Computing the Weight of a Boolean Function from Its Algebraic Normal Form

2012 ◽  
pp. 89-100 ◽  
Author(s):  
Çağdaş Çalık ◽  
Ali Doğanaksoy
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Algebraic Normal Form

Computing Walsh coefficients from the algebraic normal form of a Boolean function

2014 ◽  
Vol 6 (4) ◽  
pp. 335-358 ◽  
Author(s):  
Xinxin Gong ◽  
Bin Zhang ◽  
Wenling Wu ◽  
Dengguo Feng
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Algebraic Normal Form

Chapter 29. Theory of Quantified Boolean Formulas

2021 ◽  
Author(s):  
Hans Kleine Büning ◽  
Uwe Bubeck
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Expressive Power ◽  
Modeling Language ◽  
Complex Problems ◽  
Quantified Boolean Formulas ◽  
Horn Formulas ◽  
Boolean Formulas ◽  
Function Models ◽  
Resolution Calculus

Quantified Boolean formulas (QBF) are a generalization of propositional formulas by allowing universal and existential quantifiers over variables. This enhancement makes QBF a concise and natural modeling language in which problems from many areas, such as planning, scheduling or verification, can often be encoded in a more compact way than with propositional formulas. We introduce in this chapter the syntax and semantics of QBF and present fundamental concepts. This includes normal form transformations and Q-resolution, an extension of the propositional resolution calculus. In addition, Boolean function models are introduced to describe the valuation of formulas and the behavior of the quantifiers. We also discuss the expressive power of QBF and provide an overview of important complexity results. These illustrate that the greater capabilities of QBF lead to more complex problems, which makes it interesting to consider suitable subclasses of QBF. In particular, we give a detailed look at quantified Horn formulas (QHORN) and quantified 2-CNF (Q2-CNF).

scholarly journals On Minimum Representations of Matched Formulas

2014 ◽  
Vol 51 ◽  
pp. 707-723
Author(s):  
O. Cepek ◽  
S. Gursky ◽  
P. Kucera
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Minimization Problem ◽  
Conjunctive Normal Form ◽  
Boolean Formula ◽  
Polynomial Hierarchy ◽  
Partial Assignment ◽  
Complete Problem ◽  
Boolean Minimization ◽  
Np Complete

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. Each matched CNF is trivially satisfiable (each clause can be satisfied by its representative variable). Another property which is easy to see, is that the class of matched CNFs is not closed under partial assignment of truth values to variables. This latter property leads to a fact (proved here) that given two matched CNFs it is co-NP complete to decide whether they are logically equivalent. The construction in this proof leads to another result: a much shorter and simpler proof of the fact that the Boolean minimization problem for matched CNFs is a complete problem for the second level of the polynomial hierarchy. The main result of this paper deals with the structure of clause minimum CNFs. We prove here that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.

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