Incompleteness Theorems, Large Cardinals, and Automata over Finite Words

International audienceWe prove that one can construct various kinds of automata over finite words for which some elementary properties are actually independent from strong set theories like Tn =:ZFC + "There exist (at least) n inaccessible cardinals", for integers n ≥ 0. In particular, we prove independence results for languages of finite words generated by context-free grammars, or accepted by 2-tape or 1-counter automata. Moreover we get some independence results for weighted automata and for some related finitely generated subsemigroups of the set Z ^{3×3} of 3-3 matrices with integer entries. Some of these latter results are independence results from the Peano axiomatic system PA

Fractal Intersections and Products via Algorithmic Dimension

Algorithmic dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that a known intersection formula for Borel sets holds for arbitrary sets, and it significantly simplifies the proof of a known product formula. Both of these formulas are prominent, fundamental results in fractal geometry that are taught in typical undergraduate courses on the subject

Results on the Dimension Spectra of Planar Lines

In this paper we investigate the (effective) dimension spectra of lines in the Euclidean plane. The dimension spectrum of a line L_{a,b}, sp(L), with slope a and intercept b is the set of all effective dimensions of the points (x, ax + b) on L. It has been recently shown that, for every a and b with effective dimension less than 1, the dimension spectrum of L_{a,b} contains an interval. Our first main theorem shows that this holds for every line. Moreover, when the effective dimension of a and b is at least 1, sp(L) contains a unit interval. Our second main theorem gives lower bounds on the dimension spectra of lines. In particular, we show that for every alpha in [0,1], with the exception of a set of Hausdorff dimension at most alpha, the effective dimension of (x, ax + b) is at least alpha + dim(a,b)/2. As a consequence of this theorem, using a recent characterization of Hausdorff dimension using effective dimension, we give a new proof of a result by Molter and Rela on the Hausdorff dimension of Furstenberg sets

Projection Theorems Using Effective Dimension

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand\u27s projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorff and packing dimensions of orthogonal projections onto lines. The first shows that the conclusion of Marstrand\u27s theorem holds whenever the Hausdorff and packing dimensions agree on the set E, even if E is not analytic. Our second result gives a lower bound on the packing dimension of projections of arbitrary sets. Finally, we give a new proof of Marstrand\u27s theorem using the theory of computing

Faster FPT Algorithm for 5-Path Vertex Cover

The problem of d-Path Vertex Cover, d-PVC lies in determining a subset F of vertices of a given graph G=(V,E) such that G F does not contain a path on d vertices. The paths we aim to cover need not to be induced. It is known that the d-PVC problem is NP-complete for any d >= 2. When parameterized by the size of the solution k, 5-PVC has direct trivial algorithm with O(5^kn^{O(1)}) running time and, since d-PVC is a special case of d-Hitting Set, an algorithm running in O(4.0755^kn^{O(1)}) time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in O(4^kn^{O(1)}) time

Blocking independent sets for H-free graphs via edge contractions and vertex deletions.

Let d and k be two given integers, and let G be a graph. Can we reduce the independence number of G by at least d via at most k graph operations from some fixed set S? This problem belongs to a class of so-called blocker problems. It is known to be co-NP-hard even if S consists of either an edge contraction or a vertex deletion. We further investigate its computational complexity under these two settings: we give a sufficient condition on a graph class for the vertex deletion variant to be co-NP-hard even if d=k=1d=k=1 ; in addition we prove that the vertex deletion variant is co-NP-hard for triangle-free graphs even if d=k=1d=k=1 ; we prove that the edge contraction variant is NP-hard for bipartite graphs but linear-time solvable for trees. By combining our new results with known ones we are able to give full complexity classifications for both variants restricted to H-free graphs. D. Paulusma received support from EPSRC (EP/K025090/1)

Mathematical Logic: Proof Theory, Constructive Mathematics (hybrid meeting)

The Workshop "Mathematical Logic: Proof Theory, Constructive Mathematics" focused on proofs both as formal derivations in deductive systems as well as on the extraction of explicit computational content from given proofs in core areas of ordinary mathematics using proof-theoretic methods. The workshop contributed to the following research strands: interactions between foundations and applications; proof mining; constructivity in classical logic; modal logic and provability logic; proof theory and theoretical computer science; structural proof theory