"Modelled" can be interpreted in different ways!

Well, at perhaps the most basic level, a CNF clause $(x_1 \lor x_2 \lor \neg x_3)$ can be expressed as the IP constraint

$x_1 + x_2 + (1 - x_3) \ge 1 ~~~ (A)$.

FO is contained in $L$, LP is poly-time solvable and IP is $NP$-complete -- so there should be reductions from a given FO instance to both LP and IP. $~~$ A special case of SO, which is SO-Krom (every CNF clause has at most 2 literals) captures $NL$ -- so again, SO-Krom instances can be reduced to LP and IP.

When you write your formulation as in (A) above, if the coefficient matrix on the left side "behaves" well, your integer programming formulation could be solved in polynomial time (e.g. if the matrix is totally unimodular).

ESO (existential SO) $= NP$ by Fagin's theorem, so you should be able to reduce an ESO instance to IP.

Does this answer your question?

"Modelled" can be interpreted in different ways!

Well, at perhaps the most basic level, a CNF clause $(x_1 \lor x_2 \lor \neg x_3)$ can be expressed as the IP constraint

$x_1 + x_2 + (1 - x_3) \ge 1 ~~~ (A)$,

where each $x_i$ is an integer in $\{0, 1\}$.

FO is contained in $L$, LP is poly-time solvable and IP is $NP$-complete -- so there should be reductions from a given FO instance to both LP and IP. $~~$ A special case of SO, which is SO-Krom (every CNF clause has at most 2 literals) captures $NL$ -- so again, SO-Krom instances can be reduced to LP and IP.

When you write your formulation as in (A) above, if the coefficient matrix on the left side "behaves" well, your integer programming formulation could be solved in polynomial time (e.g. if the matrix is totally unimodular).

ESO (existential SO) $= NP$ by Fagin's theorem, so you should be able to reduce an ESO instance to IP.

Does this answer your question?