IMT Institutional Repository: No conditions. Results ordered -Date Deposited. 2023-09-29T21:06:11ZEPrintshttp://eprints.imtlucca.it/images/logowhite.pnghttp://eprints.imtlucca.it/2018-03-05T16:26:29Z2018-03-05T16:26:29Zhttp://eprints.imtlucca.it/id/eprint/3952This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/39522018-03-05T16:26:29ZInitial Algebra for a System of Right-Linear FunctorsIn 2003 we showed that right-linear systems of equations over regular expressions, when interpreted in a category of trees, have a solution whenever they enjoy a specific property that we called hierarchicity and that is instrumental to avoid critical mutual recursive definitions. In this note, we prove that a right-linear system of polynomial endofunctors on a cocartesian monoidal closed category which enjoys parameterized left list arithmeticity, has an initial algebra, provided it satisfies a property similar to hierarchicity.Anna LabellaRocco De Nicolar.denicola@imtlucca.it2011-06-13T13:10:36Z2011-07-11T14:36:27Zhttp://eprints.imtlucca.it/id/eprint/369This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/3692011-06-13T13:10:36ZA Completeness Theorem for Nondeterministic Kleene AlgebrasA 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 OHgr and par 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.Rocco De Nicolar.denicola@imtlucca.itAnna Labella2011-06-13T12:56:03Z2011-07-11T14:36:27Zhttp://eprints.imtlucca.it/id/eprint/363This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/3632011-06-13T12:56:03ZFully Abstract Models for Nondeterministic Regular ExpressionsRegular expressions and Kleene Algebras have been a direct inspiration for many constructs and axiomatizations for concurrency models. These, however, put a different stress on nondeterminism. With concurrent interpretations in mind, we study the effect of removing the idempotence law X+X=X and distribution law X·(Y+Z)=X·Y +X·Z from Kleene Algebras. We propose an operational semantics that is sound and complete w.r.t. the new set of axioms and is fully abstract w.r.t. a denotational semantic based on trees. The operational semantics is based on labelled transition systems that keep track of the performed choices and on a preorder relation (we call it resource simulation) that takes also into account the number of states reachable via every action.An important property we exhibit is that resource bisimulation equivalence can be obtained as the kernel of resource simulation.Flavio CorradiniRocco De Nicolar.denicola@imtlucca.itAnna Labella2011-06-13T12:48:41Z2011-07-11T14:36:26Zhttp://eprints.imtlucca.it/id/eprint/346This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/3462011-06-13T12:48:41ZModels of Nondeterministic Regular ExpressionsNondeterminism 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 ω-induction and obtain a complete proof system also for the full language of nondeterministic regular expressions.Flavio CorradiniRocco De Nicolar.denicola@imtlucca.itAnna Labella2011-06-13T12:42:11Z2014-10-07T14:38:28Zhttp://eprints.imtlucca.it/id/eprint/350This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/3502011-06-13T12:42:11ZTree morphisms and bisimulationsA category of (action labelled) trees is defined that can be used to model unfolding of labelled transition systems and to study behavioural relations over them. In this paper we study five different equivalences based on bisimulation for our model. One, that we called resource bisimulation, amounts essentially to three isomorphism. Another, its weak counterpart, permits abstracting from silent actions while preserving the tree structure. The other three are the well known strong, branching and weak bisimulation equivalence. For all bisimulations, but weak, canonical representatives are constructed and it is shown that they can be obtained via enriched functors over our categories of trees, with and without silent actions. Weak equivalence is more problematic; a canonical minimal representative for it cannot be denned by quotienting our trees. The common framework helps in understanding the relationships between the various equivalences and the results provide support to the claim that branching bisimulation is the natural generalization of strong bisimulation to systems with silent moves and that resource and weak resource have an interest of their own.Rocco De Nicolar.denicola@imtlucca.itAnna Labella2011-06-09T09:45:20Z2011-07-11T14:36:26Zhttp://eprints.imtlucca.it/id/eprint/344This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/3442011-06-09T09:45:20ZA finite axiomatization of nondeterministic regular expressionsAn alternative (tree-based) semantics for a class of regular expressions is proposed that assigns a central rôle to the + operator and thus to nondeterminism and nondeterministic choice. For the new semantics a consistent and complete axiomatization is obtained from the original axiomatization of regular expressions by Salomaa and by Kozen by dropping the idempotence law for + and the distribution law of • over +. Flavio CorradiniRocco De Nicolar.denicola@imtlucca.itAnna Labella2011-06-08T13:28:05Z2011-07-11T14:36:26Zhttp://eprints.imtlucca.it/id/eprint/319This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/3192011-06-08T13:28:05ZNondeterministic regular expressions as solutions of equational systemsWe define the class of the linear systems whose solution is expressible as a tuple of nondeterministic regular expressions when they are interpreted as trees of actions rather than as sets of sequences. We precisely characterize those systems that have a regular expression as "canonical" solution, and show that any regular expression can be obtained as a canonical solution of a system of the defined class.Rocco De Nicolar.denicola@imtlucca.itAnna Labella2011-06-06T13:38:51Z2011-07-11T14:36:26Zhttp://eprints.imtlucca.it/id/eprint/340This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/3402011-06-06T13:38:51ZGraded Modalities and Resource BisimulationThe 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. Flavio CorradiniRocco De Nicolar.denicola@imtlucca.itAnna Labella2011-06-06T09:41:26Z2011-07-11T14:36:26Zhttp://eprints.imtlucca.it/id/eprint/325This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/3252011-06-06T09:41:26ZAn Equational Axiomatization of Bisimulation over Regular ExpressionsWe provide a finite equational axiomatization for bisimulation equivalence of nondeterministic interpretation of regular expressions. Our axiomatization is heavily based on the one by Salomaa, that provided an implicative axiomatization for a large subset of regular expressions, namely all those that satisfy the non‐empty word property (i.e. without 1 summands at the top level) in *‐contexts. Our restriction is similar, it essentially amounts to recursively requiring that the non‐empty word property be satisfied not just at top level but at any depth. We also discuss the impact on the axiomatization of different interpretations of the 0 term, interpreted either as a null process or as a deadlock. Flavio CorradiniRocco De Nicolar.denicola@imtlucca.itAnna Labella2011-05-23T15:10:18Z2012-07-06T10:03:20Zhttp://eprints.imtlucca.it/id/eprint/272This item is in the repository with the URL: http://eprints.imtlucca.it/id/eprint/2722011-05-23T15:10:18ZTree-functors, determinacy and bisimulationsWe study the functorial characterisation of bisimulation-based equivalences over a
categorical model of labelled trees. We show that in a setting where all labels are visible,
strong bisimilarity can be characterised in terms of enriched functors by relying on the
reflection of paths with their factorisations. For an enriched functor F, this notion requires
that a path (an internal morphism in our framework) π going from F(A) to C corresponds
to a path p going from A to K, with F(K) = C, such that every possible factorisation of π
can be lifted in an appropriate factorisation of p. This last property corresponds to a
Conduch´e property for enriched functors, and a very rigid formulation of it has been used by
Lawvere to characterise the determinacy of physical systems. We also consider the setting
where some labels are not visible, and provide characterisations for weak and branching
bisimilarity. Both equivalences are still characterised in terms of enriched functors that
reflect paths with their factorisations: for branching bisimilarity, the property is the same as
the one used to characterise strong bisimilarity when all labels are visible; for weak
bisimilarity, a weaker form of path factorisation lifting is needed. This fact can be seen as
evidence that strong and branching bisimilarity are strictly related and that, unlike weak
bisimilarity, they preserve process determinacy in the sense of Milner.Rocco De Nicolar.denicola@imtlucca.itDaniele GorlaAnna Labella