Abstract
The class TFNP, of NP search problems where all instances have solutions, appears not to have complete problems. However, TFNP contains various syntactic subclasses and important problems. We introduce a syntactic class of problems that contains these known subclasses, for the purpose of understanding and classifying TFNP problems. This class is defined in terms of the search for an error in a concisely-represented formal proof. Finally, the known complexity subclasses are based on existence theorems that hold for finite structures; from Herbrand's Theorem, we note that such theorems must apply specifically to finite structures, and not infinite ones.
Highlights
The complexity class TFNP is the set of total function problems that belong to NP; that is, every input to such a nondeterministic function has at least one output, and outputs are easy to check for validity – but it may be hard to find an output
We showed that PTFNP contains the five known classes PPP, PPA, PPAD, PPADS, and PLS
Our understanding is that the present results are implicit in recent work in Bounded Arithmetic, but our system Q-EFF is of interest since it seems to allow more direct reductions
Summary
The complexity class TFNP is the set of total function problems that belong to NP; that is, every input to such a nondeterministic function has at least one output, and outputs are easy to check for validity – but it may be hard to find an output. It is known from Megiddo [21] that problems in TFNP cannot be NP-complete unless NP is equal to co-NP.
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