Stefan Milius

Generalized Eilenberg Theorem I: Local Varieties of Languages

We investigate the duality between algebraic and coalgebraic recognition of languages to derive a generalization of the local version of Eilenbergs theorem. This theorem states that the lattice of all boolean algebras of regular languages over an alphabet {Sigma} closed under derivatives is isomorphic to the lattice of all pseudovarieties of {Sigma}-generated monoids. By applying our method to different categories, we obtain three related results: one, due to Gehrke, Grigorieff and Pin, weakens boolean algebras to distributive lattices, one weakens them to join-semilattices, and the last one considers vector spaces over the binary field.

Varieties of Languages in a Category

Eilenbergs variety theorem, a centerpiece of algebraic automata theory, establishes a bijective correspondence between varieties of languages and pseudovarieties of monoids. In the present paper this result is generalized to an abstract pair of algebraic categories: we introduce varieties of languages in a category C, and prove that they correspond to pseudovarieties of monoids in a closed monoidal category D, provided that C and D are dual on the level of finite objects. By suitable choices of these categories our result uniformly covers Eilenbergs theorem and three variants due to Pin, Polák and Reutenauer, respectively, and yields new Eilenberg-type correspondences.

Killing epsilons with a dagger: A coalgebraic study of systems with algebraic label structure

We propose an abstract framework for modelling state-based systems with internal behaviour as e.g. given by silent or ϵ-transitions. Our approach employs monads with a parametrized fixpoint operator † to give a semantics to those systems and implement a sound procedure of abstraction of the internal transitions, whose labels are seen as the unit of a free monoid. More broadly, our approach extends the standard coalgebraic framework for state-based systems by taking into account the algebraic structure of the labels of their transitions. This allows to consider a wide range of other examples, including Mazurkiewicz traces for concurrent systems and non-deterministic transducers.

How to Kill Epsilons with a Dagger - A Coalgebraic Take on Systems with Algebraic Label Structure

We propose an abstract framework for modeling state-based systems with internal behavior as e.g. given by silent or ϵ -transitions. Our approach employs monads with a parametrized fixpoint operator † to give a semantics to those systems and implement a sound procedure of abstraction of the internal transitions, whose labels are seen as the unit of a free monoid. More broadly, our approach extends the standard coalgebraic framework for state-based systems by taking into account the algebraic structure of the labels of their transitions. This allows to consider a wide range of other examples, including Mazurkiewicz traces for concurrent systems.

Guard Your Daggers and Traces: Properties of Guarded (Co-)recursion

Motivated by the recent interest in models of guarded (co-)recursion, we study their equational properties. We formulate axioms for guarded fixpoint operators generalizing the axioms of iteration theories of Bloom and Esik. Models of these axioms include both standard (e.g., cpo-based) models of iteration theories and models of guarded recursion such as complete metric spaces or the topos of trees studied by Birkedal et al. We show that the standard result on the satisfaction of all Conway axioms by a unique dagger operation generalizes to the guarded setting. We also introduce the notion of guarded trace operator on a category, and we prove that guarded trace and guarded fixpoint operators are in one-to-one correspondence. Our results are intended as first steps leading, hopefully, towards future description of classifying theories for guarded recursion.

An Attack Possibility on Time Synchronization Protocols Secured with TESLA-Like Mechanisms

In network-based broadcast time synchronization, an important security goal is integrity protection linked with source authentication. One technique frequently used to achieve this goal is to secure the communication by means of the TESLA protocol or one of its variants. This paper presents an attack vector usable for time synchronization protocols that protect their broadcast or multicast messages in this manner. The underlying vulnerability results from interactions between timing and security that occur specifically for such protocols. We propose possible countermeasures and evaluate their respective advantages. Furthermore, we discuss our use of the UPPAAL model checker for security analysis and quantification with regard to the attack and countermeasures described, and report on the results obtained. Lastly, we review the susceptibility of three existing cryptographically protected time synchronization protocols to the attack vector discovered.

On finitary functors and their presentations

Finitary endofunctors of locally presentable categories are proved to have equational presentations. Special attention is being paid to the category of complete metric spaces and two endofunctors: the Hausdorff functor of all compact subsets and the Kantorovich functor of all tight measures.

On Final Coalgebras of Power-Set Functors and Saturated Trees To George Janelidze on the Occasion of His Sixtieth Birthday

The final coalgebra for the finite power-set functor was described by Worrell who also proved that the final chain converges in ω+ω steps. We describe the step ω as the set of saturated trees, a concept equivalent to the modally saturated trees introduced by K. Fine in the 1970s in his study of modal logic. And for the bounded power-set functors Pλ, where λ is an infinite regular cardinal, we prove that the construction needs precisely λ+ω steps. We also generalize Worrell’s result to M-labeled trees for a commutative monoid M, yielding a final coalgebra for the corresponding functor ℳf studied by H.-P. Gumm and T. Schröder. We describe the final chain of the power-set functor by introducing the concept of i-saturated tree for all ordinals i, and then prove that for i of cofinality ω, the i-th step in the final chain consists of all i-saturated, strongly extensional trees.

Canonical Nondeterministic Automata

For each regular language (L) we describe a family of canonical nondeterministic acceptors (nfas). Their construction follows a uniform recipe: build the minimal dfa for (L) in a locally finite variety (mathcal {V}), and apply an equivalence between the finite (mathcal {V})-algebras and a category of finite structured sets and relations. By instantiating this to different varieties we recover three well-studied canonical nfas (the atomaton, the jiromaton and the minimal xor automaton) and obtain a new canonical nfa called the distromaton. We prove that each of these nfas is minimal relative to a suitable measure, and give conditions for state-minimality. Our approach is coalgebraic, exhibiting additional structure and universal properties.

A New Foundation for Finitary Corecursion

This paper contributes to a theory of the behaviour of “finite-state” systems that is generic in the system type. We propose that such systems are modeled as coalgebras with a finitely generated carrier for an endofunctor on a locally finitely presentable category. Their behaviour gives rise to a new fixpoint of the coalgebraic type functor called locally finite fixpoint (LFF). We prove that if the given endofunctor preserves monomorphisms then the LFF always exists and is a subcoalgebra of the final coalgebra (unlike the rational fixpoint previously studied by Adamek, Milius and Velebil). Moreover, we show that the LFF is characterized by two universal properties: 1. as the final locally finitely generated coalgebra, and 2. as the initial fg-iterative algebra. As instances of the LFF we first obtain the known instances of the rational fixpoint, e.g. regular languages, rational streams and formal power-series, regular trees etc. And we obtain a number of new examples, e.g. (realtime deterministic resp. non-deterministic) context-free languages, constructively S-algebraic formal power-series (and any other instance of the generalized powerset construction by Silva, Bonchi, Bonsangue, and Rutten) and the monad of Courcelle’s algebraic trees.