Boolean Logic – Definition

Boolean Logic Definition

In mathematical concepts and mathematical logic, the term Boolean algebra refers to the branch of algebra that deals with the true value of variables.  These values are based on determining whether a number is true or false and usually assigned 1 and 0 respectively. Rather than elementary algebra, where the values are variable numbers, and the prime operations are multiplication and addition, the key operations of Boolean algebra based on conjunctions and disjunction denoted by ∧, and ∨, respectively and the negation does not have denotation as ¬.  Therefore, it is a formula used to describe logical relations in a similar way the elementary algebra describes numeric relations.

A Little More on What is Boolean Logic

The root of Boolean algebra can be traced to 1847 when George Boole introduced his first book The Mathematical Analysis of Logic (1847) and advanced it fully in his An Investigation of the Laws of Thought (1854). Huntington presents that the term “Boolean algebra” was proposed first by Sheffer in 1913 though Charles Sanders Peirce introduced the title “A Boolean Algebra with One Constant in 1880” in the first chapter of his book “The Simplest Mathematics.” The concept of Boolean algebra has considerably helped in the development of digital electronic by enhancing computer programming languages. Besides, it is also applied in set statistics and theories

While the elementary algebraic expressions mainly concern with numbers, Boolean algebraic expression concerns with truth value false and true.  The Boolean algebraic numbers are represented in either binary digits or bits which are either 0 or 1. These numbers do not have characteristics of integers 0 and 1 whereby 1+1 =2.  However, they may be identified with elements of the two-element field GF (2). In other words, the integer arithmetic modulo 2 which provides that 1+1 = 0. The multiplication and addition paly the Boolean role of XOR (exclusive=or) and AND (conjunction) respectively. On the other hand, the disjunction x∨y (inclusive-or) is described as x + y – xy.

Moreover, Boolean algebra operates with functions that have values lying between {0, 1}. The function uses a sequence of bits to generate values. One of the best examples is a subset of set E: to a subset F of E. This means that the indicator of the function will take the value 1 of F and 0 outside F. In addition, the most general example is the Boolean algebraic elements that all the preceding remains the instance thereof.

As in the case of elementary algebra, the explicitly equational part of the theory can be formed without considering the values attached to the variables

Basic operations

The Boolean algebraic operation is demonstrated below:

  • •        AND (conjunction), designated x∧y (sometimes x AND y or Kxy), contents x∧y = 1 if x = y = 1 and x∧y = 0 otherwise.
  • •        OR (disjunction), designated x∨y (sometimes x OR y or Axy), contents x∨y = 0 if x = y = 0 and x∨y = 1 otherwise.
  • •        NOT (negation), designated ¬x (sometimes NOT x, Nx or !x), contents ¬x = 0 if x = 1 and ¬x = 1 if x = 0.

Secondary operations

The above operation of Boolean algebra is considered basic operation because they can form the basis for other Boolean operations. These operations can be built up by composing the ways of combining or compounding operations.

The first operation known as the material implication is presented by, x → y, or Cxy. In this case, if x variable is true, then the value of x → y is considered to be that of y. On the other hand, if x variable is false, the value of y can be ignored; however, the operation must reveal the value of Boolean that has only two choices available. Therefore, by definition, x → y is true when x is false. (Relevance logic proposed the definition by considering the implication with a false variable as a different concept other than either true or false.

The second operation known as exclusive (XOR) is denoted by, x ⊕ y, or Jxy, is to distinguish it from inclusive disjunction. This operation excludes the likelihood of both x and y.  this operation id defined in the arithmetic as addition mod 2 that presents that 1+1 =0

The third operation, is known as the complement of exclusive or, is Boolean equality or equivalence: x ≡ y, or Exy, is considered to be true only when x and y have similar value. Therefore, the complement x ⊕ y can be understood as x ≠ y, being true only when x and y are different. The counterpart of Equivalence in arithmetic mod 2 is x + y + 1.

Given two operands, that have two possible values, 22= 4 will be generated as possible combination inputs since the output of each value can have two possible values — the sum of 24=16 possible binary Boolean operation.

References for Boolean Logic

Academic Research on Boolean Logic

  • A new formalism that combines advantages of fault-trees and Markov models: Boolean logic driven Markov processes, Bouissou, M., & Bon, J. L. (2003). Reliability Engineering & System Safety, 82(2), 149-163.
  • This paper discusses the modeling formulism that helps the operation analysist to join the inherited concepts from the Markov model, and the fault tree is a different way.   According to the author, this formalism is termed as Boolean logic Driven Markov Processes (BDMP). The author presents that this formalism has two advantages over conventional models applied in dependability assessment. First, it helps in the definition of complex dynamics models and remain readable. Secondly, it offers interesting properties of mathematical concepts that helps in an efficient procession of BDMP.
  • Error propagation in cartographic modeling using Boolean logic and continuous classification, Heuvelink, G. B., & Burrough, P. A. (1993). International Journal of Geographical Information Systems, 7(3), 231-246.
  • This paper provides an in-depth analysis of how errors in value can be distributed through continuous modeling and Boolean that involves the intersection of various maps. The analysis of the error was conducted using Monte Carlo methods on the interpolated data. The process involved block kriging to a normal grid which provides accurate predictions, standard deviation and prediction error of the attributed values.  The results of the study suggested that the Boolean method of sieve mapping are vulnerable to errors than the model of robust continuous equivalent.
  • An area-efficient carry select adder design by sharing the common Boolean logic term, Wey, I. C., Ho, C. C., Lin, Y. S., & Peng, C. C. (2012, March). Proceedings of the International Multiconference of Engineering and computer scientist, IMECS.  This paper explores the adders sharing through common Boolean logic terms.  According to the author, the process requires sharing and simplification of a partial circuit that only needs XOR and a single inverter gate.
  • Learning for quantified Boolean logic satisfiability, Giunchiglia, E., Narizzano, M., & Tacchella, A. (2002, July). In AAAI/IAAI(pp. 649-654). Understanding the ability to exploit and record some data that are not available during the search is considered to be an effective AI technique for solving problems.  This paper explores the models that are used to record and exploit data during research. According to the author, learning general purpose technique helps in improving the performance of decision procedures for Quantified Boolean Formulas (QBFs).
  • Non-volatile spin switch for Boolean and non-Boolean logic, Datta, S., Salahuddin, S., & Behin-Aein, B. (2012). Applied Physics Letters, 101(25), 252411. This paper presents that established physics of spin valves and the spin-Hall effect that was recently discovered can be used to develop read and write construct units. The author provides that these units can be integrated into one spin that has gain and fam-out, input-output isolation that can be used to write a program. He further presents that the spin switches can be interconnected without external amplification to perform logic operations.
  • An efficient SQRT architecture of carrying select adder design by common Boolean logic, Manju, S., & Sornagopal, V. (2013, January). In Emerging Trends in VLSI, Embedded System, Nano Electronics and Telecommunication System (ICEVENT), 2013 International Conference on (pp. 1-5). IEEE. This paper suggests an efficient method that can be used instead of BEC using common Boolean logic. According to the author, Carry Select Adder (CSLA) which is known to be the quickest adder existing in the Conventional adder structures. This study was based on the use of efficient Carry select adder through sharing the Common Boolean logic (CLB) term. The author further presents that, the process only needs a single OR gate and one inverter gate to carry out the process.  The result of the study revealed that the prosed architecture has various advantages in terms of delay, area, and power.
  • Developments in automatic text retrieval, Salton, G. (1991). Developments in automatic text retrieval. science, 253(5023), 974-980. This paper describes the recent development that has been achieved in storage, manipulation, and retrieval of large text files. The paper further examines the modern approaches leading to documentation and retrieval of the text items in relation to search request.
  • Backjumping for quantified Boolean logic satisfiability, Giunchiglia, E., Narizzano, M., & Tacchella, A. (2003). Artificial Intelligence, 145(1-2), 99-120. The application of effective cognitive tools for determining the satisfiability of Quantified Boolean Formulas (QBFs) is a profound issue of research in Artificial Intelligence.   This paper discusses various procedures that have been proposed for proportional satisfiability. The author shows the possibility to extend the conflict-directed backjumping schema for the SAT to the satisfiability of QBFs. According to the author, when, When conflict-directed backjumping is used,  it enables the search engine to ignore existentially quantified literals while backtracking.

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