Berman–Hartmanis conjecture

In structural complexity theory, the Berman–Hartmanis conjecture is an unsolved conjecture named after Leonard C. Berman and Juris Hartmanis that states that all NP-complete languages look alike, in the sense that they can be related to each other by polynomial time isomorphisms. An isomorphism between formal languages L1 and L2 is a bijective map f from strings in the alphabet of L1 to strings in the alphabet of L2, with the property that a string x belongs to L1 if and only if f belongs to L2, it is a polynomial time isomorphism if both f and its inverse function can be computed in an amount of time polynomial in the lengths of their arguments. Berman & Hartmanis observed that all languages known at that time to be NP-complete were p-isomorphic. More they observed that all then-known NP-complete languages were paddable, they proved that all pairs of paddable NP-complete languages are p-isomorphic. A language L is paddable if there is a polynomial time function f with a polynomial time inverse and with the property that, for all x and y, x belongs to L if and only if f belongs to L: that is, it is possible to pad the input x with irrelevant information y, in an invertible way, without changing its membership in the language.

Based on these results and Hartmanis conjectured that all NP-complete languages are p-isomorphic. Since p-isomorphism preserves paddability, there exist paddable NP-complete languages, an equivalent way of stating the Berman–Hartmanis conjecture is that all NP-complete languages are paddable. Polynomial time isomorphism is an equivalence relation, it can be used to partition the formal languages into equivalence classes, so another way of stating the Berman–Hartmanis conjecture is that the NP-complete languages form a single equivalence class for this relation. If the Berman–Hartmanis conjecture is true, an immediate consequence would be the nonexistence of sparse NP-complete languages, namely languages in which the number of yes-instances of length n grows only polynomially as a function of n; the known NP-complete languages have a number of yes-instances that grows exponentially, if L is a language with exponentially many yes-instances it cannot be p-isomorphic to a sparse language, because its yes-instances would have to be mapped to strings that are more than polynomially long in order for the mapping to be one-to-one.

The nonexistence of sparse NP-complete languages in turn implies that P ≠ NP, because if P = NP every nontrivial language in P would be NP-complete. In 1982, Steve Mahaney published his proof that the nonexistence of sparse NP-complete languages is in fact equivalent to the statement that P ≠ NP. For a relaxed definition of NP-completeness using Turing reductions, the existence of a sparse NP-complete language would imply an unexpected collapse of the polynomial hierarchy; as evidence towards the conjecture, Agrawal et al. showed that an analogous conjecture with a restricted type of reduction is true: every two languages that are complete for NP under AC0 many-one reductions have an AC0 isomorphism. Agrawal & Watanabe showed that, if there exist one-way functions that cannot be inverted in polynomial time on all inputs, but if every such function has a small but dense subset of inputs on which it can be inverted in P/poly every two NP-complete languages have a P/poly isomorphism, and Fenner, Fortnow & Kurtz found an oracle machine model in which the analogue to the isomorphism conjecture is true.

Evidence against the conjecture was provided by Kurtz, Mahaney & Royer. Joseph and Young introduced a class of NP-complete problems, the k-creative sets, for which no p-isomorphism to the standard NP-complete problems is known. Kurtz et al. showed that in oracle machine models given access to a random oracle, the analogue of the conjecture is not true: if A is a random oracle not all sets complete for NPA have isomorphisms in PA. Random oracles are used in the theory of cryptography to model cryptographic hash functions that are computationally indistinguishable from random, the construction of Kurtz et al. can be carried out with such a function in place of the oracle. For this reason, among others, the Berman–Hartmanis isomorphism conjecture is believed false by many complexity theorists

Danish Culture Canon

The Danish Culture Canon consists of 108 works of cultural excellence in eight categories: architecture, visual arts and crafts, literature, performing arts, children's culture. An initiative of Brian Mikkelsen in 2004, it was developed by a series of committees under the auspices of the Danish Ministry of Culture in 2006–2007 as "a collection and presentation of the greatest, most important works of Denmark's cultural heritage." Each category contains 12 works although music contains 12 works of score music and 12 of popular music and the literature section's 12th item is an anthology of 24 works. The committee for architecture was asked to choose 12 works covering landscaping, it was decided that works could either be in Denmark designed by one or more Danes or abroad designed by Danish architects. The committee consisted of: Lone Wiggers, Carsten Juel-Christiansen, Malene Hauxner, Lars Juel Thiis and Kent Martinussen; the committee for visual arts decided that only works of artists who had completed their oeuvre could be included.

They decided that members of the committee could each select a work they appreciated. In this way the committee first selected; the committee consisted of Hein Heinsen, Hans Edvard Nørregård-Nielsen, Bente Scavenius, Bjørn Nørgaard and Sophia Kalkau. The committee for design and crafts decided that selection should be based on works with a useful function which were relevant at the time they were created while remaining recognizable today, they should fall into an international perspective. The committee consisted of Merete Ahnfeldt-Mollerup, Erik Magnussen, Astrid Krogh, Ursula Munch-Petersen and Louise Campbell. In their selection, the committee for film focused on films reflecting Danish life with Danish actors; the included the film Sult which takes place in Oslo and has Swedish actors. The committee consisted of Susanne Bier, Vinca Wiedemann, Tivi Magnusson, Ole Michelsen and Jacob Neiiendam; the committee for literature found it important to select works with a quality, appreciated over time.

The selected works were considered to have made an important contribution both to Danish literature and to Danish culture in the widest sense. They reflect an bold artistic approach to works of value, they are worthy of being preserved for posterity as they serve as reference points in a modern global context. The committee consisted of Finn Hauberg Mortensen, Erik A. Nielsen, Mette Winge, Claes Kastholm Hansen and Jens Christian Grøndahl; the 12th item is an Anthology of lyrics consisting of the following 24 works: The committee for music explained that, taking account of the wide range of Danish music, they gave focus to individual works rather than a composer's oeuvre. They presented two lists: one for what they called score music, the other for popular music, although the two should be considered as a whole; the committee consisted of Per Erik Veng, Jørgen I. Jensen, Torben Bille, Inger Sørensen and Henrik Marstal; the 12th item titled Højskolesange consists of the following 12 songs: The 12th item Evergreens is an anthology consisting of the following works: The committee for performing arts explained that their selection was based on works of unique creativity representing something new for their time while still remaining meaningful today.

The committee consisted of Flemming Enevold, Karen-Maria Bille, Jokum Rohde, Sonja Richter and Erik Aschengreen. The committee was formed spontaneously, it is therefore not an independent selection. According to press reports, the canon has had limited impact and has been ineffective in its stated goal of fostering integration between the Danes and the immigrant communities. Berlingske pointed out that the canon will remain a milestone as a non-socialist government had dared to "simply state that some works are better than others" and assert in that "this country may well be a modern society in a globalised world but that does not mean we have no merit as a nation or no right to national pride." Erik A. Nielsen, a member of the canon's literature committee, is not surprised the literature canon has had such limited effect, faced as it is with a "tsunami of international commercial cultural interests." He points out that the only reason his students take an interest in Danish culture is that "they have to take exams in it.

If they are free to choose culture themselves, they go for films, rock music and a whole lot more, English or American in origin. "Kulturkanon", PDF Copy of the Website from 2006 Danish Ministry of Culture: Kulturkanonen PDF