[Paper Review] Learning to Count up to Symmetry

Pierre Clairambault|arXiv (Cornell University)|Jun 9, 2020

Logic, programming, and type systems8 references1 citations

TL;DR

This paper resolves a foundational issue in concurrent game semantics by introducing a refined notion of witnesses—canonical representatives under symmetry—for counting strategy configurations in weighted relational models. It proves that compositionality of weighted relational models holds under composition of strategies, provided the strategies do not deadlock and the games are representable, correcting a flawed approach in prior work that incorrectly used symmetry classes as witnesses.

ABSTRACT

In this paper we develop the theory of how to count, in thin concurrent games, the configurations of a strategy witnessing that it reaches a certain configuration of the game. This plays a central role in many recent developments in concurrent games, whenever one aims to relate concurrent strategies with weighted relational models. The difficulty, of course, is symmetry: in the presence of symmetry many configurations of the strategy are, morally, different instances of the same, only differing on the inessential choice of copy indices. How do we know which ones to count? The purpose of the paper is to clarify that, uncovering many strange phenomena and fascinating pathological examples along the way. To illustrate the results, we show that a collapse operation to a simple weighted relational model simply counting witnesses is preserved under composition, provided the strategies involved do not deadlock.

Motivation & Objective

• To resolve the foundational problem of counting strategy configurations in concurrent games when symmetry causes overcounting.
• To identify why prior approaches using symmetry classes as witnesses fail in weighted relational models.
• To develop a new, correct notion of witnesses based on canonical representatives under positive and negative reindexing.
• To prove that compositionality of weighted relational models holds under composition of strategies, provided no deadlock occurs.
• To establish representability as a necessary condition for the correctness of the witness counting mechanism.

Proposed method

• Introduces the concept of canonical representatives for symmetry classes in thin concurrent games, leveraging the split of symmetry into positive and negative reindexings.
• Defines representability as a structural condition ensuring that every symmetry class admits a canonical representative decomposable into positive and negative symmetries.
• Uses the +-covered condition to isolate relevant configurations in strategy interactions, enabling precise counting of witnesses.
• Applies a collapse operation from thin concurrent games to weighted relational models, where weights are counts of canonical witnesses.
• Employs a chain of lemmas and bijections to relate witness counts in interaction and composition, preserving weight structure.
• Establishes a formula analogous to matrix multiplication (α ◦ β)_{a,c} = Σ_b α_{a,b} · β_{b,c} for the weighted model, using canonical witnesses.

Experimental results

Research questions

• RQ1 Why does counting symmetry classes as witnesses fail in weighted relational models derived from concurrent games?
• RQ2 What structural condition ensures that witness counting remains well-defined and compositionally correct in the presence of symmetry?
• RQ3 How can one correctly define witnesses in concurrent strategies so that composition preserves the weighted relational model structure?
• RQ4 What is the role of positive and negative reindexing in defining canonical representatives for symmetry classes?
• RQ5 Does the collapse from concurrent games to weighted relational models preserve compositionality, and under what conditions?

Key findings

• The naive approach of using symmetry classes as witnesses—previously proposed in [3]—is incorrect, as demonstrated by counterexamples where distinct configurations in the same symmetry class contribute differently to composition.
• The correct witness count is not based on symmetry classes but on canonical representatives under the split of symmetry into positive and negative reindexings.
• The collapse to a weighted relational model preserves compositionality if and only if the strategies do not deadlock and the games are representable.
• The formula (α ◦ β)_{a,c} = Σ_b α_{a,b} · β_{b,c} holds for the weighted model when using canonical witnesses, with weights corresponding to the number of such representatives.
• The counterexample in Example 29 shows that |wit_{τ⊙σ}| = 2 while |wit_σ| = |wit_τ| = 1 when using symmetry classes, proving that symmetry classes overcount.
• The counterexample in Example 30 confirms the pathology in non-well-bracketed strategies, showing that canonical witnesses are essential for correct compositionality.