Danger Invariants

Static analysers search for overapproximating proofs of safety commonly known as safety invariants. Conversely, static bug finders (e.g. Bounded Model Checking) give evidence for the failure of an assertion in the form of a counterexample trace. As opposed to safety invariants, the size of a counterexample is dependent on the depth of the bug, i.e., the length of the execution trace prior to the error state, which also determines the computational effort required to find them. We propose a way of expressing danger proofs that is independent of the depth of bugs. Essentially, such danger proofs constitute a compact representation of a counterexample trace, which we call a danger invariant. Danger invariants summarise sets of traces that are guaranteed to be able to reach an error state. Our conjecture is that such danger proofs will enable the design of bug finding analyses for which the computational effort is independent of the depth of bugs, and thus find deep bugs more efficiently. As an exemplar of an analysis that uses danger invariants, we design a bug finding technique based on a synthesis engine. We implemented this technique and compute danger invariants for intricate programs taken from SV-COMP 2016

A class of homogeneous scalar-tensor cosmologies with a radiation fluid

We present a new class of exact homogeneous cosmological solutions with a radiation fluid for all scalar-tensor theories. The solutions belong to Bianchi type $VI_{h}$ cosmologies. Explicit examples of nonsingular homogeneous scalar-tensor cosmologies are also given.Comment: 7 pages, LaTex; v2 type mistakes corrected, comments adde

RGIsearch: A C++ program for the determination of Renormalization Group Invariants

RGIsearch is a C++ program that searches for invariants of a user-defined set of renormalization group equations. Based on the general shape of the $\beta$-functions of quantum field theories, RGIsearch searches for several types of invariants that require different methods. Additionally, it supports the computation of invariants up to two-loop level. A manual for the program is given, including the settings and set-up of the program, as well as a test case

Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play

Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field $\mathbb Q(\zeta_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(\zeta_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(\zeta_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_{\mathbb Q(\zeta_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a set of primes of natural density 1, but false in general. Responsible for this are irregularities in the distribution of the primes. With this survey we hope to convince the reader that the apparently dissimilar mathematical objects studied by Kummer and Ihara actually display a very similar behaviour.Comment: 20 pages, 1 figure, survey, to appear in `Irregularities in the Distribution of Prime Numbers - Research Inspired by Maier's Matrix Method', Eds. J. Pintz and M. Th. Rassia