学生-教师游戏中的并行性与适应性
Parallelism and Adaptivity in Student-Teacher Witnessing
AI总结：
本文研究了学生-教师游戏中的并行性与适应性，通过分离多项式层次中的不同类别，解决了有界算术中的两个开放问题，并扩展了电路上界和下界的不可证结论。
摘要：
学生-教师游戏是一种计算模型，在其中一个计算能力受限的学生试图生成一个满足可反驳性质的字符串，而一个全能的教师通过提供反例来反驳不正确的候选者。根据Krajíček、Pudlák和Takeuti的经典结果[KPT90]，此类游戏捕捉了有界算术中∀∃∀公式的见证过程。在本文中，我们引入了多项式层次中总搜索问题的子类，这些子类根据学生在相应层次解决这些问题所需轮次和每轮候选答案的数量进行分类。假设多项式层次不崩溃，我们分离了这些类别对于不同轮次和查询数量的情况。作为应用，我们获得了以下结果：(a) 我们研究了由细粒度长度归纳和有界集合的变体所公理化的有界算术理论。我们证明了一个一般性的见证定理，将这些理论的∀∃∀后果与相应的学生-教师游戏类别联系起来。在多项式层次不崩溃的前提下，我们分离了这些理论。(b) 这些条件分离解决了有界算术中的两个开放问题：一个是Buss和Ressayre[Bus85, CK93]关于有界集合强度的问题，另一个是Pollett[Pol97]关于严格和非严格双长度归纳之间的差异的问题。(c) 最后，我们将Krajíček和Oliveira[KO17]关于电路上界不可证的结论扩展到理论PV₁+BB(Σ^b₁)，并将Pich和Santhanam[PS21]关于强co-非确定性电路下界不可证的结论扩展到理论PV₁+LLIND(sΣ^b₁)。通过前述的分离，这两种理论在假设NP不包含于P/poly的前提下严格扩展了PV₁。
英文摘要：
Student-Teacher Games are a model of computation in which a computationally restricted Student attempts to produce a string satisfying a refutable property, while an all-powerful Teacher refutes incorrect candidates by providing counterexamples. By the classical result of Krajíček, Pudlák, and Takeuti [KPT90], such games capture the witnessing of $\forall\exists\forall$-formulas in bounded arithmetic. In this paper, we introduce subclasses of total search problems in the polynomial hierarchy, classified by the number of rounds and candidate answers per round required for a Student at the corresponding level to solve them. Assuming the polynomial hierarchy does not collapse, we separate these classes for different number of rounds and queries. As applications we obtain the following results: (a) We study theories of bounded arithmetic axiomatized by fine-grained variants of length induction and bounded collection. We prove a general witnessing theorem connecting their $\forall\exists\forall$-consequences to the appropriate classes of Student-Teacher Games. Under the non-collapse of the polynomial hierarchy, we separate these theories. (b) These conditional separations resolve two open problems in bounded arithmetic: one by Buss and Ressayre [Bus85, CK93] on the strength of bounded collection, and one by Pollett [Pol97] on the difference between strict and non-strict double length induction. (c) Finally, we extend the unprovability of circuit upper bounds due to Krajíček and Oliveira [KO17] to the theory $PV_1+BB(Σ^b_1)$, and the unprovability of strong co-nondeterministic circuit lower bounds due to Pich and Santhanam [PS21] to the theory $PV_1+LLIND(sΣ^b_1)$. By the preceding separations, both theories strictly extend $PV_1$ assuming $NP\nsubseteq P/poly$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
