Unification --- most specific unifier

Question

In unification, given a set of equations, a standard problem is to compute a most general unifier (mgu). I am interested in a somewhat reversed problem. Imagine having a set of equations that do not have an mgu, like this one:

x = a
x = b

x here is a variable, whereas a and b are terms. I am interested are there any algorithms that could find a possible replacement for a and b such that the resulting equations have mgu? In the above example, that would be a -> y, b -> y, y being a variable. Lets call this a fix. I am particularly interested in most specific fixes. I could not find anything so far, but this seems like a natural problem, or not?

Answer 1

I found out that such a thing is called anti-unification. This problem was addressed by Plotkin and Reynolds. Here is a brief overview.

Answer 2

In rewrite theory, you often want confluence of your system: if $$ u_1\leftarrow t\rightarrow u_2$$

Then there is some term $v$ such that $$ u_1\rightarrow v\leftarrow u_2$$

It is possible to tell whether a set of rewrite rules is confluent by examining the critical pairs: pairs of rules $t_1\rightarrow u_1,t_2\rightarrow u_2$ and an instance $\theta$ such that

1. $t'_1\theta = t_2\theta$
2. $t'_1$ is a subterm of $t_1$ at position $p$.

You can then add the equation $u_1\theta = t_1[u_2\theta]_p$ to make the system confluent.

This suggest an approach to your problem: look at your equations as rewrite rules, then consider the critical pairs: they tell you which equations need to hold to make your system unifiable.