Trying to understand interpretation and denotation in FOL

Question

I am going through the book "Knowledge Representation and Reasoning" by Brachman and Levesque.

So an interpretation $ F $ is defined as a pair $ \langle D,I \rangle $ mapping from a set of objects $ D$ called domain of the interpretation and $ I $ is a mapping called interpretation mapping from the non-logical symbols to functions and relations over $D$.

Then it has been written that $ I $ will assign meaning to the predicate symbols as follows:

To every predicate symbol P of arity $n$, $I[P]$ is an n-ary relation, that is, $I[P]$ is contained in $ D \times D \times D \times \dots \times D$.

Now for denotation it is written that for an interpretation $ F = \langle D,I \rangle $ we can specify which elements of $ D $ are denoted by variable free term FOL.

And hence the notion of variable assignment $ \mu $ has been introduced and the following two rules have been introduced:

Given an interpretation $ F $ and a variable assignment $ \mu $, the denotation of the term $ t $ is written as $ \|t\|_{F \mu} $ and defined as follows:

1. If $ x $ is a variable then $ \|x\|_{F \mu} = \mu [x] $.

2. If $ t_1 ,\dots,t_n$ are terms and $ f $ is a function symbol of arity $ n $ then $ \| f(t_1,\dots,t_n)\|_{F \mu} = G( d_1,\dots,d_n) $, where $ G = I[f] $, and $ d_i = \| t_i\| _{f \mu} $.

I am a bit confused about how to interpret the last two rules.

So the act of choosing elements from $ D $ to represent the arguments of the function symbol is represented by $ \mu $? Will it be right to say so?

How exactly are $ f $ and $ G $ related?

The output is evaluated by computing $ G $ for values which get substituted in the argument which is represented in $ d_i = \| t_i\| _{f \mu} $ of the second rule.

But I could not understand how are the above two rules recursive? What exactly should I conclude about the way $ I $, $ G $ , $f$ and $ \mu $ are related to each other and what exactly do they mean? Is my understanding correct?

Attaching the screenshots from the book:

Answer 1

Consider the expression $+(x,\times(y,y))$, which is the FOL way to write $x+(y\cdot y)$. If I substitute $x=3$ and $y=5$ and do the arithmetic over the integers, what do I get? I need to evaluate the expression recursively.

In order to evaluate $+(x,\times(y,y))$, I need to evaluate $x$, to evaluate $\times(y,y)$, and then to combine them using the function $+$.

Evaluating $x$ is easy, since $x$ is just a variable. My assignment says that $x=3$, so the value of the expression $x$ is $3$.

Evaluating $\times(y,y)$ is more difficult. First I need to evaluate $y$ (twice) - as before, this is easy, and gives the result $5$. Now I need to combine the two values $5,5$ using the function $\times$. I have a large table which, for every pair of integers $a,b$, tells me what $\times(a,b)$ is. Looking at the table, I find out that $\times(5,5) = 25$.

Now that I have evaluated $x$ To $3$ and $\times(y,y)$ to $25$, I can finally look at the table for $+$ and deduce that $+(3,25) = 28$. So $28$ is the value of the entire expression.

The formal definitions you quote in your question attempt to capture the preceeding process in an abstract form. It's best if you work out the correspondence on your own.