A compositional coalgebraic model of a fragment of fusion calculus

This work is a further step in exploring the labelled transitions and bisimulations of fusion calculi. We follow the approach developed by Turi and Plotkin for lifting transition systems with a syntactic structure to bialgebras and, thus, we provide a compositional model of the fusion calculus with explicit fusions. In such a model, the bisimilarity relation induced by the unique morphism to the final coalgebra coincides with fusion hyperequivalence and it is a congruence with respect to the operations of the calculus. The key novelty in our work is to give an account of explicit fusions through labelled transitions. In this short essay, we focus on a fragment of the fusion calculus without recursion and replication

A compositional coalgebraic model of a fragment of fusion calculus

This work is a further step in exploring the labelled transitions and bisimulations of fusion calculi. We follow the approach developed by Turi and Plotkin for lifting transition systems with a syntactic structure to bialgebras and, thus, we provide a compositional model of the fusion calculus with explicit fusions. In such a model, the bisimilarity relation induced by the unique morphism to the final coalgebra coincides with fusion hyperequivalence and it is a congruence with respect to the operations of the calculus. The key novelty in our work is to give an account of explicit fusions through labelled transitions. In this short essay, we focus on a fragment of the fusion calculus without recursion and replication

Abstract Semantics by Observable Contexts

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

04241 Abstracts Collection -- Graph Transformations and Process Algebras for Modeling Distributed and Mobile Systems

Recently there has been a lot of research, combining concepts of process algebra with those of the theory of graph grammars and graph transformation systems. Both can be viewed as general frameworks in which one can specify and reason about concurrent and distributed systems. There are many areas where both theories overlap and this reaches much further than just using graphs to give a graphic representation to processes. Processes in a communication network can be seen in two different ways: as terms in an algebraic theory, emphasizing their behaviour and their interaction with the environment, and as nodes (or edges) in a graph, emphasizing their topology and their connectedness. Especially topology, mobility and dynamic reconfigurations at runtime can be modelled in a very intuitive way using graph transformation. On the other hand the definition and proof of behavioural equivalences is often easier in the process algebra setting. Also standard techniques of algebraic semantics for universal constructions, refinement and compositionality can take better advantage of the process algebra representation. An important example where the combined theory is more convenient than both alternatives is for defining the concurrent (noninterleaving), abstract semantics of distributed systems. Here graph transformations lack abstraction and process algebras lack expressiveness. Another important example is the work on bigraphical reactive systems with the aim of deriving a labelled transitions system from an unlabelled reactive system such that the resulting bisimilarity is a congruence. Here, graphs seem to be a convenient framework, in which this theory can be stated and developed. So, although it is the central aim of both frameworks to model and reason about concurrent systems, the semantics of processes can have a very different flavour in these theories. Research in this area aims at combining the advantages of both frameworks and translating concepts of one theory into the other. The Dagsuthl Seminar, which took place from 06.06. to 11.06.2004, was aimed at bringing together researchers of the two communities in order to share their ideas and develop new concepts. These proceedings4 of the do not only contain abstracts of the talks given at the seminar, but also summaries of topics of central interest. We would like to thank all participants of the seminar for coming and sharing their ideas and everybody who has contributed to the proceedings

A Network-Aware Process Calculus for Global Computing and its Categorical Framework

An essential aspect of distributed systems is resource management, concerning how resources can be accessed and allocated. This aspect should also be taken into account when modeling and verifying such systems. A class of formalisms with the desired features are nominal calculi: they represent resources as atomic objects called names and have linguistic constructs to express creation of new resources. The paradigmatic nominal calculus is the π-calculus, which is well-studied and comes with models and logics. The first objective of this thesis is devising a natural and seamless extension of the π-calculus where resources are network nodes and links. The motivation is provided by a recent, successful networking paradigm called Software Defined Networks, which allows the network structure to be manipulated at runtime via software. We devise a new calculus called Network Conscious π-calculus (NCPi), where resources, namely nodes and links, are represented as names, following the π-calculus guidelines. This allows NCPi to reuse the π-calculus name-handling machinery. The semantics allows observing end-to-end routing behavior, in the form of routing paths through the network. As in the π-calculus, bisimilarity is not closed under input prefix. Interestingly, closure under parallel composition does not hold either. Taking the greatest bisimulation closed under all renamings solves the issue only for the input prefix. We conjecture that such closure yields a full congruence for the subcalculus with only guarded sums. We introduce an extension of NCPi (κNCPi) with some features that makes it closer to real-life routing. Most importantly, we add concurrency, i.e. multiple paths can be observed at the same time. Unlike the sequential version, bisimilarity is a congruence from the very beginning, due to the richer observations, so κNCPi can be considered the “right” version of NCPi when compositionality is needed. This extended calculus is used to model the peer- to-peer architecture Pastry. The second objective is constructing a convenient operational model for NCPi. We consider coalgebras, that are categorical representation of system. Coalgebras have been studied in full generality, regardless of the specific structure of systems, and algorithms and logics have been developed for them. This allows for the application of general results and techniques to a variety of systems. The main difficulty in the coalgebraic treatment of nominal calculi is the presence of name binding: it introduces α-conversion and makes SOS rules and bisimulations non-standard. The consequence is that coalgebras on sets are not able to capture these notions. The idea of the seminal paper by Fiore and Turi is resorting to coalgebras on presheaves, i.e. functors C → Set. Intuitively, presheaves allow associating to collections of names, seen as objects of C, the set of processes using those names. Fresh names generation strategies can be formalized as endofunctors on C, which are lifted to presheaves in a standard way and used to model name binding. Within this framework, a coalgebra for the π-calculus transition system is constructed: the benefit is that ordinary coalgebraic bisimulations for such coalgebra are π-calculus bisimulations. Moreover, Fiore and Turi show a technique to obtain a new coalgebra whose bisimilarity is closed under all renamings. This relation is a congruence for the π-calculus. Presheaves come with a rich theory that can help deriving new results, but coalgebras on presheaves are impractical to implement: the state space can be infinite, for instance when a process recursively creates names. However, if we restrict to a class of presheaves (according to Ciancia et al.), coalgebras admit a concrete implementation in terms of HD-automata, that are finite-state automata suitable for verification. In this thesis we adapt and extend Fiore-Turi’s approach to cope with network resources. First we provide a coalgebraic semantics for NCPi whose bisimulations are bisimulations in the NCPi sense. Then we compute coalgebras and equivalences that are closed under all renamings. The greatest such equivalence is a congruence w.r.t. the input prefix and we conjecture that, for the NCPi with only guarded sums, it is a congruence also w.r.t. parallel composition. We show that this construction applies a form of saturation. Then we prove the existence of a HD-automaton for NCPi. The treatment of network resources is non-trivial and paves the way to modeling other calculi with complex resources

Epistemic Modality and Hyperintensionality in Mathematics

This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states, by contrast to availing of modal axiom 4 (i.e. the KK principle). Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional intensions of epistemic two-dimensional semantics solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between topic-sensitive epistemic two-dimensional truthmaker semantics, the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. Chapter \textbf{9} examines the modal profile of $\Omega$-logic in set theory. Chapter \textbf{10} examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter \textbf{11} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of epistemic two-dimensional semantics in order to account for the interaction of the interpretational and objective modalities thereof. Chapter \textbf{12} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal semantics for the different types of intention and the relation of the latter to evidential decision theory. The multi-hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is applied in chapters \textbf{7}, \textbf{8}, \textbf{10}, \textbf{11}, \textbf{12}, and \textbf{14}.} *Please know that the 5 axiom was meant rather than the B axiom in ch. 10

Epistemic Modality and Hyperintensionality in Mathematics

This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states, by contrast to availing of modal axiom 4 (i.e. the KK principle). Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional intensions of epistemic two-dimensional semantics solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between topic-sensitive epistemic two-dimensional truthmaker semantics, the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. Chapter \textbf{9} examines the modal profile of $\Omega$-logic in set theory. Chapter \textbf{10} examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter \textbf{11} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of epistemic two-dimensional semantics in order to account for the interaction of the interpretational and objective modalities thereof. Chapter \textbf{12} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal semantics for the different types of intention and the relation of the latter to evidential decision theory. The multi-hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is applied in chapters \textbf{7}, \textbf{8}, \textbf{10}, \textbf{11}, \textbf{12}, and \textbf{14}.