Jean-Paul Doignon

Spaces for the assessment of knowledge

The information regarding a particular field of knowledge is conceptualized as a large, specified set of questions (or problems). The knowledge state of an individual with respect to that domain is formalized as the subset of all the questions that this individual is capable of solving. A particularly appealing postulate on the family of all possible knowledge states is that it is closed under arbitrary unions. A family of sets satisfying this condition is called a knowledge space. Generalizing a theorem of Birkhoff on partial orders, we show that knowledge spaces are in a one-to-one correspondence with AND/OR graphs of a particular kind. Two types of economical representations of knowledge spaces are analysed: bases, and Hasse systems, a concept generalizing that of a Hasse diagram of a partial order. The structures analysed here provide the foundation for later work on algorithmic procedures for the assessment of knowledge.

On realizable biorders and the biorder dimension of a relation

Abstract The paper discusses the mathematical foundations of a technique of multidimensional scaling, generalizing Guttman scaling, in which the structure of the embedding space relies only on ordinal concepts. An empirical relation is represented as an intersection of a minimal number (called bidimension) of Guttman relations. Fairly complete results are given for the cases of bidimensions 1 and 2. In the general case, the main results are based on the equivalence between the bidimension and the dimension of a certain partial order. A characterization of the bidimension as the chromatic number of some hypergraph is also provided.

Biorder families, valued relations and preference modelling

Abstract Several compatibility conditions are studied for families of interval orders or semiorders, involving for instance step-type matrices and functional representations. Our approach uses the basic notion of biorder or Guttman scale. The results answer a question raised by Roberts, who in fact treated the particular case of nested families. They provide in a very general setting various formalizations for the notion of probabilistic consistency of a subject in a binary choice process. Other possible applications are also mentioned.

A Markovian procedure for assessing the state of a system

Abstract We consider a class of systems, the states of which can be represented by particular subsets of features in a basic set. A procedure for assessing the state of a system in which the presence of a particular feature is tested on each trial is described. The feature is chosen so that the outcome of the test is as informative as possible (in a specific sense). This outcome results in a modification of a class of plausible states. The sequence of plausibilities is a finite Markov chain. Results of practical significance are obtained and cover cases in which the response data are noisy, and the state of the system may even change randomly from trial to trial. An application of primary interest to the authors is the assessment of knowledge. In this case, the systems are human subjects, the features are questions or problems, and the state of an individual is that particular subset of questions that the individual is capable of solving. The conditions investigated in the paper, even though primarily dictated by this application, are nevertheless of general scope.

How to build a knowledge space by querying an expert

Abstract A particular field of knowledge is conceptualized as a set of problems (or questions). A persons knowledge state in this domain is formalized as the subset of problems this person is capable of solving. When the family of all knowledge states is closed under union, it is called a knowledge space. Doignon and Falmagne (1985) established a 1-1 correspondence between knowledge spaces and a class of surmise systems, a slight variant of AND/OR graphs. Here we rather obtain a 1-1 correspondence with a well defined class of quasi orders on the collection of all subsets of problems. The resulting approach to knowledge spaces helps to build such spaces for particular domains. We describe a procedure which relies on the answers of an expert to a carefully chosen sequence of information requests.

Knowledge Spaces and Skill Assignments

The concept of a knowledge space is at the heart of a descriptive model of knowledge in a given body of information. Another model explains the observed knowledge of individuals by latent skills. We here reconcile these two underlying approaches by showing that each finite knowledge space can be generated from a skill assignment that is minimal and unique up to an isomorphism. Some more technicalities are required in the infinite case. Part of the results reformulate theorems from the theory of Galois lattices of relations.

Convexity in cristallographical lattices

A subset of a (cristallographical) lattice ℒn is called convex whenever it is the intersection of the lattice with a convex set of the affine space containing ℒn. We give a characterization of the convex sets which is intrinsic to the lattice and do the same for other related notions, e.g. the boundary of a convex set of ℒn. A statement analogous to Hellys theorem is also proved.

Well-graded families of relations

Any semiorder on a finite set can be reached from any other semiorder on the same set by elementary steps consisting either in the addition or in the removal of a single ordered pair, in such a way that only semiorders are generated at every step, and also that the number of steps equals the distance between the two semiorders. Similar results are also established for other families of relations (partial orders, biorders, interval orders). These combinatorial results are used in another paper to develop a stochastic theory describing the emergence and the evolution of preference relations (Falmagne and Doignon, [7]).

A polynomial time algorithm for unidimensional unfolding representations

Abstract Two conditions on a collection of simple orders - unimodality and straightness - are necessary but not jointly sufficient for unidimensional unfolding representations. From the analysis of these conditions, a polynomial time algorithm is derived for the testing of unidimensionality and for the construction of a representation when one exists.

THE REPEATED INSERTION MODEL FOR RANKINGS: MISSING LINK BETWEEN TWO SUBSET CHOICE MODELS.

Several probabilistic models for subset choice have been proposed in the literature, for example, to explain approval voting data. We show that Marley et al.s latent scale model is subsumed by Falmagne and Regenwetters size-independent model, in the sense that every choice probability distribution generated by the former can also be explained by the latter. Our proof relies on the construction of a probabilistic ranking model which we label the “repeated insertion model.” This model is a special case of Mardens orthogonal contrast model class and, in turn, includes the classical Mallows φ-model as a special case. We explore its basic properties as well as its relationship to Fligner and Verduccis multistage ranking model.