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Code of Hammurabi

The Code of Hammurabi is a Babylonian legal text composed during 1755–1750 BC. It is the longest, best-organized, and best-preserved legal text from the ancient Near East. It is written in the Old Babylonian dialect of Akkadian, purportedly by Hammurabi, sixth king of the First Dynasty of Babylon. The primary copy of the text is inscribed on a basalt stele 2.25m tall.

The stele was rediscovered in 1901 at the site of Susa in present-day Iran, where it had been taken as plunder six hundred years after its creation. The text itself was copied and studied by Mesopotamian scribes for over a millennium. The stele now resides in the Louvre Museum.

The top of the stele features an image in relief of Hammurabi with Shamash, the Babylonian sun god and god of justice. Below the relief are about 4,130 lines of cuneiform text: one fifth contains a prologue and epilogue in poetic style, while the remaining four fifths contain what are generally called the laws. In the prologue, Hammurabi claims to have been granted his rule by the gods "to prevent the strong from oppressing the weak". The laws are casuistic, expressed as "if... then" conditional sentences. Their scope is broad, including, for example, criminal law, family law, property law, and commercial law.

Modern scholars responded to the Code with admiration at its perceived fairness and respect for the rule of law, and at the complexity of Old Babylonian society. There was also much discussion of its influence on the Mosaic Law. Scholars quickly identified lex talionis—the "eye for an eye" principle—underlying the two collections. Debate among Assyriologists has since centred around several aspects of the Code: its purpose, its underlying principles, its language, and its relation to earlier and later law collections.

Despite the uncertainty surrounding these issues, Hammurabi is regarded outside Assyriology as an important figure in the history of law and the document as a true legal code. The U.S. Capitol has a relief portrait of Hammurabi alongside those of other historic lawgivers. There are replicas of the stele in numerous institutions, including the headquarters of the United Nations in New York City and the Pergamon Museum in Berlin.

Hammurabi
Hammurabi (or Hammurapi), the sixth king of the Amorite First Dynasty of Babylon, ruled from 1792 to 1750 BC (middle chronology). He secured Babylonian dominance over the Mesopotamian plain through military prowess, diplomacy, and treachery. When Hammurabi inherited his father Sin-Muballit's throne, Babylon held little local sway; the local hegemon was Rim-Sin of Larsa. Hammurabi waited until Rim-Sin grew old, then conquered his territory in one swift campaign, leaving his organisation intact. Later, Hammurabi betrayed allies in Eshnunna, Elam, and Mari to gain their territories.

Hammurabi had an aggressive foreign policy, but his letters suggest he was concerned with the welfare of his many subjects and was interested in law and justice. He commissioned extensive construction works, and in his letters, he frequently presents himself as his people's shepherd. Justice is also a theme of the prologue to the Code, and "the word translated 'justice' [ešērum]... is one whose root runs through both prologue and epilogue".

Earlier law collections
Although Hammurabi's Code was the first Mesopotamian law collection to be discovered, it was not the first written; several earlier collections survive. These collections were written in Sumerian and Akkadian. They also purport to have been written by rulers. There were almost certainly more such collections, as statements of other rulers suggest the custom was widespread. The similarities between these law collections make it tempting to assume a consistent underlying legal system. As with the Code of Hammurabi, however, it is difficult to interpret the purpose and underlying legal systems of these earlier collections, prompting numerous scholars to question whether this should be attempted. Extant collections include:

 * The Code of Ur-Nammu of Ur.
 * The Code of Lipit-Ishtar of Isin.
 * The Laws of Eshnunna (written by Bilalama or by Dadusha).
 * Another collection, which Martha Roth calls the "Laws of X", but which may simply be the end of the Code of Ur-Nammu.

There are additionally thousands of documents from the practice of law, from before and during the Old Babylonian period. These documents include contracts, judicial rulings, letters on legal cases, and reform documents such as that of Urukagina, king of Lagash in the mid-3rd millennium BC, whose reforms combatted corruption. Mesopotamia has the most comprehensive surviving legal corpus from before the Digest of Justinian, even compared to those from ancient Greece and Rome.

Louvre stele
The first copy of the text found, and still the most complete, is on a 2.25m stele. The stele is now displayed on the ground floor of the Louvre, in Room 227 of the Richelieu wing. At the top is an image of Hammurabi with Shamash, the Babylonian sun god and god of justice. Below the image are about 4,130 lines of cuneiform text: One fifth contain a prologue and epilogue, while the remaining four fifths contain what are generally called the laws. Near the bottom, seven columns of the laws, each with more than eighty lines, were polished and erased in antiquity. The stele was found in three large fragments and reconstructed. It is 225cm high, with a circumference is 165cm at the summit and 190cm at the base. Hammurabi's image is 65cm high and 60cm wide.

The Louvre stele was found at the site of the ancient Elamite city of Susa. Susa is in modern-day Khuzestan Province, Iran (Persia at the time of excavation). The stele was excavated by the French Archaeological Mission under the direction of Jacques de Morgan. Father Jean-Vincent Scheil published the initial report in the fourth volume of the Reports of the Delegation to Persia (Mémoires de la Délégation en Perse). According to Scheil, the stele's fragments were found on the tell of the Susa acropolis (l'Acropole de Suse), between December 1901 and January 1902. The few, large fragments made assembly easy.

Scheil hypothesised that the stele had been taken to Susa by the Elamite king Shutruk-Nakhunte and that he had commissioned the erasure of several columns of laws to write his legend there. It has been proposed that the relief portion of the stele, especially the beards of Hammurabi and Shamash, was reworked at the same time. Roth suggests the stele was taken as plunder from Sippar, where Hammurabi lived towards the end of his reign.

Other copies
Fragments of a second and possibly third stele recording the Code were found along with the Louvre stele at Susa. Over fifty manuscripts containing the laws are known. They were found not only in Susa but also in Babylon, Nineveh, Assur, Borsippa, Nippur, Sippar, Ur, Larsa, and more. Copies were created during Hammurabi's reign, and also after it, since the text became a part of the scribal curriculum. Copies have been found dating from one thousand years after the stele's creation, and a catalogue from the library of Neo-Assyrian king Ashurbanipal (685–631 BC) lists a copy of the "judgments of Hammurabi". The additional copies fill in most of the stele's original text, including much of the erased section.

Early scholarship
The editio princeps of the Code was published by Father Jean-Vincent Scheil in 1902, in the fourth volume of the Reports of the Delegation to Persia (Mémoires de la Délégation en Perse). After a brief introduction with details of the excavation, Scheil gave a transliteration and a free translation into French, as well as a selection of images. Editions in other languages soon followed: in German by Hugo Winckler in 1902, in English by C. H. W. Johns in 1903, and in Italian by Pietro Bonfante, also in 1903.

The Code was thought to be the earliest Mesopotamian law collection when it was rediscovered in 1902—for example, C. H. W. Johns' 1903 book was titled The Oldest Code of Laws in the World. The English writer H. G. Wells included Hammurabi in the first volume of The Outline of History, and to Wells too the Code was "the earliest known code of law". However, three earlier collections were rediscovered afterwards: the Code of Lipit-Ishtar in 1947, the Laws of Eshnunna in 1948, and the Code of Ur-Nammu in 1952. Early commentators dated Hammurabi and the stele to the 23rd century BC. However, this is an earlier estimate than even the "ultra-long chronology" would support. The Code was compiled near the end of Hammurabi's reign. This was deduced partly from the list of his achievements in the prologue.

Scheil enthused about the stele's importance and perceived fairness, calling it "a moral and political masterpiece". C. H. W. Johns called it "one of the most important monuments in the history of the human race". He remarked that "there are many humanitarian clauses and much protection is given the weak and the helpless", and even lauded a "wonderful modernity of spirit". John Dyneley Prince called the Code's rediscovery "the most important event which has taken place in the development of Assyriological science since the days of Rawlinson and Layard". Charles Francis Horne commended the "wise law-giver" and his "celebrated code". James Henry Breasted noted the Code's "justice to the widow, the orphan, and the poor", but remarked that it "also allows many of the old and naïve ideas of justice to stand". Commentators praised the advanced society they believed the Code evinced. Several singled out perceived secularism: Owen Jenkins, for example, but even Charles Souvay for the Catholic Encyclopedia, who opined that unlike the Mosaic Law the Code was "founded upon the dictates of reason". The question of the Code's influence on the Mosaic Law received much early attention. Scholars also identified Hammurabi with the Biblical figure Amraphel, but this proposal has since been abandoned.

Relief

The relief appears to show Hammurabi standing before a seated Shamash. Shamash wears the horned crown of divinity and has a solar attribute, flames, spouting from his shoulders. Contrastingly, Scheil, in his editio princeps, identified the seated figure as Hammurabi and the standing figure as Shamash. Scheil also held that the scene showed Shamash dictating to Hammurabi while Hammurabi held a scribe's stylus, gazing attentively at the god. Martha Roth lists other interpretations: "that the king is offering the laws to the god; that the king is accepting or offering the emblems of sovereignty of the rod and ring; or—most probably—that these emblems are the measuring tools of the rod-measure and rope-measure used in temple-building". Hammurabi may even be imitating Shamash. It is certain, though, that the draughtsman showed Hammurabi's close links to the divine realm, using composition and iconography.

Prologue
The prologue and epilogue together occupy one-fifth of the text. Out of around 4,130 lines, the prologue occupies 300 lines and the epilogue occupies 500. They are in ring composition around the laws, though there is no visual break distinguishing them from the laws. Both are written in poetic style, and, as William W. Davies wrote, "contain much... which sounds very like braggadocio".

The 300-line prologue begins with an etiology of Hammurabi's royal authority (1–49). Anum, the Babylonian sky god and king of the gods, granted rulership over humanity to Marduk. Marduk chose the centre of his earthly power to be Babylon, which in the real world worshipped him as its tutelary god. Marduk established the office of kingship within Babylon. Finally, Anum, along with the Babylonian wind god Enlil, chose Hammurabi to be Babylon's king. Hammurabi was to rule "to prevent the strong from oppressing the weak" (37–39: dannum enšam ana lā ḫabālim). He was to rise like Shamash over the Mesopotamians (the ṣalmāt qaqqadim, literally the "black-headed people") and illuminate the land (40–44).

Hammurabi then lists his achievements and virtues (50–291). These are expressed in noun form, in the Akkadian first person singular nominal sentence construction "[noun]... anāku" ("I am [noun]"). The first nominal sentence (50–53) is short: "I am Hammurabi, the shepherd, selected by the god Enlil" (ḫammurabi rē'ûm nibīt enlil anāku). Then Hammurabi continues for over 200 lines in a single nominal sentence with the anāku delayed to the very end (291).

Hammurabi repeatedly calls himself na'dum, "pious" (lines 61, 149, 241, and 272). The metaphor of Hammurabi as his people's shepherd also recurs. It was a common metaphor for ancient Near Eastern kings, but is perhaps justified by Hammurabi's interest in his subjects' affairs. His affinities with many different gods are stressed throughout. He is portrayed as dutiful in restoring and maintaining temples and peerless on the battlefield. The list of his accomplishments has helped establish that the text was written late in Hammurabi's reign. After the list, Hammurabi explains that he fulfilled Marduk's request to establish "truth and justice" (kittam u mīšaram) for the people (292–302), although the prologue never directly references the laws. The prologue ends "at that time:" (303: inūmišu) and the laws begin.

Epilogue
Unlike the prologue, the 500-line epilogue is explicitly related to the laws. The epilogue begins (3144'–3151'): "these are the just decisions which Hammurabi... has established" (dīnāt mīšarim ša ḫammurabi... ukinnu-ma). He exalts his laws and his magnanimity (3152'–3239'). He then expresses a hope that "any wronged man who has a lawsuit" (awīlum ḫablum ša awātam iraššû) may have the laws of the stele read aloud to him and know his rights (3240'–3256'). This would bring Hammurabi praise (3257'–3275') and divine favour (3276'–3295'). Hammurabi wishes for good fortune for any ruler who heeds his pronouncements and respects his stele (3296'–3359'). However, he invokes the wrath of the gods on any man who disobeys or erases his pronouncements (3360'–3641', the end of the text).

The epilogue contains much legal imagery, and the phrase "to prevent the strong from oppressing the weak" (3202'–3203': dannum enšam ana lā ḫabālim) is reused from the prologue. However, the king's main concern appears to be ensuring that his achievements are not forgotten and his name not sullied. The list of curses heaped upon any future defacer is 281 lines long and extremely forceful. Some of the curses are very vivid: "may the god Sin... decree for him a life that is no better than death" (3486'–3508': sîn... balāṭam ša itti mūtim šitannu ana šīmtim lišīmšum); "may he [the future defacer] conclude every day, month, and year of his reign with groaning and mourning" (3497'–3501': ūmī warḫī šanāt palēšu ina tānēḫim u dimmatim lišaqti); may he experience "the spilling of his life force like water" (3435'–3436': tabāk napištišu kīma mê). Hammurabi implores a variety of gods individually to turn their particular attributes against the defacer. For example: "may the [storm] god Adad... deprive him of the benefits of rain from heaven and flood from the springs" (3509'–3515': adad... zunnī ina šamê mīlam ina nagbim līṭeršu); "may the god [of wisdom] Ea... deprive him of all understanding and wisdom, and may he lead him into confusion" (3440'–3451': ea... uznam u nēmeqam līṭeršu-ma ina mīšītim littarrūšu). Gods and goddesses are invoked in this order:

 * 1) Anum (3387'–3394')
 * 2) Enlil (3395'–3422')
 * 3) Ninlil (3423'–3439')
 * 4) Ea (3440'–3458')
 * 5) Shamash (3459'–3485')
 * 6) Sin (3486'–3508')
 * 7) Adad (3509'–3525')
 * 8) Zababa (3526'–3536')
 * 9) Ishtar (3537'–3573')
 * 10) Nergal (3574'–3589')
 * 11) Nintu (3590'–3599')
 * 12) Ninkarrak (3600'–3619')
 * 13) All the gods (3620'–3635')
 * 14) Enlil, a second time (3636'–3641')

Laws
The Code of Hammurabi is the longest and best-organised legal text from the ancient Near East, as well as the best-preserved. The classification below (columns 1–3) is Driver & Miles', with several amendments, and Roth's translation is used. Laws represented by letters are those reconstructed primarily from documents other than the Louvre stele.

Theories of purpose
The purpose and legal authority of the Code have been disputed since the mid-20th century. Theories fall into three main categories: that it is legislation, whether a code of law or a body of statutes; that it is a sort of law report, containing records of past cases and judgments; and that it is an abstract work of jurisprudence. The jurisprudence theory has gained much support within Assyriology.

Legislation
The term "code" presupposes that the document was intended to be enforced as legislation. It was used by Scheil in his editio princeps, and widely adopted afterwards. C. H. W. Johns, one of the most prolific early commentators on the document, proclaimed that "the Code well deserves its name". Recent Assyriologists have used the term without comment, as well as scholars outside Assyriology. However, only if the text was intended as enforced legislation can it truly be called a code of law and its provisions laws.

The document, on first inspection, resembles a highly organised code similar to the Code of Justinian and the Napoleonic Code. There is also evidence that dīnātum, which in the Code of Hammurabi sometimes denote individual "laws", were enforced. One copy of the Code calls it a ṣimdat šarrim, "royal decree", which denotes a kind of enforced legislation.

However, the arguments against this view are strong. Firstly, it would make a very unusual code—Reuven Yaron called the designation "Code" a "persistent misnomer". Vital areas of society and commerce are omitted. For example, Marc Van De Mieroop observes that the Code "deals with cattle and agricultural fields, but it almost entirely ignores the work of shepherds, vital to Babylonia's economy". Then, against the legislation theory more generally, highly implausible circumstances are covered, such as threshing with goats, animals far too unruly for the task (law 270). The laws are also strictly casuistic ("if... then"); unlike in the Mosaic Law, there are no apodictic laws (general commands). These would more obviously suggest prescriptive legislation. The strongest argument against the legislation theory, however, is that most judges appear to have paid the Code no attention. This line of criticism originated with Benno Landsberger in 1950. No Mesopotamian legal document explicitly references the Code or any other law collection, despite the great scale of the corpus. Two references to prescriptions on "a stele" (narû) come closest. In contrast, numerous judgments cite royal mīšarum-decrees. Raymond Westbrook held that this strengthened the argument from silence that ancient Near Eastern legal "codes" had legal import. Furthermore, many Old Babylonian judgments run entirely counter to the Code's prescriptions.

Law report

A second theory is that the Code is a sort of law report, and as such contains records of past cases and judgments, albeit phrased abstractly. This would provide one explanation for the casuistic format of the "laws"; indeed, Jean Bottéro believed he had found a record of a case that inspired one. However, such finds are inconclusive and very rare, despite the scale of the Mesopotamian legal corpus. Furthermore, legal judgments were frequently recorded in Mesopotamia, and they recount the facts of the case without generalising them. These judgments were concerned almost exclusively with points of fact, prompting Martha Roth to comment: "I know of only one case out of thousands extant that might be said to revolve around a point of law".

Jurisprudence
A third theory, which has gained traction within Assyriology, is that the Code is not a true code but an abstract treatise on how judgments should be formulated. This led Fritz Rudolf Kraus, in an early formulation of the theory, to call it jurisprudence (Rechtssprüche). Kraus proposed that it was a work of Mesopotamian scholarship in the same category as omen collections like šumma ālu and ana ittišu. Others have provided their own versions of this theory. A. Leo Oppenheim remarked that the Code of Hammurabi and similar Mesopotamian law collections "represent an interesting formulation of social criticism and should not be taken as normative directions".

This interpretation bypasses the problem of low congruence between the Code and actual legal judgments. Secondly, the Code does bear striking similarities to other works of Mesopotamian scholarship. Key points of similarity are the list format and the order of the items, which Ann Guinan describes as a complex "serial logic". Marc Van De Mieroop explains that, in common with other works of Mesopotamian scholarship such as omen lists, king lists, and god lists, the entries of the Code of Hammurabi are arranged according to two principles. These are "opposition"—whereby a variable in one entry is altered to make another entry—and "pointillism"—whereby new conditions are added to an entry, or paradigmatic series pursued, to generate a sequence. Van De Mieroop provides the following examples:

"If a physician performs major surgery with a bronze lancet upon an [awīlum] and thus heals the [awīlum], or opens an [awīlum]'s temple with a bronze lancet and thus heals the [awīlum]'s eye, he shall take ten shekels of silver (as his fee)." "If a physician performs major surgery with a bronze lancet upon an [awīlum] and thus causes the [awīlum]'s death, or opens an [awīlum]'s temple with a bronze lancet and thus blinds the [awīlum]'s eye, they shall cut off his hand."

Laws 215 and 218 illustrate the principle of opposition: one variable of the first law, the outcome of the operations, is altered to create the second.

"If there is either a soldier or [an auxiliary] who is taken captive while serving in a royal fortress [...] if he should [...] return and get back to his city, they shall return to him his field and orchard and he himself shall perform his service obligation.

If there is either a soldier or [an auxiliary] who is taken captive in a royal fortress, and his son is able to perform the service obligation, the field and orchard shall be given to him, and he shall perform his father's service obligation.

If his son is young and is unable to perform his father's service obligation, one third of the field and orchard shall be given to his mother, and his mother shall raise him."

Here, following the principle of pointillism, circumstances are added to the first entry to create more entries. Pointillism also lets list entries be generated by following paradigmatic series common to multiple branches of scholarship. It can thus explain the implausible entries. For example, in the case of the goat used for threshing (law 270), the previous laws concern other animals that were used for threshing. The established series of domesticated beasts dictated that a goat come next.

Wolfram von Soden, who decades earlier called this way of thinking Listenwissenschaft ("list science"), often denigrated it. However, more recent writers, such as Marc Van De Mieroop, Jean Bottéro, and Ann Guinan, have either avoided value judgments or expressed admiration. Lists were central to Mesopotamian science and logic, and their distinctive structural principles let entries be generated infinitely. Linking the Code to the scribal tradition within which "list science" emerged also explains why trainee scribes copied and studied it for over a millennium. The Code appears in a late Babylonian (7th–6th century BC) list of literary and scholarly texts. No other law collection became so entrenched in the curriculum. Rather than a code of laws, then, it may be a scholarly treatise.

Much has been written on what the Code suggests about Old Babylonian society and its legal system. For example, whether it demonstrates that there were no professional advocates, or that there were professional judges. Scholars who approach the Code as a self-contained document renounce such claims.

Underlying principles
One principle widely accepted to underlie the Code is lex talionis, or "eye for an eye". Laws 196 and 200 respectively prescribe an eye for an eye and a tooth for a tooth when one man destroys another's. Punishments determined by lex talionis could be transferred to the sons of the wrongdoer. For example, law 229 states that the death of a homeowner in a house collapse necessitates the death of the house's builder. The following law 230 states that if the homeowner's son died, the builder's son must die also.

Persons were not equal before the law; not just age and profession but also class and gender dictated the punishment or remedy they received. Three main kinds of person, awīlum, muškēnum, and wardum (male)/amtum (female), are mentioned throughout the Code. A wardum/amtum was a male/female slave. As for awīlum and muškēnum, though contentious, it seems likely that the difference was one of social class, with awīlum meaning something like "gentleman" and muškēnum something like "commoner". The penalties were not necessarily stricter for a muškēnum than an awīlum: a muškēnum's life may have been cheaper, but so were some of his fines. There was also inequality within these classes: laws 200 and 202, for example, show that one awīlum could be of higher rank than another.

Martha Roth has shown that ideas of shame and honour motivated certain laws.

The above principles are distant in spirit from modern systems of common and civil law, but some may be more familiar. One such principle is the presumption of innocence; the first two laws of the stele prescribe punishments, determined by lex talionis, for unsubstantiated accusations. Written evidence was valued highly, especially in matters of contract. One crime was given only one punishment. The laws also recognized the importance of the intentions of a defendant. Lastly, the Code's establishment on public stelae was supposedly intended to increase access to justice. Whether or not this was true, suggesting that a wronged man have the stele read aloud to him (lines 3240'–3254') is a concrete measure in this direction, given the inaccessibility of scribal education in the Old Babylonian period.

The prologue asserts that Hammurabi was chosen by the gods. Raymond Westbrook observed that in ancient Near Eastern law, "the king was the primary source of legislation". However, they could delegate their god-given legal authority to judges. However, as Owen B. Jenkins observed, the prescriptions themselves bear "an astonishing absence... of all theological or even ceremonial law".

Language
The laws are written in the Old Babylonian dialect of Akkadian. Their style is regular and repetitive, and today they are a standard set text for introductory Akkadian classes. However, as A. Leo Oppenheim summarises, the cuneiform signs themselves are "vertically arranged... within boxes placed in bands side by side from right to left", an arrangement already antiquated by Hammurabi's time.

The laws are expressed in casuistic format: they are conditional sentences with the case detailed in the protasis ("if" clause) and the remedy given in the apodosis ("then" clause). The protasis begins šumma, "if", except when it adds to circumstances already specified in a previous law (e.g. laws 36, 38, and 40). The preterite is used for simple past verbs in the protasis, or possibly for a simple conditional. The perfect often appears at the end of the protasis after one or more preterites to convey sequence of action, or possibly a hypothetical conditional. The durative, sometimes called the "present" in Assyriology, may express intention in the laws. For ease of English reading, some translations give preterite and perfect verbs in the protasis a present sense. In the apodosis, the verbs are in the durative, though the sense varies between permissive—"it is permitted that x happen"—and instructive—"x must/will happen". In both protasis and apodosis, sequence of action is conveyed by suffixing verbs with -ma, "and". -ma can also have the sense "but".

The Code is relatively well-understood, but some items of its vocabulary are controversial. As mentioned, the terms awīlum and muškēnum have proved difficult to translate. They probably denote respectively a male member of a higher and lower social class. Wolfram von Soden, in his Akkadisches Handwörterbuch, proposed that muškēnum was derived from šukênum, "to bow down/supplicate". As a word for a man of low social standing, it has endured, possibly from a Sumerian root, into Arabic (miskīn), Italian (meschino), Spanish (mezquino), and French (mesquin). However, some earlier translators, also seeking to explain the muškēnum's special treatment, translated it as "leper" and even "noble". Some translators have supplied stilted readings for awīlum, such as "seignior", "elite man", and "member of the aristocracy"; others have left it untranslated. Certain legal terms have also proved difficult to translate. For example, dīnum and dīttum can denote the law in general as well as individual laws, verdicts, divine pronouncements and other phenomena. mīšarum can likewise denote the law in general as well as a kind of royal decree.

Other Mesopotamian
The Code of Hammurabi bears strong similarities to earlier Mesopotamian law collections. Many purport to have been written by rulers, and this tradition was probably widespread. Earlier law collections express their god-given legitimacy similarly. Like the Code of Hammurabi, they feature prologues and epilogues: the Code of Ur-Nammu has a prologue, the Code of Lipit-Ishtar a prologue and an epilogue, and the Laws of Eshnunna an epilogue. Also, like the Code of Hammurabi, they uphold the "one crime, one punishment" principle. The cases covered and language used are, overall, strikingly similar. Scribes were still copying earlier law collections, such as the Code of Ur-Nammu, when Hammurabi produced his own Code. This suggests that earlier collections may have not only resembled the Code but influenced it. Raymond Westbrook maintained that there was a fairly consistent tradition of "ancient Near Eastern law" which included the Code of Hammurabi, and that this was largely customary law. Nonetheless, there are differences: for example, Stephen Bertman has suggested that where earlier collections are concerned with compensating victims, the Code is concerned with physically punishing offenders. Additionally, the above conclusions of similarity and influence apply only to the law collections themselves. The actual legal practices from the context of each code are mysterious.

The Code of Hammurabi also bears strong similarities to later Mesopotamian law collections: to the casuistic Middle Assyrian Laws and to the Neo-Babylonian Laws, whose format is largely relative ("a man who..."). It is easier to posit direct influence for these later collections, given the Code's survival through the scribal curriculum. Lastly, although influence is more difficult to trace, there is evidence that the Hittite laws may have been part of the same tradition of legal writing outside Mesopotamia proper.

Mosaic, Graeco-Roman, and modern

The relationship of the Code of Hammurabi to the Mosaic Law, specifically the Covenant Code of Exodus 20:22–23:19, has been a subject of discussion since its discovery. Friedrich Delitzsch argued the case for strong influence in a 1902 lecture, in one episode of the "Babel und Bibel" ("Babel and Bible", or "Panbabylonism") debate on the influence of ancient Mesopotamian cultures on ancient Israel. However, he was met with strong resistance. There was cultural contact between Mesopotamia and the Levant, and Middle Bronze Age tablets of casuistic cuneiform law have been found at Hazor. There are also similarities between the Code of Hamurabi and the Covenant Code: in the casuistic format, in principles such as lex talionis ("eye for an eye"), and in the content of the provisions. Some similarities are striking, such as in the provisions concerning a man-goring ox (Code of Hammurabi laws 250–252, Exodus 21:28–32). Certain writers have posited direct influence: David P. Wright, for example, asserts that the Covenant Code is "directly, primarily, and throughout dependent upon the Laws of Hammurabi", "a creative rewriting of Mesopotamian sources... to be viewed as an academic abstraction rather than a digest of laws". Others posit indirect influence, such as via Aramaic or Phoenician intermediaries. The consensus, however, is that the similarities are a result of inheriting common traditions. In 1916, George A. Barton cited "a similarity of antecedents and of general intellectual outlook". More recently, David Winton Thomas has stated: "There is no ground for assuming any direct borrowing by the Hebrew from the Babylonian. Even where the two sets of laws differ little in the letter, they differ much in the spirit".

The influence of the Code of Hammurabi on later law collections is difficult to establish. Marc Van De Mieroop suggests that it may have influenced the Greek Gortyn Code and the Roman Twelve Tables. However, even Van De Mieroop acknowledges that most Roman law is not similar to the Code, or likely to have been influenced by it.

Knowing the Code's influence on modern law requires knowing its influence on Mosaic and Graeco-Roman law. Since this is contentious, commentators have restricted themselves to observing similarities and differences between the Code and, e.g., United States law and medieval law. Some have remarked that the punishments found in the Code are no more severe, and, in some cases, less so.

Law 238 stipulates that a sea captain, ship-manager, or ship charterer that saved a ship from total loss was only required to pay one-half the value of the ship to the ship-owner. In the Digesta seu Pandectae (533), the second volume of the codification of laws ordered by Justinian I (527–565) of the Eastern Roman Empire, a legal opinion written by the Roman jurist Paulus at the beginning of the Crisis of the Third Century in 235 AD was included about the Lex Rhodia ("Rhodian law") that articulates the general average principle of marine insurance established on the island of Rhodes in approximately 1000 to 800 BC as a member of the Doric Hexapolis, plausibly by the Phoenicians during the proposed Dorian invasion and emergence of the purported Sea Peoples during the Greek Dark Ages (c. 1100 – c. 750) that led to the proliferation of the Doric Greek dialect. The law of general average constitutes the fundamental principle that underlies all insurance.

Reception outside Assyriology

The Code is often referred to in legal scholarship, where its provisions are assumed to be laws, and the document is assumed to be a true code of laws. This is also true outside academia.

There is a relief portrait of Hammurabi over the doors to the House Chamber of the U.S. Capitol, along with portraits of 22 others "noted for their work in establishing the principles that underlie American law". There are replicas of the Louvre stele in institutions around the world, including the Headquarters of the United Nations in New York City and the Peace Palace in The Hague (seat of the International Court of Justice).

title: Code of Hammurabi


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Rum and Coke

Rum and Coke, or the Cuba libre (, ; literally "Free Cuba"), is a highball cocktail consisting of cola, rum, and in many recipes lime juice on ice. Traditionally, the cola ingredient is Coca-Cola ("Coke") and the alcohol is a light rum such as Bacardi; however, the drink may be made with various types of rums and cola brands, and lime juice may or may not be included.

The cocktail originated in the early 20th century in Cuba, after the country won independence in the Spanish–American War. It subsequently became popular across Cuba, the United States, and other countries. Its simple recipe and inexpensive, ubiquitous ingredients have made it one of the world's most-popular alcoholic drinks. Drink critics often consider the drink mediocre, but it has been noted for its historical significance.

History
The drink was created in Cuba in the early 1900s, but its exact origins are not certain. It became popular shortly after 1900, when bottled Coca-Cola was first imported into Cuba from the United States. Its origin is associated with the heavy U.S. presence in Cuba following the Spanish–American War of 1898; the drink's traditional name, "Cuba libre" (Free Cuba), was the slogan of the Cuban independence movement. The Cuba libre is sometimes said to have been created during the Spanish–American War. However, this predates the first distribution of Coca-Cola to Cuba in 1900. A drink called a "Cuba libre" was indeed known in 1898, but this was a mix of water and brown sugar.

Fausto Rodriguez, a Bacardi advertising executive, claimed to have been present when the drink was first poured, and produced a notarized affidavit to that effect in 1965. According to Rodriguez, this took place in August 1900, when he was a 14-year-old messenger working for a member of the U.S. Army Signal Corps in Havana. One day at a local bar, Rodriguez's employer ordered Bacardi rum mixed with Coca-Cola. This intrigued a nearby group of American soldiers, who ordered a round for themselves, giving birth to a popular new drink. Bacardi published Rodriguez's affidavit in a Life magazine ad in 1966. However, Rodriguez's status as a Bacardi executive has led some commentators to doubt the veracity of his story. Another story states that the drink was first created in 1902 at Havana's El Floridita restaurant to celebrate the anniversary of Cuban independence.

The drink became a staple in Cuba, catching on due to the pervasiveness of its ingredients. Havana was already known for its iced drinks in the 19th century, as it was one of the few warm-weather cities that had abundant stores of ice shipped down from colder regions. Bacardi and other Cuban rums also boomed after independence brought in large numbers of foreign tourists and investors, as well as new opportunities for exporting alcohol. Light rums such as Bacardi became favored for cocktails as they were considered to mix well. Coca-Cola had been a common mixer in the United States ever since it was first bottled in 1886, and it became a ubiquitous drink in many countries after it was first exported in 1900.

Rum and Coke quickly spread from Cuba to the United States. In the early 20th century the cocktail, like Coca-Cola itself, was most popular in the Southern United States. During the Prohibition era from 1922 to 1933, Coca-Cola became a favored mixer for disguising the taste of low-quality rums, as well as other liquors. In 1921 H. L. Mencken jokingly wrote of a South Carolina variant called the "jump stiddy", which consisted of Coca-Cola mixed with denatured alcohol drained from automobile radiators. After Prohibition, rum and Coke became prevalent in the northern and western U.S. as well, and in both high-brow and low-brow circles.

Rum and Coke achieved a new level of popularity during World War II. Starting in 1940, the United States established a series of outposts in the British West Indies to defend against the German Navy. The American presence created cross-cultural demand, with American servicemen and the locals developing tastes for each other's products. In particular, American military personnel took to Caribbean rum due to its inexpensiveness, while Coca-Cola became especially prevalent in the islands thanks to the company shipping it out with the military. Within the United States, imported rum became increasingly popular, as government quotas for industrial alcohol reduced the output of American distillers of domestic liquors.

In 1943, Lord Invader's Calypso song "Rum and Coca-Cola" drew further attention to the drink in Trinidad. The song was an adaptation of Lionel Belasco's 1904 composition "L'Année Passée" with new lyrics about American soldiers in Trinidad cavorting with local girls and drinking rum and Coke. Comedian Morey Amsterdam plagiarized "Rum and Coca-Cola" and licensed it to the Andrews Sisters as his own work. The Andrews Sisters' version was a major hit in 1945 and further boosted the popularity of rum and Coke, especially in the military. Lord Invader and the owners of Belasco's composition successfully sued Amsterdam for the song's rights.

During the Cuban Revolution in 1959, Bacardi fled to Puerto Rico. The following year, the U.S. placed an embargo against Cuba which prohibited the importation of Cuban products, while Cuba likewise banned the importation of American products. With Cuban-made rum unavailable in the U.S. and Coca-Cola largely unavailable in Cuba, it became difficult to make a rum and Coke with its traditional ingredients in either country.

Popularity and reception
The rum and Coke is very popular; Bacardi says that it is the world's second-most-popular alcoholic drink. Its popularity derives from the ubiquity and low cost of the main ingredients, and the fact that it is very easy to make. As it can be made with any quantity or style of rum, it is simple to prepare and difficult to ruin.

Drink critics often have a low opinion of the cocktail. Writer Wayne Curtis called it "a drink of inspired blandness", while Jason Wilson of The Washington Post called it "a lazy person's drink". Troy Patterson of Slate called it "the classic mediocre Caribbean-American highball", which "became a classic despite not being especially good".

Charles A. Coulombe considers the Cuba libre a historically important drink, writing that it is "a potent symbol of a changing world order – the marriage of rum, lubricant of the old colonial empires, and Coca-Cola, icon of modern American global capitalism". Additionally, both rum and Coca-Cola are made from Caribbean ingredients and became global commodities through European and American commerce. According to Coulombe, the drink "seems to reflect perfectly the historical elements of the modern world".

Recipe and variations

Recipes vary somewhat in measures and additional ingredients, but the main ingredients are always rum and cola. The International Bartenders Association recipe calls for 5 centiliters of light rum, 12 cl of cola, and 1 cl of fresh lime juice on ice. However, any amount and proportion of rum and cola may be used. Additionally, while light rum is traditional, dark rums and other varieties are also common.

Coca-Cola is the conventional cola in the drink, to the point that customers rarely order anything else. This dates back to the origin of the drink in Cuba and was solidified in the 1920s when Coca-Cola emerged as the primary cola brand following the bankruptcy of Pepsi and Chero-Cola, and therefore the preferred cola mixer in alcoholic drinks. Pepsi's later attempts to enter the cocktail market were unsuccessful, especially after the song "Rum and Coca-Cola" solidified the association in the public imagination.

Nonetheless, different colas are sometimes used. In Cuba, as Coca-Cola has not been imported since the U.S. embargo of 1960, the domestic TuKola is used in Cuba libres. Other common variants call for Mexican Coke (which uses cane sugar instead of high-fructose corn syrup), Moxie, Diet Coke (the Cuba Lite or rum and Diet) and Dr. Pepper (the Captain and Pepper, featuring Captain Morgan spiced rum).

Lime is traditionally included in the drink, though it is often left out, especially when the order is for just "rum and Coke". Some early recipes called for lime juice to be mixed in; others included lime only as a garnish. Other early recipes called for additional ingredients such as gin and bitters. Some sources consider lime essential for a drink to be a true Cuba libre, which they distinguish from a mere rum and Coke. However, lime is frequently included even in orders for "rum and Coke".

When aged añejo rum is used, the drink is sometimes called a Cubata, a name also used informally in Spain for any Cuba libre. Some modern recipes inspired by older ones include additional ingredients such as bitters. More elaborate variants with further ingredients include the cinema highball, which uses rum infused with buttered popcorn and mixed with cola. Another is the Mandeville cocktail, which includes light and dark rum, cola, and citrus juice along with Pernod absinthe and grenadine.

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

title: Rum and Coke


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Talk:Computability theory (computer science)

Untitled
I believe the Entscheidungsproblem was first proved unsolvable by Church, and months later by Turing. Goedel's theorems don't really talk about algorithms, so they don't directly apply. Of course, Goedel's trick of Goedel numbering and the Barber paradox is really the key to the Entscheidungsproblem and halting problem, everything else is details. --AxelBoldt

 * OK, I've added that. --LC

The context-free languages need a stack and a non-deterministic finite-state machine; deterministic won't do. --AxelBoldt

 * Yes, that's what it says. Perhaps you misread the 2nd sentence of the paragraph? --LC

 * Yup, you're right. --AxelBoldt

The languages that are accepted by a Turing machine are exactly those that are generated by formal grammars.

This is wrong. I doubt it is all formal grammars. I expect that it is context sensitive grammars the author was thinking of. Regular grammars are FSM equivalent, as I recall. -- BenBaker

No, the context sensitive grammars generate exactly the languages accepted by linear bounded nondeterministic Turing machines (Turing machines with only a linear amount of memory). The set of languages accepted by all possible Turing machines is exactly equal to the set of languages generated by all possible formal grammars. The sentence in the article is correct. See Chomsky hierarchy for more details. --LC

I moved the following question from the article. I don't know the answer. (¿Isn't it 3 parameters?)

Hi Cwitty, I'm not sure what form Turing's definition took. But your change seems unhelpful, because it puts too much weight on the word "arbitrarily".

Chaitin's constant, for example, can be approximated from below with unlimited accuracy, simply by simulating all machines and adding in a component for those which halt. It is not computable, because there is no way of telling how close the approximation is at any point.

I'm pretty sure the "incorrect" defition you changed is actually equivalent to the "correct" one. -- Pde 06:54, 8 Nov 2003 (UTC)

 * I can certainly change the definition to be precise; I didn't do so because I put a precise, correct definition in computable number a couple of months ago.

 * There is a link to Turing's original paper at the end of computable number. Note that the link includes both the original paper and a correction to it.  Turing's original definition is indeed similar to the one I removed; but in the correction, he acknowledged that this definition was flawed, and provided an alternate definition (which I can't read, due to font problems).

 * Here's an example which shows the problem of relying on "digits" in a definition of this form. Start with 1/2.  Add 2-k for every k which is an odd perfect number.  Subtract 2-k for every k which is an even number >2 that is not the sum of two primes.  This number is greater than 0.5 if the first odd perfect number is smaller than the first counterexample to Goldbach's conjecture; if Goldbach's conjecture is true, and there are no odd perfect numbers, then this number is exactly 1/2.

 * My definition (at computable number) says that this number is computable; another way to put it is that given an epsilon, we can give a number which is within epsilon of this number. However, we cannot give a single (binary or decimal) digit of this number; we do not know whether it begins 0.4 or 0.5.  (I think you might be able to produce arbitrary ternary digits of this number, though.) -- Cwitty 23:45, 8 Nov 2003 (UTC)

 * Gotcha. One could argue (at least momentarily) that it is acceptable to consider your number to be incomputable unless the Goldbach conjecture is resolvable.  The fact that the definition is not consistent across all bases makes such an argument a little untennable, though.  -- Pde 01:52, 9 Nov 2003 (UTC)

I didn't edit the page, but I believe it should read "given ANY statement in [FOL]...", not "given A statement in [FOL]"...

also, it might be helpful to add that godel's general recursive functions are also equivalent to the notion of Turing-computable functions and lambda-definable functions.

Refocus article on Computable functions
I think the article should be rewritten to focus more on computable functions instead of Turing machines. This should be done in most articles in the computability category. Of course Turing machines are easier to grasp then the abstract concept of computable function, but by focussing on the abstract concept we can eliminate the redundant discussion of lamda calculus, recursive function and Church-Turing thesis in each computability article. I guess in many cases the proofs and theorems will get shorter too. We can always add a concrete example using Turing machines if the article gets to abstract MathMartin 23:59, 9 Nov 2004 (UTC)

 * I have the opposite opinion. TM gives very intuitive and simple representation for computability. See the proofs for uncomputability of Busy Beaver functions. More of the results may be demonstarted on TM (or other programming language) examples. If you get the Quine program, it is easy to expand it to self-explorer program, and then using self-opposite behavior to proof easy non-stoping problem, Godel incompleteness theorem and other results. Such proofs are simple and don't use complex and hard for understanding mathematical notation. Skelet 19:16, 7 Dec 2004 (UTC)

 * I do not object to using Turing machines as examples, but as with any model, Turing machines only serve to illustrate certain points. Other points can be illustrated better using other models. My point is the main (technical) definition should be based on computable functions. But at the moment I am not able to devote any amount of time on this topic. So I guess for now things will stay as they are. MathMartin 19:30, 9 Dec 2004 (UTC)

What happened to this article?!
As far as I remember (and correct me if I am wrong), this article used to have a lot of content. Now it is just 8 lines. Computability theory deserves more than just a short definition and two links. This is an ENCYCLOPEDIA, not a DICTIONARY. Who destroyed the old content?! David November 13, 2005

 * Yes, this is indeed very strange! I do not know the previous article, but this topic cannot possibly be that short after 4 years of Wikipedia! Hints on what might have happened here are found at the deleted changes page. There it says that "00:11, 26 October 2005 Timwi deleted "Computability theory" (Deleted to make way for move)" Yet, I do not understand it. What moved? Where did it move to? I will go and ask Timwi. --Markus Krötzsch 21:05, 15 November 2005 (UTC)

Problematic name
It's become trendy in recent years, due to the influence of Robert I. Soare, to use the term "computability theory" to refer to what's traditionally called "recursion theory", and in fact recursion theory is a redirect here. However this article is not much about recursion theory (Turing degrees, effective descriptive set theory, that sort of thing), but more about models of in-principle-physically-realizable computation. Recursion theory is a branch of math logic, not of theory of computation.

So what do we do about this? I see there's a merge template to merge to computation. For most of the content in the current article that's probably fine. But there should also be an article about what math-logicians call computability theory or recursion theory. Perhaps we need a dab page under the name computability theory. --Trovatore 04:26, 26 November 2005 (UTC)

 * This article should be moved to theory of computation (which I can't do because I'm not an admin). This should be made in to a disambig to theory of computation and a to be written recursion theory. —R. Koot 04:40, 26 November 2005 (UTC)
 * So I've taken an awkward first step, moving the article to Computability theory (computation), which I don't propose as the permanent name, but just a good place to put this content while we sort things out. Some of the content does need to be reproduced in the (new stub) recursion theory article. Roughly speaking, recursion theory starts with what is here called "Unreasonable models of computation". Definitely the halting problem needs to be mentioned in the "recursion theory" article. --Trovatore 05:22, 26 November 2005 (UTC)

To tell the truth I'm not sure the articles should be split at all. What's clear is that, whatever article is called recursion theory (or where recursion theory redirects) has to have a very different focus from the current article here; this current article seems to be mainly about things of Turing degree 0. In the end perhaps we should have just one article, but it shouldn't spend so much time talking about Turing degree 0 in fifteen different ways, but should move quickly into the real meat of the topic, which starts with oracles. --Trovatore 06:05, 26 November 2005 (UTC)

Theory of Computation
The current article isn't really suited as the main article for all of Theory of Computation, as when I wrote it I was considered only computability theory and not the other major branch, complexity theory. So do we want separate articles on computability and complexity, or do we want to expand this article greatly to include some discussion of complexity theory as well? Clearly any article called "Theory of Computation" should at least mention P =? NP. —The preceding unsigned comment was added by Readams (talk • contribs) 17:34, 26 November 2005.
 * That's another aspect, alright. It seems we have a multidimensional mess on our hands. Interested parties should also look at the categorization question. Yesterday I created Category:Recursion theory and have been going through the articles in Category:Theory of computation to try to figure out what belongs where. Roughly speaking, the test I used was this: Would this question be considered in a math department by people who call themselves mathematical logicians, or would it be considered in a computer science department? Of course there was a very considerable overlap. Also people may disagree with my judgment as to which articles are categorized which way by this test, and may disagree with the test itself. I'd like to see some more thoughts on this. --Trovatore 18:35, 26 November 2005 (UTC)

 * There is indeed a much larger overlap than you suggested at first (I suspected this, but I know nearly nothing about recusion theory, so didn't mention it). Maybe some other people want to say something aboiut this? —The preceding unsigned comment was added by R.Koot (talk • contribs).
 * Well, I was conservative about removing the "Theory of computation" category from articles, so there may be more overlap in the current categorization than there really should be. --Trovatore 19:03, 26 November 2005 (UTC)

 * I think keeping it would be best if the article would be kept seperate. Maybe we should move this page to Computability theory (computer science)? The Theory of computation could be an introductory article wich refers to automata theory, formal language theory, computability theory (computer science) and computational complexity theory. —R. Koot 18:53, 26 November 2005 (UTC)

Chaitin
For the moment, I've removed the following:
 *  (the fact that one can follow from the other was first shown by Gregory Chaitin in his development of Chaitin's Constant, a number which can be defined but not computed)

First, it's not clear just what connection between the incompleteness theorems and the halting problem is being claimed, but whatever it is, I'd want to see a ref that Chaitin was the first one to find it. Chaitin, like Wolfram, is a bit of a self-promoter, and is good enough at self-promotion that a lot of people give him credit for things that might not have been quite as original as they think. In particular Chaitin's Omega was by no means the first non-computable number to be defined. --Trovatore 04:06, 2 December 2005 (UTC)

 * It's not clear to you, because you might not know the subject. Chaitin proved the connection to the halting problem, and anybody who knew something about Kolmogorov complexity would acknowledge that. Fact: Chaitin was one of the co-inventors of Kolmogorov complexity. The halting probability Omega contains all information required to solve the halting problem. So it acts as an oracle for the halting problem. How is that for a connection? --Exa 15:26, 22 January 2006 (UTC)

 * Omega was of course not the first oracle for the halting problem to be defined, though. The first one was, presumably, simply the set of all Gödel numbers of Turing machines that halt. You still haven't explained (or indeed even mentioned) the connection to Gödel's incompleteness theorem (which is quite distinct from the halting problem), or given a ref that Chaitin was the first one to find said connection. --Trovatore 17:13, 22 January 2006 (UTC)

 * On reflection I probably should have removed entirely the following two sentences from the version way back then:
 * Computability theory has its roots in First-order logic, and the halting problem and other non-recursive problems are closely related to Gödel's incompleteness theorem (the fact that one can follow from the other was first shown by Gregory Chaitin in his development of Chaitin's Constant, a number which can be defined but not computed). David Hilbert and Kurt Gödel laid many of the foundations for first-order logic.
 * It was a red flag to me that someone was dragging Chaitin in to a discussion of the incompleteness theorems, but really the entire first sentence is unsupported–what roots in first-order logic are being claimed, exactly? And if the first sentence goes, then the second, while certainly true, is out of context.
 * Unless someone can explain and support this passage, I'll be snipping these two sentences. --Trovatore 17:32, 22 January 2006 (UTC)

Desktop PCs are not considered to be FSMs
(Also see )

The author probably didn't read the enlightening discussion in the new edition of the Cinderella book.

It would take too much effort from me to explain it completely, but I am going to remove those uninformed assumptions that the tape of a turing machine is an infinitely long resource. In fact, at any moment, the turing machine has a finite ID, and thus the infinity talked of is potential infinity (much like the infinity of Z) Furthermore, the TM is a mathematical model. Every TM denotes a particular computation. The author probably doesn't recognize that. The potential infinity there is an idealization. It does not mean that physical machines cannot be TMs!

If you followed that naive logic, just because you can define finite state machines that would not fit into this universe, desktop pcs would not even be finite state machines.

Think about that for a while. What do I mean by "Every TM denotes a particular computation"?

I am hoping it is clear now, and now this bit about desktop pcs not being TMs is goodbye kansas. Why on earth do you think we are studying turing machines then?

The standard reason for the potentially infinite tape of the TM is to model the fact that an ongoing computation may extend the tape (allocate more memory). This is usually thought to be a property of desktop PCs as well. With some effort, it is possible to extend the PC with as much memory as you want. There are no hard limits (except physical limits which apply to EVERYTHING including FSMs). Did you ever connect two PCs with a network? Fine. Then you know what this means.

For the record, all of this can be shown rigorously.

--Exa 15:41, 22 January 2006 (UTC)
 * "It does not mean that physical machines cannot be TMs!" -- All the real HW pcs. are FSMs and, hence, are TMs.
 * "Every TM denotes a particular computation" -- The TMs are SW programs executed by particular HW FSM. --Javalenok 18:43, 9 June 2006 (UTC)

 * Except inasmuch as they behave nondeterministically, desktop machines are finite state machines. You might argue that it's more useful to treat them using the theory of Turing machines, and I'd tend to agree, but in the strictest mathematical sense they aren't Turing machines.

 * What you call "potential infinity" is what mathematicians call "countable infinity". Every particular integer is finite, but the set of integers is infinite. Each particular state of a Turing machine is finitely describable, but there are infinitely many such states. A finite state machine can only be in finitely many different states (each of which is, again, finitely describable). As the article (currently) says, a desktop computer cannot recognize the language S ::= e | a S b, because it is a finite state machine, while a Turing machine can recognize this language. You can extend a desktop computer with more memory, but the amount of memory, hence the count of states, is always finite.

 * Please don't edit the article, at least not without more discussion.

 * -- BenRG 22:05, 22 January 2006 (UTC)

 * BenRG is correct. —Ruud 22:32, 22 January 2006 (UTC)

 * Yes, I'm a computer scientist I know what countable infinity is but the discussion is philosophical. Please read what I say carefully, the potentially infinite tape in the TM models the fact that storage is extensible. This
 * is true also for ordinary desktop PCs not only in the sense that it is "useful to treat them as such", but it can be rigorously proven that you can extend a PCs memory indefinitely. Only the very physical limits apply but that applies to everything, and no, I don't think you have really read my argument. I don't know if I can make it clearer. Let's try.  You need to contend first that a desktop computer is physically extensible, it is not a static device with a hard-limit on the number of its states. It has no difference from a TM. The view in the article seems to me a Platonist,  inadequate and unwarranted reading into Turing's original paper in which his genius makes it clear that he is modelling human computers. This article comes  around and says that he failed to do what he tried to do. I don't think so. In my opinion, it is necessary to read Turing's paper in a philosophically consistent and fully scientific way and everything will be resolved.  I am perhaps guilty for trying to make a non-trivial philosophical analysis here  but it was just bad philosophy and bad science in this page that has no place. The argument I am giving is a stronger version of the argument in the Cinderella book. It is not simply "useful" to treat the PC as the realization of an r.e.  function, it is essential. This is because, I am saying this very carefully, each TM represents a particular computation, and NOT a class of computations. Since you are a mathematician, I believe you will appreciate the difference between a function, and a set of functions. So, each ongoing physical computation in a  desktop PC corresponds to a particular TM. That a desktop PC can not "instantiate" every possible TM is a no-brainer. It is due to physical limits. But the same physical limits also apply to FSMs. There are also infinitely many particular FSMs that a given desktop PC cannot "instantiate". We do not say that a desktop PC is not a particular FSM because it cannot "instantiate" all possible FSMs! Likewise, we cannot say that a desktop PC is not a particular TM because it cannot "instantiate" all possible TMs. That is to say, you might want to revise the claim here to  be precise. In the simplest language, it would read "A desktop PC that is not  extensible can be seen as either a FSM or a TM. It is interesting to note that such a machine cannot implement a universal Turing machine which could simulate any possible Turing Machine". etc. But note, such statements would really make the article descend a slippery slope, that is why I suggest to remove all references to this, in my humble opinion redundant, "PCs are finite state machines, TMs don't physically exist" argument that is best kept for redundant discussions on comp.theory . As far as authority goes, I would like to refer to the Cinderella book. However, my refutation is self-contained.  I am hoping this somewhat complicated but correct argument is appreciated. It usually isn't understood how much thought has gone into my analysis. I would be glad if you could spend the effort to read it. Best. --Exa 23:06, 22 January 2006 (UTC)

 * Could you clarify on the following points in your first post, I might be misunderstanding you.
 * Every TM denotes a particular computation. What about universal Turing machines?
 * ... can define finite state machines that would not fit into this universe, desktop pcs would not even be finite state machines. Not every finite automaton would be a desktop pc, but every desktop pc is a finite automaton?
 * Desktop PC's are not TMs in the sense that they cannot recognize any turing recognizable language, because as the input string becomes large, the amount of memory needed to reconize becomes larger as well. Every language which has a finite length wouls be regular. —Ruud 23:48, 22 January 2006 (UTC)

 * Ok, I had better read you second, and much clearer, second reply before making the comments above. I essence I do indeed agree with you, although there remains the point that physics will limit you to a "finite tape". This should be much more carefully explained in the article. —Ruud 23:58, 22 January 2006 (UTC)

 * My problem was that I was writing "Turing machine" but thinking "universal Turing machine". You (Exa) are right, a desktop computer is a particular FSM and is also a particular Turing machine. But I still don't see which part of the current article you disagree with. You wrote "this article comes around and says that [Turing] failed to do what he tried to do", but I don't see which passage you're referring to. And the subject line of your comp.theory post that brought me here was "another clueless wikipedia article". As far as I can tell the article doesn't actually say anything that's incorrect. I'm happy with extending it but a little leery of modifying what's there. -- BenRG 02:24, 23 January 2006 (UTC)

 * You may wish to read the stuff (about half of which I wrote) at Turing machine. Deco 01:46, 23 January 2006 (UTC)

 * This discussion bothers me. I don't actually see what part of this article Exa wants to change: perhaps they could publishe their exact proposed changes here in the Talk, to make the discussion more concrete.
 * That said, Exa says some things which I also find hard to accept, and sometimes adopts a rather intemperate tone.
 * The expression "desktop PC" also seems unduly specific, so I shall use RC ("Real Computer").
 * Part of the problem is the formulation "Is an RC a TM?". We may model an RC with a TM (to analyse it theoretically) or model a TM with an RC (to experiment). Can an RC be modelled by some TM? Certainly! Can an RC model any TM? Certainly not! (As becomes clearer in the Exa's 2nd posting)
 * The same goes for FSAs, except that for any FSA, an RC on current principles is conceivable, in some universe where the laws of physics permit its construction.
 * No conceivable RC can model every TM, unless capable of indefinite automatic extension during computation. Aside from physical limitations, I know of none current with an indefinitely extensible addressing model: even unbounded URLs are translated to fixed-length IP-addresses.
 * "can be rigorously proven that you can extend a PCs memory indefinitely" Do it or link to it (preferrably in Wikipedia). Arguments against in by Marissa Kay on Thurs, Feb 9 2006 7:48 (which I have just recognised in section 8 below) sound plausible.
 * "potentially infinite tape in the TM models the fact that storage is extensible". Could be taken as such, I doubt if it is so intended.
 * The question whether the tape is infinite or can grow unboundedly is irrelevant to how a TM works. Yet the mathematical model may be more easily defined using a function from Z to the symbol-space.
 * "What do I mean by "Every TM denotes a particular computation"? " What indeed?
 * There is an ambiguity in the usage of "TM": it could mean just the mechanism, in most proofs we mean the mechanism with a certain "programme" (transition table), it could even include the input tape. In the second case I suppose a real computer can be regarded as a TM, but with preset bounds on tape size. They cannot model universal TMs, i.e. those that can model any TM.
 * To say "each TM represents a particular computation, and NOT a class of computations" seems to require you to include the input tape as an immutable part.
 * "Why .. do .. we study TMs then?" Because real computers model the behaviour of TM's on certain ranges of input, and are better described by them than by other automata. One reason is that an FSA modelling an RC has enormously many states and conceals the information about their significance, while a TM acts far more locally.
 * "Platonist, inadequate and unwarranted reading" (of Turing) - please explain with examples!
 * "he is modelling human computers" .. Turing wrote "We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions q1, q2, ..., qR which will be called 'm-configurations'.", but it seems to me only a motivation for the definition of an entirely abstract machine.
 * It would also help if it were possible to provide (a link to) the indicated passage in the Cinderella book, which I do not have, and to Turing's original paper (from List_of_important_publications_in_computer_science).
 * Looking at the history since 2006-01-03, I believe that Exa's proposals have not been carried out, but that the article does not make the assertions to which he objects. Could his postings apply to a different article or to some Talk pages? BenRG's remarks seem right. The Turing machine also seems to me very useful.
 * -- PJTraill 01:04, 19 February 2006 (UTC)

structure of article, recursion theory
Some time ago I proposed splitting out recursion theory as a separate article and renaming this content to computability theory (computer science). I now think that was a mistake. I felt that non-computer-science topics were being given short shrift in the article. It now seems to me that all research topics are given short shrift.

The exposition of the elementary material is too slow, and there's not enough outline of more advanced material. The tasks of defining what it means to be computable, and establishing that there are things that aren't computable, should be described briefly, with links to computable function and Turing machine for more details. Then there should be a series of sections, one for each identifiable theme within computability theory, with a blurb on each and a link to a more-detailed main article. Themes that come to mind are complexity theory (P vs NP, PSPACE, hierarchies, etc), structure of the Turing degrees, randomness (the sort of stuff Rod Downey and his former students do), and effective descriptive set theory; computer scientists may have others they want to add. --Trovatore 16:09, 8 February 2006 (UTC)

 * I think this might be a good idea as it does appear to be the same subject, only that computer scientists and mathematical logicians have a different focus (e.g. computer scientists start with the work of Turing and Church and stop at Oracle machines, while mathematical logians start with the work of Kleene and focus on Oracle machines and beyond). —Ruud 16:54, 8 February 2006 (UTC)

 * Well, so that was more or less what I was thinking back at the time of the split (in fact, I think I wrote something very much like it) but now I'm not so sure. It seems to me that mathematical logicians are interested in, say P and NP and related hierarchies (which are all solidly within "computable"), and computer scientists are interested in random sequences, even though they can't be computable. So while there's a difference in focus, there's no clear borderline. I just think the article spends too much time on computable-vs-not-computable, which is pretty cut and dried and kind of boring. We know what's computable and what's not; what's left to say? Of course lots of things are left to say, but the article doesn't really talk about them. --Trovatore 22:28, 8 February 2006 (UTC)

I'd move the first part of this article to automata theory if this wasn't for the fact that this would leave little of the article behind. —Ruud 22:55, 8 February 2006 (UTC)

 * Ah, that's not bad. Then there'd be room to do the sort of thing I outlined, with lots of sections that point to main articles and give brief summaries of their contents. There could be one of those sections for automata theory, too. Note that I'm not personally promising to take care of the second part of the plan, at least not soon (any large time block that I commit to WP at the moment would have to be at set theory, naive set theory, axiomatic set theory, where I've won a mild consensus for an ambitious reform). --Trovatore 23:00, 8 February 2006 (UTC)

Please do not neglect the layman (obviously "interested", a modicum of intelligence/maturity also presumed) - not only should they not go away empty-handed, they should also not come to a dead end. In other words, if one comes to the article knowing nothing of computability, one should learn something about it, and also be told where to find more information if one gets stuck. At the same time I am in favour of articles covering advanced topics and current research, though I understand that Wikipedia is not intended for publishing original research. So how does one accomodate these requirements? Are there useful templates: e.g. from now on secondary school / undergraduate maths / postgraduate mathematical maturity required. Or: this presumes understanding of article X - which might get awkward if that article keeps changing level! -- PJTraill 00:23, 20 February 2006 (UTC)

All physical systems, including PC's, are FSM's
All physical systems are finite state machines (more precisely, finite state Markov processes), the number of states being given by the exponential of the entropy (in units of Boltzmann's constant) of the system (which is, in fact, how entropy is defined); and the state transition function being that generated by the system's Hamiltonian.

The Bekenstein Bound, in turn, places a hard limit on the size of the state space and on the entropy of anything occupying a given region of space (and even the space, itself), no matter what its constituency.

More on that, below.

The finitude of the state space is, in fact, a centrally important feature of statistical and thermal physics, associated with the notions of recurrence times and ergodicity.

No machine, at the present time, uses more than a tiny fraction of the (finite) state space available to it, nor will any in the foreseeable future possibly until the full development of nanotechnology enables the employment of the atomic and subatomic structure of the physical system in question.

All systems occupying a finite volume -- including space, itself, and the fields pervading it -- no matter what their composition, will be subject to the Bekenstein Bound.

The Bekenstein Bound, more precisely, provides a hard-limit on the size of the state space associated with anything in any region of space bounded by a surface of area A. It is the exponential of A, itself, with A given in units of 1/4 of the Planck area. The hard-limit on the system's entropy is kA, where k is Boltzmann's constant and A, again, in units of 1/4 the Planck area.

It turns out that for a spherical region, the limit is exactly that which is reached the moment enough information and matter is crammed into the region to produce a black hole of the same size as the region, itself. The theoretical maximum entropy of the region is then realized as none other than what has already been asserted since the 1970's to be the entropy of the black hole, itself.

(See article on Black hole thermodynamics).

So, to make a long story short, the foregoing stands as a decisive counter-point to idea of a endlessly cramming more and more into a machine. If you cram too much storage, fields, light, electricity or matter (or even sound, pressure, tension, stress or energy) into a given machine, you'll eventually reach the point when the whole thing will collapse under its own gravity and become a black hole. At that point, the theoretical maximum will be reached for the region enclosed by the black hole.

Anything more crammed in will just make the black hole larger.

(But you'll never see it get larger, unless you fall in, nor will you see anything that's too close to the event horizon at all, since time's nearly frozen to a standstill on the event horizon and everything close to it is red shifted into invisibility).

Eventually, as per Hawking, the black hole will eventually radiate all of its energy away (and all of its information contents) as thermal radiation and evaporate away. (See Hawking Radiation and Black hole information paradox, the latter having led in just the past few years to a much deeper understanding on what an event horizon is, what it is not and its close relation to the notion of Rindler horizons).

None of the foregoing, it's worth pointing out, has any bearing on whether the universe, itself, is more than a FSM. That does not follow, even with the fact that all the finite-volumes within the universe have finite state spaces.

That is, it does not follow, unless the Universe, itself, is finite. It doesn't even need to follow that the Universe have any well-defined state space at all, in the first place, finite or infinite. As Smolin has pointed out, since the underlying state space (as per Quantum Gravity) is a certain space of 3-manifolds, and since the classification problem for 3-manifolds may very well be unsolveable, it's quite possible that there may not be any effectively defineable state space for the Universe at all. (The theory of 3-manifolds is closely related to knot theory, since a knot may be regarded as the complement of a manifold).

But the one thing that can be said is that the state space associated with the sky -- that is, the past light cone of a given spacetime point -- is most likely finite. That's because the past light cone has the structure of a hypersphere. That's true, regardless of whether the Universe is finite with positive curvature, infinite and spatially flat or infinite with negative curvature. In all cases, the sky is a hypersphere that has the Big Bang at its antipode. The Big Bang is the one and only event that is directly in the line of sight in every direction from every place in the Unverse at every time (though it is somewhat obscured by the cosmic microwave background or CMB).

Now, the classical non-trivial example of an infinite state machine (ISM) is that recognizing the Dyck language, given by the grammar S -> 1; S -> u S v S (1 = empty word); which has as its minimal state diagram <-- [0] -- u --> [1] -- u --> [2] -- u --> [3] ... <-- v --    <-- v --     <-- v -- with an infinite number of states -- easily recognized as the state diagram of a particular 1-counter machine.

In theory, a quantum system like the electromagnetic field is supposed to provide a similar infinity of states, each frequency mode admitting a set of states containing 0, 1, 2, 3 or more quanta of that frequency (see article on Quantum Electrodynamics). However, with the Bekenstein Bound in place, this will invariably break down at a high enough energy (as does the rest of quantum fieid theory). The manner of breakdown, though, is still unknown. But it exists.

Other than theoretical construct of the quantum field (which, as per the above, breaks down), there is nothing that can embody even the deceptively simple automaton depicted by the state diagram above, since it has an infinite number of states. All the more so nor any of the varieties of other infinite state automata, including the Push Down automata or Turing Machines, never mind any instance of a Universal Turing Machine.

There are, of course, instances of each of these classes where the number of states that can be reached from the start state is finite. Then, since the minimal state diagram is finite, they are equally representable as finite state machines. But in the general case -- like that for the Dyck language above -- the number of states accessible from the start state is infinite, the minimal state diagram is infinite and it is simply not representable as a finite state machine.

The utility of Turing Machines or any of the other varieties of infinite state machine formalisms (push down automata, counter automata, etc.) is that they provide a more efficient means of representing finite state machines and the physical systems they may embody. Obviously, the more powerful a formalism, the greater utility it will have -- even for those things that could theoretically be represented by lesser means.

Greater utility also means greater relevance, not the other way around!

Oracles provide a more powerful mechanism, still, and despite the fact that they are strictly more powerful than TM's, they are neither no less of an idealization than any other infinite state automaton formalism, nor any less physically relevant. But theorists haven't yet made a great deal of use of them. There is, invariably, a notion of resource bounded oracles, for each variety of oracle, and this is bound to be just as relevant for finite state machines and physical systems as is that of the resource bounded TM.

The primary mode of application of any of the varieties of ISM is through the notion of "resource-boundedness". So, it is a seriously misleading notion to even assert that the exclusion of any ISM from the real of physical reality also means their irrelevance!

The whole foundation of complexity theory, in fact, is based on this notion of resource boundedness. And that lies at the center of computer science. So, of course, an unphysical model is relevant to physical devices! But neither the 1-counter machine, push down automaton, Turing Machine, nor any variety of oracle is fully realizable within the physical world as such.
 * No offense, but although the question of whether all physically constructable machines are FSMs and the associated physics are interesting, we would need a well-known existing and reputable source making these claims to add anything to the article regarding this. If I had to guess, I'd guess the literature contains statements both ways (and we can cite both). Deco 10:24, 9 February 2006 (UTC)

 * Our long-winded anonymous friend seems to be mainly concerned with showing you can't get more than a finite state machine as a physical realization, but doesn't attempt (I think; I didn't really read it all) to show that you can get even a finite state machine. I think that'd be a problem, because FSMs are deterministic, and the physical world is not. So FSMs are idealizations too; a physical realization will always have some probability of doing the thing it's not supposed to. --Trovatore 15:35, 9 February 2006 (UTC)

 * The text appears almost identical to the contribution to by Marissa Kay on Thurs, Feb 9 2006 7:48 -- PJTraill 01:11, 19 February 2006 (UTC)

 * Both had the same author. The article, itself, clearly stands on its own since the arguments made in it are sufficiently self-contained and cogent to settle the issue, subsuming prior published literature on the matter, particularly in its taking into account the finiteness imposed as a result of the Bekenstein Bound, which has been neglected in the literature and is really the critical point in deciding the issue. The question of whether the universe is deterministic or not is open. The Schroedinger equation is, in fact, deterministic and state evolution under it, likewise so. Whether the apparent non-determinism associated with "collapse" is real or only epiphenonemal and deriveable from a more fundamental, and deterministic, substrate (e.g. by decoherence, consistent histories, or some other mechanism, etc.) is really the central issue of the so-called measurement problem of quantum theory and is as-of-yet unresolved. But, the question is moot anyway: non-deterministic FSM's are not equivalent to deterministic FSM's! This equivalence only applies to deterministic vs. non-deterministic FSA's. The non-equivalence for FSM's is readily seen by the trivial counter-example of a FSM with states {S,F}, input {X}, outputs {Y,Z}, start state S, final state F and the two transitions on input X from S to F, one transition having output Y, the other having output Z. A deterministic FSM cannot yield more than one translation for any input. This one yields the two translations for the input "X": "Y" and "Z". Therefore it is not reducible to an equivalent deterministic FSM.

Universal Turing Machines
What about Universal Turing Machines? Shouldn't they be included as well? 70.111.251.203 14:42, 7 March 2006 (UTC)

Cleanup
Found use of first person. While text is clear and easy to understand, it is still bad style. 210.50.201.22 05:35, 26 May 2006 (UTC)

If you really want to try to rewrite it without the first person, and make it even remotely close to as easy to understand, be my guest. This is a very common style for proofs. --Readams 15:59, 6 June 2006 (UTC)

confusion PCs are not FSMs
How is the following statement confusing? "Actually, these are computer programs written in general-purpose programming languages which are Turing Machines (the GP languages are, therefore, called Turing-complete), while the actual computers executing the programs are FSMs." I have included it into the article even before looking into the discussion to disambiguate the popular confusion: "Poll: Are PCs Turing Machines?" Thi is the actual and vast confusion. What I see at this page proves my doubts. Please explain what is so confusing in my sentence? Peahaps it is rather wrong or inappropriate? --Javalenok 09:02, 13 June 2006 (UTC)
 * I don't know what "confusion" you're referring to. Your statement is accurate, but the English is difficult to understand, and I'm not sure how much it adds in this context. Dcoetzee 22:30, 30 January 2008 (UTC)

Church-Turing thesis
This discussion is copied from the user page.

You changed Church-Turing thesis to claim that it had been disproven. This is certainly not the case. Please feel free to discuss your changes on the talk page of the article. CMummert · talk 10:49, 12 April 2007 (UTC)

 * The Church-Turing thesis has not been disproved. However, it needs to be stately precisely which was not done in the previous version of the article.  It is very important to distinguish between computable functions and computability (computations).--71.198.216.63 16:11, 12 April 2007 (UTC)

Deciding a language
I don't understand this idea. What does it mean to decide a language? Unfree (talk) 16:32, 30 January 2008 (UTC)
 * To decide whether a given string is in the language or not. --Trovatore (talk) 20:11, 30 January 2008 (UTC)

Attempt to merge
For anyone keeping this article on their watchlist: there is currently an attempt at merging a number of articles on (non-)recursive/(un)decidable/(un)computable sets/languages/decision problems into a single, thorough article. The experiment currently sits at Recursive languages and sets. Comments are very welcome. Pichpich (talk) 00:43, 21 February 2008 (UTC)

A paradoxical situation
This is a duplicate post from Talk:Algorithm. To keep the discussion in one place, please comment there is you are interested. — Carl (CBM · talk) 20:43, 19 March 2008 (UTC)

Reorganize the Computability articles
There is a discussion currently at Talk:Recursion theory about reorganising the computability articles to get some sort of hierarchy. In particular the thinking is that Computability should be an introduction rather than a disambiguation page. Dmcq (talk) 11:26, 14 August 2009 (UTC)

 * The reorganization is underway. The centralized discussion is at Talk:Recursion theory. — Carl (CBM · talk) 14:52, 22 August 2009 (UTC)


text:
Collagen helix

In molecular biology, the collagen triple helix or type-2 helix is the main secondary structure of various types of fibrous collagen, including type I collagen. In 1954, Ramachandran & Kartha (13, 14) advanced a structure for the collagen triple helix on the basis of fiber diffraction data. It consists of a triple helix made of the repetitious amino acid sequence glycine-X-Y, where X and Y are frequently proline or hydroxyproline. Collagen folded into a triple helix is known as tropocollagen. Collagen triple helices are often bundled into fibrils which themselves form larger fibres, as in tendons.

Structure
Glycine, proline, and hydroxyproline must be in their designated positions with the correct configuration. For example, hydroxyproline in the Y position increases the thermal stability of the triple helix, but not when it is located in the X position. The thermal stabilization is also hindered when the hydroxyl group has the wrong configuration. Due to the high abundance of glycine and proline contents, collagen fails to form a regular α-helix and β-sheet structure. Three left-handed helical strands twist to form a right-handed triple helix. A collagen triple helix has 3.3 residues per turn.

Each of the three chains is stabilized by the steric repulsion due to the pyrrolidine rings of proline and hydroxyproline residues. The pyrrolidine rings keep out of each other's way when the polypeptide chain assumes this extended helical form, which is much more open than the tightly coiled form of the alpha helix. The three chains are hydrogen bonded to each other. The hydrogen bond donors are the peptide NH groups of glycine residues. The hydrogen bond acceptors are the CO groups of residues on the other chains. The OH group of hydroxyproline does not participate in hydrogen bonding but stabilises the trans isomer of proline by stereoelectronic effects, therefore stabilizing the entire triple helix.

The rise of the collagen helix (superhelix) is 2.9 Å (0.29 nm) per residue. The center of the collagen triple helix is very small and hydrophobic, and every third residue of the helix must have contact with the center. Due to the very tiny and tight space at the center, only the small hydrogen of the glycine side chain is capable of interacting with the center. This contact is impossible even when a slightly bigger amino acid residue is present other than glycine.

title: Collagen helix


text:
Cosmic censorship hypothesis

The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities arising in general relativity.

Singularities that arise in the solutions of Einstein's equations are typically hidden within event horizons, and therefore cannot be observed from the rest of spacetime. Singularities that are not so hidden are called naked. The weak cosmic censorship hypothesis was conceived by Roger Penrose in 1969 and posits that no naked singularities exist in the universe.

Basics
Since the physical behavior of singularities is unknown, if singularities can be observed from the rest of spacetime, causality may break down, and physics may lose its predictive power. The issue cannot be avoided, since according to the Penrose–Hawking singularity theorems, singularities are inevitable in physically reasonable situations. Still, in the absence of naked singularities, the universe, as described by the general theory of relativity, is deterministic: it is possible to predict the entire evolution of the universe (possibly excluding some finite regions of space hidden inside event horizons of singularities), knowing only its condition at a certain moment of time (more precisely, everywhere on a spacelike three-dimensional hypersurface, called the Cauchy surface). Failure of the cosmic censorship hypothesis leads to the failure of determinism, because it is yet impossible to predict the behavior of spacetime in the causal future of a singularity. Cosmic censorship is not merely a problem of formal interest; some form of it is assumed whenever black hole event horizons are mentioned.

The hypothesis was first formulated by Roger Penrose in 1969, and it is not stated in a completely formal way. In a sense it is more of a research program proposal: part of the research is to find a proper formal statement that is physically reasonable, falsifiable, and sufficiently general to be interesting. Because the statement is not a strictly formal one, there is sufficient latitude for (at least) two independent formulations: a weak form, and a strong form.

Weak and strong cosmic censorship hypothesis
The weak and the strong cosmic censorship hypotheses are two conjectures concerned with the global geometry of spacetimes.

The weak cosmic censorship hypothesis asserts there can be no singularity visible from future null infinity. In other words, singularities need to be hidden from an observer at infinity by the event horizon of a black hole. Mathematically, the conjecture states that, for generic initial data, the causal structure is such that the maximal Cauchy development possesses a complete future null infinity.

The strong cosmic censorship hypothesis asserts that, generically, general relativity is a deterministic theory, in the same sense that classical mechanics is a deterministic theory. In other words, the classical fate of all observers should be predictable from the initial data. Mathematically, the conjecture states that the maximal Cauchy development of generic compact or asymptotically flat initial data is locally inextendible as a regular Lorentzian manifold. Taken in its strongest sense, the conjecture suggests locally inextendibility of the maximal Cauchy development as a continuous Lorentzian manifold [very Strong Cosmic Censorship]. This strongest version was disproven in 2018 by Mihalis Dafermos and Jonathan Luk for the Cauchy horizon of an uncharged, rotating black hole.

The two conjectures are mathematically independent, as there exist spacetimes for which weak cosmic censorship is valid but strong cosmic censorship is violated and, conversely, there exist spacetimes for which weak cosmic censorship is violated but strong cosmic censorship is valid.

Example
The Kerr metric, corresponding to a black hole of mass $$M$$ and angular momentum $$J$$, can be used to derive the effective potential for particle orbits restricted to the equator (as defined by rotation). This potential looks like: $$ V_{\rm{eff}}(r,e,\ell)=-\frac{M}{r}+\frac{\ell^2-a^2(e^2-1)}{2r^2}-\frac{M(\ell-a e)^2}{r^3}, a\equiv \frac{J}{M} $$ where $$r$$ is the coordinate radius, $$e$$ and $$\ell$$ are the test-particle's conserved energy and angular momentum respectively (constructed from the Killing vectors).

To preserve cosmic censorship, the black hole is restricted to the case of $$a < 1$$. For there to exist an event horizon around the singularity, the requirement $$a < 1$$ must be satisfied. This amounts to the angular momentum of the black hole being constrained to below a critical value, outside of which the horizon would disappear.

The following thought experiment is reproduced from Hartle's Gravity:

Problems with the concept
There are a number of difficulties in formalizing the hypothesis:

 * There are technical difficulties with properly formalizing the notion of a singularity.
 * It is not difficult to construct spacetimes which have naked singularities, but which are not "physically reasonable"; the canonical example of such a spacetime is perhaps the "superextremal" $$M<|Q|$$ Reissner–Nordström solution, which contains a singularity at $$r=0$$ that is not surrounded by a horizon. A formal statement needs some set of hypotheses which exclude these situations.
 * Caustics may occur in simple models of gravitational collapse, and can appear to lead to singularities. These have more to do with the simplified models of bulk matter used, and in any case have nothing to do with general relativity, and need to be excluded.
 * Computer models of gravitational collapse have shown that naked singularities can arise, but these models rely on very special circumstances (such as spherical symmetry). These special circumstances need to be excluded by some hypotheses.

In 1991, John Preskill and Kip Thorne bet against Stephen Hawking that the hypothesis was false. Hawking conceded the bet in 1997, due to the discovery of the special situations just mentioned, which he characterized as "technicalities". Hawking later reformulated the bet to exclude those technicalities. The revised bet is still open (although Hawking died in 2018), the prize being "clothing to cover the winner's nakedness".

Counter-example
An exact solution to the scalar-Einstein equations $$R_{ab}=2\phi_a\phi_b$$ which forms a counterexample to many formulations of the cosmic censorship hypothesis was found by Mark D. Roberts in 1985: $$ds^2=-(1+2\sigma)\,dv^2+2\,dv\,dr+r(r-2\sigma v)\left(d\theta^2 + \sin^2 \theta \,d\phi^2\right),\quad \varphi = \frac{1}{2} \ln\left(1 - \frac{2\sigma v}{r}\right),$$ where $$\sigma$$ is a constant.

title: Cosmic censorship hypothesis