In this question, you are asked to model connections between actors using predicate

logic. Let $$ be the domain of every actor, alive or dead, who has ever been in any movie. Let $$ be

the co$-$star predicate on pairs of actors from $,$ defined by $(,)=$ and $$ have been in a movie

together$\mathrm{.}1$

The $$Bacon game$$ is a game where players try to connect individuals to the actor Kevin Bacon by

using the co$-$star predicate $.$ The Bacon number of an actor $($that is$,$ where $$ is an element of $)$

is the smallest number of links needed, using the co$-$star predicate $,$ to connect $$ to Kevin Bacon.

For example, Daniel Kaluuya was in the movie Sicario with Jeffrey Donovan, who in turn was in

Sleepers with Kevin Bacon. So$,$ Daniel Kaluuya has a Bacon number of $2.($This example was

generated by oracleofbacon.org, where you can determine the Bacon number for many other

actors.$)$

Let $0$ in $$ be Kevin Bacon, i$.$e$.,$ let $0$ be the special name that we give to Kevin Bacon, given the

unique nature of his role in the game.

a$.$ Write a predicate $2()$ in one variable that reflects that the Bacon number of an actor $$ in $$ is at

most two. By the definition of the Bacon number, this means that there is some actor $$ such that $$

and $$ were in a movie together and that $$ and $0$ were in a movie together. The expression for the

predicate must involve a quantifier.

b$.$ Similarly, write a predicate $3()$ in one variable that reflects that the Bacon number of an actor

$$ is at most three. This will likely involve multiple quantifiers.

c$.$ Notice for $3(),$ the predicate reflects that the Bacon number of x is at most $3.$ The predicate

does not enforce that the Bacon number is exactly $3,$ since there is a possibility that the actor $$ is$,$

e$.$g$.,$ in a movie with $0,$ or has a Bacon number of $2$ through some other links. Write a predicate for

an actor $$ in $$ to have Bacon number exactly three, called $3(),$ by excluding these other

possibilities. This predicate should be defined by using $,3$ and $2.$ Be sure to exclude the

possibility that $$ is Kevin Bacon.

d$.$ Write a full expression for $3()$ that does not use $3$ or $2,$ but still uses $.$ Make sure to

parenthesize appropriately. $($Hint: use your solution for part c$.$ and negate as necessary.$)$ Write a

short $(1-2$ sentence$)$ explanation of the entire predicate definition in natural language, verifying that

it makes sense.

$1$ In this question, we define the predicate $$ to be false when asked if an actor has been in a movie with

themselves, i$.$e$.,(,)$ is always false. We also don$$t worry about the order of elements in the predicate. It

doesn$$t matter which actor is listed first in the expression for $.$ That is$,(,)(,).$