The feasible interpolation theorem for semantic derivations from [J. Krajíček, Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, J. Symbolic Logic 62(2) (1997) 457–486] allows to derive from some short semantic derivations (e.g. in resolution) of the disjointness of two [Formula: see text] sets [Formula: see text] and [Formula: see text] a small communication protocol (a general dag-like protocol in the sense of Krajíček (1997) computing the Karchmer–Wigderson multi-function [Formula: see text] associated with the sets, and such a protocol further yields a small circuit separating [Formula: see text] from [Formula: see text]. When [Formula: see text] is closed upwards, the protocol computes the monotone Karchmer–Wigderson multi-function [Formula: see text] and the resulting circuit is monotone. Krajíček [Interpolation by a game, Math. Logic Quart. 44(4) (1998) 450–458] extended the feasible interpolation theorem to a larger class of semantic derivations using the notion of a real communication complexity (e.g. to the cutting planes proof system CP). In this paper, we generalize the method to a still larger class of semantic derivations by allowing randomized protocols. We also introduce an extension of the monotone circuit model, monotone circuits with a local oracle (CLOs), that does correspond to communication protocols for [Formula: see text] making errors. The new randomized feasible interpolation thus shows that a short semantic derivation (from a certain class of derivations larger than in the original method) of the disjointness of [Formula: see text], [Formula: see text] closed upwards, yields a small randomized protocol for [Formula: see text] and hence a small monotone CLO separating the two sets. This research is motivated by the open problem to establish a lower bound for proof system [Formula: see text] operating with clauses formed by linear Boolean functions over [Formula: see text]. The new randomized feasible interpolation applies to this proof system and also to (the semantic versions of) cutting planes CP, to small width resolution over CP of Krajíček [Discretely ordered modules as a first-order extension of the cutting planes proof system, J. Symbolic Logic 63(4) (1998) 1582–1596] (system R(CP)) and to random resolution RR of Buss, Kolodziejczyk and Thapen [Fragments of approximate counting, J. Symbolic Logic 79(2) (2014) 496–525]. The method does not yield yet lengths-of-proofs lower bounds; for this it is necessary to establish lower bounds for randomized protocols or for monotone CLOs.

中文翻译：

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具有本地预言机的随机可行插值和单调电路

语义推导的可行插值定理 [J. Krajíček，插值定理，证明系统的下界，以及有界算术的独立性结果，J. Symbolic Logic 62(2) (1997) 457–486] 允许从一些不相交的短语义推导（例如解析）中推导出两个 [Formula: see text] 集 [Formula: see text] 和 [Formula: see text] 一个小型通信协议（Krajíček (1997) 计算 Karchmer-Wigderson 多功能 [Formula :see text] 与集合相关联，并且这样的协议进一步产生了将 [Formula: see text] 与 [Formula: see text] 分开的小电路。当 [Formula: see text] 向上闭合时，协议计算单调 Karchmer –Wigderson 多功能[公式：见正文]，得到的电路是单调的。Krajíček [通过游戏进行插值，数学。逻辑夸脱。44(4) (1998) 450-458] 将可行插值定理扩展到更大类别的语义推导，使用真实通信复杂性的概念（例如，切割平面证明系统 CP）。在本文中，我们通过允许随机协议将该方法推广到更大类的语义推导。我们还介绍了单调电路模型的扩展，即带有本地预言机 (CLO) 的单调电路，它确实对应于 [公式：见文本] 出错的通信协议。因此，新的随机可行插值表明，[公式：参见文本]，[公式：参见文本]的不相交性的短语义推导（来自比原始方法中更大的某一类推导）向上闭合，为 [Formula: see text] 产生一个小的随机协议，因此产生一个小的单调 CLO 将两个集合分开。这项研究的动机是为证明系统 [公式：参见文本] 建立一个下界，该系统使用由 [公式：参见文本] 上的线性布尔函数形成的子句进行操作。新的随机可行插值适用于这个证明系统，也适用于切割平面 CP 的（语义版本），适用于 Krajíček 的 CP 上的小宽度分辨率 [离散有序模块作为切割平面证明系统的一阶扩展，J. Symbolic Logic 63(4) (1998) 1582–1596]（系统 R(CP)）和 Buss、Kolodziejczyk 和 Thapen 的随机分辨率 RR [近似计数的片段，J. Symbolic Logic 79(2) (2014) 496– 525]。该方法尚未产生证明长度的下界；