Is it possible to derive related signature? For example, the message is a json: {name: john, age: 13}, and is sealed with signature.

Now, is it possible to derive a signature for the new message: john is less than 18 ?

I feel, to achieve that, the less than operator should be well-defined.

Any known research in this field?
Here is one idea about how to do a signature that specifies a specific range (and which makes it easy to enlarge the range):

We will assume a bounded range for the 'age', say, 0 to 150 ($max$), and we assume that $\text{H}$ is a preimage-resistant hash function, and $\text{H}^k$ denotes $\text{H}$ iterated $k$ times.

Then, we have a signature that has several parts:


The values $MinAge, MaxAge$
The value $max = \text{H}^{MaxAge}( r_1 )$
The value $min = \text{H}^{max-MinAge}( r_2 )$
The values $\text{H}^{max}( r_1 ), \text{H}^{max}( r_2 )$
The signature for $\text{H}^{max}( r_1 ), \text{H}^{max}{r_2}$, and the rest of the clause (name: John).


The above signature states that John's age is between MinAge and MaxAge; the original signer (who knows the age) can generate a signature with $MinAge = MaxAge = \text{John's real age}$

This signature can be verified by verifying that $\text{H}^{max - MaxAge}(max)$ matches the $\text{H}^{max}(r_1)$ value, that $\text{H}^{MinAge}(min)$ matches the $\text{H}^{max}(r_2)$ value, and that the signature verifies.

The range that this signature denotes can be enlarged; the MaxAge can be increased to $MaxAge'$ by updating the value $MaxAge$ and replacing the value $max$ by $\text{H}^{MaxAge' - MaxAge}(max)$.  Similarly, the MinAge can be decreased to $MinAge'$ updating the value $MinAge$ and replacing the value $min$ by $\text{H}^{MinAge - MinAge'}(min)$.  Note that the range cannot be decreased, because that would require generating a preimage of $\text{H}$, which we assume is infeasible.