Formal Derivation and Verification of Coordinate Transformations in Theorem Prover Coq

2017 ◽  
Author(s):  
Zhenwei Ma ◽  
Gang Chen
Keyword(s):  
Theorem Prover ◽  
Coordinate Transformations ◽  
Formal Derivation

Evaluation Opportunities in Mechanized Theories

10.29007/7kx8 ◽  
2018 ◽  
Author(s):  
Joe Hurd
Keyword(s):  
Mathematical Theory ◽  
Theorem Prover ◽  
Mechanized Mathematics ◽  
Reasoning Tasks

This invited talk will look at logic solvers through the application lens of constructing and processing a theory library of mechanized mathematics. In fact, constructing and processing theories are two distinct applications, and each will be considered in turn. Construction is carried out by formalizing a mathematical theory using an interactive theorem prover, and logic solvers can remove much of the drudgery by automating common reasoning tasks. At the theory library level, logic solvers can provide assistance with theory engineering tasks such as compressing theories, managing dependencies, and constructing new theories from reusable theory components.

Poisson Brackets & Angular Momentum

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Generating Functions ◽  
First Integrals ◽  
Lagrangian Mechanics ◽  
Poisson Brackets ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Gauge Transformations ◽  
The Third ◽  
Point Transformations ◽  
Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

CSim 2

2021 ◽  
Vol 43 (1) ◽  
pp. 1-46
Author(s):  
David Sanan ◽  
Yongwang Zhao ◽  
Shang-Wei Lin ◽  
Liu Yang
Keyword(s):  
Programming Languages ◽  
Concurrent Systems ◽  
Theorem Prover ◽  
Compositional Techniques ◽  
Machine Code ◽  
Top Down ◽  
Hol Theorem Prover ◽  
High Level ◽  
High Degree

To make feasible and scalable the verification of large and complex concurrent systems, it is necessary the use of compositional techniques even at the highest abstraction layers. When focusing on the lowest software abstraction layers, such as the implementation or the machine code, the high level of detail of those layers makes the direct verification of properties very difficult and expensive. It is therefore essential to use techniques allowing to simplify the verification on these layers. One technique to tackle this challenge is top-down verification where by means of simulation properties verified on top layers (representing abstract specifications of a system) are propagated down to the lowest layers (that are an implementation of the top layers). There is no need to say that simulation of concurrent systems implies a greater level of complexity, and having compositional techniques to check simulation between layers is also desirable when seeking for both feasibility and scalability of the refinement verification. In this article, we present CSim 2 a (compositional) rely-guarantee-based framework for the top-down verification of complex concurrent systems in the Isabelle/HOL theorem prover. CSim 2 uses CSimpl, a language with a high degree of expressiveness designed for the specification of concurrent programs. Thanks to its expressibility, CSimpl is able to model many of the features found in real world programming languages like exceptions, assertions, and procedures. CSim 2 provides a framework for the verification of rely-guarantee properties to compositionally reason on CSimpl specifications. Focusing on top-down verification, CSim 2 provides a simulation-based framework for the preservation of CSimpl rely-guarantee properties from specifications to implementations. By using the simulation framework, properties proven on the top layers (abstract specifications) are compositionally propagated down to the lowest layers (source or machine code) in each concurrent component of the system. Finally, we show the usability of CSim 2 by running a case study over two CSimpl specifications of an Arinc-653 communication service. In this case study, we prove a complex property on a specification, and we use CSim 2 to preserve the property on lower abstraction layers.

scholarly journals Formal Derivation of Mesh Neural Networks with Their Forward-Only Gradient Propagation

2021 ◽  
Author(s):  
Federico A. Galatolo ◽  
Mario G. C. A. Cimino ◽  
Gigliola Vaglini
Keyword(s):  
Neural Networks ◽  
Formal Derivation

Structural ambiguity in Montague Grammar and categorial grammar

2015 ◽  
Vol 32 (1) ◽  
Author(s):  
Glyn Morrill
Keyword(s):  
Categorial Grammar ◽  
Theorem Prover ◽  
Montague Grammar ◽  
Structural Ambiguity

AbstractWe give a type logical categorial grammar for the syntax and semantics of Montague's seminal fragment, which includes ambiguities of quantification and intensionality and their interactions, and we present the analyses assigned by a parser/theorem prover CatLog to the examples in the first half of Chapter 7 of the classic text

Fitting Weibull Strength Data and Applying It to Stochastic Mechanical Design

1992 ◽  
Vol 114 (1) ◽  
pp. 35-41 ◽  
Author(s):  
C. R. Mischke
Keyword(s):  
Mechanical Design ◽  
Least Square ◽  
Stochastic Methods ◽  
Coordinate Transformations ◽  
Strength Data ◽  
The Third ◽  
Tension Tests ◽  
Property Data ◽  
Simple Tension ◽  
Best Fit

This is the second paper in a series relating to stochastic methods in mechanical design. The first is entitled, “Some Property Data and Corresponding Weibull Parameters for Stochastic Mechanical Design,” and the third, “Some Stochastic Mechanical Design Applications.” When data are sparse, many investigators prefer employing coordinate transformations to rectify the data string, and a least-square regression to seek the best fit. Such an approach introduces some bias, which the method presented here is intended to reduce. With mass-produced products, extensive testing can be carried out and prototypes built and evaluated. When production is small, material testing may be limited to simple tension tests or perhaps none at all. How should a designer proceed in order to achieve a reliability goal or to assess a design to see if the goal has been realized? The purpose of this paper is to show how sparse strength data can be reduced to distributional parameters with less bias and how such information can be used when designing to a reliability goal.

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