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TR23-008 | 2nd February 2023 15:57

Limits of structures and Total NP Search Problems


Authors: Ond?ej Ježil
Publication: 5th February 2023 10:30
Downloads: 120


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 Krají?ek'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 $\textbf{PPA}$) 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.

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