92 research outputs found

    There are only two paradoxes

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    Using a graph representation of classical logic, the paper shows that the liar or Yablo pattern occurs in every semantic paradox. The core graph theoretic result generalizes theorem of Richardson, showing solvability of finite graphs without odd cycles, to arbitrary graphs which are proven solvable when no odd cycles nor patterns generalizing Yablo's occur. This follows from an earlier result by a new compactness-like theorem, holding for infinitary logic and utilizing the graph representation.Comment: 7 pages, submitted to a journa

    Digraphs and cycle polynomials for free-by-cyclic groups

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    Let \phi \in \mbox{Out}(F_n) be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism ϕ\phi determines a free-by-cyclic group Γ=Fn⋊ϕZ,\Gamma=F_n \rtimes_\phi \mathbb Z, and a homomorphism α∈H1(Γ;Z)\alpha \in H^1(\Gamma; \mathbb Z). By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, α\alpha has an open cone neighborhood A\mathcal A in H1(Γ;R)H^1(\Gamma;\mathbb R) whose integral points correspond to other fibrations of Γ\Gamma whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an analog of McMullen's Teichm\"uller polynomial that computes the dilatations of all outer automorphism in A\mathcal A.Comment: 41 pages, 20 figure

    Paraconsistent resolution

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    Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models for inconsistent theories, specializing to the classical semantics for the consistent ones. Direct (instead of refutational) reasoning with classical resolution is sound and complete for this semantics, when augmented with a specific weakening which, in particular, excludes Ex Falso. Dropping all forms of weakening yields reasoning which also avoids typical fallacies of relevance

    Orbits of primitive k-homogenous groups on (N − k)-partitions with applications to semigroups

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    © 2018 American Mathematical Society. The purpose of this paper is to advance our knowledge of two of the most classic and popular topics in transformation semigroups: automorphisms and the size of minimal generating sets. In order to do this, we examine the k-homogeneous permutation groups (those which act transitively on the subsets of size k of their domain X) where |X| = n and k < n/2. In the process we obtain, for k-homogeneous groups, results on the minimum numbers of generators, the numbers of orbits on k-partitions, and their normalizers in the symmetric group. As a sample result, we show that every finite 2-homogeneous group is 2-generated. Underlying our investigations on automorphisms of transformation semigroups is the following conjecture: If a transformation semigroup S contains singular maps and its group of units is a primitive group G of permutations, then its automorphisms are all induced (under conjugation) by the elements in the normalizer of G in the symmetric group. For the special case that S contains all constant maps, this conjecture was proved correct more than 40 years ago. In this paper, we prove that the conjecture also holds for the case of semigroups containing a map of rank 3 or less. The effort in establishing this result suggests that further improvements might be a great challenge. This problem and several additional ones on permutation groups, transformation semigroups, and computational algebra are proposed at the end of the paper

    Graph Theory

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    This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results

    Extensions in graph normal form

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    Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.publishedVersio

    The modal logic of the countable random frame

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