title: Charles III of Spain

text: On 3 February 1740, King Charles issued a proclamation containing 37 paragraphs, in which Jews were formally invited to return to Sicily, from where they had been brutally expelled in 1492. This move had a little practical effect: though a few Jews did come to Sicily, though there was no legal impediment to their living there, they felt their lives insecure, and they soon went back to Turkey. Despite the King's goodwill, the Jewish community of Sicily which had flourished in the Middle East was not re-established. Still, this was a significant symbolic gesture, the King clearly repudiating a past policy of religious intolerance. Moreover, the expulsion of the Jews from Sicily had been an application of the Spanish Alhambra Decree - which would be repudiated in Spain itself only much later.


title: Charles III of Spain

text: The Kingdom of Naples remained neutral during the Seven Years' War (1756–1763). The British Prime Minister, William Pitt wanted to create an Italian league where Naples and Sardinia would fight together against Austria, but Charles refused to participate. This choice was sharply criticized by the Neapolitan Ambassador in Turin, Domenico Caraccioli, who wrote:


title: Charles III of Spain

text: "The position of Italian matters is not more beautiful; but it is worsened by the fact that the King of Naples and the King of Sardinia, adding troops to larger forces of the others, could oppose itself to the plans of their neighbors; to defend itself against the dangers of the peace of the enemies themselves they were in a way united, but they are separated by their different systems of government."


title: Charles III of Spain

text: With the Republic of Genoa in relations are stretched: Pasquale Paoli, general of Corsican pro-independence rebels, was an officer of the Neapolitan army and the Genoese one suspected that he received the assistance of the kingdom of Naples.


title: Charles III of Spain

text: He constructed a collection of palaces in and around Naples. Charles was in awe of the Palace of Versailles and the Royal Palace of Madrid in Spain (the latter being modeled on Versailles itself). He undertook and oversaw the construction of one of Europe's most lavish palaces, the Palace of Caserta ("Reggia di Caserta"). Construction ideas for the stunning palace started in 1751 when he was 35 years old. The site had previously been home to a small hunting lodge, as had Versailles, which he was fond of because it reminded him of San Ildefonso where the Royal Palace of La Granja de San Ildefonso was located in Spain. Caserta was also much influenced by his wife, the very cultured Maria Amalia of Saxony. The site of the palace was also far away from the large volcano of Mount Vesuvius, which was a constant threat to the capital, as was the sea. Charles himself laid the foundation stone of the palace amid many festivities on his 36th birthday, 20 January 1752. Other buildings he had built in his kingdom were the Palace of Portici ("Reggia di Portici"), the Teatro di San Carlo—constructed in just 270 days—and the Palace of Capodimonte ("Reggia di Capodimonte"); he also had the Royal Palace of Naples renovated. He and his wife had the Capodimonte porcelain Factory constructed in the city. He also founded the Ercolanesi Academy and the Naples National Archaeological Museum, which still operates today.


title: Charles III of Spain

text: During his rule the Roman cities of Herculaneum (1738), Stabiae and Pompeii (1748) were re-discovered. The king encouraged their excavation and continued to be informed about findings even after moving to Spain. Camillo Paderni who was in charge of excavated items at the King's Palace in Portici was also the first to attempt in reading obtained scrolls from the Villa of the Papyri in Herculaneum.


title: Charles III of Spain

text: After Charles departed for Spain, Minister Tanucci presided over the Council of Regency that ruled until Ferdinand reached 16, the age of majority.


title: Charles III of Spain

text: Charles was not expected to ascend to the throne of Spain, since his father had sons from his first wife who were more likely to rule. As the first son of his father's second wife, Charles benefited from his mother's ambition that he have a kingdom to rule, an experience that served him well when he ascended to the throne of Spain and ruled the Spanish Empire.


title: Charles III of Spain

text: At the end of 1758, Charles's half brother Ferdinand VI was displaying the same symptoms of depression that their father used to suffer from. Ferdinand lost his devoted wife, Barbara of Portugal, in August 1758, and fell into deep mourning for her. He named Charles his heir presumptive on 10 December 1758 before leaving Madrid to stay at Villaviciosa de Odón, where he died on 10 August 1759.


title: Charles III of Spain

text: At that point, Charles was proclaimed King of Spain under the name of Charles III of Spain. He gained the title, respecting the third Treaty of Vienna that stated he would not be able to join the Neapolitan and Sicilian territories to the Spanish throne.


title: Charles III of Spain

text: Charles was later given the title of "Lord of the Two Sicilies". The Treaty of Aix-la-Chapelle, which Charles had not ratified, foresaw the eventuality of his accession to Spain; thus Naples and Sicily went to his brother Philip, Duke of Parma, while the possessions of the latter were divided between Maria Theresa (Parma and Guastalla) and the King of Sardinia (Plaisance).


title: Charles III of Spain

text: Determined to maintain the hold of his descendants on the court of Naples, Charles undertook lengthy diplomatic negotiations with Maria Theresa, and in 1758 the two signed the Fourth Treaty of Versailles, by which Austria formally renounced the Italian Duchies. Charles Emmanuel III of Sardinia, however, continued to pressure on the possible gain of Plaisance and even threatened to occupy it.


title: Charles III of Spain

text: In order to defend the Duchy of Parma from Charles Emmanuel's threats, Charles deployed troops on the borders of the Papal States. Thanks to the mediation of Louis XV, Charles Emmanuel renounced his claims to Plaisance in exchange for financial compensation. Charles thus assured the succession of one of his sons and, at the same time, reduced Charles Emmanuel's ambitions. According to Domenico Caracciolo, this was "a fatal blow to the hopes and designs of the king of Sardinia".


title: Charles III of Spain

text: The eldest son of Charles, Infante Philip, Duke of Calabria, had learning difficulties and was thus taken out of the line of succession to any throne; he died in Portici, where he had been born, in 1747. The title of Prince of Asturias was given to Charles, the second-born. The right of succession to Naples and Sicily was reserved for his third son, Ferdinand; he would stay in Italy while his father was in Spain. Charles formally abdicated the crowns of Naples and Sicily on 6 October 1759 in favor of Ferdinand. Charles left his son's education and care to a regency council composed of eight members, which would govern the kingdom until the young king was 16 years old. Charles and his wife arrived in Barcelona on 7 October 1759.


title: Charles III of Spain

text: His twenty years in the Italian Peninsula had been very fruitful, and he came to the throne of Spain with significant experience. Internal politics, as well as diplomatic relationships with other countries, underwent complete reform. Charles represented a new type of ruler, who followed "Enlightened absolutism". This was a form of absolute monarchy or despotism in which rulers embraced the principles of the Enlightenment, especially its emphasis upon rationality, and applied them to their territories. They tended to allow religious toleration, freedom of speech and the press, and the right to hold private property. Most fostered the arts, sciences, and education. Charles shared these ideals with other monarchs, including Maria Theresa of Austria, her son Joseph, and Catherine the Great of Russia.


title: Charles III of Spain

text: The principles of the Enlightenment were applied to his rule in Naples, and he intended to do the same in Spain though on a much larger scale. Charles went about his reform along with the help of the Marquis of Esquilache, Count of Aranda, Count of Campomanes, Count of Floridablanca, Ricardo Wall and the Genoan aristocrat Jerónimo Grimaldi.


title: Charles III of Spain

text: Under Charles's reign, Spain began to be recognized as a nation state rather than a collection of kingdoms and territories with a common sovereign. This was a long process that his Bourbon predecessors had initiated. Philip V had abolished the special privileges ("fueros") of the Kingdoms of Aragon and Valencia, subordinating them to the Crown of Castile and ruled by the Council of Castile. In the Nueva Planta decrees, Philip V also disbanded the Generalitat de Catalunya, abolished its Constitutions, banned the Catalan language from any official use and mandated the use of Castilian Spanish in legal affairs. He incorporated these formerly privileged entities into the Cortes of Castile, in effect, the Cortes of Spain. When Charles III became king of Spain, he further solidified the standing of the nation as a single political entity. He created the national anthem and a flag, a capital city worthy of the name, and the construction of a network of coherent roads converging on Madrid. On 3 September 1770 Charles III declared that the "Marcha Real" was to be used in official ceremonies. It was Charles who chose the colors of the present flag of Spain: two red stripes above and below a central yellow stripe double in width and the arms of Castile and León. The flag of the military navy was introduced by the king on 28 May 1785. Until then, Spanish vessels sported the white flag of the Bourbons with the arms of the sovereign. Charles replaced it due to his concern that it looked too similar to the flags of other nations.


title: Charles III of Spain

text: Bourbon Spain, like their Habsburg predecessors, were drawn into European conflicts, not necessarily to Spain's benefit. The traditional friendship with Bourbon France brought about the idea that the power of Great Britain would decrease and that of Spain and France would do the opposite; this alliance was marked by a "Family Compact" signed on 15 August 1761 (called the "Treaty of Paris"). Charles had become deeply concerned that British success in the Seven Years War would upset the balance of power, and they would soon seek to declare war against the Spanish Empire as well. The French government ceded its largest territory in North America, New France to Britain as a result of the conflict.


title: Charles III of Spain

text: In early 1762, Spain entered the war. The major Spanish objectives to invade Portugal and capture Jamaica were both failures. Britain and Portugal not only repulsed the Spanish attack on Portugal, but captured the cities of Havana, Cuba, a strategic port for all of Spanish America, and Manila, in the Philippines, Spain's stronghold for its Asian trade and colony of strategic islands. Charles III wanted to keep fighting the following year, but he was persuaded by the French leadership to stop. In the 1763 Treaty of Paris, Spain ceded Florida to Great Britain in exchange for the return of Havana and Manila. This was partly compensated by the acquisition of a portion of Louisiana given to Spain by France as compensation for Spain's war losses. Britain's easy victories in capturing Spanish ports prompted Spain to create a standing army and local militias in key parts of Spanish America and fortify vulnerable forts.


title: Charles III of Spain

text: In the Falklands Crisis of 1770 the Spanish came close to war with Great Britain after expelling the British garrison of the Falkland Islands. However, Spain was forced to back down when the British Royal Navy was mobilized and France declined to support Spain.


title: Charles III of Spain

text: The invasion of Algiers in 1775 was ordered by Charles, who was attempting to demonstrate to the Barbary States the power of the revitalized Spanish military after the disastrous Spanish experience in the Seven Years' War. The assault was also meant to demonstrate that Spain would defend its North African territories against any Ottoman or Moroccan encroachment.


title: Charles III of Spain

text: Continuing territorial disputes with Portugal led to the First Treaty of San Ildefonso, on 1 October 1777, in which Spain got Colonia del Sacramento, in present-day Uruguay, and the Misiones Orientales, in present-day Brazil, but not the western regions of Brazil, and also the Treaty of El Pardo, on 11 March 1778, in which Spain again conceded that Portuguese Brazil had expanded far west of the longitude specified in the Treaty of Tordesillas, and in return Portugal ceded present-day Equatorial Guinea to Spain.


title: Charles III of Spain

text: Concerns about the intrusions of British and Russian merchants into Spain's colonies in California prompted the extension of Franciscan missions to Alta California, as well as presidios.


title: Charles III of Spain

text: The rivalry with Britain also led him to support the American revolutionaries in their war of independence (1776-1783), despite his misgivings about the example it would set for Spain's overseas territories. During the war, Spain recovered Menorca and West Florida in several military campaigns, but failed in their attempt to capture Gibraltar. Spanish military operations in West Florida and on the Mississippi River helped the Thirteen Colonies secure their southern and western frontiers during the war. The capture of Nassau in the Bahamas enabled Spain to also recover East Florida during peace negotiations. The Treaty of Paris of 1783 confirmed the recovery of the Floridas and Menorca and restricted the actions of British commercial interests in Central America.


title: Charles III of Spain

text: Charles had able and enlightened ministers who helped craft his reform policies. During his early rule in Spain, he appointed Italians, including the Marquess of Esquilache and Duke of Grimaldi, who supported reforms by the Count of Campomanes. The Count of Floridablanca was an important minister late in Charles's reign, who was carried over as minister after Charles's death. Also important was the Count of Aranda, who dominated the Council of Castile (1766-1773).


title: Charles III of Spain

text: His internal government was, on the whole, beneficial to the country. He began by compelling the people of Madrid to give up emptying their slops out of the windows, and when they objected he said they were like children who cried when their faces were washed. At the time of his accession to Spain, Charles named secretary to the Finances and Treasurer the Marquis of Esquillache and both realized many reforms. The Spanish Army and Navy were reorganized despite the losses from the Seven Years War.


title: Charles III of Spain

text: Charles also eliminated the tax on flour and generally liberalized most commerce. Despite this action, it provoked the overlord to charge high prices because of the "monopolizers" speculating on the bad harvests of the previous years. On 23 March 1766, his attempt to force the "madrileños" to adopt French dress for public security reasons was the excuse for a riot ("Motín de Esquilache"), during which he did not display much personal courage. For a long time after, he remained at Aranjuez, leaving the government in the hands of his minister the Count of Aranda. Not all his reforms were of this formal kind.


title: Charles III of Spain

text: The Count of Campomanes tried to show Charles that the true leaders of the revolt against Esquilache were the Jesuits. The wealth and power of the Jesuits were very great; and by the royal decree of 27 February 1767, known as the "Pragmatic Penalty of 1767", the Jesuits were expelled from Spain, and all their possessions were confiscated. His quarrel with the Jesuits, and the memory of those with the Pope while he was King of Naples, turned him towards a general policy of restriction of what he saw as the overgrown power of the Church. The number of reputedly idle clergy, and more particularly of the monastic orders, was reduced, and the Spanish Inquisition, though not abolished, was rendered torpid.


title: Charles III of Spain

text: In the meantime, much-antiquated legislation that tended to restrict trade and industry was abolished, and roads, canals, and drainage works were established. Many of his paternal ventures led to little more than the wasting of monies, or the creation of hotbeds of jobbery; on the whole, however, the country prospered. The result was largely due to the king who, even when ill-advised, did at least work steadily at his task of government.


title: Charles III of Spain

text: Charles also sought to reform Spanish colonial policy, in order to make Spain's colonies more competitive with the plantations of the French Antilles (particularly the French colony of Saint-Domingue) and Portuguese Brazil. This resulted in the creation of the ""Códigos Negros Españoles"", or Spanish Black Codes. The Black Codes, which were partly based on the French Code Noir and 13th-century Castilian Siete Partidas, aimed to establish greater legal control over slaves in the Spanish colonies, in order to expand agricultural production. The first code was written for the city of Santo Domingo in 1768, while the second code was written for the recently acquired Spanish territory of Louisiana in 1769. The third code, which was named the ""Código Negro Carolino"" after Charles himself, divided the freed black and slave populations of Santo Domingo into strictly stratified socio-economic classes.


title: Charles III of Spain

text: In Spain, he continued with his work trying to improve the services and facilities of his people. He created the Luxury Porcelain factory under the name of Real Fábrica del Buen Retiro in 1760; Crystal followed at the "Real Fábrica de Cristales de La Granja" and then there was the "Real Fábrica de Platería Martínez" in 1778. During his reign, the areas of Asturias and Catalonia industrialized quickly and produced much revenue for the Spanish economy. He then turned to the foreign economy looking towards his colonies in the Americas. In particular, he looked at the finances of the Philippines and encouraged commerce with the United States, starting in 1778. He also carried out a number of public works; he had the "Imperial Canal of Aragon" constructed, as well a number of routes that led to the capital of Madrid, which is located in the center of Spain. Other cities were improved during his reign; Seville for example saw the introduction of many new structures such as hospitals and the Archivo General de Indias. In Madrid, he was nicknamed "el Mejor Alcalde de Madrid" ("the Best Mayor of Madrid"). Charles was responsible for granting the title "Royal University" to the University of Santo Tomás in Manila, which is the oldest in Asia.


title: Charles III of Spain

text: In the capital, he also had the famous Puerta de Alcalá constructed along with the statue of Alcachofa fountain, and moved and redesigned the Real Jardín Botánico de Madrid. He had the present National Art Museum of Queen Sofia (named in honor of the present Queen of Spain, born Princess Sophia of Greece and Denmark) built, as well as the renowned Museo del Prado. At Aranjuez he added wings to the palace.


title: Charles III of Spain

text: He created the Spanish Lottery and introduced Christmas cribs following Neapolitan models. During his reign, the movement to found "Economic Societies" (an early form of Chamber of Commerce) was born.


title: Charles III of Spain

text: The Royal Palace of Madrid underwent many alterations under his rule. It was in his reign that the huge "Comedor de gala" (Gala Dining room) was built during the years of 1765–1770; the room took the place of the old apartments of Queen Maria Amalia. He died in the palace on 14 December 1788.


title: Charles III of Spain

text: The Spanish Empire has been called "improbable," since Iberian Spain had been poor and did not have natural endowments, but its empire was huge and far-flung, starting in the late fifteenth century. Most of the European portions of the empire, which had come under the control of the Spanish monarchy when the first Habsburg monarch Charles I of Spain became king in 1516. Those territories were separated from it, leaving huge possessions in Spanish America and the Philippines, which Charles III ruled.


title: Charles III of Spain

text: The policies that centralized the Spanish state on the Iberian peninsula were extended to its overseas territories, especially after the end of the Seven Years' War, when Havana and Manila were captured (1762–63) by the British. Charles's predecessors on the throne had begun reforming the relationship between the Iberian metropole Spanish American and Philippine possessions, to create a centralized and unified empire. The Seven Years' War had demonstrated to Charles that Spain's military was insufficient for a war with Britain. Military defense of the empire was a top priority, an expensive but necessary undertaking. With the 1763 peace treaty ending the Seven Years' War, Spain regained its ports of Havana, Cuba, and Manila, in the Philippines. Esquilache needed to find revenue to support the establishment of a standing military and fortification of ports. To raise funds, the sales tax alcabala was increased from 2% to 5%. To increase trade, Havana and other Caribbean ports were allowed to trade with other ports within the Spanish empire, not full trade, but "comercio libre" was freer trade. With the expansion, Spain hoped to undermine Britain's secret trade with Spanish America, and gain more revenue for the Spanish crown.


title: Charles III of Spain

text: Charles sent José de Gálvez as inspector general ("visitor") to New Spain in 1765 to find ways to extract further revenue from its richest overseas possession and to observe conditions. The position gave sweeping powers to its holder, sometimes greater than that of the viceroy. Following his return to Spain in 1771, Gálvez became Minister of the Indies and proceeded with sweeping administrative changes, replacing the old system of governance with administrative districts ("intendancies") and strengthening centralized crown control.


title: Charles III of Spain

text: Charles's Italian minister Esquilache was hated in Spain, seen as a foreigner, and responsible for policies that many Spaniards opposed. Bread riots in 1766, known as the Esquilache Riots, pinned the blame on the minister, but behind the uprising, the Society of Jesus was seen as the real culprit. After exiling Esquilache, Charles expelled the Jesuits from Spain and its empire in 1767. In Spanish America, the impact was significant, since the Jesuits were a wealthy and powerful religious order, owning lucrative haciendas that produced revenue funding its missions on the frontier and its educational institutions. For American-born Spaniards, at a stroke, the most wealthy and prestigious religious order that educated their sons and accepted a chosen few into their ranks were consigned to Italian exile. Jesuit properties, included thriving haciendas, were confiscated, the "colleges" educating their sons closed, and frontier missions were turned over to other religious orders. Politically, culturally, and economically the expulsion was a blow into the fabric of the empire.


title: Charles III of Spain

text: The government of Spain, in an effort to streamline the operation of its colonial empire, began introducing what became known as the Bourbon Reforms throughout South America. In 1776, as part of these reforms, it created the Viceroyalty of the Río de la Plata by separating Upper Peru (modern Bolivia) and the territory that is now Argentina from the Viceroyalty of Peru. These territories included the economically important silver mines at Potosí, whose economic benefits began to flow to Buenos Aires in the east, instead of Cuzco and Lima to the west. The economic hardship this introduced to parts of the Altiplano combined with systemic oppression of Indian and mestizo underclasses created an environment in which a large-scale uprising could occur. In 1780 an indigenous insurrection of mostly Aymara and Quechua peoples took place against the colonial rulers of the Viceroyalty of Peru, led by Túpac Amaru II. Túpac Amaru's uprising was simultaneous with the rebellion of Túpac Katari in colonial-era Upper Peru.


title: Charles III of Spain

text: Charles received the strict and structured education of a Spanish Infante; he was very pious and was often in awe of his domineering mother, who according to many contemporaries, he resembled greatly. The Alvise Giovanni Mocenigo, Doge of Venice and Ambassador of Venice to Naples declared that "...he received an education removed from all studies and all applications in order to be able to govern himself" ("...tenne sempre un'educazione lontanissima da ogni studio e da ogni applicazione per diventare da sé stesso capace di governo").


title: Charles III of Spain

text: He was educated in printmaking (remaining an enthusiastic etcher), painting, and a wide range of physical activities, including a future favourite of his, hunting. Sir Horatio Mann, a British diplomat in Florence noted that he was greatly impressed at the fondness Charles had for the sport.


title: Charles III of Spain

text: His physical appearance was dominated by the Bourbon nose that he had inherited from his father's side of the family. He was described as "a brown boy, who has a lean face with a bulging nose", and was known for his happy and exuberant character.


title: Charles III of Spain

text: Charles's mother Elisabeth Farnese sought potential brides for her son, when he was formally recognized as King of Naples and Sicily. It was impossible to get an Archduchess of Austria as a bride, so she looked to Poland, choosing Princess Maria Amalia of Saxony, a daughter of the newly elected Polish king Augustus III and his (ironically) Austrian wife Maria Josepha of Austria. Maria Josepha was a niece of Emperor Charles; the marriage was seen as the only alternative to an Austrian marriage. Maria Amalia was only 13 when she was informed of her proposed marriage. The marriage date was confirmed on 31 October 1737. Maria Amalia was married by proxy at Dresden in May 1738, with her brother Frederick Christian of Saxony representing Charles. This marriage was looked upon favorably by the Holy See and effectively ended its diplomatic disagreement with Charles. The couple met for the first time on 19 June 1738 at Portella, a village on the frontier of the kingdom near Fondi. At court, festivities lasted till 3 July. As part of the celebration, Charles created the Order of Saint Januarius—the most prestigious order of chivalry in the kingdom. He later had the Order of Charles III created in Spain on 19 September 1771.


title: Charles III of Spain

text: The first crisis that Charles had to deal with as King of Spain was the death of his beloved wife Maria Amalia. She died unexpectedly at the Palace of Buen Retiro on the eastern outskirts of Madrid, aged 35, on 27 September 1760. She was buried at the El Escorial in the royal crypt.


title: Charles III of Spain

text: The example of his actions and works was not without effect on other Spanish nobles. In his domestic life, King Charles was regular, and was a considerate master, though he had a somewhat caustic tongue and took a rather cynical view of humanity. He was passionately fond of hunting. During his later years he had some trouble with his eldest son and daughter-in-law.


title: Charles III of Spain

text: The rule of Charles III has been considered the "apogee of empire" and not sustained after his death. Charles III ascended the throne of Spain with considerable experience in governance, and enacted significant reforms to revivify Spain's economy and strengthen its empire. Although there were European conflicts to contend with, he died in 1788, months before the eruption of the French Revolution in July 1789. Charles III did not equip his son and heir, Charles IV with skills or experience in governance. Charles IV continued a number of policies of his more distinguished father, but was forced to abdicate by his son Ferdinand VII of Spain and then imprisoned by Napoleon Bonaparte who invaded Spain in 1808.


title: Charles III of Spain

text: The arms used by Charles while King of Spain were used until 1931 when his great-great-great grandson Alphonso XIII lost the crown, and the Second Spanish Republic was proclaimed (there was also a brief interruption from 1873 to 1875). Felipe VI of Spain, Spain's current monarch, is a direct male-line descendant of Charles the and a descendant by four of his great-great-grandparents. Felipe VI is also a descendant of Maria Theresa of Austria.


title: Charles III of Spain

text: Charles III University of Madrid, established in 1989 and one of the world's top 300 Universities, is named after him.


title: Buttermilk

text: Buttermilk is a fermented dairy drink. Traditionally, it was the liquid left behind after churning butter out of cultured cream. As most modern butter in western countries is not made with cultured cream but uncultured sweet cream, most modern buttermilk in western countries is cultured separately. It is common in warm climates where unrefrigerated milk sours quickly.


title: Buttermilk

text: Buttermilk can be drunk straight, and it can also be used in cooking. In making soda bread, the acid in buttermilk reacts with the raising agent, sodium bicarbonate, to produce carbon dioxide which acts as the leavening agent. Buttermilk is also used in marination, especially of chicken and pork.


title: Buttermilk

text: Originally, buttermilk referred to the liquid left over from churning butter from cultured or fermented cream. Traditionally, before the advent of homogenization, the milk was left to sit for a period of time to allow the cream and milk to separate. During this time, naturally occurring lactic acid-producing bacteria in the milk fermented it. This facilitates the butter churning process, since fat from cream with a lower pH coalesces more readily than that of fresh cream. The acidic environment also helps prevent potentially harmful microorganisms from growing, increasing shelf life.


title: Buttermilk

text: Traditional buttermilk is still common in many Arabic, Indian, Nepalese, Pakistani, Finnish, Polish, and Dutch households, but rarely found in other Western countries. It is a common drink in many Indian and Nepalese homes, and often served with roasted maize. In the Arab world, buttermilk is a common beverage to be sold ice cold with other dairy products. It is popular during Ramadan, where it is consumed during iftar and .


title: Buttermilk

text: Cultured buttermilk was first commercially introduced in the United States in the 1920s. Commercially available cultured buttermilk is milk that has been pasteurized and homogenized, and then inoculated with a culture of "Lactococcus lactis" or "Lactobacillus bulgaricus" plus "Leuconostoc citrovorum" to simulate the naturally occurring bacteria in the old-fashioned product. The tartness of cultured buttermilk is primarily due to lactic acid produced by lactic acid bacteria while fermenting lactose, the primary sugar in milk. As the bacteria produce lactic acid, the pH of the milk decreases and casein, the primary milk protein, precipitates, causing the curdling or clabbering of milk. This process makes buttermilk thicker than plain milk. While both traditional and cultured buttermilk contain lactic acid, traditional buttermilk tends to be less viscous than cultured buttermilk.


title: Buttermilk

text: When introduced in America, cultured buttermilk was popular among immigrants, and was viewed as a food that could slow aging. It reached peak annual sales of 517,000,000 kg (1.14 billion lbs.) in 1960. Buttermilk's popularity has declined since then, despite an increasing population, and annual sales in 2012 reached less than half that number.


title: Buttermilk

text: However, condensed buttermilk and dried buttermilk remain important in the food industry. Liquid buttermilk is used primarily in the commercial preparation of baked goods and cheese. Buttermilk solids are used in ice cream manufacturing, as well as being added to pancake mixes to make buttermilk pancakes.


title: Buttermilk

text: "Acidified buttermilk" is a substitute made by adding a food-grade acid such as vinegar or lemon juice to milk. It can be produced by mixing 1 tablespoon () of acid with 1 cup () of milk and letting it sit until it curdles after about 10 minutes. Any level of fat content for the milk ingredient may be used, but whole milk is usually used for baking.


title: Buttermilk

text: Commercially produced buttermilk is comparable to regular milk in terms of food energy and fat. One cup (237 mL) of whole milk contains and 8.9 grams of fat. One cup of whole buttermilk contains and 8.1 grams of total fat. Low-fat buttermilk is also available. Buttermilk contains vitamins, potassium, calcium, and traces of phosphorus.


title: Philippine Airlines

text: Philippine Airlines (PAL), a trade name of PAL Holdings, Inc. (PSE: PAL) (Philippine Air Lines until 1970), is the flag carrier airline of the Philippines. Headquartered at the PNB Financial Center in Pasay, the airline was founded in 1941 and is the first and oldest commercial airline in Asia.


title: Philippine Airlines

text: The airline's main flight operations are located at Ninoy Aquino International Airport in Metro Manila. Its subsidiary PAL Express mainly operates regional routes while PAL operates both domestic (Cebu, Davao, General Santos, Kalibo, Laoag, Manila, and Zamboanga) and international routes.


title: Philippine Airlines

text: Philippine Airlines was founded on . Its predecessor company, Philippine Aerial Taxi Company, was founded on December 3, 1930.


title: Philippine Airlines

text: Formerly one of the largest airlines in Asia, PAL was severely affected by the 1997 Asian financial crisis. In one of the Philippines' biggest corporate failures, PAL downsized its international operations by ending flights to Europe and the Middle East, cutting virtually all domestic flights except routes operated from Manila, reducing the size of its fleet, and laying off thousands of employees.


title: Philippine Airlines

text: The airline was placed under receivership in 1998 but later restored operations to many destinations. After PAL's exit from receivership in 2007, the airline has frequently revamped its management, seeking to re-establish itself as one of Asia's premier carriers.


title: Philippine Airlines

text: Philippine Airlines is owned by PAL Holdings (), a holding company responsible for the airline's operations. PAL Holdings is part of a group of companies owned by business tycoon Lucio Tan. ANA Holdings, the holding company of All Nippon Airways, has a 9.5% stake in PAL Holdings. In 2017, PAL was the ninth-largest corporation in the Philippines in terms of gross revenue, as ranked by "BusinessWorld".


title: Philippine Airlines

text: As of December 2018, PAL had 6,689 employees, including 999 pilots and 2,647 cabin crew. In 2007, PAL was the 61st largest airline in the world in terms of revenue passenger kilometers flown, with over sixteen million miles flown for 21 million available seat kilometers, an average load factor of 76 percent. PAL flew approximately twelve million passengers in 2014 and sixteen million in 2016.


title: Philippine Airlines

text: For the fiscal year ending on March 31, 2007, Philippine Airlines reported a net income of US$140.3 million, the largest profit in its 76-year history. This allowed it to exit receivership in October 2007. PAL forecasted its net profits would reach $32.32 million in the fiscal year, a profit goal of $26.28 million in 2009, and an expected profit of $47.41 million for 2010. However, these profit levels proved difficult to achieve, and the airlines announced a large loss in early 2009.


title: Philippine Airlines

text: Philippine Airlines reported a comprehensive income of $20.4 million for 2014, the company's first profit in four years. The company continued its financial turnaround, reporting a net income growth of 4430.04% for 2015. However, PAL reported a loss in 2016, with a net income growth of -38.80%.


title: Philippine Airlines

text: Like other airlines, Philippine Airlines was severely affected by the COVID-19 pandemic and its impact on the aviation industry. By 2020, Philippine Airlines reported a net loss of $1.46 billion. It later filed for Chapter 11 bankruptcy protection in the following year, but managed to complete its financial restructuring in three months, which is a must faster period than any other global airlines that underwent the same bankruptcy process. By 2022, the airline returned to profitability by reporting a net income of $71 million for the first half of that year.


title: Philippine Airlines

text: In 1989, Philippine Airlines operated to destinations in Europe, such as Amsterdam, Athens, Frankfurt, London–Gatwick and Rome–Fiumicino. Following the 1997 Asian financial crisis, PAL dropped some of its long-haul flights, with Cathay Pacific temporarily taking over PAL's routes for fourteen days in 1998.


title: Philippine Airlines

text: In 2010, the European Union (EU) banned Philippine carriers until 2013, despite the safety assessment by the International Civil Aviation Organization. After the EU lifted its ban, Philippine Airlines resumed its flights to Europe, including Amsterdam, London, Madrid, Paris, and Rome. On November 4, 2013, after a fifteen-year absence, PAL returned to London–Heathrow using a Boeing 777-300ER. On October 29, 2018, the airline launched a non-stop flight to New York–JFK using an Airbus A350-900; this was the ninth longest flight in the world. On April 29, 2021, Philippine Airlines announced its plans to fly to Israel by October, using its Airbus A350-900. The airline previously operated in Israel from the mid-1940s to the 1950s.


title: Philippine Airlines

text: As of 2021, Philippine Airlines operates 43 international routes and 31 domestic routes. Its subsidiary, PAL Express, serves most domestic flights in the Philippines.


title: Philippine Airlines

text: , there are 45 aircraft registered in the PAL mainline fleet, including a mix of Airbus narrow-body and wide-body jets and Boeing wide-body jets.


title: Philippine Airlines

text: The Philippine Airlines logo changed four times since the company's founding. Its first logo incorporated a blue oval with "PAL" superimposed in white letters, a four-pointed star with points intersecting behind the "A" in PAL, and a wing whose orientation varied depending on the location of the logo. The wing pointed to the right if located on the left side of the plane, and to the left if on the right side. A variant of this logo featured a globe instead of the blue oval with the superimposed PAL initials. It was used from the 1950s through the mid-1960s.


title: Philippine Airlines

text: The second logo adopted a blue triangle with the bottom point missing and a red triangle superimposed upon it, enclosed by a circle. This was meant to evoke a vertically-displayed national flag with white forming by the negative space between the two triangles. In the mid-1970s, a third logo was introduced that removed the circle and simplified the shape. The typeface used in the third logo was later applied to the second logo and remained the official PAL logo until 1986.


title: Philippine Airlines

text: The current PAL logo features the same two blue and red sail triangles used in the second and third logos. An eight-rayed yellow sunburst was superimposed on top of the blue triangle, and a new Helvetica typeface was used.


title: Philippine Airlines

text: PAL liveries have undergone many incarnations. The first PAL aircraft bore a simple white-top, silver-bottom livery separated by solid straight cheatlines, with a small Philippine flag superimposed on the tail. The name "Philippine Air Lines" was superimposed on the upper forward portion of the fuselage and the PAL logo was located at the back. Later variants of the livery, especially on PAL jet aircraft, made use of an extended Philippine flag as cheatlines, with the PAL logo superimposed on the tail. By this time, the name "Philippine Airlines" was used in the livery.


title: Philippine Airlines

text: Another variant of the original livery used by PAL is somewhat similar to the current livery. However, it uses PAL's third logo on the tail with blue, white, and red cheatlines running the center of the fuselage. Later on, the bottom half of the fuselage was also painted white


title: Philippine Airlines

text: The current "Eurowhite" livery, first used with the Short 360, was adopted in 1986 following PAL's corporate rebranding. This livery, (designed by Landor Associates) has "Philippines" superimposed on the forward portion of the fuselage in italics (using the PAL logo typeface), while the tail is painted with the logo and the Philippine flag is visible near the rear of the aircraft. The PAL logo is also painted on the winglets of aircraft that have them. The name "Philippines", instead of "Philippine Airlines", is to denote that PAL is the primary flag carrier of the Philippines. However, this sometimes leads to confusion that a PAL plane, especially when chartered by the President for official or state visits, is, in fact, the official air transport of the Philippine head of state. Any PAL aircraft with the flight number "PR/PAL 001" and the callsign "PHILIPPINE ONE" is a special plane operated by Philippine Airlines to transport the President or Vice President of the Philippines. The flight number "PR/PAL 002" and the callsign "PHILIPPINE TWO" are used if the Vice President travels simultaneously with the President. As such, the presidential seals are patched on or near the L1 and R1 doors of any PAL aircraft chartered by the president.


title: Philippine Airlines

text: For the airline's 70th anniversary in 2011, a special decal was placed on all of its aircraft. The sticker featured a stylized "70" and the words, "Asia's first, shining through". For its 75th anniversary in 2016, a special decal was put on the back of every aircraft. The sticker features a stylized "75".PAL also placed a 4-star Skytrax sticker on its aircraft to celebrate its new rating.


title: Philippine Airlines

text: In February 2019, Philippine Airlines rolled out its fifth Airbus A350 aircraft with a LoveBus decal that represents the 40th anniversary of its partnership with Airbus and to celebrate the airline's signature heartfelt service. The kiss-marked LoveBus logo was also placed in 1979 on one of PAL's Airbus A300 that represented their first year of partnership with Airbus. Its LoveBus A350 was rolled out from the paint shop and was accepted on February 14, 2019, Valentine's Day. PAL took delivery of it three days later, and held a welcoming ceremony at Ninoy Aquino International Airport Terminal 2.


title: Philippine Airlines

text: Mabuhay Miles is Philippine Airlines' frequent-flyer program. It was established in 2002 after merging all of PAL's existing frequent flyer programs before the Asian financial crisis. Mabuhay Miles are divided into multiple tiers: Classic, Elite, Premiere Elite, and Million Miler. Philippine National Bank issues co-branded credit cards, debit cards, and prepaid cards that offer benefits such as free mileage points, travel insurance, priority check-in, access to a Mabuhay Lounge, and discounts when booking flights on the Philippine Airlines website.


title: Philippine Airlines

text: The Mabuhay Lounge is the airport lounge for Philippine Airlines. Business Class and Elite Members of Mabuhay Miles can use the lounge. These lounges have open bars, food catering, Wi-Fi, and charging ports for personal electronic devices. On June 27, 2018, Philippine Airlines unveiled a new Mabuhay Lounge for international business class, million millers, premier elite, and elite passengers at Terminal 2 of Mactan–Cebu International Airport.


title: Philippine Airlines

text: The Mabuhay Lounge can be found at the following airports: Ninoy Aquino International Airport Terminal 2, Puerto Princesa International Airport, Bacolod–Silay Airport, Mactan–Cebu International Airport, Laguindingan Airport, Francisco Bangoy International Airport, Iloilo International Airport, and San Francisco International Airport.


title: Philippine Airlines

text: Philippine Airlines currently offers three classes of service: Business, Premium Economy, and Economy depending on the aircraft. Its Airbus A330s and Airbus A350s offer three classes, while its other aircraft offer two classes. On September 15, 2022, the airline rebranded its premium economy service to Comfort Class for its domestic flights.


title: Philippine Airlines

text: In 2017, PAL reconfigured the cabin layout of eight A330s, from a single class 414-seater into a 309-seater tri-class with Business, Premium Economy, and Economy sections. The reconfigured A330s were rolled out within seven months in 2017. The IFEs were Zodiac's RAVE system. The seats were designed by Lift Strategic Design, and Lufthansa Technik Philippines performed the reconfiguration.


title: Philippine Airlines

text: The previous aircraft acquired had no embedded in-flight entertainment (IFE), except for Boeing 777-300ERs. Instead, they offered rentable (Economy) or complementary (Business) iPad Minis with OnAir's wireless IFE solution, OnAir Play. New aircraft have embedded IFE, due to the failure of PAL's wireless IFE program for long-haul flights. They still offer wireless IFE on all aircraft.


title: Philippine Airlines

text: Business Class is available on all aircraft. It offers increased legroom and lie-flat seats on the Airbus A330, Boeing 777, Airbus A350, select Airbus A321neo. Philippine Airlines is the only Philippine carrier to offer business class on domestic flights. On medium-haul and long-haul flights, Philippine Airlines provides amenity kits from L'Occitane en Provence.


title: Philippine Airlines

text: Seats on the early Boeing 777s feature angled-flat seats manufactured by Recaro, while some of the latest aircraft feature lie-flat seats by Zodiac Aerospace (now Safran), arranged in a 2-3-2 configuration. Reconfigured Airbus A330s and the Airbus A350 feature lie-flat seats manufactured by Thompson Aero Seating, in a seating configuration of 1-2-1. A350 and A330 Business Class seats also feature Lantal air cushions, a four-way headrest, a storage shelf for personal belongings, a headphone hook, and a padded inner shell that absorbs noise.


title: Philippine Airlines

text: Seats feature 15-inch (Boeing 777), 18.5-inch (Airbus A330 and Airbus A350), and 15.4-inch (Airbus A321neo) personal in-flight entertainment monitors with video on demand, as well as in-seat power. The seats feature a USB port where passengers can charge mobile devices. Passengers are also given noise-cancelling headphones. Select Airbus A321neos also feature lie-flat seats manufactured by Rockwell Collins (now Collins Aerospace), arranged in a 2-2 layout. They have a 60-inch seat pitch and a maximum seat width of 23 inches when fully flat. Business Class seats on Airbus A321ceos recline and have a seat pitch of . They feature a laptop power supply (both AC and USB).


title: Philippine Airlines

text: Premium Economy (known as Comfort Class for domestic flights since September 2022) is available on the Airbus A330 and Airbus A350, as well as PAL Express flights using two-class Airbus A320s . The seats are similar in design to standard economy class seats but feature at least four to five inches of additional legroom, providing a minimum legroom of 34 to 36 inches.


title: Philippine Airlines

text: Reconfigured A330s, as well as A350s, have a different seat design with a layer of extra padding. It is 9.55" wide, pitched at 38 inches with eight inches of recline. The seats also feature a 13.3-inch screen-flight entertainment unit with video on demand, a headphone jack, a USB port, and in-seat power. On aircraft without IFE, passengers are also treated to complimentary iPads.


title: Philippine Airlines

text: Economy class is available on all aircraft. Tray tables are found in the back of the seat in front, except for bulkhead and exit seats, where tray tables are embedded in the seats. They have four-way headrests. The seats offer a pitch that varies between 31 and 34 inches. Each seat offers video on demand and is equipped with 9-inch (Boeing 777) or 10-inch (Airbus A330, Airbus A321neo) monitors with a headphone jack and USB port for charging. They have four-way headrests and six inches of recline. An articulating seat bottom cushion comes with extra foam under the seat cover.


title: Philippine Airlines

text: Although Philippine Airlines aircraft have been involved in a string of accidents since its founding in 1941, the majority of airline accidents have occurred with propeller aircraft during the airline's early years of operations. Few PAL jet aircraft have been involved in accidents, the most notable being the explosion onboard Philippine Airlines Flight 434, masterminded by al-Qaeda through Project Bojinka.


title: Philippine Airlines

text: In February 2007, PAL was the first airline in the Philippines to be accredited by the International Air Transport Association, passing the IATA Operational Safety Audit (IOSA). Philippine Airlines has the highest safety rating of 7/7 according to AirlineRatings.com and was rated a safer airline than some of its Southeast Asian counterparts.


title: Philippine Airlines

text: PAL experienced significant financial losses in the late 2000s. On March 31, 2006, PAL's consolidated total assets amounted to 100,984,477 PHP, an 11% decrease from March 31, 2005. On March 31, 2007, the company's consolidated assets continued to diminish by 8%, an amount equivalent to 92,837,849 PHP, compared to 2006 figures. The decline of PAL's assets was primarily due to a net decrease in property and equipment and advance payments to aircraft and engine manufacturers. On March 31, 2007, other current and noncurrent assets fell by 29% to 2,960.4 million PHP and by 20% to 2,941.7 million PHP "due to the effect of re-measurement to the fair value of certain financial assets and derivative instruments". After carrying 17% more passengers in 2009 due to acquisition of additional aircraft and growth in the local market, PAL annual income report showed an increase in revenues of US$1.634 billion from US$1.504 billion in 2008. Despite this, PAL expenses escalated as a result of more flight operations and higher maintenance costs aggravated by fuel price fluctuations; forty-four percent (44%) of PAL income operating expenditures is utilized for fuel consumption.


title: Philippine Airlines

text: PAL has a history of labor relations problems. On June 15, 1998, PAL retrenched 5,000 employees, including more than 1,400 flight attendants and stewards to reduce costs and alleviate the financial downturn in the airline industry. Represented by the Flight Attendants and Stewards Association of the Philippines (FASAP), retrenched employee sought remedy through the judicial process, filing a complaint on the grounds of unfair labor practices and illegal retrenchment. It took a decade before the suit was finally settled. It passed the Labor Arbiter to the National Labor Relations Commission, then on to the Court of Appeals, and the Supreme Court. The Philippine Highest Tribunal favored the aggrieved party and on July 22, 2008, in its 32-page decision ordered PAL to: Reinstate the cabin crew personnel who were covered by the retrenchment of and demotion scheme of June 15, 1998, made effective on July 15, 1998, without loss of seniority right and other privileges, and to pay them full back wages, inclusive of allowances and other monetary benefits computed from the time of their separation up to the time of actual reinstatement, provided that those who have received their respective separation pay, the number of payments shall be deducted from their back wages. The Supreme Court ruling explained that there was a failure on the part of PAL to substantiate its claims of actual and imminent substantial losses. Although the Asian financial downturn severely affected the airline, PAL's defense of bankruptcy and rehabilitation was considered untenable; hence, the retrenchment policy was deemed unjustified. On March 26, 2018, the Supreme Court en banc voted in favor of Philippine Airlines, which affirms the 2006 Court of Appeals decision that says Philippine Airlines is not required to consult FASAP for its criteria for its retrenchment program.


title: Philippine Airlines

text: For more than twenty years, PAL monopolized the air transport industry in the Philippines. This ended in 1995 through the passage of Executive Order No. 219 which permitted the entry of new airlines into the industry. The liberalization and deregulation of the Philippine airline industry brought competition into the domestic air transport industry resulting in lower airfares, improvements in the quality of services, and efficiency in the industry in general. At present, three airlines are competing in international and major domestic routes: PAL, Cebu Pacific, and PAL Express. Several airlines serve the minor and short-distance routes including Philippines AirAsia and Cebgo.


title: Halting problem

text: In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program–input pairs cannot exist.


title: Halting problem

text: For any program that might determine whether programs halt, a "pathological" program , called with some input, can pass its own source and its input to "f" and then specifically do the opposite of what "f" predicts "g" will do. No "f" can exist that handles this case. A key part of the proof is a mathematical definition of a computer and program, which is known as a Turing machine; the halting problem is "undecidable" over Turing machines. It is one of the first cases of decision problems proven to be unsolvable. This proof is significant to practical computing efforts, defining a class of applications which no programming invention can possibly perform perfectly.


title: Halting problem

text: Jack Copeland attributes the introduction of the term "halting problem" to the work of Martin Davis in the 1950s.


title: Halting problem

text: The halting problem is a decision problem about properties of computer programs on a fixed Turing-complete model of computation, i.e., all programs that can be written in some given programming language that is general enough to be equivalent to a Turing machine. The problem is to determine, given a program and an input to the program, whether the program will eventually halt when run with that input. In this abstract framework, there are no resource limitations on the amount of memory or time required for the program's execution; it can take arbitrarily long and use an arbitrary amount of storage space before halting. The question is simply whether the given program will ever halt on a particular input.


title: Halting problem

text: While deciding whether these programs halt is simple, more complex programs prove problematic. One approach to the problem might be to run the program for some number of steps and check if it halts. But if the program does not halt, it is unknown whether the program will eventually halt or run forever. Turing proved no algorithm exists that always correctly decides whether, for a given arbitrary program and input, the program halts when run with that input. The essence of Turing's proof is that any such algorithm can be made to produce contradictory output and therefore cannot be correct.


title: Halting problem

text: Some infinite loops can be quite useful. For instance, event loops are typically coded as infinite loops. However, most subroutines are intended to finish. In particular, in hard real-time computing, programmers attempt to write subroutines that are not only guaranteed to finish, but are also guaranteed to finish before a given deadline


title: Halting problem

text: but attempt to write in a restricted style—such as MISRA C or SPARK—that makes it easy to prove that the resulting subroutines finish before the given deadline.


title: Halting problem

text: Other times these programmers apply the rule of least power—they deliberately use a computer language that is not quite fully Turing-complete. Frequently, these are languages that guarantee all subroutines finish, such as Coq.


title: Halting problem

text: The difficulty in the halting problem lies in the requirement that the decision procedure must work for all programs and inputs. A particular program either halts on a given input or does not halt. Consider one algorithm that always answers "halts" and another that always answers "does not halt". For any specific program and input, one of these two algorithms answers correctly, even though nobody may know which one. Yet neither algorithm solves the halting problem generally.


title: Halting problem

text: There are programs (interpreters) that simulate the execution of whatever source code they are given. Such programs can demonstrate that a program does halt if this is the case: the interpreter itself will eventually halt its simulation, which shows that the original program halted. However, an interpreter will not halt if its input program does not halt, so this approach cannot solve the halting problem as stated; it does not successfully answer "does not halt" for programs that do not halt.


title: Halting problem

text: The halting problem is theoretically decidable for linear bounded automata (LBAs) or deterministic machines with finite memory. A machine with finite memory has a finite number of configurations, and thus any deterministic program on it must eventually either halt or repeat a previous configuration:


title: Halting problem

text: However, a computer with a million small parts, each with two states, would have at least 2 possible states:


title: Halting problem

text: It can also be decided automatically whether a nondeterministic machine with finite memory halts on none, some, or all of the possible sequences of nondeterministic decisions, by enumerating states after each possible decision.


title: Halting problem

text: The halting problem is historically important because it was one of the first problems to be proved undecidable. In April 1936, Alonzo Church published his proof of the undecidability of a problem in the lambda calculus. Turing's proof was published later, in January 1937. Since then, many other undecidable problems have been described.


title: Halting problem

text: In none of his work did Turing use the word "halting" or "termination". Turing's biographer Hodges does not have the word "halting" or words "halting problem" in his index. The earliest recorded use of the words "halting problem" is in a proof by Davis in 1958:


title: Halting problem

text: Davis adds no attribution for his proof, so one infers that it is original with him. But Davis has said that Kleene stated the proof informally. Copeland states that:


title: Halting problem

text: In his original proof Turing formalized the concept of "algorithm" by introducing Turing machines. However, the result is in no way specific to them; it applies equally to any other model of computation that is equivalent in its computational power to Turing machines, such as Markov algorithms, Lambda calculus, Post systems, register machines, or tag systems.


title: Halting problem

text: What is important is that the formalization allows a straightforward mapping of algorithms to some data type that the algorithm can operate upon. For example, if the formalism lets algorithms define functions over strings (such as Turing machines) then there should be a mapping of these algorithms to strings, and if the formalism lets algorithms define functions over natural numbers (such as computable functions) then there should be a mapping of algorithms to natural numbers. The mapping to strings is usually the most straightforward, but strings over an alphabet with "n" characters can also be mapped to numbers by interpreting them as numbers in an "n"-ary numeral system.


title: Halting problem

text: The conventional representation of decision problems is the set of objects possessing the property in question. The "halting set"


title: Halting problem

text: This set is recursively enumerable, which means there is a computable function that lists all of the pairs ("i", "x") it contains. However, the complement of this set is not recursively enumerable.


title: Halting problem

text: There are many equivalent formulations of the halting problem; any set whose Turing degree equals that of the halting problem is such a formulation. Examples of such sets include:


title: Halting problem

text: Christopher Strachey outlined a proof by contradiction that the halting problem is not solvable. The proof proceeds as follows: Suppose that there exists a total computable function "halts(f)" that returns true if the subroutine "f" halts (when run with no inputs) and returns false otherwise. Now consider the following subroutine:


title: Halting problem

text: "halts(g)" must either return true or false, because "halts" was assumed to be total. If "halts(g)" returns true, then "g" will call "loop_forever" and never halt, which is a contradiction. If "halts(g)" returns false, then "g" will halt, because it will not call "loop_forever"; this is also a contradiction. Overall, "g" does the opposite of what "halts" says "g" should do, so "halts(g)" can not return a truth value that is consistent with whether "g" halts. Therefore, the initial assumption that "halts" is a total computable function must be false.


title: Halting problem

text: The concept above shows the general method of the proof, but the computable function "halts" does not directly take a subroutine as an argument; instead it takes the source code of a program. Moreover, the definition of "g" is self-referential. A rigorous proof addresses these issues. The overall goal is to show that there is no total computable function that decides whether an arbitrary program "i" halts on arbitrary input "x"; that is, the following function "h" (for "halts") is not computable:


title: Halting problem

text: Here "program i" refers to the "i" th program in an enumeration of all the programs of a fixed Turing-complete model of computation.


title: Halting problem

text: The proof proceeds by directly establishing that no total computable function with two arguments can be the required function "h". As in the sketch of the concept, given any total computable binary function "f", the following partial function "g" is also computable by some program "e":


title: Halting problem

text: Because "g" is partial computable, there must be a program "e" that computes "g", by the assumption that the model of computation is Turing-complete. This program is one of all the programs on which the halting function "h" is defined. The next step of the proof shows that "h"("e","e") will not have the same value as "f"("e","e").


title: Halting problem

text: In either case, "f" cannot be the same function as "h". Because "f" was an "arbitrary" total computable function with two arguments, all such functions must differ from "h".


title: Halting problem

text: This proof is analogous to Cantor's diagonal argument. One may visualize a two-dimensional array with one column and one row for each natural number, as indicated in the table above. The value of "f"("i","j") is placed at column "i", row "j". Because "f" is assumed to be a total computable function, any element of the array can be calculated using "f". The construction of the function "g" can be visualized using the main diagonal of this array. If the array has a 0 at position ("i","i"), then "g"("i") is 0. Otherwise, "g"("i") is undefined. The contradiction comes from the fact that there is some column "e" of the array corresponding to "g" itself. Now assume "f" was the halting function "h", if "g"("e") is defined ("g"("e") = 0 in this case), "g"("e") halts so "f"("e,e") = 1. But "g"("e") = 0 only when "f"("e,e") = 0, contradicting "f"("e,e") = 1. Similarly, if "g"("e") is not defined, then halting function "f"("e,e") = 0, which leads to "g"("e") = 0 under "g"'s construction. This contradicts the assumption of "g"("e") not being defined. In both cases contradiction arises. Therefore any arbitrary computable function "f" cannot be the halting function "h".


title: Halting problem

text: For example, there cannot be a general algorithm that decides whether a given statement about natural numbers is true or false. The reason for this is that the proposition stating that a certain program will halt given a certain input can be converted into an equivalent statement about natural numbers. If an algorithm could find the truth value of every statement about natural numbers, it could certainly find the truth value of this one; but that would determine whether the original program halts.


title: Halting problem

text: Rice's theorem generalizes the theorem that the halting problem is unsolvable. It states that for "any" non-trivial property, there is no general decision procedure that, for all programs, decides whether the partial function implemented by the input program has that property. (A partial function is a function which may not always produce a result, and so is used to model programs, which can either produce results or fail to halt.) For example, the property "halt for the input 0" is undecidable. Here, "non-trivial" means that the set of partial functions that satisfy the property is neither the empty set nor the set of all partial functions. For example, "halts or fails to halt on input 0" is clearly true of all partial functions, so it is a trivial property, and can be decided by an algorithm that simply reports "true." Also, this theorem holds only for properties of the partial function implemented by the program; Rice's Theorem does not apply to properties of the program itself. For example, "halt on input 0 within 100 steps" is "not" a property of the partial function that is implemented by the program—it is a property of the program implementing the partial function and is very much decidable.


title: Halting problem

text: Gregory Chaitin has defined a halting probability, represented by the symbol Ω, a type of real number that informally is said to represent the probability that a randomly produced program halts. These numbers have the same Turing degree as the halting problem. It is a normal and transcendental number which can be defined but cannot be completely computed. This means one can prove that there is no algorithm which produces the digits of Ω, although its first few digits can be calculated in simple cases.


title: Halting problem

text: Since the negative answer to the halting problem shows that there are problems that cannot be solved by a Turing machine, the Church–Turing thesis limits what can be accomplished by any machine that implements effective methods. However, not all machines conceivable to human imagination are subject to the Church–Turing thesis (e.g. oracle machines). It is an open question whether there can be actual deterministic physical processes that, in the long run, elude simulation by a Turing machine, and in particular whether any such hypothetical process could usefully be harnessed in the form of a calculating machine (a hypercomputer) that could solve the halting problem for a Turing machine amongst other things. It is also an open question whether any such unknown physical processes are involved in the working of the human brain, and whether humans can solve the halting problem.


title: Halting problem

text: Turing's proof shows that there can be no mechanical, general method (i.e., a Turing machine or a program in some equivalent model of computation) to determine whether algorithms halt. However, each individual instance of the halting problem has a definitive answer, which may or may not be practically computable. Given a specific algorithm and input, one can often show that it halts or does not halt, and in fact computer scientists often do just that as part of a correctness proof. There are some heuristics that can be used in an automated fashion to attempt to construct a proof, which frequently succeed on typical programs. This field of research is known as automated termination analysis.


title: Halting problem

text: Some results have been established on the theoretical performance of halting problem heuristics, in particular the fraction of programs of a given size that may be correctly classified by a recursive algorithm. These results do not give precise numbers because the fractions are uncomputable and also highly dependent on the choice of program encoding used to determine "size". For example, consider classifying programs by their number of states and using a specific "Turing semi-infinite tape" model of computation that errors (without halting) if the program runs off the left side of the tape. Then formula_3, over programs formula_4 chosen uniformly by number of states. But this result is in some sense "trivial" because these decidable programs are simply the ones that fall off the tape, and the heuristic is simply to predict not halting due to error. Thus a seemingly irrelevant detail, namely the treatment of programs with errors, can turn out to be the deciding factor in determining the fraction of programs.


title: Halting problem

text: To avoid these issues, several restricted notions of the "size" of a program have been developed. A dense Gödel numbering assigns numbers to programs such that each computable function occurs a positive fraction in each sequence of indices from 1 to n, i.e. a Gödelization φ is dense iff for all formula_5, there exists a formula_6 such that formula_7. For example, a numbering that assigns indexes formula_8 to nontrivial programs and all other indices the error state is not dense, but there exists a dense Gödel numbering of syntactically correct Brainfuck programs. A dense Gödel numbering is called optimal if, for any other Gödel numbering formula_9, there is a 1-1 total recursive function formula_10 and a constant formula_11 such that for all formula_5, formula_13 and formula_14. This condition ensures that all programs have indices not much larger than their indices in any other Gödel numbering. Optimal Gödel numberings are constructed by numbering the inputs of a universal Turing machine. A third notion of size uses universal machines operating on binary strings and measures the length of the string needed to describe the input program. A universal machine "U" is a machine for which every other machine "V" there exists a total computable function h such that formula_15. An optimal machine is a universal machine that achieves the Kolmogorov complexity invariance bound, i.e. for every machine "V", there exists "c" such that for all outputs "x", if a "V"-program of length "n" outputs "x", then there exists a "U"-program of at most length formula_16 outputting "x".


title: Halting problem

text: We consider partial computable functions (algorithms) formula_17. For each formula_18 we consider the fraction formula_19 of errors among all programs of size metric at most formula_18, counting each program formula_4 for which formula_17 fails to terminate, produces a "don't know" answer, or produces a wrong answer, i.e. formula_4 halts and formula_24 outputs codice_3, or formula_4 does not halt and formula_24 outputs codice_4. The behavior may be described as follows, for dense Gödelizations and optimal machines:


title: Halting problem

text: The complex nature of these bounds is due to the oscillatory behavior of formula_19. There are infrequently occurring new varieties of programs that come in arbitrarily large "blocks", and a constantly growing fraction of repeats. If the blocks of new varieties are fully included, the error rate is at least formula_35, but between blocks the fraction of correctly categorized repeats can be arbitrarily high. In particular a "tally" heuristic that simply remembers the first N inputs and recognizes their equivalents allows reaching an arbitrarily low error rate infinitely often.


title: Halting problem

text: Many variants of the halting problem can be found in computability textbooks. Typically, these problems are RE-complete and describe sets of complexity formula_36 in the arithmetical hierarchy, the same as the standard halting problem. The variants are thus undecidable, and the standard halting problem reduces to each variant and vice-versa. However, some variants have a higher degree of unsolvability and cannot be reduced to the standard halting problem. The next two examples are common.


title: Halting problem

text: The "universal halting problem", also known (in recursion theory) as "totality", is the problem of determining whether a given computer program will halt "for every input" (the name "totality" comes from the equivalent question of whether the computed function is total).


title: Halting problem

text: This problem is not only undecidable, as the halting problem is, but highly undecidable. In terms of the arithmetical hierarchy, it is formula_37-complete.


title: Halting problem

text: There are many programs that, for some inputs, return a correct answer to the halting problem, while for other inputs they do not return an answer at all.


title: Halting problem

text: However the problem "given program "p", is it a partial halting solver" (in the sense described) is at least as hard as the halting problem.


title: Halting problem

text: To see this, assume that there is an algorithm PHSR ("partial halting solver recognizer") to do that. Then it can be used to solve the halting problem,


title: Halting problem

text: To test whether input program "x" halts on "y", construct a program "p" that on input ("x","y") reports "true" and diverges on all other inputs.


title: Halting problem

text: harder problems such as "halting on all inputs" can also be reduced, implying that PHS recognition is not only undecidable, but higher in the arithmetical hierarchy, specifically formula_37-complete.


title: Halting problem

text: A "lossy Turing machine" is a Turing machine in which part of the tape may non-deterministically disappear. The halting problem is decidable for a lossy Turing machine but non-primitive recursive.


title: Halting problem

text: A machine with an oracle for the halting problem can determine whether particular Turing machines will halt on particular inputs, but they cannot determine, in general, whether machines equivalent to themselves will halt.


title: MD5

text: The MD5 message-digest algorithm is a widely used hash function producing a 128-bit hash value. MD5 was designed by Ronald Rivest in 1991 to replace an earlier hash function MD4, and was specified in 1992 as RFC 1321.


title: MD5

text: MD5 can be used as a checksum to verify data integrity against unintentional corruption. Historically it was widely used as a cryptographic hash function; however it has been found to suffer from extensive vulnerabilities. It remains suitable for other non-cryptographic purposes, for example for determining the partition for a particular key in a partitioned database, and may be preferred due to lower computational requirements than more recent Secure Hash Algorithms.


title: MD5

text: MD5 is one in a series of message digest algorithms designed by Professor Ronald Rivest of MIT (Rivest, 1992). When analytic work indicated that MD5's predecessor MD4 was likely to be insecure, Rivest designed MD5 in 1991 as a secure replacement. (Hans Dobbertin did indeed later find weaknesses in MD4.)


title: MD5

text: In 1993, Den Boer and Bosselaers gave an early, although limited, result of finding a "pseudo-collision" of the MD5 compression function; that is, two different initialization vectors that produce an identical digest.


title: MD5

text: In 1996, Dobbertin announced a collision of the compression function of MD5 (Dobbertin, 1996). While this was not an attack on the full MD5 hash function, it was close enough for cryptographers to recommend switching to a replacement, such as SHA-1 (also compromised) or RIPEMD-160.


title: MD5

text: The size of the hash value (128 bits) is small enough to contemplate a birthday attack. MD5CRK was a distributed project started in March 2004 to demonstrate that MD5 is practically insecure by finding a collision using a birthday attack.


title: MD5

text: MD5CRK ended shortly after 17 August 2004, when collisions for the full MD5 were announced by Xiaoyun Wang, Dengguo Feng, Xuejia Lai, and Hongbo Yu. Their analytical attack was reported to take only one hour on an IBM p690 cluster.


title: MD5

text: On 1 March 2005, Arjen Lenstra, Xiaoyun Wang, and Benne de Weger demonstrated construction of two X.509 certificates with different public keys and the same MD5 hash value, a demonstrably practical collision. The construction included private keys for both public keys. A few days later, Vlastimil Klima described an improved algorithm, able to construct MD5 collisions in a few hours on a single notebook computer. On 18 March 2006, Klima published an algorithm that could find a collision within one minute on a single notebook computer, using a method he calls tunneling.


title: MD5

text: In 2009, the United States Cyber Command used an MD5 hash value of their mission statement as a part of their official emblem.


title: MD5

text: On 24 December 2010, Tao Xie and Dengguo Feng announced the first published single-block (512-bit) MD5 collision. (Previous collision discoveries had relied on multi-block attacks.) For "security reasons", Xie and Feng did not disclose the new attack method. They issued a challenge to the cryptographic community, offering a US$10,000 reward to the first finder of a different 64-byte collision before 1 January 2013. Marc Stevens responded to the challenge and published colliding single-block messages as well as the construction algorithm and sources.


title: MD5

text: In 2011 an informational RFC 6151 was approved to update the security considerations in MD5 and HMAC-MD5.


title: MD5

text: One basic requirement of any cryptographic hash function is that it should be computationally infeasible to find two distinct messages that hash to the same value. MD5 fails this requirement catastrophically; such collisions can be found in seconds on an ordinary home computer. On 31 December 2008, the CMU Software Engineering Institute concluded that MD5 was essentially "cryptographically broken and unsuitable for further use". The weaknesses of MD5 have been exploited in the field, most infamously by the Flame malware in 2012. , MD5 continues to be widely used, despite its well-documented weaknesses and deprecation by security experts.


title: MD5

text: The security of the MD5 hash function is severely compromised. A collision attack exists that can find collisions within seconds on a computer with a 2.6 GHz Pentium 4 processor (complexity of 2). Further, there is also a chosen-prefix collision attack that can produce a collision for two inputs with specified prefixes within seconds, using off-the-shelf computing hardware (complexity 2).


title: MD5

text: The ability to find collisions has been greatly aided by the use of off-the-shelf GPUs. On an NVIDIA GeForce 8400GS graphics processor, 16–18 million hashes per second can be computed. An NVIDIA GeForce 8800 Ultra can calculate more than 200 million hashes per second.


title: MD5

text: These hash and collision attacks have been demonstrated in the public in various situations, including colliding document files and digital certificates. As of 2015, MD5 was demonstrated to be still quite widely used, most notably by security research and antivirus companies.


title: MD5

text: As of 2019, one quarter of widely used content management systems were reported to still use MD5 for password hashing.


title: MD5

text: In 1996, a flaw was found in the design of MD5. While it was not deemed a fatal weakness at the time, cryptographers began recommending the use of other algorithms, such as SHA-1, which has since been found to be vulnerable as well.


title: MD5

text: In 2004 it was shown that MD5 is not collision-resistant. As such, MD5 is not suitable for applications like SSL certificates or digital signatures that rely on this property for digital security. Researchers additionally discovered more serious flaws in MD5, and described a feasible collision attack -- a method to create a pair of inputs for which MD5 produces identical checksums. Further advances were made in breaking MD5 in 2005, 2006, and 2007. In December 2008, a group of researchers used this technique to fake SSL certificate validity.


title: MD5

text: As of 2010, the CMU Software Engineering Institute considers MD5 "cryptographically broken and unsuitable for further use", and most U.S. government applications now require the SHA-2 family of hash functions. In 2012, the Flame malware exploited the weaknesses in MD5 to fake a Microsoft digital signature.


title: MD5

text: In 1996, collisions were found in the compression function of MD5, and Hans Dobbertin wrote in the RSA Laboratories technical newsletter, "The presented attack does not yet threaten practical applications of MD5, but it comes rather close ... in the future MD5 should no longer be implemented ... where a collision-resistant hash function is required."


title: MD5

text: In 2005, researchers were able to create pairs of PostScript documents and X.509 certificates with the same hash. Later that year, MD5's designer Ron Rivest wrote that "md5 and sha1 are both clearly broken (in terms of collision-resistance)".


title: MD5

text: On 30 December 2008, a group of researchers announced at the 25th Chaos Communication Congress how they had used MD5 collisions to create an intermediate certificate authority certificate that appeared to be legitimate when checked by its MD5 hash. The researchers used a PS3 cluster at the EPFL in Lausanne, Switzerland to change a normal SSL certificate issued by RapidSSL into a working CA certificate for that issuer, which could then be used to create other certificates that would appear to be legitimate and issued by RapidSSL. VeriSign, the issuers of RapidSSL certificates, said they stopped issuing new certificates using MD5 as their checksum algorithm for RapidSSL once the vulnerability was announced. Although Verisign declined to revoke existing certificates signed using MD5, their response was considered adequate by the authors of the exploit (Alexander Sotirov, Marc Stevens, Jacob Appelbaum, Arjen Lenstra, David Molnar, Dag Arne Osvik, and Benne de Weger). Bruce Schneier wrote of the attack that "we already knew that MD5 is a broken hash function" and that "no one should be using MD5 anymore". The SSL researchers wrote, "Our desired impact is that Certification Authorities will stop using MD5 in issuing new certificates. We also hope that use of MD5 in other applications will be reconsidered as well."


title: MD5

text: In 2012, according to Microsoft, the authors of the Flame malware used an MD5 collision to forge a Windows code-signing certificate.


title: MD5

text: MD5 uses the Merkle–Damgård construction, so if two prefixes with the same hash can be constructed, a common suffix can be added to both to make the collision more likely to be accepted as valid data by the application using it. Furthermore, current collision-finding techniques allow to specify an arbitrary "prefix": an attacker can create two colliding files that both begin with the same content. All the attacker needs to generate two colliding files is a template file with a 128-byte block of data, aligned on a 64-byte boundary, that can be changed freely by the collision-finding algorithm. An example MD5 collision, with the two messages differing in 6 bits, is:


title: MD5

text: The difference between the two samples is that the leading bit in each nibble has been flipped. For example, the 20th byte (offset 0x13) in the top sample, 0x87, is 10000111 in binary. The leading bit in the byte (also the leading bit in the first nibble) is flipped to make 00000111, which is 0x07, as shown in the lower sample.


title: MD5

text: Later it was also found to be possible to construct collisions between two files with separately chosen prefixes. This technique was used in the creation of the rogue CA certificate in 2008. A new variant of parallelized collision searching using MPI was proposed by Anton Kuznetsov in 2014, which allowed finding a collision in 11 hours on a computing cluster.


title: MD5

text: In April 2009, an attack against MD5 was published that breaks MD5's preimage resistance. This attack is only theoretical, with a computational complexity of 2 for full preimage.


title: MD5

text: MD5 digests have been widely used in the software world to provide some assurance that a transferred file has arrived intact. For example, file servers often provide a pre-computed MD5 (known as md5sum) checksum for the files, so that a user can compare the checksum of the downloaded file to it. Most unix-based operating systems include MD5 sum utilities in their distribution packages; Windows users may use the included PowerShell function "Get-FileHash", install a Microsoft utility, or use third-party applications. Android ROMs also use this type of checksum.


title: MD5

text: As it is easy to generate MD5 collisions, it is possible for the person who created the file to create a second file with the same checksum, so this technique cannot protect against some forms of malicious tampering. In some cases, the checksum cannot be trusted (for example, if it was obtained over the same channel as the downloaded file), in which case MD5 can only provide error-checking functionality: it will recognize a corrupt or incomplete download, which becomes more likely when downloading larger files.


title: MD5

text: Historically, MD5 has been used to store a one-way hash of a password, often with key stretching. NIST does not include MD5 in their list of recommended hashes for password storage.


title: MD5

text: MD5 is also used in the field of electronic discovery, to provide a unique identifier for each document that is exchanged during the legal discovery process. This method can be used to replace the Bates stamp numbering system that has been used for decades during the exchange of paper documents. As above, this usage should be discouraged due to the ease of collision attacks.


title: MD5

text: MD5 processes a variable-length message into a fixed-length output of 128 bits. The input message is broken up into chunks of 512-bit blocks (sixteen 32-bit words); the message is padded so that its length is divisible by 512. The padding works as follows: first, a single bit, 1, is appended to the end of the message. This is followed by as many zeros as are required to bring the length of the message up to 64 bits fewer than a multiple of 512. The remaining bits are filled up with 64 bits representing the length of the original message, modulo 2.


title: MD5

text: The main MD5 algorithm operates on a 128-bit state, divided into four 32-bit words, denoted , , , and . These are initialized to certain fixed constants. The main algorithm then uses each 512-bit message block in turn to modify the state. The processing of a message block consists of four similar stages, termed "rounds"; each round is composed of 16 similar operations based on a non-linear function , modular addition, and left rotation. Figure 1 illustrates one operation within a round. There are four possible functions; a different one is used in each round:


title: MD5

text: Instead of the formulation from the original RFC 1321 shown, the following may be used for improved efficiency (useful if assembly language is being used – otherwise, the compiler will generally optimize the above code. Since each computation is dependent on another in these formulations, this is often slower than the above method where the nand/and can be parallelised):


title: MD5

text: The 128-bit (16-byte) MD5 hashes (also termed "message digests") are typically represented as a sequence of 32 hexadecimal digits. The following demonstrates a 43-byte ASCII input and the corresponding MD5 hash:


title: MD5

text: Even a small change in the message will (with overwhelming probability) result in a mostly different hash, due to the avalanche effect. For example, adding a period to the end of the sentence:


title: MD5

text: The MD5 algorithm is specified for messages consisting of any number of bits; it is not limited to multiples of eight bits (octets, bytes). Some MD5 implementations such as md5sum might be limited to octets, or they might not support "streaming" for messages of an initially undetermined length.


title: Baby (Justin Bieber song)

text: "Baby" is a song by Canadian singer Justin Bieber, featuring American rapper Ludacris. It was released as the lead single on Bieber's debut album, "My World 2.0". The track was written by Bieber, Ludacris, Christina Milian, Tricky Stewart (who worked with Bieber on a previous single "One Time"), and R&B singer The-Dream, with production helmed by the latter two.


title: Baby (Justin Bieber song)

text: It was available for digital download on January 18, 2010. The song received airplay directly after release, officially impacting mainstream and rhythmic radio on January 26, 2010.


title: Baby (Justin Bieber song)

text: The song has received positive reviews from critics who complimented the song's effective lyrics and chorus and commended Ludacris' part and the song's ability to have an urban twist.


title: Baby (Justin Bieber song)

text: The song was a commercial success, peaking at number one in France and Scotland, and charting in the top ten of the United Kingdom, Canada, the United States, Australia, Norway, Japan, Ireland, Hungary, Belgium (Flanders), Slovakia and New Zealand. "Baby" was the fourth most successful single of 2010 in France. The accompanying music video takes place in a mall/bowling-alley setting. The video features several cameos such as Drake, Lil Twist and Tinashe, in which Bieber chases after a girl. As of May 2013, the song has sold 3.9 million digital downloads in the United States and is certified Diamond in the country.


title: Baby (Justin Bieber song)

text: The official music video is the 40th most-viewed video on YouTube and, as of July 2021, is the fourth-most-disliked YouTube video and the second most-disliked YouTube music video. It had been the most-viewed from July 12, 2010, when it surpassed the music video for "Bad Romance" by Lady Gaga, until November 24, 2012, when it was surpassed by the music video for "Gangnam Style" by Psy. Bieber performed the song several times, including "Saturday Night Live", and the ninth season of "American Idol". Bieber also performed an acoustic rendition of the remix version with Drake at the 2010 Juno Awards. The official remix in the United Kingdom features British hip-hop artist Chipmunk.


title: Baby (Justin Bieber song)

text: "Baby" was written by Bieber with Christopher "Tricky" Stewart and Terius "The-Dream" Nash, both of whom worked with Bieber on "One Time", R&B singer and The-Dream's then-wife Christina Milian and labelmate, Ludacris. When asked how the collaboration initially came about, Bieber said, "[Ludacris and I] both live in Atlanta. I met him a year prior to this and we figured it was a perfect collaboration for him, so we invited him out to do it." Bieber first premiered the song performing on MuchMusic on December 28, 2009. A few days before the release of the single, Bieber posted an acoustic version of the song to his YouTube with his guitarist Dan Kanter, similar to the way he did with "Favorite Girl". "Billboard" said that: "Bieber delivers his characteristically clean vocals with conviction: "My first love broke my heart for the first time / and I'm like, 'baby, baby, baby, no!' I thought you'd always be mine." Also in a review of the acoustic version Bill Lamb of About.com said that, "I think the vocals here will convince some that Justin Bieber really does have the vocal chops."


title: Baby (Justin Bieber song)

text: The song is predominantly upbeat, featuring Bieber's R&B vocals over a backdrop containing a dance infused beat, full of keyboard and "disco string" synths. The song is played at a tempo of 130 BPM and is composed in the key of E♭ major with Bieber's vocal range spanning from the low-note of G to the high-note of C. According to Jody Rosen of "Rolling Stone", the song "blends winks at Fifties doo-wop with hip-hop chants", comparing the style and the lyrics "My first love broke my heart for the first time/And I was like/Baby, baby, baby, ooooh/I thought you'd always be mine" to fifties ballads like "Tears on My Pillow", "Why Do Fools Fall in Love" and "Earth Angel". Lyrically, Bieber's lines explain his distress over his lost love, and promise to get it back, featured in lines like, "And I wanna play it cool/But I'm losin' you…/I'm in pieces/So come and fix me…". The chorus features the distinct and repetitive "baby, baby, baby, ohhhh (nooooo)" hook. After the second verse, Ludacris comes in with the verse-rap, an anecdote of young love when he was thirteen, as it runs "When I was 13/I had my first love/She had me going crazy/Oh, I was star-struck/She woke me up daily/Don't need no Starbucks…".


title: Baby (Justin Bieber song)

text: Nick Levine of Digital Spy felt the song was "no great departure from what he's given us before", but felt the production of Tricky and The-Dream helped give the song a "simple", "big" chorus about "puppy love" that works.


title: Baby (Justin Bieber song)

text: Melanie Bertoldi of "Billboard" said, "The midtempo number's undeniably contagious chorus should keep Bieber's tween fan base satisfied, and Ludacris' brief cameo adds a welcome urban twist." Bertoli went on to say, "The matchup adds a layer of maturity to Bieber's repertoire and should further solidify his growing presence on the charts."


title: Baby (Justin Bieber song)

text: "Rap-Up" magazine said that "the sweet pop fare gets a little street cred courtesy of the teen sensation's labelmate Ludacris."


title: Baby (Justin Bieber song)

text: Jody Rosen of "Rolling Stone" appreciated the song's vintage doo-wop and fifties aesthetic and hip-hop chants, and said the song included "one of the catchiest choruses concocted by the-Dream and Tricky Stewart, the duo behind "Umbrella" and "Single Ladies."


title: Baby (Justin Bieber song)

text: Luke O'Neill of "Boston Globe" had mixed thoughts about the song calling Ludacris' cameo "goofy", and said, the song "effects a musical anachronism, albeit a catchy one, but in this sped-up recycling moment all styles all at once are grist for the mill."


title: Baby (Justin Bieber song)

text: In the United States, "Baby" debuted at number five on the "Billboard" Hot 100, becoming Bieber's highest-charting single in the US. Bieber then went on to surpass this chart position when "Boyfriend" debuted at number two in April 2012. The song was also Ludacris's biggest song to date, solo or featuring, since 2007's "Glamorous". "Baby" was beaten out as the week's highest debut by Taylor Swift's "Today Was a Fairytale", which debuted at number two. The debuting of the pair of songs was just the third time in history that the Hot 100 had two new top-five debuts. The instance had last happened in 2003 when "American Idol"<nowiki>'</nowiki>s Ruben Studdard and Clay Aiken's debut singles charted at number one and two.


title: Baby (Justin Bieber song)

text: During its first week of radio impact, the song had over 1400 spins, and was the top gainer for mainstream and rhythmic radio formats.


title: Baby (Justin Bieber song)

text: The song debuted at thirty-three on the Pop Songs chart on the week labeled February 13, 2010. The song missed out becoming the greatest gainer by one spot, which was Kesha's "Blah Blah Blah". However the following week, the song did collect greatest gainer recognition, jumping to twenty-five on the chart, and has since peaked at sixteen. On August 2, 2010, the song was certified double-platinum by the Recording Industry Association of America for shipments of 2,000,000 units.


title: Baby (Justin Bieber song)

text: As of May 2013, the song had sold 3.9 million digital downloads in the United States. With the inclusion of streaming in the RIAA certifications in 2013, the song became the highest certificated single of all time at 12× Platinum (surpassing "Candle in the Wind 1997" by Elton John at 11× Platinum).


title: Baby (Justin Bieber song)

text: "Baby" debuted and peaked on the Canadian Hot 100 at number three, becoming his highest-charting single at the time. On the issue dated February 8, 2010, the song debuted in Australia on the official ARIA Singles Chart at number thirty-seven. After twelve weeks of ascending and descending the charts the single reached a peak of three. "Baby" has since been certified platinum by the Australian Recording Industry Association (ARIA) for shipments of 70,000 units.


title: Baby (Justin Bieber song)

text: In New Zealand, the song entered the chart at nineteen. After weeks of ascending and descending the charts, "Baby" reached a peak of four. The song dropped to the number five position the following week and after weeks of fluctuating around the chart it attained the number four position once again. It has since been certified platinum by the Recording Industry Association of New Zealand (RIANZ).


title: Baby (Justin Bieber song)

text: The song peaked at number three on the UK Singles Chart on the issue date March 14, 2010. The following week, "Baby" fell to number four, but climbed back to its previous peak position of three on the issue date March 28, 2010. In the process of climbing the charts, "Baby" gained a new peak of two on the R&B Singles Chart.