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

Coalgebraic Trace Semantics for Continuous Probabilistic Transition Systems

Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with arbitrary (possibly uncountable) state spaces. We consider the sub-probability monad and the probability monad (Giry monad) on the category of measurable spaces and measurable functions. Our main contribution is that the existence of a final coalgebra in the Kleisli category of these monads is closely connected to the measure-theoretic extension theorem for sigma-finite pre-measures. In fact, we obtain a practical definition of the trace measure for both finite and infinite traces of PTS that subsumes a well-known result for discrete probabilistic transition systems. Finally we consider two example systems with uncountable state spaces and apply our theory to calculate their trace measures

Coalgebraic Behavior Analysis — From Qualitative To Quantitative Analyses

In order to specify and analyze the behavior of systems (computer programs, circuits etc.) it is important to have a suitable specification language. Although it is possible to define such a language separately for each type of system, it is desirable to have a standard toolbox that allows to do this in a generic way for various – possibly quite different – systems. Coalgebra, a concept of category theory, has proven to be a suitable framework to model transition systems. This class of systems includes many well-known examples like deterministic automata, nondeterministic automata or probabilistic systems. All these systems are coalgebras and their behavior can be analyzed via the notion of final coalgebra or other category theoretic constructions. This thesis investigates how to improve and build upon existing results to explore the expressive power of category theory and in particular coalgebra in behavioral analysis. The three main parts of the thesis all have a different focus but are strongly connected by the coalgebraic concepts used. Part one discusses adjunctions in the context of coalgebras. Here, well-known automata constructions such as the powerset-construction are (re)discovered as liftings of simple and well-known basic adjunctions. The second part deals with continuous generative probabilistic systems. It is shown that their trace semantics can be captured by a final coalgebra in a category of stochastic relations. The final contribution is a shift from qualitative to quantitative reasoning. Via the development of methods to lift functors on the category of sets and functions to functors on pseudometric spaces and nonexpansive functions it is possible to define a canonical, coalgebraic framework for behavioral pseudometrics.Um das Verhalten von Systemen (Computerprogrammen, Schaltkreisen etc.) zu spezifizieren und zu analysieren, ist es wichtig eine geeignete Spezifizierungssprache zu finden. Obwohl es möglich ist, eine solche Sprache separat für jeden Typ von System zu definieren, ist es erstrebenswert einen standardisierten Ansatz zu haben, welcher es ermöglicht, dies in einer generischen Weise für diverse – möglicherweise stark unterschiedliche – Systeme zu tun. Koalgebra, ein Konzept der Kategorientheorie, hat sich als geeignetes Modell zur Modellierung von Transitionssystemen herausgestellt. Diese Klasse von Systemen umfasst viele bekannte Beispiele wie deterministische, nichtdeterministische oder probabilistische Automaten. Alle diese Systeme sind Koalgebren und ihr Verhalten kann mittels finaler Koalgebren oder anderer kategorientheoretischer Konstruktionen analysiert werden. Diese Dissertation beschäftigt sich mit der Frage, wie existierende Resultate verbessert und erweitert werden können, um die Ausdrucksmächtigkeit von Kategorientheorie und besonders Koalgebra in der Verhaltensanalyse zu untersuchen. Die drei Hauptteile dieser Arbeit haben alle eine unterschiedliche Ausrichtung, sind aber durch die verwendeten koalgebraischen Methoden stark miteinander verbunden. Der erste Teil behandelt Adjunktionen im Kontext von Koalgebren. Hierbei werden übliche Automatenkonstruktionen wie die Potenzmengenkonstruktion als Lifting einfacher und wohlbekannter Basisadjunktionen (wieder-)entdeckt. Im zweiten Teil werden kontinuierliche, generative probabilististische Systeme betrachtet. Es wird gezeigt, dass deren lineares Verhalten von einer finalen Koalgebra in einer Kategorie stochastischer Relationen erfasst werden kann. Der letzte Beitrag ist ein Wechsel von qualitativer zu quantitativer Argumentation. Durch die Entwicklung von Methoden, die dazu dienen Funktoren von der Kategorie der Mengen und Funktionen zu Funktoren auf pseudometrischen Räumen und nicht-expansiven Funktionen zu erweitern, ist es möglich eine kanonische, koalgebraische Herangehensweise für Verhaltensmetriken zu entwerfen

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, by identifying conditions under which 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 in 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

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 /F 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 /F-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

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

cFLIPL Inhibits Tumor Necrosis Factor-related Apoptosis-inducing Ligand-mediated NF-κB Activation at the Death-inducing Signaling Complex in Human Keratinocytes

Human keratinocytes undergo apoptosis following treatment with tumor necrosis factor-related apoptosis-inducing ligand (TRAIL) via surface-expressed TRAIL receptors 1 and 2. In addition, TRAIL triggers nonapoptotic signaling pathways including activation of the transcription factor NF-kappaB, in particular when TRAIL-induced apoptosis is blocked. The intracellular protein cFLIP(L) interferes with TRAIL-induced apoptosis at the death-inducing signaling complex (DISC) in many cell types. To study the role of cFLIP(L) in TRAIL signaling, we established stable HaCaT keratinocyte cell lines expressing varying levels of cFLIP(L). Functional analysis revealed that relative cFLIP(L) levels correlated with apoptosis resistance to TRAIL. Surprisingly, cFLIP(L) specifically blocked TRAIL-induced NF-kappaB activation and TRAIL-dependent induction of the proinflammatory target gene interleukin-8. Biochemical characterization of the signaling pathways involved showed that apoptosis signaling was inhibited at the DISC in cFLIP(L)-overexpressing keratinocytes, although cFLIP(L) did not significantly impair enzymatic activity of the receptor complex. In contrast, recruitment and modification of receptor-interacting protein was blocked in cFLIP(L)-overexpressing cells. Taken together, our data demonstrate that cFLIP(L) is not only a central antiapoptotic modulator of TRAIL-mediated apoptosis but also an inhibitor of TRAIL-induced NF-kappaB activation and subsequent proinflammatory target gene expression. Hence, cFLIP(L) modulation in keratinocytes may not only influence apoptosis sensitivity but may also lead to altered death receptor-dependent skin inflammation

Becker Naevus Syndrome of the Lower Body: A New Case and Review of the Literature

Becker naevus syndrome is a rare epidermal naevus syndrome defined by the co-occurrence of a Becker naevus with various cutaneous, muscular and skeletal anomalies. In the majority of cases, abnormalities exclusively consist of ipsilateral hypoplasia of the breast, areola and/or nipple in addition to the naevus. Here, we report on a 42-year-old woman with an extensive Becker naevus reaching from the left buttock to the left calf verified on histological examination. In addition, there was marked hypoplasia of the fatty tissue of the left thigh confirmed by magnetic resonance imaging in contrast to hyperplasia of the fatty tissue of the left gluteal area. Underlying muscles and bones were not affected. There was no difference in leg lengths. In addition, we review and discuss the features of Becker naevus syndrome with emphasis on 10 reported cases with involvement of the lower body