Oracle machine

In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, able to solve certain decision problems in a single operation; the problem can be of any complexity class. Undecidable problems, such as the halting problem, can be used. An oracle machine can be conceived as a Turing machine connected to an oracle; the oracle, in this context, is an entity capable of solving some problem, which for example may be a decision problem or a function problem. The problem does not have to be computable; the oracle is a "black box", able to produce a solution for any instance of a given computational problem: A decision problem is represented as a set A of natural numbers. An instance of the problem is an arbitrary natural number; the solution to the instance is "YES" if the number is in the set, "NO" otherwise. A function problem is represented by a function f from natural numbers to natural numbers.

An instance of the problem is an input x for f. The solution is the value f. An oracle machine can perform all of the usual operations of a Turing machine, can query the oracle to obtain a solution to any instance of the computational problem for that oracle. For example, if the problem is a decision problem for a set A of natural numbers, the oracle machine supplies the oracle with a natural number, the oracle responds with "yes" or "no" stating whether that number is an element of A. There are many equivalent definitions of oracle Turing machines; the one presented here is from van Melkebeek. An oracle machine, like a Turing machine, includes: a work tape: a sequence of cells without beginning or end, each of which may contain a B or a symbol from the tape alphabet. In addition to these components, an oracle machine includes: an oracle tape, a semi-infinite tape separate from the work tape; the alphabet for the oracle tape may be different from the alphabet for the work tape. An oracle head which, like the read/write head, can move left or right along the oracle tape reading and writing symbols.

From time to time, the oracle machine may enter the ASK state. When this happens, the following actions are performed in a single computational step: the contents of the oracle tape are viewed as an instance of the oracle's computational problem; the effect of changing to the ASK state is thus to receive, in a single step, a solution to the problem instance, written on the oracle tape. There are many alternative definitions to the one presented above. Many of these are specialized for the case. In this case: Some definitions, instead of writing the answer to the oracle tape, have two special states YES and NO in addition to the ASK state; when the oracle is consulted, the next state is chosen to be YES if the contents of the oracle tape are in the oracle set, chosen to the NO if the contents are not in the oracle set. Some definitions eschew the separate oracle tape; when the oracle state is entered, a tape symbol is specified. The oracle is queried with the number of times. If that number is in the oracle set, the next state is the YES state.

Another alternative definition makes the oracle tape read-only, eliminates the ASK and RESPONSE states entirely. Before the machine is started, the indicator function of the oracle set is written on the oracle tape using symbols 0 and 1; the machine is able to query the oracle by scanning to the correct square on the oracle tape and reading the value located there. These definitions are equivalent from the point of view of Turing computability: a function is oracle-computable from a given oracle under all of these definitions if it is oracle-computable under any of them; the definitions are not equivalent, from the point of view of computational complexity. A definition such as the one by van Melkebeek, using an oracle tape which may have its own alphabet, is required in general; the complexity class of decision problems solvable by an algorithm in class A with an oracle for a language L is called AL. For example, PSAT is the class of problems solvable in polynomial time by a deterministic Turing machine with an oracle for the Boolean satisfiability problem.

The notation AB can be extended to a set of languages B, by using the following definition: A B = ⋃ L ∈ B A L When a language L is complete for some class B AL=AB provided that machines in A can execute reductions used in the completeness definition of class B. In particular, since SAT is NP-complete with respect to polynomial time reductions, PSA

Pete Pfitzinger

Peter Dickson Pfitzinger is an American former distance runner, who became an author and exercise physiologist. He is best known for his accomplishments in the marathon, an event in which he represented the United States in two Summer Olympic Games: the Los Angeles Olympics and the 1988 Seoul Olympics. In the 1984 Olympic Marathon Team Trials in Buffalo, New York, Pfitzinger became known among American marathoners by taking the lead halfway through the race, relinquishing it in the final mile storming past the favored Alberto Salazar in the final fifty yards to win the race, in a time of 2:11:43. In the 1988 Olympic Marathon Team Trials, held in Jersey City, New Jersey, Pfitzinger finished 3rd in a time of 2:13:09, to qualify for his second Olympic Games. In other marathons apart from the Olympic Trials and Olympic Games, Pfitzinger won the Syracuse marathon in 1981, the Wiri marathon in 1983, the San Francisco Marathon in 1983 and 1986, he was 2nd at the Montreal marathon in 1983. He was 3rd at the Nike OTC Marathon in 1981 and at the New York City Marathon in 1987.

He was a consistent performer: All of his 13 career marathons were run in times between 2:11:43 - 2:15:21. He won 5 of his 13 marathons, finished 2nd or 3rd in 4 others. Apart from the Olympic Games marathons, his only other finishes outside the top 3 were at the New York City Marathon in 1986 and at the 1985 Marathon World Cup, where he finished 18th. Pfitzinger is a 1979 graduate of Cornell University, he holds a Master of Business Administration from Cornell's Johnson Graduate School of Management and a Master of Arts in exercise science from the University of Massachusetts at Amherst. Pfitzinger is the co-author of two popular training books for distance runners - Advanced Marathoning and Road Racing for Serious Runners, he is a senior writer for Running Times Magazine. Pfitzinger's wife Christine is a former world-class runner; the Pfitzingers have lived in New Zealand - Chrissey's country of origin - since the mid-1990s. Pete Pfitzinger at Olympics at

Thomas Frei

Thomas Frei is a Swiss road bicycle racer. Frei was Swiss Junior champion in road cycling in 2002. Between 2004 and 2006 he cycled for the Bürgis Cycling Team and won the Mountain Championship for U23 cyclists. In the 2007 and 2008 seasons, Frei rode for UCI ProTour team Astana before moving to BMC Racing Team for 2009 on a two-year contract. There he was domestique for future Tour de France champion Cadel Evans. In his second season with BMC, Frei tested positive for EPO, he was withdrawn from competition by his team in April 2010 during the Giro del Trentino when the doping violation was revealed, before the team released him altogether. As of August 2011, Frei was training full-time with the intention of returning to professional cycling upon the completion of his two-year suspension from competition. Official Site Thomas Frei at Cycling Archives Thomas Frei on Twitter