A thermostat for a heater may have two belief states: off and heating. The environment may have three states: cold, comfortable, and hot. There are thus six states corresponding to the different combinations of belief and environment states. These states may not fully describe the world, but they are adequate to describe what a thermostat should do. The thermostat should move to, or stay in, heating if the environment is cold and move to, or stay in, off if the environment is hot. If the environment is comfortable, the thermostat should stay in its current state. The thermostat agent keeps the heater on in the heating state and keeps the heater off in the off state.

An agent that has to look after a house may have to reason about whether light bulbs are broken. It may have features for the position of each switch, the status of each switch (whether it is working okay, whether it is shorted, or whether it is broken), and whether each light works. The feature $\text{𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛}\mathit{}\mathrm{\_}\mathit{}{s}_{\mathrm{2}}$ may be a feature that has value up when switch $s_{\mathrm{2}}$ is up and has value down when the switch is down. The state of the house’s lighting may be described in terms of values for each of these features. These features depend on each other, but not in arbitrarily complex ways; for example, whether a light is on may just depend on whether it is okay, whether the switch is turned on, and whether there is electricity.

Consider an agent that has to recognize letters of the alphabet. Suppose the agent observes a binary image, a $\mathrm{30}\mathrm{\times }\mathrm{30}$ grid of pixels, where each of the 900 grid points is either black or white. The action is to determine which of the letters $\mathrm{\{}a\mathrm{,}\mathrm{\dots }\mathrm{,}z\mathrm{\}}$ is drawn in the image. There are ${\mathrm{2}}^{\mathrm{900}}$ different possible states of the image, and so ${\mathrm{26}}^{{\mathrm{2}}^{\mathrm{900}}}$ different functions from the image state into the characters $\mathrm{\{}a\mathrm{,}\mathrm{\dots }\mathrm{,}z\mathrm{\}}$. We cannot even represent such functions in terms of the state space. Instead, handwriting recognition systems define features of the image, such as line segments, and define the function from images to characters in terms of these features. Modern implementations learn the features that are useful.

The agent that looks after a house in Example 1.6 could have the lights and switches as individuals, and relations position and connected_to. Instead of the feature $\text{𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛}\mathit{}\mathrm{\_}\mathit{}{s}_{\mathrm{2}}\mathrm{=}\text{𝑢𝑝}$, it could use the relation $\text{𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛}\mathit{}\mathrm{(}{s}_{\mathrm{2}}\mathrm{,}\text{𝑢𝑝}\mathrm{)}$. This relation enables the agent to reason about all switches or for an agent to have general knowledge about switches that can be used when the agent encounters a switch.

If an agent is enrolling students in courses, there could be a feature that gives the grade of a student in a course, for every student–course pair where the student took the course. There would be a passed feature for every student–course pair, which depends on the grade feature for that pair. It may be easier to reason in terms of individual students, courses and grades, and the relations grade and passed. By defining how passed depends on grade once, the agent can apply the definition for each student and course. Moreover, this can be done before the agent knows of any of the individuals and so before it knows any of the features.

The two-argument relation passed, with 1000 students and 100 courses can represent $\mathrm{1000}\mathrm{*}\mathrm{100}\mathrm{=}\mathrm{100000}$ propositions and so ${\mathrm{2}}^{\mathrm{100000}}$ states.