# Comparing symbolic and analog descriptions

I've never seen the following comparison before. Let me start with a specific example:

Given a finite structure with two symmetric binary relations, i.e. a graph $G$ with one vertex set $V$ and two edge sets $E_1$, $E_2$.

Giving an explicit description of $G$ in a formal language can be seen as a special case of defining a function $f$ from a set $T$ with a betweenness relation $B$ to the set $V\ \cup \big( E_1 \times \lbrace E_1 \rbrace \big) \cup \big( E_2 \times \lbrace E_2 \rbrace \big)$ such that $(v,w) \in E_i$ iff

$$(\exists x,y,z) f(x) = v \wedge f(y) = ((v,w),E_i) \wedge f(z) = w \wedge B(x,y,z)$$

Such a tuple $(x,y,z)$ represents the "sentence" that $(v,w) \in E_i$. The sentences may overlap and reuse "symbols", but this can be avoided. For example ($T = \mathbb{N}$):

````|u|v|vwE|uwE|w| | | |...
````

represents the sentences $(u,w) \in E$ and $(v,w) \in E$. But also does:

````|u|uwE|w|v|vwE|w| | | |...
````

In fact, it's enough to consider functions $f$ from $T$ to $V\ \cup \lbrace E_1 \rbrace \cup \lbrace E_2 \rbrace$ such that $(v,w) \in E_i$ iff

$$(\exists x,y,z) f(x) = v \wedge f(y)=E_i \wedge f(z) = w \wedge B(x,y,z)$$

This comes closer to the normal usage of formal languages. So

````|u|E|w|v|E|w| | | |...
````

represents the sentences $(u,w) \in E$ and $(v,w) \in E$, which are oftenly written as $uEw$ and $vEw$.

This is what "symbolic" description essentially is: a "structure preserving" function from a medium to the structure. (Using intermediate symbols from an alphabet isn't essential: the elements of the structure may symbolize themselves.)

In contrast, "analog" description essentially is a "structure preserving" function from the structure to a medium:

Consider a function $f$ from the set $V\ \cup \big( E_1 \times \lbrace E_1 \rbrace \big) \cup \big( E_2 \times \lbrace E_2 \rbrace \big)$ to a set $T$ with a betweenness relation $B$ such that $(v,w) \in E_i$ iff $f((v,w),E_i))$ is between $f(v)$ and $f(w)$, or:

$$(\exists x,y,z) f(v) = x \wedge f(((v,w),E_i)) = y \wedge f(w) = z \wedge B(x,y,z)$$

For example ($T = \mathbb{N}^2$):

````|u|uwE|w|
| |vwE| |
|v|   | |
````

Again, we can ignore the specific edges and write/draw for short:

````|u|E|w|
| |E| |
|v| | |
````

We even can ignore the specific vertices and write/draw for short (note, how this looks like a unlabelled graph!):

````|V|E|V|
| |E| |
|V| | |
````

Analog description can be generalized for relations of arbitrary arity, but that's a bit tedious. In principle, it works.

Is this comparison in any sense enlightening, or is it just baublery?

As I said before, I've never seen this comparison made explicit. So can any references be given?

-

– Qiaochu Yuan Feb 25 '11 at 12:34
Below is what I understand, considering the first example. Your $T$ encodes bits of sentences and $f$ decodes them. What other data $T$ contains? $B$ distinguish a tuple of bits of some syntactically correct sentence. Here the question “what is a sentence, precisely?” arise. If it is a logical formula, like in first-order logic, then $B$ is too rigid to handle it. $B$ checks only 3 bits. On the contrary, carriers of term algebras are sets of terms. BTW, you can simplify the formula in the second example $B(f(v), f(((v,w),E_i)), f(w))$. – beroal Feb 25 '11 at 17:21
I wanted to show the similarity between the two formulas. – Hans Stricker Feb 25 '11 at 19:23
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