# computational complexity theory

> theoretical computer science and mathematics theory that classifies problems according to their inherent difficulty, and relates those classes to each other

**Wikidata**: [Q205084](https://www.wikidata.org/wiki/Q205084)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Computational_complexity_theory)  
**Source**: https://4ort.xyz/entity/computational-complexity-theory

## Summary  
Computational complexity theory is the branch of theoretical computer science and mathematics that classifies computational problems according to their inherent difficulty and studies the relationships between those difficulty classes. It provides a formal framework for understanding how much time, space, or other resources an algorithm needs to solve a problem.

## Key Facts  
- **Discipline**: An *academic discipline* (instance_of) that belongs to both *mathematics* and *theoretical computer science* (part_of).  
- **Core Focus**: Classifies problems by inherent difficulty and relates the resulting *complexity classes* to one another (wikidata_description).  
- **Sub‑field Relationships**: A *subclass* of **computability theory** and of **complexity theory**; closely linked to **quantum complexity theory** (subclass_of, related).  
- **Identifiers**: GND ID 4120591‑1; Freebase ID /m/023z_; MathWorld ID ComplexityTheory; Library of Congress ID sh85029473.  
- **Aliases**: Also known as *complexity theory*, *teoría de la complejidad computacional*, *complexité des algorithmes*, *計算複雑性*, among many others (aliases).  
- **Coverage**: Listed under the *algorithm* facet (facet_of) and maintained by **WikiProject Mathematics** (maintained_by_wikiproject).  
- **Online Presence**: 39 Wikipedia language editions (sitelink_count = 39) and dedicated Stack Exchange tags for both Stack Overflow and CS.SE (stack_exchange_tag).  
- **Related Topics**: Directly studies *quantum supremacy* and broader *computational complexity* (is_the_study_of).  

## FAQs  
### Q: What does computational complexity theory study?  
**A:** It studies how much computational resource—such as time or memory—is required to solve problems, grouping problems into *complexity classes* that reflect their intrinsic difficulty.  

### Q: How is it different from computability theory?  
**A:** Computability theory asks *whether* a problem can be solved at all, while computational complexity theory asks *how efficiently* it can be solved, measuring resources needed for solvable problems.  

### Q: Why is quantum complexity theory mentioned?  
**A:** Quantum complexity theory extends the same classification ideas to algorithms that run on quantum computers, exploring how quantum resources affect problem difficulty.  

## Why It Matters  
Computational complexity theory underpins every decision about algorithm design, optimization, and feasibility. By revealing the minimal resources required for a problem, it guides engineers to choose realistic solutions, informs cryptographers about the security of encryption schemes (which rely on hard problems), and shapes the limits of what can be efficiently computed. Its insights also drive advances in emerging fields like quantum computing, where understanding *quantum supremacy* hinges on comparing classical and quantum complexity classes. In academia, the theory provides a rigorous language for proving lower bounds and for classifying problems such as P, NP, and PSPACE, which are central to both theoretical research and practical technology development.

## Notable For  
- **Formal Classification**: Introduced a systematic way to group problems into complexity classes (e.g., P, NP).  
- **Bridge Between Disciplines**: Sits at the intersection of mathematics, theoretical computer science, and algorithm analysis.  
- **Quantum Extension**: Directly leads to *quantum complexity theory*, assessing the power of quantum algorithms.  
- **Extensive Identifier Network**: Recognized across major catalogues (GND, Freebase, MathWorld, Library of Congress).  
- **Community Support**: Actively maintained by WikiProject Mathematics and has a broad multilingual Wikipedia presence (39 language editions).  

## Body  

### Definition and Scope  
Computational complexity theory examines the *resource consumption* of algorithms—primarily time and space—and categorizes decision problems based on the asymptotic growth of these resources. It treats problems as *languages* or *functions* and studies the *worst‑case* behavior of any algorithm solving them.

### Historical Context  
- Originated as a refinement of **computability theory**, which established which problems are solvable.  
- Early work introduced the first major classes, such as **P** (polynomial time) and **NP** (nondeterministic polynomial time).  

### Core Concepts  

- **Complexity Classes**: Sets of problems sharing similar resource bounds (e.g., P, NP, PSPACE, EXP).  
- **Reductions**: Transformations that preserve difficulty, used to prove that a problem is at least as hard as another.  
- **Time/Space Hierarchies**: Theorems showing strict inclusions between classes when more resources are allowed.  
- **Complete Problems**: Representative hardest problems within a class (e.g., SAT for NP‑complete).  

### Relationship to Other Theories  

- **Computability Theory**: Provides the baseline “solvable vs. unsolvable” distinction; complexity adds quantitative depth.  
- **Quantum Complexity Theory**: Extends the same classification to quantum algorithms, defining classes like **BQP** (bounded‑error quantum polynomial time).  

### Applications  

- **Algorithm Design**: Guides the search for efficient algorithms or proves that none exist under current assumptions.  
- **Cryptography**: Relies on problems believed to be intractable (e.g., factoring in NP‑intermediate).  
- **Complexity‑Based Lower Bounds**: Establishes limits for circuit size, communication, and query complexity.  

### Resources and Community  

- **Academic Discipline**: Recognized as a formal field of study (instance_of).  
- **Identifiers**: GND 4120591‑1, Freebase /m/023z_, MathWorld ComplexityTheory, Library of Congress sh85029473.  
- **Online Presence**: 39 Wikipedia language editions; dedicated Stack Exchange tags; maintained by WikiProject Mathematics.  

## Schema Markup  
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  "@type": "Thing",
  "name": "Computational complexity theory",
  "description": "Theoretical computer science and mathematics theory that classifies problems according to their inherent difficulty, and relates those classes to each other.",
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## References

1. [Nuovo soggettario](https://thes.bncf.firenze.sbn.it/termine.php?id=2244)
2. Nuovo soggettario
3. Freebase Data Dumps. 2013
4. Quora
5. National Library of Israel
6. [OpenAlex](https://docs.openalex.org/download-snapshot/snapshot-data-format)