What Juris Hartmanis taught me about Reductions
Neil Immerman
TL;DR
The paper examines how very weak reductions, notably first-order projections and their variants (fop and qfp), illuminate connections between descriptive and classical computational complexity. It explains Fagin's Theorem, showing that $NP$ coincides with existential second-order logic ($NP = \mathrm{SO}\exists$) on finite ordered structures with numeric symbols, and demonstrates how first-order reductions can preserve completeness (e.g., $3$-SAT $\leq_{\mathrm{fo}} 3$-COLOR). It then establishes that $REACH$ is qfp-complete for $NL$, $REACH_d$ for $L$, and $REACH_a$ for $P$, illustrating how transitive-closure operators capture space-bounded computation. Most notably, it presents a proof that for all $s(n) \ge \log n$, $NSPACE[s(n)] = co-NSPACE[s(n)]$, achieved by a qfp reduction from $\overline{REACH}$ to $REACH$ and a counting argument, thereby solving a long-standing open problem. The work underscores how Hartmanis' focus on very weak reductions provides a powerful, general approach to separating and relating complexity classes through descriptive methods.
Abstract
I was a student of Juris Hartmanis at Cornell in the late 1970's. He believed that there was great potential in studying restricted reductions. I describe here some of his influences on me and, in particular, how his insights concerning reductions helped me to prove that nondeterministic space is closed under complementation.