Simon Vilmin's research while affiliated with Université Clermont Auvergne and other places

The notion of functional dependencies (FDs) can be used by data scientists and domain experts to confront background knowledge against data. To overcome the classical, too restrictive, satisfaction of FDs, it is possible to replace equality with more meaningful binary predicates, and use a coverage measure such as the \(g_3\)-error to estimate the degree to which a FD matches the data. It is known that the \(g_3\)-error can be computed in polynomial time if equality is used, but unfortunately, the problem becomes NP-complete when relying on more general predicates instead. However, there has been no analysis of which class of predicates or which properties alter the complexity of the problem, especially when going from equality to more general predicates. In this work, we provide such an analysis. We focus on the properties of commonly used predicates such as equality, similarity relations, and partial orders. These properties are: reflexivity, transitivity, symmetry, and antisymmetry. We show that symmetry and transitivity together are sufficient to guarantee that the \(g_3\)-error can be computed in polynomial time. However, dropping either of them makes the problem NP-complete.Keywordsfunctional dependencies\(g_3\)-errorpredicates

The notion of functional dependencies (FDs) can be used by data scientists and domain experts to confront background knowledge against data. To overcome the classical, too restrictive, satisfaction of FDs, it is possible to replace equality with more meaningful binary predicates, and use a coverage measure such as the $g_3$-error to estimate the degree to which a FD matches the data. It is known that the $g_3$-error can be computed in polynomial time if equality is used, but unfortunately, the problem becomes NP-complete when relying on more general predicates instead. However, there has been no analysis of which class of predicates or which properties alter the complexity of the problem, especially when going from equality to more general predicates. In this work, we provide such an analysis. We focus on the properties of commonly used predicates such as equality, similarity relations, and partial orders. These properties are: reflexivity, transitivity, symmetry, and antisymmetry. We show that symmetry and transitivity together are sufficient to guarantee that the $g_3$-error can be computed in polynomial time. However, dropping either of them makes the problem NP-complete.

This paper deals with the problem of finding the preferred extensions of an argumentation framework by means of a bijection with the naive extensions of another framework. First we consider the case where an argumentation framework is naive-realizable: its naive and preferred extensions are equal. Recognizing naive-realizable argumentation frameworks is hard, but we show that it is tractable for frameworks with bounded in-degree. Next, we give a bijection between the preferred extensions of an argumentation framework being admissible-closed (the intersection of two admissible sets is admissible) and the naive extensions of another framework on the same set of arguments. On the other hand, we prove that identifying admissible-closed argumentation frameworks is coNP-complete. At last, we introduce the notion of irreducible self-defending sets as those that are not the union of others. It turns out there exists a bijection between the preferred extensions of an argumentation framework and the naive extensions of a framework on its irreducible self-defending sets. Consequently, the preferred extensions of argumentation frameworks with some lattice properties can be listed with polynomial delay and polynomial space.

We study the well-known problem of translating between two representations of closure systems, namely implicational bases and meet-irreducible elements. Albeit its importance, the problem is open. In this paper, we introduce splits of an implicational base. It is a partitioning operation of the implications which we recursively apply to obtain a binary tree representing a decomposition of the implicational base. We show that this decomposition can be conducted in polynomial time and space in the size of the input implicational base. Focusing on the case of acyclic splits, we obtain a recursive characterization of the meet-irreducible elements of the associated closure system. We use this characterization and hypergraph dualization to derive new results for the translation problem in acyclic convex geometries.

Knowledge Space Theory (KST) is a field of mathematical psychology which aims to assess and represent students knowledge. Its core structures, knowledge spaces, are equivalent to closure systems (or lattices). Apart from KST, closure systems are used in numerous fields of computer science such as Formal Concept Analysis, propositional logic, database theory, combinatorial optimization or argumentation theory for instance. Because of their size, closure systems are often encoded with compact representations such as implications or meet-irreducible elements. In this thesis, we focus on two problems regarding closure systems and their representations.We begin with the problem of translating between the two representations of a closure system. This famous open problem generalizes hypergraph dualization. Our approach here is to hierarchically decompose a set of implications with partitioning operations called (acyclic) splits. We deduce a recursive characterization of the meet-irreducible elements of the associated closure system. As a consequence,we obtain new types of closure systems for which the translation can be done in output-quasipolynomial time using hypergraph dualization.Next, we study forbidden sets in closure systems. Here, the tasks we handle is the enumeration of the closed sets (the sets in the closure system) that are admissible and preferred (minimal or maximal admissible) with respect to a family of forbidden sets. With the help of dualization in lattices, we obtain several intractability results. On the positive side, we derive output-polynomial time algorithms under various restrictions concerning the Carathéodory number, forbidden pairs and forbidden co-pairs of elements.

It is well known that every closure system can be represented by an implicational base, or by the set of its meet-irreducible elements. In Horn logic, these are respectively known as the Horn expressions and the characteristic models. In this paper, we consider the problem of translating between the two representations in acyclic convex geometries. Quite surprisingly, we show that the problem in this context is already harder than the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem for which the existence of an output quasi-polynomial time algorithm is open. In light of this result, we consider a proper subclass of acyclic convex geometries, namely ranked convex geometries, as those that admit a ranked implicational base analogous to that of ranked posets. For this class, we provide output quasi-polynomial time algorithms based on hypergraph dualization for translating between the two representations.

Given an implicational base and an inconsistency binary relation over a finite set, we are interested in the problem of enumerating all maximal consistent closed sets (denoted by MCCEnum for short). We show that MCCEnum cannot be solved in output-polynomial time unless \({{\,\mathrm{ {\textsf {P}}}\,}}= {{\,\mathrm{ {\textsf {NP}}}\,}}\), even for lower bounded lattices. We give an incremental-polynomial time algorithm to solve MCCEnum for closure systems with constant Carathéodory number. Finally we prove that in biatomic atomistic closure systems MCCEnum can be solved in output-quasipolynomial time if minimal generators obey an independence condition, which holds in atomistic modular lattices. For closure systems closed under union (i.e. distributive), MCCEnum is solved by a polynomial delay algorithm [23, 26].