How to fit large complexity classes into TFNP
Neil Thapen
TL;DR
The paper develops two complementary approaches to carving TFNP into meaningful subclasses based on objects outside TFNP: (i) a game-based construction from complexity classes via Inspector–Adversary axioms, yielding robust correspondences like $\text{PPA}$ from parity and links to $\text{Frege}$; and (ii) a counterexample-reducibility framework from TF$\Sigma^p_2$ problems, yielding $\text{PLS}$ from $P^\text{NP}$ and $\text{PPADS}$ from Empty, while organizing a polynomial-hierarchy-like structure inside TFNP via Local-$\Gamma_{\mathrm{TQBF}}^k$. The article further connects Ramsey and approximate counting to $\text{APPROX}$, provides direct reductions from Ramsey to Approximate counting, and situates these TFNP classes within propositional and bounded-arithmetic proof frameworks (LK/Frege, $T^k_2$, $U^1_2$), clarifying how totality corresponds to proof complexity. Collectively, these results illuminate how TFNP can reflect properties of outside complexity-theoretic objects, enabling new characterizations, hierarchies, and reductions that bridge computational, logical, and combinatorial viewpoints. The work advances understanding of how natural principles like counting and Ramsey interact with TFNP structure and demonstrates robust, axiomatizable pathways to explore total search within a rich logical and computational landscape.
Abstract
Subclasses of TFNP (total functional NP) are usually defined by specifying a complete problem, which is necessarily in TFNP, and including all problems many-one reducible to it. We study two notions of how a TFNP problem can be reducible to an object, such as a complexity class, outside TFNP. This gives rise to subclasses of TFNP which capture some properties of that outside object. We show that well-known subclasses can arise in this way, for example PPA from reducibility to parity P and PLS from reducibility to P^NP. We study subclasses arising from PSPACE and the polynomial hierarchy, and show that they are characterized by the propositional proof systems Frege and constant-depth Frege, extending the known pairings between natural TFNP subclasses and proof systems. We study approximate counting from this point of view, and look for a subclass of TFNP that gives a natural home to combinatorial principles such as Ramsey which can be proved using approximate counting. We relate this to the recently-studied Long choice and Short choice problems.