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We investigate three formalisms to specify graph languages, i.e. sets of graphs, based on type graphs. First, we are interested in (pure) type graphs, where the corresponding language consists of all graphs that can be mapped homomorphically to a given type graph. In this context, we also study languages specified by restriction graphs and their relation to type graphs. Second, we extend this basic approach to a type graph logic and, third, to type graphs with annotations. We present decidability results and closure properties for each of the formalisms., (v2): -Fixed some typos -Added more references

Behavioural equivalences can be characterized via bisimulation, modal logics, and spoiler-duplicator games. In this paper we work in the general setting of coalgebra and focus on generic algorithms for computing the winning strategies of both players in a bisimulation game. The winning strategy of the spoiler (if it exists) is then transformed into a modal formula that distinguishes the given non-bisimilar states. The modalities required for the formula are also synthesized on-the-fly, and we present a recipe for re-coding the formula with different modalities, given by a separating set of predicate liftings. Both the game and the generation of the distinguishing formulas have been implemented in a tool called T-BEG., Comment: Long version of CMCS 2020 paper

We introduce a variant of transition systems, where activation of transitions depends on conditions of the environment and upgrades during runtime potentially create additional transitions. Using a cornerstone result in lattice theory, we show that such transition systems can be modelled in two ways: as conditional transition systems (CTS) with a partial order on conditions, or as lattice transition systems (LaTS), where transitions are labelled with the elements from a distributive lattice. We define equivalent notions of bisimilarity for both variants and characterise them via a bisimulation game.\ud \ud \ud \ud We explain how conditional transition systems are related to featured transition systems for the modelling of software product lines. Furthermore, we show how to compute bisimilarity symbolically via BDDs by defining an operation on BDDs that approximates an element of a Boolean algebra into a lattice. We have implemented our procedure and provide runtime results.\ud \ud \ud \ud This is an extended version of the TASE 2017 paper [1], including all proofs, additional examples, an extension of the formalism to account for deactivation of updates and detailed runtime results.

We present a programming library for the rapid development of graph tools, with applications in graph transformation and related fields. Features include working with graphs, graph morphisms, basic categorical constructions such as computing pushouts and pushout complements or enumerating all morphisms with certain properties, but also applications such as executing graph transformation steps. Additionally, we offer graphical user interface widgets for visualization and manipulation of graphs, morphisms and categorical diagrams. Our objective is to allow users to quickly develop graph tools for both simple and complex problems, to allow easy embedding into existing software, and to have comprehensible code especially for the main algorithms. Existing tools that demonstrate the versatility and ease of use of the library include: DPOdactic (a didactic tool for teaching double-pushout graph transformation), DrAGoM (a tool to handle multiply annotated type graphs for abstract graph rewriting), and Grez (termination analysis of graph transformation systems). © 2020, Springer Nature Switzerland AG.

In this article, we generalize Courcelle's recognizable graph languages and results on monadic second-order logic to more general structures.First, we give a category-theoretical characterization of recognizability. A recognizable subset of arrows in a category is defined via a functor into the category of relations on finite sets. This can be seen as a straightforward generalization of finite automata. We show that our notion corresponds to recognizable graph languages if we apply the theory to the category of cospans of graphs.In the second part of the paper, we introduce a simple logic that allows to quantify over the subobjects of a categorical object. Again, we show that, for the category of graphs, this logic is equally expressive as monadic second-order graph logic (msogl). Furthermore, we show that in the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Courcelle's result that everymsogl-expressible property is recognizable. This is done by giving an inductive translation of formulas of our logic into automaton functors.

Graph transformation systems (GTSs) can be seen as well-structured transition systems (WSTSs) and via well-structuredness it is possible to obtain decidability results for certain classes of GTSs. We present a generic framework, parameterized over the well-quasi-order (wqo), in which several types of GTSs can be seen as (restricted) WSTSs. We instantiate this framework with three orders: the minor ordering, the subgraph ordering and the induced subgraph ordering. Furthermore we consider two case studies where we apply the theory to analyze a leader election protocol and a simple access rights management system with our tool Uncover.

Many analysis and verifications tasks, such as static program analyses and model-checking for temporal logics reduce to the solution of systems of equations over suitable lattices. Inspired by recent work on lattice-theoretic progress measures, we develop a game-theoretical approach to the solution of systems of monotone equations over lattices, where for each single equation either the least or greatest solution is taken. A simple parity game, referred to as fixpoint game, is defined that provides a correct and complete characterisation of the solution of equation systems over continuous lattices, a quite general class of lattices widely used in semantics. For powerset lattices the fixpoint game is intimately connected with classical parity games for $\mu$-calculus model-checking, whose solution can exploit as a key tool Jurdzi\'nski's small progress measures. We show how the notion of progress measure can be naturally generalised to fixpoint games over continuous lattices and we prove the existence of small progress measures. Our results lead to a constructive formulation of progress measures as (least) fixpoints. We refine this characterisation by introducing the notion of selection that allows one to constrain the plays in the parity game, enabling an effective (and possibly efficient) solution of the game, and thus of the associated verification problem. We also propose a logic for specifying the moves of the existential player that can be used to systematically derive simplified equations for efficiently computing progress measures. We discuss potential applications to the model-checking of latticed $\mu$-calculi and to the solution of fixpoint equations systems over the reals.

The fuzzy modality `probably` is interpreted over probabilistic type spaces by taking expected truth values. The arising probabilistic fuzzy description logic is invariant under probabilistic bisimilarity; more informatively, it is non-expansive wrt. a suitable notion of behavioural distance. In the present paper, we provide a characterization of the expressive power of this logic based on this observation: We prove a probabilistic analogue of the classical van Benthem theorem, which states that modal logic is precisely the bisimulation-invariant fragment of first-order logic. Specifically, we show that every formula in probabilistic fuzzy first-order logic that is non-expansive wrt. behavioural distance can be approximated by concepts of bounded rank in probabilistic fuzzy description logic. For a modal logic perspective on the same result, see arXiv:1810.04722., Comment: arXiv admin note: text overlap with arXiv:1810.04722

We consider the permutation flow shop scheduling problem and aim to obtain smoothness of jobs' completion times, by minimising the variance or the variability of inter-completion times. This problem, including an efficient heuristics, was introduced in Leisten and Rajendran (2015). Here we solve an open problem from that paper and show that the problem for more than two machines is NP-complete.

We present a fuzzy (or quantitative) version of the van Benthem theorem, which characterizes propositional modal logic as the bisimulation-invariant fragment of first-order logic. Specifically, we consider a first-order fuzzy predicate logic along with its modal fragment, and show that the fuzzy first-order formulas that are non-expansive w.r.t. the natural notion of bisimulation distance are exactly those that can be approximated by fuzzy modal formulas.

We present a generic algorithm, generalizing partition refinement, for deciding behavioural equivalences for various types of transition systems. In order to achieve this generality, we work with coalgebra, which offers a general framework for modelling different types of state-based systems. The underlying idea of the algorithm is to work on the so-called final chain and to factor out redundant information. If the algorithm terminates, the result of the construction is a representative of the given coalgebra that is not necessarily unique and that allows to precisely answer questions about behavioural equivalence. We instantiate the algorithm to the particularly interesting case of weighted automata over semirings in order to obtain a procedure for checking language equivalence for a large number of semirings. We use fuzzy automata with weights from an l-monoid as a case study.

The large online encyclopedia “Wikipedia” has become a valuable information resource. However, its large size and the interconnectedness of its pages can make it easy to get lost in detail and difficult to gain a good overview of a topic. As a solution we propose a procedure to extract, summarize, and visualize large categories of historic Wikipedia articles. At the heart of this procedure we apply the method of main path analysis—originally developed for citation networks—to a modified network of linked Wikipedia articles. Beside the aggregation method itself, we describe our data mining process of the Wikipedia datasets and the considerations that guided the visualization of the article networks. Finally, we present our web app that allows to experiment with the procedure on an arbitrary Wikipedia category.

When specifying graph languages via type graphs, cores are a convenient way to minimize the type graph without changing the graph language, i.e. the set of graphs typed over the type graph. However, given a type graph, the problem of finding its core is NP-hard. Using the Tool CoReS, we automatically encode all required properties into SAT- and SMT-formulas to iteratively compute cores by employing the corresponding solvers. We obtain runtime results to evaluate and compare the two encoding approaches.

We show how up-to techniques for (bi-)similarity can be used in the setting of weighted systems. The problems we consider are language equivalence, language inclusion and the threshold problem (also known as universality problem) for weighted automata. We build a bisimulation relation on the fly and work up-to congruence and up-to similarity. This requires to determine whether a pair of vectors (over a semiring) is in the congruence closure of a given relation of vectors. This problem is considered for rings and l-monoids, for the latter we provide a rewriting algorithm and show its confluence and termination. We then explain how to apply these up-to techniques to weighted automata and provide runtime results.

Tools that implement automated verification techniques can be used to fruitfully analyze and validate complex software systems. Developing such tools is an active research area that has produced several promising techniques in the last decade: however, many challenges lie ahead. We briefly review the research area and summarize four papers selected from the Eighteenth International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS 2012).

The paper presents a case study on the synthesis of labelled transition systems (ltss) for process calculi, choosing as testbed Milner's Calculus of Communicating System (ccs). The proposal is based on a graphical encoding: each ccs process is mapped into a graph equipped with suitable interfaces, such that the denotation is fully abstract with respect to the usual structural congruence. Graphs with interfaces are amenable to the synthesis mechanism proposed by Ehrig and König and based on borrowed contexts (bcs), an instance of relative pushouts originally introduced by Milner and Leifer. The bc mechanism allows the effective construction of an lts that has graphs with interfaces as both states and labels, and such that the associated bisimilarity is automatically a congruence. Our paper focuses on the analysis of the lts distilled by exploiting the encoding of ccs processes: besides offering major technical contributions towards the simplification of the bc mechanism, a key result of our work is the proof that the bisimilarity on processes obtained via bcs coincides with the standard strong bisimilarity for ccs. © 2008 Elsevier Inc. All rights reserved.

Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a simple inequational deduction system, based on term graphs, for inferring inclusions of derived relations in a multi-algebra, and we show that term graph rewriting provides a sound and complete implementation of it. © 2007 Elsevier B.V. All rights reserved.

We consider recognizable languages of cospans in adhesive categories defined via automaton functors, of which recognizable graph languages are a special case.There are three contributions in this paper: we first show that the notion of recognizable languages is robust in the sense that also semi-functors, i.e., functors that do not necessarily preserve identities, characterize recognizable languages. This is done by converting semi-functors to functors with a procedure akin to epsilon elimination for non-deterministic finite automata.Second, relying on this result, we show that recognizable languages are closed under concatenation, i.e. under cospan composition, by providing a concrete construction that creates a concatenation automaton from two given automata. The construction is considerably more complex than the corresponding construction for finite automata. We conclude by showing negative closure properties for Kleene star and substitution. We show that recognizable languages are closed under concatenation by constructing an automaton accepting the concatenation.Using counter-examples, we show that recognizable languages are not closed under Kleene-star and substitution.We introduce the notion of semi-automata.We show that (for suitable categories) semi-automata accept the same class of languages as automata.

We investigate a general framework which can be instantiated in order to obtain type systems for graph rewriting, allowing us to statically infer behavioural properties of a graph. We describe conditions such as the subject reduction property and compositionality that should be satisfied by such a framework. We present a methodology for proving these conditions, specifically we prove that it is sufficient to show properties that are local to graph transformation rules. In order to show the applicability of this framework, we describe in several case studies how to integrate existing type systems (for the ?-calculus and the ?-calculus) and a system for typing acyclic graphs.

We propose a framework for the verication of behavioural properties of systems modelled as graph transformation systems. The properties can be expressed in a temporal logic which is basically a -calculus where the state predicates are formulae of a monadic second order logic, describing graph properties. The verication technique relies on an algorithm for the construction of nite over-approximations of the unfolding of a graph transformation system.

We investigate the relationship between regular languages and syntactic monoid size. In particular, we consider the transformation monoids of n-state (minimal) deterministic finite automata. We show tight upper and lower bounds on the syntactic monoid size depending on the number of generators (input alphabet size) used. It turns out, that the two generator case is the most involved one. There we show a lower bound of nn (1 - 2/√n) for the size of the syntactic monoid of a language accepted by an n-state deterministic finite automaton with binary input alphabet. Moreover, we prove that for every prime n ≥ 7, the maximal size semigroup w.r.t. its size among all (transformation) semigroups which can be generated with two generators, is generated by a permutation with two cycles (of appropriate lengths) and a non-bijective mapping merging elements from these two cycles. As a by-product of our investigations we determine the maximal size among all semigroups generated by two transformations, where one is a permutation with a single cycle and the other is a non-bijective mapping.

We define a construction operation on hypergraphs based on a colimit and show that its expressiveness is equal to the graph expressions of Bauderon and Courcelle. We also demonstrate that by closing a set of rewrite rules under graph construction we obtain a notion of rewriting equivalent to the double-pushout approach of Ehrig. The usefulness of our approach for the compositional modelling of concurrent systems is then demonstrated by giving a semantics of process graphs (corresponding to a process calculus with mobility) and of Petri nets. We introduce on the basis if a hypergraph construction, a method for the static analysis of process graphs, related to type systems.

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.

Coalgebra offers a general framework for modelling different types of state-based systems. Our aim is to present generic algorithms to decide behavioural equivalence for coalgebras which generalize partition refinement. The underlying idea of the algorithms is to work on the final chain and to factor out redundant information. If the algorithm terminates, the result of the construction is a representative of the given coalgebra that is not necessarily unique and that allows to precisely answer questions about behavioural equivalence. We apply the algorithm to weighted automata over semirings in order to obtain a procedure for checking language equivalence for a large number of semirings.

Graph transformation systems (GTSs) can be seen as well-structured transition systems (WSTSs), thus obtaining decidability results for certain classes of GTSs. In earlier work it was shown that well-structuredness can be obtained using the minor ordering as a well-quasi-order. In this paper we extend this idea to obtain a general framework in which several types of GTSs can be seen as (restricted) WSTSs. We instantiate this framework with the subgraph ordering and the induced subgraph ordering and apply it to analyse a simple access rights management system.

Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with arbitrary (possibly uncountable) state spaces. We consider the sub-probability monad and the probability monad (Giry monad) on the category of measurable spaces and measurable functions. Our main contribution is that the existence of a final coalgebra in the Kleisli category of these monads is closely connected to the measure-theoretic extension theorem for sigma-finite pre-measures. In fact, we obtain a practical definition of the trace measure for both finite and infinite traces of PTS that subsumes a well-known result for discrete probabilistic transition systems. Finally we consider two example systems with uncountable state spaces and apply our theory to calculate their trace measures. © Henning Kerstan and Barbara König.

We will revisit the categorical notion of cospan decompositions of graphs and compare it to the well-known notions of path decomposition and tree decomposition from graph theory. More specifically, we will define several types of cospan decompositions with appropriate width measures and show that these width measures coincide with pathwidth and treewidth. Such graph decompositions of small width are used to efficiently decide graph properties, for instance via graph automata. Hence we will give an application by defining graph-accepting tree automata, thus integrating previous work by Courcelle into the setting of cospan decompositions. Furthermore we will show that regardless of whether we consider path or tree decompositions, we arrive at the same notion of recognizability.

A contextual net is a Petri net extended with read arcs, which allows transitions to check for tokens without consuming them. Contextual nets allow for better modelling of concurrent read access than Petri nets, and their unfoldings can be exponentially more compact than those of a corresponding Petri net. A constructive but abstract procedure for generating those unfoldings was proposed in previous work. However, it remained unclear whether the approach was useful in practice and which data structures and algorithms would be appropriate to implement it. Here, we address this question. We provide two concrete methods for computing contextual unfoldings, with a view to efficiency. We report on experiments carried out on a number of benchmarks. These show that not only are contextual unfoldings more compact than Petri net unfoldings, but they can be computed with the same or better efficiency, in particular with respect to alternative approaches based on encodings of contextual nets into Petri nets. © 2012 Elsevier B.V. All rights reserved. OA embargo

Given a transition system and a partial order on its states, the coverability problem is the question to decide whether a state can be reached that is larger than some given state. For graphs, a typical such partial order is the minor ordering, which allows to specify "bad graphs" as those graphs having a given graph as a minor. Well-structuredness of the transition system enables a finite representation of upward-closed sets and gives rise to a backward search algorithm for deciding coverability. It is known that graph tranformation systems without negative application conditions form well-structured transition systems (WSTS) if the minor ordering is used and certain condition on the rules are satisfied. We study graph transformation systems with negative application conditions and show under which conditions they are well-structured and are hence amenable to a backwards search decision procedure for checking coverability.

Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with arbitrary (possibly uncountable) state spaces. We consider the sub-probability monad and the probability monad (Giry monad) on the category of measurable spaces and measurable functions. Our main contribution is that the existence of a final coalgebra in the Kleisli category of these monads is closely connected to the measure-theoretic extension theorem for sigma-finite pre-measures. In fact, we obtain a practical definition of the trace measure for both finite and infinite traces of PTS that subsumes a well-known result for discrete probabilistic transition systems.

Process calculi and graph transformation systems provide models of reactive systems with labelled transition semantics (LTS). While the semantics for process calculi is compositional, this is not the case for graph transformation systems, in general. Hence, the goal of this article is to obtain a compositional semantics for graph transformation system in analogy to the structural operational semantics (SOS) for Milner's Calculus of Communicating Systems (CCS). The paper introduces an SOS style axiomatization of the standard labelled transition semantics for graph transformation systems that is based on the idea of minimal reaction contexts as labels, due to Leifer and Milner. In comparison to previous work on inductive definitions of similarly derived LTSs, the main feature of the proposed axiomatization is a composition rule that captures the communication of sub-systems so that it can feature as a counterpart to the communication rule of CCS.

We introduce graph automata as a more automata-theoretic view on (bounded) automaton functors and we present how automaton-based techniques can be used for invariant checking in graph transformation systems. Since earlier related work on graph automata suffered from the explosion of the size of the automata and the need of approximations due to the non-determinism of the automata, we here employ symbolic bdd-based techniques and recent antichain algorithms for language inclusion to overcome these issues. We have implemented techniques for generating, manipulating and analyzing graph automata and perform an experimental evaluation.

We consider conditional reactive systems, a general abstract framework for rewriting, in which reactive systems a la Leifer and Milner are enriched with (nested) application conditions. We study the problem of deriving labelled transitions and bisimulation congruences from a reduction semantics. That is, we synthesize interactions with the environment in order to obtain a compositional semantics. Compared to earlier work we not only address the problem of deriving information about the (minimal) context needed to obtain a full left-hand side and thus be able to perform a reduction, but also generate conditions on the remaining context.

International audience; It is a known fact that the subobjects of an object in an adhesive category form a distributive lattice. Building on this observation, in the paper we show how the representation theorem for finite distributive lattices applies to subobject lattices. In particular, we introduce a notion of irreducible object in an adhesive category, and we prove that any finite object of an adhesive category can be obtained as the colimit of its irreducible subobjects. Furthermore we show that every arrow between finite objects in an adhesive category can be interpreted as a lattice homomorphism between subobject lattices and, conversely, we characterize those homomorphisms between subobject lattices which can be seen as arrows.

We introduce an analysis method for graph transformation systems which checks that certain forbidden graphs are not reachable from the start graph. These forbidden graphs are specified by a contextfree graph grammar. The technique is based on the approximation of graph transformation systems by Petri nets and on semilinear sets of markings. Especially we exploit Parikh's theorem which says that the Parikh image of a context-free grammar is semilinear. An important application is deadlock analysis for interaction nets and we specifically show how to apply the technique to an infinite-state dining philosopher's system.

We introduce a simple logic that allows to quantify over the subobjects of a categorical object. We subsequently show that, for the category of graphs, this logic is equally expressive as second-order monadic graph logic (msogl). Furthermore we show that for the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Courcelle’s result that every msogl-expressible property is recognizable. This is done by giving an inductive translation of formulas of our logic into so-called automaton functors which accept recognizable languages of cospans.

We generalize the unfolding semantics, previously developed for concrete formalisms such as Petri nets and graph grammars, to the abstract setting of (single pushout) rewriting over adhesive categories. The unfolding construction is characterized as a coreflection, i.e. 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 mono-morphisms. Compared to 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.

In recent years there have been several approaches for the automatic derivation of labels from an unlabeled reactive system. This can be done in such a way that the resulting bisimilarity is automatically a congruence. One important aspect that has not been studied so far is the treatment of reduction rules with negative application conditions. That is, a rule may only be applied if certain patterns are absent in the vicinity of a left-hand side. Our goal in this paper is to extend the borrowed context framework to label derivation with negative application conditions and to show that bisimilarity remains a congruence. An important application area is graph transformation and we will present a small example in order to illustrate the theory.

We describe an approach for the verification of attributed graph transformation systems (AGTS). AGTSs are graph transformation systems where graphs are labelled over an algebra. We base our verification procedure on so-called approximated unfoldings combined with counterexample-guided abstraction refinement. Both techniques were originally developed for non-attributed systems. With respect to refinement we focus especially on detecting whether the spurious counterexample is caused by structural over-approximation or by an abstraction of the attributes which is too coarse. The technique is implemented in the verification tool Augur 2 and a leader election protocol has been successfully verified.

We show how to view certain subclasses of (single-pushout) graph transformation systems as well-structured transition systems, which leads to decidability of the covering problem via a backward analysis. As the well-quasi order required for a well-structured transition system we use the graph minor ordering. We give an explicit construction of the backward step and apply our theory in order to show the correctness of a leader election protocol.

In a seminal paper, McMillan proposed a technique for constructing a finite complete prefix of the unfolding of bounded (i.e., finite-state) Petri nets, which can be used for verification purposes. Contextual nets are a generalisation of Petri nets suited to model systems with read-only access to resources. When working with contextual nets, a finite complete prefix can be obtained by applying McMillan's construction to a suitable encoding of the contextual net into an ordinary net. However, it has been observed that if the unfolding is itself a contextual net, then the complete prefix can be significantly smaller than the one obtained with the above technique. A construction for generating such a contextual complete prefix has been proposed for a special class of nets, called read-persistent. In this paper, we propose an algorithm that works for arbitrary semi-weighted, bounded contextual nets. The construction explicitly takes into account the fact that, unlike in ordinary or read-persistent nets, an event can have several different histories in general contextual net computations.

In this paper we give a category-based characterization of recognizability. A recognizable subset of arrows is defined via a functor into the category of relations on sets, which can be seen as a straightforward generalization of a finite automaton. In the second part of the paper we apply the theory to graphs, and we show that our approach is a generalization of Courcelle's recognizable graph languages.

The unfolding of a system represents in a single branching structure all its possible computations: it is the cornerstone both of semantical constructions and of efficient partial order verification techniques. In this paper we survey the contributions we elaborated in the last decade with Ugo Montanari and other colleagues, concerning the unfolding of graph transformation systems, and its use in the definition of a Winskel style functorial semantics and in the development of methodologies for the verification of finite and infinite state systems.

Behavior preservation, namely the fact that the behavior of a model is not altered by the transformations, is a crucial property in refactoring. The most common approaches to behavior preservation rely basically on checking given models and their refactored versions. In this paper we introduce a more general technique for checking behavior preservation of refactorings defined by graph transformation rules. We use double pushout (DPO) rewriting with borrowed contexts, and, exploiting the fact that observational equivalence is a congruence, we show how to check refactoring rules for behavior preservation. When rules are behavior-preserving, their application will never change behavior, i.e., every model and its refactored version will have the same behavior. However, often there are refactoring rules describing intermediate steps of the transformation, which are not behavior-preserving, although the full refactoring does preserve the behavior. For these cases we present a procedure to combine refactoring rules to behavior-preserving concur- rent productions in order to ensure behavior preservation. An example of refactoring for finite automata is given to illustrate the theory.