View source for Boolean differential calculus

A branch of mathematics dealing with the concepts of differentials and derivatives of Boolean functions (cf. [[Boolean function|Boolean function]]) and the manner of using these in the study of such functions.

Boolean differential calculus originated from the treatment of electrical engineering problems in the areas of error-correcting codes (cf. [[Error-correcting code|Error-correcting code]]) and of design and testing of switching circuits; development into a self-contained mathematical theory was achieved in 1959 [[#References|[a1]]], [[#References|[a2]]], and continued in the time thereafter [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]]. Boolean differential calculus has also found other engineering applications: e.g., it can be used as a unifying framework for the modeling and investigation of finite automata (cf. [[Automaton, finite|Automaton, finite]]) and of discrete event dynamical systems [[#References|[a7]]] (cf. also [[Discrete event system|Discrete event system]]), i.e., dynamical systems with discrete states and changes of states called events; such systems arise e.g. in digital network communication protocols.

Many concepts in Boolean differential calculus are in analogy to those of classical [[Differential calculus|differential calculus]] for real-valued functions of one or more real variables; such are, e.g., the concept of a differential, describing the change of the value of a function and variables, and the concept of a derivative, describing how the value of a function depends on changes of its argument(s).

The simplest and (with regard to applications) most important case is based on the two-element Boolean algebra with carrier set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b1107601.png" />, on Boolean or binary variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b1107602.png" />, and on vectors of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b1107603.png" /> in a Boolean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b1107604.png" />. A Boolean function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b1107605.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b1107606.png" />, and a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b1107607.png" /> functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b1107608.png" /> can be represented as a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b1107609.png" />.

A Boolean equation of the general form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076010.png" /> can always be written in homogeneous form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076011.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076012.png" />, and a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076013.png" /> simultaneous equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076014.png" /> can always be combined into one single equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076015.png" />. Here and below, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076016.png" /> denotes addition modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076017.png" />, or the operation of exclusive or, and the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076020.png" /> stand for disjunction, conjunction and negation, respectively.

==Derivatives.==
Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076022.png" />. The (Boolean) derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076023.png" /> of a Boolean function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076024.png" /> with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076025.png" /> is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076026.png" /> given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076027.png" /></td> </tr></table>

or, equivalently,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076028.png" /></td> </tr></table>

It has the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076029.png" /> if and only if a change in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076030.png" /> changes the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076031.png" />.

The maximum and the minimum of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076032.png" /> with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076033.png" /> is defined as:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076034.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076035.png" /></td> </tr></table>

Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076037.png" />. The derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076038.png" /> of a Boolean function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076039.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076040.png" /> variables in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076041.png" /> is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076042.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076043.png" /></td> </tr></table>

Maxima and minima of a function with respect to more than one of its variables are defined accordingly.

==Differentials.==
The variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076044.png" /> defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076045.png" /></td> </tr></table>

is called the differential of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076046.png" />, and describes changes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076047.png" />. Likewise, the vector

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076048.png" /></td> </tr></table>

is called the differential of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076049.png" />, and describes the changes that occur in the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076050.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076051.png" /> changes to some other value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076052.png" />; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076053.png" /> denotes component-wise exclusive-or. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076055.png" /> is a point, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076056.png" /> is the direction from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076057.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076058.png" />. The (total) differential of a Boolean function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076059.png" /> is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076060.png" /></td> </tr></table>

the partial differential of a Boolean function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076061.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076062.png" /> is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076063.png" /></td> </tr></table>

and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076065.png" />th partial differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076066.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076067.png" /> is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076068.png" /></td> </tr></table>

Other useful operators include the various differential minima and maxima that can be derived from the various differentials of functions by replacing  ""  with  ""  or  "+" .

Boolean differential equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076069.png" />, as well as Boolean equations, can be solved and investigated with the aid of differential operators. Numerical tools may operate on the solution sets of equations rather than on the equations themselves. A compact representation of solution sets uses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076070.png" />-, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110760/b11076071.png" />-, and  "do-not-care" -elements in ternary-valued tables.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.B. Akers,   "On a theory of Boolean functions"  ''SIAM J.'' , '''7'''  (1959)  pp. 487–498</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.D. Talantsev,   "On the analysis and synthesis of certain electrical circuits by means of special logical operators"  ''Avtomat. i Telemeh.'' , '''20'''  (1959)  pp. 898–907  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Thayse,   "Boolean differential calculus"  ''Philips Res. Rep.'' , '''26'''  (1971)  pp. 229–246</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Bochmann,   "Boolean differential calculus (a survey)"  ''Engin. Cybernet.'' , '''15''' :  5  (1977)  pp. 67–75</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Bochmann,   C. Posthoff,   "Binäre dynamische Systeme" , R. Oldenbourg  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Thayse,   "Boolean calculus of differences" , ''Lecture Notes in Computer Science'' , '''101''' , Springer  (1981)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R. Scheuring,   H. Wehlan,   "On the design of discrete event dynamic systems by means of the Boolean differential calculus"  D. Franke (ed.)  F. Kraus (ed.) , ''Design Methods of Control Systems'' , '''2''' , Pergamon  (1991)  pp. 723–728</TD></TR></table>