The Boolean Differential Calculus and Differential Equations
Subject Areas : Statistics
1 -
Keywords: حساب دیفرانسیل بولی, جبر بولی, تابع منطقی, معادلات دیفرانسیل بولی,
Abstract :
The Boolean Differential Calculus is a powerful theory that based on definitions of differentials of Boolean variables, differentials of Boolean functions and their derivatives, and extended the Boolean algebra significantly. Definitions and theorems that the Boolean differential calculus based on them, allow us to model the changes of function values together with the changes of the values of variables and many other properties of Boolean functions. In this paper, first we describe the theory of the Boolean differential calculus and present several examples. Then, we introduce the Boolean differential equations. These equations contain the Boolean derivatives of an unknown Boolean function. We show that different from a Boolean equation, the solution of a Boolean differential equation is a set of Boolean functions, and therefore, Boolean differential equations allow us to describe and handle the sets of Boolean functions. In this paper, first we describe the theory of the Boolean differential calculus and present several examples. Then, we introduce the Boolean differential equations. These equations contain the Boolean derivatives of an unknown Boolean function. We show that different from a Boolean equation, the solution of a Boolean differential equation is a set of Boolean functions, and therefore, Boolean differential equations allow us to describe and handle the sets of Boolean functions.
[1] D. Bochmann, C. Posthoff. Binäre Dynamische Systeme. Akademie-Verlag, Berlin (1981)
[2] I. I. Bucur, M. Dragoicea, N. Constantin. Boolean Differential Calculus-Applied in Logic Testing. Studies in Informatics and Control. 16: 209-216 (2007)
[3] D. A. Huffman. Solvability criterion for simultaneous logical equations. Quarterly Progress Report. 1: 87-88 (1958)
[4] C. Posthoff, B. Steinbach. Logic Functions and Equations-Binary Models for Computer Science. Springer, Dordrecht, The Netherlands (2004)
[5] I. S. Reed. A class of multiple-error-correcting codes and the decoding scheme. Transactions of the IRE Professional Group on Information Theory. 4: 38-49 (1954)
[6] J. Sheldon, B. Akers. On a theory of Boolean functions. Journal of the Society for Industrial and Applied Mathematics (SIAM). 7: 487-498 (1959)
[7] B. Steinbach. Lösung binrer Differentialgleichungen und ihre Anwendung auf binäre Systeme. Dissertation A (PhD-thesis), TH Karl-Marx-Stadt (1981)
[8] B. Steinbach, Ch. Posthoff. The Boolean Differential Calculus-Introduction and Examples. Proceedings–Reed-Muller Workshop 2009, Naha, Okinawa, Japan. 107-117 (2009)
[9] B. Steinbach, Ch. Posthoff. Boolean Differential Calculus. in: Sasao, T. and Butler, J. T. Progress in Application of Boolean Functions. Morgan & Claypool Publishers, San Rafael, CA, USA (2010)
[10] B. Steinbach, Ch. Posthoff. Boolean Differential Calculus–Theory and Applications. Journal of Computational and Theoretical Nanoscience. 7: 933-981 (2010)
[11] B. Steinbach, Ch. Posthoff. Boolean Differential Equations. Proceedings of the 20th International Workshops on Post-Binary ULSI Systems. Tuusula, Finnland. 46-53 (2011)
[12] B. Steinbach, Ch. Posthoff. Boolean Differential Equations–A Common Model for Classes, Lattices, and Arbitrary Sets of Boolean Functions. Facta Universitatis. 28: 51-76 (2015)
