbitrary forcing into a space that is fully comprehensible by itself of a logical scheme. Consider for example the introduction of some fuzzy scheme. It would in any case introduce, even in softer way, the adoption of some form of classification and would therefore implicitly imply that the evolutionary target is the improvement of the competence making use of a form of fuzzy logic. This would be evidently absurd. The utility of information exchange between agents, as that suggested in our model is the only tool for simplifying the computation tasks of interacting agents once subsumption (which, as we have shown, can be reduced itself to a formal computation) is substituted to another form of computation, mirroring formal logic in its most rigid or softer forms. The system does not need to “know” what is the state of affairs, it must simply “produce” a competent action.

_{Next Page: Consequences of subsumption on classification}

References: Brooks' original papers here