Wednesday, August 11, 2010

Vinay Deolalikar, a mathematician who works for HP Labs, claims to have proven that P is not equal to NP. The problem is the greatest unsolved problem in theoretical computer science and is one of seven problems in which the Clay Mathematics Institute has offered million dollar prizes to the solutions.

The question of whether P equals NP essentially asks whether there exist problems which take a long time to solve but whose solutions can be checked quickly. More formally, a problem is said to be in P if there is a program for a Turing machine, an ideal theoretical computer with unbounded amounts of memory, such that running instances of the problem through the program will always answer the question in polynomial time — time always bounded by some fixed polynomial power of the length of the input. A problem is said to be in NP, if the problem can be solved in polynomial time when instead of being run on a Turing machine, it is run on a non-deterministic Turing machine, which is like a Turing machine but is able to make copies of itself to try different approaches to the problem simultaneously.

Mathematicians have long believed that P does not equal NP, and the question has many practical implications. Much of modern cryptography, such as the RSA algorithm and the Diffie-Hellman algorithm, rests on certain problems, such as factoring integers, being in NP and not in P. If it turned out that P=NP, these methods would not work but many now difficult problems would likely be easy to solve. If P does not equal NP then many natural, practical problems such as the traveling salesman problem are intrinsically difficult.

In 2000, the Clay Foundation listed the “Clay Millenium Problems,” seven mathematical problems each of which they would offer a million dollars for a correct solution. One of these problems was whether P equaled NP. Another of theseseven, the Poincaré conjecture, was solved in 2002 by Grigori Perelman who first made headlines for solving the problem and then made them again months later for refusing to take the prize money.

On August 7, mathematician Greg Baker noted on his blog that he had seen a draft of a claimed proof by Deolalikar although among experts a draft had apparently been circulating for a few days. Deolalikar’s proof works by connecting certain ideas in computer science and finite model theory to ideas in statistical mechanics. The proof works by showing that if certain problems known to be in NP were also in P then those problems would have impossible statistical properties. Computer scientists and mathematicians have expressed a variety of opinions about Deolalikar’s proof, ranging from guarded optimism to near certainty that the proof is incorrect. Scott Aaronson of the Massachusetts Institute of Technology has expressed his pessimism by stating that he will give $200,000 of his own money to Deolalikar if the proof turns out to be valid. Others have raised specific technical issues with the proof but noted that the proof attempt presented interesting new techniques that might be relevant to computer science whether or not the proof turns out to be correct. Richard Lipton, a professor of computer science at Georgia Tech, has said that “the author certainly shows awareness of the relevant obstacles and command of literature supporting his arguments.” Lipton has listed four central objections to the proof, none of which are necessarily fatal but may require more work to address. On August 11, 2010, Lipton reported that consensus of the reviewers was best summarized by mathematician Terence Tao, who expressed the view that Deolalikar’s paper probably did not give a proof that P!=NP even after major changes, unless substantial new ideas are added.

Friday, June 26, 2015

Annabel Goldie, Scottish Conservative Party leader from 2005 to 2011, has announced she will stand down as an MSP at the next elections in 2016. Goldie, who has been an MSP for the West Scotland (previously West of Scotland) electoral region since the Scottish Parliament’s formation in 1999, said she intends to focus on her role in the House of Lords, where she has been a peer since 2013.File:Annabel Goldie.jpg

In a statement today, Goldie said leading the party was an “enormous honour” for her. She also said: “It has afforded me both satisfaction and pleasure to serve my constituents and to serve the parliament and I will look back with great happiness at my time as an MSP. I am grateful to friends and colleagues from all parties for their support. Sometimes we found common ground, sometimes we disagreed but never I hope with rancour nor disrespect. Politics is a rough trade but we have built a strong parliament in Scotland of which we can all be rightly proud.” She said because of Ruth Davidson, her successor as Scottish Conservative leader, the party is now “in fine fettle and stands a great chance of making real progress in the years ahead,” concluding by saying: “I look forward to continuing to work as part of that effort in the House of Lords in the years to come.”

Davidson responded to the news by calling Goldie an “unstoppable force”, adding: “She has been an inspiration to a whole generation of Scottish Conservatives, and she has been a tremendous mentor, support and friend to me. In Holyrood, she has fostered both affection and respect from all members – regardless of their political affiliation – and her retirement from the Scottish Parliament will leave an Annabel-sized hole which won’t ever quite be filled. She is unique.” Meanwhile, David Cameron, UK Conservative leader and UK Prime Minister, said: “Annabel is one of those rare breeds in Scottish politics, somebody known by her first name alone. When she was Scottish Conservative leader, I valued her sage advice. She has been a towering strength to our party in Scotland, a doughty debater in the TV studios and Scottish Parliament and has one of the sharpest wits around. I wish her a long and happy retirement after 17 years unstinting service at Holyrood – but look forward to seeing her on the red benches of the Lords for years to come.”

Goldie, the Scottish Conservatives’ first ever female leader, was elected unopposed. She took up the role in the aftermath of David McLetchie’s resignation from the role in an expenses usage controversy and subsequent resignation of Brian Monteith from his Conservative whip role in the Scottish Parliament for briefing the media against him. Meanwhile, as Scottish Conservatives won 18 seats in the Scottish Parliament in 1999 and 2003, the party had been less successful in UK general elections in Scotland; Conservatives went up from zero out of a possible 72 UK MPs in Scotland in 1997 to one in 2001. This led to Goldie remarking in her inaugural speech in 2005 that: “The wheels are back on the wagon – and I’m the nag hitched up to tow it.” She also said: “The party is still way ahead of where it was in 1997. And my first task is to take it forward to 2007.” However, under Goldie’s leadership, the number of seats the Scottish Conservatives won in the Scottish Parliament slightly decreased from 18 in 2003 to 17 in 2007 and to 15 in 2011. At the same time, the number of Conservative MPs stood at one out of a possible 59 after the 2010 UK general election.

In the aforementioned 2005 speech, she also said the party could be trusted with devolution in Scotland, adding: “making devolution work better means real devolution: not the lumbering and cripplingly expensive array of government departments, government advisers, consultants, quangos, quasi-quangos and agencies with all their expensive appendages, but devolving down to people and their communities, their right to make their own decisions about their lives, how for example they procure healthcare and how they educate their children.” Goldie would go on to sit on the advisory board for the Smith Commission, which was set up to examine which further political powers should be devolved to Scotland following the 2014 Scottish independence referendum. More recently, Goldie supported a reduction in the voting age for Scottish Parliament elections from 18 to 16 in a vote earlier this month, commenting: “I think it is an opportunity for them to continue their high level of engagement in topical affairs that we saw with the independence referendum.”

Goldie, a member of the Salvation Army’s West of Scotland Advisory Board and a Church of Scotland elder, is not the only Scottish Conservative MSP intending to stand down in 2016. Mary Scanlon, Gavin Brown, Alex Fergusson and Nanette Milne all reportedly intend to leave the Scottish Parliament next year.