[论文解读] Inductive Definition and Domain Theoretic Properties of Fully Abstract Models for PCF and PCF

Vladimir Sazonov|arXiv (Cornell University)|Jul 21, 2007

Computability, Logic, AI Algorithms被引用 4

一句话总结

本文提出了一种新颖的、逐级构建的PCF和PCF+完全抽象模型的方法，基于顺序与非确定性计算策略。尽管这些模型并非ω-完备或标准的dcpos，仍建立了完全抽象性以及关键的域论性质——如顺序完备性、自然连续性与自然代数性。

ABSTRACT

A construction of fully abstract typed models for PCF and PCF+ (i.e., PCF+ “parallel conditional function”), respectively, is presented. It is based on general notions of sequential computational strategies and wittingly consistent non-deterministic strategies introduced by the author in the seventies. Although these notions of strategies are old, the definition of the fully abstract models is new, in that it is given level-by-level in the finite type hierarchy. To prove full abstraction and non-dcpo domain theoretic properties of these models, a theory of computational strategies is developed. This is also an alternative and, in a sense, an analogue to the later game strategy semantics approaches of Abramsky, Jagadeesan, and Malacaria; Hyland and Ong; and Nickau. In both cases of PCF and PCF+ there are definable universal (surjective) functionals from numerical functions to any given type, respectively, which also makes each of these models unique up to isomorphism. Although such models are non-omega-complete and therefore not continuous in the traditional terminology, they are also proved to be sequentially complete (a weakened form of omega-completeness), “naturally” continuous (with respect to existing directed “pointwise”, or “natural” lubs) and also “naturally” omega-algebraic and “naturally” bounded complete—appropriate generalisation of the ordinary notions of domain theory to the case of non-dcpos.

研究动机与目标

• 开发一种新颖的、逐级构建的PCF与PCF+完全抽象模型的方法
• 为这些模型建立完全抽象性与非dcpos的域论性质
• 将标准域论概念（如ω-完备性、代数性）推广至非dcpos，使用‘自然’的完备性与连续性概念
• 通过可定义的通用泛函证明这些模型在同构意义下的唯一性

提出的方法

• 使用作者早期工作中提出的通用顺序与非确定性计算策略概念
• 在有限类型层次结构中逐级构建模型
• 发展一种新的计算策略理论，以证明完全抽象性与域论性质
• 引入‘自然’连续、‘自然’ω-代数与‘自然’有界完备结构，作为标准域论概念在非dcpos上的推广
• 证明每个模型均存在一个从数值函数到任意给定类型的可定义的通用满射泛函
• 在非dcpos环境中，将顺序完备性确立为ω-完备性的弱化形式

实验结果

研究问题

• RQ1 能否在有限类型层次结构中，通过逐级方法构建PCF与PCF+的完全抽象模型？
• RQ2 如何将域论性质推广至非标准dcpos或非ω-完备的模型？
• RQ3 可定义的通用泛函在确保模型在同构意义下的唯一性中起什么作用？
• RQ4 在缺乏传统连续性的情况下，计算策略如何支持完全抽象性？
• RQ5 在非dcpos模型中，‘自然’形式的连续性与完备性由什么构成？

主要发现

• 所提出的模型对PCF与PCF+均为完全抽象
• 这些模型并非ω-完备，因此在传统意义上不连续，但具有顺序完备性
• 这些模型相对于现有的定向‘逐点’或‘自然’上确界，表现出‘自然’连续性
• 这些模型为‘自然’ω-代数结构与‘自然’有界完备结构，将标准域论概念推广至非dcpos
• 由于存在从数值函数到任意给定类型的可定义通用泛函，每个模型在同构意义下唯一
• 计算策略理论为后来的游戏语义方法（如Abramsky、Hyland与Ong的研究）提供了一种替代途径