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Building models that comply with the invariances inherent to different domains, such as invariance under translation or rotation, is a key aspect of applying machine learning to real world problems like molecular property prediction,…
In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial…
We present simple new Hoare logics and refinement calculi for hybrid systems in the style of differential dynamic logic. (Refinement) Kleene algebra with tests is used for reasoning about the program structure and generating verification…
Computer algebra systems are really good at factoring polynomials, i.e. writing f as a product of irreducible factors. It is relatively easy to verify that we have a factorisation, but verifying that these factors are irreducible is a much…
In this paper, we utilize Isabelle/HOL to develop a formal framework for the basic theory of double-pushout graph transformation. Our work includes defining essential concepts like graphs, morphisms, pushouts, and pullbacks, and…
We consider the problem of minimizing a polynomial $f$ over the binary hypercube. We show that, for a specific set of polynomials, their binary non-negativity can be checked in a polynomial time via minimum cut algorithms, and we construct…
In this paper, we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems. Starting from a polynomial system with an isolated singular zero, by computing the derivatives of the input…
Fully homomorphic encryption (FHE) allows an untrusted party to evaluate arithmetic cir- cuits, i.e., perform additions and multiplications on encrypted data, without having the decryp- tion key. One of the most efficient class of FHE…
Quantum homomorphic encryption (QHE) is an encryption method that allows quantum computation to be performed on one party's private data with the program provided by another party, without revealing much information about the data nor the…
Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of…
In this paper, we tackle the parametric complete multiplicity problem for a univariate polynomial. Our approach to the parametric complete multiplicity problem has a significant difference from the classical method, which relies on repeated…
Informal arguments that cryptographic protocols are secure can be made rigorous using inductive definitions. The approach is based on ordinary predicate calculus and copes with infinite-state systems. Proofs are generated using…
We describe our ongoing project of formalization of algebraic methods for geometry theorem proving (Wu's method and the Groebner bases method), their implementation and integration in educational tools. The project includes formal…
In this preliminary work, we propose to use a polynomial approach in order to study the stability of switched systems. The proposed strategy is based on the Bernstein interpolation method that may transform a switched system into a…
We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial $p$ of degree $d$ with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for…
We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal)…
In this paper we present an efficient approach to implementing model checking in the Higher Order Logic (HOL) of Isabelle. This is a non-trivial task since model checking is restricted to finite state sets. By restricting our scope to…
Motivated by certain applications from physics, biochemistry, economics, and computer science, in which the objects under investigation are not accessible because of various limitations, we propose a trial-and-error model to examine…
In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real…
We offer a digital signature scheme using Boolean automorphisms of a multivariate polynomial algebra over integers. Verification part of this scheme is based on the approximation of the number of zeros of a multivariate Boolean function.