generalized Super Mario Bros. is computationally hard

image credit: Christopher Furniss

An interesting paper by G. Aloupis, E.D. Demaine, and A. Guo has shown NP-completeness of generalized Super Mario and Donkey Kong Country. Also interesting is a paper by "V.V. Vargomax". Among the two, G. Aloupis et al., in their paper entitled "Classic Nintendo Games are (NP-)Hard" have shown a more serious proof; actually, we can set aside the work of "V.V. Vargomax".

The paper "Classic Nintendo Games are (NP-)Hard", which came out in March 2012, has shown at least the NP-hardness of certain classic Nintendo games. They have used a general framework for the design of a particular game level to which 3-SAT, an NP-complete problem, is reducible in polynomial time. NP-complete is a class of mathematical problems, and 3-SAT is a Boolean satisfiability problem, which answers the question about whether a particular Boolean expression returns true. What do these results tell us? They tell us that determining whether a level design of a particular game has a solution, that it can be finished from the starting point, is computationally hard. So if a computer will try to check deterministically whether there is a solution to a particular level, e.g. through brute-force method, it cannot do it in polynomial time unless P=NP. P is a class of mathematical problems that can be solved deterministically in polynomial time, while NP is a class of problems that can be solved non-deterministically in polynomial time. Therefore, obtaining a solution to the given problem is computationally expensive under the current theoretical conditions in which, of course, the practical consequences depend.

The results at least show that the difficulty encountered by the game designers is justified since their work revolves around how the obstacles should be arranged such that the level can be challenging enough for the players but a solution still exists. So in the case of generalized Super Mario Brothers, how can a game designer arrange the blocks, Goombas, Koopas, Firebars, and others such that it can be an interestingly difficult but playable level? No wonder it can be such a struggle to finish the levels of Super Mario. It is a game that requires creativity and ingenuity on the part of the players, and perhaps this is one of the reasons why the game is considered a classic. Right now, several versions of this game exist, but they owe it to the success of the earlier versions. In fact, arguably, Super Mario Bros. 3 is still considered to be one of the best versions of the game even though the newer versions beat it when it comes to the technical aspects such as graphics, world features, and the things that Mario can do.

talking about more ice

 

photo credit: Dave Marcus

photo credit: Dave Marcus

Moving further to the north direction above Iceland is another land called Greenland. Interesting name. Greenland is almost like Antarctica because of its polar climate. So does this mean that there's more vegetation here than in Iceland? With over around 80% of its area covered by the ice sheet, Greenland seems more icelandic or "ice-ish" than Iceland. But why call it "Greenland"? According to the Icelandic sagas, an Icelander was exiled from Iceland for murder. It was said that during his exile, he set out with his constituents to find a rumored land in the northwest. They found it, settled there, and named the land Grœnland. The name is translated as "Greenland". It was named as such in order to attract settlers. Hmm, also interesting. Were they successful? I'm not sure, but for a land with an area of about 2.2 million square kilometers, including the nearby islands, the population is a little bit more than 56,000 as of mid 2013. It is sparsely populated, but it can also be a feat already considering the freezing climate all year round.

Greenland is the largest island in the world in terms of land area since it is not considered to be a continent (think about Australia or Antarctica). However, considering the ice sheet, which is said to depress the central region to altitudes below sea level, Greenland was determined by a survey in 1951 to be composed of three large islands. We don't see it right now, but global warming can reveal that physically.

From climate reports, it was said that the average temperature all year round is getting higher. As a consequence, new islands are getting exposed; and in 2007, a new island called "Uunartoq Qeqertaq" or "Warming Island" was announced when a glacier had melted completely. Well that's probably another way of creating an island after a previous method called "series of volcanic eruptions". So we now have "glacier melting" and "series of volcanic eruptions". Kidding aside, melting of glacier is indeed a sign of a serious climate situation. Just like Antarctica, Greenland is also experiencing the pressure of climate change.

There is something green in that island indeed:

photo credit: Susan Liepa

on the way to Surtsey

 

 

photo credit: Le Jhe

 

photo credit: Bruce McAdam

 

photo credit: David in Derbyshire

Here we go, Surtsey. It's Surtsey for Surtsey Island, but wait, it is not open to visitors. It's only open to scientists, particularly those who are searching for life forms. How about visitors with scientific inclination? If it's a sort of scientific laboratory where cutting-edge technology is currently being developed with the race for some patent, then I don't think casual visitors are allowed. There's even confidentiality agreement in this type of research. So probably cutting-edge and crucial is brewing in that island, not to mention probably dangerous and unstable. This sounds like an isotope during the development of the atomic bomb. Anyway, we don't want to exaggerate things such as thinking about an island to be on the same level as the atomic bomb, but there are indeed novel things in that island.

As one may have read about this island, Surtsey is a brand new island in Iceland. Yes, there is this thing as a "brand new" island, and moreover a "geologically young land" description also exists. So that makes Surtsey Island a brand new island within the territory of a geologically young land called Iceland. From science books, a geologically young land during the early years of the planet Earth is characterized by boiling and molten substances. No wonder Iceland is characterized by geysers and active volcanoes. That's a sharp contrast to its sub-polar climate and name as "Ice" land. Also add to the mixture the names Hekla, Eldgjá, Herðubreið and Eldfell, which are the notable volcanoes in the territory. Given these, Iceland is indeed another interesting place on Earth.

Surtsey Island was formed in 1963 during a series of volcano eruptions from late 1963 to mid 1968. The geological developments in the island sort of surprised the scientists since varied land features like sandy beaches, lagoons, cliffs, and canyons were created in less than a decade. This is in contrast to what was believed to take thousands of years to develop geologically.

stigmergy: follow the leader

WARNING: This is an attempt to be a little bit technical.

We now describe the stigmergy among ants, and we present three models: A, B, and C.  The first model, model A, is a set of equations.  This model is considered scientific while models B and C are non-scientific (not sure if B and C fall under the "artistic" category though).

A:
Model A is an abstraction of the foraging behavior of the ants called the ant system.  Imagine that there is an undirected graph G = (V,E) such that V is a set of vertices or points and E is a set of edges.  The state transition rule is given by the following equation:

Equation 1 is the probability that the ant chooses to move to point s given its current position at point r. Note that r and s are elements of V with (r,s) is an element of E.

Through these equations, there is an indirect form of communication through the amount of pheromone on a particular trail, a form of memory not locally stored within the individual ants, and this communication is called stigmergy.

B:

C:
Instrumental track: "The Odd Dawdler" (03:40), 2013

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sail, sail, sail

Instrumental track: "In Peril" (04:18), 2013

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Way of the Wind

A tree on a high cliff faced a strong wind resulting to the dispersal of its leaves into the countryside.  The leaves moved according to the way of the wind and glided above a waterfall, a dense forest, and then a vast plain.  Some of them landed on a river, and the river carried them for several miles until they reached the sea.  The story can go on and on until probably a surviving leaf has circumnavigated the whole world.  So this is an attempt to capture the moment of the tree's confrontation with the wind.

Instrumental track: "Way of the Wind" (03:11), 2013

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