# Computational complexity theory

**computational complexitycomplexity theorycomplexityintractableasymptotic complexitycomputationally intractableintractabilitytractabletractable problemtime complexity**

Computational complexity theory focuses on classifying computational problems according to their inherent difficulty, and relating these classes to each other.wikipedia

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### Model of computation

**models of computationmodelmodels of computing**

The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage.

In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input.

### Travelling salesman problem

**traveling salesman problemTSPtraveling salesman**

To further highlight the difference between a problem and an instance, consider the following instance of the decision version of the traveling salesman problem: Is there a route of at most 2000 kilometres passing through all of Germany's 15 largest cities?

In the theory of computational complexity, the decision version of the TSP (where, given a length L, the task is to decide whether the graph has any tour shorter than L) belongs to the class of NP-complete problems.

### Computational complexity

**complexitycomputational burdenamount of computation**

The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage.

The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory.

### Computability theory

**recursion theorycomputablecomputability**

Closely related fields in theoretical computer science are analysis of algorithms and computability theory.

Although there is considerable overlap in terms of knowledge and methods, mathematical recursion theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages.

### Analysis of algorithms

**computational complexitycomplexity analysiscomputationally expensive**

Closely related fields in theoretical computer science are analysis of algorithms and computability theory.

Algorithm analysis is an important part of a broader computational complexity theory, which provides theoretical estimates for the resources needed by any algorithm which solves a given computational problem.

### P versus NP problem

**P = NPP ≠ NPP=NP**

The P versus NP problem, one of the seven Millennium Prize Problems, is dedicated to the field of computational complexity.

The relation between the complexity classes P and NP is studied in computational complexity theory, the part of the theory of computation dealing with the resources required during computation to solve a given problem.

### Decision problem

**decision procedureundecidabledecision problems**

Decision problems are one of the central objects of study in computational complexity theory. A function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem—that is, the output isn't just yes or no.

In computability theory and computational complexity theory, a decision problem is a problem that can be posed as a yes-no question of the input values.

### Function problem

**FPfunction problems**

A function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem—that is, the output isn't just yes or no.

In computational complexity theory, a function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem.

### Communication complexity

Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

Note that, unlike in computational complexity theory, communication complexity is not concerned with the amount of computation performed by Alice or Bob, or the size of the memory used, as we generally assume nothing about the computational power of either Alice or Bob.

### Algorithm

**algorithmsalgorithm designcomputer algorithm**

A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.

Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters".

### Formal language

**formal language theoryformal languageslanguage**

A decision problem can be viewed as a formal language, where the members of the language are instances whose output is yes, and the non-members are those instances whose output is no.

In computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages that can be parsed by machines with limited computational power.

### Alternating Turing machine

**alternationalternating Turing machinesexistential state**

Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines.

In computational complexity theory, an alternating Turing machine (ATM) is a non-deterministic Turing machine (NTM) with a rule for accepting computations that generalizes the rules used in the definition of the complexity classes NP and co-NP.

### Church–Turing thesis

**Church-Turing thesisChurch's thesisTuring's Thesis**

Indeed, this is the statement of the Church–Turing thesis.

These variations are not due to Church or Turing, but arise from later work in complexity theory and digital physics.

### DTIME

**deterministicDTIME(f(''n''))DTIME(f(n))**

For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).

In computational complexity theory, DTIME (or TIME) is the computational resource of computation time for a deterministic Turing machine.

### Decision tree model

**algebraic decision treequery complexitydecision tree complexity**

Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

In computational complexity the decision tree model is the model of computation in which an algorithm is considered to be basically a decision tree, i.e., a sequence of branching operations based on comparisons of some quantities, the comparisons being assigned unit computational cost.

### Turing machine

**deterministic Turing machineTuring machinesuniversal computer**

Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines.

Studying their abstract properties yields many insights into computer science and complexity theory.

### P (complexity)

**Ppolynomial timePTIME**

This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time.

In computational complexity theory, P, also known as PTIME or DTIME(n O(1) ), is a fundamental complexity class.

### Big O notation

**Obig-O notationlittle-o notation**

The complexity of an algorithm is often expressed using big O notation.

In computer science, big O notation is used to classify algorithms according to how their running time or space requirements grow as the input size grows.

### Complexity

**complexdetailcomplexity theory**

Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource.

Ho and Basu present a set of complexity measures for binary classification problems.

### Boolean circuit

**Boolean circuitscircuitcircuits**

In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for digital logic circuits.

### Counting problem (complexity)

**counting problemcounting problemscounting complexity**

In computational complexity theory and computability theory, a counting problem is a type of computational problem.

### BPP (complexity)

**BPPBounded-error Probabilistic Polynomialbounded error probability in polynomial time**

Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines.

In computational complexity theory, bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded away from 1/3 for all instances.

### FP (complexity)

**FPthe class of polynomial-time functions**

The corresponding set of function problems is FP.

In computational complexity theory, the complexity class FP is the set of function problems which can be solved by a deterministic Turing machine in polynomial time; it is the function problem version of the decision problem class P.

### Circuit complexity

**non-uniformcircuitmonotone circuit**

Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.

In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of Boolean circuits that compute them.

### Promise problem

**promisedPromise-UPrestricted**

In computational complexity theory, a promise problem is a generalization of a decision problem where the input is promised to belong to a particular subset of all possible inputs.