Satisfiability in Big Boolean Algebras via Boolean-Equation Solving

This paper studies Satisfiability (SAT) in finite atomic Boolean algebras larger than the two-valued one B2, which are named big Boolean algebras. Unlike the formula ݃(ࢄ (in the SAT problem over B2, which is either satisfiable or unsatisfiable, this formula for the SAT problem over a big Boolean algebra could be unconditionally satisfiable, conditionally satisfiable, or unsatisfiable depending on the nature of the consistency condition of the Boolean equation {݃(ࢄ = (1}, since this condition could be an identity, a genuine equation, or a contradiction. The paper handles this latter SAT problem by using a conventional method and a novel one for deriving parametric general solutions, and subsequently utilizing expansion trees for generating all particular solutions of the aforementioned Boolean equation. Each of these two methods could be cast in pure algebraic form, but becomes much easier to visualize and comprehend when presented via the natural map of a big Boolean algebra, which (for historical reasons) is called the variable-entered Karnaugh map (VEKM). In the classical method, the number of parameters used is minimized and compact solutions are obtained. However, the parameters belong to the underlying big Boolean algebra. By contrast, the novel method does not attempt to minimize the number of parameters used, as it uses independent parameters belonging to the two-valued Boolean algebra B2 for each asserted atom in the Boole-Shannon expansion of the formula ݃(ࢄ .(Though the method produces non-compact expressions, it is much quicker in generating particular solutions. The two methods are demonstrated via two detailed examples.

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Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras

International Journal of Mathematical Engineering and Management Sciences ◽

10.33889/ijmems.2018.3.4-029 ◽

2018 ◽

Vol 3 (4) ◽

pp. 404-428 ◽

Cited By ~ 4

Author(s):

Ali Muhammad Ali Rushdi ◽

Waleed Ahmad

Keyword(s):

Boolean Algebra ◽

Circuit Design ◽

Digital Circuit ◽

Boolean Algebras ◽

Binary Vectors ◽

Equation Solving ◽

Digital Circuit Design ◽

Boolean Equation ◽

Novel Method ◽

Independent Parameters

A task frequently encountered in digital circuit design is the solution of a two-valued Boolean equation of the form h(X,Y,Z)=1, where h: B_2^(k+m+n)→ B_2 and X,Y, and Z are binary vectors of lengths k, m, and n, representing inputs, intermediary values, and outputs, respectively. The resultant of the suppression of the variables Y from this equation could be written in the form g(X,Z)=1 where g: B_2^(k+n)→ B_2. Typically, one needs to solve for Z in terms of X, and hence it is unavoidable to resort to ‘big’ Boolean algebras which are finite (atomic) Boolean algebras larger than the two-valued Boolean algebra. This is done by reinterpreting the aforementioned g(X,Z) as g(Z): B_(2^K)^n→ B_(2^K ), where B_(2^K ) is the free Boolean algebra FB(X_1,X_2…….X_k ), which has K= 2^k atoms, and 2^K elemnets. This paper describes how to unify many digital specifications into a single Boolean equation, suppress unwanted intermediary variables Y, and solve the equation g(Z)=1 for outputs Z (in terms of inputs X) in the absence of any information about Y. The paper uses a novel method for obtaining the parametric general solutions of the ‘big’ Boolean equation g(Z)=1. The parameters used do not belong to B_(2^K ) but they belong to the two-valued Boolean algebra B_2, also known as the switching algebra or propositional algebra. To achieve this, we have to use distinct independent parameters for each asserted atom in the Boole-Shannon expansion of g(Z). The concepts and methods introduced herein are demonsrated via several detailed examples, which cover the most prominent type among basic problems of digital circuit design.

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Boolean Curve Fitting with the Aid of Variable-Entered Karnaugh Maps

International Journal of Mathematical Engineering and Management Sciences ◽

10.33889/ijmems.2019.4.6-102 ◽

2019 ◽

Vol 4 (6) ◽

pp. 1287-1306

Author(s):

Ali Muhammad Ali Rushdi ◽

Ahmed Said Balamesh

Keyword(s):

Boolean Algebra ◽

Boolean Function ◽

Unique Solution ◽

Curve Fitting ◽

Multiple Solutions ◽

Consistency Condition ◽

Visual Perspective ◽

Boolean Space ◽

Potential Applications ◽

Karnaugh Map

The Variable-Entered Karnaugh Map is utilized to grant a simpler view and a visual perspective to Boolean curve fitting (Boolean interpolation); a topic whose inherent complexity hinders its potential applications. We derive the function(s) through m points in the Boolean space B^(n+1) together with consistency and uniqueness conditions, where B is a general ‘big’ Boolean algebra of l≥1 generators, L atoms (2^(l-1)<L≤2^l) and 2^L elements. We highlight prominent cases in which the consistency condition reduces to the identity (0=0) with a unique solution or with multiple solutions. We conjecture that consistent (albeit not necessarily unique) curve fitting is possible if, and only if, m=2^n. This conjecture is a generalization of the fact that a Boolean function of n variables is fully and uniquely determined by its values in the {0,1}^n subdomain of its B^n domain. A few illustrative examples are used to clarify the pertinent concepts and techniques.

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CHOICE-FREE STONE DUALITY

Journal of Symbolic Logic ◽

10.1017/jsl.2019.11 ◽

2019 ◽

Vol 85 (1) ◽

pp. 109-148

Author(s):

NICK BEZHANISHVILI ◽

WESLEY H. HOLLIDAY

Keyword(s):

Boolean Algebra ◽

Boolean Algebras ◽

Stone Space ◽

Spectral Space ◽

Topological Representation ◽

Stone Duality ◽

Closed Sets ◽

Spectral Spaces ◽

Basic Facts ◽

Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

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On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem Löwenheim theorems and compactness of related quantifiers

Journal of Symbolic Logic ◽

10.2307/2273187 ◽

1980 ◽

Vol 45 (2) ◽

pp. 265-283 ◽

Cited By ~ 7

Author(s):

Matatyahu Rubin ◽

Saharon Shelah

Keyword(s):

Boolean Algebra ◽

Automorphism Groups ◽

Order Logic ◽

Boolean Algebras ◽

First Order Logic ◽

Elementary Equivalence ◽

First Order ◽

Finite Models ◽

Countably Compact

AbstractTheorem 1. (◊ℵ1,) If B is an infinite Boolean algebra (BA), then there is B1, such that ∣ Aut (B1) ≤∣B1∣ = ℵ1 and 〈B1, Aut (B1)〉 ≡ 〈B, Aut(B)〉.Theorem 2. (◊ℵ1) There is a countably compact logic stronger than first-order logic even on finite models.This partially answers a question of H. Friedman. These theorems appear in §§1 and 2.Theorem 3. (a) (◊ℵ1) If B is an atomic ℵ-saturated infinite BA, Ψ Є Lω1ω and 〈B, Aut (B)〉 ⊨Ψ then there is B1, Such that ∣Aut(B1)∣ ≤ ∣B1∣ =ℵ1, and 〈B1, Aut(B1)〉⊨Ψ. In particular if B is 1-homogeneous so is B1. (b) (a) holds for B = P(ω) even if we assume only CH.

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On the Representation of Boolean Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-1962-006-6 ◽

1962 ◽

Vol 5 (1) ◽

pp. 37-41 ◽

Cited By ~ 1

Author(s):

Günter Bruns

Keyword(s):

Boolean Algebra ◽

Boolean Algebras ◽

Two Systems ◽

Image Position

Let B be a Boolean algebra and let ℳ and n be two systems of subsets of B, both containing all finite subsets of B. Let us assume further that the join ∨M of every set M∊ℳ and the meet ∧N of every set N∊n exist. Several authors have treated the question under which conditions there exists an isomorphism φ between B and a field δ of sets, satisfying the conditions:

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Wreath Product Action on Generalized Boolean Algebras

The Electronic Journal of Combinatorics ◽

10.37236/4831 ◽

2015 ◽

Vol 22 (2) ◽

Author(s):

Ashish Mishra ◽

Murali K. Srinivasan

Keyword(s):

Finite Group ◽

Boolean Algebra ◽

Wreath Product ◽

Boolean Algebras ◽

Product Action ◽

Finite Set ◽

A Finite Group ◽

Multiplicity Free ◽

Generalized Boolean Algebra ◽

Generalized Boolean Algebras

Let $G$ be a finite group acting on the finite set $X$ such that the corresponding (complex) permutation representation is multiplicity free. There is a natural rank and order preserving action of the wreath product $G\sim S_n$ on the generalized Boolean algebra $B_X(n)$. We explicitly block diagonalize the commutant of this action.

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Boolean Algebra Retracts

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-034-3 ◽

1971 ◽

Vol 23 (2) ◽

pp. 339-344

Author(s):

Timothy Cramer

Keyword(s):

Boolean Algebra ◽

Dual Problem ◽

Sufficient Conditions ◽

Boolean Algebras ◽

Necessary And Sufficient Conditions ◽

Infinite Cardinal ◽

Necessary And Sufficient

A Boolean algebra B is a retract of an algebra A if there exist homomorphisms ƒ: B → A and g: A → B such that gƒ is the identity map B. Some important properties of retracts of Boolean algebras are stated in [3, §§ 30, 31, 32]. If A and B are a-complete, and A is α-generated by B, Dwinger [1, p. 145, Theorem 2.4] proved necessary and sufficient conditions for the existence of an α-homomorphism g: A → B such that g is the identity map on B. Note that if a is not an infinite cardinal, B must be equal to A. The dual problem was treated by Wright [6]; he assumed that A and B are σ-algebras, and that g: A → B is a σ-homomorphism, and gave conditions for the existence of a homomorphism ƒ:B → A such that gƒ is the identity map.

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Densities of Ultraproducts of Boolean Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1995-007-0 ◽

1995 ◽

Vol 47 (1) ◽

pp. 132-145

Author(s):

Sabine Koppelberg ◽

Saharon Shelah

Keyword(s):

Boolean Algebra ◽

Boolean Algebras ◽

Cardinal Invariants ◽

The Third ◽

Topological Density

AbstractWe answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density πA resp. the topological density cL4 of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.

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Recursive isomorphism types of recursive Boolean algebras

Journal of Symbolic Logic ◽

10.2307/2273757 ◽

1981 ◽

Vol 46 (3) ◽

pp. 572-594 ◽

Cited By ~ 49

Author(s):

J. B. Remmel

Keyword(s):

Boolean Algebra ◽

Recursive Function ◽

Boolean Algebras ◽

Isomorphism Type ◽

Turing Degrees ◽

Main Question ◽

Partial Recursive Function ◽

Recursive Isomorphism ◽

Infinite Set ◽

The Ideal

A Boolean algebra is recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Given two Boolean algebras and , we write if is isomorphic to and if is recursively isomorphic to , that is, if there is a partial recursive function f: B1 → B2 which is an isomorphism from to . will denote the set of atoms of and () will denote the ideal generated by the atoms of .One of the main questions which motivated this paper is “To what extent does the classical isomorphism type of a recursive Boolean algebra restrict the possible recursion theoretic properties of ?” For example, it is easy to see that must be co-r.e. (i.e., N − is an r.e. set), but can be immune, not immune, cohesive, etc? It follows from a result of Goncharov [4] that there exist classical isomorphism types which contain recursive Boolean algebras but do not contain any recursive Boolean algebras such that is recursive. Thus the classical isomorphism can restrict the possible Turing degrees of , but what is the extent of this restriction? Another main question is “What is the recursion theoretic relationship between and () in a recursive Boolean algebra?” In our attempt to answer these questions, we were led to a wide variety of recursive isomorphism types which are contained in the classical isomorphism type of any recursive Boolean algebra with an infinite set of atoms.

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