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3.4 Modeling with relations  (Page 3/2)

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(Blank Abstract)

Modeling with relations

Note that the nhbr relation can actually represent an arbitrarily weird board, such aslocations that look adjacent on the map but actually aren't, boards which wrap around a cylinder or toroid , or a location with a tunnel connecting it toa location far across the board (like the secret passages in the game Clue, or the harrowing sub trip through Naboo in Star Wars: The Phantom Menace .) One-way passages can be encoded as well(meaning the nhbr relation need not be symmetric). Actually, any graph can be represented!

How shall we encode concepts such as

location A has 3 dangerous neighbors
, using relations?

A good first guess might be to say we have a function which returns the number of pirates next to a given location.That is,

piratesNear A 3
. However,
piratesNear
doesn't qualify as a relation. Why not?

To work around this, we could propose a binary relation along the lines of

piratesNear A 3
. This is better, but it requires our domainto be not only board locations, but also numbers. And to be able to talk about numbers, we'd need more axioms,as well as numeric relations such as>.
Hmmm, what is the arity of
>
?
While this approach is feasible, and ultimately might be what we want, for now, let's stick with relations involvingonly locations, not numbers.

Okay, the third time's the charm: we'll implement the concept

A neighbors three pirates
as a relation has-3 A being true. To cover the cases when there are exactly two neighboring pirates,we'll use a whole new separate relation,
has-2
; has-2 A would be false on any board where has-3 A is true (at least, in our standard interpretation).

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Proofs otherwise unchanged.Note that we might express our rules as

for any locations x and y , we have the following axiom: has-3 x nhbr x y safe y
. Really, note that there's something else going on here: x and y are symbols which can represent any location: they are variables, whose value can be any element of the domain.

For the domain of types-of-vegetables, the relation yummy is a useful one to know, when cooking. In case you weren't sure, yummy Brussels sprouts , and yummy carrots .

Suppose we had a second relation, yucky . Is it conceivable that we could model a vegetable that's neitheryucky nor yummy, using these relations? Sure! (Iceberg lettuce, perhaps.)In fact, we could even have a vegetable which is both yummy and yucky — radishes!

A quick digression on a philosophical nuance: the domain for the above problem is not vegetables; it's types-of-vegetables.That is, we talk about whether or not carrots are yummy; this is different than talking the yumminess ofthe carrot I dropped under the couch yesterday, or the carrot underneath the chocolate sauce.In computer science, this often manifests itself as the difference between values, and types of values.As examples, we distinguish between 3 and the set of all integers, and we distinguish between particular carrots and the abstract idea of carrots.(Some languages even include types as values.) Philosophers enjoy debating how particular instances definethe abstract generalization, but for our purposes we'll take each both vegetables andtypes-of-vegetables as given.

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Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
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