• bisimilarity
• coalgebras
• graph rewriting
• Logic Programming
• process calculi
• reactive systems

The operational behavior of interactive systems is usually given in terms of transition systems
labeled with actions, which, when visible, represent both observations and interactions with the external world. The abstract semantics is given in terms of behavioral equivalences, which depend on the action labels and on the amount of branching structure considered. Behavioural equivalences are often congruences with respect to the operations of the language, and this property expresses the compositionality of the abstract semantics.
A simpler approach, inspired by classical formalisms like pi-calculus, Petri nets, term and graph rewriting, and pioneered by the Chemical Abstract Machine [13], defines operational semantics by means of structural axioms and reaction rules. Process calculi representing complex systems, in particular those able to generate and communicate names, are often defined in this way, since structural axioms give a clear idea of the intended structure of the states while reaction rules, which are often non-conditional, give a direct account of the possible steps. Transitions caused by reaction rules, however, are not labeled, since they represent evolutions of the system without interactions with the external world. Thus reduction semantics in itself is neither abstract nor compositional.
One standard solution, pioneered in [89], is that of defining a saturated transition system as
follows:
a process p can do a move with label C[-] and become q, iff C[p]--> q.
Saturated semantics, i.e., the abstract semantics defined over the saturated transition system, are
always congruences, but they are usually untractable since they have to tackle all possible contexts of which there are usually an infinite number. Moreover, in several paradigmatic cases, saturated semantics are too coarse. For example, in Milner's Calculus of Communicating Systems (CCS), saturated bisimilarity cannot distinguish "always divergent processes" and for this reason
Milner and Sangiorgi introduced barbs. These are observations on the states representing the ability to interact over some channels.
Sewell introduced a different approach that consists in deriving a transition system
where labels are not all contexts but just the minimal ones allowing a system to reach a rule. In
such a way, one obtains two advantages: firstly one avoids considering all contexts, and secondly,
labels precisely represent interactions, i.e., the portion of environment that is really needed to
react. This idea was then refined by Leifer and Milner in the theory of reactive systems, where
the categorical notion of idem relative pushout precisely captures this idea of minimal context.
In this thesis, we show that in some cases this approach works well (e.g., CCS) but often, the
resulting abstract semantics are too strict. In our opinion, they are not really observational since the observer can know exactly how much structure a process needs to reach a specific rule, and thus the observation depends on the rules. One result of the thesis is that
of providing evidence of this through several interesting formalisms modeled as reactive systems: Logic Programming, a fragment of open pi-calculus, and an interactive version of Petri nets.
Moreover, we introduce two alternative definitions of bisimilarity that efficiently characterize
saturated bisimilarity, namely semi-saturated bisimilarity and symbolic bisimilarity. These
allow us to reason about saturated semantics without considering all contexts, but saturated semantics are in several cases too coarse. In order to have a framework that is suitable for many formalisms, we add to the above approach observations. Indeed, in our opinion, labels cannot represent both interactions and observations, because these two concepts are in general different, like for example, in the asynchronous calculi where receiving is not observable. Thus, we believe that some notion of observation, either on transitions or on states (e.g. barbs), is necessary.

A further result of the thesis is that of providing a generalization of the above theory starting not just from purely reaction rules, but from transition systems labeled with observations. Here we can easily reuse saturated transition systems by defining them as follows:
a process p can do a move with context C[-] and observation o and become q iff C[p] --o--> q.
Again, saturated semantics, i.e. abstract semantics defined over the above transition systems, are congruences. Analogously to the case of reactive systems, we can define semi-saturated bisimilarity and symbolic bisimilarity as efficient characterizations of saturated semantics. The definition of symbolic bisimilarity which arises from this generalization is similar to the abstract semantics of several works. Here we consider open and asynchronous pi-calculus, by showing that their abstract semantics are instances of our general concepts of saturated and symbolic semantics. We also apply our approach to open Petri nets (that are an interactive version of P/T Petri nets) obtaining a new symbolic semantics for them, that efficiently characterizes their abstract semantics.

We round up the thesis with a coalgebraic characterization for saturated, semi-saturated and
symbolic bisimilarity.
Universal Coalgebra provides a categorical framework where abstract semantics of interactive systems are described as morphisms to their minimal representatives. More precisely, if the
category of coalgebras has final object 1, then the unique morphisms from a certain coalgebra to 1
equates all the bisimilar states. In other words, the final object can be seen as a universe of abstract behaviors and the unique morphism as a function assigning to each system its abstract behavior.
This characterization of abstract semantics is not only theoretically interesting, but also pragmat-
ically useful, since it suggests an algorithm which can check the equivalence: one computes the
image of some coalgebras through the unique morphism (that for the finite lts corresponds to the list partitioning algorithm by Kanellakis and Smolka), and these coalgebras are behaviorally
equivalent if their images are the same.
Ordinary labeled transition systems can be represented as coalgebras, and the resulting abstract semantics exactly coincides with canonical bisimilarity. Then, providing a coalgebraic characterization of saturated bisimilarity is almost straightforward. The case of semi-saturated and symbolic bisimilarities are more complicated because their definitions are asymmetric. In order to properly characterizes semi-saturated and symbolic cases, we first introduce a new notion of redundancy on transitions and then normalized coalgebras: a special kind of coalgebras without redundant
transitions. We prove that the category of normalized coalgebras is isomorphic to the category of saturated coalgebras (the coalgebras containing all the redundant transitions), where the large saturated transition system can be directly modelled. In doing this, we use the notions of normalization that throws away all the redundant transitions, and of saturation that adds all the redundant transitions. Both are natural transformations between the endofunctors (defining the two categories of coalgebras) and one is the inverse of the other. As a corollary of the isomorphism theorem, saturated bisimilarity can be characterized as bisimilarity in the category of normalized coalgebras, i.e., abstracting away from redundant transitions. This is interesting because, on the one hand, it provides us with a canonical representatives for ~S without redundant transitions
(and then much smaller with respect to the saturated ones), on the other hand, it suggests a
minimization algorithm for "efficiently" computing ~S.