scholarly journals How to prove your calculus is decidable: practical applications of second-order algebraic theories and computation

2017 ◽  
Vol 1 (ICFP) ◽  
pp. 1-28 ◽  
Author(s):  
Makoto Hamana
Keyword(s):  
Second Order ◽  
Practical Applications ◽  
Algebraic Theories ◽  
Order Algebraic

How to prove decidability of equational theories with second-order computation analyser SOL

2019 ◽  
Vol 29 ◽  
Author(s):  
MAKOTO HAMANA
Keyword(s):  
Programming Language ◽  
Equational Theory ◽  
Second Order ◽  
Language Theory ◽  
Analysis Tool ◽  
Monoidal Categories ◽  
Equational Theories ◽  
Algebraic Theories ◽  
Order Algebraic ◽  
Π Calculus

Abstract We present a general methodology of proving the decidability of equational theory of programming language concepts in the framework of second-order algebraic theories. We propose a Haskell-based analysis tool, i.e. Second-Order Laboratory, which assists the proofs of confluence and strong normalisation of computation rules derived from second-order algebraic theories. To cover various examples in programming language theory, we combine and extend both syntactical and semantical results of the second-order computation in a non-trivial manner. We demonstrate how to prove decidability of various algebraic theories in the literature. It includes the equational theories of monad and λ-calculi, Plotkin and Power’s theory of states and bits, and Stark’s theory of π-calculus. We also demonstrate how this methodology can solve the coherence of monoidal categories.

scholarly journals DIFFERENTIATION OF POLYNOMIALS IN SEVERAL VARIABLES OVER GALOIS FIELDS OF FUZZY CARDINALITY AND APPLICATIONS TO REED-MULLER CODES

2018 ◽  
Vol 18 (3) ◽  
pp. 339-348
Author(s):  
V. M. Deundyak ◽  
N. S. Mogilevskaya
Keyword(s):  
Coding Theory ◽  
Communication Systems ◽  
Communication Channels ◽  
Galois Field ◽  
Second Order ◽  
Galois Fields ◽  
Practical Applications ◽  
Several Variables ◽  
Confidential Data ◽  
Partial Distortion

Introduction. Polynomials in several variables over Galois fields provide the basis for the Reed-Muller coding theory, and are also used  in a number of cryptographic problems. The properties of such polynomials specified over the derived Galois fields of fuzzy cardinality are studied. For the results obtained,  two  real-world  applications  are  proposed: partitioning scheme and Reed-Muller code decoder.Materials and Methods. Using linear algebra, theory of Galois fields, and general theory of polynomials in several variables, we have obtained results related to the differentiation and integration  of polynomials  in  several  variables  over  Galois fields of fuzzy cardinality. An analog of the differentiation operator is constructed and studied for vectors.Research Results. On the basis of the obtained results on the differentiation and integration of polynomials, a new decoder for Reed-Muller codes of the second order is given, and a scheme for organizing the partitioned transfer of confidential data is proposed. This is a communication system in which the source data on the sender is divided into several parts and, independently of one  another,  transmitted  through  different communication channels, and then, on the receiver, the initial data is restored of the parts retrieved. The proposed scheme feature is that it enables to protect data, both from the nonlegitimate access, and from unintentional errors; herewith, one  and  the  same  mathematical  apparatus  is  used  in  both cases. The developed decoder for the second-order Reed-Muller codes prescribed over the derived odd Galois field may have a constraint to the recoverable error level; however, its use is advisable for a number of the communication channels.Discussion    and    Conclusions.    The    proposed    practical applications   of   the   results   obtained   are   useful   for   the organization of reliable communication systems. In future, it is planned  to  study  the  restoration  process  of  the  original polynomial by its derivatives, in case of their partial distortion, and the development of appropriate applications.

Biquadratic Digital Phase-Compensator Design with Stability-Margin Controllability

2019 ◽  
Vol 28 (04) ◽  
pp. 1950068 ◽  
Author(s):  
Tian-Bo Deng
Keyword(s):  
Stability Margin ◽  
Transformation Method ◽  
Second Order ◽  
Approximation Accuracy ◽  
Parameter Transformation ◽  
Practical Applications ◽  
Compensator Design ◽  
Novel Method ◽  
Design Idea ◽  
The Stability

This paper proposes a novel method for the design of a recursive second-order (biquadratic) all-pass phase compensator with controllable stability margin. The design idea stems from the generalized stability triangle (GST) derived by the author for the second-order biquadratic digital filter. Based on the GST, a parameter-transformation method is proposed on the transformations of the denominator coefficients of the transfer function of the biquadratic phase compensator. The transformations convert the original denominator coefficients to other new parameters, and any values of those new parameters can guarantee that the GST condition is always satisfied. Optimizing the new parameters yields a biquadratic phase compensator that definitely meets a prespecified stability margin. That is, a biquadratic all-pass phase compensator can be designed to have an arbitrarily specified stability margin. This in turn avoids the occurrence that a recursive phase compensator may become unstable in the practical applications. Thus, the resulting biquadratic phase compensator has robust stability, which is extremely important during the practical filtering operations. A design example is given to show the stability margin guarantee as well as the approximation accuracy.

Analysis of Flexibility and Stability of Crane Telescopic Boom with Elastic Restraint and Second-Order Effect

2013 ◽  
Vol 774-776 ◽  
pp. 109-113
Author(s):  
Liang Du ◽  
Nian Li Lu ◽  
Peng Lan
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Effect ◽  
Second Order ◽  
Analysis Model ◽  
Characteristic Equations ◽  
Lateral Deflection ◽  
Practical Applications ◽  
Vertical Stability

The cylinder support crane telescopic booms deformation and stability analysis model in the lifting plane is equivalent with the multistep column with elastic restraint. To analyze the lateral flexibility and vertical stability of the telescopic booms with elastic restraint accurately, this paper established the deflection differential equations of multi-sectioned telescopic booms with second-order effect, introduced proper boundary conditions, obtained the precise recurrence lateral deflection differential equations and the buckling characteristic equations of arbitrary sectioned telescopic booms, and some practical applications of the buckling characteristic equations were presented. Took certain five-sectioned telescopic booms as example, by comparing the results with ANSYS method, the accuracy of the equations deduced in this paper was verified.

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