29 research outputs found
Fast Gr\"obner Basis Computation for Boolean Polynomials
We introduce the Macaulay2 package BooleanGB, which computes a Gr\"obner
basis for Boolean polynomials using a binary representation rather than
symbolic. We compare the runtime of several Boolean models from systems in
biology and give an application to Sudoku
An Algebraic Model For Quorum Systems
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 Groebner bases. The nice nature of Boolean Groebner 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.Comment: 15 pages, 3 algorithm
Computer Science 2019 APR Self-Study & Documents
UNM Computer Science APR self-study report and review team report for Spring 2019, fulfilling requirements of the Higher Learning Commission
Internship report MPRI 2 Reverse engineering on arithmetic proofs
International audiencededukti is a logical framework that implements the λΠ− modulo theory, an extension of the simply typed lambda calculus with dependent types and rewriting rules. It aims to be a back-end for other proof checkers by compiling proofs from these proof checkers to dedukti. This may also increase re-usability of proofs between proof checkers. However if a logic is more powerful than an other, a theorem in the first logic may not be a theorem in the second. During this internship, we consider arithmetic theorems since many proof checker are able to check arithmetic proofs. One problem that we study in this master thesis is to translate arithmetic proofs coming from a powerful proof checker, -- in our casematita -- to a less powerful proof checker -- HOL-- . This translation needs to modify the logic used in proofs and that is why dedukti is handy here. But a lot of arithmetic theorems are proved also by automatic provers. Indeed, today a lot of easy arithmetic theorems are proved by this kind of tool. But most of them do not give a proof if it claims to prove a theorem. Since for these kind of tool, constructing a full proof may be tiresome, they prefer to give a certificate , a sketch of a proof. However, any automatic prover can implement its own certificate format. To answer this problem, Zakaria Chihani & Dale Miller proposed a certificate framework: Foundational Proof Certificate (FPC) [CMR13]. This framework aims to provide a certificate format shared by many automatic provers so that from the latter, a full proof might be reconstructed.However, for now, no certificate format is given for arithmetic proofs. A second problem addressed in this internship is to answer what kind of certificate is needed for arithmetic proofs (arithmetic without multiplication)
PaMpeR: Proof Method Recommendation System for Isabelle/HOL
Deciding which sub-tool to use for a given proof state requires expertise
specific to each ITP. To mitigate this problem, we present PaMpeR, a Proof
Method Recommendation system for Isabelle/HOL. Given a proof state, PaMpeR
recommends proof methods to discharge the proof goal and provides qualitative
explanations as to why it suggests these methods. PaMpeR generates these
recommendations based on existing hand-written proof corpora, thus transferring
experienced users' expertise to new users. Our evaluation shows that PaMpeR
correctly predicts experienced users' proof methods invocation especially when
it comes to special purpose proof methods.Comment: An anonymized version of this paper has been submitted to a Computer
Science conference in April 201
A proof system for graph (non)-isomorphism verification
In order to apply canonical labelling of graphs and isomorphism checking in
interactive theorem provers, these checking algorithms must either be
mechanically verified or their results must be verifiable by independent
checkers. We analyze a state-of-the-art algorithm for canonical labelling of
graphs (described by McKay and Piperno) and formulate it in terms of a formal
proof system. We provide an implementation that can export a proof that the
obtained graph is the canonical form of a given graph. Such proofs are then
verified by our independent checker and can be used to confirm that two given
graphs are not isomorphic