This is a commentary on, and critique of, Latif Salum's paper titled "Tractability of Onein-three 3SAT: P = NP." Salum purports to give a polynomial-time algorithm that solves the NP-complete problem X3SAT, thereby claiming P = NP. The algorithm, in short, fixes the polarity of a variable, carries out simplifications over the resulting formula to decide whether to keep the value assigned or flip the polarity, and repeats with the remaining variables. One thing this algorithm does not do is backtrack. We give an illustrative counterexample showing why the lack of backtracking makes this algorithm flawed. * Supported in part by NSF grant CCF-2006496. † Supported in part by NSF grant CCF-2030859 to the Computing Research Association for the CIFellows Project. 1 Salum calls the ⊙ symbol the "exactly-1 disjunction" which suggests it is an operator. Nevertheless, as an operator the semantics would be flawed as it is neither binary nor ternary. Instead, we use it in this critique as a purely notational element, which is essentially the role this symbol actually plays in Salum's paper.

Cai and Hemachandra used iterative constant-setting to prove that Few ⊆ ⊕P (and thus that FewP ⊆ ⊕P). In this paper, we note that there is a tension between the nondeterministic ambiguity of the class one is seeking to capture, and the density (or, to be more precise, the needed "nongappy"ness) of the easy-to-find "targets" used in iterative constant-setting. In particular, we show that even less restrictive gap-size upper bounds regarding the targets allow one to capture ambiguity-limited classes. Through a flexible, metatheorem-based approach, we do so for a wide range of classes including the logarithmic-ambiguity version of Valiant's unambiguous nondeterminism class UP. Our work lowers the bar for what advances regarding the existence of infinite, P-printable sets of primes would suffice to show that restricted counting classes based on the primes have the power to accept superconstant-ambiguity analogues of UP. As an application of our work, we prove that the Lenstra-Pomerance-Wagstaff Conjecture implies that all O(log log n)-ambiguity NP sets are in the restricted counting class RC PRIMES .

Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also important. While there is one single definition of rank that completely captures the complexity of matrices (and thus linear transformations), there is no definitive analog for tensors. Rather, many notions of tensor rank have been defined over the years, each with their own set of uses. In this paper we survey the popular notions of tensor rank. We give a brief history of their introduction, motivating their existence, and discuss some of their applications in computer science. We also give proof sketches of recent results by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between three key notions of tensor rank over finite fields with at least three elements.