WP9 Innovation Activities

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
Graphs are everywhere

• Use of diagrams / graphs is pervasive to Computer
  Science

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
Encoding computational formalisms From algebraic
to graph-based syntax
5
The goal sound and complete encoding
6
Main complication the representation gap
7
The proposed solution graph algebras
one to one
8
From graphs to graph algebras

• 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

9
Towards more structured graphsA 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
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
The Algebra of Graphs with Nesting syntax and
some terms, with their intuitive meaning
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(No Transcript)
13
Hierarchical (layered) graphs
14
Interpreting AGN over Hierarchical Graphs
Main result soundness, completeness and
surjectivity of the interpretation
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
Conclusions

• ... after Ugo's talk...

17
CHARM's axioms
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GS-monoidal theory an axiomatization of term
graphs