Many well-known logical identities are naturally written as equivalences between contextual formulas. A simple example is the Boole-Shannon expansion $c[p] \equiv (p \wedge c[\mathrm{true}] ) \vee (\neg\, p \wedge c[\mathrm{false}] )$, where $c$ denotes an arbitrary formula with possibly multiple occurrences of a "hole", called a context, and $c[\varphi]$ denotes the result of "filling" all holes of $c$ with the formula $\varphi$. Another example is the unfolding rule $\mu X. c[X] \equiv c[\mu X. c[X]]$ of the modal $\mu$-calculus. We consider the modal $\mu$-calculus as overarching temporal logic and, as usual, reduce the problem whether $\varphi_1 \equiv \varphi_2$ holds for contextual formulas $\varphi_1, \varphi_2$ to the problem whether $\varphi_1 \leftrightarrow \varphi_2$ is valid . We show that the problem whether a contextual formula of the $\mu$-calculus is valid for all contexts can be reduced to validity of ordinary formulas. Our first result constructs a canonical context such that a formula is valid for all contexts if{}f it is valid for this particular one. However, the ordinary formula is exponential in the nesting-depth of the context variables. In a second result we solve this problem, thus proving that validity of contextual formulas is EXP-complete, as for ordinary equivalences. We also prove that both results hold for CTL and LTL as well. We conclude the paper with some experimental results. In particular, we use our implementation to automatically prove the correctness of a set of six contextual equivalences of LTL recently introduced by Esparza et al. for the normalization of LTL formulas. While Esparza et al. need several pages of manual proof, our tool only needs milliseconds to do the job and to compute counterexamples for incorrect variants of the equivalences.

Kein CA

Abstract The paper develops an abstract (over-approximating) semantics for double-pushout rewriting of graphs and graph-like objects. The focus is on the so-called materialization of left-hand sides from abstract graphs, a central concept in previous work. The first contribution is an accessible, general explanation of how materializations arise from universal properties and categorical constructions, in particular partial map classifiers, in a topos. Second, we introduce an extension by enriching objects with annotations and give a precise characterization of strongest post-conditions, which are effectively computable under certain assumptions.

Abstract Estimating the cost of a multi-component application (e.g., its resource or energy consumption) is fundamental in nowadays enterprise IT, especially if we consider that current pricing models are mainly pay per-use. While this is still manageable on small applications, it is really hard to manually estimate the cost of large-scale enterprise applications involving hundreds of interdependent application components. In this article, we formalise the problem of estimating costs of multi-component applications, by representing the structure of an application as a typed directed graph, and by allowing to associate different types of costs with different application components. We show that costs can be fully customised, and that associating different costs with the same application leads to different cost estimation problems defined on that application.We then present an approach for solving cost estimation problems on multi-component applications, which is based on terminating and confluent graph transformations. We also present a prototype implemenation of our approach, which we use to run a case study based on a third-party application.

Event structures are a well-accepted model of concurrency. In a seminal paper by Nielsen, Plotkin and Winskel, they are used to establish a bridge between the theory of domains and the approach to concurrency proposed by Petri. A basic role is played by an unfolding construction that maps (safe) Petri nets into a subclass of event structures, called prime event structures, where each event has a uniquely determined set of causes. Prime event structures, in turn, can be identified with their domain of configurations. At a categorical level, this is nicely formalised by Winskel as a chain of coreflections. Contrary to prime event structures, general event structures allow for the presence of disjunctive causes, i.e., events can be enabled by distinct minimal sets of events. In this paper, we extend the connection between Petri nets and event structures in order to include disjunctive causes. In particular, we show that, at the level of nets, disjunctive causes are well accounted for by persistent places. These are places where tokens, once generated, can be used several times without being consumed and where multiple tokens are interpreted collectively, i.e., their histories are inessential. Generalising the work on ordinary nets, Petri nets with persistence are related to a new subclass of general event structures, called locally connected, by means of a chain of coreflections relying on an unfolding construction.

Stable event structures, and their duality with prime algebraic domains (arising as partial orders of configurations), are a landmark of concurrency theory, providing a clear characterisation of causality in computations. They have been used for defining a concurrent semantics of several formalisms, from Petri nets to linear graph rewriting systems, which in turn lay at the basis of many visual frameworks. Stability however is restrictive for dealing with formalisms where a computational step can merge parts of the state, like graph rewriting systems with non-linear rules, which are needed to cover some relevant applications (such as the graphical encoding of calculi with name passing). We characterise, as a natural generalisation of prime algebraic domains, a class of domains that is well-suited to model the semantics of formalisms with fusions. We then identify a corresponding class of event structures, that we call connected event structures, via a duality result formalised as an equivalence of categories.We show that connected event structures are exactly the class of event structures that arise as the semantics of nonlinear graph rewriting systems. Interestingly, the category of general unstable event structures coreflects into our category of domains, so that our result provides a characterisation of the partial orders of configurations of such event structures.

$\mathcal{M}$ -adhesive categories provide an abstract framework for a large variety of specification frameworks for modelling distributed and concurrent systems. They extend the well-known frameworks of adhesive and weak adhesive HLR categories and integrate high-level constructs such as attribution as in the case of typed attributed graphs. In the current paper, we investigate $\mathcal{M}$ -adhesive transformation systems including negative application conditions (NACs) for transformation rules, which are often used in applications. For such systems, we propose an original equivalence on transformation sequences, calledpermutation equivalence, that is coarser than the classical switch equivalence. We also present a general construction of deterministic processes for $\mathcal{M}$ -adhesive transformation systems based on subobject transformation systems. As a main result, we show that the process obtained from a transformation sequence identifies its equivalence class of permutation-equivalent transformation sequences. Moreover, we show how the analysis of this process can be reduced to the analysis of the reachability graph of a generated Place/Transition Petri net. This net encodes the dependencies between rule applications of the transformation sequence, including the inhibiting effects of the NACs.

We generalise both the notion of a non-sequential process and the unfolding construction (which was previously developed for concrete formalisms such as Petri nets and graph grammars) to the abstract setting of (single pushout) rewriting of objects in adhesive categories. The main results show that processes are in one-to-one correspondence with switch-equivalent classes of derivations, and that the unfolding construction can be characterised as a coreflection, that is, the unfolding functor arises as the right adjoint to the embedding of the category of occurrence grammars into the category of grammars. As the unfolding represents potentially infinite computations, we need to work in adhesive categories with ‘well-behaved’ colimits of ω-chains of monos. Compared with previous work on the unfolding of Petri nets and graph grammars, our results apply to a wider class of systems, which is due to the use of a refined notion of grammar morphism.

We propose a framework for the specification of behaviour-preserving reconfigurations of systems modelled as Petri nets. The framework is based on open nets, a mild generalisation of ordinary Place/Transition nets suited to model open systems which might interact with the surrounding environment and endowed with a colimit-based composition operation. We show that natural notions of bisimilarity over open nets are congruences with respect to the composition operation. The considered behavioural equivalences differ for the choice of the observations, which can be single firings or parallel steps. Additionally, we consider weak forms of such equivalences, arising in the presence of unobservable actions. We also provide an up-to technique for facilitating bisimilarity proofs. The theory is used to identify suitable classes of reconfiguration rules (in the double-pushout approach to rewriting) whose application preserves the observational semantics of the net.

In order to model the behaviour of open concurrent systems by means of Petri nets, we introduce open Petri nets, a generalisation of the ordinary model where some places, designated as open, represent an interface between the system and the environment. Besides generalising the token game to reflect this extension, we define a truly concurrent semantics for open nets by extending the Goltz–Reisig process semantics of Petri nets. We introduce a composition operation over open nets, characterised as a pushout in the corresponding category, suitable for modelling both interaction through open places and synchronisation of transitions. The deterministic process semantics is shown to be compositional with respect to such a composition operation. If a net . Technically, our result is similar to the amalgamation theorem for data-types in the framework of algebraic specification. A possible application field of the proposed constructions and results is the modelling of interorganisational workflows, recently studied in the literature. This is illustrated by a running example.

We first give a new definition of graph grammars, which, although following the algebraic double-pushout approach, is more general than the classical one because of the use of a graph of types where all involved graphs are mapped to. Then, we develop

Using a categorical semantics that has been developed recently as a basis, we study composition and refinement as horizontal and vertical structuring techniques for typed graph transformation systems. Composition of graph transformation systems with respect to common subsystems is shown to be compatible with the semantics,i.e., the semantics of the composed system is obtained as the composition of the semantics of the component systems. Moreover, the structure of a composed graph transformation system is preserved during a refinement step in the sense that compatible refinements of the components induce a refinement of the composition. The concepts and results are illustrated by a sample development of a small information system using entity relationship modelling techniques.

Abstract There is an increasing interest in the study of software architectures; however, it still unclear which kind of formalisms and techniques should be used in their design. We study the suitability of a rule‐based, parallel logic language in the specification of the architecture of a complex software system, i.e. a software development environment. We have used as a case study SMILE, an environment for programming‐in‐the‐large. Because of the declarative, concurrent and object‐oriented features of parallel logic programming, we have been able to design a software architecture that emphasizes the dynamics of co‐ordination inside the software development environment. The result of this experience shows the usefulness and some weaknesses of logic languages for specifying and prototyping the software architecture of a distributed interactive system.