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Boolean satisfiability problem

In logic and computer science, the Boolean satisfiability problem is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make = TRUE. In contrast, "a AND NOT a" is unsatisfiable. SAT is the first problem, proven to be NP-complete; this means that all problems in the complexity class NP, which includes a wide range of natural decision and optimization problems, are at most as difficult to solve as SAT. There is no known algorithm that efficiently solves each SAT problem, it is believed that no such algorithm exists.

As of 2007, heuristic SAT-algorithms are able to solve problem instances involving tens of thousands of variables and formulas consisting of millions of symbols, sufficient for many practical SAT problems from, e.g. artificial intelligence, circuit design, automatic theorem proving. A propositional logic formula called Boolean expression, is built from variables, operators AND, OR, NOT, parentheses. A formula is said to be satisfiable if it can be made TRUE by assigning appropriate logical values to its variables; the Boolean satisfiability problem is, given a formula. This decision problem is of central importance in many areas of computer science, including theoretical computer science, complexity theory, algorithmics and artificial intelligence. There are several special cases of the Boolean satisfiability problem in which the formulas are required to have a particular structure. A literal is either a variable, called positive literal, or the negation of a variable, called negative literal.

A clause is a disjunction of literals. A clause is called a Horn clause. A formula is in conjunctive normal form. For example, x1 is a positive ¬ x2 is a negative literal, x1 ∨ ¬ x2 is a clause; the formula ∧. The formula is satisfiable, by choosing x1 = FALSE, x2 = FALSE, x3 arbitrarily, since ∧ ∧ ¬FALSE evaluates to ∧ ∧ TRUE, in turn to TRUE ∧ TRUE ∧ TRUE. In contrast, the CNF formula a ∧ ¬a, consisting of two clauses of one literal, is unsatisfiable, since for a=TRUE or a=FALSE it evaluates to TRUE ∧ ¬TRUE or FALSE ∧ ¬FALSE, respectively. For some versions of the SAT problem, it is useful to define the notion of a generalized conjunctive normal form formula, viz. as a conjunction of arbitrarily many generalized clauses, the latter being of the form R for some boolean operator R and literals li. Different sets of allowed boolean operators lead to different problem versions; as an example, R is a generalized clause, R ∧ R ∧ R is a generalized conjunctive normal form. This formula is used below, with R being the ternary operator, TRUE just when one of its arguments is.

Using the laws of Boolean algebra, every propositional logic formula can be transformed into an equivalent conjunctive normal form, which may, however, be exponentially longer. For example, transforming the formula ∨ ∨... ∨ into conjunctive normal form yields ∧ ∧ ∧ ∧... ∧ ∧ ∧ ∧. SAT was the first known NP-complete problem, as proved by Stephen Cook at the University of Toronto in 1971 and independently by Leonid Levin at the National Academy of Sciences in 1973; until that time, the concept of an NP-complete problem did not exist. The proof shows how every decision problem in the complexity class NP can be reduced to the SAT problem for CNF formulas, sometimes called CNFSAT. A useful property of Cook's reduction is. For example, deciding whether a given graph has a 3-coloring is another problem in NP. NP-completeness only refers to the run-time of the worst case instances. Many of the instances that occur in practical applications can be solved much more quickly. See Algorithms for solving SAT below.

SAT is trivial if the formulas are restricted to those in disjunctive normal form, that is, they are disjunction of conjunctions of literals. Such a formula is indeed satisfiable if and only if at least one of its conjunctions is satisfiable, a conjunction is satisfiable if and on

Murder of Amanda Duffy

Amanda Duffy, a 19-year-old Scottish student, was killed in 1992. The main suspect, Francis Auld, was tried for murder in the High Court of Justiciary in Glasgow and was acquitted when the jury returned a majority verdict of "not proven". A bid by prosecutors to try Auld for a second time on the basis of new evidence was rejected by the courts in 2016. Auld died of cancer in July 2017; the outcome of Auld's trial prompted a national conversation around the continued existence of the "not proven" verdict and around double jeopardy rules. Duffy, a 19-year-old student at Motherwell College, went missing in the early hours of the morning of Saturday 30 May 1992 after a night out with friends, celebrating the fact she had been called to audition at the Royal Scottish Academy of Music and Drama, her body was found by passers-by that evening in an area of waste ground near a car park at Miller Street, South Lanarkshire. Duffy was found "lying on her back, naked from the waist down, with her face and head covered in blood" and branches and twigs "had been inserted into her mouth and vagina".

According to a post mortem examination, she had died between 1.30 am and 1.30 pm, having suffered extensive blunt force injuries to the head and neck, as well as asphyxia and injuries to the anus and rectum. 20-year-old Francis Auld was tried for Duffy's murder in 1992. Witnesses had seen Auld with Duffy between 1 am. A bite to Duffy's right breast, which would have been "excruciatingly painful" and was inflicted within an hour prior to death, matched Auld's dental features. However, Auld said that he had left Duffy in the company of someone named "Mark", never identified; the jury returned a majority verdict of "not proven" in November 1992. In 1994, Auld was convicted of making threatening phone calls to two former friends, sentenced to 240 hours community service, he admitted telling one of them "Patrick, you thought. Well, you're next, after Caroline." In 1995, Duffy's parents and Kathleen, sued Auld in civil court, where the standard of evidence is lower. Auld did not contest the lawsuit and the couple were awarded a £50,000 payout.

This amount was never paid. Following the verdict in the criminal trial, Duffy's parents launched a high-profile campaign for the "not proven" verdict to be abolished in Scots law. A national petition was launched at an event in Glasgow addressed by Joe Duffy. In 1993, the couple's Member of Parliament, George Robertson, launched the Criminal Procedure Bill in the House of Commons to scrap the verdict, though its likelihood of success was considered slim; the Duffys' campaign increased pressure on the Scottish Office, which launched a consultation on scrapping the "not proven" verdict in 1994. MP John Home Robertson, in a 1995 bid to scrap the verdict, praised "Kathleen and Joe Duffy for the thoughtful and constructive campaign that they have been waging"; when the Scottish Parliament debated scrapping the verdict in 2016, the Duffy case was cited by MSP Michael McMahon in support of scrapping the verdict. After the introduction of the Double Jeopardy Act 2011, which allows for a person to be tried twice for the same crime in certain circumstances, Strathclyde Police reopened an investigation into Duffy's murder in 2012 because they believed that "certain people have information in relation to Amanda's murder that they are withholding from a sense of misguided loyalty".

Police Scotland's cold case unit re-examined the crime on the instruction of the Crown Office. In 2015, prosecutors launched a bid under the Act to re-try Auld for the murder. However, the bid was rejected by judges in February 2016. Jim Govan, the chief forensic scientist at the original trial went on public record, saying the jury got the verdict wrong and there was more than enough evidence to convict. Auld died of cancer in July 2017. List of unsolved deaths

Francis Barber

Francis Barber, born Quashey, was the Jamaican manservant of Samuel Johnson in London from 1752 until Johnson's death in 1784. Johnson made him his residual heir, with £70 a year to be given him by Trustees, expressing the wish that he move from London to Lichfield, Johnson's native city. After Johnson's death, Barber did this, marrying a local woman. Barber was bequeathed Johnson's books and papers, a gold watch. In years he had acted as Johnson's assistant in revising his famous Dictionary of the English Language and other works. Barber was an important source for Boswell concerning Johnson's life in the years before Boswell himself knew Johnson. Barber was born a slave in Jamaica on a sugarcane plantation belonging to the Bathurst family, his original name was Quashey, a common name for men of Coromantee origin. At the age of about 15, he was brought to England by his owner, Colonel Richard Bathhurst, whose son called Richard, was a close friend of Johnson. Barber was sent to school in Yorkshire.

Johnson's wife Elizabeth died in 1752, plunging Johnson into a depression that Barber vividly described to James Boswell. The Bathursts sent Barber to Johnson as a valet. Although the legal validity of slavery in England was ambiguous at this time, when the elder Bathurst died two years he gave Barber his freedom in his will, with a small legacy of £12. Johnson himself was an outspoken opponent of slavery, not just in England but in the American colonies as well. Barber went to work for an apothecary in Cheapside but kept in touch with Johnson, he signed up as a sailor for the Navy. He served as a "landman" aboard various ships, received regular pay and good reports, saw the coast of Britain from Leith to Torbay, acquired a taste for tobacco, he was discharged "three days before George II died", in other words on 22 October 1760, with a pipe between his teeth and a roll in his gait, he returned to London and to Johnson to be his servant. Barber's brief maritime career is known from James Boswell's Life of Johnson: Later Johnson put Barber, by in his early thirties, in a school so that he could act as Johnson's assistant.

From Boswell's Life: Barber is mentioned in James Boswell's Life of Johnson and other contemporary sources, there are at least two versions of a portrait, one now in Dr. Johnson's House, which may be of him. Most recent art historians thought it was painted by James Northcote, or by Northcote's master Sir Joshua Reynolds, one of Barber's Trustees under the will. An alternative view expressed on a BBC programme, is that it is by Reynolds himself, but of his own black servant, not Barber; when making his will, Johnson asked Sir John Hawkins his first biographer, what provision he should make for Barber. Sir John said. I shall be "noblissimus" replied Johnson, give him 70. Hawkins disapproved, after Johnson's death criticised his "ostentatious bounty favour to negroes"; the bequest was indeed covered in the press. Johnson, in fact, left £750 in the trust of his friend Bennet Langton from which he was expected to pay an annuity. Barber moved with his family to a rented terrace house in Lichfield, Johnson's birthplace, where – as a Gentleman's Magazine correspondent reported – he spent his time "in fishing, cultivating a few potatoes, a little reading".

He opened up a small village school in nearby Burntwood. The money from his inheritance did not last and Barber sold off his store of Johnson memorabilia to defray his debts. Johnson's biographers and Hester Piozzi, were critical of Barber's marriage to a white woman, he died in Stafford on 13 January 1801 due to an unsuccessful operation at Staffordshire Royal Infirmary. He was survived by his son, Samuel Barber, his daughter and his wife, Elizabeth. Samuel became a Methodist lay preacher, while Ann set up a small school. Both Samuel and Ann married white partners, his descendants still farm near Lichfield. Francis Barber's marriage to Elizabeth Ball is featured in the short silent animation The Trouble with Francis, whilst Barber appears as a character in the 2015 play Mr Foote's Other Leg. Edward Thaddeus Barleycorn Barber Black British Historical immigration to Great Britain References SourcesJames Boswell: Life of Johnson Sir John Hawkins: Life of Johnson 100 Great black britons BBC feature Transcript of Johnson's Will Life, including portrait Lecture on Johnson and the abolition of Slavery at the Wayback Machine