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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
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 nam ...
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 functi ...
s.
Boolean differential calculus concepts are analogous to those of classical
differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
, notably studying the changes in functions and variables with respect to another/others.
[H. Wehlan]
Boolean Algebra in ''Encyclopedia of Mathematics''
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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 by nature of the ergodic theory, ergodicity of dynamic systems. When differ ...
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 with close connections to cognitive science and mathematical l ...
on 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 ...
* Petri net theory
History and applications
Originally inspired by the design and testing of switching circuits and the utilization of error-correcting code
In computing, telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels.
The centra ...
s in electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, the roots for the development of what later would evolve into the Boolean differential calculus were initiated by works of Irving S. Reed,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 co ...
.
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 ...
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)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 variation of a physical quantity. The protocol defines the rules, syntax, semantics (computer science), sem ...
s.
Meanwhile, BDC has seen extensions to multi-valued
In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional p ...
variables and functions
Overview
Boolean differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
s play a significant role in BDC. They allow the application of differentials 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 (38 ...
to be extended to logical functions.
The differentials of a Boolean variable models the relation:
:
There are no constraints in regard to the nature, the causes and consequences of a change.
The differentials are binary. They can be used just like common binary variables.
See also
* 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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
* Boole's expansion theorem
* Ramadge–Wonham framework
References
Further reading
* (14 pages)
* (462 pages)
* (9 pages) Translation of: (9 pages)
* (18 pages)
* (NB. Also: Chemnitz, Technische Universität, Dissertation.) (147 pages)
* (15 pages)
* (392 pages)
* (xxii+232 pages
(NB. Per this hardcover edition has been rereleased as softcover edition in 2010.)
* (49 pages)
* (24 of 153 pages)
External links
*
* with
{{Authority control
Boolean algebra
Automata (computation)
Mathematical logic
Order theory
Set theory