[论文解读] The Quantitative Collapse of Concurrent Games with Symmetry

Pierre Clairambault|arXiv (Cornell University)|Jul 7, 2021

Logic, programming, and type systems参考文献 31被引用 1

一句话总结

本文提出了一种新颖的、保持解释的坍缩函子，从具有对称性的细并发游戏映射到连续半环上的加权关系模型，通过引入‘正见证’的精细概念，解决了在坍缩至加权关系模型时计数无限对称见证的挑战。主要贡献在于构建了一个组合上正确的计数机制，该机制保持函子性和语义，从而实现对定量 PCF 的完全充分性，并可扩展至概率模型及其他加权模型。

ABSTRACT

Quantitative semantics are those denotational semantics that inherit from linear logic [Jean-Yves Girard, 1987] a sensitivity to the multiplicity of resources involved in computation. Those include the relational model [Jean-Yves Girard, 1987] and its numerous variations (such as finiteness spaces [Thomas Ehrhard, 2005], weighted relational models [Jim Laird et al., 2013] and their extensions [Thomas Ehrhard et al., 2011; Thomas Ehrhard, 2002], generalized species of structure [Fiore et al., 2008], span models [Paul-André Melliès, 2019; Pierre Clairambault and Simon Forest, 2023], etc), as well as related syntactic methods such as non-idempotent intersection types [Daniel de Carvalho, 2018] and Taylor expansion of lambda-terms [Thomas Ehrhard and Laurent Regnier, 2003]. Interactive semantics are usually also quantitative, but in addition they present the interactive behaviour of proofs and programs, generally organized chronologically - those include the many variants of game semantics (starting with [J. M. E. Hyland and C.-H. Luke Ong, 2000; Samson Abramsky et al., 2000]), and other frameworks such as Geometry of Interaction [Girard, 1989] or ludics [Jean-Yves Girard, 2001]. Both families are cornerstones of modern denotational semantics, and both have associated Alonzo Church awards: game semantics in 2017, and quantitative semantics (in particular, differential linear logic and the differential λ-calculus) in 2024. It has more or less always been clear to the experts that the two, sharing an origin in linear logic, are conceptually related. Yet there are differences, which seem fundamental: in particular, while quantitative models compose relationally, the composition of strategies follows an intricate "parallel interaction plus hiding" process inspired from concurrency theory [Abramsky, 1997]. The two families of models have also historically targeted different kinds of languages: whereas quantitative semantics focused on theoretical calculi (and the λ-calculus in particular), game semantics is known for fully abstract models for languages with elaborate combinations of effects including local state [Samson Abramsky and Guy McCusker, 1996], control operators [James Laird, 1997], and concurrent primitives [Dan R. Ghica and Andrzej S. Murawski, 2008]. Early on, researchers have explored the relationship between the two [Thomas Ehrhard, 1996; Patrick Baillot et al., 1997], and investigations on this question have spanned decades [Pierre Boudes, 2009; Ana C. Calderon and Guy McCusker, 2010; Takeshi Tsukada and C.-H. Luke Ong, 2016; C.-H. Luke Ong, 2017]. In particular, Melliès' work on asynchronous games [Paul-André Melliès, 2006; Paul-André Melliès, 2005] made significant conceptual contributions, showing that the issue was enlightened by adopting a positional formulation of game semantics, where points in the relational model simply arise as certain positions. This talk surveys recent developments in this line of work, shedding light on the connection between those two families. Our work is set in so-called "thin concurrent games" [Simon Castellan et al., 2019; Pierre Clairambault, 2024], an extension with symmetry of Rideau and Winskel’s concurrent games on event structures [Silvain Rideau and Glynn Winskel, 2011]. Event structures being one of the main "truly concurrent" models of concurrency [Glynn Winskel, 1986], it is perhaps expected that thin concurrent games can model concurrent languages: they provide a truly concurrent refinement of Ghica and Murawski’s fully abstract model of Idealized Concurrent Algol [Simon Castellan and Pierre Clairambault, 2024; Pierre Clairambault, 2024]. But beyond the semantics of concurrency, thin concurrent games are also a deep reworking on game semantics built from causal principles, inheriting from asynchronous games a positional flavour. In thin concurrent games, strategies have a dual nature: an event-based nature where they appear as certain event structures composed via parallel interaction plus hiding; or a positional nature where they appear as certain spans of groupoids, composed by pullback (modulo a technical condition on strategies called visibility) - they can be regarded both as a games and a relational model! Leveraging this dual nature, in a sequence of papers with Castellan, de Visme, Olimpieri and Paquet, we have been able to link the single framework of thin concurrent games with numerous other models. This includes various traditional alternating or non-alternating games models [Simon Castellan and Pierre Clairambault, 2024; Pierre Clairambault, 2024], the weighted relational model [Pierre Clairambault and Hugo Paquet, 2021], the quantum relational model [Pierre Clairambault and Marc de Visme, 2020], generalized species of structure [Pierre Clairambault et al., 2023], and - going beyond quantitative semantics - the linear Scott model [Clairambault, 2025], a linear decomposition of standard Scott domain semantics [Thomas Ehrhard, 2012]. All these distinct models are obtained by projecting away certain aspects of thin concurrent games, giving some support to the claim that thin concurrent games are a Rosetta stone for interactive and quantitative semantics.

研究动机与目标

• 解决在非仿射并发游戏中，将策略坍缩至加权关系模型时，计数无限对称配置的根本性挑战。
• 在任意连续半环（包括 N∪{+∞} 和概率模型）上，定义从具有对称性的细并发游戏到加权关系模型的语义保持函子。
• 证明标准配置见证的概念在对称性下失效，并提出‘正见证’作为正确替代。
• 证明在复合策略中，尽管对称性存在复杂相互作用，坍缩函子在复合下仍保持函子性。
• 将框架扩展至支持递归，并实现对 R-PCF 的完全充分性，确保坍缩保持关键的逻辑与计算结构。

提出的方法

• 引入‘正见证’作为在对称性下配置的等价类，通过消去策略中无限对称副本实现计数。
• 在对象上，通过将每个游戏映射为其配置对称类的网（web）来定义坍缩函子。
• 在态射上，将解释定义为加权关系，其中对称类的系数为该策略中所有正见证的权重（来自半环）之和。
• 利用对称性的组合分析，确保见证复合尊重张量结构与函子性。
• 利用连续半环中的整数除法概念，将构造扩展至一般加权模型，而不仅限于 N∪{+∞}。
• 证明坍缩函子的连续性及对 R-PCF 的充分性，表明坍缩与递归和类型规则可交换。

实验结果

研究问题

• RQ1 如何在细并发游戏的策略中计数无限对称配置，使其与加权关系模型中的系数相匹配？
• RQ2 为何朴素的配置见证概念在对称性存在时无法保持函子性？
• RQ3 何种组合意义上的‘见证’概念能尊重对称性并实现语义保持的坍缩？
• RQ4 坍缩函子能否从 N∪{+∞} 扩展至任意连续半环，包括概率模型与最坏情况模型？
• RQ5 坍缩是否保持关键结构特征，如递归、复合，以及范畴模型中的相对 Seely 结构？

主要发现

• 标准配置见证概念因无限对称副本的存在而无法产生函子性坍缩，即使在计数配置时亦然。
• 引入‘正见证’——即在对称性下配置的精细等价类——可实现正确且保持解释的坍缩至加权关系模型。
• 坍缩函子保持复合性且连续，确保对 R-PCF 的充分性及递归定义的正确性。
• 该构造可扩展至任意具有整数除法的连续半环，涵盖概率模型、最坏情况模型及其他定量模型。
• 坍缩是一个相对 Seely ∼-函子，保持模型的完整范畴结构，包括 ⊸ 和 & 连接词。
• 关键洞见在于：见证复合必须同时考虑策略端与游戏端的对称性，因而需要对对称作用进行详尽的组合分析。