Wednesday, September 1, 2010
Monday, August 30, 2010
Wodge, Wadge a lumpy bundle or mass, 1860.
|In 1991, tragedy struck the people of this planet. An erupting volcano on the Big Island of Hawai'i spewed molten chunks of GRINDCORE and 1980's DEATH METAL into the atmosphere. The NOISE given off as it tore through the earth's CRUST was totally HARDCORE; the force of the blast so extreme it sent magma all the way to Canada where it crashed through the roof of a simple house in a quiet country town. Before hitting the ground, this lava morphed into a series of musical objects: a drum machine, a BC Rich Bitch, a bass guitar and a microphone. The stunned resident was slack-jawed, though really for him, this was nothing new. Extending his left index finger to touch one of the buttons on this bizarre drum making box, the fellow scalded his digit on the visibly red hot device. What a dumb-ass. After these instruments finally cooled, the dullard tried his best to think what this all could mean. "This must be something," he barely thought. Perhaps it's a sign that the tiki gods had chosen him to be the One. Wiping away the drool, he picked up these primitive tools and began beating himself over the head with them. The sounds he created quickly soothed this savage beast. Using a coconut to record the outcome, he made a demo tape and sent it all over the world. Only two people ever wrote him back. But this was enough to prompt him to continue on for all of these wasted years. This, my friends, is the sad but true story of how WADGE came to be.|
The Wadge game is a simple infinite game discovered by William Wadge (pronounced "wage"). It is used to investigate the notion of continuous reduction for subsets of Baire space. Wadge had analyzed the structure of the Wadge hierarchy for Baire space with games by 1972, but published these results only much later in his PhD thesis. In the Wadge game G(A,B), player I and player II each in turn play integers which may depend on those played before. The outcome of the game is determined by checking whether the sequences x and y generated by players I and II are contained in the sets A and B, respectively. Player II wins if the outcome is the same for both players, i.e. x is in A if and only if y is in B. Player I wins if the outcome is different. Sometimes this is also called the Lipschitz game, and the variant where player II has the option to pass (but has to play infinitely often) is called the Wadge game.