The symmetric Post Correspondence Problem, and errata for the freeness problem for matrix semigroups

We define the symmetric Post Correspondence Problem (PCP) and prove that it is undecidable. As an application we show that the original proof of undecidability of the freeness problem for 3-by-3 integer matrix semigroups works for the symmetric PCP, but not for the PCP in general.Comment: 12 pages. Small corrections and clarifications were added in version

Extending the Ehresmann-Schein-Nambooripad Theorem

We extend the `join-premorphisms' part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (1991) for the `morphisms' part. However, it is so-called `meet-premorphisms' which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for meet-premorphisms in the case of two-sided restriction semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case.Comment: 23 pages; final section on Szendrei expansions removed; further reordering of materia

Quotient Complexity of Regular Languages

The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formal-language terms as the number of distinct quotients of the language, and to call it "quotient complexity". The problem of finding the quotient complexity of a language f(K,L) is considered, where K and L are regular languages and f is a regular operation, for example, union or concatenation. Since quotients can be represented by derivatives, one can find a formula for the typical quotient of f(K,L) in terms of the quotients of K and L. To obtain an upper bound on the number of quotients of f(K,L) all one has to do is count how many such quotients are possible, and this makes automaton constructions unnecessary. The advantages of this point of view are illustrated by many examples. Moreover, new general observations are presented to help in the estimation of the upper bounds on quotient complexity of regular operations

A Two-Locus Model of the Evolution of Insecticide Resistance to Inform and Optimise Public Health Insecticide Deployment Strategies

We develop a flexible, two-locus model for the spread of insecticide resistance applicable to mosquito species that transmit human diseases such as malaria. The model allows differential exposure of males and females, allows them to encounter high or low concentrations of insecticide, and allows selection pressures and dominance values to differ depending on the concentration of insecticide encountered. We demonstrate its application by investigating the relative merits of sequential use of insecticides versus their deployment as a mixture to minimise the spread of resistance. We recover previously published results as subsets of this model and conduct a sensitivity analysis over an extensive parameter space to identify what circumstances favour mixtures over sequences. Both strategies lasted more than 500 mosquito generations (or about 40 years) in 24% of runs, while in those runs where resistance had spread to high levels by 500 generations, 56% favoured sequential use and 44% favoured mixtures. Mixtures are favoured when insecticide effectiveness (their ability to kill homozygous susceptible mosquitoes) is high and exposure (the proportion of mosquitoes that encounter the insecticide) is low. If insecticides do not reliably kill homozygous sensitive genotypes, it is likely that sequential deployment will be a more robust strategy. Resistance to an insecticide always spreads slower if that insecticide is used in a mixture although this may be insufficient to outperform sequential use: for example, a mixture may last 5 years while the two insecticides deployed individually may last 3 and 4 years giving an overall ‘lifespan’ of 7 years for sequential use. We emphasise that this paper is primarily about designing and implementing a flexible modelling strategy to investigate the spread of insecticide resistance in vector populations and demonstrate how our model can identify vector control strategies most likely to minimise the spread of insecticide resistance