Citation Context

... sections to the setting of supported pre-categories pose no real difficulties, it only requires a lot of footwork. For want of space, we omit that footwork here; the interested reader is referred to =-=[16]-=-. We do, however, make the following comments. A sorting on pure abstract bigraphs induces a sorting on the corresponding concrete bigraphs, found by taking the pullback of that sorting along the supp...

Citation Context

...mplete graph embedding problem). Therefore we decided early on to study the matching problem quite formally and base our prototype implementation on a provably correct specification. In previous work =-=[1, 4]-=-, we gave a sound and complete inductive characterization of the matching problem for bigraphs. Our inductive characterization was based on normal form theorems for binding bigraphs [9]. In the presen...

Citation Context

...s work on variable binding in a linear setting [26], whose relation to the present work remains unclear to us. A number of papers have been devoted to better understanding bigraphs, be it as sortings =-=[7]-=-, as cospans over graphs [25], as a compact closed category [13], or as a language with variable binding [5]. We appear to be the first to reconcile a full treatment of scope (Theorem 4) with algebrai...

Citation Context

...blem. More precisely, it is reducible to the sub-DAG isomorphism problem, which is also NP-complete [6]. The time complexity of Match(_, _) is dominated by the running time of sub-routine Sub_iso(_, _), which is O(|V T ||V P |). Note that the introduction of sharing does not involve an increase in complexity in the general case. We have chosen to define our matching algorithm as a reduction to the sub-DAG isomorphism problem for several reasons. The first one is that algorithms defined by structural induction on the algebraic representation of the input bigraphs (e.g. the algorithm defined in [7]) are not suited for the DAG structure of place graphs with sharing since the amount of unnecessary blind search introduced during the inference process dramatically increases, thus negatively affecting the algorithm’s performance. Second, a reduction to a standard problem tends to produce faster algorithms because it fully exploits established solvers which are very efficient due to the use of advanced heuristics. A typical example of this approach is the encoding of hard problems as instances of the boolean satisfiability problem (SAT) [8,9]. Finally, we want our algorithm to support concret...

Citation Context

...) Merge (merge) merge 2 (generalized to merge n) 0 1 Closure (� � x y ) x X y (generalized to [x,w] X [y,z]) Substitution (△ y X ) y / [x_1,...,x_n] (also y / x and y /) x y y ... x1x2...xn Swap (γ) @=-=[1,0]-=- (generalized to @[3,4,1,0,2]) 1 0 Fusion (▽ Y x ) [y_1,...,y_n] x (also y x and x) Ion (K x y (l)) K[-y_1,...,-y_n,+x_1,...,+x_m] (the names in the list can be written in any order) y1 y2...ym ... ...

Citation Context

... working on a prototype implementation of BRSs. Normalisation of binding bigraphs has been implemented. The next step is to implement an algorithm for matching (like the examples we saw in chapter 4) =-=[BDGM06]-=-. Perhaps, it is an idea to implement an add-on for local bigraphs. This tool will be crucial for simulation purposes because realistic examples easily become too large to handle manually, let alone r...