A Note on the NP-Hardness of PARTITION Via First-Order Projections

In the article ‘‘On the (Non) NP-Hardness of Computing Circuit Complexity’’, Murray and Williams imply the PARTITION decision problem is not known to be NP-hard via $2^{n^{o(1)}}$-size AC0 reductions. In this note, we show PARTITION is NP-hard via first-order projections. Basically, we slightly modify well-known reductions from 3SAT to SUBSET-SUM and from SUBSET-SUM to PARTITION, but do so in the context of descriptive computational complexity, i.e., we use first-order logical formulas to define them. Hardness under polynomial-size AC0 reductions follows because first-order reductions are a particular type of them. Thus, this note fills a gap in the literature.