# Meet Over All Paths Solution

The desired solution to a set of simultaneous equations.

Given a global data flow analysis problem [gdfap], *the* *meet* *over* *all* *paths* solution for a program can be interpreted informally as the calculation for each statement in the program of the maximum information, relevant to the gdfap, which is true along every possible execution path from the starting point of the program to that particular statement.

There is proof of nonexistence algorithm to compute the meet over all paths for monotone frameworks. There is proof that the MOP exists for monotone frameworks which meet the *distributivity* condition. Instead of computing the MOP for monotone frameworks, the maximal fixed point is computed.

It appears generally true that what one searches for in a data flow problem is what we shall call the *meet* *over* *all* *paths* (MOP) solution. That is let $\text{PATH}(n)$ denote the set of paths from the initial node to the node $n$ in some flow graph. Then we really want $\underset{P\in \text{PATH}(n)}{\bigwedge}{f}_{p}(0)$ for each $n$. It is this function, the MOP solution, that in any practical data flow problem we can think of, express the desired information.

[0] Monotone Data Flow Analysis Frameworks. John B, Kam and Jeffrey D. Ullman.