Up-To Techniques for Behavioural Metrics via Fibrations

Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to techniques for behavioural metrics between systems modelled as coalgebras and we provide abstract results to prove their soundness in a compositional way. In order to obtain a general framework, we need a systematic way to lift functors: we show that the Wasserstein lifting of a functor, introduced in a previous work, corresponds to a change of base in a fibrational sense. This observation enables us to reuse existing results about soundness of up-to techniques in a fibrational setting. We focus on the fibrations of predicates and relations valued in a quantale, for which pseudo-metric spaces are an example. To illustrate our approach we provide an example on distances between regular languages

Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad

Generalized metric spaces are obtained by weakening the requirements (e.g., symmetry) on the distance function and by allowing it to take values in structures (e.g., quantales) that are more general than the set of non-negative real numbers. Quantale-valued metric spaces have gained prominence due to their use in quantitative reasoning on programs/systems, and for defining various notions of behavioral metrics. We investigate imprecision and robustness in the framework of quantale-valued metric spaces, when the quantale is continuous. In particular, we study the relation between the robust topology, which captures robustness of analyses, and the Hausdorff-Smyth hemi-metric. To this end, we define a preorder-enriched monad $\mathsf{P}_S$, called the Hausdorff-Smyth monad, and when $Q$ is a continuous quantale and $X$ is a $Q$-metric space, we relate the topology induced by the metric on $\mathsf{P}_S(X)$ with the robust topology on the powerset $\mathsf{P}(X)$ defined in terms of the metric on $X$.Comment: 19 pages, 1 figur

Combining Semilattices and Semimodules

We describe the canonical weak distributive law $\delta \colon \mathcal S \mathcal P \to \mathcal P \mathcal S$ of the powerset monad $\mathcal P$ over the $S$-left-semimodule monad $\mathcal S$, for a class of semirings $S$. We show that the composition of $\mathcal P$ with $\mathcal S$ by means of such $\delta$ yields almost the monad of convex subsets previously introduced by Jacobs: the only difference consists in the absence in Jacobs's monad of the empty convex set. We provide a handy characterisation of the canonical weak lifting of $\mathcal P$ to $\mathbb{EM}(\mathcal S)$ as well as an algebraic theory for the resulting composed monad. Finally, we restrict the composed monad to finitely generated convex subsets and we show that it is presented by an algebraic theory combining semimodules and semilattices with bottom, which are the algebras for the finite powerset monad $\mathcal P_f$

From enhanced coinduction towards enhanced induction

International audienceThere exist a rich and well-developed theory of enhancements of the coinduction proof method, widely used on behavioural relations such as bisimilarity. We study how to develop an analogous theory for inductive behaviour relations, i.e., relations defined from inductive observables. Similarly to the coinductive setting, our theory makes use of (semi)-progressions of the form R->F(R), where R is a relation on processes and F is a function on relations, meaning that there is an appropriate match on the transitions that the processes in R can perform in which the process derivatives are in F(R). For a given preorder, an enhancement corresponds to a sound function, i.e., one for which R->F(R) implies that R is contained in the preorder; and similarly for equivalences. We introduce weights on the observables of an inductive relation, and a weight-preserving condition on functions that guarantees soundness. We show that the class of functions contains non-trivial functions and enjoys closure properties with respect to desirable function constructors, so to be able to derive sophisticated sound functions (and hence sophisticated proof techniques) from simpler ones. We consider both strong semantics (in which all actions are treated equally) and weak semantics (in which one abstracts from internal transitions). We test our enhancements on a few non-trivial examples

Fuzzy Algebraic Theories and M,N-adhesive categories

This thesis deals with two quite unrelated subjects in Computer Science: one is the relationship between algebraic theories and monads, the other one is the study of adhesivity properties of categories. The first part of the thesis begins by revisiting some basic facts regarding monads. Specifically, we review the correspondence between monads, with rank, on the category of sets and functions, and algebraic theories in which the operations’ arity is bounded by some regular cardinal. Next, we move to the heart of this part of the thesis: the extension of this correspondence to the category Fuz(H) of fuzzy sets. This result is obtained by means of a formal system for fuzzy algebraic reasoning. We define a sequent calculus based on two types of propositions: those that establish the equality of terms, and those that assert the membership degree of such terms. We establish a sound semantics for this calculus, and demonstrate the existence of a notion of free model for any theory in the system. This, in turn, allows us, to prove a completeness result: a formula is derivable from a given theory if and only if it is satisfied by all models of the theory. Moreover, we also prove that, under certain restrictions, it is possible to recover models of a given theory as Eilenberg-Moore algebras for a monad on Fuz(H). Finally, leveraging the work of Milius and Urbat, we provide a HSP-like characterizations of subcategories of algebras that are categories of models of specific types of theories. The second part of the thesis is devoted to the study of adhesivity properties of various categories. Adhesive and quasiadhesive categories, and other generalizations such as M,N-adhesive ones, marked a watershed moment for the algebraic approaches to the rewriting of graph-like structures, since they provide an abstract framework where many general results (on, e.g., parallelism) could be recast and uniformly proved. However, often checking that a model satisfies the adhesivity properties is far from immediate. After having recalled, the basic definitions, we present a new criterion giving a sufficient condition for M,N-adhesivity. It is known that in a quasiadhesive category the join of any two regular subobjects is also a regular subobject. Conversely, if regular monomorphisms are adhesive, the existence of a regular join for every pair of regular subobjects implies quasiadhesivity. Furthermore, (quasi)adhesive categories can be embedded in a Grothendieck topos via a functor that preserves pullbacks and pushouts along (regular) monomorphisms. In this paper, we extend these results to M,N-adhesive categories. To achieve this, we introduce the concept of an N-(pre)adhesive morphism, which enables us to express M,N-adhesivity as a condition on the poset of subobjects. Additionally, N-adhesive morphisms allow us to demonstrate how a M,N-adhesive category can be embedded into a Grothendieck topos, preserving pullbacks and M,N-pushouts. Finally, we exploit the previous results to establish adhesivity properties of several existing categories of graph-like structures, including hypergraphs, various kinds of hierarchical graphs (a formalism that is notoriously difficult to fit in the mould of algebraic approaches to rewriting), and combinations of them