Integrating the Specication Techniques of
Graph Transformation and Temporal Logic ?
Reiko Heckel1 , Hartmut Ehrig1, Uwe Wolter1, Andrea Corradini2??
1 TU Berlin, FR 6-1, Franklinstrasse 28/29, 10587 Berlin, Germany
freiko, ehrig, [email protected]
2 CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
[email protected]
Abstract. The aim of this paper is an integration of graph grammars
with dierent kinds of behavioural constraints, in particular with temporal logic constraints. Since the usual algebraic semantics of graph
transformation systems is not able to express constrained behaviour we
introduce - in analogy to other approaches - a coalgebraic semantics
which associates with each system a category of models (deterministic
transition systems). Such category has a nal object, which includes
all nite and innite transition sequences. The coalgebraic framework
makes it possible to introduce a general notion of `logic of behavioural
constraints'. Instances include, for example, graphical consistency constraints and temporal logic constraints. We show that the considered
semantics can be restricted to a nal coalgebra semantics for systems
with behavioural constraints. This result can be instantiated in order to
obtain a nal coalgebra semantics for graph grammars with temporal
logic constraints.
1 Introduction
The theory of graph transformation basically studies a variety of formalisms
which extend the theory of formal languages in order to deal with structures
more general than strings, like graphs and maps. A graph transformation system allows one to describe nitely a (possibly innite) collection of graphs, i.e.,
those which can be obtained from a start graph through repeated applications
of graph productions. Each production can be applied to a graph by replacing
an occurrence of its left-hand side with its right-hand side. The form of graph
productions and the mechanisms stating how a production can be applied to a
graph and what the resulting graph is, depend on the specic formalism. In this
paper we use the \algebraic, Double Pushout (DPO) approach" [1,2].
The classical theory of the DPO approach is mainly concerned with structural properties and the sequential and parallel composition of transformation
Research partially supported by the German Research Council (DFG) and the TMR
network GETRATS
?? A. Corradini is on leave from Dipartimento di Informatica, Pisa. He is also supported
by the EC Fixed Contribution Contract n. EBRFMBICT960840
?
steps. More recently, also distributed graph transformations and means of synchronisation have been investigated, see [2] for a recent survey. It has been shown
that graph transformations are well-suited for modelling the state transformation aspect of software systems. In a purely rule-based framework, however, it
is dicult to control the order and frequency of rule applications. Such control
aspects are most important for the specication of software systems, in particular of concurrent and distributed ones. For this reason we study behavioural
constraints for graph grammars.
Instead of inventing new techniques for specifying the behavioural aspects
of a system we shall provide an interface for the integration of graph transformation with existing specication techniques. In this paper we shall elaborate
in some detail a temporal logic for graph grammars, extending the approach
of [3]. The advantage of this approach is two-fold. First, it combines graphical
(visual) and logical means which makes it much easier to work with than purely
textual expressions (at least for non-experts). Second, all graph-specic aspects
are formulated categorically, which makes the approach applicable also to other
kinds of structures.
The integration of graph grammars and behavioural constraints is done both
syntactically and semantically. On the syntactical level, a notion of behavioural
constraint is assumed together with a satisfaction relation for derivation sequences. On the semantical level, a coalgebraic semantics of graph grammars
provides a model for the restricted behaviour of systems. This coalgebraic semantics is dened in analogy to a recently developed coalgebraic semantics for
graph grammars in [4], where graph transitions are used as a kind of \loose
transformation step". Since, the coalgebraic idea is orthogonal to the kind of
transformation one considers, we may reuse the constructions of [4] without additional eort. The proofs for the results of this paper can also be found there.
The coalgebraic loose semantics allows us to handle in a natural way a large
class of constraints imposed on the behaviour of a graph transformation system:
It is shown that for each set of behavioural constraints, the restriction to the
behaviours that satisfy these constraints is a cofree construction. This implies the
existence of a nal (maximal) model for each graph grammar with behavioural
constraints.
2 Basic Notions
This section reviews basic notions and denitions of the algebraic double-pushout
(DPO) approach to the transformation of typed graphs [2,5].
Let Graph be the category of graphs and graph morphisms. Given a graph
TG 2 jGraphj, the category GraphTG of typed graphs over TG and typed
graph morphisms is the comma category (Graph # TG). If not stated otherwise, graphs and graph morphisms will be assumed to be typed over TG in the
following.
A graph production p : s is composed of a production name p and of a span
r
of injective graph morphisms s = (L l? K ?!
R), called production span. A
(typed) graph transformation system G = hTG; P; i consists of a type graph TG,
a set of production names P, and a mapping associating with each production
name p a production span (p). A (typed) graph grammar GG = hG; G0i is a
typed graph transformation system G = hTG; P; i together with a start graph
G0 2 jGraphTG j.
Derivation steps in the DPO approach are dened as double pushout constructions: A double-pushout d is a diagram like below, where top and bottom
r
are production spans and (1) and (2) are pushouts. If p : (L l? K ?!
R) is a
p=d
production, a derivation step from G to H is denoted by G =) H. We also write
p=d if G and H are understood, and denote by In; Out, and pn the projections
In(p=d) = G; Out(p=d) = H, and pn(p=d) = p. The set of all derivation steps in
G is G).
L
l
(1)
dL
G
l
K
dK
D
r
R
(2)
dR
r
H
A derivation in a graph transformation system G is a nite or innite se=d2
=d1
G2 : : : where p1; p2; : : : are producG1 p=2 )
quence of derivation steps G0 p=1 )
tion names of G. A derivation in a grammar GG = hG; G0i is a derivation in
G that starts in G0.
HookOn
☎
L1
K1
☎ ☎
(1)
TG Ph
☎
Ring
Ring
R1
☎
K2
R2
Ring
HookOff
Ring
HookOff
☎
☎
G1
☎ ☎
L2
(2)
HookOn
Phone
HookOn HookOff
HookOff
D2
Ring
HookOff
☎
G2
☎
D3
H3
Fig. 1. Telephone example.
Our sample graph transformation system GPh = hTGPh ; fHookOff;
Muteg; Ph i models (part of) the user's interaction with a telephone (see [6]
for the full case study). Type graph TGPh and productions HookOff : (L1
K1 ! R1 ) and Mute : (L2 K2 ! R2) are shown in Figure 1. The typing
is indicated by the inscription of vertices and the phone icon. Using production
HookOff, the user may change the hook status of the phone, while Mute models the phone's reaction of turning the bell o. A derivation step using HookOff
is given by the pushout diagrams (1) and (2).
3 Coalgebraic Semantics of Graph Grammars
In analogy to [4] we present now a coalgebraic loose semantics for graph grammars. A model for a graph grammar GG is a deterministic transition system
with terminal states that at each step outputs a derivation step. Such models are
dened as coalgebras over a suitable functor, and they form a category having a
nal object, the full transition system over G, that includes all nite and innite
derivation sequences.
Let T be an endofunctor on a category C. A T-coalgebra (see e.g., [7]) is a
pair hC; i consisting of an object C and a morphism : C ! T(C) in C. A
morphism f : hC; i ! hC 0; 0i of coalgebras is a morphism f : C ! C 0 in C
such that T(f) = f 0. This denes category CT , called the category of
T-coalgebras.
Denition 1 (category of graph transition systems). Let TG : Set !
Set be the endofunctor dened for each graph transformation system G by
TG (S) = (G) S) + 1 on objects and TG (f) = (idG) f) + id1 on arrows (recall that G) is the set of all derivation steps in G, while 1 denotes the nal object
f?g in Set). For each TG -coalgebra T = hS; step : S ! (G) S)+1i, the partial
functions first and next are dened for all states s 2 S where step(s) = ht; s0 i,
and in this case next(s) = s0 and first(s) = t. The ith iteration nexti of next
is dened by next0 = idS and nexti+1 = next nexti.
A graph transition system over G is a TG -coalgebra such that for all s 2 S,
Out(first(s)) = In(first(next(s))), i.e., the output graph of the rst step equals
the input graph of the second, etc. If GG = hG; G0i is graph grammar, a graph
transition system over GG is a transition system over G where for all s 2 S
there is n 2 IlN (including zero) and s0 2 S such that In(first(s0 )) = G0 and
nextn(s0 ) = s, i.e., all sequences are reachable from the start graph G0.
The category GraTS(GG) of graph transition systems over GG is the full
subcategory of the category SetTG of TG -coalgebras having all graph transition
systems over GG as objects.
4
Intuitively, a graph transition system T = hS; stepi is a deterministic automaton with nal states hS; next : S ! S; first : S ! G) i where step(s) = ?
represents the case where both next(s) and first(s) are undened, i.e., the termination of the automaton [7]. A state transition from s to s0 in S requires no
input, but produces a derivation step in G) as output (observation). Due to
the absence of input, the future behaviour of the system is fully determined by
its current state.
Theorem 2 (nal coalgebra semantics). For each graph grammar GG, the
category GraTS(GG) has a nal object, the full transition system T S (GG).
The unique morphism for some T 2 jGraTS(GG)j is denoted by T : T !
T S (GG).
The category GraTS(GG) of all graph transition systems over the grammar GG is regarded as the category of its \models". In contrast to this loose
semantics, the full transition system T S (GG) is the canonical model of GG
representing all nite and innite derivation sequences in GG.
For any transition system T, the morphism induced by the nality of
T S (GG) is denoted by T : T ! T S (GG). If s0 2 S 0 is a state of T, then
T(s0 ) is a derivation sequence representing the behaviour of s0 . Two states with
the same behaviour are bisimilar [8]. For the full transition system T S (GG)
this means that if two derivation sequences are bisimilar, then they are already
equal, i.e., T S (GG) provides a minimal representation of all possible behaviours
of GG.
It is worth summarising here some advantages of having dened the category
of models GraTS(GG) via coalgebraic instead of algebraic techniques, obtaining
for example an initial model by a free construction, as in [9,10]. First of all, the
free construction in the mentioned papers only generates nite sequences, while
the full transition system contains both nite and innite sequences. But more
importantly, in the algebraic approach all models have to include a homomorphic
image of all the computations of the initial model, thus there are no models
corresponding to some kind of restriction of behaviour. On the contrary, it is wellknown (see e.g., [7,8]) that the coalgebraic framework allows for the denition
of various expressive techniques for considering models that realize only part of
the behaviours of the nal model. Such techniques, based on various kinds of
constraints, will be described in more detail in the next section.
4 Behavioural Constraints for Graph Grammars
In this section we introduce logics of behavioural constraints as syntactical interfaces for the integration of graph grammars with behavioural specication
techniques. The main result of this section (see Theorem 4) shows that the
full transition system can be restricted to those derivation sequences satisfying
the constraints, such that we obtain a nal coalgebra semantics with behavioural
constraints. Particular instances of behavioural constraints, like graphical consistency constraints and temporal logic constraints are studied in the next section.
Denition 3 (logic of behavioural constraints). A logic of behavioural
constraints for graph grammars LOBC = hConstr; j=i is given by a class
Constr(GG) of behavioural constraints and a satisfaction relation j=GG jGraTS(GG)j Constr(GG) for each graph grammar GG, such that for each
c 2 Constr(GG) the empty graph transition system satises c, and satisfaction
is closed under homomorphic images and union of subcoalgebras.1
The satisfaction relation is extended to sets of behavioural constraints C Constr(GG) in the obvious way.
4
A graph grammar with behavioural constraints GC = hGG; C i consists
of a graph grammar GG together with a set of behavioural constraints
1 A transition system T0 = h 0
0 i is a subcoalgebra of T = h
i, written
0
0
0
to . The homomorphic
T T, if and
= jS0 is the restriction of
image of a coalgebra T under a morphism : T ! T0 is the subcoalgebra (T) T0
S ; step
S
S
step
S; step
step
step
f
S
f
determined by f (S ), the set-theoretical image of S . Coalgebras are closed under settheoretical union just like algebras are closed under set-theoretical intersection. It
has been shown in [4] that this also holds for transition systems over grammars.
C Constr(GG). The category of (constrained) graph transition systems
GraTS(GC) over GC is the full subcategory of GraTS(GG) where for each
T 2 jGraTS(GC)j we have that T j=GG C.
The nal coalgebra semantics can be lifted to graph grammars with constraints:
Theorem 4 (nal coalgebra semantics with beh. constraints).
For each graph grammar with constraints GC = hGG; C i, the inclusion functor EGC : GraTS(GC) ! GraTS(GG) has a right adjoint
jC : GraTS(GG) ! GraTS(GC). Consequently, GraTS(GC) has a nal
object, the constrained transition system T S (GG)jC .
4
Roughly speaking, T S (GG)jC is obtained as the largest subcoalgebra of
T S (GG) where all derivation sequences satisfy the constraints C.
In the next section, examples of behavioural constraints are dened following
a generic scheme: Given a graph grammar GG = hG; G0i, constraints and their
satisfaction are dened for derivation sequences in G rst. Then, a logic of
behavioural constraints is derived from this in one of three possible ways, by
universal quantication either (I) over all sequences, (II) over runs (i.e., maximal
sequences), or (III) over initial runs (i.e., runs that start in the start graph of a
grammar). Hence, the same constraint may become more and more \permissive"
when the logic is dened according to II or III instead of I.
Let for each graph transformation system v S S denote the usual prex
order on derivation sequences in G. If T = hST ; stepT i is a transition system over
G, some s 2 ST is called run if for all s0 2 ST , T(s) v T(s0 ) implies that s and
s0 are bisimilar, i.e., T (s) = T (s0 ). Let GG = hG; G0i be a graph grammar
and T 2 jGraTS(GG)j. Then, a run s in T is initial if In(first(s)) = G0.
Proposition 5 (constraints for derivation sequences). Assume for each
graph transformation system G a class of behavioural constraints Constr(G)
together with a satisfaction relation j=seq
G S Constr(G) for derivation sequences (i.e., S is the set of all derivation sequences in G). Then, there is a
logic of behavioural constraints for graph grammars LOBC I = hConstrI ; j=I i
where ConstrI (GG) = Constr(G) and j=IGG jGraTS(GG)j ConstrI (GG)
is dened by T j=IGG c () for all s 2 ST : T (s) j=seq
G c.
Moreover, LOBC II = hConstrII ; j=II i forms a logic of behavioural con(GG)j straints where ConstrII (GG) = Constr(G) and j=II
GG jGraTSseq
Constr(GG) is given by T j=II
c
()
for
all
runs
r
in
T
:
(r)
j
=
T
GG
G c.
III
Finally, a logic of behavioural constraints LOBC = hConstrIII ; j=III i
is dened by ConstrIII (GG) = Constr(G) and j=III
GG)j GG jGraTS(seq
Constr(GG) where T j=III
GG c () for all initial runs r in T: T(r) j=G c. 4
In order to dene a particular logic it is now sucient to provide a notion of
constraints and satisfaction for derivation sequences and to specify if the logic
shall be of type I, II, or III.
5 Temporal Logic Constraints and Other Examples
In this section several logics of behavioural constraints, especially temporal logic
constraints, are dened and illustrated by the telephone example.
Denition 6 (graphical consistency constraints). Assume a graph transformation system G = hTG; P; i and denote by S the set of all derivation sequences in G. Let Constr(G) be the set of all graphical consistency constraints
[11], i.e., injective morphisms c : X ! Y 2 GraphTG . An assignment for X in
a graph G is an injective morphism a : X ! G. It is a solution for c if there is
an injective b : Y ! G such that b c = a. Now, dene j=seq
G S Constr(G)
by s j=seq
c
i
each
assignment
a
:
X
!
In(first(s))
is
a
solution for c. Then,
G
the logic of consistency constraints is dened by CC = hConstrI ; j=I i according
to Proposition 5.
4
For the condition c on the left of Figure 2, a graph satises c if for each
occurrence of X there is also an occurrence of Y , i.e., if a phone is ringing then
the hook must be on. Hence, c is satised by the graph G1 of Figure 1 but not
by the derived graph G2.
c
HookOn
☎
Ring
☎
c
P*
Ph
Ring
X
Y
☎
Ring
☎
X
Gn
c
☎
L
Y
Y
l
(1)
an
+1
Dn
k
r
+1
Gn
+1
an
Ring
X
Fig.2. Graphical consistency constraint, temporal logic constraint with
;
(2)
, and derivation step (bottom span) with assignments.
L
=
H ookOf f M ute
Another logic of behavioural constraints of type I could be dened by positive
and negative application conditions for graph productions in the sense of [12],
where all steps in a derivation sequence would have to satisfy the application
conditions of their corresponding productions.
In the rest of this section we dene temporal logic [13] as a logic of behavioural
constraints using case II and III of Proposition 5. Therefore, let G = hTG; P; i
be a graph transformation system and GG = hG; G0i be a graph grammar.
If Q is a set of atomic propositions, the language of propositional temporal
formulas over a graph transformation system G has the form
::= Q j : j 1 ^ 2 j [L] j hLi
where Q 2 Q, and L P is a set of sequences over production names. As usual,
we abbreviate : _ as ) . The intended meaning of [L] is, that after
every application of a sequence of productions w 2 L, holds. Dually, hLi
means that there exists an application of a production sequence w 2 L such that
holds.
This propositional language is, of course, not able to express any property
of the graphs in a derivation sequences. Hence, we have to combine it with a
calculus like graphical consistency constraints. The resulting language shall be a
graphical temporal logic where consistency constraints for graphs are connected
by the propositional temporal operators dened above.
Assume an interpretation I of all atomic propositions Q 2 Q into graphical
consistency constraints I(Q) over G (cf. Denition 6). Then, for each graph
X 2 jGraphTG j the class ConstrX (G) of interpreted temporal formulas over X
is given by all those formulas of the propositional temporal language where
for all atomic proposition Q in the interpretation of Q is a constraint I(Q) :
X ! Y with domain X.
A sample temporal logic constraint is shown in Figure 2 in the middle. It says
that every state reachable from the initial state by productions in PPh has to
satisfy the following condition: If the phone is ringing, then HookO is applied
before Mute and the phone will stop ringing. In this way we are able to specify,
in a loose, axiomatic way, the eect of the application of a production and the
order in which productions shall be applied.
Temporal formulas are evaluated over transition systems. For each graph
transformation system G = hTG; P; i and each graph X 2 jGraphTG j, the
w
labelled transition system TX (G) = (AX ; (?!
)w2P ) of G over X consists of
{ the set AX of all pairs hs; a0i of a derivation sequence s of G and an assignment (i.e., an injective morphisms) a0 : X ! G0 into the rst graph of the
sequence.
w
{ for each sequence of production names w 2 P a transition relation ?!
AX AX , which is dened inductively by hs; a0 i ?! hs; a0i (where denotes
n+1
the empty sequence) for all hs; a0i 2 AX , and hs; a0 i wp?!
hs0 ; an+1i for
w
00
w = p0 : : :pn 2 P i hs; a0i ?! hs ; ani and for step n + 1 of the sequence
there exists k : X ! Dn+1 such that (1) and (2) in Figure 2 on the right
commute.
Notice that the assignment an+1, if it exists, is uniquely determined by a0 and
the commutativity of (1) and (2).
Now we are able to dene the satisfaction of interpreted temporal formulas
by derivation sequences in G:
w
Denition 7 (logic of temporal constraints). Let TX (G) = (AX ; (?!
)w2P ) be the labelled transition system of a graph transformation system G
over X 2 jGraphTG j and let I be an interpretation of the atomic propositions Q 2 Q. We dene the satisfaction of X-interpreted temporal formulas
2 ConstrX (G) by hs; a0i 2 AX inductively as
{ hs; a0i j=G;I Q i a0 is a solution for I(Q) (cf. Denition 6),
{ hs; a0i j=G;I : i hs; a0i 6j=G;I ,
{ hs; a0i j=G;I 1 ^ 2 i hs; a0 i j=G;I 1 and hs; a0 iwj=G;I 2,
{ hs; a0i j=G;I [L] i for all hs0 ; ani 2 AX , hs; a0i ?! hs0; ani for some w 2 L
implies that hs0 ; ani j=G;I ,
w
{ hs; a0i j=G;I hLi i there exists hs0 ; ani 2 AX with hs; a0i ?!
hs0 ; ani for
0
some w 2 L and hs ; ani j=G;I .
Now, a derivation sequence s satises an X-interpreted temporal formula ,
written s j=seq
G;I , i hs; a0i j=G;I for all assignments a : X ! G0 into the rst
graph of s.
According to Proposition 5, this satisfaction relation denes two logics of behavioural constraints TempII = hConstrII ; j=II i and TempIII =
hConstrIII ; j=III i.
4
By Denition 7 and Theorem 4 this implies that for each graph grammar
with temporal logic constraints GC = hGG; C i, the restriction T S (GG)jC of
the full transition system T S (GG) is a nal object in GraTS(GC).
In TempII the constraints are evaluated w.r.t. behaviours of maximal length
only. It can be shown that this property is closed under reachability, i.e., if s is
maximalin a transition system T, then also next(s) is maximal in T. This means
that all formulas are implicitly quantied over all subsequent states, i.e., there
is the sound inference rule that T j=II implies [P ]T j=II . Contrastingly
for logic TempIII , the evaluation of formulas is restricted to those maximal
sequences that begin in the start graph G0 of a grammar, i.e., the constraints
are less restrictive than in TempII .
6 Conclusion
In this paper graph grammars have been integrated with behavioural constraints.
A nal coalgebraic semantics for graph transformation systems with constraints
in general has been developed and specialised to temporal logic constraints. It
shall be stressed that these notions and results can be generalised to arbitrary
HLR categories [14] in order to transfer them to the transformation other kinds
of objects like hypergraphs, attributed graphs, and relational structures.
Behavioural constraints, as an abstract framework, are comparable to control
conditions in GRACE, a graph and rule based specication language that is
being developed independent of the particular graph transformation approach
[15]. It could be an interesting future topic to extend our approach of coalgebraic
semantics to this language.
Moreover, the coalgebraic semantics has still to be extended to graph grammar morphisms. There are various notions of such morphisms to contend with
in literature ([9,16,10,6]). We only mention that in the category proposed in [6],
pullbacks are used for describing a sort of parallel composition with synchronisation of systems. Since the nal coalgebra semantics can be characterised via a
cofree construction (that preserves pullbacks) this semantics would be compositional with respect to such parallel composition.
The authors are greatful to Fabio Gadducci, Martin Groe-Rhode and
Manuel Koch for their valuable remarks and contributions.
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