An Algorithm for Approximating the Satisfiability Problem of High-level Conditions

AbstractThe satisfiability problem is the fundamental problem in proving the conflict-freeness of specifications, or in finding a counterexample for an invalid statement. In this paper, we present a non-deterministic, monotone algorithm for this undecidable problem on graphical conditions that is both correct and complete, but in general not guaranteed to terminate. For a fragment of high-level conditions, the algorithm terminates, hence it is able to decide. Instead of enumerating all possible objects of a category to approach the problem, the algorithm uses the input condition in a constructive way to progress towards a solution. To this aim, programs over transformation rules with external interfaces are considered. We use the framework of weak adhesive HLR categories. Consequently, the algorithm is applicable to a number of replacement capable structures, such as Petri-Nets, graphs or hypergraphs

Symbolic model generation for graph properties

Graphs are ubiquitous in Computer Science. For this reason, in many areas, it is very important to have the means to express and reason about graph properties. In particular, we want to be able to check automatically if a given graph property is satisfiable. Actually, in most application scenarios it is desirable to be able to explore graphs satisfying the graph property if they exist or even to get a complete and compact overview of the graphs satisfying the graph property. We show that the tableau-based reasoning method for graph properties as introduced by Lambers and Orejas paves the way for a symbolic model generation algorithm for graph properties. Graph properties are formulated in a dedicated logic making use of graphs and graph morphisms, which is equivalent to first-order logic on graphs as introduced by Courcelle. Our parallelizable algorithm gradually generates a finite set of so-called symbolic models, where each symbolic model describes a set of finite graphs (i.e., finite models) satisfying the graph property. The set of symbolic models jointly describes all finite models for the graph property (complete) and does not describe any finite graph violating the graph property (sound). Moreover, no symbolic model is already covered by another one (compact). Finally, the algorithm is able to generate from each symbolic model a minimal finite model immediately and allows for an exploration of further finite models. The algorithm is implemented in the new tool AutoGraph.Peer ReviewedPostprint (author's final draft

Automated reasoning for attributed graph properties

Graphs are ubiquitous in computer science. Moreover, in various application fields, graphs are equipped with attributes to express additional information such as names of entities or weights of relationships. Due to the pervasiveness of attributed graphs, it is highly important to have the means to express properties on attributed graphs to strengthen modeling capabilities and to enable analysis. Firstly, we introduce a new logic of attributed graph properties, where the graph part and attribution part are neatly separated. The graph part is equivalent to first-order logic on graphs as introduced by Courcelle. It employs graph morphisms to allow the specification of complex graph patterns. The attribution part is added to this graph part by reverting to the symbolic approach to graph attribution, where attributes are represented symbolically by variables whose possible values are specified by a set of constraints making use of algebraic specifications. Secondly, we extend our refutationally complete tableau-based reasoning method as well as our symbolic model generation approach for graph properties to attributed graph properties. Due to the new logic mentioned above, neatly separating the graph and attribution parts, and the categorical constructions employed only on a more abstract level, we can leave the graph part of the algorithms seemingly unchanged. For the integration of the attribution part into the algorithms, we use an oracle, allowing for flexible adoption of different available SMT solvers in the actual implementation. Finally, our automated reasoning approach for attributed graph properties is implemented in the tool AutoGraph integrating in particular the SMT solver Z3 for the attribute part of the properties. We motivate and illustrate our work with a particular application scenario on graph database query validation.Peer ReviewedPostprint (author's final draft

An Algorithm for Approximating the Satisfiability Problem of High-level Conditions (Long Version)

The satisfiability problem is the fundamental problem in proving the conflict-freeness of specifications, or in finding a counterexample for an invalid statement. In this paper, we present a non-deterministic, monotone algorithm for this undecidable problem on graphical conditions that is both correct and complete, but in general not guaranteed to terminate. For a fragment of high-level conditions, the algorithm terminates, hence it is able to decide. Instead of enumerating all possible objects of a category to approach the problem, the algorithm uses the input condition in a constructive way to progress towards a solution. To this aim, programs over transformation rules with external interfaces are considered. We use the framework of weak adhesive HLR categories. Consequently, the algorithm is applicable to a number of replacement capable structures, such as Petri-Nets, graphs or hypergraphs