A Blockchain-Based Microgrid Data Disaster Backup Scheme in Edge Computing

In view of the problems of low security, poor reliability, inability to backup automatically, and overreliance on the third party in traditional microgrid data disaster backup schemes based on cloud backup, the edge computing is used to preprocess power big data, and a microgrid data disaster backup scheme based on blockchain in edge computing environment is proposed in this paper. First, the honey encryption (HE) technology and advanced encryption standard (AES) are combined to propose a new encryption algorithm HE-AES, which is used to encrypt the preprocessed data. Second, the Kademlia algorithm is embedded in the edge server to realize the distributed storage and automatic recovery of microgrid data. Finally, the traditional proof of authority (PoA) consensus mechanism is improved partially, and the improved PoA is used to make each node reach consensus and pack blocks on the chain. The scheme can not only realize the data disaster backup automatically but also has high efficiency of data processing, which can provide a new idea for improving the current data disaster backup schemes.

TacticToe: Learning to Reason with HOL4 Tactics

Techniques combining machine learning with translation to automated reasoning have recently become an important component of formal proof assistants. Such “hammer” techniques complement traditional proof assistant automation as implemented by tactics and decision procedures. In this paper we present a unified proof assistant automation approach which attempts to automate the selection of appropriate tactics and tactic-sequences combined with an optimized small-scale hammering approach. We implement the technique as a tactic-level automation for HOL4: TacticToe. It implements a modified A*-algorithm directly in HOL4 that explores different tactic-level proof paths, guiding their selection by learning from a large number of previous tactic-level proofs. Unlike the existing hammer methods, TacticToe avoids translation to FOL, working directly on the HOL level. By combining tactic prediction and premise selection, TacticToe is able to re-prove 39% of 7902 HOL4 theorems in 5 seconds whereas the best single HOL(y)Hammer strategy solves 32% in the same amount of time.

Inference rules for probability logic

Gentzen?s and Prawitz?s approach to deductive systems, and Carnap?s and Popper?s treatment of probability in logic were two fruitful ideas of logic in the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized by means of inference rules, we introduce a system of inference rules based on the traditional proof-theoretic principles enabling to work with each form of probabilized propositional formulae. Namely, for each propositional connective, we define at least one introduction and one elimination rule, over the formulae of the form A[a,b] with the intended meaning that ?the probability c of truthfulness of a sentence A belongs to the interval [a,b] ?[0,1]?. It is shown that our system is sound and complete with respect to the Carnap-Poper-type probability models.

Type classes for mathematics in type theory

The introduction of first-class type classes in the Coq system calls for a re-examination of the basic interfaces used for mathematical formalisation in type theory. We present a new set of type classes for mathematics and take full advantage of their unique features to make practical a particularly flexible approach that was formerly thought to be unfeasible. Thus, we address traditional proof engineering challenges as well as new ones resulting from our ambition to build upon this development a library of constructive analysis in which any abstraction penalties inhibiting efficient computation are reduced to a minimum.The basis of our development consists of type classes representing a standard algebraic hierarchy, as well as portions of category theory and universal algebra. On this foundation, we build a set of mathematically sound abstract interfaces for different kinds of numbers, succinctly expressed using categorical language and universal algebra constructions. Strategic use of type classes lets us support these high-level theory-friendly definitions, while still enabling efficient implementations unhindered by gratuitous indirection, conversion or projection.Algebra thrives on the interplay between syntax and semantics. The Prolog-like abilities of type class instance resolution allow us to conveniently define a quote function, thus facilitating the use of reflective techniques.

Teaching Algebra and Geometry Concepts by Modeling Telescope Optics

Preparation and delivery of high school mathematics lessons that integrate mathematics and astronomy through The Geometer's Sketchpad models, traditional proof, and inquiry-based activities. The lessons were created by a University of Texas UTeach preservice teacher as part of a project-based field experience in which high school students construct a working Dobsonian telescope. Eleven investigations with questions and answers are included.

Transitivity is Not Necessary to Show That Indifference Curves Cannot Intersect, a Note

Professors in intermediate and advanced microeconomic theory courses often propose a proof that indifference curves cannot intersect that relies on the transitivity and monotonicity of preferences. In the interest of stimulating thought on the topic, we derive an elementary proof that indifference curves cannot intersect which relies on fewer assumptions than the traditional proof. We conclude that transitivity is essential in constructing a theory of rational choice; but transitivity is not essential in showing that indifference curves cannot intersect.

Normalization and the Yoneda embedding

We show how to solve the word problem for simply typed λβη-calculus by using a few well-known facts about categories of presheaves and the Yoneda embedding. The formal setting for these results is [Pscr ]-category theory, a version of ordinary category theory where each hom-set is equipped with a partial equivalence relation. The part of [Pscr ]-category theory we develop here is constructive and thus permits extraction of programs from proofs. It is important to stress that in our method we make no use of traditional proof-theoretic or rewriting techniques. To show the robustness of our method, we give an extended treatment for more general λ-theories in the Appendix.