🤖 AI Summary
This paper addresses the challenge of determining optimality (minimality or maximality) of fixed points in semantics defined recursively or iteratively. Methodologically, it introduces the first structured verification framework grounded in gs-monotone categories, elevating fixed-point checking to the monadic categorical level. It identifies the lax gs-monotone functor structure of approximation maps and extends this to co-algebraic behavioral metrics via Wasserstein liftings. Technically, the framework integrates gs-monotonicity theory, lax functor modeling, and functional circuit-style composition to yield UDEfix—a compositional, modular verification tool. The approach unifies verification across diverse semantic notions, including bisimilarity, termination probabilities in stochastic games, and behavioral metrics. By embedding fixed-point reasoning within a principled categorical setting, the framework significantly enhances both the expressivity and practical applicability of fixed-point verification in programming language semantics.
📝 Abstract
Fixpoints are ubiquitous in computer science as they play a central role in providing a meaning to recursive and cyclic definitions. Bisimilarity, behavioural metrics, termination probabilities for Markov chains and stochastic games are defined in terms of least or greatest fixpoints. Here we show that our recent work which proposes a technique for checking whether the fixpoint of a function is the least (or the largest) admits a natural categorical interpretation in terms of gs-monoidal categories. The technique is based on a construction that maps a function to a suitable approximation. We study the compositionality properties of this mapping and show that under some restrictions it can naturally be interpreted as a (lax) gs-monoidal functor. This guides the development of a tool, called UDEfix that allows us to build functions (and their approximations) like a circuit out of basic building blocks and subsequently perform the fixpoints checks. We also show that a slight generalisation of the theory allows one to treat a new relevant case study: coalgebraic behavioural metrics based on Wasserstein liftings.
