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The world states of a cyclical 2 × 2 grid world í µí±Š í µí± , where changes to the world are due to an agent moving either up, down, left, or right. The position of the agent in the world is represented by the position of the circled A.

The world states of a cyclical 2 × 2 grid world í µí±Š í µí± , where changes to the world are due to an agent moving either up, down, left, or right. The position of the agent in the world is represented by the position of the circled A.

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In this paper, we propose a framework to extract the algebra of the transformations of worlds from the perspective of an agent. As a starting point, we use our framework to reproduce the symmetry-based representations from the symmetry-based disentangled representation learning (SBDRL) formalism proposed by [1]; only the algebra of transformations...

Contexts in source publication

Context 1
... consider a cyclical 2 × 2 grid world, denoted by í µí²² í µí± , containing an agent as shown in Figure 1. The transformations of í µí²² í µí± are due to an agent moving either up (í µí±ˆ), down (í µí°·), left (í µí°¿), right (í µí± ), or doing nothing (1). ...
Context 2
... transformations of í µí²² í µí± are due to an agent moving either up (í µí±ˆ), down (í µí°·), left (í µí°¿), right (í µí± ), or doing nothing (1). The possible world states of í µí²² í µí± are shown in Figure 1. í µí²² í µí± , and variations of it, is used as a running example to illustrate the concepts presented in this paper. ...
Context 3
... say the world being cyclical means that if the agent performs the same action enough times, then the agent will return to its starting position; for example, for the world í µí²² í µí± if the agent performs the action í µí±ˆ twice when the world is in state í µí±¤ 0 in Figure 1 then the world will transition into the state í µí±¤ 0 (i.e., í µí±ˆ 2 * í µí±¤ 0 = í µí±¤ 0 ). The transition due to performing each action in each state can be found in Table 1. 1 í µí±ˆ í µí°· í µí°¿ í µí± í µí±¤ 0 í µí±¤ 0 í µí±¤ 2 í µí±¤ 2 í µí±¤ 1 í µí±¤ 1 í µí±¤ 1 í µí±¤ 1 í µí±¤ 3 í µí±¤ 3 í µí±¤ 0 í µí±¤ 0 í µí±¤ 2 í µí±¤ 2 í µí±¤ 0 í µí±¤ 0 í µí±¤ 3 í µí±¤ 3 í µí±¤ 3 í µí±¤ 3 í µí±¤ 1 í µí±¤ 1 í µí±¤ 2 í µí±¤ 2 Table 1: Each entry in this table shows the outcome state of the agent performing the action given in the column label when in the world state given by the row label. ...
Context 4
... is not a consumable in every state, therefore if the agent performs the consume action in any state except í µí±¤ 1 then the effect will be the same as if the agent had performed the no-op action 1 ∈ í µí°´/∼ (see Figure 11). The action Cayley table for world í µí²² í µí±í µí±œí µí±›í µí± í µí±¢í µí±ší µí±Ží µí±í µí±™í µí±’ with restricted actions treated as identity actions contains 64 elements. ...
Context 5
... will now apply the masking treatment of restricted actions to the world í µí²² í µí±í µí±œí µí±›í µí± í µí±¢í µí±ší µí±Ží µí±í µí±™í µí±’ (see Figure 10 for world states); for example, if í µí²² í µí±í µí±œí µí±›í µí± í µí±¢í µí±ší µí±Ží µí±í µí±™í µí±’ is not in state í µí±¤ 1 then the consume action is undefined. The action Cayley table for í µí²² í µí±í µí±œí µí±›í µí± í µí±¢í µí±ší µí±Ží µí±í µí±™í µí±’ with the masked treatment of the walls contains 20 elements. ...

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