ENTAILMENT MESHES AS NON-DIRECTED GRAPHS

Although THOUGHTSTICKER is more flexible than traditional AI knowledge representation systems, its underlying Lp logic has a number of inherent shortcomings. The meaning of a coherence is not very clear and is likely to be interpreted differently by different users. Moreover, Lp's structure is too loose to support "inferences" in the stronger sense of logical deductions showing the truth or falsity of certain expressions. Entailment meshes also do not have any obvious connection with the--admittedly little--knowledge we have about cognition and brain functioning. The most important limitation, however, seems to me the inherent non-directionality of its entailment meshes.

Pask's motivation for non-directionality was to replace the traditional hierarchical structures--where higher order concepts are always reduced to (non-grounded) primitives--by a heterarchical ("bootstrapping") structure--where concepts mutually produce each other. Directionality does not imply hierarchy or reducibility, though. When I started working on my structural language (Heylighen, 1990a), I made it intrinsically directional in order to model processes and their "arrow of time", not hierarchical dependencies. The advantage of directed structures is that they encompass non-directed structures as a special case, whereas the converse is not true. In the mathematics of relations, non-directionality corresponds to symmetry. A relation R is symmetric if for every pair for which the relation holds, (a, b) R, the inverse pair also satisfies the relation: (b, a) R. General relations are asymmetric, though, which means that the inverse pair can either be there, or not. So, a non-directed structure or graph can always be created by limiting the allowed relations to symmetric ones. On the other hand, there is no clear way to create a directed structure if only symmetric relations are available.

Let me show how such a relational representation generalizes the structure underlying entailment meshes. Two topics a, b in a coherence are connected by the reflexive and symmetric relation C: "belongs to the same coherence as", which we might also read as "is coherent with". Symmetry means that for all a and b, a C b => b C a. Moreover, within one coherence the relation is also transitive: if a C b and b C c, then a C c. However, the relation is no longer transitive if we consider topics belonging to different but overlapping coherences. Consider the two coherences <a, b, c> and <c, d, e>. In that case we have a C c, and c C d, but not a C d. The combination of reflexivity, symmetry and transitivity defines an equivalence relation. The individual coherence clusters could then be viewed as the equivalence classes induced by the relation C. Since C is only locally transitive, its equivalence classes can overlap, instead of partitioning the set of all topics into separate clusters, like a full equivalence relation would.

The representation of an entailment mesh through its collection of coherences is thus equivalent to its representation by the symmetric relation C defined on the set of all topics T = {a, b, c, ...}: C T x T. The fundamental requirement of the avoidance of structural ambiguity can then be formulated in the following way. Every topic a is derived from the other topics it is coherent with: {x != a | a C x} = I(a). Ambiguity is the situation where we have two distinct topics a != b but such that I(a) = I(b).