DeepAI

Proper Semirings and Proper Convex Functors

Esik and Maletti introduced the notion of a proper semiring and proved that some important (classes of) semirings -- Noetherian semirings, natural numbers -- are proper. Properness matters as the equivalence problem for weighted automata over a semiring which is proper and finitely and effectively presented is decidable. Milius generalised the notion of properness from a semiring to a functor. As a consequence, a semiring is proper if and only if its associated "cubic functor" is proper. Moreover, properness of a functor renders soundness and completeness proofs for axiomatizations of equivalent behaviour. In this paper we provide a method for proving properness of functors, and instantiate it to cover both the known cases and several novel ones: (1) properness of the semirings of positive rationals and positive reals, via properness of the corresponding cubic functors; and (2) properness of two functors on (positive) convex algebras. The latter functors are important for axiomatizing trace equivalence of probabilistic transition systems. Our proofs rely on results that stretch all the way back to Hilbert and Minkowski.

• 7 publications
• 3 publications
05/04/2021

Switching 3-edge-colorings of cubic graphs

The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switch...
10/28/2019

Weak equivalence of higher-dimensional automata

This paper introduces a notion of weak equivalence for higher-dimensiona...
04/19/2021

On sublinear approximations for the Petersen coloring conjecture

If f:ℕ→ℕ is a function, then let us say that f is sublinear if lim_...
09/02/2021

Coalgebras for Bisimulation of Weighted Automata over Semirings

Weighted automata are a generalization of nondeterministic automata that...
09/25/2017

Intensional Constructed Numbers: Towards Formalizing the Notion of Algorithm

This work is meant to be a step towards the formal definition of the not...
08/31/2022

Geometrical tilings : distance, topology, compactness and completeness

We present the different distances on tilings of Rd that exist in the li...
12/01/2021

Lévy copulas: a probabilistic point of view

There is a one-to-one correspondence between Lévy copulas and proper cop...