For an infinite class of finite graphs of unbounded size, we define a limit object, to be called a $\textit{wide limit}$, relative to some computationally restricted class of functions. The limit object is a first order Boolean-valued structure. The first order properties of the wide limit then reflect how a computationally restricted viewer "sees" a generic member of the class. The construction uses arithmetic forcing with random variables [Kraj\'i\v{c}ek, Forcing with random variables and proof complexity 2011]. We give 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. To illustrate the concept we give an example in which the wide limit relates to total NP search problems. In particular, we take the wide limit of all maps from $\{0,\dots,k-1\}$ to $\{0,\dots,\lfloor k/2\rfloor-1\}$ to obtain a model of $\forall \text{PV}_1(f)$ where the problem $\textbf{RetractionWeakPigeon}$ is total but $\textbf{WeakPigeon}$, the complete problem for $\textbf{PWPP}$, is not. Thus, we obtain a new proof of this unprovability and show it implies that $\textbf{WeakPigeon}$ is not many-one reducible to $\textbf{RetractionWeakPigeon}$ in the oracle setting.

翻译：对于一类规模无界的无限有限图族，我们定义了一个极限对象，称为$\textit{宽极限}$，该定义相对于某个计算受限的函数类。该极限对象是一个一阶布尔值结构。宽极限的一阶性质反映了计算受限的观察者如何“看待”该图族中的一般成员。此构造使用了带随机变量的算术力迫法[Kraj\'i\v{c}ek, Forcing with random variables and proof complexity 2011]。我们给出了全称与存在语句在极限中有效的充分条件，提供了若干示例，并证明了此类极限对象可扩展为弱算术模型。为阐明这一概念，我们给出一个宽极限与全NP搜索问题相关的示例。具体而言，我们取所有从$\{0,\dots,k-1\}$到$\{0,\dots,\lfloor k/2\rfloor-1\}$的映射的宽极限，得到一个$\forall \text{PV}_1(f)$模型，其中问题$\textbf{RetractionWeakPigeon}$是全的，但$\textbf{PWPP}$的完全问题$\textbf{WeakPigeon}$并非全的。由此，我们获得了该不可证性的新证明，并表明其蕴含在预言机设置下$\textbf{WeakPigeon}$不能多一归约到$\textbf{RetractionWeakPigeon}$。