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.

翻译：TFNP（全函数NP）的子类通常通过指定一个必然属于TFNP的完全问题，并包含所有可多一归约到该问题的问题来定义。我们研究了TFNP问题可归约到TFNP外部对象（如复杂性类）的两种概念。这产生了能捕捉该外部对象某些性质的TFNP子类。我们证明著名的子类可以此方式产生，例如PPA源自对奇偶性P的归约，PLS源自对P^NP的归约。我们研究了由PSPACE和多项式层次结构产生的子类，并证明它们可由命题证明系统Frege和常深度Frege刻画，从而扩展了已知的自然TFNP子类与证明系统之间的配对关系。我们从这一视角研究近似计数，并寻找一个TFNP子类，为诸如Ramsey等可使用近似计数证明的组合原理提供自然归宿。我们将此与近期研究的Long choice和Short choice问题联系起来。