The variable Class_time may denote the starting time for a particular class. The domain of Class_time may be the following set of possible times:

The variable Height_joe may refer to the height of a particular person at a particular time and have as its domain the set of real numbers, in some range, that represent the height in centimeters. Raining may be a random variable with domain $\mathrm{\{}\text{𝑡𝑟𝑢𝑒}$, $\text{𝑓𝑎𝑙𝑠𝑒}\mathrm{\}}$, which has value true if it is raining at a particular time.

The assignment $\mathrm{\{}\mathit{\text{Class\_time}}\mathrm{=}\mathrm{11}\mathrm{,}\mathit{\text{Height\_joe}}\mathrm{=}\mathrm{165}\mathrm{,}\text{𝑅𝑎𝑖𝑛𝑖𝑛𝑔}\mathrm{=}\text{𝑓𝑎𝑙𝑠𝑒}\mathrm{\}}$ specifies that the class starts at 11, Joe is 165cm tall and it is not raining.

In the electrical environment of Figure 1.8, there may be a variable for the position of each switch that specifies whether the switch is up or down. There may be a variable for each light that specifies whether it is lit or not. There may be a variable for each component specifying whether it is working properly or if it is broken. Some variables that the following examples use include:

• •

  $S_1\mathrm{\_}\text{𝑝𝑜𝑠}$ is a binary variable denoting the position of switch $s_1$ with domain $\{\text{𝑢𝑝}$, $\text{𝑑𝑜𝑤𝑛}\}$, where $S_1\mathrm{\_}\text{𝑝𝑜𝑠}=\text{𝑢𝑝}$ means switch $s_1$ is up, and $S_1\mathrm{\_}\text{𝑝𝑜𝑠}=\text{𝑑𝑜𝑤𝑛}$ means switch $s_1$ is down.

• •

  $S_1\mathrm{\_}\text{𝑠𝑡}$ is a discrete variable denoting the status of switch $s_1$ with domain $\{\text{𝑜𝑘},$ $\mathit{\text{upside\_down}},$ $\text{𝑠ℎ𝑜𝑟𝑡},$ $\text{𝑖𝑛𝑡𝑒𝑟𝑚𝑖𝑡𝑡𝑒𝑛𝑡},$ $\text{𝑏𝑟𝑜𝑘𝑒𝑛}\}$, where $S_1\mathrm{\_}\text{𝑠𝑡}=\text{𝑜𝑘}$ means switch $s_1$ is working normally, $S_1\mathrm{\_}\text{𝑠𝑡}=\mathit{\text{upside\_down}}$ means it is installed upside down, $S_1\mathrm{\_}\text{𝑠𝑡}=\text{𝑠ℎ𝑜𝑟𝑡}$ means it is shorted and it allows electricity to flow whether it is up or down, $S_1\mathrm{\_}\text{𝑠𝑡}=\text{𝑖𝑛𝑡𝑒𝑟𝑚𝑖𝑡𝑡𝑒𝑛𝑡}$ means it is working intermittently, and $S_1\mathrm{\_}\text{𝑠𝑡}=\text{𝑏𝑟𝑜𝑘𝑒}n$ means it is broken and does not allow electricity to flow.

• •

  Number_of_broken_switches is an integer-valued variable denoting the number of switches that are broken.

• •

  ${\mathit{\text{Current\_w}}}_1$ is a real-valued variable denoting the current, in amps, flowing through wire $w_1$. ${\mathit{\text{Current\_w}}}_1=1.3$ means there are $1.3$ amps flowing through wire $w_1$. We also allow inequalities between variables and constants; for example, ${\mathit{\text{Current\_w}}}_1\ge 1.3$ is true when there are at least $1.3$ amps flowing through wire $w_1$.

A world specifies the position of every switch, the status of every device, and so on. For example, a world may be described as switch 1 is up, switch 2 is down, fuse 1 is okay, wire 3 is broken, etc.

A classic example of a constraint satisfaction problem is a crossword puzzle. There are two different representations of crossword puzzles in terms of variables:

• •

  In one representation, the variables are the numbered squares with the direction of the word (down or across), and the domains are the set of possible words that can be used. For example, one_across could be a variable with domain $\{$’ant’, ’big’, ’bus’, ’car’, ’has’$\}$. A possible world corresponds to an assignment of a word for each of the variables.

• •

  In another representation of a crossword, the variables are the individual squares and the domain of each variable is the set of letters in the alphabet. For example, the top-left square could be a variable $p00$ with domain $\{a,\mathrm{\dots },z\}$. A possible world corresponds to an assignment of a letter to each square.

A trading agent, in planning a trip for a group of tourists, may be required to schedule a given set of activities. There could be two variables for each activity: one for the date, for which the domain is the set of possible days for the activity, and one for the location, for which the domain is the set of possible towns where it may occur. A possible world corresponds to an assignment of a date and a town for each activity.

An alternative representation may have the days as the variables, with domains the set of possible activity–location pairs.

If there are two variables, $A$ with domain $\mathrm{\{}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{\}}$ and $B$ with domain $\mathrm{\{}\text{𝑡𝑟𝑢𝑒}\mathrm{,}\text{𝑓𝑎𝑙𝑠𝑒}\mathrm{\}}$, there are six possible worlds, which we name $w_{\mathrm{0}}\mathrm{,}\mathrm{\dots }\mathrm{,}{w}_{\mathrm{5}}$ as follows

• •

  $w_0=\{A=0,B=\text{𝑡𝑟𝑢𝑒}\}$

• •

  $w_1=\{A=0,B=\text{𝑓𝑎𝑙𝑠𝑒}\}$

• •

  $w_2=\{A=1,B=\text{𝑡𝑟𝑢𝑒}\}$

• •

  $w_3=\{A=1,B=\text{𝑓𝑎𝑙𝑠𝑒}\}$

• •

  $w_4=\{A=2,B=\text{𝑡𝑟𝑢𝑒}\}$

• •

  $w_5=\{A=2,B=\text{𝑓𝑎𝑙𝑠𝑒}\}$