Boolean differential calculus (BDC) (German: (BDK)) is a subject field of

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...

discussing changes of

Boolean variable In computer science, the Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted ''true'' and ''false'') which is intended to represent the two truth values of logic and Boolean algebra. It is named ...

s and

Boolean function In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function ...

s. Boolean differential calculus concepts are analogous to those of classical differential calculus, notably studying the changes in functions and variables with respect to another/others.H. Wehlan
Boolean Algebra in ''Encyclopedia of Mathematics''
/ref> The Boolean differential calculus allows various aspects of

dynamical systems theory Dynamical systems theory is an area of mathematics used to describe the behavior of complex systems, complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theo ...

such as *

automata theory Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word ''automata'' comes from the Greek word αὐτόματο� ...

finite automata A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number o ...

Petri net theory A Petri net, also known as a place/transition (PT) net, is one of several mathematical modeling languages for the description of distributed systems. It is a class of discrete event dynamic system. A Petri net is a directed bipartite graph that h ...

supervisory control theory The supervisory control theory (SCT), also known as the Ramadge–Wonham framework (RW framework), is a method for automatically synthesizing supervisors that restrict the behavior of a plant such that as much as possible of the given specification ...

(SCT) to be discussed in a united and closed form, with their individual advantages combined.

History and applications

Originally inspired by the design and testing of

switching circuit Switching circuit theory is the mathematical study of the properties of networks of idealized switches. Such networks may be strictly combinational logic, in which their output state is only a function of the present state of their inputs; or may ...

s and the utilization of

error-correcting code In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea i ...

s in electrical engineering, the roots for the development of what later would evolve into the Boolean differential calculus were initiated by works of

Irving S. Reed Irving Stoy Reed (November 12, 1923 – September 11, 2012) was an American mathematician and engineer. He is best known for co-inventing a class of algebraic error-correcting and error-detecting codes known as Reed–Solomon codes in collabora ...

David E. Muller David Eugene Muller (November 2, 1924 – April 27, 2008) was an American mathematician and computer scientist. He was a professor of mathematics and computer science at the University of Illinois (1953–92), after which he became an emerit ...

David A. Huffman David Albert Huffman (August 9, 1925 – October 7, 1999) was an American pioneer in computer science, known for his Huffman coding. He was also one of the pioneers in the field of mathematical origami. Education Huffman earned his bachelor's de ...

, Sheldon B. Akers Jr. and (, ) between 1954 and 1959, and of Frederick F. Sellers Jr., Mu-Yue Hsiao and

Leroy W. Bearnson Leroy or Le Roy may refer to: People * Leroy (name), a given name and surname * Leroy (musician), American musician * Leroy (sailor), French sailor Places United States * Leroy, Alabama * Le Roy, Illinois * Le Roy, Iowa * Le Roy, Kansas * Le Roy ...

in 1968. Since then, significant advances were accomplished in both, the theory and in the application of the BDC in switching circuit design and

logic synthesis In computer engineering, logic synthesis is a process by which an abstract specification of desired circuit behavior, typically at register transfer level (RTL), is turned into a design implementation in terms of logic gates, typically by a comp ...

. Works of ,

Marc Davio Marc or MARC may refer to: People * Marc (given name), people with the first name * Marc (surname), people with the family name Acronyms * MARC standards, a data format used for library cataloging, * MARC Train, a regional commuter rail system of ...

and in the 1970s formed the basics of BDC on which , and further developed BDC into a self-contained mathematical theory later on. A complementary theory of Boolean integral calculus (German: ) has been developed as well. BDC has also found uses in

discrete event dynamic system In control engineering, a discrete-event dynamic system (DEDS) is a discrete-state, event-driven system of which the state evolution depends entirely on the occurrence of asynchronous discrete events over time. Although similar to continuous-variab ...

s (DEDS) in

digital network A computer network is a set of computers sharing resources located on or provided by network nodes. The computers use common communication protocols over digital interconnections to communicate with each other. These interconnections are m ...

communication protocol A communication protocol is a system of rules that allows two or more entities of a communications system to transmit information via any kind of variation of a physical quantity. The protocol defines the rules, syntax, semantics and synchroniza ...

s. Meanwhile, BDC has seen extensions to

multi-valued In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...

variables and functions as well as to

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornam ...

s of Boolean functions.

Overview

Boolean differential operators play a significant role in BDC. They allow the application of

differential Differential may refer to: Mathematics * Differential (mathematics) comprises multiple related meanings of the word, both in calculus and differential geometry, such as an infinitesimal change in the value of a function * Differential algebra * ...

s as known from classical

analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

to be extended to logical functions. The differentials

dx_i

of a Boolean variable

x_i

models the relation: :

dx_i = \begin
  0,  & \text x_i\\
  1, & \text x_i
\end

There are no constraints in regard to the nature, the causes and consequences of a change. The differentials

dx_i

are binary. They can be used just like common

binary variables Binary data is data whose unit can take on only two possible states. These are often labelled as 0 and 1 in accordance with the binary numeral system and Boolean algebra. Binary data occurs in many different technical and scientific fields, wher ...

References

External links

* * with {{Authority control Algebra Automata (computation) Mathematical logic Order theory Set theory

History and applications

Overview

See also

References

Further reading

External links