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Blockchain technology started as the backbone for cryptocurriencies and it has emerged as one of the most interesting technologies of the last decade. It is a new paradigm able to modify the way how industries transact. Today, the industries’ concern is about their ability to handle a high volume of data transactions per second while preserving both decentralization and security. Both decentralization and security are guaranteed by the mathematical strength of cryptographic primitives. There are two main approaches to achieve consensus: the Proof-of-Work based blockchains—PoW—and the Proof-of-Stake—PoS. Both of them come with some pros and drawbacks, but both rely on cryptography. In this survey, we present a review of the main consensus procedures, including the new consensus proposed by Algorand: Pure Proof-of-Stake—Pure PoS. In this article, we provide a framework to compare the performances of PoW, PoS and the Pure PoS, based on throughput and scalability.

Texts in Transit addresses the question what happened to texts during their production in printing houses in the fifteenth century. Lotte Hellinga finds some answers by exploring printer's copy and proofs in diverse printing houses, covering the period 1459 -1496.

In this paper we prove that no packing of unit balls in Euclidean space ℝ⁸ has density greater than that of the E₈-lattice packing.

In this paper a framework to distinguish in a Fregean manner between sense and denotation of \\(\\)-term-annotated derivations will be applied to a bilateralist sequent calculus displaying two derivability relations, one for proving and one for refuting. Therefore, a two-sorted typed \\(\\)-calculus will be used to annotate this calculus and a Dualization Theorem will be given, stating that for any derivable sequent expressing a proof, there is also a derivable sequent expressing a refutation and vice versa. By having joint \\(\\)-term annotations for proof systems in natural deduction and sequent calculus style, a comparison with respect to sense and denotation between derivations in those systems will be feasible, since the annotations elucidate the structural correspondences of the respective derivations. Thus, we will have a basis for determining in which cases, firstly, derivations expressing a proof vs. derivations expressing a refutation and, secondly, derivations in natural deduction vs. in sequent calculus can be identified and on which level.

Mathematicians often intentionally leave gaps in their proofs. Based on interviews with mathematicians about their refereeing practices, this paper examines the character of intentional gaps in published proofs. We observe that mathematicians’ refereeing practices limit the number of certain intentional gaps in published proofs. The results provide some new perspectives on the traditional philosophical questions of the nature of proof and of what grounds mathematical knowledge.

Undergraduate students are expected to produce and comprehend constructive existence proofs; yet, these proofs are notoriously difficult for students. This study investigates students’ thinking about these proofs by asking students to validate two arguments for the existence of a mathematical object. The first argument featured a common structural error while the second was a valid argument of the claim. We found that the students often considered the logical structures of the arguments when validating them. They provided reasons for their evaluations, including why they thought the structure of the first argument functioned to prove the claim and why they thought the structure of the second argument did not function to prove the claim. We discuss how these reasons provide insights into why constructive existence proofs might be challenging for students. We end the paper with implications for the teaching and learning of constructive existence proofs and their proof frameworks.

Proof plays a central role in developing, establishing and communicating mathematical knowledge. Nevertheless, it is not such a central element in school mathematics. This article discusses some issues involving mathematical proof in school, intending to characterize the understanding of mathematical proof in school, its function and the meaning and relevance attributed to the notion of simple proof. The main conclusions suggest that the idea of addressing mathematical proof at all levels of school is a recent idea that is not yet fully implemented in schools. It requires an adaptation of the understanding of proof to the age of the students, reducing the level of formality and allowing the students to experience the different functions of proof and not only the function of verification. Among the different functions of proof, the function of explanation deserves special attention due to the illumination and empowerment that it can bring to the students and their learning. The way this function of proof relates to the notion of simple proof (and the related aesthetic issues) seems relevant enough to make it, in the future, a focus of attention for the teachers who address mathematical proof in the classroom. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

An account of mathematical understanding should account for the differences between theorems whose proofs are “easy” to discover, and those whose proofs are difficult to discover. Though Hilbert seems to have created proof theory with the idea that it would address this kind of “discovermental complexity”, much more attention has been paid to the lengths of proofs, a measure of the difficulty of verifying of a given formal object that it is a proof of a given formula in a given formal system. In this paper we will shift attention back to discovermental complexity, by addressing a “topological” measure of proof complexity recently highlighted by Alessandra Carbone ( 2009 ). Though we will contend that Carbone’s measure fails as a measure of discovermental complexity, it forefronts numerous important formal and epistemological issues that we will discuss, including the structure of proofs and the question of whether impure proofs are systematically simpler than pure proofs.

Adverse outcome pathways (AOPs) are a recent toxicological construct that connects, in a formalized, transparent and quality-controlled way, mechanistic information to apical endpoints for regulatory purposes. AOP links a molecular initiating event (MIE) to the adverse outcome (AO) via key events (KE), in a way specified by key event relationships (KER). Although this approach to formalize mechanistic toxicological information only started in 2010, over 200 AOPs have already been established. At this stage, new requirements arise, such as the need for harmonization and re-assessment, for continuous updating, as well as for alerting about pitfalls, misuses and limits of applicability. In this review, the history of the AOP concept and its most prominent strengths are discussed, including the advantages of a formalized approach, the systematic collection of weight of evidence, the linkage of mechanisms to apical end points, the examination of the plausibility of epidemiological data, the identification of critical knowledge gaps and the design of mechanistic test methods. To prepare the ground for a broadened and appropriate use of AOPs, some widespread misconceptions are explained. Moreover, potential weaknesses and shortcomings of the current AOP rule set are addressed (1) to facilitate the discussion on its further evolution and (2) to better define appropriate vs. less suitable application areas. Exemplary toxicological studies are presented to discuss the linearity assumptions of AOP, the management of event modifiers and compensatory mechanisms, and whether a separation of toxicodynamics from toxicokinetics including metabolism is possible in the framework of pathway plasticity. Suggestions on how to compromise between different needs of AOP stakeholders have been added. A clear definition of open questions and limitations is provided to encourage further progress in the field.

We introduce proof systems for propositional logic that admit short proofs of hard formulas as well as the succinct expression of most techniques used by modern SAT solvers. Our proof systems allow the derivation of clauses that are not necessarily implied, but which are redundant in the sense that their addition preserves satisfiability. To guarantee that these added clauses are redundant, we consider various efficiently decidable redundancy criteria which we obtain by first characterizing clause redundancy in terms of a semantic implication relationship and then restricting this relationship so that it becomes decidable in polynomial time. As the restricted implication relation is based on unit propagation—a core technique of SAT solvers—it allows efficient proof checking too. The resulting proof systems are surprisingly strong, even without the introduction of new variables—a key feature of short proofs presented in the proof-complexity literature. We demonstrate the strength of our proof systems on the famous pigeon hole formulas by providing short clausal proofs without new variables.