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Model of computation
computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal language can be
Boolean_circuit
Programming language construct
Short-circuit evaluation, minimal evaluation, or McCarthy evaluation (after John McCarthy) is the semantics of some Boolean operators in some programming
Short-circuit_evaluation
Computational problem
The circuit value problem (or circuit evaluation problem) is the computational problem of computing the output of a given Boolean circuit on a given input
Circuit_value_problem
Model of computational complexity
In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according
Circuit_complexity
Model of computation
this kind provide a generalization of Boolean circuits and a mathematical model for digital logic circuits. Circuits are defined by the gates they contain
Circuit_(computer_science)
Mathematical topics based on the works of George Boole
digital logical circuits. Boolean expression, an expression in a programming language that produces a Boolean value when evaluated Boolean function, a function
Boolean
Algebraic manipulation of "true" and "false"
algebra as the two-element Boolean algebra. In modern circuit engineering settings, there is little need to consider other Boolean algebras, thus "switching
Boolean_algebra
Function returning one of only two values
function Boolean formulas can also be displayed as a graph: Propositional directed acyclic graph Digital circuit diagram of logic gates, a Boolean circuit And-inverter
Boolean_function
Branch of computational complexity theory
weighted Boolean circuit problems: Input: a Boolean circuit Parameter: k Output: Whether there exists a weight-k input such that the circuit outputs True
Parameterized_complexity
Testing method for testing the randomness of pseudo-random number generators
i } {\displaystyle C=\{C_{k}^{i}\}} is a collection of boolean circuits, such that each circuit C k i {\displaystyle C_{k}^{i}} has less than P C ( k )
Next-bit_test
Expression in a computer program
propositional formulas in logic and are associated to Boolean circuits. Most programming languages have the Boolean operators OR, AND and NOT; in C and some languages
Boolean_expression
Classic NP-complete problem in 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
Circuit satisfiability problem
Circuit_satisfiability_problem
Cryptographic protocol for two-party computation
party. In the garbled circuit protocol, the function has to be described as a Boolean circuit. The history of garbled circuits is complicated. The invention
Garbled_circuit
Process in digital electronics and integrated circuit design
metallic structures on an integrated circuit. In terms of Boolean algebra, the optimization of a complex Boolean expression is a process of finding a
Logic_optimization
Type of digital logic implemented by Boolean circuits
time-independent logic) is a type of digital logic that is implemented by Boolean circuits, where the output is a pure function of the present input only. This
Combinational_logic
Class in computational complexity theory
the circuit family must be uniform (see below). Equivalently, NC can be defined as those decision problems decidable by a uniform Boolean circuit (which
NC_(complexity)
Set of problems in computational complexity theory
computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and quantum computers). The study of the relationships between complexity
Complexity_class
Boolean function
A majority gate is a logical gate used in circuit complexity and other applications of Boolean circuits. A majority gate returns true if and only if
Majority_function
Computational input that relies on the length but not content of the input
polynomial size Boolean circuit A(n) deciding the problem, we can use a Turing machine that interprets the advice string as a description of the circuit. Then,
Advice_(complexity)
Complexity class used in circuit complexity
hierarchy of TC classes. TC0 contains all languages which are decided by Boolean circuits with constant depth and polynomial size, containing only unbounded
TC0
Algorithmic complexity class
example, some graphs can be succinctly described by a small Boolean circuit. The circuit has 2 n {\displaystyle 2n} inputs, 1 output and p o l y ( n )
EXPTIME
Topics referred to by the same term
paths Boolean circuit, a mathematical model for digital logic circuits Integer circuit, a mathematical object of computational complexity Circuit complexity
Circuit
System with multiple networked computers
executed by each computer. Models such as Boolean circuits and sorting networks are used. A Boolean circuit can be seen as a computer network: each gate
Distributed_computing
Subfield of cryptography
evaluated. The function is viewed as a Boolean circuit, with inputs in binary of fixed length. A Boolean circuit is a collection of gates connected with
Secure multi-party computation
Secure_multi-party_computation
Algebraic structure modeling logical operations
In mathematics, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties
Boolean_algebra_(structure)
Russian mathematician (1932–2006)
finite-state automata, Boolean circuits and multi-valued logic circuits. Ingo Wegener, in his book The Complexity of Boolean Functions, credits O. B
Oleg_Lupanov
Formal language concept
( ℓ ) ) {\displaystyle O(\ell ^{2}\log(\ell ))} , and by a uniform Boolean circuit of depth O ( log ℓ ) {\displaystyle O(\log \ell )} . For two nondeterministic
Nested_word
Difficulty measures for computer science problems
hierarchy can be described as the class of Boolean circuits over NP predicates. A collapse of the Boolean hierarchy would imply a collapse of the polynomial
Boolean_hierarchy
Mathematical study of switched networks
Number One Electronic Switching System Boolean circuit Boolean differential calculus C-element Circuit complexity Circuit minimization Karnaugh map Logic design
Switching_circuit_theory
Measure of complexity of a Boolean circuit
complexity theory, especially circuit complexity theory, the weft of a Boolean circuit is a measure of its complexity. A Boolean circuit is a directed acyclic
Weft_(circuit)
Cirquent calculus (circuit sequent calculus) is a proof calculus that combines aspects of sequent calculus and boolean circuits. Its proof-objects are
Cirquent_calculus
Type of conservative logic circuit
processes in physics. This model can be used to simulate Boolean circuits in which the wires of the circuit correspond to paths on which one of the balls may
Billiard-ball_computer
Standard model in theoretical computer science
the differences between the study of arithmetic circuits and the study of Boolean circuits. In Boolean complexity, one is mostly interested in computing
Arithmetic_circuit_complexity
Subject field of Boolean algebra discussing changes of Boolean variables and functions
circuits and the use of error-correcting codes in electrical engineering, the roots for the development of what later would evolve into the Boolean differential
Boolean_differential_calculus
(FHEW) scheme for Boolean circuit evaluation with optimizations Chillotti-Gama-Georgieva-Izabachene (TFHE) scheme for Boolean circuit evaluation with extensions
PALISADE_(software)
on the size of constant-depth Boolean circuits. It was first introduced by Johan Håstad to prove that AC0 Boolean circuits of depth k require size exp
Switching_lemma
Graphical representation of an electrical circuit
equipment. In computer science, circuit diagrams are useful when visualizing expressions using Boolean algebra. Circuit diagrams are pictures with symbols
Circuit_diagram
Formal concept in theoretical computer science and cryptography
(unproven) circuit lower bounds in computational complexity theory. Hence the construction of pseudorandom generators for the class of Boolean circuits of a
Pseudorandom_generator
Device performing a Boolean function
of the algorithms and mathematics that can be described with Boolean logic. Logic circuits include such devices as multiplexers, registers, arithmetic
Logic_gate
Complexity class
polynomial-time reduction to the PIGEON problem, defined as follows: Given a Boolean circuit C {\displaystyle C} having the same number n {\displaystyle n} of input
PPP_(complexity)
Ordering of binary values, used for positioning and error correction
n-ary Gray code, also known as a non-Boolean Gray code. As the name implies, this type of Gray code uses non-Boolean values in its encodings. For example
Gray_code
Function in Boolean algebra
function is notable for its role in theoretical investigation of circuit complexity of Boolean functions. The output of the parity function is the parity bit
Parity_function
Operation in Boolean circuit theory
transformation, takes as input an arbitrary combinatorial logic circuit and produces an equisatisfiable boolean formula in conjunctive normal form (CNF). The length
Tseytin_transformation
Cryptographical test for pseudo-randomness
collection C = { C k } {\displaystyle C=\{C_{k}\}} is a collection of boolean circuits of size less than P C ( k ) {\displaystyle P_{C}(k)} . Let p k , S
Yao's_test
Electronic circuit formed on a small, flat piece of semiconductor material
and microcontrollers, use boolean algebra to process "one" and "zero" signals. Among the most advanced integrated circuits are the microprocessors or
Integrated_circuit
Two-dimensional Turing machine with emergent behavior
In 2000, Gajardo et al. showed a construction that calculates any boolean circuit using the trajectory of a single instance of Langton's ant. Greg Turk
Langton's_ant
Cryptographic software library
(FHEW) scheme for Boolean circuit evaluation with optimizations Chillotti–Gama–Georgieva–Izabachene (TFHE) scheme for Boolean circuit evaluation with extensions
OpenFHE
On collapse of the polynomial hierarchy if NP is in non-uniform polynomial time class
Karp–Lipton theorem states that if the Boolean satisfiability problem (SAT) can be solved by Boolean circuits with a polynomial number of logic gates
Karp–Lipton_theorem
Puzzle video game genre
constructive, a method to quickly convert any Boolean circuit into such a grid that is possible if and only if the circuit is satisfiable; membership in NP is established
Minesweeper_(video_game)
Problem of determining if a Boolean formula could be made true
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY
Boolean satisfiability problem
Boolean_satisfiability_problem
Boolean circuit on which the key generation algorithm would be applied. The key generation algorithm runs Yao's garbling procedure over this Boolean circuit
Verifiable_computing
circuit complexity, TC (Threshold Circuit) is a complexity class of decision problems that can be recognized by threshold circuits, which are Boolean
TC_(complexity)
avoid trivial failures. DLOGTIME-uniformity is used in circuit complexity. A Boolean circuit family C 0 , C 1 , … {\displaystyle C_{0},C_{1},\dots }
DLOGTIME
English mathematician and philosopher (1815–1864)
with operations resembling logical ones Boolean circuit, a mathematical model for digital logical circuits. Boolean data type is a data type, having two
George_Boole
Type of geometry processing
Boolean operations on polygons are a set of Boolean operations (AND, OR, NOT, XOR, ...) operating on one or more sets of polygons in computer graphics
Boolean operations on polygons
Boolean_operations_on_polygons
Digital logic gate type
gate, the OR gate is one of three basic logic gates from which any Boolean circuit may be constructed. All other logic gates may be made from these three
OR_gate
Class of models and problems in circuit complexity
results, so-called circuit lower bounds, can be proved. Informally, ACC0 models the class of computations realised by Boolean circuits of constant depth
ACC0
Logic gate implementing negation
1s. The NOT gate is one of three basic logic gates from which any Boolean circuit may be built up. Together with the AND gate and the OR gate, any function
Inverter_(logic_gate)
Set of problems solved by small circuits
description of a Boolean circuit having n inputs, and that a Turing Machine for the language merely evaluates the given Boolean circuit on inputs of length
P/poly
Electrical circuit with active components
binary '0'. Digital circuits make extensive use of transistors, interconnected to create logic gates that provide the functions of Boolean logic: AND, NAND
Electronic_circuit
(AOI) gate, it reduces the boolean expression ABCD + EFGH + EXPAND. When configured as AND-OR (AO) gate, it reduces the boolean expression ABCD + EFGH +
List of 4000-series integrated circuits
List_of_4000-series_integrated_circuits
graph (PDAG) is a data structure that is used to represent a Boolean function. A Boolean function can be represented as a rooted, directed acyclic graph
Propositional directed acyclic graph
Propositional_directed_acyclic_graph
Class of problems solvable in polynomial time
uniform family of Boolean circuits. A language L is in P if and only if there exists a polynomial-time uniform family of Boolean circuits { C n : n ∈ N }
P_(complexity)
Electronic circuits that utilize digital signals
are often packaged into integrated circuits. Complex devices may have simple electronic representations of Boolean logic functions. The binary number
Digital_electronics
Swedish computer scientist
bounds on the size of constant-depth Boolean circuits for the parity function. After Andrew Yao proved that such circuits require exponential size, Håstad
Johan_Håstad
Cryptographic primitive
proofs is relatively small; however, transforming a statement into a boolean circuit incurs considerable overhead. Proof systems under the sub-group hiding
Non-interactive zero-knowledge proof
Non-interactive_zero-knowledge_proof
Unsolved problem in computer science
in NP can be transformed mechanically into a Boolean satisfiability problem in polynomial time. The Boolean satisfiability problem is one of many NP-complete
P_versus_NP_problem
In circuit complexity, AC is a complexity class hierarchy. Each class, ACi, consists of the languages recognized by Boolean circuits with depth O ( log
AC_(complexity)
Standard forms of Boolean functions
simplification of Boolean functions, which is of great importance in the optimization of Boolean formulas in general and digital circuits in particular.
Canonical_normal_form
Topics referred to by the same term
Short-circuit evaluation, a form of Boolean evaluation in programming Short-circuit test Short (disambiguation) Circuit (disambiguation) Open circuit (disambiguation)
Short circuit (disambiguation)
Short_circuit_(disambiguation)
Logic constructed only from NAND gates
NAND(x,x). In the field of digital electronic circuits, this implies that it is possible to implement any Boolean function using just NAND gates. The mathematical
NAND_logic
, there is a Boolean circuit of size polynomial in n {\displaystyle n} that implements a verifier for the problem. That is, the circuit computes a function
NP/poly
Unconventional logic circuit
its clock input acts randomly and with probability p = 1/2. Unlike Boolean circuits, which behave deterministically, a random flip-flop behaves non-deterministically
Random_flip-flop
Unproven computational hardness assumption
by Impagliazzo & Paturi (1999). It states that satisfiability of 3-CNF Boolean formulas (3-SAT) cannot be solved in subexponential time, 2 o ( n ) {\displaystyle
Exponential_time_hypothesis
Pair of logical equivalences
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid
De_Morgan's_laws
Logic puzzle
constructing puzzles equivalent to arbitrary Boolean circuits, which shows NP-completeness because of the Boolean satisfiability problem. Fertin, Jamshidi
Tentai_Show
Computing by new or unusual methods
billiard balls to perform computations. In this model, the wires of a Boolean circuit are represented by paths for the balls to travel on, the presence or
Unconventional_computing
Collection of molecular regulators
the functionalities of feedback circuits is determinant for the attractors' number and size in pathway-like Boolean networks". Scientific Reports. 7
Gene_regulatory_network
Graphical method to simplify Boolean expressions
Karnaugh map (KM or K-map) is a diagram that can be used to simplify a Boolean algebra expression. Maurice Karnaugh introduced the technique in 1953 as
Karnaugh_map
Concept in mathematical logic
In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true. In logic
Boolean_domain
Russian mathematician
Prize (1990) for introducing the "approximation method" in proving Boolean circuit lower bounds of some essential algorithmic problems, Erdős Lecturer
Alexander_Razborov
Master's thesis by C. E. Shannon
arrangements of relays to solve Boolean algebra problems. His thesis laid the foundations for all digital computing and digital circuits. The utilization of the
A Symbolic Analysis of Relay and Switching Circuits
A_Symbolic_Analysis_of_Relay_and_Switching_Circuits
Inherent difficulty of computational problems
based on non-deterministic Turing machines, Boolean circuits, quantum Turing machines, monotone circuits, etc. The resource (or resources) that is being
Computational complexity theory
Computational_complexity_theory
Theorem in Boolean algebra
In Boolean algebra, the consensus theorem or rule of consensus is the identity: x y ∨ x ¯ z ∨ y z = x y ∨ x ¯ z {\displaystyle xy\vee {\bar {x}}z\vee
Consensus_theorem
Provides lower bounds on the circuit complexity of boolean functions
lower bounds on the circuit complexity of boolean functions. A natural proof shows, either directly or indirectly, that a boolean function has a certain
Natural_proof
Logical connective OR
will come.' Affirming a disjunct Boolean algebra (logic) Boolean algebra topics Boolean domain Boolean function Boolean-valued function Conjunction/disjunction
Logical_disjunction
Form of encryption that allows computation on ciphertexts
encrypted data. The computations are represented as either Boolean or arithmetic circuits. Some common types of homomorphic encryption are partially homomorphic
Homomorphic_encryption
Process by which desired circuit behavior is turned into a schematic of logic gates
now termed Boolean algebra. In 1938, Claude Shannon showed that the two-valued Boolean algebra can describe the operation of switching circuits. In the early
Logic_synthesis
Graph representing an implementation of the logical functionality of a network
rarely structurally efficient for large circuits, but is an efficient representation for manipulation of boolean functions. Typically, the abstract graph
And-inverter_graph
Succinct versions of many graph problems, with graphs represented as Boolean circuits, ordered binary decision diagrams or other related representations:
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
Computer science concept
_{2}^{\mathrm {P} }} is circuit minimization: given a number k and a circuit A computing a Boolean function f, determine if there is a circuit with at most k gates
Polynomial_hierarchy
Type of cryptographic software obfuscation
the following two statements: Completeness or Functionality: For any Boolean circuit C of input length n and input x ∈ { 0 , 1 } n {\displaystyle x\in \{0
Indistinguishability obfuscation
Indistinguishability_obfuscation
Technical treatment of Boolean algebras
Boolean algebras are models of the equational theory of two values; this definition is equivalent to the lattice and ring definitions. Boolean algebra
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
Algorithm that outputs all solutions to a problem
disjunctive normal form, a binary decision diagram such as an OBDD, or a Boolean circuit in restricted classes studied in knowledge compilation, e.g., NNF.
Enumeration_algorithm
Determining all voltages and currents within an electrical network
analysed using Boolean algebra by assigning the two states ("on"/"off", "positive"/"negative" or whatever states are being used) to the Boolean constants "0"
Network analysis (electrical circuits)
Network_analysis_(electrical_circuits)
Logical gate whose output is false if all its inputs are true
NOR logic. Boolean algebra Flash memory Functional completeness Logic gate symbols NAND logic Sheffer stroke Smith, J.S. "Digital circuits, sizing, output
NAND_gate
Class in computational complexity theory
_{m}^{{\mathsf {NC}}^{k}}L'} if and only if there exists a L-uniform NCk Boolean circuit family that together computes a function f : { 0 , 1 } ∗ → { 0 , 1
P-complete
simple circuit like this can be used: In digital electronics a diode-OR circuit is used to derive a simple Boolean logic function. This kind of circuit was
Diode-or_circuit
Method for increasing reliability
set of specified Boolean function. If there are no circuit failures, the outputs of the three circuits are identical. But due to circuit failures, the outputs
Triple_modular_redundancy
Theorem in Boolean algebra
{\displaystyle F=x\cdot F_{x}+x'\cdot F_{x'}} , where F {\displaystyle F} is any Boolean function, x {\displaystyle x} is a variable, x ′ {\displaystyle x'} is
Boole's_expansion_theorem
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