On the Computation of Elimination Ideals of Boolean Polynomial Rings

Sato, Yosuke; Nagai, Akira; Inoue, Shinji

doi:10.1007/978-3-540-87827-8_29

Cited by 7 publications

(3 citation statements)

References 7 publications

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“…Our results will heavily rely on the algebraic background that we will introduce in this section. Many of the definitions and proofs can be found in [7,Section 4.3] and [27,11,12], we are hereby giving the result without proofs.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

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An Algebraic Model For Quorum Systems

Pellegrini,

Zanolini

2020

Preprint

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Quorum systems are a key mathematical abstraction in distributed fault-tolerant computing for capturing trust assumptions. A quorum system is a collection of subsets of all processes, called quorums, with the property that each pair of quorums have a non-empty intersection. They can be found at the core of many reliable distributed systems, such as cloud computing platforms, distributed storage systems and blockchains. In this paper we give a new interpretation of quorum systems, starting with classical majority-based quorum systems and extending this to Byzantine quorum systems. We propose an algebraic representation of the theory underlying quorum systems making use of multivariate polynomial ideals, incorporating properties of these systems, and studying their algebraic varieties. To achieve this goal we will exploit properties of Boolean Gröbner bases. The nice nature of Boolean Gröbner bases allows us to avoid part of the combinatorial computations required to check consistency and availability of quorum systems. Our results provide a novel approach to test quorum systems properties from both algebraic and algorithmic perspectives.

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Section: Preliminaries and Notationmentioning

confidence: 99%

“…Exploiting the properties of γ we define the Boolean polynomial γ(ϕ(F ), T ) for every F ∈ F such that its zero locus are the points representing the powerset 2 F . We can thus express the consistency property in (27) like in the previous section, i.e., by defining an ideal and checking its variety's properties.…”

Section: Dissemination Quorum Systemmentioning

confidence: 99%

An Algebraic Model For Quorum Systems

Pellegrini,

Zanolini

2020

Preprint

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“…As pointed out in [17,18,24] there is some strong relation between comprehensive Gröbner bases and Göbner bases over (von Neumann) regular rings [22]. Since we also have Gröbner bases over (finite) regular rings implemented in JAS, we can check, for example, if a comprehensive Gröbner base is indeed also a Gröbner base over a suitable regular ring.…”

Section: Examples and Gröbner Bases Over Regular Ringsmentioning

confidence: 99%

Comprehensive Gröbner Bases in a Java Computer Algebra System

Kredel

2014

Computer Mathematics

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We present an implementation of the algorithms for computing comprehensive Gröbner bases in a Java computer algebra system (JAS). Contrary to approaches to implement comprehensive Gröbner bases with minimal requirements to the computer algebra system, we aim to provide and utilize all necessary algebraic structures occurring in the algorithm. In the implementation of a condition we aim at the maximal semantic exploitation of the occurring algebraic structures: the set of equations equal zero are implemented as an ideal (with Gröbner base computation) and the set of inequalities are implemented as a multiplicative set which is simplified to polynomials of minimal degrees using, for example, square-free decomposition. With our approach we can also make the transition of a comprehensive Gröbner system to a polynomial ring over a (commutative, finite, von Neuman) regular coefficient ring and test or compute Gröbner bases in such polynomial rings.

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A Mathematical Hierarchy of Sudoku Puzzles and Its Computation by Boolean Gröbner Bases

Inoue

Sato

2014

Artificial Intelligence and Symbolic Computation

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On the Computation of Elimination Ideals of Boolean Polynomial Rings

Cited by 7 publications

References 7 publications

An Algebraic Model For Quorum Systems

An Algebraic Model For Quorum Systems

Comprehensive Gröbner Bases in a Java Computer Algebra System

A Mathematical Hierarchy of Sudoku Puzzles and Its Computation by Boolean Gröbner Bases

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