17 research outputs found
Quantitative Hennessy-Milner Theorems via Notions of Density
The classical Hennessy-Milner theorem is an important tool in the analysis of concurrent processes;
it guarantees that any two non-bisimilar states in finitely branching labelled transition systems can
be distinguished by a modal formula. Numerous variants of this theorem have since been established
for a wide range of logics and system types, including quantitative versions where lower bounds on
behavioural distance (e.g. in weighted, metric, or probabilistic transition systems) are witnessed
by quantitative modal formulas. Both the qualitative and the quantitative versions have been
accommodated within the framework of coalgebraic logic, with distances taking values in quantales,
subject to certain restrictions, such as being so-called value quantales. While previous quantitative
coalgebraic Hennessy-Milner theorems apply only to liftings of set functors to (pseudo)metric spaces,
in the present work we provide a quantitative coalgebraic Hennessy-Milner theorem that applies more
widely to functors native to metric spaces; notably, we thus cover, for the first time, the well-known
Hennessy-Milner theorem for continuous probabilistic transition systems, where transitions are given
by Borel measures on metric spaces, as an instance of such a general result. In the process, we also
relax the restrictions imposed on the quantale, and additionally parametrize the technical account
over notions of closure and, hence, density, providing associated variants of the Stone-Weierstraß
theorem; this allows us to cover, for instance, behavioural ultrametrics.publishe
Expressive Logics for Coinductive Predicates
The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata
Kantorovich Functors and Characteristic Logics for Behavioural Distances
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 the actual values
distances take, thus covering, e.g., two-valued equivalences, (pseudo-)metrics,
and probabilistic (pseudo-)metrics. Coalgebraic behavioural distances have been
based either on liftings of SET-functors to categories of metric spaces, or on
lax extensions of SET-functors to categories of quantitative relations. Every
lax extension induces a functor lifting but not every lifting comes from a lax
extension. It was shown recently that every lax extension is Kantorovich, i.e.
induced by a suitable choice of monotone predicate liftings, implying via a
quantitative coalgebraic Hennessy-Milner theorem that behavioural distances
induced by lax extensions can be characterized by quantitative modal logics.
Here, we essentially show the same in the more general setting of behavioural
distances induced by functor liftings. In particular, we show that every
functor lifting, and indeed every functor on (quantale-valued) metric spaces,
that preserves isometries is Kantorovich, so that the induced behavioural
distance (on systems of suitably restricted branching degree) can be
characterized by a quantitative modal logic
Hennessy-Milner Theorems via Galois Connections
We introduce a general and compositional, yet simple, framework that allows to derive soundness and expressiveness results for modal logics characterizing behavioural equivalences or metrics (also known as Hennessy-Milner theorems). It is based on Galois connections between sets of (real-valued) predicates on the one hand and equivalence relations/metrics on the other hand and covers a part of the linear-time-branching-time spectrum, both for the qualitative case (behavioural equivalences) and the quantitative case (behavioural metrics). We derive behaviour functions from a given logic and give a condition, called compatibility, that characterizes under which conditions a logically induced equivalence/metric is induced by a fixpoint equation. In particular, this framework allows to derive a new fixpoint characterization of directed trace metrics
Explicit Hopcroft's Trick in Categorical Partition Refinement
Algorithms for partition refinement are actively studied for a variety of
systems, often with the optimisation called Hopcroft's trick. However, the
low-level description of those algorithms in the literature often obscures the
essence of Hopcroft's trick. Our contribution is twofold. Firstly, we present a
novel formulation of Hopcroft's trick in terms of general trees with weights.
This clean and explicit formulation -- we call it Hopcroft's inequality -- is
crucially used in our second contribution, namely a general partition
refinement algorithm that is \emph{functor-generic} (i.e. it works for a
variety of systems such as (non-)deterministic automata and Markov chains).
Here we build on recent works on coalgebraic partition refinement but depart
from them with the use of fibrations. In particular, our fibrational notion of
-partitioning exposes a concrete tree structure to which Hopcroft's
inequality readily applies. It is notable that our fibrational framework
accommodates such algorithmic analysis on the categorical level of abstraction
Hennessy-Milner Theorems via Galois Connections
We introduce a general and compositional, yet simple, framework that allows
us to derive soundness and expressiveness results for modal logics
characterizing behavioural equivalences or metrics (also known as
Hennessy-Milner theorems). It is based on Galois connections between sets of
(real-valued) predicates on the one hand and equivalence relations/metrics on
the other hand and covers a part of the linear-time-branching-time spectrum,
both for the qualitative case (behavioural equivalences) and the quantitative
case (behavioural metrics). We derive behaviour functions from a given logic
and give a condition, called compatibility, that characterizes under which
conditions a logically induced equivalence/metric is induced by a fixpoint
equation. In particular this framework allows us to derive a new fixpoint
characterization of directed trace metrics
Coinduction in Flow: The Later Modality in Fibrations
This paper provides a construction on fibrations that gives access to the so-called later modality, which allows for a controlled form of recursion in coinductive proofs and programs. The construction is essentially a generalisation of the topos of trees from the codomain fibration over sets to arbitrary fibrations. As a result, we obtain a framework that allows the addition of a recursion principle for coinduction to rather arbitrary logics and programming languages. The main interest of using recursion is that it allows one to write proofs and programs in a goal-oriented fashion. This enables easily understandable coinductive proofs and programs, and fosters automatic proof search.
Part of the framework are also various results that enable a wide range of applications: transportation of (co)limits, exponentials, fibred adjunctions and first-order connectives from the initial fibration to the one constructed through the framework. This means that the framework extends any first-order logic with the later modality. Moreover, we obtain soundness and completeness results, and can use up-to techniques as proof rules. Since the construction works for a wide variety of fibrations, we will be able to use the recursion offered by the later modality in various context. For instance, we will show how recursive proofs can be obtained for arbitrary (syntactic) first-order logics, for coinductive set-predicates, and for the probabilistic modal mu-calculus. Finally, we use the same construction to obtain a novel language for probabilistic productive coinductive programming. These examples demonstrate the flexibility of the framework and its accompanying results
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science