[Paper Review] Interaction Graphs: Graphings

Thomas Seiller|arXiv (Cornell University)|May 24, 2014

Logic, programming, and type systems21 references4 citations

TL;DR

This paper introduces graphings—measure-theoretic generalizations of directed weighted graphs—as a new framework for modeling multiplicative-additive linear logic (MALL), extending Girard's Geometry of Interaction. By defining models via weight monoids and microcosms of measurable transformations, it constructs a hierarchy of models that generalize existing GoI constructions and successfully account for second-order quantification without prior limitations.

ABSTRACT

In two previous papers, we exposed a combinatorial approach to the program of Geometry of Interaction, a program initiated by Jean-Yves Girard. The strength of our approach lies in the fact that we interpret proofs by simpler structures - graphs - than Girard's constructions, while generalizing the latter since they can be recovered as special cases of our setting. This third paper extends this approach by considering a generalization of graphs named graphings, which is in some way a geometric realization of a graph. This very general framework leads to a number of new models of multiplicative-additive linear logic which generalize Girard's geometry of interaction models and opens several new lines of research. As an example, we exhibit a family of such models which account for second-order quantification without suffering the same limitations as Girard's models.

Motivation & Objective

• To generalize Girard's Geometry of Interaction models by introducing graphings as a geometric realization of graphs on measure spaces.
• To overcome limitations in existing GoI models that fail to adequately account for second-order quantification.
• To establish a systematic hierarchy of MALL models parameterized by weight monoids and microcosms of measurable transformations.
• To provide a unified framework that recovers and extends previous GoI constructions, including those based on hyperfinite factors.
• To explore connections between computational complexity and dynamical systems via microcosm-based models.

Proposed method

• Define graphings as directed, weighted graphs whose vertices are measurable subsets of a measure space and edges are realized by non-singular measurable maps.
• Introduce two key parameters: a weight monoid Ω for edge weights and a microcosm 𝔪 — a monoid of non-singular measurable transformations on a measure space (X,ℬ,μ).
• Construct interpretations of MALL proofs via projective operators, using distributivity maps and tensor products to model logical connectives.
• Define proof interpretation recursively: for ⊗, use tensor product with identity maps; for ⊕, use projective implementations; for cut, use a generalized composition operation denoted ⊛.
• Leverage the trefoil property as a foundational geometric-algebraic structure underlying all GoI constructions.
• Use localization via variable enumeration to extend local MALL models to full MALL², enabling second-order quantification.

Experimental results

Research questions

• RQ1 Can a general framework be developed to model multiplicative-additive linear logic that extends and unifies existing Geometry of Interaction constructions?
• RQ2 How can second-order quantification be modeled in GoI-style frameworks without the limitations present in Girard’s original models?
• RQ3 What is the role of microcosms — monoids of measurable transformations — in shaping the computational power and logical expressiveness of GoI models?
• RQ4 How do the hierarchies of weight monoids and microcosms relate to complexity classes such as regular languages and logspace predicates?
• RQ5 Can the theory of dynamical systems and ergodic invariants (e.g., ℓ²-Betti numbers) be used to analyze or classify computational complexity within this GoI framework?

Key findings

• The paper constructs a hierarchy of models for MALL based on graphings, parameterized by weight monoids and microcosms, generalizing all prior GoI constructions.
• Second-order quantification is successfully modeled in the framework, overcoming limitations of Girard’s original GoI models that could not handle it uniformly.
• The interpretation of proofs is defined via projective operators and distributivity maps, ensuring soundness: if π is a valid derivation in locMALL², then IΦ(π) is a successful project in the interpretation of its conclusion.
• Full MALL² soundness is established by localizing proofs via variable enumeration, showing that IΦ(πᵉ) is a successful project in the localized interpretation.
• The framework enables connections to computational complexity: larger microcosms correspond to larger classes of definable predicates, such as regular languages and logspace predicates.
• The approach opens new avenues for modeling quantum computation by restricting microcosms to specific bases of unitary gates, allowing analysis of computational effects of gate choice.