1. Modeling a Service and Session Calculus with Hierarchical Graph Transformation Roberto Bruni, Andrea Corradini, Ugo Montanari (based on joint work with Fabio Gadducci and Alberto Lluch Lafuente) GraMoT, International Colloquium on Graph and Model Transformation Berlin, February 11 - 12, 2010

2. 2 Graphs are everywhere  Use of diagrams / graphs is pervasive to Computer Science

3. 3 Graph-based approaches  Some key features of graph-based approaches  help to convey ideas visually  ability to represent in a direct way relevant topological features  to make "links", "connection", "separation", "structure" explicit  ability to model systems at the “right” level of abstraction  ability to represent systems “up to isomorphism”  irrelevant details can be omitted (e.g. names of states in FSA)  important body of theory available  Graph grammars  Graph transformation systems  DPO, SPO, SHR  Layout algorithms

4. 4 Encoding computational formalisms: From algebraic to graph-based syntax

5. 5 The goal: sound and complete encoding

6. 6 Main complication: the representation gap

7. 7 The proposed solution: graph algebras one to one

8. 8  Start with a given class (category?) of graphs  Define an equational signature,  operators correspond to operations on graphs  axioms describe their properties  Prove once and for all soundness and completeness of the axioms with respect to the interpretation on graphs, as well as surjectivity  Next, you can safely use the algebra as an alternative, more handy syntax for the graphs  Example of graph algebra:  CHARM [CMR'94]  states correspond to monos of hypergraphs (graph with interface)  rewriting rules correspond to spans, reductions to dpo steps From graphs to graph algebras

9. 9 Towards more structured graphs A classical example: Term Graphs  First-order terms are an ingredient of the syntax of several formalisms:  term rewriting  logic programming  functional programming  narrowing ...  A graphical modeling of this family of formalisms is easier when using graphs which embody a notion of operator applied to arguments: Term Graphs  Term Graphs were introduced in a variety of equivalent ways (directed graphs, jungles, linear systems of equations)  They also have a sound and complete axiomatization:  GS-monoidal theories

10. 10 Modeling Sensoria calculi: Need of hierarchical graphs  Several process calculi have been proposed in the framework of the Sensoria project  Such calculi use names, and include a notion of nesting, for modeling sessions, transactions,...  To reduce the representation gap, we introduce an algebra of nested graphs which, besides the CHARM operators and axioms, provides primitives for representing  names  nesting of structures  The algebra is coupled with a new definition of hierarchical (layered) graphs  An interpretation of the terms of the algebra as hierarchical graphs is provided  Soundness and completeness of the axioms with respect to the encoding has been proved  The encoding is proved to be surjective

11. 11 The Algebra of Graphs with Nesting: syntax and some terms, with their intuitive meaning

12. 12 The Algebra of Graphs with Nesting: Axioms Optional for certain L

13. 13 Hierarchical (layered) graphs

14. 14 Interpreting AGN over Hierarchical Graphs Main result: soundness, completeness and surjectivity of the interpretation

15. 15 Encoding AGN into Term Graphs  A different graphical representation of AGN is obtained by interpreting it over Term Graphs  The interpretation is defined using the GS-monoidal theory  The encoding is sound and complete, but not surjective

16. 16 Conclusions... after Ugo's talk...

17. 17 CHARM's axioms

18. 18 GS-monoidal theory: an axiomatization of term graphs