Operational semantics can become repetitive if every expression form needs separate rules saying where the next step may occur. An evaluation context factors out that control information.

A context $E[;]$ is an expression with one hole. For call-by-value arithmetic, contexts might include

$$E ::= [;] \mid E+e \mid v+E.$$

The first form is the hole itself. The second says evaluation may occur in the left side of an addition; the third allows the right side only after the left has become a value.

A general congruence rule can then say

$$\frac{e\to e'}{E[e]\to E[e']}.$$

Only the truly computational redex rules need to be stated separately. For example,

$$n_1+n_2\to n_3$$

performs arithmetic, while contexts determine where such a redex may be reduced.

Evaluation contexts make evaluation order explicit while avoiding a large family of nearly identical rules. They scale particularly well to function calls, exceptions, mutable state and control operators, and they expose a useful decomposition of a program into “the next reducible operation” plus “the surrounding continuation of the computation.”