In this paper we introduce the notions of nested constraints and application conditions, short nested conditions. For a category associated with a graphical representation such as graphs, conditions are a graphical and intuitive, yet precise, formalism that is well suited to describing structural properties. We show that nested graph conditions are expressively equivalent to first-order graph formulas. A part of the proof includes transformations between two satisfiability notions of conditions, namely-satisfiability and-satisfiability. We consider a number of transformations on conditions that can be composed to construct constraint-guaranteeing and constraint-preserving application conditions, weakest preconditions and strongest postconditions. The restriction of rule applications by conditions can be used to correct transformation systems by pruning transitions leading to states violating given constraints. Weakest preconditions and strongest postconditions can be used to verify the correctness of transformation systems with respect to pre- and postconditions.

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Graph transformation has many application areas in computer science, such as software engineering or the design of concurrent and distributed systems. Being a visual modeling technique, graph transformation has the potential to play a decisive role in the development of increasingly larger and complex systems. However, the use of visual modeling techniques alone does not guarantee the correctness of a design. In context of rising standards for trustworthy systems, there is a growing need for the verification of graph transformation systems and programs. The research of appropriate methods for this purpose is the topic of this thesis.The primary goal is to obtain the capability to decide graphical program specifications. These specifications consists of a graphical precondition, a graph program, and a graphical postcondition. As usual, such a specification is said to be correct, if all those system states satisfy the postcondition that are reachable by applying the program on a start state satisfying the precondition. In the considered programs, the selection, deletion, addition and deselection of a graph's nodes and edges are the elementary constructs that can be composed to more complex programs by non-deterministic choice, sequential composition and iteration. The resulting programming language is computationally complete and is able to model transactions that deal with an unbounded number of nodes and edges. As language for the specification of state properties, graph conditions are investigated and used. We show that graph conditions provide an intuitive formalism for first-order structural properties and are suited to infer knowledge about the behavior of graph transformation systems and programs.According to Dijkstra, the correctness of program specifications can be shown by constructing a weakest precondition of the program relative to the postcondition and checking whether the specified precondition implies the weakest precondition. Hence the correctness problem of program specifications is reduced to an implication problem of conditions. In this thesis, it is shown how to construct weakest preconditions for graph programs and graph conditions. Following a dual approach, a sound and complete satisfiability iv Development of Correct Graph Transformation Systems algorithm for graph conditions is investigated and a fragment of conditions is identified, for which the algorithm decides. On the other hand, a resolutionbased calculus for graph conditions is presented and its soundness is proven. Implementations of the aforementioned deciders for conditions are compared with existing theorem provers and satisfiability solvers for first-order logic by verifying three case studies: a railroad control, an access control for computer systems, and, as an external example, a car platoon maneuver protocol.The research is done within the framework of the so-called weak adhesive high-level replacement categories. Therefore, the results will be applicable to different kinds of graph replacement systems and Petri n...

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Abstract. The tautology problem is the problem to prove the validity of statements. In this paper, we present a calculus for this undecidable problem on graphical conditions, prove its soundness, investigate the necessity of each deduction rule, and discuss practical aspects concerning an implementation. As we use the framework of weak adhesive HLR categories, the calculus is applicable to a number of replacement capable structures, such as Petri-Nets, graphs or hypergraphs.

To the memory of Marcus Ratzke Atom economy (AE) or atom utilization was one of the first defining terms in the sustainable chemistry movement. In contrast to the often-cited twelve (qualitative) principles of green chemistry, AE represents a metric for quantification purposes. The theoretical efficiency of a reaction expressed by its stoichiometric equation can be determined by AE Product [g/mol]/(Substrate 1 Substrate 2 ...) [g/mol] and compared with synthetic alternatives. Of course, the atom economy will be of limited use, if starting materials differ much in complexity, i.e., in the degree of refinement. In these cases, their syntheses have to be taken in consideration, too. But, the further the retrospect goes and the more preceding synthesis steps ramify, the more complex the calculation gets. To overcome this limitation, we introduce a stepwise approach that is enabled by a simple modification of the above formula (P product; S substrate; Syn. synthesis): AE P [g/mol]/((S1/AE(Syn. of S1)) (S2/AE(Syn. of S2)) ...) [g/mol]. To illustrate this equation, which is derived mathematically, the convergent multistep synthesis of the natural product trans-chrysanthemic acid is subjected to a stepwise method of calculation. The equation can be understood as a general expression for related ratios, i.e., there are corresponding modified equations for yield, selectivity, etc. In terms of the yield, it is no longer necessary to decide between the chains of the convergent synthesis, when possibly forced to ignore significant parts of the sequence. For demonstration purposes, the yield of the convergent synthesis of the natural product peridinin has been determined with a correspondingly modified equation.

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