Anna Labella

Equational Properties of Iteration in Algebraically Complete Categories

The main result is the following completeness theorem: If the fixed point operation over a category is defined by initiality, then the equations satisfied by the fixed point operation are exactly those of iteration theories. Thus, in such categories, the equational axioms of iteration theories provide a sound and complete axiomatization of the equational properties of the fixed point operation.

Rewriting P systems with conditional communication

A membrane system (P system) is a model of computation inspired by some basic features of the structure and behaviour of living cells. In this paper we consider systems with string-objects processed by rewriting, with the communication controlled by conditions on the contents of the strings. Symbols, snbstrings (in an arbitrary place, or as a prefix/suffix), or the shape of the whole string are used as permitting and as forbidding conditions when moving strings from a membrane to a neighboring membrane. Many of the obtained variants lead to new characterizations of recursively enumerable languages (as expected, these characterizations indicate a trade-off between the number of membranes and the strength of the communication conditions used). Several open problems are also formulated.

Categories enriched on two sides

Abstract We introduce morphisms V → W of bicategories, more general than the original ones of Benabou. When V = 1 , such a morphism is a category enriched in the bicategory W . Therefore, these morphisms can be regarded as categories enriched in bicategories “on two sides”. There is a composition of such enriched categories, leading to a tricategory Caten of a simple kind whose objects are bicategories. It follows that a morphism from V to W in Caten induces a 2-functor V - Cat → W - Cat , while an adjunction between V and W in Caten induces one between the 2-categories V - Cat and W - Cat . Left adjoints in Caten are necessarily homomorphisms in the sense of Benabou, while right adjoints are not. Convolution appears as the internal hom for a monoidal structure on Caten . The 2-cells of Caten are functors; modules can also be defined, and we examine the structures associated with them.

A Completeness Theorem fro Nondeterministic Kleene Algebras

A generalization of Kleene Algebras (structures with +·*, 0 and 1 operators) is considered to take into account possible nondeterminism expressed by the + operator. It is shown that essentially the same complete axiomatization of Salomaa is obtained except for the elimination of the distribution P·(Q + R) = P·Q + P·R and the idempotence law P + P = P. The main result is that an algebra obtained from a suitable category of labelled trees plays the same role as the algebra of regular events. The algebraic semantics and the axiomatization are then extended by adding Ω and ∥ operator, and the whole set of laws is used as a touchstone for starting a discussion over the laws for deadlock, termination and divergence proposed for models of concurrent systems.

MADCOW: a visual interface for annotating web pages

The use of the Web and the diffusion of knowledge management systems makes it possible to base discussions upon a vast set of documents, many of which also include links to multimedia material, such as images or videos. This perspective could be exploited by allowing a team to collaborate by exchanging and retrieving annotated multimedia documents (text, images, audio and video). We designed and developed a digital annotation system, MADCOW, to assist users in constructing, disseminating, and retrieving multimedia annotations of documents, supporting collaborative activities to build a web of decision-related documents. We made a strong effort in designing the user interface and we tested it with 24 users. We describe a scenario in which annotation plays a crucial role, where the object of the collaboration is a politically and artistically important palace of Rome, for which the availability of images and historical documentation is fundamental in order to take informed decisions. We demonstrate the MADCOW interface and its use in the restoration team. The annotations can be used to support teamwork as well as to offer the public some reasoned integration and guide to the available material.

Models of Nondeterministic Regular Expressions

Nondeterminism is a direct outcome of interactions and is, therefore a central ingredient for modelling concurrent systems. Trees are very useful for modelling nondeterministic behaviour. We aim at a tree-based interpretation of regular expressions and study the effect of removing the idempotence law X+X=X and the distribution law X?(Y+Z)=X?Y+X?Z from Kleene algebras. We show that the free model of the new set of axioms is a class of trees labelled over A. We also equip regular expressions with a two-level behavioural semantics. The basic level is described in terms of a class of labelled transition systems that are detailed enough to describe the number of equal actions a system can perform from a given state. The abstract level is based on a so-called resource bisimulation preorder that permits ignoring uninteresting details of transition systems. The three proposed interpretations of regular expressions (algebraic, denotational, and behavioural) are proven to coincide. When dealing with infinite behaviours, we rely on a simple version of the ?-induction and obtain a complete proof system also for the full language of nondeterministic regular expressions.

Enriched categorial semantics for distributed calculi

Abstract Algebraic models for distributed computations, synchronization and concurrency describe computations distributed among agents performing ‘local’ tasks and their interactions as well, which allow sharing and exchanging information for the solution of a common problem. Many calculi have been introduced for this purpose, like for instance Milners CCS and SCCS. In our approach, we describe composition of computations as a tensor product in a suitable monoidal category, generalizing the free monoid generated by the alphabet A of elementary actions. The interactive aspect, namely the synchronization operation, is given by a monoidal functor ∗, which by its very definition exhibits the interchange law between the two operations. Several examples of this kind of monoidal category are introduced. The typical one, treated in full detail, is the topos BT of budding trees , constructed out of the topos of labelled A-trees T / F ( A ) and the (base) monoidal closed category of nice trees NT , which provides the semantics of computations , and several process semantics as categories enriched over it. Good semantics are characterized in terms of a notion of hereditary fullness and computations are synchronized considering the ‘effect’ of the monoidal functor ∗. It is then possible to describe operators such as relabelling , restriction and insertion-deletion of an idle move , showing that our set-up covers the classical semantics for synchronization calculi, like for instance Milners CCS and Bergstra and Klops synchronization algebras. Further, we introduce a distinction between those structures (and properties) which make sense at the level of computations and those which occur at the level of process semantics. Although the stress is generally on the enriched-categorical structure, nice topos-theoretical aspects arise, connected with Benabous work on the topos of trees.

CoPuppet : Collaborative Interaction in Virtual Puppetry

Puppetry is one of the most ancient forms of representation, diffused all over the world in different shapes, degrees of freedom in movements and forms of manipulation. Puppets make an ideal environment for creative collaboration, inspiring the development of supporting technologies (from carpentry to digital worlds). The CoPuppet project explores the possibilities offered by multimodal and cooperative interaction, in which performers, or even audience members, are called to affect different parts of a puppet through gestures and voice. In particular, we exploit an existing architecture for the creation of multimodal interfaces to develop the CoPuppet framework for designing, deploying and interacting during performances in which virtual puppets are steered by multiple multimodal controls. The paper illustrates the steps needed to define performances, also showing the integration of digital and real puppetry for the case of wayang shadowplay.

Observational trees as models for concurrency

Given an automaton, its behaviour can be modelled as the sets of strings over an alphabet A that can be accepted from any of its states. When considering concurrent systems, we can see a concurrent agent as an automaton, where non-determinism derives from the fact that its states can offer a different behaviour at different moments in time. Non-deterministic computations between a pair of states can then no longer be described as a ‘set’ of strings in a free monoid. Consequently, between two states we will have a labelled structured set of computations, where the structure describes the possibility of two computations parting from each other while maintaining the same observable steps. In this paper, we shall consider different kinds of observation domains and related structured sets of computations. Structured sets of computations will be organised as a category of generalised trees built over a meet-semilattice monoid formalizing the observation domain. Theorems allowing us to introduce the usual concurrency operators in the models and relating different models will then be obtained by first considering ordinary functors (on and between the observation domains), and then lifting them to the categories of structured sets of computations.

Graded Modalities and Resource Bisimulation

The logical characterization of the strong and the weak (ignoring silent actions) versions of resource bisimulation are studied. The temporal logics we introduce are variants of Hennessy-Milner Logics that use graded modalities instead of the classical box and diamond operators. The considered strong bisimulation induces an equivalence that, when applied to labelled transition systems, permits identifying all and only those systems that give rise to isomorphic unfoldings. Strong resource bisimulation has been used to provide nondeterministic interpretation of finite regular expressions and new axiomatizations for them. Here we generalize this result to its weak variant.