Limits of structures and Total NP Search Problems

01/31/2023

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by   Ondřej Ježil, et al.

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For a class of finite graphs, we define a limit object relative to some computationally restricted class of functions. The properties of the limit object then reflect how a computationally restricted viewer "sees" a generic instance from the class. The construction uses Krajíček's forcing with random variables [7]. We prove sufficient conditions for universal and existential sentences to be valid in the limit, provide several examples, and prove that such a limit object can then be expanded to a model of weak arithmetic. We then take the limit of all finite pointed paths to obtain a model of arithmetic where the problem OntoWeakPigeon is total but Leaf (the complete problem for PPA) is not. This can be viewed as a logical separation of the oracle classes of total NP search problems, which in our setting implies standard nonreducibility of Leaf to OntoWeakPigeon.