A game that is 0-dimensional in the cardinality of its game play means that the player has no degrees of freedom to control the movement of game entities. Initially, the idea is rather confusing since it bears asking what it is a player does? Clearly, the player must be able to do something, and it must be in some way related to whatever feedback and information he is receiving from the game.
In practical terms, games that exhibit this cardinality are usually those in which the player does not control movement on the screen and must time key presses to actions that are happening. Pressing a button at the right moment implies success while not doing so means failure.
Dance Dance Revolution
In Dance Dance Revolution, the player must time keypresses to coincide with the overlapping of fixed arrows at the top of the screen with arrows that move steadily upwards from the bottom of the screen. The player cannot perform any movements nor modify the steady stream of arrows that ascends. Which arrows, and the speed at which they ascend is dependent on the music that is being played.
There is no avatar that the player controls in Guitar Hero [RedOctane, 2005]. The game resolves solely around pushing the right buttons using a guitar-shaped controller. There gameworld and avatars exists purely for cosmetic purpose and play no role in gameplay.
In Vib-Ribbon, the player must carefully time keypresses so that a rabbit-like character does not get tripped up by distortions that move towards him. If the appropriate keypress is not timed correctly, the rabbit takes a hit and could possibly lose a life. The keypress that is required is dependent on the type of distortion that is approaching the rabbit. Player does not control de movement nor the speed of movement that the rabbit makes.
This is considered a weak example because when the player presses a key, the rabbit jumps. In this sense, the player is controlling the movement of the rabbit along 1 dimension (up-down). However, the player cannot control the height of the jump or anything else related to it, thus the jump is serves more as a presentation tool rather than an actually degree of freedom regarding movement.
In Gitaroo Man the player presses buttons either by following a line with the analog stick and pressing a button in sync with marked parts of the line or pressing particular face buttons when they come flying into the centre of the screen from four directions. During the first set the onscreen character will play his guitar to send lasers at his opponent, and during the second he will dodge the lasers sent back at him. How well he fires lasers/dodges depends on how well-timed the button-presses are, thus making this a weak example. However, the power of the lasers nor the kind of dodging can be controlled by the player, just if it happens or not. Like Vib-Ribbon, then, the action is more of a way to show how well you did rather than giving you control over the hero.