Skip to main content Accessibility help
×
Home
Hostname: page-component-6c8bd87754-hvdfp Total loading time: 0.227 Render date: 2022-01-18T05:41:42.249Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

2 - Boolean equations

from Part I - Foundations

Published online by Cambridge University Press:  01 June 2011

Yves Crama
Affiliation:
Université de Liège, Belgium
Peter L. Hammer
Affiliation:
Rutgers University, New Jersey
Get access

Summary

The solution of Boolean equations is arguably the most fundamental problem arising in the theory of Boolean functions. Actually, the quote at the beginning of Chapter 1 shows that an important aspect of Boole's original research program was essentially to reduce logic to the solution of Boolean equations. Although his hopes eventually proved overly optimistic, it will become clear in subsequent chapters of this book that Boolean equations often arise as subproblems to be solved in the course of tackling more complex problems. Therefore, their solution is a cornerstone of many Boolean algorithms.

In this chapter, we present some representative models involving Boolean equations and describe various algorithmic procedures for their solution: branching, variable elimination, the consensus method, and mathematical programming approaches. In view of the importance of this topic, we spend quite a lot of time discussing the details of classical procedures, their interrelations, respective merits, and complexity. In the last section, we generalize the basic consistency-testing problem in several ways: We examine the problems of counting and of generating all solutions of a Boolean equation and briefly discuss the maximum satisfiability (Max Sat) problem.

Definitions and applications

Definition 2.1.A Boolean equation is an equation of the form ϕ(X) = ψ(X), where X = (x1,x2,…, xn) is a vector of Boolean variables, and ϕ,ψ are Boolean expressions in these variables. A solution of the equation is a point X* ∈ Bn such that ϕ(X*) = ψ(X*).

Type
Chapter
Information
Boolean Functions
Theory, Algorithms, and Applications
, pp. 67 - 122
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Send book to Kindle

To send this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Boolean equations
  • Yves Crama, Université de Liège, Belgium, Peter L. Hammer, Rutgers University, New Jersey
  • Book: Boolean Functions
  • Online publication: 01 June 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511852008.003
Available formats
×

Send book to Dropbox

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox.

  • Boolean equations
  • Yves Crama, Université de Liège, Belgium, Peter L. Hammer, Rutgers University, New Jersey
  • Book: Boolean Functions
  • Online publication: 01 June 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511852008.003
Available formats
×

Send book to Google Drive

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive.

  • Boolean equations
  • Yves Crama, Université de Liège, Belgium, Peter L. Hammer, Rutgers University, New Jersey
  • Book: Boolean Functions
  • Online publication: 01 June 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511852008.003
Available formats
×