Quantalic Behavioural Distances

02/14/2022
βˆ™
by   Sergey Goncharov, et al.
βˆ™
0
βˆ™

Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest in generic approaches to behavioural distances. In particular, coalgebraic methods capture variations in the system type (nondeterministic, probabilistic, game-based etc.), and the notion of quantale abstracts over actual values distances take, thus covering, e.g., two-valued equivalences, metrics, and probabilistic metrics. Coalgebraic behavioural distances have variously been based on liftings of 𝖲𝖾𝗍-functors to categories of metric spaces; on modalities modeled as predicate liftings, via a generalised Kantorovich construction; and on lax extensions of 𝖲𝖾𝗍-functors to categories of quantitative relations. Every lax extension induces a functor lifting in a straightforward manner. Moreover, it has recently been shown that every lax extension is Kantorovich, i.e. induced by a suitable choice of monotone predicate liftings. In the present work, we complete this picture by determining, in coalgebraic and quantalic generality, when a functor lifting is induced by a class of predicate liftings or by a lax extension. We subsequently show coincidence of the respective induced notions of behavioural distances, in a unified approach via double categories that applies even more widely, e.g. to (quasi)uniform spaces.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
βˆ™ 07/02/2020

Characteristic Logics for Behavioural Metrics via Fuzzy Lax Extensions

Behavioural distances provide a fine-grained measure of equivalence in s...
research
βˆ™ 01/27/2018

Quantitative Behavioural Reasoning for Higher-order Effectful Programs: Applicative Distances (Extended Version)

This paper studies the quantitative refinements of Absramsky's applicati...
research
βˆ™ 07/19/2022

Quantitative Hennessy-Milner Theorems via Notions of Density

The classical Hennessy-Milner theorem is an important tool in the analys...
research
βˆ™ 06/02/2023

Quantitative Graded Semantics and Spectra of Behavioural Metrics

Behavioural metrics provide a quantitative refinement of classical two-v...
research
βˆ™ 07/27/2023

Graded Semantics and Graded Logics for Eilenberg-Moore Coalgebras

Coalgebra, as the abstract study of state-based systems, comes naturally...
research
βˆ™ 11/27/2017

Distances between States and between Predicates

This paper gives a systematic account of various metrics on probability ...
research
βˆ™ 12/16/2022

Weakly weighted generalised quasi-metric spaces and semilattices

Motivated by recent applications to entropy theory in dynamical systems,...

Please sign up or login with your details

Forgot password? Click here to reset