Executable Multivariate Polynomials
نویسندگان
چکیده
We define multivariate polynomials over arbitrary (ordered) semirings in combination with (executable) operations like addition, multiplication, and substitution. We also define (weak) monotonicity of polynomials and comparison of polynomials where we provide standard estimations like absolute positiveness or the more recent approach of [3]. Moreover, it is proven that strongly normalizing (monotone) orders can be lifted to strongly normalizing (monotone) orders over polynomials. Our formalization was performed as part of the IsaFoR/CeTA-system [4] which contains several termination techniques. The provided theories have been essential to formalize polynomial-interpretations [1, 2].
منابع مشابه
Towards abstract and executable multivariate polynomials in Isabelle
This work in progress report envisions a library for multivariate polynomials developed jointly by experts from computer theorem proving (CTP) and computer algebra (CA). The urgency of verified algorithms has been recognised in the field of CA, but the cultural gap to CTP is considerable; CA users expect high usability and efficiency. This work collects the needs of CA experts and reports on th...
متن کاملGrey Box Implementation of Block Ciphers Preserving the Confidentiality of their Design
In 1997, Patarin and Goubin introduce new asymmetric cryptosystems based on the diÆculty of recovering two systems of multivariate polynomials from their composition. We make a di erent use of this diÆcult algorithmic problem to obtain a way of representing block ciphers concealing their design but leaving them executable. We show how to implement our solution giving a compact representation wi...
متن کاملEchelon Form
In this work we present the formalization of an algorithm to compute the Echelon Form of a matrix. We have proved its existence over Bezout domains and we have made it executable over Euclidean domains, such as Z and K[x]. This allows us to compute determinants, inverses and characteristic polynomials of matrices. The work is based on the HOL-Multivariate Analysis library, and on both the Gauss...
متن کاملParametric Linear Arithmetic over Ordered Fields in Isabelle/HOL
We use higher-order logic to verify a quantifier elimination procedure for linear arithmetic over ordered fields, where the coefficients of variables are multivariate polynomials over another set of variables, we call parameters. The procedure generalizes Ferrante and Rackoff’s algorithm for the non-parametric case. The formalization is based on axiomatic type classes and automatically carries ...
متن کاملMultivariate Bernoulli and Euler polynomials via Lévy processes
By a symbolic method, we introduce multivariate Bernoulli and Euler polynomials as powers of polynomials whose coefficients involve multivariate Lévy processes. Many properties of these polynomials are stated straightforwardly thanks to this representation, which could be easily implemented in any symbolic manipulation system. A very simple relation between these two families of multivariate po...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Archive of Formal Proofs
دوره 2010 شماره
صفحات -
تاریخ انتشار 2010