Jean-Paul Doignon

Jean-Paul Doignon
Université Libre de Bruxelles | ULB · Department of Mathematics

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121
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January 2006 - present
January 1997 - present

Publications

Publications (121)
Preprint
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The Multiple Choice Polytope (MCP) is the prediction range of a random utility model due to Block and Marschak (1960). Fishburn (1998) offers a nice survey of the findings on random utility models at the time. A complete characterization of the MCP is a remarkable achievement of Falmagne (1978). Apart for a recognition of the facets by Suck (2002),...
Article
Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry. Contrary to what happens for similar constructions–compounds of hypergra...
Article
In knowledge space theory, the (latent) knowledge state of a student consists of the subset of test items that he masters in principle. Even at a given stage of apprenticeship, the student’s knowledge state may vary in a given collection of subsets. The collection of all possible states of all potential students forms a knowledge structure. In the...
Preprint
Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry. Contrary to what happens for similar constructions -- compounds of hyper...
Chapter
Among the real-valued representations of nested families of biorders some representations reflect the nestedness of the family in a simple way. Calling them chain representations, we prove their existence in the finite and countably infinite cases. For the general case, we obtain chain representations in a well-chosen linearly ordered set. Although...
Preprint
Full-text available
Among the real-valued representations of nested families of biorders some representations reflect the nestedness of the family in a simple way. Calling them chain representations, we prove their existence in the finite and countably infinite cases. For the general case, we obtain chain representations in a well-chosen linearly ordered set. Although...
Preprint
We consider a natural combinatorial optimization problem on chordal graphs, the class of graphs with no induced cycle of length four or more. A subset of vertices of a chordal graph is (monophonically) convex if it contains the vertices of all chordless paths between any two vertices of the set. The problem is to find a maximum-weight convex subset...
Article
Full-text available
For any given finite group, Schulte and Williams (2015) produce a convex polytope whose combinatorial automorphisms form a group isomorphic to the given group. We provide here a shorter proof for a stronger result: the convex polytope we build for the given finite group is binary, and even combinatorial in the sense of Naddef and Pulleyblank (1981)...
Article
Full-text available
Mathematical psychology has a long tradition of modeling probabilistic choice via distribution-free random utility models and associated random preference models. For such models, the predicted choice probabilities often form a bounded and convex polyhedral set, or polytope. Polyhedral combinatorics have thus played a key role in studying the mathe...
Article
Let P be a finite poset. By definition, the linear extension polytope of P has as vertices the characteristic vectors of all linear extensions of P. In case P is an antichain, it is the linear ordering polytope. The linear extension polytope appears in combinatorial optimization in the context of scheduling with precedence constraints, see e.g. [A....
Article
For any given finite group G, we construct a convex polytope and an antimatroid whose automorphism groups are both isomorphic to G. The convex polytope is combinatorial in the sense of Naddef and Pulleyblank (1981), in particular it is binary; the diameter of its skeleton is at most 2; any automorphism of the polytope skeleton is the restriction of...
Article
Mixture models on order relations play a central role in recent investigations of transitivity in binary choice data. In such a model, the vectors of choice probabilities are the convex combinations of the characteristic vectors of all order relations of a chosen type. The five prominent types of order relations are linear orders, weak orders, semi...
Article
Knowledge assessment should be a central component in any computerized course. Procedures for knowledge assessment have been designed in the theoretical setting of knowledge spaces and learning spaces. They allow for a better outcome than just a crude number, giving deeper information on what is mastered together with advices for further study. Kno...
Article
Unlike poset antimatroids, chordal graph shelling antimatroids have received little attention as regard their structures, optimization properties and associated circuits. Here we consider a special case of those antimatroids, namely the split graph shelling antimatroids. We establish a connection between the structure of split graph shelling antima...
Article
Full-text available
We consider the covariance matrix for dichotomous Guttman items under a set of uniformity conditions, and obtain closed-form expressions for the eigenvalues and eigenvectors of the matrix. In particular, we describe the eigenvalues and eigenvectors of the matrix in terms of trigonometric functions of the number of items. Our results parallel those...
Article
Full-text available
How to design automated procedures which (i) accurately assess the knowledge of a student, and (ii) efficiently provide advices for further study? To produce well-founded answers, Knowledge Space Theory relies on a combinatorial viewpoint on the assessment of knowledge, and thus departs from common, numerical evaluation. Its assessment procedures f...
Article
Full-text available
Mixture models on order relations play a central role in recent investigations of transitivity in binary choice data. In such a model, the vectors of choice probabilities are the convex combinations of the characteristic vectors of all order relations of a chosen type. The five prominent types of order relations are linear orders, weak orders, semi...
Conference Paper
Full-text available
In Knowledge Space Theory (KST), a knowledge structure encodes a body of information as a domain, consisting of all the relevant pieces of information, together with the collection of all possible states of knowledge, identified with specific subsets of the domain. Knowledge spaces and learning spaces are defined through pedagogically natural requi...
Chapter
The core of an educational software based on learning space theory, such as the ALEKS system, is a combinatorial structure representing the cognitive organization of a particular curriculum, like beginning algebra or 4th grade arithmetic. This structure consists in a family K of subsets of a basic set Q. The elements of Q are the types of problems...
Article
Full-text available
Let a finite semiorder, or unit interval order, be given. When suitably defined, its numerical representations are the solutions of a system of linear inequalities. They thus form a convex polyhedron. We show that the facets of the representation polyhedron correspond to the noses and hollows of the semiorder. Our main result is to prove that the s...
Article
Full-text available
A fixed, interval order is considered on a finite set of elements. When appropriately defined, its representations form a convex polyhedron. Our results describe the geometricstructure of the polyhedron. The facets are in a one-to-one correspondence with the objects of oneof four types: the minimal elements, the contractible elements as well as the no...
Chapter
An application of either QUERY or its extension PS-QUERY results in a knowledge space, that is, a structure closed under union which is not necessarily a learning space. In many practical situations, however, the essential properties of learning spaces are regarded as crucial. In particular, the fringes enables a compact, precise delineation of any...
Chapter
Suppose that some complex system is assessed by an expert, who checks for the presence or absence of some revealing features. Ultimately, the state of the system is described by the subset of features, from a possibly large set, which are detected by the expert. This concept is very general, and becomes powerful only on the background of specific a...
Chapter
This chapter discusses an assessment procedure that is similar in spirit to those described in Chapter 13, but different in a key aspect: it is based on a finite Markov chain rather than on a Markov process with an uncountable set of Markov states. As a consequence, the procedure requires less storage and computation. It can thus be implemented on...
Chapter
Suppose that, having applied the techniques described in the preceding chapters1, we have obtained a particular knowledge structure. We now ask: how can we uncover, by appropriate questioning, the knowledge state of a particular individual? Two broad classes of stochastic assessment procedures are described in this chapter and the next one.
Chapter
The theory described in this monograph has led to a number of applications, the most prominent ones being the ALEKS and the RATH educational softwares1,2. The focus of this chapter is on the ALEKS system. A large scale statistical analysis of the validity of its assessments was recently reported by Cosyn et al. (2010). We summarize these results wh...
Chapter
A ‘medium’ is a collection of transformations on a set of states, specified by two constraining axioms. The term ‘medium’ stems from the original intuition suggesting such a structure, which is that of a system exposed to a bombardment of bits of information, each of which is capable of modifying its state in a minute way 1997. The system could be,...
Article
Full-text available
A fixed, interval order is considered on a finite set of elements. When appropriately defined, its representations form a convex polyhedron. Our results describe the geometric structure of the polyhedron. The facets are in a one-to-one correspondence with the objects of one of four types: the minimal elements, the contractible elements as well as t...
Book
Learning spaces offer a rigorous mathematical foundation for various practical systems of knowledge assessment. An example is offered by the ALEKS system (Assessment and LEarning in Knowledge Spaces), a software for the assessment of mathematical knowledge. From a mathematical standpoint, learning spaces as well as knowledge spaces (which made the...
Chapter
In the vein of Theorem 7.4.1, derive the meaningful collection, satisfying the translation equation, whose initial code is $$F(x,\ y)={{\left( {{x}^{\frac{1}{\vartheta }}}+{{c}^{\frac{1}{\vartheta }}y} \right)}^{\vartheta }}.
Chapter
In the two preceding chapters, we have described assessment procedures for uncovering the knowledge state of a student in a scholarly topic. Such a knowledge state is one among possibly many states forming a knowledge structure for the topic. We now turn to the problem of building a knowledge structure in practice. In this chapter, we deal with the...
Chapter
A student is facing a teacher, who is probing his1 knowledge of high school mathematics. The student, a new recruit, is freshly arrived from a foreign country, and important questions must be answered. To which grade should the student be assigned? What are his strengths and weaknesses? Should the student take a remedial course in some subject? Whi...
Article
Full-text available
In the context of the relativistic Doppler effect [DE] and the Lorentz–Fitzgerald contraction [LF], we investigate the consequences of two abstract axioms [R] and [M] expressed in terms of an operation $${\oplus}$$ generalizing the addition of velocities and a function $${L:(\lambda, v)\mapsto L(\lambda, v)}$$ . The latter can represent either the...
Article
A graph is alpha-critical if its stability number increases whenever an edge is removed from its edge set. The class of alpha-critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a remarkable result of Lov\'asz (1978) is the...
Article
Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph vertices, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalitie...
Article
Full-text available
The formula for the relativistic Doppler effect is investigated in the context of two compelling invariance axioms. The axioms are expressed in terms of an abstract operation generalizing the relativistic addition of velocities. We prove the following results. (1) If the standard representation for the operation is not assumed a priori, then each o...
Article
The binary choice polytope appeared in the investigation of the binary choice prob-lem formulated by Guilbaud (1953) and Block and Marschak (1960). It is nowadays known to be the same as the linear ordering polytope from operations research (Grötschel, Jünger and Reinelt, 1985). The central problem is to find facet-defining linear inequalities for...
Article
The m-subspace polytope is defined as the convex hull of the characteristic vectors of all m-dimensional subspaces of a finite affine space. The particular case of the hyperplane polytope has been investigated by Maurras (1993) and Anglada and Maurras (2003), who gave a complete characterization of the facets. The general m-subspace polytope that w...
Article
Full-text available
Finding a sequence of transpositions that transforms a given permutation into the identity permutation and is of the shortest possible length is an important problem in bioinformatics. Here, a transposition consists in exchanging two contiguous intervals of the permutation. Bafna and Pevzner introduced the cycle graph as a tool for working on this...
Conference Paper
This paper is adapted from a book and many scholarly articles. It reviews the main ideas of a theory for the assessment of a student's knowledge in a topic and gives details on a practical implementation in the form of a software. The basic concept of the theory is the ‘knowledge state,' which is the complete set of problems that an individual is c...
Conference Paper
Full-text available
Adaptations from a book and many scholarly articles are presented. It reviews the main ideas of a novel theory for the assessment of a student's knowledge in a topic and gives details on a practical implementation in the form of a software system available on the Internet. The basic concept of the theory is the 'knowledge state,' which is the compl...
Article
We introduce a class of weighted graphs which generalize α-critical graphs. These graphs are in correspondence with certain facets of the linear ordering polytope.
Article
Some classical models of clustering (hierarchies, pyramids, etc.) are related to interval hypergraphs. In this paper we study clustering models related to hypertrees which are an extension of interval hypergraphs. We first prove that a hypertree can ...
Article
Characterizing the size-independent model for approval voting of Falmagne and Regenwetter (J. Math. Psychol. 40 (1996) 152) was shown by Doignon and Regenwetter (J. Math. Psychol. 41 (1997) 171) to be equivalent to determining all facets of the approval-voting polytope. Here, we prove that the facets of this polytope correspond in a natural way to...
Article
Characterizing the size-independent model for approval voting of Falmagne and Regenwetter (J. Math. Psychol. 40 (1996) 152) was shown by Doignon and Regenwetter (J. Math. Psychol. 41 (1997) 171) to be equivalent to determining all facets of the approval-voting polytope. Here, we prove that the facets of this polytope correspond in a natural way to...
Article
Full-text available
Imbens and Angrist (1994) were the first to exploit a monotonicity condition in order to identify a local average treatment effect parameter using instrumental variables. More recently, Heckman and Vytlacil (1999) suggested the estimation of a variety of treatment effect parameters using a local version of their approach. We investigate the sensiti...
Article
Biorders, also called Ferrers relations, formalize Guttman scales. Irreflexive biorders on a set are exactly the interval orders on that set. The biorder polytope is the convex hull of the characteristic matrices of biorders. Its definition is thus similar to the definition of other order polytopes, the linear ordering polytope being the proeminent...
Article
Full-text available
Biorders were introduced first as Guttman scales and then as Ferrers relations. They are now well recognized in combinatorics and its applications. However, it seems that no procedure besides plain enumeration was made available for obtaining the number of biorders from an m-element set to an n-element set. We establish first a double-recurrence fo...
Article
Full-text available
A transitivity part of a relation on a set X is any subset of X on which the restric- tion of the relation is transitive. What can be recovered of a relation from the sole knowledge of its transitivity parts? In general, the relation itself cannot be recov- ered, because it has the same transitivity parts as its converse. In certain situations, the...
Article
Full-text available
Doignon and Fiorini (2003)determine all facets of the approval-voting polytope, thus offering a characterization of the size-independent model for approval voting of Falmagne and Regenwetter (1996). The present paper is a follow-up. It first provides an alternate proof of the basic result, which is more direct and at the same time constructive. The...
Article
Falmagne and Regenwetter (1996) proposed a probabilistic choice model of approval voting called the size-independent model. This model was investigated geometrically by Doignon and Regenwetter (1997), who introduced the approval-voting polytope associated with the choice model. In an effort to get closer to a full linear description of the polytope...
Article
One of the standard axioms for semiorders states that no three-point chain is incomparable to a fourth point. We refer to asymmetric relations satisfying this axiom as ‘almost connected orders’ or ‘ac-orders.’ It turns out that any relation lying between two weak orders, one of which covers the other for inclusion, is an ac-order (albeit of a speci...
Article
Full-text available
. The weak order polytopes are studied in Gurgel and Wakabayashi (1997), Gurgel and Wakabayashi (1996), and Fiorini and Fishburn (1999). We make use of their natural, affine projection onto the partition polytopes to determine several new families of facets for them. It turns out that not all facets of partition polytopes are lifted into facets of...
Article
The classical notion of dimension of a partial order can be extended to the valued setting, as was indicated in a particular case by Ovchinnikov (1984) (Ovchinnikov, S.V., 1984. Representations of transitive fuzzy relations. In: Skala, H.J., Termini, S., Trillas, E. (Eds.), Aspects of vagueness. Reidel, Boston, pp. 105–118). Relying on Valverde's r...
Article
Any partial order is an intersection of linear orders, generally in many ways. The smallest number of linear orders required in such an intersection is called the dimension of the given partial order. This concept of dimension can be trivially extended to quasi orders (i.e. reflexive and transitive relations), by using weak orders instead of linear...
Article
Two recent developments in random utility theory are reviewed, with special attention devoted to their combinatoric and geometric underpinnings. One concerns a new class of stochastic models describing the evolution of preferences, and the other some probabilistic models for approval voting. After recalling various commonly used preference relation...
Chapter
Suppose that, having applied the techniques described in the preceding chapters, we have obtained a particular knowledge structure. We now ask: how can we uncover, by appropriate questioning, the knowledge state of a particular individual? Two broad classes of stochastic assessment procedures are described in this chapter and the next one.
Chapter
When a knowledge structure is a quasi ordinal space, it can be faithfully represented by its surmise relation (cf. Theorem 1.49). In fact, as illustrated by Example 1.46, an ordinal space is completely recoverable from the Hasse diagram of the surmise relation. However, for knowledge structures in general, and even for knowledge spaces, the informa...
Chapter
In vaxious preceding chapters, several one-to-one correspondences were established between particular collections of mathematical structures. For instance, Birkhoff’s Theorem 1.49 asserts the existence of a one-to-one correspondence between the collection of all quasi ordinal spaces on a domain Q, and the collection of all quasi orders on Q. All th...
Chapter
The concept of a knowledge structure is a deterministic one. As such, it does not provide realistic predictions of subjects’ responses to the problems of a test. There are two ways in which probabilities must enter in a realistic model. For one, the knowledge states will certainly occur with different frequencies in the population of reference. It...
Chapter
So far, cognitive interpretations of our mathematical concepts have been limited to the use of mildly evocative words such as ‘knowledge state’, ‘learning path’ or ‘gradation.’ This makes sense since, as suggested by our Examples in 0.9, 0.10 and 0.11, many of our results are potentially applicable to widely different fields. It must be realized, h...
Chapter
In Chapter 5, we established the equivalence of two seemingly quite different concepts: on the one hand the knowledge spaces, and on the other hand the entailments for Q. Recall that the latter are the relations {IE275-1} that satisfy the following two conditions: for all q ∈ Q and A, B ∈ 2Q\{∅}, (1) if q ∈ A, then {IE274-2}; (2) if {IE274-3} and {...
Chapter
The knowledge state of an individual may vary over time. For example, the following learning scheme is reasonable. A novice student is in the empty state and thus knows nothing at all. Then, one or a few items are mastered; next, another batch is absorbed, etc., up to the eventual mastery of the full domain of the knowledge structure. There may be...
Chapter
The stochastic theory presented in this chapter is more ambitious than those examined in Chapter 7. The description of the learning process is more complete and takes place in real time, rather than in a sequence of discrete trials. This theory also contains a provision for individual differences. Nevertheless, its basic intuition is similar, in th...
Chapter
This chapter presents an assessment procedure that is similar in spirit to those described in Chapter 10, but different in a key aspect: it is based on a finite Markov chain rather than on a Markov process with an uncountable set of Markov states. As a consequence, the procedure requires less storage and computation and can thus be implemented on a...
Chapter
Suppose that some complex system is assessed by an expert, who checks for the presence or absence of some revealing features. Ultimately, the state of the system is described by the subset of features (from a possibly large set) which are detected by the expert. This concept is very general, and becomes powerful only on the background of specific a...
Chapter
In practice, how can we build a knowledge structure for a specific body of information? The first step is to select the items forming a domain Q. For real-life applications, we will typically assume this domain to be finite. The second step is then to construct a list of all the subsets of Q that are knowledge states. To secure such a list, we coul...
Chapter
A student is facing a teacher, who is probing her1 knowledge of high school mathematics. The student, a new recruit, is freshly arrived from a foreign country, and important questions must be answered: to which grade should the student be assigned? What are her strengths and weaknesses? Should the student take a remedial course in some subject? Whi...
Chapter
How can we economically describe a state in a knowledge structure? The question is inescapable because realistic states will typically be quite large. In such cases, it is impractical to describe a state by giving the full list of items that it contains. It is also unnecessary: because of the redundancy in many real-life knowledge structures1, a st...
Article
Two probabilistic models for subset choices are compared. The first one, due to Marley (1993), was dubbed the latent-scale model by Regenwetter, Marley, and Joe (1996). The second one is Falmagne and Regenwetter's (1996) size-independent model of approval voting. We show that for up to five choice alternatives, the choice probabilities generated by...
Book
We have learned from Theorem 2.2.4 that any learning space is a knowledge space, that is, a knowledge structure closed under union. The ∪-closure property is critical for the following reason. Certain knowledge spaces, and in particular the finite ones, can be faithfully summarized by a subfamily of their states. To wit, any state of the knowledge...
Article
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...
Article
A probabilistic model of approval voting on n alternatives generates a collection of probability distributions on the family of all subsets of the set of alternatives. Focusing on the size-independent model proposed by Falmagne and Regenwetter, we recast the problem of characterizing these distributions as the search for a minimal system of linear...
Article
Following up on previous results by Falmagne, this paper investigates possible mechanisms explaining how preference relations are created and how they evolve over time. We postulate a preference relation which is initially empty and becomes increasingly intricate under the influence of a random environment delivering discrete tokens of information...
Article
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.
Article
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...
Chapter
An easy and common way of assessing a student’s knowledge consists of a written examination. A list of questions is presented, the student’s answers are collected, and finally the examiner returns an appreciation, which usually boils down to a single number or percentage. Table 1.1 presents an excerpt of such a test in elementary arithmetics and wi...
Article
Patrick Suppes, professor of philosophy, statistics, education, and psychology at Stanford University (California), president of Computer Curriculum Corporation (Palo Alto, California), and recipient of the American Psychological Association's (APA's) Distinguished Scientific Contribution Award, is author of the chapter "Philosophy and the Sciences...
Book
Sometime in the late sixties, one of the editors of this volume realized that the mathematica psychologists in Europe-an odd lot mostly concentrated in Germany, the Netherlands, France, England and Belgium-were suffering from an acute sense of isolation. The papers that they presented at meetings of their national or regional societies had to be 's...
Chapter
In decision making under multiple criteria, the weighting of attributes should agree with the relative importance of attribute coalitions. Thus, the choice of weights amounts to a representation problem in the sense of measurement theory. Conditions for the existence and the uniqueness of the weights can be derived from similar results on the repre...
Chapter
There are few, if any, genuine applications of geometry in the behavioral sciences. Factor analysis, or multidimensional scaling methods, cannot be used as examples since the basic concepts of geometry — such as the concepts of ‘line’ or ‘direction’ — do not play any role in the representation. (Typically, the lines obtained in factor analytic or m...
Article
Gives a comprehensive description of a theory for the efficient assessment of knowledge. The essential concept is that the knowledge state of a subject with regard to a specified field of information can be represented by a particular subset of questions or problems that the subject is capable of solving. The family of all knowledge states forms t...
Article
Full-text available
A particular field of knowledge is conceptualized as a set of problems (or questions). A person's 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 corre...
Article
A knowledge space was defined as a family of subsets of questions closed under union. This concept provides a formal background for the design of knowledge assessment algorithms. We investigate here various numerical parameters measuring in some sense the intricacy of such a structure. Two sources are considered for these parameters: axiomatic conv...
Article
Full-text available
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 specifi...
Article
Defines a Markovian class of stochastic assessment procedures and investigates the properties of such a system. The knowledge state of an individual with respect to a particular body of information is conceptualized as the set of all the questions that the S is capable of solving. The goal of an assessment procedure is to identify, by a sequence of...
Article
Interval orders and semiorders ore models of preference structures involving a discrimination threshold. Recently two kinds of generalizations were described: one leads to the corresponding partial structures, the other covers the case of multiple levels of preference. Here the two extensions are combined. Numerical representations of the ensuing s...
Article
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 provid...
Article
Any element S in a family ψ of subsets of a finite set X can be specified by a sequence of statements such as: x ∈ S, y ∉ S, t ∉ S,…, zϵS. This sequence can be coded as a “word” and a complete set of such words forms a “descriptive language” for the family ψ. This class of languages is defined precisely, and some connections between such languages...
Article
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 possib...
Article
We survey sone recent, yet unpublished results that generalize interval and semi orders in three directions: (i) multiple thresholds, (ii) partial structures, (iii) infinite underlying set.

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How many objects of size n are there with property P? Is there a formula? What does their distribution look like for large values of n?
Project
This work is in collaboration with Eric Cosyn, Chris Doble, Jean-Paul Doignon, and Andrea Spotto. The main concepts of learning space theory can be adapted to computerized medical diagnosis. The essential difference lies in the fact that while a learning space is equipped with a unique maximal state, a diagnostic space may have several maximal states, corresponding to different variants of a disease or even different diseases. We begin by stating the axioms of a diagnostic space . We then observe that when a unique maximal state exists, then the diagnostic space becomes a learning space. We also give a detailed application of diagnostic space theory to the obsessive-compulsive disorder.