Abstract For any $n<\omega $ we construct an infinite $(n+1)$ -generated Heyting algebra whose n -generated subalgebras are of cardinality $\leq m_n$ for some positive integer $m_n$ . From this we conclude that for every $n<\omega $ there exists a variety of Heyting algebras which contains an infinite $(n+1)$ -generated algebra, but which contains only finite n -generated algebras. For the case $n=2$ this provides a negative answer to a question posed by G. Bezhanishvili and R. Grigolia in [4].

A bi-Heyting algebra validates the Gödel-Dummett axiom (p→q)∨(q→p) iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension bi-GD of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice Λ(bi-GD) of extensions of bi-GD. We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of bi-GD. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of bi-GD. We introduce a sequence of co-trees, called the finite combs, and show that a logic in Λ(bi-GD) is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest nonlocally tabular extension of bi-GD and consequently, a unique pre-locally tabular extension of bi-GD. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.

Recent developments in Generative Deep Learning have fostered new engineering methods for protein design. Although deep generative models trained on protein sequence can learn biologically meaningful representations, the design of proteins with optimised properties remains a challenge. We combined deep learning architectures with evolutionary computation to steer the protein generative process towards specific sets of properties to address this problem. The latent space of a Variational Autoencoder is explored by evolutionary algorithms to find the best candidates. A set of single-objective and multi-objective problems were conceived to evaluate the algorithms' capacity to optimise proteins. The optimisation tasks consider the average proteins' hydrophobicity, their solubility and the probability of being generated by a defined functional Hidden Markov Model profile. The results show that Evolutionary Algorithms can achieve good results while allowing for more variability in the design of the experiment, thus resulting in a much greater set of possibly functional novel proteins.

Through real-life experiments, it has been proved that spoofing is a practical threat to applications using the free civil service provided by Global Navigation Satellite Systems (GNSS). In this paper, we demonstrate a prototype that can verify the integrity of GNSS civil signals. By integrity we intuitively mean that civil signals originate from a GNSS satellite without having been artificially interfered with. Our prototype provides interfaces that can incorporate existing spoofing detection methods whose results are then combined into an overall evaluation of the signal's integrity, which we call integrity level. Considering the various security requirements from different applications, integrity levels can be calculated in many ways determined by their users. We also present an application scenario that deploys our prototype and offers a public central service -- localisation assurance certification. Through experiments, we successfully show that our prototype is not only effective but also efficient in practice.

Through real-life experiments, it has been proved, not only in theory but also in practice, that civil signals of Global Navigation Satellite Systems (GNSS) can be spoofed. Consequently, a number of spoofing detection techniques have been proposed to verify the integrity of GNSS signals. In this paper, we develop a novel trust framework based on subjective logic to evaluate the integrity of received GNSS civil signals. We formally define signal integrity for the first time in the framework and use it to precisely characterise different spoofing detection methods. Our framework captures the uncertainty during the inference of signal integrity which has been largely ignored or not explicitly specified in the literature. Our framework also gives rise to several natural ways to combine the outputs of various spoofing detection methods on signal integrity. We validate our framework through experiments using both real and simulated signals and the results show that our framework is effective.