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Type-two polynomial-time and restricted lookahead
Theoretical Computer Science ( IF 1.1 ) Pub Date : 2019-07-08 , DOI: 10.1016/j.tcs.2019.07.003
Bruce M. Kapron , Florian Steinberg

This paper provides an alternate characterization of type-two polynomial-time computability, with the goal of making second-order complexity theory more approachable. We rely on the usual oracle machines to model programs with subroutine calls. In contrast to previous results, the use of higher-order objects as running times is avoided, either explicitly or implicitly. Instead, regular polynomials are used. This is achieved by refining the notion of oracle-polynomial-time introduced by Cook. We impose a further restriction on the oracle interactions to force feasibility. Both the restriction as well as its purpose are very simple: it is well-known that Cook's model allows polynomial depth iteration of functional inputs with no restrictions on size, and thus does not guarantee that polynomial-time computability is preserved. To mend this we restrict the number of lookahead revisions, that is the number of times a query can be asked that is bigger than any of the previous queries. We prove that this leads to a class of feasible functionals and that all feasible problems can be solved within this class if one is allowed to separate a task into efficiently solvable subtasks. Formally put: the closure of our class under lambda-abstraction and application includes all feasible operations. We also revisit the very similar class of strongly polynomial-time computable operators previously introduced by Kawamura and Steinberg. We prove it to be strictly included in our class and, somewhat surprisingly, to have the same closure property. This can be attributed to properties of the limited recursion operator: It is not strongly polynomial-time computable but decomposes into two such operations and lies in our class.



中文翻译:

第二类多项式时间和受限先行

本文提供了第二类型多项式时间可计算性的另一种刻画,其目的是使二阶复杂度理论更易于理解。我们依靠常规的oracle计算机来通过子例程调用对程序进行建模。与以前的结果相反,避免了显式或隐式使用高阶对象作为运行时间。而是使用正则多项式。这是通过完善Cook提出的oracle-polynomial-time的概念来实现的。我们对oracle交互施加了进一步的限制,以强制可行性。限制及其目的都非常简单:众所周知,Cook模型允许函数输入的多项式深度迭代,而对大小没有限制,因此不能保证保留多项式时间的可计算性。为了解决这个问题,我们限制了前瞻性修订的数量,即可以查询的次数大于以前的任何查询的次数。我们证明,这导致了一类可行的功能,并且如果允许将一个任务分解为可有效解决的子任务,则所有此类可行问题都可以在此类中解决。正式提出:在lambda抽象和应用程序下关闭我们的课程,包括所有可行的操作。我们还回顾了Kawamura和Steinberg以前引入的非常相似的一类强多项式时间可计算算子。我们证明它是严格包含在我们的类中的,并且有些令人惊讶的是,它具有相同的关闭属性。这可以归因于有限递归运算符的属性:

更新日期:2019-07-08
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