Conceptual Space

	G	objects (tokens)
	M	attributes (types)
	V	views (??)
	H	incidence (classification)
	E	instance-of, extent (??)
	I	abstraction, intent (??)

Conceptual Base (Classification)	hG,M,Hi
Conceptual Space (??)	hhG,M,Hi,hV,E,Iii

residuation	inclusion	derivation
incidence constraints
I = E\H E = H/I	"v2V,m2M   vIm  iff  EvvHm "g2G,v2V   gEv  iff  gHwvI	"v2V   vI=(Ev)' "v2V   Ev=(vI)'

Discussion

• We start out by specifying the set V to be a collection of names, with each v 2 V representing a subset of attributes (types) vI₀ 2 PM. We use (two) residuation operators, which are a generalized form of the derivation operators of formal concept analysis, in order to define the notion of a "collective formal concept." We define E to be the target (right) residuation of H along I₀:

  E = H / I₀ = { (g,v) | "_m2M (vI₀m ) gHm) }.

  so that E o I₀ v H. We define I to be the source (left) residuation of H along E:

  I = E \ H = { (v,m) | "_g2G (gEv ) gHm) }.

  From these definitions it is straightforward to show that E is the target (right) residuation, E = H / I, of H along I: First of all, since I₀ v E \ H = I, by contravariance of residuation E = H / I₀ w H / I; and secondly, since E o I = E o (E \ H) v H, E v H / I.
• The preorder V can be regarded to be a collective formal concept, where E is its extent and I is its intent. In terms of formal concept analysis, the two residuations generalize derivation: the source (left) residuation along the incidence H maps a collective subset of G (a G-source matrix) to a (dual) collective subset of M (a M-target matrix), and the target (right) residuation along the incidence H maps a (dual) collective subset of M (a M-target matrix) to a collective subset of G (a G-source matrix). The definition of a collective formal concept generalizes the usual definition of a formal concept.
• We form the disjoint union (coproduct) V + M and the copairing relation [E,H] as in the commuting diagram on the right. For any v 2 V + M, note that gHm for all m in vI. So v is the meet v = u vI in V + M of the set of attributes (types) vI. Thus, the injective map i_V factors through a sub-conjunction infomorphism on V + M.

Example: The Living Concept Space

The "living" concept space is a tiny dataset, which exists within a conceptual universe of living organisms. This space consists of eight organisms (plants and animals), and nine of their properties. The organisms are the objects of the space, and the properties are the attributes. The living conceptual universe contains 19 formal concepts. The living space names five of these concepts. Named formal concepts are called conceptual views. Conceptual knowledge is represented by: 3 orders on objects, attributes and views; the membership or instantiation relation between objects and views whose columns record the extent of all the conceptual views; the abstraction relation between views and attributes whose rows record the intent of all the conceptual views; and the incidence relation between objects and attributes. In the WAVE system conceptual knowledge is represented by the notion of a concept space. In particular, the living conceptual knowledge is represented by the frame of the living concept space. A concept space is an algebraic object which can be generated by a combinatorial object (reduced data structure) called the concept space input format. In the WAVE system an input format is used to initialize a concept space. In particular, the living input format is used to initialize the living concept space. Conceptual browsing over the living space, in either extensional or intensional mode, allows a better understanding of this dataset.

Please send questions, comments and suggestions about this page to: Robert E. Kent rekent@eecs.wsu.edu Last modification date: October 1998