Two-valued Boolean Algebra Research Articles

Note on the signed occurrences of propositional variables

This note concerns the positive and negative occurrences of propositional variables. Just like the theory of infectious truth-values provides an algebraic understanding of the position according to which identity of subject-matter between two formulas can approximated syntactically by the identity of propositional variables occurring in these formulas, we develop an algebraic understanding of the similar position which considers signed occurrence (i.e., positive or negative) instead of mere occurrence. We apply our framework to classical logic, yielding this first (to our knowledge) semantic characterisation of the logic called SCL by Hornischer (2020). Moreover, we settle two conjectures by Humberstone (2014) which use signed occurrences to study the equational logic of the power algebra of the two-valued Boolean algebra.

Real Functions in Stochastic Dependence

Abstract In a fuzzified probability theory, random events are modeled by measurable functions into [0,1] and probability measures are replaced with probability integrals. The transition from Boolean two-valued logic to Lukasiewicz multivalued logic results in an upgraded probability theory in which we define and study asymmetrical stochastic dependence/independence and conditional probability based on stochastic channels and joint experiments so that the classical constructions follow as particular cases. Elementary categorical methods enable us to put the two theories into a perspective.

Switching-Algebraic Analysis of Multi-State System Reliability

Multi-State systems are systems whose outputs are multi-valued (due to multiple levels of capacity or performance) and (possibly) whose inputs are also multi-valued (due to multiple performance levels or multiple modes of failure). These systems are a generalization of binary or dichotomous systems that have binary or two-valued outputs and inputs. The multi-state reliability model generalizes and adapts many of the concepts and techniques of the binary reliability model, and naturally ends up with sophisticated concepts and techniques of its own. This paper explores the possibility of simply analyzing a multi-state system by reformulating or encoding its inputs in terms of binary inputs and evaluating each of its multiple output levels as an individual binary output of these alternative inputs. This means that we dispense with multiple-valued logic in the analysis of a multi-state system, since this system is now analyzed solely via switching algebra (two-valued Boolean algebra). The wealth of tools and techniques of switching algebra are now used (without any modification or adaptation) in the analysis of the multi-state system (at the cost of an expanded input domain). The paper makes its point though the analysis of a standard commodity-supply system, whose multi-valued inputs are expressed in terms of physically-meaningfully binary inputs. The analysis is made possible through the use of advanced techniques for deriving probability–ready expressions together with the employment of large-size Karnaugh maps and utilization of multiplication tables, symmetric switching functions, and Boolean quotients. Though the system studied involves twelve binary input variables, its manual analysis is completed successfully herein, yielding results that exactly agree with those obtained earlier via automated methods, and are possibly less prone to the notorious effects of round-off errors.

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

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.

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.

Rewriting systems over similarity and generalized pseudometric spaces and their properties

We present a study of confluence and related properties of fuzzy relations defined over similarity spaces. The ordinary confluence is an essential property of relations connected to the idea of rewriting and substituting which appears in abstract rewriting systems. This paper is a continuation of our previous paper, where we have introduced analogous notions related to substitutability in graded setting using residuated lattices as structures of truth degrees, leaving the ordinary notions a particular case when the underlying structure is the two-valued Boolean algebra. In this paper, we further extend our previous results by developing the notions of confluence and related properties respecting a given similarity relation. We also present definitions of confluence and related properties of relations on a generalized pseudometric spaces. Using the well-known link between generalized pseudometrics and similarities, we describe the connection of the notions defined on generalized pseudometric spaces to the corresponding notions on similarity spaces. The introduced notions are further illustrated by example of rewriting based on graded if–then rules.

Similarity issues of confluence of fuzzy relations

We study similarity issues of confluence and related properties of fuzzy relations. The ordinary notions of divergence, convergence, convertibility, and confluence are essential properties of relations which appear in abstract rewriting systems. In a graded setting, we can introduce analogous notions related to the idea of approximate rewriting. This paper is a continuation of our previous paper (Belohlavek et al. 2010), where we have introduced such notions using residuated structures of truth degrees, leaving the ordinary notions a particular case when the underlying structure of truth degrees is the two-valued Boolean algebra. In this paper, we focus on similarity issues of confluence and present formulas which provide approximations of degrees of confluence and related properties based on similarity of the original fuzzy relations.

Confluence and termination of fuzzy relations

Confluence and termination are essential properties connected to the idea of rewriting and substituting which appear in abstract rewriting systems. The aim of the present paper is to investigate confluence, termination, and related properties from the point of view of fuzzy logic leaving the ordinary notions a particular case when the underlying structure of truth degrees is two-valued Boolean algebra. The main motivation of this study is the fact that in several natural situations, the notion of substitutability is inherently fuzzy rather than crisp.

Linguistic Fuzzy-Logic Social Game of Cooperation

I develop a new game-theoretic approach based, not on conventional Boolean two-valued logic, but instead on linguistic fuzzy logic which admits linguistic truth values. A linguistic fuzzy game is defined with linguistic fuzzy strategies, linguistic fuzzy preferences, and the rules of reasoning and inferences of the game operate according to linguistic fuzzy logic, not Boolean logic. This leads to the introduction of a new notion of fuzzy domination and Nash equilibrium which is based not on the usual ‘greater than’ relation ordering but rather on a more general form of relation termed linguistic fuzzy relation. Each agent models others as linguistic fuzzy rational agents and tries to find a linguistic fuzzy Nash equilibrium that will achieve the highest linguistic fuzzy payoff. If the linguistic fuzzy relation is simplified into a crisp two-valued logic, the linguistic fuzzy game reduces to the conventional game. In this article I apply the new approach to situations of non-cooperative twoplayer games such as a 2 × 2 prisoner's dilemma (PD) game and a social game of cooperation with N players. I find that there is always an optimum strong Nash equilibrium which is Pareto optimal, thereby lifting many of the dilemmas that emerge in crisp game theory in two-player and social games.

Linguistic Fuzzy-Logic Game Theory

The author develops a new game-theoretic approach, anchored not in Boolean two-valued logic but instead in linguistic fuzzy logic. The latter is characterized by two key features. First, the truth values of logical propositions span a set of linguistic terms such as true, very true, almost false, very false, and false. Second, the logic allows logical categories to overlap in contrast to Boolean logic, where the two possible logical categories, “true” and “false,” are sharply distinct. A game becomes a linguistic fuzzy logic game by turning strategies into linguistic fuzzy strategies, players’ preferences into linguistic fuzzy preferences, and the rules of reasoning and inferences of the game into linguistic fuzzy reasoning operating according to linguistic fuzzy logic. This leads to the introduction of a new notion of linguistic fuzzy domination and linguistic Nash equilibrium.

Physical and Information-Dynamical (Cognitive) Picture of the World

After presenting by extensive quotations the views of prominent philosophers, White-head, Schopenhauer, Wittgenstein and Popper, concerning human picture of the world, the picture of the physical world represented in language is discussed. It is shown that the categorical language based on Boolean two-valued logic is not sufficient for the description of such basic physical concepts as energy, force, entropy (information), etc. Such modal concepts as possibility, necessity, potentiality, probability, are needed in addition. Relativity physics, as well as quantum mechanics and field theory, shifted physical theory more and more in the direction of nonclassical modal logic. But if we pass from physics to general information dynamics of open systems (cognitive dynamics), where, except of the physical world, also mental and cultural worlds (Popperian world 1, world 2, and world 3) are considered, the classical scheme of information channel is not sufficient. We define, therefore, a generalized language scheme in which the three sets fixing the general meaning, denotation (D), connotation (C), and emotional aspect (E), are essential. At the end some classifications of sentences and verbs (predicates) are presented, as examples of primary syntactic, semantic, and pragmatic categories.

Algebraic Kripke Sheaf Semantics for Non-Classical Predicate Logics

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke semantics. We introduce algebraic Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Godel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic.