Related papers: Verifying Buchberger's Algorithm in Reduction Ring…
This paper discusses limitations of reflexive and diagonal arguments as methods of proof of limitative theorems (e.g. G\"odel's theorem on Entscheidungsproblem, Turing's halting problem or Chaitin-G\"odel's theorem). The fact, that a formal…
We give a new theoretical solution to a leading-edge experimental challenge, namely to the verification of quantum computations in the regime of high computational complexity. Our results are given in the language of quantum interactive…
A Comprehensive Grobner system for a parametric ideal I in K(A)[X] represents the collection of all Grobner bases of the ideals I' in K[X] obtained as the values of the parameters A vary in K. The recent algorithms for computing them…
Generalized quasi-cyclic (GQC) codes form a wide and useful class of linear codes that includes thoroughly quasi-cyclic codes, finite geometry (FG) low density parity check (LDPC) codes, and Hermitian codes. Although it is known that the…
In this paper we describe how an idea centered on the concept of self-saturation allows several improvements in the computation of Groebner bases via Buchberger's Algorithm.
Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced…
Proving statements about linear operators expressed in terms of identities often leads to finding elements of certain form in noncommutative polynomial ideals. We illustrate this by examples coming from actual operator statements and…
The present paper studies multiple failure and signature analysis of coherent systems using the theory of monomial ideals. While system reliability has been studied using Hilbert series of monomial ideals, this is not enough to understand…
It is well known that we can use structural proof theory to refine, or generalize, existing paradigmatic computational primitives, or to discover new ones. Under such a point of view we keep developing a programme whose goal is establishing…
Higher-order unification has been shown to be undecidable. Miller discovered the pattern fragment and subsequently showed that higher-order pattern unification is decidable and has most general unifiers. We extend the algorithm to…
F\"ur die freien orthogonalen Quantengruppen $A_o(n)$ wird ein vollst\"a ndiges Reduktionssystem angegeben und verifiziert. F\"ur den Fall $n = 2$ wird ein endlicher Automat angegeben, der s\"amtliche der Basiselemente findet. Weiterhin…
Recursive techniques have recently been introduced into quantum programming so that a variety of large quantum circuits and algorithms can be elegantly and economically programmed. In this paper, we present a proof system for formal…
We propose to analyse quantum protocols by applying formal verification techniques developed in classical computing for the analysis of communicating concurrent systems. One area of successful application of these techniques is that of…
We consider ideals involving the maximal minors of a polynomial matrix. For example, those arising in the computation of the critical values of a polynomial restricted to a variety for polynomial optimisation. Gr\"obner bases are a…
This paper presents a new refutation procedure for multimodular systems of integer constraints that commonly arise when verifying cryptographic protocols. These systems, involving polynomial equalities and disequalities modulo different…
Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is one example of such a structure: given a system of weights…
Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with…
We perform formal verification of quantum circuits by integrating several techniques specialized to particular classes of circuits. Our verification methodology is based on the new notion of a reversible miter that allows one to leverage…
In recent years, G\"odel's ontological proof and variations of it were formalized and analyzed with automated tools in various ways. We supplement these analyses with a modeling in an automated environment based on first-order logic…
The program Reverse Mathematics in the foundations of mathematics seeks to identify the minimal axioms required to prove theorems of ordinary mathematics. One always assumes the base theory, a logical system embodying computable…