Abstract. We study languages over infinite alphabets equipped with some structure that can be tested by recognizing automata. We develop a framework for studying such alphabets and the ensuing automata theory, where the key role is played by an automorphism group of the alphabet. In the process, we generalize nominal sets due to Gabbay and Pitts.

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This paper contains two results on timed extensions of pushdown automata (PDA). As our first result we prove that the model of dense-timed PDA of Abdulla et al. collapses: it is expressively equivalent to dense-timed PDA with timeless stack. Motivated by this result, we advocate the framework of firstorder definable PDA, a specialization of PDA in sets with atoms, as the right setting to define and investigate timed extensions of PDA. The general model obtained in this way is Turing complete. As our second result we prove NEXPTIME upper complexity bound for the non-emptiness problem for an expressive subclass. As a byproduct, we obtain a tight EXPTIME complexity bound for a more restrictive subclass of PDA with timeless stack, thus subsuming the complexity bound known for dense-timed PDA.

Logical relations and their generalisations are a fundamental tool in proving properties of lambda calculi, for example, for yielding sound principles for observational equivalence. We propose a natural notion of logical relations that is able to deal with the monadic types of Moggi's computational lambda calculus. The treatment is categorical, and is based on notions of subsconing, mono factorisation systems and monad morphisms. Our approach has a number of interesting applications, including cases for lambda calculi with non-determinism (where being in a logical relation means being bisimilar), dynamic name creation and probabilistic systems.

A notion of alternating timed automata is proposed. It is shown that such automata with only one clock have decidable emptiness problem over finite words. This gives a new class of timed languages that is closed under boolean operations and which has an effective presentation. We prove that the complexity of the emptiness problem for alternating timed automata with one clock is nonprimitive recursive. The proof gives also the same lower bound for the universality problem for nondeterministic timed automata with one clock. We investigate extension of the model with epsilon-transitions and prove that emptiness is undecidable. Over infinite words, we show undecidability of the universality problem.

We study Turing machines over sets with atoms, also known as nominal sets. Our main result is that deterministic machines are weaker than nondeterministic ones; in particular, P =NP in sets with atoms. Our main construction is closely related to the Cai-Fürer-Immerman graphs used in descriptive complexity theory.

Background: Transposable elements constitute a significant fraction of plant genomes. The PIF/Harbinger superfamily includes DNA transposons (class II elements) carrying terminal inverted repeats and producing a 3 bp target site duplication upon insertion. The presence of an ORF coding for the DDE/DDD transposase, required for transposition, is characteristic for the autonomous PIF/Harbinger-like elements. Based on the above features, PIF/Harbinger-like elements were identified in several plant genomes and divided into several evolutionary lineages. Availability of a significant portion of Medicago truncatula genomic sequence allowed for mining PIF/Harbinger-like elements, starting from a single previously described element MtMaster.

Multimodal oncological strategies which combine chemotherapy or radiotherapy with hyperthermia, have a potential of improving the efficacy of the non-surgical methods of cancer treatment. Hyperthermia engages the heat-shock response (HSR) mechanism, the main component of which are heat-shock proteins. Cancer cells have already partially activated HSR, thereby hyperthermia may be more toxic to them relative to normal cells. On the other hand, HSR triggers thermotolerance, i.e. hyperthermia-treated cells show an impairment in their susceptibility to a subsequent heat-induced stress. This poses questions about efficacy and optimal strategy for anti-cancer therapy combined with hyperthermia treatment. To address these questions, we adapt our previous HSR model and propose its stochastic extension. We formalize the notion of a HSP-induced thermotolerance. Next, we estimate the intensity and the duration of the thermotolerance. Finally, we quantify the effect of a multimodal therapy based on hyperthermia and a cytotoxic effect of bortezomib, a clinically approved proteasome inhibitor. Consequently, we propose an optimal strategy for combining hyperthermia and proteasome inhibition modalities. In summary, by a mathematical analysis of HSR, we are able to support the common belief that the combination of cancer treatment strategies increases therapy efficacy.