MOCIÓ CONSENSUADA PEL GRUP COMPROMÍS PER PATERNA, GRUP

1. r : K→R is monic and

2. for each Y such that

b) the following square is a pullback:

K

K

R

Y

(PB)

we have the following diagram:

W X R Y (PB) K r (=)

Now, we are ready to formalize compass steps. In particular, start- ing from a given compass and a given control action, we provide separate constructions for the positive and the negative pole, respec- tively, which together yield the result of the compass step. In addition to agnosticism described above, we consider a further possibility for retaining a negative pole element during a step: if the application of a rule ρ could create that pole element by applying it at a structure for- bidden by current non-applicability conditions, then we can conclude that it remains impossible for this structure to appear in the output graph. However, for similar reasons as for agnosticism, we have to restrict ρ to not creating nodes. (This time, creation is the relevant direction as we have to argue over inverse applications of ρ, cf. the proof of Theorem7.2.)

Definition 7.7 (Compass Step). The relation≡_{V⊆ C × (R}∗×2R) × C is a labeled relation on compasses s.t.

(Π, ¯Π) (ρ,N)

≡≡≡V (Π0, ¯Π0) if

1. X0 ∈ Π0 iff there exists a diagram in CanGraph Can(L) Can(K) X l G · · · ¯ X1 X¯|_Π˜| 6 6 m D (PO) f k Can(R) X0 r n g (PO) x where a) X ∈Π, b) ¯Xi ∈Π¯ (i=1 . . .|Π¯|),

c) ∀p∈ N : G 6=⇒p and

d) the pair(x, m)is jointly epimorphic.

2. ¯X0 ∈ Π¯0_{iff ¯X}0 _{is connected and}

a) ρ is agnostic to ¯X0, where i. ¯X0 ∈_Π¯

or

ii. ¯X0 = Can(Lp) for some p ∈ N where p does not delete nodes

or

b) ¯X0 ∈ _{Π, ρ does not create nodes, Can}¯ (Lp)=⇒δ X¯0 for a rule p∈ N not deleting nodes, with δ in CanGraph s.t. ρ(δ) =Can(ρ), m(δ) =Can(L),→m Can(Lp).

As an example, consider a step over p_Search (thus, with an empty N) on min(Csearch2) as specified at the end of the previous section. As for the evolution of the positive pole, LSearch can be joined with G_f in various jointly epimorphic ways, in particular, by injectively embedding Gf into LSearch, yielding LSearch itself (as we are canonical in compasses). Thus, RSearch is in the result of the positive pole of the compass step. Note, however, that due to p_Search not being agnostic to LSuccess nor to LSuccess2, as expected, we lose our negative compass elements as the assumption of Csearch2 does not hold anymore. If, instead, we consider a step (from an adequate compass immaterial here) over the action(pUcP,{pSearch})as in Prep, LSearch appears in the resulting negative pole, correctly expressing that the search rule does not match in any topology after that action, as there are no morphisms between the left- and right-hand sides of pUcP and LSearch due to an edge type mismatch.

The compass transition system is defined analogously to the RePro transition system (Definition5.2): whenever the control process com-

ponent of a state is able to perform an action, the compass component also performs a step over the same action. (Note that, in contrast to the RePro transition system, this behavior is non-blocking: Definition7.7

never results in an empty positive pole, as there is always at least one option, namely disjoint composition, for combining a previously existing element with the left-hand side of the rule.)

Definition 7.8 (Compass Transition System). The compass transition system is an LTS(P × [C],(R∗×2R),−→_C), where−→_Cis the least relation satisfying the following rule:

compass

P−−−→(ρ,N) P0 min([C]) (ρ,N) ≡≡≡VC0

c o m pa s s l a n g ua g e s. For providing an abstract verification ap- proach through compasses and abstract interpretation, we do not intend to capture and involve any graph languages, but only those which actually arise during rule applications.

As a first simplification, compasses intentionally describe only infi- nite graph languages: Although the output set of a graph-rewriting system might be finite, such graph languages cannot be characterized by smaller (either in number or in size) positive or negative patterns in general. Having noted this, we identify the study of classes of finite graph languages admitting a “quantitative reduction” as future work.

As for infinite graph languages, languages which defy any “regular” (recurring) pattern characteristics arguably go beyond the expressive power of graph-rewriting systems and, thus, require completely differ- ent formalization and analysis approaches than those in the present thesis.

Therefore, in the thesis, we confine ourselves to graph languages having such regularities, i.e., exactly ones described by compasses. We argue that many relevant application examples fall into this cate- gory: in graph-based system modeling, one often formulates language characteristics by requiring the presence or absence of positive or negative patterns, respectively. For example, in the context of our WSN scenario, we might reason about classes of topologies such as: topologies not containing an active triangle or containing an unexplored virtual link. More precisely, compass-definable languages capture those graph languages which arise from applying a GTS to any arbitrary graph, as we will demonstrate below.

Definition 7.9 (Compass-Definable Graph Language). A graph lan- guageG⊆ |Graph|is compass-definable (abbreviated as CGL) if∃[C] ∈ [C]:L([C]) =G.

The set of all compass-definable graph languages is denotedLC and also

referred to as CGL.

To conclude this section, we investigate some general properties of CGL, inspired by literature on abstract graph language specifi- cations [20, 24, 106]. In particular, inspited by the general formal

characterization criteria of Corradini et al. [20], we reason about some

basic closure and decidability properties.

As for the latter, luckily, the definition of compasses (Definition7.1)

allows for an easy decision process for basic membership, as summa- rized below.

Proposition 7.5. Given a compass C and a graph G (typed over the same