Term Graph Representations for Cyclic Lambda-Terms

We study various representations for cyclic lambda-terms as higher-order or as first-order term graphs. We focus on the relation between `lambda-higher-order term graphs' (lambda-ho-term-graphs), which are first-order term graphs endowed with a well-behaved scope function, and their representations as `lambda-term-graphs', which are plain first-order term graphs with scope-delimiter vertices that meet certain scoping requirements. Specifically we tackle the question: Which class of first-order term graphs admits a faithful embedding of lambda-ho-term-graphs in the sense that: (i) the homomorphism-based sharing-order on lambda-ho-term-graphs is preserved and reflected, and (ii) the image of the embedding corresponds closely to a natural class (of lambda-term-graphs) that is closed under homomorphism? We systematically examine whether a number of classes of lambda-term-graphs have this property, and we find a particular class of lambda-term-graphs that satisfies this criterion. Term graphs of this class are built from application, abstraction, variable, and scope-delimiter vertices, and have the characteristic feature that the latter two kinds of vertices have back-links to the corresponding abstraction. This result puts a handle on the concept of subterm sharing for higher-order term graphs, both theoretically and algorithmically: We obtain an easily implementable method for obtaining the maximally shared form of lambda-ho-term-graphs. Also, we open up the possibility to pull back properties from first-order term graphs to lambda-ho-term-graphs. In fact we prove this for the property of the sharing-order successors of a given term graph to be a complete lattice with respect to the sharing order. This report extends the paper with the same title (http://arxiv.org/abs/1302.6338v1) in the proceedings of the workshop TERMGRAPH 2013.Comment: 35 pages. report extending proceedings article on arXiv:1302.6338 (changes with respect to version v2: added section 8, modified Proposition 2.4, added Remark 2.5, added Corollary 7.11, modified figures in the conclusion

Behavioural equivalences for timed systems

Timed transition systems are behavioural models that include an explicit treatment of time flow and are used to formalise the semantics of several foundational process calculi and automata. Despite their relevance, a general mathematical characterisation of timed transition systems and their behavioural theory is still missing. We introduce the first uniform framework for timed behavioural models that encompasses known behavioural equivalences such as timed bisimulations, timed language equivalences as well as their weak and time-abstract counterparts. All these notions of equivalences are naturally organised by their discriminating power in a spectrum. We prove that this result does not depend on the type of the systems under scrutiny: it holds for any generalisation of timed transition system. We instantiate our framework to timed transition systems and their quantitative extensions such as timed probabilistic systems

CCS Dynamic Bisimulation is Progressing

Weak Observational Congruence (woc) defined on CCS agents is not a bisimulation since it does not require two states reached by bisimilar computations of woc agents to be still woc, e.g.\ $\alpha.\tau.\beta.nil$ and $\alpha.\beta.nil$ are woc but $\tau.\beta.nil$ and $\beta.nil$ are not. This fact prevents us from characterizing CCS semantics (when $\tau$ is considered invisible) as a final algebra, since the semantic function would induce an equivalence over the agents that is both a congruence and a bisimulation. In the paper we introduce a new behavioural equivalence for CCS agents, which is the coarsest among those bisimulations which are also congruences. We call it Dynamic Observational Congruence because it expresses a natural notion of equivalence for concurrent systems required to simulate each other in the presence of dynamic, i.e.\ run time, (re)configurations. We provide an algebraic characterization of Dynamic Congruence in terms of a universal property of finality. Furthermore we introduce Progressing Bisimulation, which forces processes to simulate each other performing explicit steps. We provide an algebraic characterization of it in terms of finality, two characterizations via modal logic in the style of HML, and a complete axiomatization for finite agents. Finally, we prove that Dynamic Congruence and Progressing Bisimulation coincide for CCS agents. Thus the title of the paper

Dynamic Congruence vs. Progressing Bisimulation for CCS

Weak Observational Congruence (woc) defined on CCS agents is not a bisimulation since it does not require two states reached by bisimilar computations of woc agents to be still woc, e.g. \alpha.\tau.\beta.nil and \alpha.\beta.nil are woc but \tau.\beta.nil and \beta.nil are not. This fact prevent us from characterizing CCS semantics (when \tau is considered invisible) as a final algebra, since the semantic function would induce an equivalence over the agents that is both a congruence and a bisimulation. In the paper we introduce a new behavioural equivalence for CCS agents, which is the coarsest among those bisimulations which are also congruences. We call it Dynamic Observational Congruence because it expresses a natural notion of equivalence for concurrent systems required to simulate each other in the presence of dynamic, i.e. run time, (re)configurations. We provide an algebraic characterization of Dynamic Congruence in terms of a universal property of finality. Furthermore we introduce Progressing Bisimulation, which forces processes to simulate each other performing explicit steps. We provide an algebraic characterization of it in terms of finality, two logical characterizations via modal logic in the style of HML and a complete axiomatization for finite agents (consisting of the axioms for Strong Observational Congruence and of two of the three Milner's $\tau$-laws). Finally, we prove that Dynamic Congruence and Progressing Bisimulation coincide for CCS agents

Towards a Theory of Glue

We propose and study the notions of behaviour type and composition operator making a first step towards the definition of a formal framework for studying behaviour composition in a setting sufficiently general to provide insight into how the component-based systems should be modelled and compared. We illustrate the proposed notions on classical examples (Traces, Labelled Transition Systems and Coalgebras). Finally, the definition of memoryless glue operators, takes us one step closer to a formal understanding of the separation of concerns principle stipulating that computational aspects of a system should be localised within its atomic components, whereas coordination layer responsible for managing concurrency should be realised by memoryless glue operators.Comment: In Proceedings ICE 2012, arXiv:1212.345

An algebraic characterization of observational equivalence

AbstractWe show that observational equivalence can be characterized by saturating homomorphisms (with respect to Hennessy-Milner logic), thus bringing together results developed independently by Castellani and by Arnold and Dicky on characterizations of transition system equivalences. We take this opportunity to compare Castellani's abstraction homomorphisms and Arnold-Dicky's saturating homomorphisms. It turns out that they are very similar notions: their difference in formulation is partly due to the fact that abstraction homomorphisms were defined on a restricted class of transition systems

The Power of Convex Algebras

Probabilistic automata (PA) combine probability and nondeterminism. They can be given different semantics, like strong bisimilarity, convex bisimilarity, or (more recently) distribution bisimilarity. The latter is based on the view of PA as transformers of probability distributions, also called belief states, and promotes distributions to first-class citizens. We give a coalgebraic account of the latter semantics, and explain the genesis of the belief-state transformer from a PA. To do so, we make explicit the convex algebraic structure present in PA and identify belief-state transformers as transition systems with state space that carries a convex algebra. As a consequence of our abstract approach, we can give a sound proof technique which we call bisimulation up-to convex hull.Comment: Full (extended) version of a CONCUR 2017 paper, to be submitted to LMC