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In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be decided by a family of Boolean circuits, one circuit for each possible input length. Boolean circuits are defined in terms of the logic gates they contain.
Boolean function; Boolean-valued function; Boolean-valued model; Boolean satisfiability problem; Boolean differential calculus; Indicator function (also called the characteristic function, but that term is used in probability theory for a different concept) Espresso heuristic logic minimizer; Logical matrix; Logical value; Stone duality; Stone ...
It has been called the "fundamental theorem of Boolean algebra". [1] Besides its theoretical importance, it paved the way for binary decision diagrams (BDDs), satisfiability solvers, and many other techniques relevant to computer engineering and formal verification of digital circuits.
A law of Boolean algebra is an identity such as x ∨ (y ∨ z) = (x ∨ y) ∨ z between two Boolean terms, where a Boolean term is defined as an expression built up from variables and the constants 0 and 1 using the operations ∧, ∨, and ¬. The concept can be extended to terms involving other Boolean operations such as ⊕, →, and ≡ ...
The circuit on the left is satisfiable but the circuit on the right is not. In theoretical computer science, the circuit satisfiability problem (also known as CIRCUIT-SAT, CircuitSAT, CSAT, etc.) is the decision problem of determining whether a given Boolean circuit has an assignment of its inputs that makes the output true. [1]
In digital electronics and computer science (fields of applied logic engineering and mathematics), truth tables can be used to reduce basic Boolean operations to simple correlations of inputs to outputs, without the use of logic gates or code. For example, a binary addition can be represented with the truth table:
Boolean functions play a basic role in questions of complexity theory as well as the design of processors for digital computers, where they are implemented in electronic circuits using logic gates. The properties of Boolean functions are critical in cryptography, particularly in the design of symmetric key algorithms (see substitution box).
Therefore, the theorems of the Boolean algebra applies to ADD, notably the Boole's expansion theorem. [ 1 ] Each node of is labeled by a Boolean variable and has two outgoing edges: a 1-edge which represents the evaluation of the variable to the value TRUE, and a 0-edge for its evaluation to FALSE.