Coalgebraic Behavioral Metrics

We study different behavioral metrics, such as those arising from both branching and linear-time semantics, in a coalgebraic setting. Given a coalgebra $\alpha\colon X \to HX$ for a functor $H \colon \mathrm{Set}\to \mathrm{Set}$, we define a framework for deriving pseudometrics on $X$ which measure the behavioral distance of states. A crucial step is the lifting of the functor $H$ on $\mathrm{Set}$ to a functor $\overline{H}$ on the category $\mathrm{PMet}$ of pseudometric spaces. We present two different approaches which can be viewed as generalizations of the Kantorovich and Wasserstein pseudometrics for probability measures. We show that the pseudometrics provided by the two approaches coincide on several natural examples, but in general they differ. If $H$ has a final coalgebra, every lifting $\overline{H}$ yields in a canonical way a behavioral distance which is usually branching-time, i.e., it generalizes bisimilarity. In order to model linear-time metrics (generalizing trace equivalences), we show sufficient conditions for lifting distributive laws and monads. These results enable us to employ the generalized powerset construction

Towards Trace Metrics via Functor Lifting

We investigate the possibility of deriving metric trace semantics in a coalgebraic framework. First, we generalize a technique for systematically lifting functors from the category Set of sets to the category PMet of pseudometric spaces, showing under which conditions also natural transformations, monads and distributive laws can be lifted. By exploiting some recent work on an abstract determinization, these results enable the derivation of trace metrics starting from coalgebras in Set. More precisely, for a coalgebra on Set we determinize it, thus obtaining a coalgebra in the Eilenberg-Moore category of a monad. When the monad can be lifted to PMet, we can equip the final coalgebra with a behavioral distance. The trace distance between two states of the original coalgebra is the distance between their images in the determinized coalgebra through the unit of the monad. We show how our framework applies to nondeterministic automata and probabilistic automata

Fixpoint Games on Continuous Lattices

Many analysis and verifications tasks, such as static program analyses and model-checking for temporal logics reduce to the solution of systems of equations over suitable lattices. Inspired by recent work on lattice-theoretic progress measures, we develop a game-theoretical approach to the solution of systems of monotone equations over lattices, where for each single equation either the least or greatest solution is taken. A simple parity game, referred to as fixpoint game, is defined that provides a correct and complete characterisation of the solution of equation systems over continuous lattices, a quite general class of lattices widely used in semantics. For powerset lattices the fixpoint game is intimately connected with classical parity games for $\mu$-calculus model-checking, whose solution can exploit as a key tool Jurdzi\'nski's small progress measures. We show how the notion of progress measure can be naturally generalised to fixpoint games over continuous lattices and we prove the existence of small progress measures. Our results lead to a constructive formulation of progress measures as (least) fixpoints. We refine this characterisation by introducing the notion of selection that allows one to constrain the plays in the parity game, enabling an effective (and possibly efficient) solution of the game, and thus of the associated verification problem. We also propose a logic for specifying the moves of the existential player that can be used to systematically derive simplified equations for efficiently computing progress measures. We discuss potential applications to the model-checking of latticed $\mu$-calculi and to the solution of fixpoint equations systems over the reals

Beneficial Bacteria Isolated from Grapevine Inner Tissues Shape Arabidopsis thaliana Roots

We investigated the potential plant growth-promoting traits of 377 culturable endophytic bacteria, isolated from Vitis vinifera cv. Glera, as good biofertilizer candidates in vineyard management. Endophyte ability in promoting plant growth was assessed in vitro by testing ammonia production, phosphate solubilization, indole-3-acetic acid (IAA) and IAA-like molecule biosynthesis, siderophore and lytic enzyme secretion. Many of the isolates were able to mobilize phosphate (33%), release ammonium (39%), secrete siderophores (38%) and a limited part of them synthetized IAA and IAA-like molecules (5%). Effects of each of the 377 grapevine beneficial bacteria on Arabidopsis thaliana root development were also analyzed to discern plant growth-promoting abilities (PGP) of the different strains, that often exhibit more than one PGP trait. A supervised model-based clustering analysis highlighted six different classes of PGP effects on root architecture. A. thaliana DR5::GUS plantlets, inoculated with IAA-producing endophytes, resulted in altered root growth and enhanced auxin response. Overall, the results indicate that the Glera PGP endospheric culturable microbiome could contribute, by structural root changes, to obtain water and nutrients increasing plant adaptation and survival. From the complete cultivable collection, twelve promising endophytes mainly belonging to the Bacillus but also to Micrococcus and Pantoea genera, were selected for further investigations in the grapevine host plants towards future application in sustainable management of vineyards

Bisimilarity and Behaviour-Preserving Reconfigurations of Open Petri Nets

We propose a framework for the specification of behaviour-preserving reconfigurations of systems modelled as Petri nets. The framework is based on open nets, a mild generalisation of ordinary Place/Transition nets suited to model open systems which might interact with the surrounding environment and endowed with a colimit-based composition operation. We show that natural notions of bisimilarity over open nets are congruences with respect to the composition operation. The considered behavioural equivalences differ for the choice of the observations, which can be single firings or parallel steps. Additionally, we consider weak forms of such equivalences, arising in the presence of unobservable actions. We also provide an up-to technique for facilitating bisimilarity proofs. The theory is used to identify suitable classes of reconfiguration rules (in the double-pushout approach to rewriting) whose application preserves the observational semantics of the net.Comment: To appear in "Logical Methods in Computer Science", 41 page

Behavioral Metrics via Functor Lifting

We study behavioral metrics in an abstract coalgebraic setting. Given a coalgebra alpha: X -> FX in Set, where the functor F specifies the branching type, we define a framework for deriving pseudometrics on X which measure the behavioral distance of states. A first crucial step is the lifting of the functor F on Set to a functor in the category PMet of pseudometric spaces. We present two different approaches which can be viewed as generalizations of the Kantorovich and Wasserstein pseudometrics for probability measures. We show that the pseudometrics provided by the two approaches coincide on several natural examples, but in general they differ. Then a final coalgebra for F in Set can be endowed with a behavioral distance resulting as the smallest solution of a fixed-point equation, yielding the final coalgebra in PMet. The same technique, applied to an arbitrary coalgebra alpha: X -> FX in Set, provides the behavioral distance on X. Under some constraints we can prove that two states are at distance 0 if and only if they are behaviorally equivalent.Comment: to be published in: Proceedings of FSTTCS 201

verifying a behavioural logic for graph transformation systems

We propose a framework for the verication of behavioural properties of systems modelled as graph transformation systems. The properties can be expressed in a temporal logic which is basically a -calculus where the state predicates are formulae of a monadic second order logic, describing graph properties. The verication technique relies on an algorithm for the construction of nite over-approximations of the unfolding of a graph transformation system

Fixpoint Theory -- Upside Down

Knaster-Tarski's theorem, characterising the greatest fixpoint of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The dual principle, used for showing that an element is above the least fixpoint, is related to inductive invariants. In this paper we provide proof rules which are similar in spirit but for showing that an element is above the greatest fixpoint or, dually, below the least fixpoint. The theory is developed for non-expansive monotone functions on suitable lattices of the form $\mathbb{M}^Y$, where $Y$ is a finite set and $\mathbb{M}$ an MV-algebra, and it is based on the construction of (finitary) approximations of the original functions. We show that our theory applies to a wide range of examples, including termination probabilities, metric transition systems, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games

Impact of phenylpropanoid compounds on heat stress tolerance in carrot cell cultures

The phenylpropanoid and flavonoid families include thousands of specialized metabolites that influence a wide range of processes in plants, including seed dispersal, auxin transport, photoprotection, mechanical support and protection against insect herbivory. Such metabolites play a key role in the protection of plants against abiotic stress, in many cases through their well-known ability to inhibit the formation of reactive oxygen species (ROS). However, the precise role of specific phenylpropanoid and flavonoid molecules is unclear. We therefore investigated the role of specific anthocyanins (ACs) and other phenylpropanoids that accumulate in carrot cells cultivated in vitro, focusing on their supposed ability to protect cells from heat stress. First we characterized the effects of heat stress to identify quantifiable morphological traits as markers of heat stress susceptibility. We then fed the cultures with precursors to induce the targeted accumulation of specific compounds, and compared the impact of heat stress in these cultures and unfed controls. Data modeling based on Projection to Latent Structures (PLS) regression revealed that metabolites containing coumaric or caffeic acid, including ACs, correlate with less heat damage. Further experiments suggested that one of the cellular targets damaged by heat stress and protected by these metabolites is the actin microfilament cytoskeleton