First order convergence of matroids

The model theory based notion of the first order convergence unifies the notions of the left-convergence for dense structures and the Benjamini-Schramm convergence for sparse structures. It is known that every first order convergent sequence of graphs with bounded tree-depth can be represented by an analytic limit object called a limit modeling. We establish the matroid counterpart of this result: every first order convergent sequence of matroids with bounded branch-depth representable over a fixed finite field has a limit modeling, i.e., there exists an infinite matroid with the elements forming a probability space that has asymptotically the same first order properties. We show that neither of the bounded branch-depth assumption nor the representability assumption can be removed.Comment: Accepted to the European Journal of Combinatoric

Even Delta-Matroids and the Complexity of Planar Boolean CSPs

The main result of this paper is a generalization of the classical blossom algorithm for finding perfect matchings. Our algorithm can efficiently solve Boolean CSPs where each variable appears in exactly two constraints (we call it edge CSP) and all constraints are even $\Delta$-matroid relations (represented by lists of tuples). As a consequence of this, we settle the complexity classification of planar Boolean CSPs started by Dvorak and Kupec. Using a reduction to even $\Delta$-matroids, we then extend the tractability result to larger classes of $\Delta$-matroids that we call efficiently coverable. It properly includes classes that were known to be tractable before, namely co-independent, compact, local, linear and binary, with the following caveat: we represent $\Delta$-matroids by lists of tuples, while the last two use a representation by matrices. Since an $n\times n$ matrix can represent exponentially many tuples, our tractability result is not strictly stronger than the known algorithm for linear and binary $\Delta$-matroids.Comment: 33 pages, 9 figure

Tree automata and pigeonhole classes of matroids -- I

Hlineny's Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of F-representable matroids with bounded branch-width (where F is a finite field). If each matroid in a class can be decomposed by a subcubic tree in such a way that only a bounded amount of information flows across displayed separations, then the class has bounded decomposition-width. We introduce the pigeonhole property for classes of matroids: if every subclass with bounded branch-width also has bounded decomposition-width, then the class is pigeonhole. An efficiently pigeonhole class has a stronger property, involving an efficiently-computable equivalence relation on subsets of the ground set. We show that Hlineny's Theorem extends to any efficiently pigeonhole class. In a sequel paper, we use these ideas to extend Hlineny's Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, where H is any finite group. We also give a characterisation of the families of hypergraphs that can be described via tree automata: a family is defined by a tree automaton if and only if it has bounded decomposition-width. Furthermore, we show that if a class of matroids has the pigeonhole property, and can be defined in monadic second-order logic, then any subclass with bounded branch-width has a decidable monadic second-order theory.Comment: Slightly extending the main theorem to cover a more expressive logi

Structure identification in relational data

This paper presents several investigations into the prospects for identifying meaningful structures in empirical data, namely, structures permitting effective organization of the data to meet requirements of future queries. We propose a general framework whereby the notion of identifiability is given a precise formal definition similar to that of learnability. Using this framework, we then explore if a tractable procedure exists for deciding whether a given relation is decomposable into a constraint network or a CNF theory with desirable topology and, if the answer is positive, identifying the desired decomposition. Finally, we address the problem of expressing a given relation as a Horn theory and, if this is impossible, finding the best k-Horn approximation to the given relation. We show that both problems can be solved in time polynomial in the length of the data

Qualitative Reasoning about Relative Directions : Computational Complexity and Practical Algorithm

Qualitative spatial reasoning (QSR) enables cognitive agents to reason about space using abstract symbols. Among several aspects of space (e.g., topology, direction, distance) directional information is useful for agents navigating in space. Observers typically describe their environment by specifying the relative directions in which they see other objects or other people from their point of view. As such, qualitative reasoning about relative directions, i.e., determining whether a given statement involving relative directions is true, can be advantageously used for applications, for example, robot navigation, computer-aided design and geographical information systems. Unfortunately, despite the apparent importance of reasoning about relative directions, QSR-research so far could not provide efficient decision procedures for qualitative reasoning about relative directions. Accordingly, the question about how to devise an efficient decision procedure for qualitative reasoning about relative directions has meanwhile turned to the question about whether an efficient decision procedure exists at all. Answering the latter existential question, which requires a formal analysis of relative directions from a computational complexity point of view, has remained an open problem in the field of QSR. The present thesis solves the open problem by proving that there is no efficient decision procedure for qualitative reasoning about relative directions, even if only left or right relations are involved. This is surprising as it contradicts the early premise of QSR believed by many researchers in and outside the field, that is, abstracting from an infinite domain to a finite set of relations naturally leads to efficient reasoning. As a consequence of this rather negative result, efficient reasoning with any of the well-known relative direction calculi (OPRAm, DCC, DRA, LR) is impossible. Indeed, the present thesis shows that all the relative direction calculi belong to one and the same class of ∃R-complete problems, which are the problems that can be reduced to the NP-hard decision problem of the existential theory of the reals, and vice versa. Nevertheless, in practice, many interesting computationally hard AI problems can be tackled by means of approximative algorithms and heuristics. In the same vein, the present thesis shows that qualitative reasoning about relative directions can also be tackled with approximative algorithms. In the thesis we develop the qualitative calculus SVm which allows for a practical algorithm for qualitative reasoning about relative directions. SVm also provides an effective semi-decision procedure for the OPRAm calculus, the most versatile one among the relative direction calculi. In this thesis we substantiate the usefulness of SVm by applying it in the marine navigation domain