11 research outputs found
Coalgebraic Behavior Analysis â 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 â possibly quite different â 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 fĂŒr jeden Typ von System zu definieren, ist es erstrebenswert einen standardisierten Ansatz zu haben, welcher es ermöglicht, dies in einer generischen Weise fĂŒr diverse â möglicherweise stark unterschiedliche â 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 beschÀftigt sich mit der Frage, wie existierende Resultate verbessert und erweitert werden können, um die AusdrucksmÀchtigkeit 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 ĂŒbliche 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 RĂ€umen und nicht-expansiven Funktionen zu erweitern, ist es möglich eine kanonische, koalgebraische Herangehensweise fĂŒr Verhaltensmetriken zu entwerfen
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 for a functor , we define a framework for deriving pseudometrics on which
measure the behavioral distance of states.
A crucial step is the lifting of the functor on to a
functor on the category 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 has a final coalgebra, every lifting 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
Contraherent cosheaves
Contraherent cosheaves are globalizations of cotorsion (or similar) modules
over commutative rings obtained by gluing together over a scheme. The category
of contraherent cosheaves over a scheme is a Quillen exact category with exact
functors of infinite product. Over a quasi-compact semi-separated scheme or a
Noetherian scheme of finite Krull dimension (in a different version - over any
locally Noetherian scheme), it also has enough projectives. We construct the
derived co-contra correspondence, meaning an equivalence between appropriate
derived categories of quasi-coherent sheaves and contraherent cosheaves, over a
quasi-compact semi-separated scheme and, in a different form, over a Noetherian
scheme with a dualizing complex. The former point of view allows us to obtain
an explicit construction of Neeman's extraordinary inverse image functor
for a morphism of quasi-compact semi-separated schemes . The latter approach
provides an expanded version of the covariant Serre-Grothendieck duality theory
and leads to Deligne's extraordinary inverse image functor (which we
denote by ) for a morphism of finite type between Noetherian schemes.
Semi-separated Noetherian stacks, affine Noetherian formal schemes, and
ind-affine ind-schemes (together with the noncommutative analogues) are briefly
discussed in the appendices.Comment: LaTeX 2e with pb-diagram, xy-pic, and mathx fonts; 257 pages, 1
commutative diagram. v.5: a paragraph inserted in the introduction,
improvements and additions in sections C.2-C.3 and D.1, section D.2
rewritten, new section D.3 inserted, several references added; v.6: details
of arguments added in sections A.2 and A.5, several misprints correcte
Lifting of operations in modular monadic semantics
Monads have become a fundamental tool for structuring denotational semantics and programs by abstracting a wide variety of computational features such as side-effects, input/output, exceptions, continuations and non-determinism. In this setting, the notion of a monad is equipped with operations that allow programmers to manipulate these computational effects. For example, a monad for side-effects is equipped with operations for setting and reading the state, and a monad for exceptions is equipped with operations for throwing and handling exceptions.
When several effects are involved, one can employ the incremental approach to mod- ular monadic semantics, which uses monad transformers to build up the desired monad one effect at a time. However, a limitation of this approach is that the effect-manipulating operations need to be manually lifted to the resulting monad, and consequently, the lifted operations are non-uniform. Moreover, the number of liftings needed in a system grows as the product of the number of monad transformers and operations involved.
This dissertation proposes a theory of uniform lifting of operations that extends the incremental approach to modular monadic semantics with a principled technique for lifting operations. Moreover the theory is generalized from monads to monoids in a monoidal category, making it possible to apply it to structures other than monads.
The extended theory is taken to practice with the implementation of a new extensible monad transformer library in Haskell, and with the use of modular monadic semantics to obtain modular operational semantics