NC (complexity)¶

In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10927
Domain group: Applied Sciences & Engineering
Origin domain: Computer Science & Software Engineering
Subdomain: Computational Complexity Theory → Computer Science & Software Engineering

Core Idea¶

NC (complexity) is treated here as the recurring mathematics and formal science identity summarized by this source-grounded definition: In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors.

In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem with input size n is in NC if there exist constants c and k such that it can be solved in time using parallel processors. Stephen Cook coined the name "Nick's class" after Nick Pippenger, who had done extensive research on circuits with polylogarithmic depth and polynomial size.

As in the case of circuit complexity theory, usually the class has an extra constraint that the circuit family must be uniform (see below). Just as the class P can be thought of as the tractable problems (Cobham's thesis), so NC can be thought of as the problems that can be efficiently solved on a parallel computer. NC is a subset of P because polylogarithmic parallel computations can be simulated by polynomial-time sequential ones.

For NC (complexity), the abstraction is narrower than the article's general subject matter: a positive case must preserve In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics and formal science, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — For instance, it implies that the majority function can be computed by a family of branching programs of constant width and polynomial size, while intuition might suggest that to achieve polynomial size, one needs a linear number of states.
Constitutive relation — Recursively applying such property, it is possible to build a binary tree of length O(\log(n)) in which every sum between two bits x_i and x_j is expressible by means of basic logical operators, e.g. through the Boolean expression (x_i \land \neg x_j) \lor (\neg x_i \land x_j) .
Operating condition — NC i is the class of decision problems decidable by uniform Boolean circuits with a polynomial number of gates of at most two inputs and depth , or the class of decision problems solvable in time O((log n) i ) on a parallel computer with a polynomial number of processors.
Recognition evidence — The definition of NC is not affected by the choice of how the PRAM handles simultaneous access to a single bit by more than one processor.
Admissible variation — As with P, by a slight abuse of language, one might classify function problems and search problems as being in NC.
Characteristic consequence — Often algorithms for those problems had to be separately invented and could not be naïvely adapted from well-known algorithms – Gaussian elimination and Euclidean algorithm rely on operations performed in sequence.
Failure boundary — The problem consists in counting the number of 1s in a string made of 1 and 0.

What It Is Not¶

Not the whole field of mathematics and formal science. The node requires the specific identity stated by In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors.
Not an over-broad reading. Often algorithms for those problems had to be separately invented and could not be naïvely adapted from well-known algorithms – Gaussian elimination and Euclidean algorithm rely on operations performed in sequence.
Not an over-broad reading. With different levels of constraints, we would obtain possibly different complexity classes, with a more stringent constraint leading to a possibly smaller complexity class.
Not an over-broad reading. However, for k \geq 2 , NC k -uniform NC k and LOGSPACE-uniform NC k are equal, and both are equivalent to the following definition: The family is decided by an alternating Turing machine.
Not automatically Complete (complexity). Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

NC (complexity) applies literally inside mathematics and formal science wherever the source-defined carrier and relation can be established. Its documented habitats include:

Problems in NC. As with P, by a slight abuse of language, one might classify function problems and search problems as being in NC.
The NC hierarchy. The smallest class, NC 0 , is the class of functions definable by Boolean circuits with constant depth and bounded fan-in.
Barrington's theorem. Each of the instructions is a tuple (i, p, q) where i is the index of variable to check (1 ≤ i ≤ n), and p and q are functions from {1, 2, ..., k} to {1, 2, ..., k}.
Barrington's theorem. The function mapping an input to a final state of the program is called the yield of the program (more precisely, the yield on an input is the function mapping any initial state to the corresponding final state).
Barrington's theorem. The program accepts a set A \subseteq 2^n of variable values when there is some set of functions F \subseteq k^k such that a variable sequence x \in 2^n is in A precisely when its yield is in F.
Barrington's theorem. For instance, it implies that the majority function can be computed by a family of branching programs of constant width and polynomial size, while intuition might suggest that to achieve polynomial size, one needs a linear number of states.

Outside mathematics and formal science, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of NC (complexity) names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. The strongest recognition evidence in the frozen account is: The definition of NC is not affected by the choice of how the PRAM handles simultaneous access to a single bit by more than one processor. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification Often algorithms for those problems had to be separately invented and could not be naïvely adapted from well-known algorithms – Gaussian elimination and Euclidean algorithm rely on operations performed in sequence. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

NC (complexity) compresses multiple mathematics and formal science details into a stable diagnostic relation. The source shows both the central mechanism—recursively applying such property, it is possible to build a binary tree of length O(\log(n)) in which every sum between two bits x_i and x_j is expressible by means of basic logical operators, e.g. through the Boolean expression (x_i \land \neg x_j) \lor (\neg x_i \land x_j) .—and the practical consequence—often algorithms for those problems had to be separately invented and could not be naïvely adapted from well-known algorithms – Gaussian elimination and Euclidean algorithm rely on operations performed in sequence. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics and formal science entities to which the claim applies.
State the relation. Use the source-grounded identity: In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors.
Check operation and conditions. NC i is the class of decision problems decidable by uniform Boolean circuits with a polynomial number of gates of at most two inputs and depth , or the class of decision problems solvable in time O((log n) i ) on a parallel computer with a polynomial number of processors.
Demand recognition evidence. The definition of NC is not affected by the choice of how the PRAM handles simultaneous access to a single bit by more than one processor.
Test variation. Change an implementation or setting while preserving as with P, by a slight abuse of language, one might classify function problems and search problems as being in NC.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about NC (complexity) transfers literally when a new case preserves the same carrier type, relation, and recognition test. As with P, by a slight abuse of language, one might classify function problems and search problems as being in NC. The smallest class, NC 0 , is the class of functions definable by Boolean circuits with constant depth and bounded fan-in.

Beyond the home domain. No canonical parent is asserted for NC (complexity). An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

This is true for both the uniform and nonuniform case (DLOGTIME-uniformity suffices). This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors; recognition evidence → The definition of NC is not affected by the choice of how the PRAM handles simultaneous access to a single bit by more than one processor

Applied / In Practice¶

It is widely believed that (1) is the case, although no proof as to the truth of either statement has yet been discovered. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Uniformity; invariant → In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors; boundary → the case exits the class when often algorithms for those problems had to be separately invented and could not be naïvely adapted from well-known algorithms – Gaussian elimination and Euclidean algorithm rely on operations performed in sequence

Structural Tensions¶

T1 — Stable identity versus admissible variation. Often algorithms for those problems had to be separately invented and could not be naïvely adapted from well-known algorithms – Gaussian elimination and Euclidean algorithm rely on operations performed in sequence. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. With different levels of constraints, we would obtain possibly different complexity classes, with a more stringent constraint leading to a possibly smaller complexity class. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. However, for k \geq 2 , NC k -uniform NC k and LOGSPACE-uniform NC k are equal, and both are equivalent to the following definition: The family is decided by an alternating Turing machine. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. One major open question in complexity theory is whether or not every containment in the NC hierarchy is proper. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. For instance, it implies that the majority function can be computed by a family of branching programs of constant width and polynomial size, while intuition might suggest that to achieve polynomial size, one needs a linear number of states. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate NC (complexity) literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. Recursively applying such property, it is possible to build a binary tree of length O(\log(n)) in which every sum between two bits x_i and x_j is expressible by means of basic logical operators, e.g. through the Boolean expression (x_i \land \neg x_j) \lor (\neg x_i \land x_j) . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does NC (complexity) distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

NC (complexity) is structural-leaning. Its structural side is the repeatable organization summarized by In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. Its framed side is the mathematics and formal science vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: NC i is the class of decision problems decidable by uniform Boolean circuits with a polynomial number of gates of at most two inputs and depth , or the class of decision problems solvable in time O((log n) i ) on a parallel computer with a polynomial number of processors. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: For instance, it implies that the majority function can be computed by a family of branching programs of constant width and polynomial size, while intuition might suggest that to achieve polynomial size, one needs a linear number of states. Recursively applying such property, it is possible to build a binary tree of length O(\log(n)) in which every sum between two bits xi and xj is expressible by means of basic logical operators, e.g. through the Boolean expression (xi \land \neg xj) \lor (\neg xi \land xj) . It further constrains recognition and variation through: NC i is the class of decision problems decidable by uniform Boolean circuits with a polynomial number of gates of at most two inputs and depth , or the class of decision problems solvable in time O((log n) i ) on a parallel computer with a polynomial number of processors. The definition of NC is not affected by the choice of how the PRAM handles simultaneous access to a single bit by more than one processor.

What is domain-bound. mathematics and formal science supplies the operative entities, technical vocabulary, warrants, and exceptions that make NC (complexity) literal. Its documented scope includes the condition that As with P, by a slight abuse of language, one might classify function problems and search problems as being in NC. Another bounded application condition is that The smallest class, NC 0 , is the class of functions definable by Boolean circuits with constant depth and bounded fan-in. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—As with P, by a slight abuse of language, one might classify function problems and search problems as being in NC.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Complexity Class.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for NC (complexity). The reviewed identity is: In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction NC (complexity) Domain-specific

Parents (1) — more general patterns this builds on

NC (complexity) is a kind of Complexity Class Domain-specific

NC (complexity) is a domain-specific kind of complexity class under the frozen identity and differentia. Complete-catalog comparison found the corresponding live broader identity.

Neighborhood in Abstraction Space¶

NC (complexity) sits in a crowded region of the domain-specific corpus (35^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Computation Models & Complexity Classes (37 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors?
Complete (complexity). Complete (complexity) denotes notion of the "hardest" or "most general" problem in a complexity class in computing and information systems. Tell: Which entry's carrier, operation, and failure condition are satisfied?
NTIME. NTIME(f(n)) is the complexity class of decision problems solvable by a nondeterministic Turing machine within O(f(n)) steps, with NTIME denoting the corresponding time-bounded nondeterministic hierarchy. Tell: Which entry's carrier, operation, and failure condition are satisfied?
TC (Complexity). The threshold-circuit hierarchy of languages recognized by polynomial-size, polylogarithmic-depth circuit families with unbounded-fan-in majority or threshold gates. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would NC (complexity) remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics and formal science lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/NC_(complexity) (revision 1367249415).
Preserved source candidate: http://citeseerx.ist.psu.edu/showciting?cid=1672592
Preserved source candidate: https://web.archive.org/web/20220310162138/http://citeseerx.ist.psu.edu/showciting?cid=1672592
Preserved source candidate: https://www.infona.pl//resource/bwmeta1.element.ieee-art-000004568025
Preserved source candidate: https://lin-web.clarkson.edu/~alexis/PCMI/Notes/lectureB02.pdf
Preserved source candidate: https://complexityzoo.net/Complexity_Zoo:T
Preserved source candidate: https://dl.acm.org/doi/10.1145/1706591.1706594
Preserved source candidate: https://dx.doi.org/10.1016/0022-0000%2890%2990022-D
Preserved source candidate: http://www.cs.umass.edu/~barring/publications/bwbp.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.