A Short Note on Undirected Fitch Graphs

The symmetric version of Fitch's xenology relation coincides with class of complete multipartite graph and thus cannot convey any non-trivial phylogenetic information

A structural Markov property for decomposable graph laws that allows control of clique intersections

We present a new kind of structural Markov property for probabilistic laws on decomposable graphs, which allows the explicit control of interactions between cliques, so is capable of encoding some interesting structure. We prove the equivalence of this property to an exponential family assumption, and discuss identifiability, modelling, inferential and computational implications.Comment: 10 pages, 3 figures; updated from V1 following journal review, new more explicit title and added section on inferenc

Generalized Fitch Graphs III: Symmetrized Fitch maps and Sets of Symmetric Binary Relations that are explained by Unrooted Edge-labeled Trees

Binary relations derived from labeled rooted trees play an import role in mathematical biology as formal models of evolutionary relationships. The (symmetrized) Fitch relation formalizes xenology as the pairs of genes separated by at least one horizontal transfer event. As a natural generalization, we consider symmetrized Fitch maps, that is, symmetric maps $\varepsilon$ that assign a subset of colors to each pair of vertices in $X$ and that can be explained by a tree $T$ with edges that are labeled with subsets of colors in the sense that the color $m$ appears in $\varepsilon(x,y)$ if and only if $m$ appears in a label along the unique path between $x$ and $y$ in $T$. We first give an alternative characterization of the monochromatic case and then give a characterization of symmetrized Fitch maps in terms of compatibility of a certain set of quartets. We show that recognition of symmetrized Fitch maps is NP-complete. In the restricted case where $|\varepsilon(x,y)|\leq 1$ the problem becomes polynomial, since such maps coincide with class of monochromatic Fitch maps whose graph-representations form precisely the class of complete multi-partite graphs

Improving legibility of natural deduction proofs is not trivial

In formal proof checking environments such as Mizar it is not merely the validity of mathematical formulas that is evaluated in the process of adoption to the body of accepted formalizations, but also the readability of the proofs that witness validity. As in case of computer programs, such proof scripts may sometimes be more and sometimes be less readable. To better understand the notion of readability of formal proofs, and to assess and improve their readability, we propose in this paper a method of improving proof readability based on Behaghel's First Law of sentence structure. Our method maximizes the number of local references to the directly preceding statement in a proof linearisation. It is shown that our optimization method is NP-complete.Comment: 33 page