3,603 research outputs found
Equational reasoning with context-free families of string diagrams
String diagrams provide an intuitive language for expressing networks of
interacting processes graphically. A discrete representation of string
diagrams, called string graphs, allows for mechanised equational reasoning by
double-pushout rewriting. However, one often wishes to express not just single
equations, but entire families of equations between diagrams of arbitrary size.
To do this we define a class of context-free grammars, called B-ESG grammars,
that are suitable for defining entire families of string graphs, and crucially,
of string graph rewrite rules. We show that the language-membership and
match-enumeration problems are decidable for these grammars, and hence that
there is an algorithm for rewriting string graphs according to B-ESG rewrite
patterns. We also show that it is possible to reason at the level of grammars
by providing a simple method for transforming a grammar by string graph
rewriting, and showing admissibility of the induced B-ESG rewrite pattern.Comment: International Conference on Graph Transformation, ICGT 2015. The
final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-21145-9_
Ten virtues of structured graphs
This paper extends the invited talk by the first author about the virtues
of structured graphs. The motivation behind the talk and this paper relies on our
experience on the development of ADR, a formal approach for the design of styleconformant,
reconfigurable software systems. ADR is based on hierarchical graphs
with interfaces and it has been conceived in the attempt of reconciling software architectures
and process calculi by means of graphical methods. We have tried to
write an ADR agnostic paper where we raise some drawbacks of flat, unstructured
graphs for the design and analysis of software systems and we argue that hierarchical,
structured graphs can alleviate such drawbacks
The three dimensions of proofs
In this document, we study a 3-polygraphic translation for the proofs of SKS,
a formal system for classical propositional logic. We prove that the free
3-category generated by this 3-polygraph describes the proofs of classical
propositional logic modulo structural bureaucracy. We give a 3-dimensional
generalization of Penrose diagrams and use it to provide several pictures of a
proof. We sketch how local transformations of proofs yield a non contrived
example of 4-dimensional rewriting.Comment: 38 pages, 50 figure
On Term-Graph Rewrite Strategies
AbstractWe tackle the problem of cyclic term-graph rewriting. We first revisit the classical algorithmic approach to term-graph rewriting by providing a definition of rewrite rules of the form lhs→rhs where the left-hand sides are term-graphs and the right-hand sides are sequences of actions. Such actions, which specify how to rewrite a term-graph in a stepwise manner, contribute to simplify substantially the definition of cyclic term-graph rewriting. Then we define a new class of term-graph rewrite systems which are confluent over the so-called admissible term-graphs. Finally, we provide an efficient rewrite strategy which contracts only needed redexes and give pointers to other results regarding optimal rewrite strategies of admissible term-graphs
Categorified cyclic operads
In this paper, we introduce a notion of categorified cyclic operad for
set-based cyclic operads with symmetries. Our categorification is obtained by
relaxing defining axioms of cyclic operads to isomorphisms and by formulating
coherence conditions for these isomorphisms. The coherence theorem that we
prove has the form "all diagrams of canonical isomorphisms commute". Our
coherence results come in two flavours, corresponding to the "entries-only" and
"exchangeable-output" definitions of cyclic operads. Our proof of coherence in
the entries-only style is of syntactic nature and relies on the coherence of
categorified non-symmetric operads established by Do\v{s}en and Petri\'c. We
obtain the coherence in the exchangeable-output style by "lifting" the
equivalence between entries-only and exchangeable-output cyclic operads, set up
by the second author. Finally, we show that a generalisation of the structure
of profunctors of B\' enabou provides an example of categorified cyclic operad,
and we exploit the coherence of categorified cyclic operads in proving that the
Feynman category for cyclic operads, due to Kaufmann and Ward, admits an odd
version.Comment: 57 page
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