18

Open Problems

We have gathered a number of problems in the course of our investigations,which we have left unsolved. We list them below for the interested reader,with relevant references when available.

18.1 Knowledge Spaces and ∪-Closed Families

18.1.1 On large bases. Let Q be a domain with a finite number m of items.Consider all the bases of knowledge spaces on Q. What is the largest cardinal-ity of a base as a function of m ? What are all the knowledge bases on Q havinga base of that cardinality? A manuscript of R.T. Johnson and T.P. Vaughancontains results for small values of m. (It is the extended preprint version ofJohnson and Vaughan, 1998).

18.1.2 Defining a knowledge spaces by a language. Let K be a discrim-inating knowledge space. Does there always exist some assessment language(in the sense of 9.2.3) describing K but no other knowledge space? Or better,describing no other knowledge structure?

18.1.3 Projections and bases. Let K be a knowledge space on the domainQ and for any nonempty subset Q′ of Q denote by K′ the projection of K onQ′ (cf. Theorem 13.7.4). Find necessary and sufficient (interesting) conditionson the knowledge space K implying that the space K′ always has a base.Conditions that are only sufficient would also be of interest if they cover awide variety of examples (among which the finitary spaces).

18.1.4 Uniqueness of a Hasse system. Characterize efficiently the gran-ular knowledge spaces that admit a unique Hasse system (see end of Sec-tion 5.5).

J.-C. Falmagne, J.-P. Doignon, Learning Spaces, DOI 10.1007/978-3-642-01039-2_18, © Springer-Verlag Berlin Heidelberg 2011

376 18 Open Problems

18.1.5 Frankl’s Conjecture. A famously difficult problem asks whether forany finite union-closed family K with ∪K finite and K 6= {∅}, there alwaysexists some element in ∪K that belongs to at least half of the subsets in K.This is often referred to as Frankl’s conjecture. A good access to the literatureis the Wikipedia entry for “Union-closed sets conjecture.” It should be com-plemented with Johnson and Vaughan (1998). (Warning: the risk of wastingprecious research time is high.)

18.2 Wellgradedness and the Fringes

18.2.1 Strengthening [L1] and [L2]. Lemma 2.2.7 asserts that any well-graded partially union-closed family is a partial learning space, and that theconverse implication does not hold. Find axioms that strengthen (or at leastare in the spirit of) Axioms [L1] and [L2] that characterize well-graded par-tially union-closed families.

18.2.2 About the fringes economy. The fringes of a state in a learningspace were defined in 4.1.6. The concept was introduced informally earlier,in Section 1.1.5, and presented there as a device permitting an economicalrepresentation of the states. Consider the following parameter for measuringthe overall economy realized by such a representation: the sum of the sizes ofall the states minus the sum of the sizes of all the fringes. (Another parameteris obtained when we divide that difference by the number of states.) It is notdifficult to find examples in which the fringe representation of states is noteconomical at all (that is, in which the parameter takes a negative value). Onthe other hand, for what kind of learning space is the economy:

(i) maximal in the sense that for a fixed number of items, the parameter takesthe largest possible value)?

(ii) minimal (for a fixed number of items, the parameter takes the smallestpossible value)?

18.2.3 Characterize fringes mappings. Characterize those mappingsK 7→ (KI,KO), K 7→ KI ∪KO, K 7→ KI, etc. arising from learning spaces.Which learning spaces are completely specified by any such mappings (one ormore of them)? The same problem can be raised for well-graded families.

18.2.4 Characterize well-graded spans. Theorem 4.5.8 characterizes thosefamilies whose span† is well-graded. However the characterization refers ex-plicitly to the span†. Find a characterization solely in terms of the spanningfamily.

18.4 Miscellaneous 377

18.3 About Granularity

18.3.1 Dropping the granularity assumption. Does the conclusion ofTheorem 5.5.6 still hold when granularity is not assumed? (cf. Remark 5.5.7(a)).

18.3.2 Characterize granular attributions. In 8.5.2, we defined the con-cept of a granular attribution by the property that such an attribution pro-duces a granular knowledge space. So far, we do not have a direct characteri-zation of this concept.

18.4 Miscellaneous

18.4.1 The width and the dimension of a surmise system. Surmisesystems and AND/OR graphs (cf. Definitions 5.1.2 and 5.3.1) are two aspectsof a same generalization of partially ordered sets. Concepts which are classicalfor partially ordered sets can, in principle, be extended to surmise systems.This generates a large collection of problems. For instance, what would be ap-propriate extensions of classical concepts such as the ‘width’, the ‘dimension’,etc. of a partial order? Do central theorems about these concepts remain truefor the extended situation? (A first pass at some of these problems was madein Doignon and Falmagne, 1988.)

18.4.2 About the set differences for projections. Under wich conditionson a knowledge structure (Q̊, K̊) are all the differences S(a,K)\S(a,K ∪{b})empty, for all the projections (Q,K) of (Q̊, K̊)? (Cf. Example 12.7.2).