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Inductive counting certifies complete reachable sets while retaining only a count and a few vertex indices. For a directed graph with $N$ vertices and start $s$, let $R_k$ contain the vertices reachable in at most $k$ steps and write $r_k=|R_k|$.

Suppose the exact count $r_k$ is known. A nondeterministic computation can enumerate each vertex, guess which belong to $R_k$, and justify every positive guess by a path of length at most $k$. Requiring exactly $r_k$ justified vertices prevents omissions: false positives have no valid path certificate. To certify $v\notin R_{k+1}$, also check $v\ne s$ and that none of those enumerated vertices has an edge to $v$.

Starting from $r_0=1$, certify membership or nonmembership for each vertex and accumulate $r_{k+1}$. After $N-1$ stages every reachable vertex has been counted, since cycles can be removed from a reaching path. Loop counters and the current vertex use only logarithmic space; guessed paths are checked one edge at a time, and the full reachable set is never stored. This is the counting mechanism behind the Immerman–Szelepcsényi theorem.^[ref]

Neil Immerman proved closure of nondeterministic space under complementation for space bounds at least logarithmic, independently of the result associated with the other name in the Immerman–Szelepcsényi theorem. In particular, $\mathrm{NL}=\mathrm{coNL}$. His paper's proof uses inductive counting of reachable configurations.^[ref]

The obstacle is that guessing a path certifies reachability, while failure of one guessed path does not certify nonreachability. Counting the complete reachable set and verifying its enumeration supplies the missing certificate within a small space bound. This makes complementation possible without storing every reached configuration at once. The result concerns nondeterministic space, and does not establish the analogous equality $\mathrm{NP}=\mathrm{coNP}$ for polynomial time.