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Dudley Metropolitan Borough Council

Dudley Metropolitan Borough Council, also known as Dudley Council, is the local authority for the Metropolitan Borough of Dudley in the West Midlands, England. The town of Dudley had been a borough since the thirteenth century, being reformed on numerous occasions. Since 1974 the council has been a metropolitan borough council. It provides the majority of local government services in the borough. The council has been a member of the West Midlands Combined Authority since 2016.

The council has been under no overall control since the 2024 election, with the leader of the council being a Conservative. It is based at Dudley Council House.

History
The town of Dudley had been a seigneurial borough from the thirteenth century, under the control of the lord of the manor. More modern forms of local government for the town began in 1791 when a body of improvement commissioners was established to pave, light and clean the streets, and supply water. The commissioners were replaced in 1853 with an elected local board.

The town was incorporated as a municipal borough in 1865, governed by a body formally called the "mayor, aldermen and burgesses of the borough of Dudley", generally known as the corporation, town council or borough council. The old local board's functions passed to the new borough council, which also replaced the ancient borough corporation.

When elected county councils were established in 1889, Dudley was considered large enough to provide its own county-level functions, and so it was made a county borough, independent from Worcestershire County Council, whilst remaining part of the geographical county of Worcestershire (despite being an exclave detached from the rest of the county). The County Borough of Dudley was enlarged on several occasions, notably in 1966 when it absorbed the majority of the abolished urban districts of Brierley Hill, Coseley and Sedgeley, alongside boundary adjustments with several other neighbours. As part of the 1966 reforms the borough was transferred to the geographical county of Staffordshire.

The modern metropolitan borough and its council were established in 1974 under the Local Government Act 1972, as one of seven boroughs in the new metropolitan county of the West Midlands. The new borough covered the combined area of the old county borough of Dudley plus the municipal boroughs of Halesowen and Stourbridge (the latter having absorbed the main part of the abolished Amblecote Urban District in the 1966 reforms). The enlarged district was named Dudley, and the borough status previously held by the county borough passed to the new district on its creation, allowing the chair of the council to take the title of mayor, continuing Dudley's series of mayors dating back to at least the sixteenth century.

From 1974 until 1986 the council was a lower-tier authority, with upper-tier functions provided by the West Midlands County Council. The county council was abolished in 1986 and its functions passed to the county's seven borough councils, including Dudley, with some services provided through joint committees.

Since 2016 the council has been a member of the West Midlands Combined Authority, which has been led by the directly elected Mayor of the West Midlands since 2017. The combined authority provides strategic leadership and co-ordination for certain functions across the county, but Dudley Council continues to be responsible for most local government functions.

The council bid for the borough to be awarded city status in 2011 and again in 2021, but was unsuccessful on both occasions.

Governance
Dudley Metropolitan Borough Council provides both county-level and district-level services, with some functions across the West Midlands provided via joint committees with the other West Midlands authorities, overseen by the combined authority and mayor. There are no civil parishes in the borough, which is an unparished area.

Political control
The council has been under no overall control since 2024.

Political control of the council since 1974 has been as follows:

Leadership
The role of mayor is largely ceremonial in Dudley. Political leadership is instead provided by the leader of the council. The leaders since 1998 have been:

Composition
Following the 2024 election, the composition of the council was:

Following the election, after a week of negotiation it was announced that Patrick Harley would remain leader of the council while Labour would be in charge of the majority of scrutiny committee chairmanships and the mayoralty. As Leader of the Council, Cllr Harley was in charge of appointing the Council's cabinet, meaning Dudley Council became a Conservative minority administration. The appointed cabinet was decided on 16 May 2024. It was subsequently criticised by the council's Labour councillors for a lack of diversity, with only one woman in its line-up and no councillors from ethnic minorities.

Premises
The council has its main offices at Dudley Council House on Priory Road, which was built in phases between 1928 and 1935 for the old county borough council. The main customer services reception is in a converted shop in the town centre at 259 Castle Street.

Elections
Since the last boundary changes in 2004 the council has comprised 72 councillors representing 24 wards, with each ward electing three councillors. Elections are held three years out of every four, with a third of the council (one councillor for each ward) being elected each time for a four-year term of office. Elections for the Mayor of the West Midlands are held in the fourth year of the cycle when there are no borough council elections. New ward boundaries are being prepared to take effect from the 2024 election, which will require all seats to be contested at that election.

Wards
The 24 wards of the Dudley Borough are each represented by 3 councillors. The council groups wards together into 5 Community Forums to enable community engagement under the banner 'Your home, your forum'.

Members of parliament
Following the 2023 review of constituencies and the July 2024 UK general election, the members of parliament for constituencies within Dudley MBC area are:

Mayor of Dudley and Civic Awards
The Mayor presides over meetings of the full Council to ensure that business is carried out properly and efficiently, with due regard to the rights of Councillors and the interest of the Community. The Mayor of the Borough is elected at the Annual Meeting of the Council (usually in May each year) from the existing elected councillors.

The Mayor also nominates charities they wish to support during their mayoral year and hosts the annual Mayors Ball and Civic Awards. The Civic Awards aim to recognise individuals and groups who make a difference in the borough. Each award is named for a local personality in that field.
 * The Frank Foley award for community spirit
 * The William Shenstone award for environment
 * The Duncan Edwards award for sport
 * The Cedric Hardwicke award for arts
 * The Thomas Attwood award for education
 * The Mike Holder award for business

Civic arms and motto
Dudley’s coat of arms was designed in 1975. It symbolises each of the authorities that came together to form the present borough. Key themes on the civic arms reflect the area’s pride in its industrial past.

The council adopted “Unity and Progress” as its motto in 1974.

title: Dudley Metropolitan Borough Council


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Very low-density lipoprotein

Very-low-density lipoprotein (VLDL), density relative to extracellular water, is a type of lipoprotein made by the liver. VLDL is one of the five major groups of lipoproteins (chylomicrons, VLDL, intermediate-density lipoprotein, low-density lipoprotein, high-density lipoprotein) that enable fats and cholesterol to move within the water-based solution of the bloodstream. VLDL is assembled in the liver from triglycerides, cholesterol, and apolipoproteins. VLDL is converted in the bloodstream to low-density lipoprotein (LDL) and intermediate-density lipoprotein (IDL). VLDL particles have a diameter of 30–80 nanometers (nm). VLDL transports endogenous products, whereas chylomicrons transport exogenous (dietary) products. In the early 2010s both the lipid composition and protein composition of this lipoprotein were characterised in great detail.

Physical properties
Very-low-density lipoprotein size is variable, with diameters ranging from approximately 35 to 70 nm. Some researchers further classify VLDL particles into VLDL1 and VLDL2 based on size (as measured by Svedberg flotation units), where VLDL1 particles are larger and contain more triglycerides, while other researchers create a tripartite system by subdividing VLDL1 into larger VLDL1 and smaller VLDL2 and relabeling VLDL2 as VLDL3.

Function
Very-low-density lipoproteins transport endogenous triglycerides, phospholipids, cholesterol, and cholesteryl esters. They function as the body's internal transport mechanism for lipids. In addition they can assist in long-range transport of hydrophobic intercellular messengers, like the morphogen Indian hedgehog (protein).

Changes during circulation
Nascent VLDL released from the liver contains apolipoprotein B100, apolipoprotein C1 (apoC1), apolipoprotein E (apoE), cholesterol, cholesteryl esters, and triglycerides. As it circulates in blood, it picks up apolipoprotein C-II (apoC-II) and additional apoE donated from high-density lipoprotein (HDL). At this point, nascent VLDL becomes a mature VLDL. Once in circulation, VLDL will come in contact with lipoprotein lipase (LPL) in the capillary beds in the body (adipose, cardiac, and skeletal muscle). LPL will remove triglycerides from VLDL for storage or energy production. VLDL now meets back up with HDL where apoC-II is transferred back to HDL (but keeps apoE). HDL also transfers cholesteryl esters to the VLDL in exchange for phospholipids and triglycerides via cholesterylester transfer protein (CETP). As more and more triglycerides are removed from the VLDL because of the action of LPL and CETP enzymes, the composition of the molecule changes, and it becomes intermediate-density lipoprotein (IDL).

Fifty percent of IDLs are recognized by receptors in the liver cells because of the apolipoprotein B-100 (apoB-100) and apoE they contain and are endocytosed. The other 50% of IDL lose apoE; when their cholesterol content becomes greater than the content of triglyceride, they become LDL, with apoB-100 as the primary apolipoprotein. The LDL is taken into a cell via the LDL receptor via endocytosis, where the contents are either stored, used for cell membrane structure, or converted into other products such as steroid hormones or bile acids.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Very low-density lipoprotein


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Yuki, Tokushima

Yuki (由岐町) was a town located in Kaifu District, Tokushima Prefecture, Japan.

As of 2003, the town had an estimated population of 3,356 and a density of 144.91 persons per km2. The total area was 23.16 km2. On March 31, 2006, Yuki, along with the town of Hiwasa (also from Kaifu District), was merged to create the town of Minami.

title: Yuki, Tokushima


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Hiwasa, Tokushima

Hiwasa (日和佐町) was a town located in Kaifu District, Tokushima Prefecture, Japan.

As of 2003, the town had an estimated population of 5,563 and a density of 47.27 persons per km2. The total area was 117.69 km2.

On March 31, 2006, Hiwasa, along with the town of Yuki (also from Kaifu District), was merged to create the town of Minami.

title: Hiwasa, Tokushima


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Mugi, Tokushima

Mugi (牟岐町) is a town located in Kaifu District, Tokushima Prefecture, Japan. , the town had an estimated population of 3,734 in 1971 households and a population density of 66 persons per km2. The total area of the town is 56.62 sqkm.

Geography
Mugi is located along the southeastern coast of Tokushima Prefecture on the island of Shikoku. The Mugi River runs through the center of the town forming a basin suitable for agriculture. The town faces the Pacific Ocean along a series of cliffs to the east, and is subject to frequent typhoons. There are a number islands offshore, including Oshima, the largest uninhabited island in Tokushima Prefecture, and Teba Island, which is inhabited. Parts of the town are within the borders of the Muroto-Anan Kaigan Quasi-National Park.

Neighbouring municipalities

 * Minami
 * Kaiyō

Climate
Mugi has a humid subtropical climate (Köppen Cfa) characterized by warm summers and cool winters with light snowfall. The average annual temperature in Mugi is 16.2 °C. The average annual rainfall is 2330 mm with September as the wettest month. The temperatures are highest on average in August, at around 26.0 °C, and lowest in January, at around 6.6 °C.

Demographics
Per Japanese census data, the population of Mugi has been declining rapidly since the 1960s, and is now roughly half of what it was a century ago.

History
As with all of Tokushima Prefecture, the area of Mugi was part of ancient Awa Province. It was noted for ocean-borne shipping, especially of timber, from Tosa Province to the Kinai region. During the Edo period, the area was part of the holdings of Tokushima Domain ruled by the Hachisuka clan from their seat at Tokushima Castle. In 1830, Australian convicts heading for Canton, China laid anchor off the coast of Mugi in a convict transport vessel, the brig Cyprus, which they had seized from their masters while being transported to Macquarie Harbour Penal Station in the former British penal colony of Van Diemen's Land, now Tasmania, an incident known as the Cyprus mutiny. The convicts were attacked and sent away by the Japanese, in line with the isolationist policy of the time. This may be the first incidence of an Australian aboriginal boomerang being gifted to Japanese people by Australian visitors. Other trading and a toast to the recently deceased wife of one of the crew was also held.

The village of Mugi was established within Kaifu District, Tokushima with the creation of the modern municipalities system on October 1, 1889. It was raised to town status on November 10, 1915. The 1946 Nankai earthquake caused great damage to the town, killing 53 people and completely destroying 374 homes.

Government
Mugi has a mayor-council form of government with a directly elected mayor and a unicameral town council of eight members. Mugi, together with the other municipalities of Kaifu District, contributes two members to the Tokushima Prefectural Assembly. In terms of national politics, the town is part of Tokushima 1st district of the lower house of the Diet of Japan.

Economy
Mugi has an economy based on agriculture and commercial fishing.

Education
Mugi has one public elementary school and one public middle school operated by the town government. The town does not have a high school.

Railway
Shikoku Railway Company – Mugi Line
 * Hegawa - Mugi

Sister city relations

 * - Puyan, Changhua, Taiwan, sister city since 1983.

title: Mugi, Tokushima


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Kainan, Tokushima

Kainan (海南町) was a town located in Kaifu District, Tokushima Prefecture, Japan.

As of 2003, the town had an estimated population of 5,967 and a density of 28.52 persons per km2. The total area was 209.22 km2.

On March 31, 2006, Kainan, along with the towns of Kaifu and Shishikui (all from Kaifu District), was merged to create the town of Kaiyō.

title: Kainan, Tokushima


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Kaifu, Tokushima

Kaifu (海部町) was a town located in Kaifu District, Tokushima, Japan.

As of 2003, the town had an estimated population of 2,423 and a density of 91.92 persons per km2. The total area was 26.36 km2.

On March 31, 2006, Kaifu, with the towns of Kainan and Shishikui (all from Kaifu District), was merged to create the town of Kaiyō.

title: Kaifu, Tokushima


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Shishikui, Tokushima

Shishikui (宍喰町) was a town located in Kaifu District, Tokushima Prefecture, Japan.

As of 2003, the town had an estimated population of 3,550 and a density of 38.59 persons per km2. The total area was 92.00 km2.

On March 31, 2006, Shishikui, along with the towns of Kaifu and Kainan (all from Kaifu District), was merged to create the town of Kaiyō.

title: Shishikui, Tokushima


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Kaifu District, Tokushima

Kaifu (海部郡) is a district located in Tokushima Prefecture, Japan.

As of June 1, 2019, the district has an estimated population of 18,863 and a population density of 35.9 PD/km2. The total area is 525.07 km2.

Towns and villages

 * Mugi
 * Kaiyō
 * Minami

Municipal timeline

 * March 31, 2006: The towns of Kainan, Kaifu and Shishikui merged to form the new town of Kaiyō.  On the same date, the towns of Hiwasa and Yuki merged to form the new town of Minami.

title: Kaifu District, Tokushima


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Matsushige, Tokushima

Matsushige (松茂町) is a town located in Itano District, Tokushima Prefecture, Japan. , the town had an estimated population of 14,770 in 6794 households and a population density of 1000 persons per km². The total area of the town is 14.24 sqkm. Matsushige is the location of Tokushima Airport, from where domestic flights depart for various other Japanese cities. The town also hosts a Japan Maritime Self-Defense Force base.

Geography
Matsushige is located between Tokushima and Naruto cities, facing the Kii Channel to the east side. It is situated on the delta of the Yoshino River.

Neighbouring municipalities
Tokushima Prefecture
 * Tokushima
 * Naruto
 * Kitajima

Climate
Matsushige has a Humid subtropical climate (Köppen Cfa) characterized by warm summers and cool winters with light snowfall. The average annual temperature in Matsushige is 16.2 °C. The average annual rainfall is 1637 mm with September as the wettest month. The temperatures are highest on average in August, at around 26.7 °C, and lowest in January, at around 6.3 °C.

Demographics
Per Japanese census data, the population of Matsushige has been growing steadily since the 1940s.

History
As with all of Tokushima Prefecture, the area of Matsushige was part of ancient Awa Province. It was noted for ocean-borne shipping, especially of timber, from Tosa Province to the Kinai region. During the Edo period, the area was part of the holdings of Tokushima Domain ruled by the Hachisuka clan from their seat at Tokushima Castle. The village of Matsushige was established within Itano District, Tokushima with the creation of the modern municipalities system on 1 October 1889. It was raised to town status on 1 August 1961.

Government
Matsushige has a mayor-council form of government with a directly elected mayor and a unicameral town council of 12 members. Matsushige, together with the other municipalities of Itano District, contributes four members to the Tokushima Prefectural Assembly. In terms of national politics, the town is part of Tokushima 2nd district of the lower house of the Diet of Japan.

The JMSDF Air Training Group Tokushima and the JGSDF 14th Aviation Unit are located in Matsushige.

Economy
Matsushige has a mixed economy based on agriculture, commercial fishing and light manufacturing.

Education
Matsushige has three public elementary schools and one public middle school operated by the town government. The town does not have a high school.

Airports

 * Tokushima Airport

Railway
Matsushige has no passenger railway service. The closest stations are Kyokaimae Station and Kompiramae Station on the JR Shikoku Naruto Line; however, Tokushima Station is often used for convenience.

Highways

 * JP Expressway E11.svg Tokushima Expressway

Sister city relations

 * 🇺🇸 - Mount Vernon, Washington, United States

title: Matsushige, Tokushima


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Kitajima, Tokushima

Kitajima (北島町) is a town located in Itano District, Tokushima Prefecture, Japan. , the town had an estimated population of 23,680 in 10340 households and a population density of 2700 persons per km2. The total area of the town is 8.74 sqkm. It has the smallest area among local governments in Tokushima prefecture and is also the most densely populated town or village in Shikoku.

Geography
Kitajima is located in the flatlands of northeastern Tokushima Prefecture on the island of Shikoku. It is located on the delta at the mouth of the Yoshino River. The Kyuyoshino River runs on the north side of the town, and the Imagire River runs on the south side of the town, and both rivers form a sandbar shaped like a gourd.

Neighbouring municipalities
Tokushima Prefecture
 * Tokushima
 * Naruto
 * Aizumi
 * Matsushige

Climate
Kitajima has a Humid subtropical climate (Köppen Cfa) characterized by warm summers and cool winters with light snowfall. The average annual temperature in Kitajima is 16.2 °C. The average annual rainfall is 1637 mm with September as the wettest month. The temperatures are highest on average in August, at around 26.7 °C, and lowest in January, at around 6.3 °C.

Demographics
Per Japanese census data, the population of Kitajima has been growing steadily for the past century.

History
As with all of Tokushima Prefecture, the area of Kitajima was part of ancient Awa Province. The village of Kitajima was established within Itano District, Tokushima with the creation of the modern municipalities system on October 1, 1889. It was raised to town status on February 11, 1940.

Government
Kitajima has a mayor-council form of government with a directly elected mayor and a unicameral town council of 13 members. Kitajima, together with the other municipalities of Itano District, contributes four members to the Tokushima Prefectural Assembly. In terms of national politics, the town is part of Tokushima 2nd district of the lower house of the Diet of Japan.

Economy
Kitajima once prospered as an industrial company town with large factories by Nisshinbo, Toagosei, and Toho Rayon, but in recent years it been transformed into a commercial center and a commuter town for neighboring Tokushima city.

Education
Kitajima has three public elementary schools and two public middle schools operated by the town government and one public high school operated by the Tokushima Prefectural Department of Education. The Tokushima Prefectural Fire Academy is located in Kitajima.

Railway
The Shikoku Railway Company Kōtoku Line passes through the town, but there is no station. The nearest passenger stations are Shōzui in Aizumi or Yoshinari in Tokushima.

Highways

 * JP Expressway E11.svg Tokushima Expressway

Notable people from Kitajima

 * Seiji Honda, footballer

title: Kitajima, Tokushima


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Complexity class

In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly analyzed resources are time and memory.

In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. counting problems and function problems) and using other models of computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and quantum computers).

The study of the relationships between complexity classes is a major area of research in theoretical computer science. There are often general hierarchies of complexity classes; for example, it is known that a number of fundamental time and space complexity classes relate to each other in the following way: L⊆NL⊆P⊆NP⊆PSPACE⊆EXPTIME⊆NEXPTIME⊆EXPSPACE (where ⊆ denotes the subset relation). However, many relationships are not yet known; for example, one of the most famous open problems in computer science concerns whether P equals NP. The relationships between classes often answer questions about the fundamental nature of computation. The P versus NP problem, for instance, is directly related to questions of whether nondeterminism adds any computational power to computers and whether problems having solutions that can be quickly checked for correctness can also be quickly solved.

Background
Complexity classes are sets of related computational problems. They are defined in terms of the computational difficulty of solving the problems contained within them with respect to particular computational resources like time or memory. More formally, the definition of a complexity class consists of three things: a type of computational problem, a model of computation, and a bounded computational resource. In particular, most complexity classes consist of decision problems that can be solved by a Turing machine with bounded time or space resources. For example, the complexity class P is defined as the set of decision problems that can be solved by a deterministic Turing machine in polynomial time.

Computational problems
Intuitively, a computational problem is just a question that can be solved by an algorithm. For example, "is the natural number $$n$$ prime?" is a computational problem. A computational problem is mathematically represented as the set of answers to the problem. In the primality example, the problem (call it $$\texttt{PRIME}$$) is represented by the set of all natural numbers that are prime: $$\texttt{PRIME} = \{ n \in \mathbb{N} | n \text{ is prime}\}$$. In the theory of computation, these answers are represented as strings; for example, in the primality example the natural numbers could be represented as strings of bits that represent binary numbers. For this reason, computational problems are often synonymously referred to as languages, since strings of bits represent formal languages (a concept borrowed from linguistics); for example, saying that the $$\texttt{PRIME}$$ problem is in the complexity class NP is equivalent to saying that the language $$\texttt{PRIME}$$ is in NP.

Decision problems
The most commonly analyzed problems in theoretical computer science are decision problems—the kinds of problems that can be posed as yes–no questions. The primality example above, for instance, is an example of a decision problem as it can be represented by the yes–no question "is the natural number $$n$$ prime". In terms of the theory of computation, a decision problem is represented as the set of input strings that a computer running a correct algorithm would answer "yes" to. In the primality example, $$\texttt{PRIME}$$ is the set of strings representing natural numbers that, when input into a computer running an algorithm that correctly tests for primality, the algorithm answers "yes, this number is prime". This "yes-no" format is often equivalently stated as "accept-reject"; that is, an algorithm "accepts" an input string if the answer to the decision problem is "yes" and "rejects" if the answer is "no".

While some problems cannot easily be expressed as decision problems, they nonetheless encompass a broad range of computational problems. Other types of problems that certain complexity classes are defined in terms of include:

 * Function problems (e.g. FP)
 * Counting problems (e.g. #P)
 * Optimization problems
 * Promise problems (see section "Other types of problems")

Computational models
To make concrete the notion of a "computer", in theoretical computer science problems are analyzed in the context of a computational model. Computational models make exact the notions of computational resources like "time" and "memory". In computational complexity theory, complexity classes deal with the inherent resource requirements of problems and not the resource requirements that depend upon how a physical computer is constructed. For example, in the real world different computers may require different amounts of time and memory to solve the same problem because of the way that they have been engineered. By providing an abstract mathematical representations of computers, computational models abstract away superfluous complexities of the real world (like differences in processor speed) that obstruct an understanding of fundamental principles.

The most commonly used computational model is the Turing machine. While other models exist and many complexity classes are defined in terms of them (see section "Other models of computation"), the Turing machine is used to define most basic complexity classes. With the Turing machine, instead of using standard units of time like the second (which make it impossible to disentangle running time from the speed of physical hardware) and standard units of memory like bytes, the notion of time is abstracted as the number of elementary steps that a Turing machine takes to solve a problem and the notion of memory is abstracted as the number of cells that are used on the machine's tape. These are explained in greater detail below.

It is also possible to use the Blum axioms to define complexity classes without referring to a concrete computational model, but this approach is less frequently used in complexity theory.

Deterministic Turing machines
A Turing machine is a mathematical model of a general computing machine. It is the most commonly used model in complexity theory, owing in large part to the fact that it is believed to be as powerful as any other model of computation and is easy to analyze mathematically. Importantly, it is believed that if there exists an algorithm that solves a particular problem then there also exists a Turing machine that solves that same problem (this is known as the Church–Turing thesis); this means that it is believed that every algorithm can be represented as a Turing machine.

Mechanically, a Turing machine (TM) manipulates symbols (generally restricted to the bits 0 and 1 to provide an intuitive connection to real-life computers) contained on an infinitely long strip of tape. The TM can read and write, one at a time, using a tape head. Operation is fully determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1; if the symbol seen is 1, change into state 17; in state 17, if the symbol seen is 0, write a 1 and change to state 6". The Turing machine starts with only the input string on its tape and blanks everywhere else. The TM accepts the input if it enters a designated accept state and rejects the input if it enters a reject state. The deterministic Turing machine (DTM) is the most basic type of Turing machine. It uses a fixed set of rules to determine its future actions (which is why it is called "deterministic").

A computational problem can then be defined in terms of a Turing machine as the set of input strings that a particular Turing machine accepts. For example, the primality problem $$\texttt{PRIME}$$ from above is the set of strings (representing natural numbers) that a Turing machine running an algorithm that correctly tests for primality accepts. A Turing machine is said to recognize a language (recall that "problem" and "language" are largely synonymous in computability and complexity theory) if it accepts all inputs that are in the language and is said to decide a language if it additionally rejects all inputs that are not in the language (certain inputs may cause a Turing machine to run forever, so decidability places the additional constraint over recognizability that the Turing machine must halt on all inputs). A Turing machine that "solves" a problem is generally meant to mean one that decides the language.

Turing machines enable intuitive notions of "time" and "space". The time complexity of a TM on a particular input is the number of elementary steps that the Turing machine takes to reach either an accept or reject state. The space complexity is the number of cells on its tape that it uses to reach either an accept or reject state.

Nondeterministic Turing machines

The deterministic Turing machine (DTM) is a variant of the nondeterministic Turing machine (NTM). Intuitively, an NTM is just a regular Turing machine that has the added capability of being able to explore multiple possible future actions from a given state, and "choosing" a branch that accepts (if any accept). That is, while a DTM must follow only one branch of computation, an NTM can be imagined as a computation tree, branching into many possible computational pathways at each step (see image). If at least one branch of the tree halts with an "accept" condition, then the NTM accepts the input. In this way, an NTM can be thought of as simultaneously exploring all computational possibilities in parallel and selecting an accepting branch. NTMs are not meant to be physically realizable models, they are simply theoretically interesting abstract machines that give rise to a number of interesting complexity classes (which often do have physically realizable equivalent definitions).

The time complexity of an NTM is the maximum number of steps that the NTM uses on any branch of its computation. Similarly, the space complexity of an NTM is the maximum number of cells that the NTM uses on any branch of its computation.

DTMs can be viewed as a special case of NTMs that do not make use of the power of nondeterminism. Hence, every computation that can be carried out by a DTM can also be carried out by an equivalent NTM. It is also possible to simulate any NTM using a DTM (the DTM will simply compute every possible computational branch one-by-one). Hence, the two are equivalent in terms of computability. However, simulating an NTM with a DTM often requires greater time and/or memory resources; as will be seen, how significant this slowdown is for certain classes of computational problems is an important question in computational complexity theory.

Resource bounds
Complexity classes group computational problems by their resource requirements. To do this, computational problems are differentiated by upper bounds on the maximum amount of resources that the most efficient algorithm takes to solve them. More specifically, complexity classes are concerned with the rate of growth in the resources required to solve particular computational problems as the input size increases. For example, the amount of time it takes to solve problems in the complexity class P grows at a polynomial rate as the input size increases, which is comparatively slow compared to problems in the exponential complexity class EXPTIME (or more accurately, for problems in EXPTIME that are outside of P, since $$\mathsf{P}\subseteq\mathsf{EXPTIME}$$).

Note that the study of complexity classes is intended primarily to understand the inherent complexity required to solve computational problems. Complexity theorists are thus generally concerned with finding the smallest complexity class that a problem falls into and are therefore concerned with identifying which class a computational problem falls into using the most efficient algorithm. There may be an algorithm, for instance, that solves a particular problem in exponential time, but if the most efficient algorithm for solving this problem runs in polynomial time then the inherent time complexity of that problem is better described as polynomial.

Time bounds
The time complexity of an algorithm with respect to the Turing machine model is the number of steps it takes for a Turing machine to run an algorithm on a given input size. Formally, the time complexity for an algorithm implemented with a Turing machine $$M$$ is defined as the function $$t_M: \mathbb{N} \to \mathbb{N}$$, where $$t_M(n)$$ is the maximum number of steps that $$M$$ takes on any input of length $$n$$.

In computational complexity theory, theoretical computer scientists are concerned less with particular runtime values and more with the general class of functions that the time complexity function falls into. For instance, is the time complexity function a polynomial? A logarithmic function? An exponential function? Or another kind of function?

Space bounds
The space complexity of an algorithm with respect to the Turing machine model is the number of cells on the Turing machine's tape that are required to run an algorithm on a given input size. Formally, the space complexity of an algorithm implemented with a Turing machine $$M$$ is defined as the function $$s_M: \mathbb{N} \to \mathbb{N}$$, where $$s_M(n)$$ is the maximum number of cells that $$M$$ uses on any input of length $$n$$.

Basic definitions
Complexity classes are often defined using granular sets of complexity classes called DTIME and NTIME (for time complexity) and DSPACE and NSPACE (for space complexity). Using big O notation, they are defined as follows:
 * The time complexity class $$\mathsf{DTIME}(t(n))$$ is the set of all problems that are decided by an $$O(t(n))$$ time deterministic Turing machine.
 * The time complexity class $$\mathsf{NTIME}(t(n))$$ is the set of all problems that are decided by an $$O(t(n))$$ time nondeterministic Turing machine.
 * The space complexity class $$\mathsf{DSPACE}(s(n))$$ is the set of all problems that are decided by an $$O(s(n))$$ space deterministic Turing machine.
 * The space complexity class $$\mathsf{NSPACE}(s(n))$$ is the set of all problems that are decided by an $$O(s(n))$$ space nondeterministic Turing machine.

P and NP
P is the class of problems that are solvable by a deterministic Turing machine in polynomial time and NP is the class of problems that are solvable by a nondeterministic Turing machine in polynomial time. Or more formally,
 * $$\mathsf{P} = \bigcup_{k\in\mathbb{N}} \mathsf{DTIME}(n^k) $$
 * $$\mathsf{NP} = \bigcup_{k\in\mathbb{N}} \mathsf{NTIME}(n^k) $$

P is often said to be the class of problems that can be solved "quickly" or "efficiently" by a deterministic computer, since the time complexity of solving a problem in P increases relatively slowly with the input size.

An important characteristic of the class NP is that it can be equivalently defined as the class of problems whose solutions are verifiable by a deterministic Turing machine in polynomial time. That is, a language is in NP if there exists a deterministic polynomial time Turing machine, referred to as the verifier, that takes as input a string $$w$$ and a polynomial-size certificate string $$c$$, and accepts $$w$$ if $$w$$ is in the language and rejects $$w$$ if $$w$$ is not in the language. Intuitively, the certificate acts as a proof that the input $$w$$ is in the language. Formally:
 * NP is the class of languages $$L$$ for which there exists a polynomial-time deterministic Turing machine $$M$$ and a polynomial $$p$$ such that for all $$w \in \{0,1\}^*$$, $$w$$ is in $$L$$ if and only if there exists some $$c \in \{0,1\}^{p(|w|)}$$ such that $$M(w,c)$$ accepts.

This equivalence between the nondeterministic definition and the verifier definition highlights a fundamental connection between nondeterminism and solution verifiability. Furthermore, it also provides a useful method for proving that a language is in NP—simply identify a suitable certificate and show that it can be verified in polynomial time.

The P versus NP problem
While there might seem to be an obvious difference between the class of problems that are efficiently solvable and the class of problems whose solutions are merely efficiently checkable, P and NP are actually at the center of one of the most famous unsolved problems in computer science: the P versus NP problem. While it is known that $$\mathsf{P} \subseteq \mathsf{NP}$$ (intuitively, deterministic Turing machines are just a subclass of nondeterministic Turing machines that don't make use of their nondeterminism; or under the verifier definition, P is the class of problems whose polynomial time verifiers need only receive the empty string as their certificate), it is not known whether NP is strictly larger than P. If P=NP, then it follows that nondeterminism provides no additional computational power over determinism with regards to the ability to quickly find a solution to a problem; that is, being able to explore all possible branches of computation provides at most a polynomial speedup over being able to explore only a single branch. Furthermore, it would follow that if there exists a proof for a problem instance and that proof can be quickly be checked for correctness (that is, if the problem is in NP), then there also exists an algorithm that can quickly construct that proof (that is, the problem is in P). However, the overwhelming majority of computer scientists believe that $$\mathsf{P}\neq\mathsf{NP}$$, and most cryptographic schemes employed today rely on the assumption that $$\mathsf{P}\neq\mathsf{NP}$$.

EXPTIME and NEXPTIME
EXPTIME (sometimes shortened to EXP) is the class of decision problems solvable by a deterministic Turing machine in exponential time and NEXPTIME (sometimes shortened to NEXP) is the class of decision problems solvable by a nondeterministic Turing machine in exponential time. Or more formally,
 * $$\mathsf{EXPTIME} = \bigcup_{k\in\mathbb{N}} \mathsf{DTIME}(2^{n^k}) $$
 * $$\mathsf{NEXPTIME} = \bigcup_{k\in\mathbb{N}} \mathsf{NTIME}(2^{n^k}) $$

EXPTIME is a strict superset of P and NEXPTIME is a strict superset of NP. It is further the case that EXPTIME$$\subseteq$$NEXPTIME. It is not known whether this is proper, but if P=NP then EXPTIME must equal NEXPTIME.

L and NL
While it is possible to define logarithmic time complexity classes, these are extremely narrow classes as sublinear times do not even enable a Turing machine to read the entire input (because $$\log n < n $$). However, there are a meaningful number of problems that can be solved in logarithmic space. The definitions of these classes require a two-tape Turing machine so that it is possible for the machine to store the entire input (it can be shown that in terms of computability the two-tape Turing machine is equivalent to the single-tape Turing machine). In the two-tape Turing machine model, one tape is the input tape, which is read-only. The other is the work tape, which allows both reading and writing and is the tape on which the Turing machine performs computations. The space complexity of the Turing machine is measured as the number of cells that are used on the work tape.

L (sometimes lengthened to LOGSPACE) is then defined as the class of problems solvable in logarithmic space on a deterministic Turing machine and NL (sometimes lengthened to NLOGSPACE) is the class of problems solvable in logarithmic space on a nondeterministic Turing machine. Or more formally,
 * $$\mathsf{L} = \mathsf{DSPACE}(\log n)$$
 * $$\mathsf{NL} = \mathsf{NSPACE}(\log n)$$

It is known that $$\mathsf{L}\subseteq\mathsf{NL}\subseteq\mathsf{P}$$. However, it is not known whether any of these relationships is proper.

PSPACE and NPSPACE
The complexity classes PSPACE and NPSPACE are the space analogues to  P and  NP. That is, PSPACE is the class of problems solvable in polynomial space by a deterministic Turing machine and NPSPACE is the class of problems solvable in polynomial space by a nondeterministic Turing machine. More formally,
 * $$\mathsf{PSPACE} = \bigcup_{k\in\mathbb{N}} \mathsf{DSPACE}(n^k)$$
 * $$\mathsf{NPSPACE} = \bigcup_{k\in\mathbb{N}} \mathsf{NSPACE}(n^k)$$

While it is not known whether P=NP, Savitch's theorem famously showed that PSPACE=NPSPACE. It is also known that $$\mathsf{P} \subseteq \mathsf{PSPACE}$$, which follows intuitively from the fact that, since writing to a cell on a Turing machine's tape is defined as taking one unit of time, a Turing machine operating in polynomial time can only write to polynomially many cells. It is suspected that P is strictly smaller than PSPACE, but this has not been proven.

EXPSPACE and NEXPSPACE
The complexity classes EXPSPACE and NEXPSPACE are the space analogues to EXPTIME and NEXPTIME. That is, EXPSPACE is the class of problems solvable in exponential space by a deterministic Turing machine and NEXPSPACE is the class of problems solvable in exponential space by a nondeterministic Turing machine. Or more formally,
 * $$\mathsf{EXPSPACE} = \bigcup_{k\in\mathbb{N}} \mathsf{DSPACE}(2^{n^k})$$
 * $$\mathsf{NEXPSPACE} = \bigcup_{k\in\mathbb{N}} \mathsf{NSPACE}(2^{n^k})$$

Savitch's theorem showed that EXPSPACE=NEXPSPACE. This class is extremely broad: it is known to be a strict superset of PSPACE, NP, and P, and is believed to be a strict superset of EXPTIME.

Closure
Complexity classes have a variety of closure properties. For example, decision classes may be closed under negation, disjunction, conjunction, or even under all Boolean operations. Moreover, they might also be closed under a variety of quantification schemes. P, for instance, is closed under all Boolean operations, and under quantification over polynomially sized domains. Closure properties can be helpful in separating classes—one possible route to separating two complexity classes is to find some closure property possessed by one class but not by the other.

Each class X that is not closed under negation has a complement class co-X, which consists of the complements of the languages contained in X (i.e. $$\textsf{co-X} = \{L| \overline{L} \in \mathsf{X} \}$$). co-NP, for instance, is one important complement complexity class, and sits at the center of the unsolved problem over whether co-NP=NP.

Closure properties are one of the key reasons many complexity classes are defined in the way that they are. Take, for example, a problem that can be solved in $$O(n)$$ time (that is, in linear time) and one that can be solved in, at best, $$O(n^{1000})$$ time. Both of these problems are in P, yet the runtime of the second grows considerably faster than the runtime of the first as the input size increases. One might ask whether it would be better to define the class of "efficiently solvable" problems using some smaller polynomial bound, like $$O(n^3)$$, rather than all polynomials, which allows for such large discrepancies. It turns out, however, that the set of all polynomials is the smallest class of functions containing the linear functions that is also closed under addition, multiplication, and composition (for instance, $$O(n^3) \circ O(n^2) = O(n^6)$$, which is a polynomial but $$O(n^6)>O(n^3)$$). Since we would like composing one efficient algorithm with another efficient algorithm to still be considered efficient, the polynomials are the smallest class that ensures composition of "efficient algorithms". (Note that the definition of P is also useful because, empirically, almost all problems in P that are practically useful do in fact have low order polynomial runtimes, and almost all problems outside of P that are practically useful do not have any known algorithms with small exponential runtimes, i.e. with $$O(c^n)$$ runtimes where $c$ is close to 1.)

Reductions
Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem, i.e. a reduction takes inputs from one problem and transforms them into inputs of another problem. For instance, you can reduce ordinary base-10 addition $$x+y$$ to base-2 addition by transforming $$x$$ and $$y$$ to their base-2 notation (e.g. 5+7 becomes 101+111). Formally, a problem $$X$$ reduces to a problem $$Y$$ if there exists a function $$f$$ such that for every $$x \in \Sigma^* $$, $$x \in X$$ if and only if $$f(x) \in Y$$.

Generally, reductions are used to capture the notion of a problem being at least as difficult as another problem. Thus we are generally interested in using a polynomial-time reduction, since any problem $$X$$ that can be efficiently reduced to another problem $$Y$$ is no more difficult than $$Y$$. Formally, a problem $$X$$ is polynomial-time reducible to a problem $$Y$$ if there exists a polynomial-time computable function $$p$$ such that for all $$x \in \Sigma^*$$, $$x \in X$$ if and only if $$ p(x) \in Y$$.

Note that reductions can be defined in many different ways. Common reductions are Cook reductions, Karp reductions and Levin reductions, and can vary based on resource bounds, such as polynomial-time reductions and log-space reductions.

Hardness
Reductions motivate the concept of a problem being hard for a complexity class. A problem $$X$$ is hard for a class of problems C if every problem in C can be polynomial-time reduced to $$X$$. Thus no problem in C is harder than $$X$$, since an algorithm for $$X$$ allows us to solve any problem in C with at most polynomial slowdown. Of particular importance, the set of problems that are hard for NP is called the set of NP-hard problems.

Completeness
If a problem $$X$$ is hard for C and is also in C, then $$X$$ is said to be complete for C. This means that $$X$$ is the hardest problem in C (since there could be many problems that are equally hard, more precisely $$X$$ is as hard as the hardest problems in C).

Of particular importance is the class of NP-complete problems—the most difficult problems in NP. Because all problems in NP can be polynomial-time reduced to NP-complete problems, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.

Savitch's theorem
Savitch's theorem establishes the relationship between deterministic and nondetermistic space resources. It shows that if a nondeterministic Turing machine can solve a problem using $$f(n)$$ space, then a deterministic Turing machine can solve the same problem in $$f(n)^2$$ space, i.e. in the square of the space. Formally, Savitch's theorem states that for any $$f(n) > n $$,

 * $$\mathsf{NSPACE}\left(f\left(n\right)\right) \subseteq \mathsf{DSPACE}\left(f\left(n\right)^2\right).$$

Important corollaries of Savitch's theorem are that PSPACE = NPSPACE (since the square of a polynomial is still a polynomial) and EXPSPACE = NEXPSPACE (since the square of an exponential is still an exponential).

These relationships answer fundamental questions about the power of nondeterminism compared to determinism. Specifically, Savitch's theorem shows that any problem that a nondeterministic Turing machine can solve in polynomial space, a deterministic Turing machine can also solve in polynomial space. Similarly, any problem that a nondeterministic Turing machine can solve in exponential space, a deterministic Turing machine can also solve in exponential space.

Hierarchy theorems
By definition of DTIME, it follows that $$\mathsf{DTIME}(n^{k_1})$$ is contained in $$\mathsf{DTIME}(n^{k_2})$$ if $$k_1 \leq k_2 $$, since $$O(n^{k_1}) \subseteq O(n^{k_2})$$ if $$k_1 \leq k_2$$. However, this definition gives no indication of whether this inclusion is strict. For time and space requirements, the conditions under which the inclusion is strict are given by the time and space hierarchy theorems, respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. The hierarchy theorems enable one to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

The time hierarchy theorem states that
 * $$\mathsf{DTIME}\big(f(n) \big) \subsetneq \mathsf{DTIME} \big(f(n) \sdot \log^{2}(f(n)) \big)$$.

The space hierarchy theorem states that
 * $$\mathsf{DSPACE}\big(f(n)\big) \subsetneq \mathsf{DSPACE} \big(f(n) \sdot \log(f(n)) \big)$$.

The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem establishes that P is strictly contained in EXPTIME, and the space hierarchy theorem establishes that L is strictly contained in PSPACE.

Other models of computation
While deterministic and non-deterministic Turing machines are the most commonly used models of computation, many complexity classes are defined in terms of other computational models. In particular,
 * A number of classes are defined using probabilistic Turing machines, including the classes BPP, PP, RP, and ZPP
 * A number of classes are defined using interactive proof systems, including the classes IP, MA, and AM
 * A number of classes are defined using Boolean circuits, including the classes P/poly and its subclasses NC and AC
 * A number of classes are defined using quantum Turing machines, including the classes BQP and QMA

These are explained in greater detail below.

Randomized computation
A number of important complexity classes are defined using the probabilistic Turing machine, a variant of the Turing machine that can toss random coins. These classes help to better describe the complexity of randomized algorithms.

A probabilistic Turing machine is similar to a deterministic Turing machine, except rather than following a single transition function (a set of rules for how to proceed at each step of the computation) it probabilistically selects between multiple transition functions at each step. The standard definition of a probabilistic Turing machine specifies two transition functions, so that the selection of transition function at each step resembles a coin flip. The randomness introduced at each step of the computation introduces the potential for error; that is, strings that the Turing machine is meant to accept may on some occasions be rejected and strings that the Turing machine is meant to reject may on some occasions be accepted. As a result, the complexity classes based on the probabilistic Turing machine are defined in large part around the amount of error that is allowed. Formally, they are defined using an error probability $$\epsilon$$. A probabilistic Turing machine $$M$$ is said to recognize a language $$L$$ with error probability $$\epsilon$$ if:
 * 1) a string $$w$$ in $$L$$ implies that $$\text{Pr}[M \text{ accepts } w] \geq 1 - \epsilon$$
 * 2) a string $$w$$ not in $$L$$ implies that $$\text{Pr}[M \text{ rejects } w] \geq 1 - \epsilon$$

Important complexity classes

The fundamental randomized time complexity classes are ZPP, RP, co-RP, BPP, and PP.

The strictest class is ZPP (zero-error probabilistic polynomial time), the class of problems solvable in polynomial time by a probabilistic Turing machine with error probability 0. Intuitively, this is the strictest class of probabilistic problems because it demands no error whatsoever.

A slightly looser class is RP (randomized polynomial time), which maintains no error for strings not in the language but allows bounded error for strings in the language. More formally, a language is in RP if there is a probabilistic polynomial-time Turing machine $$M$$ such that if a string is not in the language then $$M$$ always rejects and if a string is in the language then $$M$$ accepts with a probability at least 1/2. The class co-RP is similarly defined except the roles are flipped: error is not allowed for strings in the language but is allowed for strings not in the language. Taken together, the classes RP and co-RP encompass all of the problems that can be solved by probabilistic Turing machines with one-sided error.

Loosening the error requirements further to allow for two-sided error yields the class BPP (bounded-error probabilistic polynomial time), the class of problems solvable in polynomial time by a probabilistic Turing machine with error probability less than 1/3 (for both strings in the language and not in the language). BPP is the most practically relevant of the probabilistic complexity classes—problems in BPP have efficient randomized algorithms that can be run quickly on real computers. BPP is also at the center of the important unsolved problem in computer science over whether P=BPP, which if true would mean that randomness does not increase the computational power of computers, i.e. any probabilistic Turing machine could be simulated by a deterministic Turing machine with at most polynomial slowdown.

The broadest class of efficiently-solvable probabilistic problems is PP (probabilistic polynomial time), the set of languages solvable by a probabilistic Turing machine in polynomial time with an error probability of less than 1/2 for all strings.

ZPP, RP and co-RP are all subsets of BPP, which in turn is a subset of PP. The reason for this is intuitive: the classes allowing zero error and only one-sided error are all contained within the class that allows two-sided error, and PP simply relaxes the error probability of BPP. ZPP relates to RP and co-RP in the following way: $$\textsf{ZPP}=\textsf{RP}\cap\textsf{co-RP}$$. That is, ZPP consists exactly of those problems that are in both RP and co-RP. Intuitively, this follows from the fact that RP and co-RP allow only one-sided error: co-RP does not allow error for strings in the language and RP does not allow error for strings not in the language. Hence, if a problem is in both RP and co-RP, then there must be no error for strings both in and not in the language (i.e. no error whatsoever), which is exactly the definition of ZPP.

Important randomized space complexity classes include BPL, RL, and RLP.

Interactive proof systems
A number of complexity classes are defined using interactive proof systems. Interactive proofs generalize the proofs definition of the complexity class NP and yield insights into cryptography, approximation algorithms, and formal verification.

Interactive proof systems are abstract machines that model computation as the exchange of messages between two parties: a prover $$P$$ and a verifier $$V$$. The parties interact by exchanging messages, and an input string is accepted by the system if the verifier decides to accept the input on the basis of the messages it has received from the prover. The prover $$P$$ has unlimited computational power while the verifier has bounded computational power (the standard definition of interactive proof systems defines the verifier to be polynomially-time bounded). The prover, however, is untrustworthy (this prevents all languages from being trivially recognized by the proof system by having the computationally unbounded prover solve for whether a string is in a language and then sending a trustworthy "YES" or "NO" to the verifier), so the verifier must conduct an "interrogation" of the prover by "asking it" successive rounds of questions, accepting only if it develops a high degree of confidence that the string is in the language.

Important complexity classes
The class NP is a simple proof system in which the verifier is restricted to being a deterministic polynomial-time Turing machine and the procedure is restricted to one round (that is, the prover sends only a single, full proof—typically referred to as the certificate—to the verifier). Put another way, in the definition of the class NP (the set of decision problems for which the problem instances, when the answer is "YES", have proofs verifiable in polynomial time by a deterministic Turing machine) is a proof system in which the proof is constructed by an unmentioned prover and the deterministic Turing machine is the verifier. For this reason, NP can also be called dIP (deterministic interactive proof), though it is rarely referred to as such.

It turns out that NP captures the full power of interactive proof systems with deterministic (polynomial-time) verifiers because it can be shown that for any proof system with a deterministic verifier it is never necessary to need more than a single round of messaging between the prover and the verifier. Interactive proof systems that provide greater computational power over standard complexity classes thus require probabilistic verifiers, which means that the verifier's questions to the prover are computed using probabilistic algorithms. As noted in the section above on randomized computation, probabilistic algorithms introduce error into the system, so complexity classes based on probabilistic proof systems are defined in terms of an error probability $$\epsilon$$.

The most general complexity class arising out of this characterization is the class IP (interactive polynomial time), which is the class of all problems solvable by an interactive proof system $$(P,V)$$, where $$V$$ is probabilistic polynomial-time and the proof system satisfies two properties: for a language $$L \in \mathsf{IP}$$
 * 1) (Completeness) a string $$w$$ in $$L$$ implies $$\Pr[V \text{ accepts }w \text{ after interacting with } P] \ge \tfrac{2}{3}$$
 * 2) (Soundness) a string $$w$$ not in $$L$$ implies $$\Pr[V \text{ accepts }w \text{ after interacting with } P] \le \tfrac{1}{3}$$

An important feature of IP is that it equals PSPACE. In other words, any problem that can be solved by a polynomial-time interactive proof system can also be solved by a deterministic Turing machine with polynomial space resources, and vice versa.

A modification of the protocol for IP produces another important complexity class: AM (Arthur–Merlin protocol). In the definition of interactive proof systems used by IP, the prover was not able to see the coins utilized by the verifier in its probabilistic computation—it was only able to see the messages that the verifier produced with these coins. For this reason, the coins are called private random coins. The interactive proof system can be constrained so that the coins used by the verifier are public random coins; that is, the prover is able to see the coins. Formally, AM is defined as the class of languages with an interactive proof in which the verifier sends a random string to the prover, the prover responds with a message, and the verifier either accepts or rejects by applying a deterministic polynomial-time function to the message from the prover. AM can be generalized to AM[k], where k is the number of messages exchanged (so in the generalized form the standard AM defined above is AM[2]). However, it is the case that for all $$k \geq 2$$, AM[k]=AM[2]. It is also the case that $$\mathsf{AM}[k]\subseteq\mathsf{IP}[k]$$.

Other complexity classes defined using interactive proof systems include MIP (multiprover interactive polynomial time) and QIP (quantum interactive polynomial time).

Boolean circuits
An alternative model of computation to the Turing machine is the Boolean circuit, a simplified model of the digital circuits used in modern computers. Not only does this model provide an intuitive connection between computation in theory and computation in practice, but it is also a natural model for non-uniform computation (computation in which different input sizes within the same problem use different algorithms).

Formally, a Boolean circuit $$C$$ is a directed acyclic graph in which edges represent wires (which carry the bit values 0 and 1), the input bits are represented by source vertices (vertices with no incoming edges), and all non-source vertices represent logic gates (generally the AND, OR, and NOT gates). One logic gate is designated the output gate, and represents the end of the computation. The input/output behavior of a circuit $$C$$ with $$n$$ input variables is represented by the Boolean function $$f_C:\{0,1\}^n \to \{0,1\}$$; for example, on input bits $$x_1,x_2,...,x_n$$, the output bit $$b$$ of the circuit is represented mathematically as $$b = f_C(x_1,x_2,...,x_n)$$. The circuit $$C$$ is said to compute the Boolean function $$f_C$$.

Any particular circuit has a fixed number of input vertices, so it can only act on inputs of that size. Languages (the formal representations of decision problems), however, contain strings of differing lengths, so languages cannot be fully captured by a single circuit (this contrasts with the Turing machine model, in which a language is fully described by a single Turing machine that can act on any input size). A language is thus represented by a circuit family. A circuit family is an infinite list of circuits $$(C_0,C_1,C_2,...)$$, where $$C_n$$ is a circuit with $$n$$ input variables. A circuit family is said to decide a language $$L$$ if, for every string $$w$$, $$w$$ is in the language $$L$$ if and only if $$C_n(w)=1$$, where $$n$$ is the length of $$w$$. In other words, a string $$w$$ of size $$n$$ is in the language represented by the circuit family $$(C_0,C_1,C_2,...)$$ if the circuit $$C_n$$ (the circuit with the same number of input vertices as the number of bits in $$w$$) evaluates to 1 when $$w$$ is its input.

While complexity classes defined using Turing machines are described in terms of time complexity, circuit complexity classes are defined in terms of circuit size — the number of vertices in the circuit. The size complexity of a circuit family $$(C_0,C_1,C_2,...)$$ is the function $$f:\mathbb{N} \to \mathbb{N}$$, where $$f(n)$$ is the circuit size of $$C_n$$. The familiar function classes follow naturally from this; for example, a polynomial-size circuit family is one such that the function $$f$$ is a polynomial.

Important complexity classes
The complexity class P/poly is the set of languages that are decidable by polynomial-size circuit families. It turns out that there is a natural connection between circuit complexity and time complexity. Intuitively, a language with small time complexity (that is, requires relatively few sequential operations on a Turing machine), also has a small circuit complexity (that is, requires relatively few Boolean operations). Formally, it can be shown that if a language is in $$\mathsf{DTIME}(t(n))$$, where $$t$$ is a function $$t:\mathbb{N} \to \mathbb{N}$$, then it has circuit complexity $$O(t^2(n))$$. It follows directly from this fact that $\mathsf{\color{Blue}P}\subset\textsf{P/poly}$. In other words, any problem that can be solved in polynomial time by a deterministic Turing machine can also be solved by a polynomial-size circuit family. It is further the case that the inclusion is proper, i.e. $$\textsf{P}\subsetneq \textsf{P/poly}$$ (for example, there are some undecidable problems that are in P/poly).

P/poly has a number of properties that make it highly useful in the study of the relationships between complexity classes. In particular, it is helpful in investigating problems related to P versus NP. For example, if there is any language in NP that is not in P/poly, then $$\mathsf{P}\neq\mathsf{NP}$$. P/poly is also helpful in investigating properties of the polynomial hierarchy. For example, if NP ⊆ P/poly, then PH collapses to $$\Sigma_2^{\mathsf P}$$. A full description of the relations between P/poly and other complexity classes is available at "Importance of P/poly". P/poly is also helpful in the general study of the properties of Turing machines, as the class can be equivalently defined as the class of languages recognized by a polynomial-time Turing machine with a polynomial-bounded advice function.

Two subclasses of P/poly that have interesting properties in their own right are NC and AC. These classes are defined not only in terms of their circuit size but also in terms of their depth. The depth of a circuit is the length of the longest directed path from an input node to the output node. The class NC is the set of languages that can be solved by circuit families that are restricted not only to having polynomial-size but also to having polylogarithmic depth. The class AC is defined similarly to NC, however gates are allowed to have unbounded fan-in (that is, the AND and OR gates can be applied to more than two bits). NC is a notable class because it can be equivalently defined as the class of languages that have efficient parallel algorithms.

Quantum computation
The classes BQP and QMA, which are of key importance in quantum information science, are defined using quantum Turing machines.

Other types of problems
While most complexity classes studied by computer scientists are sets of decision problems, there are also a number of complexity classes defined in terms of other types of problems. In particular, there are complexity classes consisting of counting problems, function problems, and promise problems. These are explained in greater detail below.

Counting problems
A counting problem asks not only whether a solution exists (as with a decision problem), but asks how many solutions exist. For example, the decision problem $$\texttt{CYCLE}$$ asks whether a particular graph $$G$$ has a simple cycle (the answer is a simple yes/no); the corresponding counting problem $$\#\texttt{CYCLE}$$ (pronounced "sharp cycle") asks how many simple cycles $$G$$ has. The output to a counting problem is thus a number, in contrast to the output for a decision problem, which is a simple yes/no (or accept/reject, 0/1, or other equivalent scheme).

Thus, whereas decision problems are represented mathematically as formal languages, counting problems are represented mathematically as functions: a counting problem is formalized as the function $$f:\{0,1\}^* \to \mathbb{N}$$ such that for every input $$w \in \{0,1\}^*$$, $$f(w)$$ is the number of solutions. For example, in the $$\#\texttt{CYCLE}$$ problem, the input is a graph $$G \in \{0,1\}^*$$ (a graph represented as a string of bits) and $$f(G)$$ is the number of simple cycles in $$G$$.

Counting problems arise in a number of fields, including statistical estimation, statistical physics, network design, and economics.

Important complexity classes
#P (pronounced "sharp P") is an important class of counting problems that can be thought of as the counting version of NP. The connection to NP arises from the fact that the number of solutions to a problem equals the number of accepting branches in a nondeterministic Turing machine's computation tree. #P is thus formally defined as follows:
 * #P is the set of all functions $$f:\{0,1\}^* \to \mathbb{N}$$ such that there is a polynomial time nondeterministic Turing machine $$M$$ such that for all $$w \in \{0,1\}^*$$, $$f(w)$$ equals the number of accepting branches in $$M$$'s computation tree on $$w$$.

And just as NP can be defined both in terms of nondeterminism and in terms of a verifier (i.e. as an interactive proof system), so too can #P be equivalently defined in terms of a verifier. Recall that a decision problem is in NP if there exists a polynomial-time checkable certificate to a given problem instance—that is, NP asks whether there exists a proof of membership (a certificate) for the input that can be checked for correctness in polynomial time. The class #P asks how many such certificates exist. In this context, #P is defined as follows:
 * #P is the set of functions $$f: \{0,1\}^* \to \mathbb{N}$$ such that there exists a polynomial $$p: \mathbb{N} \to \mathbb{N}$$ and a polynomial-time Turing machine $$V$$ (the verifier), such that for every $$w \in \{0,1\}^*$$, $$f(w)=\Big| \big\{c \in \{0,1\}^{p(|w|)} : V(w,c)=1 \big\}\Big| $$. In other words, $$f(w)$$ equals the size of the set containing all of the polynomial-size certificates for $$w$$.

Function problems
Counting problems are a subset of a broader class of problems called function problems. A function problem is a type of problem in which the values of a function $$f:A \to B$$ are computed. Formally, a function problem $$f$$ is defined as a relation $$R$$ over strings of an arbitrary alphabet $$\Sigma$$:
 * $$ R \subseteq \Sigma^* \times \Sigma^*$$

An algorithm solves $$f$$ if for every input $$x$$ such that there exists a $$y$$ satisfying $$(x, y) \in R$$, the algorithm produces one such $$y$$. This is just another way of saying that $$f$$ is a function and the algorithm solves $$f(x)$$ for all $$x \in \Sigma^*$$.

Important complexity classes
An important function complexity class is FP, the class of efficiently solvable functions. More specifically, FP is the set of function problems that can be solved by a deterministic Turing machine in polynomial time. FP can be thought of as the function problem equivalent of P. Importantly, FP provides some insight into both counting problems and P versus NP. If #P=FP, then the functions that determine the number of certificates for problems in NP are efficiently solvable. And since computing the number of certificates is at least as hard as determining whether a certificate exists, it must follow that if #P=FP then P=NP (it is not known whether this holds in the reverse, i.e. whether P=NP implies #P=FP).

Just as FP is the function problem equivalent of P, FNP is the function problem equivalent of NP. Importantly, FP=FNP if and only if P=NP.

Promise problems
Promise problems are a generalization of decision problems in which the input to a problem is guaranteed ("promised") to be from a particular subset of all possible inputs. Recall that with a decision problem $$L \subseteq \{0,1\}^*$$, an algorithm $$M$$ for $$L$$ must act (correctly) on every $$w \in \{0,1\}^*$$. A promise problem loosens the input requirement on $$M$$ by restricting the input to some subset of $$\{0,1\}^*$$.

Specifically, a promise problem is defined as a pair of non-intersecting sets $$(\Pi_{\text{ACCEPT}},\Pi_{\text{REJECT}})$$, where:
 * $$\Pi_{\text{ACCEPT}} \subseteq \{0,1\}^*$$ is the set of all inputs that are accepted.
 * $$\Pi_{\text{REJECT}} \subseteq \{0,1\}^*$$ is the set of all inputs that are rejected.

The input to an algorithm $$M$$ for a promise problem $$(\Pi_{\text{ACCEPT}},\Pi_{\text{REJECT}})$$ is thus $$\Pi_{\text{ACCEPT}} \cup \Pi_{\text{REJECT}}$$, which is called the promise. Strings in $$\Pi_{\text{ACCEPT}} \cup \Pi_{\text{REJECT}}$$ are said to satisfy the promise. By definition, $$\Pi_{\text{ACCEPT}}$$ and $$\Pi_{\text{REJECT}}$$ must be disjoint, i.e. $$\Pi_{\text{ACCEPT}} \cap \Pi_{\text{REJECT}} = \emptyset$$.

Within this formulation, it can be seen that decision problems are just the subset of promise problems with the trivial promise $$\Pi_{\text{ACCEPT}} \cup \Pi_{\text{REJECT}} = \{0,1\}^*$$. With decision problems it is thus simpler to simply define the problem as only $$\Pi_{\text{ACCEPT}}$$ (with $$\Pi_{\text{REJECT}}$$ implicitly being $$\{0,1\}^* / \Pi_{\text{ACCEPT}}$$), which throughout this page is denoted $$L$$ to emphasize that $$\Pi_{\text{ACCEPT}}=L$$ is a formal language.

Promise problems make for a more natural formulation of many computational problems. For instance, a computational problem could be something like "given a planar graph, determine whether or not..." This is often stated as a decision problem, where it is assumed that there is some translation schema that takes every string $$s \in \{0,1\}^*$$ to a planar graph. However, it is more straightforward to define this as a promise problem in which the input is promised to be a planar graph.

Relation to complexity classes
Promise problems provide an alternate definition for standard complexity classes of decision problems. P, for instance, can be defined as a promise problem:
 * P is the class of promise problems that are solvable in deterministic polynomial time. That is, the promise problem $$(\Pi_{\text{ACCEPT}},\Pi_{\text{REJECT}})$$ is in P if there exists a polynomial-time algorithm $$M$$ such that:
 * For every $$x \in \Pi_{\text{ACCEPT}}, M(x)=1$$
 * For ever $$x \in \Pi_{\text{REJECT}}, M(x)=0$$

Classes of decision problems—that is, classes of problems defined as formal languages—thus translate naturally to promise problems, where a language $$L$$ in the class is simply $$L= \Pi_{\text{ACCEPT}}$$ and $$\Pi_{\text{REJECT}}$$ is implicitly $$\{0,1\}^* / \Pi_{\text{ACCEPT}}$$.

Formulating many basic complexity classes like P as promise problems provides little additional insight into their nature. However, there are some complexity classes for which formulating them as promise problems have been useful to computer scientists. Promise problems have, for instance, played a key role in the study of SZK (statistical zero-knowledge).

Summary of relationships between complexity classes
The following table shows some of the classes of problems that are considered in complexity theory. If class X is a strict subset of Y, then X is shown below Y with a dark line connecting them. If X is a subset, but it is unknown whether they are equal sets, then the line is lighter and dotted. Technically, the breakdown into decidable and undecidable pertains more to the study of computability theory, but is useful for putting the complexity classes in perspective.

title: Complexity class


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Aizumi, Tokushima

Aizumi (藍住町) is a town located in Itano District, in northeastern Tokushima Prefecture, Japan. , the town had an estimated population of 35,512 in 15198 households and a population density of 2200 persons per km2. The total area of the town is 16.27 sqkm.

Geography
Aizumi is located in the northern part of Tokushima Prefecture, on the island of Shikoku. It is situated on the north side of the downstream area of the Yoshino River. The area is an alluvial fan which was once subject to frequent flooding before Edo period flood control projects shifted the course of the Yoshino River from its original stream to its current stream some three kilometers south. At 5.17 meters average elevation above sea level the area is a very flat area with no hilly or mountainous areas found within the confines of the town district.

Neighbouring municipalities
Tokushima Prefecture
 * Tokushima
 * Naruto
 * Itano
 * Kitajima
 * Kamiita
 * Ishii

Climate
Aizumi has a Humid subtropical climate (Köppen Cfa) characterized by warm summers and cool winters with light snowfall. The average annual temperature in Aizumi is 15.9 °C. The average annual rainfall is 1637 mm with September as the wettest month. The temperatures are highest on average in August, at around 26.6 °C, and lowest in January, at around 4.6 °C. The annual mean rainfall is 1,586.6 mm, and the average monthly discharge is 132.2 mm. During winter, the average monthly discharge is 50 mm throughout the season, with a touch of Seto Inland Sea-style climate being present.

Demographics
Per Japanese census data, the population of Aizumi has been growing rapidly since the 1960s. With a population growth rate of 10% over the past 10 years, it is one of the most developed towns in Tokushima Prefecture.

History
As with all of Tokushima Prefecture, the area of Aizumi was part of ancient Awa Province. From the Muromachi period, Shōzui Castle was the seat of the Hosokawa clan and later the Miyoshi clan who ruled as shugo of the province. Chōsokabe Motochika seized control of the province in the Sengoku period and Shōzui Castle fell into ruins. He was subsequently defected by the forces of Toyotomi Hideyoshi. During the Edo period, the area was part of the holdings of Tokushima Domain ruled by the Hachisuka clan from their seat at Tokushima Castle, and Aizumi became an agricultural area noted for its production of indigo. The villages of Aizono (藍園村) and Sumiyoshi (住吉村) were established within Itano District, Tokushima with the creation of the modern municipalities' system on October 1, 1889. The two village merged on April 29, 1955, to form the town of Aizumi.

Government
Aizumi has a mayor-council form of government with a directly elected mayor and a unicameral town council of 16 members. Aizumi, together with the other municipalities of Itano District, contributes four members to the Tokushima Prefectural Assembly. In terms of national politics, the town is part of Tokushima 2nd district of the lower house of the Diet of Japan.

Economy
Aizumi has a mixed economy of agriculture and light manufacturing. The production of indigo dye and miso fermented bean paste are traditional local industries. Due to its geographical location and abundance of flat land for housing development, Aizumi is also developing into a commuter town for neighboring Tokushima.

Education
Aizumi has four public elementary schools and two public middle schools operated by the town government, and one public high school operated by the Tokushima Prefectural Department of Education.

Railway
Shikoku Railway Company — Kōtoku Line
 * Shōzui

Highways

 * JP Expressway E11.svg Tokushima Expressway
 * The Shikoku Saburo Bridge across the Yoshino River links Aizumi with Tokushima city.

Local attractions

 * Shōzui Castle ruins, National Historic Site

title: Aizumi, Tokushima


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Itano, Tokushima

Itano (板野町) is a town located in Itano District, Tokushima Prefecture, Japan. , the town had an estimated population of 13,105 in 5790 households and a population density of 360 persons per km². The total area of the town is 36.22 sqkm.

Geography
Itano is located in northeastern Tokushima Prefecture on the island of Shikoku. The northern part of the town is occupied by the Sanuki Mountains, and the southern part is the delta area of the Yoshino River. The Japan Median Tectonic Line fault zone, which is an active fault, runs from east to west at the boundary between the mountains and the plains.

Neighbouring municipalities
Tokushima Prefecture Kagawa Prefecture
 * Naruto
 * Kamiita
 * Aizumi
 * Higashikagawa

Climate
Itano has a Humid subtropical climate (Köppen Cfa) characterized by warm summers and cool winters with light snowfall. The average annual temperature in Itano is 15.9 °C. The average annual rainfall is 1637 mm with September as the wettest month. The temperatures are highest on average in August, at around 26.7 °C, and lowest in January, at around 5.7 °C.

Demographics
Per Japanese census data, the population of Itano has remained relatively stable for the past 60 years.

History
As with all of Tokushima Prefecture, the area of Itano was part of ancient Awa Province. The village of Itanishi (板西村) was established within Itano District, Tokushima with the creation of the modern municipalities system on October 1, 1889. It was raised to town status on July 29, 1908. Itanishi merged with neighboring villages of Matsusaka and Sakae on February 11, 1911 and was renamed Itano.

Government
Itano has a mayor-council form of government with a directly elected mayor and a unicameral town council of 13 members. Itano, together with the other municipalities of Itano District, contributes four members to the Tokushima Prefectural Assembly. In terms of national politics, the town is part of Tokushima 2nd district of the lower house of the Diet of Japan.

Economy
Itano has primarily an agricultural economy, with rice and carrots as the main crops.

Education
Itano has three public elementary schools and one public middle schools operated by the town government and one public high school operated by the Tokushima Prefectural Department of Education. The Tokushima College of Technology is located in Itano.

Railway
Shikoku Railway Company – Kōtoku Line
 * Awa-Kawabata - Itano - Awa-Ōmiya

Highways

 * JP Expressway E11.svg Takamatsu Expressway

Local attractions

 * Konsen-ji, 3rd temple on the Shikoku Pilgrimage
 * Dainichi-ji, 4th temple on the Shikoku Pilgrimage
 * Jizō-ji, 5th temple on the Shikoku Pilgrimage
 * Tokushima Prefectural Buried Cultural Properties Research Centre

Noted people from Itano

 * Angela Aki, singer, songwriter

title: Itano, Tokushima


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Kamiita, Tokushima

Kamiita (上板町) is a town located in Itano District, Tokushima Prefecture, Japan. , the town had an estimated population of 11,725 in 4951 households and a population density of 340 persons per km2. The total area of the town is 34.58 sqkm.

Geography
Kamiita is located in northeastern Tokushima Prefecture on the island of Shikoku. It is sandwiched between the Sanuki Mountains to the north and the Yoshino River to the south. The Japan Median Tectonic Line runs from east to west in the center of the town, separating the mountainous area to the north and the plains to the south. The highest elevation of the town is Oyama, which is 691 meters above sea level. Numerous small rivers which originate in the Sanuki Mountains form multiple alluvial fans and have been inhabited by people from ancient times.

Neighbouring municipalities
Tokushima Prefecture Kagawa Prefecture
 * Yoshinogawa
 * Awa
 * Itano
 * Aizumi
 * Ishii
 * Higashikagawa

Climate
Kamiita has a Humid subtropical climate (Köppen Cfa) characterized by warm summers and cool winters with light snowfall. The average annual temperature in Kamiita is 15.2 °C. The average annual rainfall is 1637 mm with September as the wettest month. The temperatures are highest on average in August, at around 26.4 °C, and lowest in January, at around 4.5 °C.

Demographics
Per Japanese census data, the population of Kamiita has been relatively steady for the past 70 years.

History
As with all of Tokushima Prefecture, the area of Kammita was part of ancient Awa Province. The area is one of the oldest inhabited areas of Shikoku, and stone tools from the Japanese Paleolithic period, Jomon pottery, dotaku from the Yayoi period, burial mounds from the Kofun period have been found. Itano District was one of the seven counties of Awa established by the Taika Reform. During the Edo period, the area was part of the holdings of Tokushima Domain ruled by the Hachisuka clan from their seat at Tokushima Castle, and Kamiita became an agricultural area noted for its production of indigo. The villages of Oyama (大山村) and Matsushima (松島町) within Itano District, Tokushima and the village of Takashi (高志村) within Myōzai District, Tokushima were established with the creation of the modern municipalities system on October 1, 1889. The three municipalities merged on March 31, 1955 to form the town of Kamiita.

Government
Kamiita has a mayor-council form of government with a directly elected mayor and a unicameral town council of 13 members. Kamiita, together with the other municipalities of Itano District, contributes four members to the Tokushima Prefectural Assembly. In terms of national politics, the town is part of Tokushima 2nd district of the lower house of the Diet of Japan.

Economy
Since the Edo period, the area has been known for production of indigo and sugar, which remain important heritage products to this day. The local economy is strongly agricultural, with rice and market crops as carrots, lettuce, spinach, and strawberries, and fruit trees such as persimmons and peaches predominating. It is also the largest dairy area in Tokushima Prefecture.

Education
Kamiita has four public elementary schools and one public middle school operated by the town government. The town does not have a high school.

Railway
Kamiita has not had passenger rail service since the abolition of the JNR Kajiyabara Line in 1972. Currently, the nearest passenger station is Itano on the JR Shikoku Kōtoku Line

Highways

 * JP Expressway E11.svg Tokushima Expressway

Local attractions

 * Anraku-ji, 6th temple on the Shikoku Pilgrimage
 * Taisan-ji

title: Kamiita, Tokushima


text:
Sortilege

Sortilege may refer to:

 * Cleromancy, divination by casting lots
 * Sortes (ancient Rome)
 * Sortilège (band), a French heavy metal band
 * Sortilège (EP), 1983
 * Sortilège, a Canadian brand of maple liqueur
 * Sortilèges (film), a 1945 French film

title: Sortilege


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Yoshino, Tokushima

Yoshino (吉野町) was a town located in Itano District, Tokushima Prefecture, Japan.

As of 2003, the town had an estimated population of 8,397 and a density of 630.41 persons per km2. The total area was 13.32 km2.

On April 1, 2005, Yoshino, along with the towns of Awa (former) and Ichiba (both from Awa District), and the town of Donari (also from Itano District), was merged to create the city of Awa.

title: Yoshino, Tokushima


text: Template:1972 districts of Northern Ireland


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Donari, Tokushima

Donari (土成町) was a town located in Itano District, Tokushima Prefecture, Japan.

As of 2003, the town had an estimated population of 8,255 and a density of 145.82 persons per km2. The total area was 56.61 km2.

On April 1, 2005, Donari, along with the towns of Awa (former) and Ichiba (both from Awa District), and the town of Yoshino (also from Itano District), was merged to create the city of Awa.

title: Donari, Tokushima


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Lavinium

Lavinium was a port city of Latium, 6 km to the south of Rome, midway between the Tiber river at Ostia and Antium. The coastline then, as now, was a long strip of beach. Lavinium was on a hill at the southernmost edge of the Silva Laurentina, a dense laurel forest, and the northernmost edge of the Pontine Marshes, a vast malarial tract of wetlands. The basis for the port, the only one between Ostia and Antium, was evidently the mouth of the Numicus river.

The location of Lavinium has never been lost to historians nor does there appear to have been any significant break in its habitation. Today's settlement remains a walled village of medieval design, Pratica di Mare, in the comune of Pomezia. The latter is a city constructed in 1939 and settled according to a plan of Benito Mussolini, whose engineers completed the millennia-long task of draining and filling the marsh, now the Pontine fields. A brief strip of field separates the large and flourishing city from the village. One Roman gate allows entry into the narrow streets of the village past the Castello Borghese, originally a fortification, purchased along with the village in 1617 by Marcantonio Borghese. The castle and the village were periodically renovated. All that remains of the river that once partly surrounded the village is a small stream, the Fosso di Pratica.

Pratica di Mare is about 6 km from the Tyrrhenian Sea near the top of a slope descending to an alluvial shelf on which the Pratica di Mare Air Force Base has been placed. It has the historical distinction of being the airfield from which Otto Skorzeny flew Mussolini to safety in Germany after his rescue from imprisonment in a mountain villa. Today the base is both a secure airport for the protection of distinguished visitors to the Rome region and a home for air shows of advanced aircraft. The Fosso di Pratica was re-routed around the end of a runway; however, today's small brook is in no way compatible with the concept of a port. The sea may well have formerly extended up to the base of the hill, as sites further north, such as Ostia, appear to have retreated one or two miles inland. Ancient Roman seaside villas are no longer on the beach.

Archaeology
Pratica di Mare is observably smaller than ancient Lavinium, whose remains crop out in the surrounding fields. Recent archaeological excavations performed to the south date Lavinium to well before the legendary foundation of Rome. It was already fortified in the 7th century BC and flourishing in the 6th. Lavinium was assimilated by Republican Rome. It was connected to Rome in the north and Ardea to the south by the Via Laurentina. Under the empire it was combined with the mysterious Laurentum, where many wealthy Romans maintained a winter villa, to become Laurolavinium. The nature of the union remains ambiguous.

A number of kilns have been identified within the perimeter of the city walls. Outside the city was a sanctuary dedicated to Sol Indiges and a vast sanctuary with numerous altars, where the bronze inscribed plaque records that the Dioscuri were being venerated at one of numerous altars.

Legend and history
According to Roman mythology, which links Lavinium more securely to Rome, the city was named by Aeneas in honor of Lavinia, daughter of Latinus, king of the Latins, and his wife, Amata. Aeneas reached Italy and there fought a war against Turnus, the leader of the local Rutuli people. Aeneas founded not Rome but rather Lavinium, the main centre of the Latin league, from which the people of Rome sprang. Aeneas thus links the royal house of Troy with the early Roman royal house.

The foundation of Lavinium and the Rutulian war are both mentioned prominently in Virgil's Aeneid.

In ancient times Lavinium had a close association with the nearby Laurentum. According to Livy, in the eighth century BC, when Romulus and Titus Tatius jointly ruled Rome, the ambassadors of the Laurentes came to Rome, but were beaten by Tatius' relatives. The Laurentes complained, but Tatius accorded more weight to the influence of his relatives than to the injury done the Laurentes. When Tatius afterwards visited Lavinium to celebrate an anniversary sacrifice, he was slain in a tumult. Romulus declined to go to war and instead renewed the treaty between Rome and Lavinium.

In 509 BC, after the overthrow of the Roman monarchy, one of Rome's first two consuls Lucius Tarquinius Collatinus was coerced into leaving Rome because of his relation to the kings. He voluntarily went into exile in Lavinium.

In around 488 BC, Lavinium was captured by an invading army of the Volsci, led by Gaius Marcius Coriolanus and Attius Tullus Aufidius.

title: Lavinium


text:
Clyde R. Hoey

Clyde Roark Hoey (December 11, 1877 –May 12, 1954) was an American Democratic politician from North Carolina. He served in both houses of the state legislature and served briefly in the U.S. House of Representatives from 1919 to 1921. He was North Carolina's governor from 1937 to 1941. He entered the U.S. Senate in 1945 and served there until his death in 1954, only days before the Brown v. Board of Education decision. He was a segregationist.

Biography
Hoey (HOO-ee) was born to Captain Samuel Alberta Hoey, a Confederate States Army officer, and Mary Charlotte Roark. He attended school until age eleven. He worked on his family's farm and bought a weekly newspaper when he was 16. He was elected to the state legislature when he was twenty. He served as a state representative and then as a state senator. He was elected in a special election to the United States House of Representatives to fill the vacancy caused by the resignation of Edwin Y. Webb who had accepted a federal judgeship. He defeated a Republican who opposed United States support for the League of Nations. He served from 1919 to 1921. He prosecuted the leaders of the 1929 Loray Mill strike for the murder of the Gastonia police chief.

He was the 59th governor of the U.S. state of North Carolina from 1937 to 1941. In July 1937, he pardoned Luke Lea, a Tennessee politician and former U.S. senator, who had been paroled a year earlier. His appointment of a black man to the board of trustees of a black college set a precedent. Following the 1938 Gaines Supreme Court decision on racial segregation in higher education, he asked the North Carolina legislature to provide for segregated higher education for blacks. Though opposed to integrated education, he said that the people of the state "do believe in equality of opportunity in their respective fields of service" and that "the white race cannot afford to do less than simple justice to the Negro."

In 1940, Hoey quietly opposed a third term for FDR. When he believed that President Franklin D. Roosevelt would not seek a third term, Hoey rejected the favorite son role for which the state legislature had recommended him and supported the presidential candidacy of Secretary of State Cordell Hull.

Hoey won election to the U.S. Senate in 1944. He served from 1945 until his death in 1954.

Hoey's politics were those of a conservative Democrat. He opposed Harry S. Truman's attempt to make the Fair Employment Practices Commission (FEPC) permanent. He promised to filibuster the effort as an attack on "the rights of every businessman in America." He supported the President's threats against striking railroad workers in December 1946. In the 1948 election, he supported Truman over the alternative, Strom Thurmond.

He supported President Truman's refusal to allow Congress access to records of government employees' loyalty investigations.

In 1950, Hoey opposed statehood for Hawaii because he thought it "inconceivable" to allow a territory with "only a small percentage of white people" to become a state. He advocated independence for Hawaii and cited U.S. treatment of Cuba and the Philippines as precedents.

From 1949 to 1952 he headed the Investigations Subcommittee of the Committee on Expenditures in Executive Departments. He conducted hearings into the role of "five percenters", government influence peddlers. In 1950 he chaired an investigation that resulted in a report, known as the Hoey Report, released in December of that year that said all of the government's intelligence agencies "are in complete agreement that sex perverts [meaning, primarily, gay men] in Government constitute security risks.". Douglas Charles characterizes Hoey's involvement in the committee as reluctant, due to fears that the issue could become hyperbolic, leaving chief counsel Francis Flanagan as the actual driving force behind the Hoey Report The 1957 Crittenden Report, a review by the U.S. Navy, criticized it: "No intelligence agency, as far as can be learned, adduced any factual data before that committee with which to support these opinions."

Hoey married Bessie Gardner, sister of North Carolina Governor O. Max Gardner. They had three children. His wife died in 1942. He was a lifelong member of the Methodist Episcopal Church, South and taught Sunday school classes. He was also a member of the Freemasons, Odd Fellows, Woodmen of the World, and the Knights of Pythias.

Hoey died at his desk in his Washington, D.C., office. Sam Ervin was appointed to his seat in June 1954.

Legacy
In 1974, journalist Jonathan Daniels assessed Hoey's politics as "always satisfactory to conservative interests without being abrasive to New Dealers."

Three university buildings in North Carolina were named for Hoey, but have been renamed. The first renaming was in July 2019, when, given Hoey's history of segregationist advocacy and use of racist language in a public address, his name was removed from North Carolina Central University's administration building and replaced with that of the university's African-American founder, James E. Shepard. Hoey Hall, a dormitory at Appalachian State University, and Hoey Auditorium, on the campus of Western Carolina University, were renamed in June 2020, as part of the name changes due to the George Floyd protests. According to a unanimous vote of the trustees of Western Carolina, "Hoey's espoused racist views are contrary to this university's core values of diversity and equality."

title: Clyde R. Hoey