Algebraic Approaches to Graph Transformation – Part I: Basic Concepts and Double Pushout Approach

Abstract

The algebraic approaches to graph transformation are based on the concept of gluing of graphs, modelled by pushouts in suitable categories of graphs and graph morphisms. This allows one not only to give an explicit algebraic or set theoretical description of the constructions, but also to use concepts and results from category theory in order to build up a rich theory and to give elegant proofs even in complex situations. In this chapter we start with an overwiev of the basic notions common to the two algebraic approaches, the double-pushout (DPO) approach and the singlepushout (SPO) approach; next we present the classical theory and some recent development of the double-pushout approach. The next chapter is devoted instead to the single-pushout approach, and it is closed by a comparison between the two approaches.

Typing of Graph Transformation Units

“…If G =⇒ r H, one says that G directly derives H by applying r. There are many possibilities to choose rules and their applications. Types of rules may vary from the more restrictive ones, like edge replacement [4] or node replacement [8], to the more general ones, like double-pushout rules [2], single-pushout rules [6], or PROGRES rules [17].…”

“…All components in the body must be consistent with the base type of the unit. 2 The body of tgtu is a system body = (I, U, R, C, T ) where I and T are graph tuple class expressions over BT , U is a set of imported typed graph transformation units, R is a tuple of rule sets (R 1 , . .…”

“…It has as input graphs a string graph and a deterministic finite state graph and as output graph a boolean value. The mapping in of the declaration part of recognize is defined by in : [2] → [3] with in(1) = 1 and in(2) = 2. We use the more intuitive tuple notation (string, dfsg, −) for this.…”

“…where a1 = (−, −, −, recognize (1,2), recognize (1, 3), −) and a2 = (−, −, −, is-true, is-true, set-to-true) terminal: (string , dfsg, dfsg, bool , bool , bool ) The unit recognize-intersection imports the above unit recognize and has as local rules is-true and set -to-true where is-true can be applied to the fourth and the fifth component of the current graph tuples and set -to-true to the sixth component. The control condition requires the following.…”

“…In the second point recognize must be applied to the input string graph and to the second deterministic finite state graph. These two transformations can be performed in parallel within one and the same action denoted by the tuple (−, −, −, recognize (1,2),…”