Limits of structures and Total NP Search Problems

For an infinite class of finite graphs of unbounded size, we define a limit object, to be called wide limit, relative to some computationally restricted class of functions. The properties of the wide limit then reflect how a computationally restricted viewer "sees" a generic instance from the class. The construction uses arithmetic forcing with random variables [10]. 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 wide 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 logical separation of the oracle classes of total NP search problems in our setting implies that Leaf is not reducible to OntoWeakPigeon even if some errors are allowed in the reductions.