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Concurrency Theorems for Non-linear Rewriting Theories

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Abstract

Sesqui-pushout (SqPO) rewriting along non-linear rules and for monic matches is well-known to permit the modeling of fusing and cloning of vertices and edges, yet to date, no construction of a suitable concurrency theorem was available. The lack of such a theorem, in turn, rendered compositional reasoning for such rewriting systems largely infeasible. We develop in this paper a suitable concurrency theorem for non-linear SqPO-rewriting in categories that are quasi-topoi (subsuming the example of adhesive categories) and with matches required to be regular monomorphisms of the given category. Our construction reveals an interesting "backpropagation effect" in computing rule compositions. We derive in addition a concurrency theorem for non-linear double pushout (DPO) rewriting in rm-adhesive categories. Our results open non-linear SqPO and DPO semantics to the rich static analysis techniques available from concurrency, rule algebra and tracelet theory.
Concurrency Theorems for
Non-linear Rewriting Theories?
Nicolas Behr1Q, Russ Harmer2, and Jean Krivine1
1Universit´e de Paris, CNRS, IRIF
8 Place Aur´elie Nemours, 75205 Paris Cedex 13, France
nicolas.behr@irif.fr jean.krivine@irif.fr
2Universit´e de Lyon, ENS de Lyon, UCBL, CNRS, LIP
46 all´ee d’Italie, 69364 Lyon Cedex 07, France
russell.harmer@ens-lyon.fr
Abstract. Sesqui-pushout (SqPO) rewriting along non-linear rules and
for monic matches is well-known to permit the modeling of fusing and
cloning of vertices and edges, yet to date, no construction of a suitable
concurrency theorem was available. The lack of such a theorem, in turn,
rendered compositional reasoning for such rewriting systems largely in-
feasible. We develop in this paper a suitable concurrency theorem for
non-linear SqPO-rewriting in categories that are quasi-topoi (subsuming
the example of adhesive categories) and with matches required to be reg-
ular monomorphisms of the given category. Our construction reveals an
interesting “backpropagation effect” in computing rule compositions. We
derive in addition a concurrency theorem for non-linear double pushout
(DPO) rewriting in rm-adhesive categories. Our results open non-linear
SqPO and DPO semantics to the rich static analysis techniques available
from concurrency, rule algebra and tracelet theory.
1 Introduction
Sesqui-pushout (SqPO) graph transformation was introduced [16] as an exten-
sion of single-pushout rewriting that accommodates the possibility of non-input-
linear3rules. The result of such a rewrite is specified abstractly by the notion
of final pullback complement (FPC) [20], a categorical generalization of the no-
tion of set difference: the FPC of two composable arrows, f:ABand
g:BDis the largest, i.e. least general, Ctogether with arrows g0:AC
and f0:CDfor which the resulting square is a pullback (PB). The exten-
sion of graph transformation to input-non-linear rules allows for the expression
of the natural operation of the cloning of a node, or an edge (when the lat-
ter is meaningful), as explained in [16,17,13] . More recently, such rules have
?This is an extended version (containing additional technical appendices) of a paper
with the same tittle accepted for ICGT 2021.
3In this paper, we follow the conventions of compositional rewriting theory [8], i.e.,
we speak of “input”/“output” motifs of rules, as opposed to “left”/“right” motifs in
the traditional literature [21].
arXiv:2105.02842v1 [cs.LO] 6 May 2021
2 N. Behr, R. Harmer and J. Krivine
also been used to express operations such as concept refinement in schemata for
graph databases [10] and, more generally, in graph-based knowledge representa-
tion [29]. In combination with output-non-linear rules, as for (non-linear) double-
or single-pushout rewriting, SqPO thus allows the expression of all the natural
primitive operations on graphs: addition and deletion of nodes and edges; and
cloning and merging of nodes and edges.
In this paper, we study the categorical structure required in order to support
SqPO rewriting and establish that quasi-topoi [1,14,15,33,26] naturally possess
all the necessary structure to express the effect of SqPO rewriting and to prove
the concurrency theorem for fully general non-linear rules. This significantly gen-
eralizes previous results on concurrency theorems for linear SqPO-rewriting over
adhesive categories [2] and for linear SqPO-rewriting for linear rules with condi-
tions in M-adhesive categories [8,7]. In terms of SqPO-rewriting for generic rules,
previous results were rather sparse and include work on polymorphic SqPO-
rewriting [36] and on reversible SqPO rewriting [18,30], where [30] in particular
introduced a synthesis (but not an analysis) construction for reversible non-linear
SqPO rules without application conditions which motivated the present paper.
An interesting technical aspect of basing our constructions on quasi-topoi
concerns the rewriting of simple directed graphs, which constitutes one of the
running examples in this paper: unlike the category of directed multigraphs
(which constitutes one of the prototypical examples of an adhesive category [34]),
the category of simple graphs is neither adhesive nor quasi-adhesive [33], but it
is in fact only a quasi-topos [1,33], and as such also an example of an rm-quasi-
adhesive [26] and of an M-adhesive category [23,22,31,24].
Our proof of the concurrency theorem relies on the existence of certain struc-
tures in quasi-topoi that, to the best of our knowledge, have not been previously
noted in the literature (cf. Section 2.2): restricted notions of multi-sum and multi-
pushout complement (mPOC), along the lines of the general theory of multi-(co-)
limits due to Diers [19], and a notion of FPC-pushout-augmentation (FPA). The
notion of multi-sum provides a generalization of the property of effective unions
(in adhesive categories) that guarantees that all necessary monos are regular.
The notions of mPOC and FPA handle the “backward non-determinism” intro-
duced by non-linear rules: given a rule and a matching from its output motif, we
cannot—unlike with linear or reversible non-linear rules—uniquely determine a
matching from the input motif of the rule.
Related work Conditions under which FPCs are guaranteed to exist have been
studied in [20], and more concretely and of particular relevance to our approach
in [17], which provides a direct construction assuming the existence of appropri-
ate partial map classifiers [31,15]. We make additional use of these partial map
classifiers in order to construct mPOCs in a quasi-topos (Section 2.2). Our con-
struction is a mild, but necessary for our purposes, generalization of the notion
of minimal pushout complement defined in [13] that requires the universal prop-
erty with respect to a larger class of encompassing pushouts (POs)—precisely
analogous to the definition of FPC. However, there is the additional complexity
Concurrency Theorems for Non-linear Rewriting Theories 3
that, for our purposes, PO complements are not uniquely determined, and we
must therefore specify a family of solutions that collectively satisfy this universal
property (`a la Diers [19]). We also exploit the epi-regular mono factorization [1]
in quasi-topoi in order to construct multi-sums—with respect to co-spans of
regular monos—and FPAs. Our overall approach relates closely to the work of
Garner and Lack on rm-quasi-adhesive categories [26], which provide an abstract
setting for graph transformation that accommodates the technical particulari-
ties of simple graphs—notably the fact that the ‘exactness’ direction of the van
Kampen condition fails in general for cubes where the vertical arrows, between
the two PO faces, are not regular.
2 Quasi-topoi
In this section, we will demonstrate that quasi-topoi form a natural setting
within which non-linear sesqui-pushout (SqPO) rewriting is well-posed. Quasi-
topoi have been considered in the context of rewriting theories as a natural
generalization of adhesive categories in [35]. While several adhesive categories
of interest to rewriting are topoi, including in particular the category Graph
of directed multigraphs (cf. Definition 4), it is not difficult to find examples of
categories equally relevant to rewriting theory that fail to be topoi. A notable
such example is the category SGraph of directed simple graphs (cf. Definition 5).
We will demonstrate that quasi-topoi combine all technical properties nec-
essary such as to admit the construction of non-linear sesqui-pushout semantics
over them. We will first list these abstract properties, and illustrate them via
the two aforementioned paradigmatic examples of topoi and quasi-topoi.
Let us first recall a number of results from the work of Cockett and Lack [14,15]
on restriction categories. We will only need a very small fragment of their theory,
namely the definition and existence guarantees for M-partial map classifiers, so
we will follow mostly [17]. We will in particular not be concerned with the notion
of M-partial maps itself.
Definition 1 ([14], Sec. 3.1). For a category C, a stable system of monics
Mis a class of monomorphisms of Cthat (i)includes all isomorphisms, (ii)is
stable under composition, and (iii)is stable under pullbacks (i.e., if (f0, m0)is
a pullback of (m, f )with m∈ M, then m0∈ M). Throughout this paper, we will
reserve the notation for monics in M, and for generic monics.
Definition 2 ([17], Sec. 2.1; compare [15], Sec. 2.1). For a stable system
of monics Min a category C, an M-partial map classifier (T, η)is a functor
T:CCand a natural transformation η:IDC
.
Tsuch that
1. for all Xobj(C),ηX:XT(X)is in M
2. for each span (Am
Xf
B)with m∈ M, there exists a unique morphism
Aϕ(m,f)
T(B)such that (m, f)is a pullback of (ϕ(m, f), ηB).
4 N. Behr, R. Harmer and J. Krivine
Proposition 1 ([17], Prop. 6). For every M-partial map classifier (T, η ),
Tpreserves pullbacks, and ηis Cartesian, i.e., for each Xf
Y,(ηx, f )is a
pullback of (T(f), ηY).
Definition 3 ([33], Def. 9). A category Cisaquasi-topos iff
1. it has finite limits and colimits
2. it is locally Cartesian closed
3. it has a regular-subobject-classifier.
Based upon a variety of different results from the rich literature on quasi-
topoi, we will now exhibit that quasi-topoi indeed possess all technical properties
required in order for non-linear SqPO-rewriting to be well-posed:
Corollary 1. Every quasi-topos Cenjoys the following properties:
It has (by definition) a stable system of monics M=rm(C)(the class of reg-
ular monos), which coincides with the class of extremal monomorphisms [1,
Cor. 28.6], i.e., if m=fefor mrm(C)and eepi(C), then eiso(C).
It has (by definition) a M-partial map classifier (T, η).
It is rm-quasi-adhesive, i.e., it has pushouts along regular monomorphisms,
these are stable under pullbacks, and pushouts along regular monos are pull-
backs [26].
It is M-adhesive [31, Lem. 13].
For all pairs of composable morphisms Af
Band Bm
Cwith m∈ M,
there exists a final pullback-complement (FPC) An
Fg
C, and with
n∈ M ([17, Thm. 1]; cf. Theorem 2).
It possesses an epi-M-factorization [1, Prob. 28.10]: each morphism Af
B
factors as f=me, with morphisms Ae
¯
Bin epi(C)and ¯
Bm
Ain M
(uniquely up to isomorphism in ¯
B).
It possesses a strict initial object obj(C)[32, A1.4], i.e., for every
object Xobj(C), there exists a morphism iX:X, and if there exists
a morphism X, then X
=.
If in addition the strict initial object is M-initial, i.e., if for all objects
Xobj(C)the unique morphism iX:Xis in M, then Chas disjoint
coproducts, i.e., for all X, Y obj(C), the pushout of the M-span XY
is XX+YY(cf. [37, Thm. 3.2], which also states that this condition is
equivalent to requiring Ctobeasolid quasi-topos), and the coproduct injections
are M-morphisms as well. Finally, if pushouts along regular monos of Care
van Kampen, Cisarm-adhesive category [26, Def. 1.1].
2.1 The categories of directed multi- and simple graphs
Throughout this paper, we will illustrate our constructions with two prototypical
examples of (quasi-)topoi, namely categories of two types of directed graphs.
Concurrency Theorems for Non-linear Rewriting Theories 5
Definition 4. The category Graph of directed multigraphs is defined as the
presheaf category Graph := (Gop Set), where G:= (·)is a category with
two objects and two morphisms [34]. Objects G= (VG, EG, sG, tG)of Graph are
given by a set of vertices VG, a set of directed edges EGand the source and target
functions sG, tG:EGVG. Morphisms of Graph between G, H obj(Graph)
are of the form ϕ= (ϕV, ϕE), with ϕV:VGVHand ϕE:EGEHsuch
that ϕVsG=sHϕEand ϕVtG=tHϕE.
Definition 5. The category SGraph of directed simple graphs4is defined as
the category of binary relations BRel
=Set // ∆ [33]. Here, :Set Set is
the pullback-preserving diagonal functor defined via ∆X := X×X, and Set // ∆
denotes the full subcategory of the slice category Set/∆ defined via restriction
to objects m:X∆X that are monomorphisms. More explicitly, an object of
Set // ∆ is given by S= (V, E , ι), where Vis a set of vertices, Eis a set of
directed edges, and where ι:EV×Vis an injective function. A morphism
f= (fV, fE)between objects Sand S0is a pair of functions fV:VV0and
fE:EE0such that ι0fE= (fV×fV)ι(see (2)).
These two categories satisfy the following well-known properties:
Theorem 1. The category Graph is an adhesive category and (by definition)
apresheaf topos [34] (and thus in particular a quasi-topos), with strict-initial
object = (,,∅→∅,∅→∅)the empty graph, and with the following additional
properties:
Morphisms are in the classes mono(Graph)/epi(Graph)/iso(Graph)if they
are component-wise injective/surjective/bijective functions, respectively. All
monos in Graph are regular, and Graph therefore possesses an epi-mono-
factorization.
For each Gobj(Graph)[17, Sec. 2.1], ηG:GT(G)is defined as the
embedding of Ginto T(G), where T(G)is defined as the graph with vertex
set V0
G:= VG] {}and edge set EG]E0
G. Here, E0
Gcontains one directed
edge en,p :vnvpfor each pair of vertices (vn, vp)V0
G×V0
G.
The category SGraph is not adhesive, but it is a quasi-topos [33], and with the
following additional properties:
In SGraph [33] (compare [13, Prop. 9]), morphisms f= (fV, fE)are monic
(epic) if fVis monic (epic), while isomorphisms satisfy that both fVand
fEare bijective. Regular monomorphisms in SGraph are those for which
(ι, fE)is a pullback of ((fV), ι0)[33, Lem. 14(ii)], i.e., a monomorphism is
regular iff it is edge-reflecting. As is the case for any quasi-topos,SGraph
possesses an epi-regular mono-factorization.
4Some authors prefer to not consider directly the category BRel, but rather define
SGraph as some category equivalent to BRel, where simple graphs are of the form
hV, E iwith EV×V. This is evidently equivalent to directly considering BRel,
whence we chose to not make this distinction in this paper.
6 N. Behr, R. Harmer and J. Krivine
The regular mono-partial map classifier (T, η)of SGraph is defined as fol-
lows [1, Ex. 28.2(3)]: for every object S= (V, E, ι)obj(SGraph),
T(S) := (V?=V] {}, E?=E](V× {})]({} × V)] {(, )}, ι?),(1)
where ι?is the evident inclusion map, and moreover ηS:ST(S)is the
(by definition edge-reflecting) inclusion of Sinto T(S).
SGraph possesses a regular mono-initial object = (,,∅→∅).
Proof. While most of these results are standard, we briefly demonstrate that the
epi-regular mono-factorization of SGraph [33] is “inherited” from the epi-mono-
factorization of the adhesive category Set. To this end, given an arbitrary mor-
phism f= (fV, fE) in SGraph as on the left of (2), the epi-mono-factorization
fV=mVeVlifts via application of the diagonal functor to a decomposition
of the morphism fV×fV. Pulling back ((mv), ι0) results in a span (˜ι, f00
E) and
(by the universal property of pullbacks) an induced morphism f0
Ethat makes
the diagram commute. By stability of monomorphisms under pullbacks, ˜ιis a
monomorphism, thus the square marked () precisely constitutes the data of a
regular monomorphism in SGraph, while the square marked () is an epimor-
phism in SGraph (since eVepi(Set)).
E E0E˜
EE0
V×V V 0×V0V×Vim(fV)×im(fV)V0×V0
V V 0Vim(fV)V0
eVmV
ι
eV×eVmV×mV
!f0
Ef00
E
ι0
˜ι
fE
fV
fV×fV
fV
fE
ιι0PB
()
()
(2)
2.2 FPCs, M-multi-POCs, M-multi-sums and FPAs
Compared to compositional SqPO-type rewriting for M-linear rules [2], in the
generic SqPO-type setting we require both a generalization of the concept of
pushout complements that forgoes uniqueness, as well as a certain form of FPC-
augmentation. To this end, it will prove useful to recall from [17] the following
constructive result:
Theorem 2 ([17], Thm. 1). For a category Cwith M-partial map classifier
(T, η), the final pullback complement (FPC) of a composable sequence of arrows
Af
Band Bm
Cwith m∈ M is guaranteed to exist, and is constructed via
the following algorithm:
1. Let ¯m:= ϕ(m, idB)(i.e., the morphism that exists by the universal property
of (T, η), cf. square (1) below).
Concurrency Theorems for Non-linear Rewriting Theories 7
2. Construct T(A)¯n
Fg
Cas the pullback of T(A)T(f)
T(B)¯m
C(cf.
square (2) below); by the universal property of pullbacks, this in addition
entails the existence of a morphism An
F.
Then (n, g)is the FPC of (f, m), and nis in M.
A B
A B
FCB
C
T(A)T(B)B
f
m
f
m
!n
g
¯n
T(f)
¯m
ηA
ηb
m
(1)
(2)
PB PB
(3)
This guarantee for the existence of FPCs will prove quintessential for con-
structing M-multi-pushout complements, which are defined as follows:
Definition 6. For a category Cwith an M-partial map classifier, the M-multi-
pushout complement (M-multi-POC) P(f, b)of a composable sequence of mor-
phisms Af
Band Bb
Dwith b∈ M is defined as
P(f, b) := {(Aa
P , P d
D)mor(C)2|a M ∧ (d, b) = PO(a, f)}.(4)
Proposition 2. In a quasi-topos Cand for M=rm(C)the class of regular
monomorphisms, let P(f, b)be an M-multi-POC.
Universal property of P(f, b): for every diagram such as in (5)(i) where
(1) + (2) is a pushout along an M-morphism n, and where m=m0b
for some m0, b ∈ M, there exists an element (a, d)of P(f, b)and an M-
morphism p∈ M such that the diagram commutes and (2) is a pushout.
Moreover, for any p0∈ M and for any other element (a0, d0)of P(f , b)with
the same property, there exists an isomorphism δiso(C)such that δa=a0
and d0δ=d.
Algorithm to compute P(f, b):
1. Construct (n, g)in diagram (5)(ii) by taking the FPC of (f , b).
2. For every pair of morphisms (a, p)such that a∈ M and ap=n,
take the pushout (1), which by universal property of pushouts induces an
arrow De
C; if eiso(C),(a, d)is a contribution to the M-multi-POC
of (f, b).
A B A B
PDPD
Q E FC
(i) (ii)
f
n
aPO
d
p
b
e
m
g
f
b!a
!d
m0
q
n m
!p
(1)
(2)
(1)
(2)
(5)
8 N. Behr, R. Harmer and J. Krivine
Proof. The universal property of P(f, b) follows from pushout-pullback decom-
position:pushouts along M-morphisms are pullbacks, so (1) + (2) is a pullback;
taking the pullback (p, d) of (q, m0) yields by the universal property of pullbacks a
morphism a(which is unique up to isomorphism), and thus by pullback-pullback
decomposition that (1) and (2) are pullbacks. By stability of M-morphisms un-
der pullbacks, both aand pare in M, and finally by pushout-pullback decom-
position, both (1) and (2) are pushouts. This proves that (a, d) is in P(f, b).
To prove that the algorithm provided indeed computes P(f, b), note first that
by the universal property of FPCs, whenever in a diagram as in (5)(ii) we have
that D
=Cand b∈ M, since pushouts along M-morphisms are pullbacks,
square (1) is a pullback, which entails by the universal property of FPCs that
there exists a morphism psuch that pa=n.Bystability of M-morphisms under
pullbacks, we find that amust be in M, so indeed every possible contribution
to P(f, b) must give rise to a diagram as in (5)(ii), which proves the claim.
An example of an M-multi-POC construction both in SGraph and in Graph
is given in the diagram below. Note that in Graph, the M-multi-POC does not
contain the FPC contribution (since in Graph the pushout of the relevant span
would yield to a graph with a multi-edge).
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B
<latexit sha1_base64="/B1XSKVJvrHoLKjblLBbR7G85Ss=">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</latexit>
A
<latexit sha1_base64="XOesLas+EHE8gkjtGzbnI6d2z7s=">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</latexit>
?
<latexit sha1_base64="+9t7lfzG4m39xKwQx+QGWNhDGvE=">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</latexit>
B
<latexit sha1_base64="9GACVUnEyIndndtYcl1Nzl5VkoE=">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</latexit>
A
<latexit sha1_base64="XOesLas+EHE8gkjtGzbnI6d2z7s=">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</latexit>
?
<latexit sha1_base64="C+zZ5TkOmS0L/IzYSdAVwmbrSR0=">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</latexit>
¯n
<latexit sha1_base64="lmCMb+5nFpnzSgh3eVIznRUNlNw=">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</latexit>
T(f)
<latexit sha1_base64="kOrmCjedly7mupjHaquchnGp/qI=">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</latexit>
n
PBPB
FPC
POC
POC
<latexit sha1_base64="kEXi/4JbMwFIWRtZUhOAfb1thUw=">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</latexit>
C
(1)(2)
<latexit sha1_base64="lVTyXMpW+Sp0rGtym6mBjAXb8S0=">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</latexit>
f
<latexit sha1_base64="E0uc/qJp9at1SOW/Ck7VFUNtSf4=">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</latexit>
g
<latexit sha1_base64="+9t7lfzG4m39xKwQx+QGWNhDGvE=">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</latexit>
B
<latexit sha1_base64="npH7tXZjA5DYXKBTMyajVTP4FXM=">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</latexit>
B
<latexit sha1_base64="/B1XSKVJvrHoLKjblLBbR7G85Ss=">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</latexit>
A
<latexit sha1_base64="XOesLas+EHE8gkjtGzbnI6d2z7s=">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</latexit>
?
<latexit sha1_base64="+9t7lfzG4m39xKwQx+QGWNhDGvE=">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</latexit>
B
<latexit sha1_base64="9GACVUnEyIndndtYcl1Nzl5VkoE=">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</latexit>
A
<latexit sha1_base64="XOesLas+EHE8gkjtGzbnI6d2z7s=">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</latexit>
?
<latexit sha1_base64="C+zZ5TkOmS0L/IzYSdAVwmbrSR0=">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</latexit>
¯n
<latexit sha1_base64="lmCMb+5nFpnzSgh3eVIznRUNlNw=">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</latexit>
T(f)
<latexit sha1_base64="kOrmCjedly7mupjHaquchnGp/qI=">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</latexit>
n
PBPB
FPC
POC
POC
<latexit sha1_base64="kEXi/4JbMwFIWRtZUhOAfb1thUw=">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</latexit>
C
(1)(2)
POC
(6)
Definition 7 (M-FPC-augmentations). In a quasi-topos5Cwith M=rm(C),
consider a pushout square along an M-morphism such as square (1) in the dia-
gram below (where α, ¯α∈ M):
A B
C D
F E
α
a
¯a
¯α
e
e¯α
n
f
nα
(2)
PO
(1)
(7)
We define an M-FPC augmentation (FPA) of the pushout square (1) as a di-
agram formed from an epimorphism eepi(C)and that satisfies the following
properties:
5As demonstrated in [25, Fact 3.4], every finitary M-adhesive category Cpossesses
an (extremal E,M)-factorization, so if Cis known to possess FPCs as required
by the construction, this might allow to generalize the M-FPC-PO-augmentation
construction to this setting.
Concurrency Theorems for Non-linear Rewriting Theories 9
The morphism e¯αis an M-morphism.
(¯α, idB)is a pullback of (e, e ¯α).
Square (1) + (2) is an FPC, and the induced morphism nthat exists6by the
universal property of FPCs, here w.r.t. the FPC (nα, f)of (a, e ¯α), is an
M-morphism.
For a pushout as in (1), we denote by FPA(α, a)its class of FPAs:
FPA(α, a) := {(n, f, e)|eepi(C)e¯α, n ∈ M∧(f, nα) = FPC(a, e¯α)}(8)
As induced by the properties of pushouts and of FPCs, FPAs are defined up to
universal isomorphisms (in D,Eand F), and for a given pushout square there
will in general exist multiple non-isomorphic such augmentations.
The final technical ingredient for our rewriting theoretic constructions is a
notion of multi-sum adapted to the setting of quasi-topoi, a variation on the
general theory of multi-(co-)limits due to Diers [19].
Definition 8. In a quasi-topos C, the multi-sum PM(A, B)of two objects
A, B obj(C)is defined as a family of cospans of regular monomorphisms
Af
Yg
Bwith the following universal property: for every cospan Aa
Zb
B
with a, b rm(C), there exists an element Af
Yg
Bin PM(A, B )and a
regular monomorphism Yy
Zsuch that a=yfand b=yg, and moreover
(f, g)as well as yare unique up to universal isomorphisms.
X
A B A B
A+B A +B
Y Y P
Q
Z Z
(i) (ii)
inAinB
ab
e
m
yAyB
[a,b]
a
yA
m
e
b
pA
pB
inAinB
yB
ιAιB
xAxB
ιX
eP
q
mQ
qB
qA
!z
(9)
6Note that square (1) pasted with the pullback square formed by the morphisms
α, idB, e, e ¯αyields a pullback square that is indeed of the right form to warrant
the existence of a morphism ninto the FPC square (1) + (2).
10 N. Behr, R. Harmer and J. Krivine
Lemma 1. If Cis a quasi-topos, the multi-sum PM(A, B)arises from the epi-
M-factorization of C(for M=rm(C); compare [29]).
Existence: Let AinA
A+BinB
Bbe the disjoint union of Aand B.
Then for any cospan Aa
Zb
Bwith a, b ∈ M, the epi-M-factorization of
the induced arrow A+B[a,b]
Zinto an epimorphism A+Be
Yand an
M-morphism Ym
Zyields a cospan (yA=einA, yB=einB), which by
the decomposition property of M-morphisms is a cospan of M-morphisms
(cf. (9)(i)).
Construction: For objects A, B obj(C), every element AqA
QqB
B
in PM(A, B)is obtained from a pushout of some span AxA
XxB
Bwith
xA, xB∈ M and a morphism Pq
Qin mono(C)epi(C)(cf. (9)(ii)).
Proof. See Appendix B.
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G
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H
<latexit sha1_base64="PAePFhWLDknrShoufrjOi3bvy10=">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</latexit>
S0
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S1
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S2
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S3
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S4
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I
Since in an adhesive category all monos are regular [34],
in this case the multi-sum construction simplifies to the
statement that every monic cospan can be uniquely fac-
torized as a cospan obtained as the pushout of a monic
span composed with a monomorphism. It is however worth-
while emphasizing that for generic quasi-topoi Cone may
have M 6=mono(C), as is the case in particular for the
quasi-topos SGraph of simple graphs. We illustrate this
phenomenon in the diagram on the right via presenting the
multi-sum construction for A=B=. Note in particular
the monic-epis that extend the two-vertex graph S0into
the graphs S1,S2and S3, all of which have the same vertices as S0(recalling
that a morphism in SGraph is monic/epic if it is so on vertices), yet additional
edges, so that in particular none of the morphisms S0Sjfor j= 1,2,3 is
edge-reflecting.
3 Non-linear sesqui- and double-pushout rewriting
In much of the traditional work on graph- and categorical rewriting theories [21],
while it was appreciated early in its development that in particular SqPO-
rewriting permits the cloning of subgraphs [16], and that both SqPO- and DPO-
semantics permit the fusion of subgraphs (i.e. via input-linear, but output-non-
linear rules), the non-uniqueness of pushout complements along non-monic mor-
phisms for the DPO- and the lack of a concurrency theorem in the SqPO-case
in general has prohibited a detailed development of non-linear rewriting theo-
ries to date. Interestingly, the SqPO-type concurrency theorem for linear rules
as developed in [2] exhibits the same obstacle for the generalization to non-
linear rewriting as the DPO-type concurrency theorem, i.e., the non-uniqueness
Concurrency Theorems for Non-linear Rewriting Theories 11
of certain pushout complements. Our proof for non-linear rules identifies in ad-
dition a new and highly non-trivial “backpropagation effect”, which will be high-
lighted in Section 4. It may be worthwhile emphasizing that there exists previ-
ous work that aimed at circumventing some of the technical obstacles of non-
linear rewriting either via specializing the semantics e.g. from double pushout
to a version based upon so-called minimal pushout complements [13], or from
sesqui-pushout to reversible SqPO-semantics [18,30] or other variants such as
AGREE-rewriting [17]. In contrast, we will in the following introduce the “true”
extensions of both SqPO- and DPO-rewriting to the non-linear setting, with our
constructions based upon multi-sums, multi-POCs and FPAs.
Definition 9. General SqPO-rewriting semantics over a quasi-topos C:
The set of SqPO-admissible matches of a rule rule r= (OKI)
span(C)into an object Xobj(C)is defined as
MSqP O
r(X) := {Im
X|mrm(C)}.(10)
ASqPO-type direct derivation7of Xobj(C)with rule ralong m
MSqP O
r(X)is defined as a diagram in (11), where (1) is formed as an FPC,
while (2) is formed as a pushout.
O K I
rm(X)¯
XX
m
o
¯m
¯o¯
i
i
m
(2) (1)
(11)
The set of SqPO-type admissible matches of rules r2, r1span(C)(also
referred to in the literature as dependency relations) is defined as
MSqP O
r2(r1) := {(j2, j1,¯
j1,¯o1,¯
¯
j1,¯
¯
i1, ι21)|
(j2, j1)XM(I2, O1)(¯
j1,¯o1)∈ P(o1, j1)
(¯
¯
j1,¯
¯
i1, ι21)FPA(¯
j1, i1)},
(12)
where equivalence is defined up to the compatible universal isomorphisms of
multi-sums, multi-POCs and FPAs (see below).
An SqPO-type rule composition of two general rules r1, r2span(C)along
an admissible match µ∈ MSqP O
r2(r1)is defined via a diagram as in (13)
below, where (going column-wise from the left) squares (22),(6), and (4) are
pushouts, (11)is the multi-POC element specified as part of the data of the
match, (21)and (3) form an FPA-diagram as per the data of the match, and
7Note that this part of the definition of general SqPO-semantics coincides precisely
with the original definition of [16].
12 N. Behr, R. Harmer and J. Krivine
finally (12)and (5) are FPCs:
O2K2I2O1K1I1
O21 K2J21 K1I21
O21 K2¯
J21 K1I21
i1
o1
i2
o2
j2j1
j21
¯
j2
¯
i2
j
2
¯o2
ι
21 ¯
¯
j2
¯
¯
i2
¯
¯o2
¯o1
¯
j1
¯
i1
j
1
¯
¯o1¯
¯
i1
¯
¯
j1ι21
ι1
(12)
(22)(11)(21)
(3)
(4)(5)
(6)
(13)
We then define the composite rule via span composition:
r2
µ
^r1:= (O21 K2J21)(J21 K1I21 ) (14)
Definition 10. General DPO-rewriting semantics over an rm-adhesive category
C:
The set of DPO-admissible matches of a rule rule r= (OKI)
span(C)into an object Xobj(C)is defined as
MDP O
r(X) := {(m, ¯m,¯
i)|mrm(C)( ¯m,¯
i)∈ P(i, m)}.(15)
ADPO-type direct derivation of Xobj(C)with rule ralong mMDPO
r(X)
is defined as a diagram in (11), where (1) is the multi-POC element chosen
as part of the data of the match, while (2) is formed as a pushout.
The set of DPO-type admissible matches of rules r2, r1span(C)(also
referred to as dependency relations) is defined as
MDP O
r2(r1) := {(j2, j1,¯
j2,¯
i2,¯
j1,¯o1)|
(j2, j1)XM(I2, O1)
(¯
j2,¯
i2)∈ P(i2, j2)(¯
j1,¯o1)∈ P(o1, j1)},
(16)
where equivalence is defined up to the compatible universal isomorphisms of
multi-sums and multi-POCs (see below).
ADPO-type rule composition of two general rules r1, r2span(C)along an
admissible match µ∈ MDP O
r2(r1)is defined via a diagram as in (17) below,
where (12)and (11)are the multi-POC elements chosen as part of the data
of the match, while (22)and (21)are pushouts:
O2K2I2O1K1I1
O21 K2J21 K1I21
i1
o1
i2
o2
j2j1
¯
j2
¯
i2
j
2
¯o2¯o1
¯
j1
¯
i1
j
1
(12)
(22)(11)(21)
(17)
We then define the composite rule via span composition:
r2
µ
Jr1:= (O21 K2J21)(J21 K1I21 ) (18)
Concurrency Theorems for Non-linear Rewriting Theories 13
The precise reasons for the definitions of SqPO- and DPO-semantics for generic
rules and regular monos as matches will only become evident via the concurrency
theorems that will be developed in the following sections.
Let us illustrate the notion of SqPO-type rule composition, as given in Def-
inition 9, with the following example in the setting of directed multi-graphs.
Note that, since this is an adhesive category, all monos are automatically reg-
ular and we therefore have no need to restrict matches to being edge-reflecting
monomorphisms.
(19)
In this example, we have two rules. The first clones a node8, but not its
incident edge, then adds a new edge between the original node and its clone and
merges the blue node with the original node. The second rule deletes a node
and then merges two nodes. The given applications to the graphs X0and X1
illustrate some of the idiosyncrasies of SqPO-rewriting:
Since the node of X0that is being cloned possesses a self-loop, the result of
cloning is two nodes, each with a self-loop, with one edge going each way
between them.
In the application of the second rule to X1, we see the side-effect whereby
all edges incident to the deleted node are themselves deleted (as also occurs
in SPO-, but not in DPO-rewriting).
The overall effect of the two rewrites can be seen in X2; as usual, this depends
on the overlap between the images of O1and I2in X1. This overlap is precisely
8Note that we have drawn the rule from right to left so that the input, sometimes
called the left-hand side, of the rule is the topmost rightmost graph. Note also that
the structure of the homomorphisms may be inferred from the node positions, with
the exception of the vertex clonings that are explicitly mentioned in the text.
14 N. Behr, R. Harmer and J. Krivine
the multi-sum element J21. Since our example is set in an adhesive category,
this can be most easily computed by taking the PB of m
1and m2and then the
PO of the resulting span. The PO that defines the rewrite from X0to X1can
now be factorized by computing the PB of j21 and the arrow from X0to X1;
this determines K1and its universal arrow from K1with consequence that (11)
and (21) are both POs. Let us note that K1is the appropriate member of the
multi-POC, as determined by the particular structure of X0.
The PO (31) induces a universal arrow from I21 to X0; but an immediate
inspection reveals that this homomorphism is not a mono (nor an epi in this
case). As such, we cannot hope to use I21 as the input/left hand side of the
composite rule. Furthermore, we find that the square (41) is neither a PB nor a
PO. However, the FPA I21 resolves these problems by enabling a factorization
of this square, giving rise to a monomorphism m21 into X0, where (400
1) and
(31) + (40
1) are PBs and indeed FPCs. This factorization, as determined by e21,
can now be back-propagated to factorize (21) into POs (20
1) and (200
1) which gives
rise to an augmented version J21 of the multi-sum object in the middle. Note
moreover that the effect of back-propagation concerns also the contribution of
the second rule in the composition: the final output motif contains an extra self-
loop (compared to the motif O21 defined by the PO (32)), which is induced by
the extra self-loop of J21 that appears due to back-propagation.
We may then compute the composite rule via taking a pullback to obtain
K21, yielding in summary the rule O21 K21 I21 . Performing the remaining
steps of the “synthesis” construction of the concurrency theorem (compare Ap-
pendix C.1) then amounts to constructing the commutative cube in the middle
of the diagram, yielding the FPC (71) and the PO (72), and thus finally the
one-step SqPO-type direct derivation from X0to X2along the composite rule
O21 K21 I21.
Let us finally note, as a general remark, that if the first rule in an SqPO-
type rule composition is output- (or right-) linear then the POC is uniquely
determined; and if it is input- (or left-) linear then the PO (31) is also an FPC
and (41) is a PB, by Lemma 2(h) of [2]. In this case, the FPA is trivial, and
consequently so is the back-propagation process. Our rule composition can thus
be seen as a conservative extension of that defined for linear rules in [2].
4 Concurrency theorem for non-linear SqPO rewriting
Part of the reason that a concurrency theorem for generic SqPO-rewriting had
remained elusive in previous work concerns the intricate nature of the inter-
play between multi-sums, multi-POCs and FPAs as seen from the definition of
rule compositions according to Definition 9, which is justified via the following
theorem, constituting the first main result of the present paper:
Theorem 3. Let Cbeaquasi-topos, let X0obj(C)be an object, and let
r2, r1span(C)be two (generic) rewriting rules.
Concurrency Theorems for Non-linear Rewriting Theories 15
1. Synthesis: For every pair of admissible matches m1MSqP O
r1(X0)and m2
MSqP O
r2(r1m1(X0)), there exist an admissible match µ∈ MSqP O
r2(r1)and an
admissible match m21 MSqP O
r21 (X0)(for r21 the composite of r2with r1
along µ) such that r21m21 (X0)
=r2m2(r1m1(X0)).
2. Analysis: For every pair of admissible matches µ∈ MSqP O
r2(r1)and m21
MSqP O
r21 (X0)(for r21 the composite of r2with r1along µ), there exists a pair
of admissible matches m1MSqP O
r1(X0)and m2MSqP O
r2(r1m1(X0)) such
that r2m2(r1m1(X0))
=r21m21 (X0).
3. Compatibility: If in addition Cis finitary [25, Def. 2.8], i.e., if for every
object of Cthere exist only finitely many regular subobjects up to isomor-
phisms, the sets of pairs of matches (m1, m2)and (µ, m21)are isomorphic
if they are suitably quotiented by universal isomorphisms, i.e., by univer-
sal isomorphisms of X1=r1m1(X0)and X2=r2m2(X1)for the set of pairs
(m1, m2), and by the universal isomorphisms of multi-sums, multi-POCs and
FPAs for the set of pairs (µ, m21), respectively.
Proof. See Appendix C
5 Concurrency theorem for non-linear DPO-rewriting
The well-known and by now traditional results on concurrency in DPO-type
semantics by Ehrig et al. were formulated for M-linear rules in M-adhesive cat-
egories (albeit possibly for non-monic matches; cf. [21, Sec. 5] for the precise
details), and notably the non-uniqueness of pushout complements along non-
linear morphisms posed the main obstacle for extending this line of results to
non-linear DPO rewriting. As we will demonstrate in this section, taking ad-
vantage of multi-sums and multi-POCs, and if the underlying category Cis
an rm-adhesive category [26, Def. 1.1], one may lift this restriction and obtain a
fully well-posed semantics for DPO-rewriting along generic rules, and for regular
monic matches:
Theorem 4. Let Cbe an rm-adhesive category, let X0obj(C)be an object,
and let r2, r1span(C)be (generic) spans in C.
Synthesis: For every pair of admissible matches m1MDPO
r1(X0)and m2
MDP O
r2(r1m1(X0)), there exist an admissible match µ∈ MDPO
r2(r1)and an
admissible match m21 MDP O
r21 (X0)(for r21 the composite of r2with r1
along µ) such that r21m21 (X0)
=r2m2(r1m1(X0)).
Analysis: For every pair of admissible matches µ∈ MDP O
r2(r1)and m21
MDP O
r21 (X0)(for r21 the composite of r2with r1along µ), there exists a pair
of admissible matches m1MDP O
r1(X0)and m2MSqP O
r2(r1m1(X0)) such
that r2m2(r1m1(X0))
=r21m21 (X0).
Compatibility: If in addition Cis finitary, the sets of pairs of matches
(m1, m2)and (µ, m21)are isomorphic if they are suitably quotiented by uni-
versal isomorphisms, i.e., by universal isomorphisms of X1=r1m1(X0)and
X2=r2m2(X1)for the set of pairs of matches (m1, m2), and by the universal
16 N. Behr, R. Harmer and J. Krivine
isomorphisms of multi-sums and multi-POCs for the set of pairs of matches
(µ, m21), respectively.
Proof. See Appendix D.
It is worthwhile noting that for an adhesive category C(in which every
monomorphism is regular) and if we consider linear rules (i.e., spans of monomor-
phisms), the characterization of multi-sums according to Lemma 1 permits to
verify that DPO-type rule compositions as in Theorem 4 specialize in this setting
precisely to the notion of DPO-type D-concurrent compositions [35, Sec. 7.2].
This is because, in this case, each multi-sum element is precisely characterized
as the pushout of a monic span (referred to as a D-dependency relation between
rules in [35]), so one finds indeed that Theorem 4 conservatively generalizes
the traditional DPO-type concurrency theorem to the non-linear setting. Un-
like for the generic SqPO-type setting however, quasi-topoi are not sufficient for
generic DPO-rewriting, since in the “analysis” part of the proof of the DPO-
type concurrency theorem the van Kampen property of pushouts along regular
monomorphisms is explicitly required (cf. Appendix D).
6 Conclusion and outlook
We have defined an abstract setting for SqPO graph transformation in quasi-
topoi that captures the important concrete cases of (directed) multi-graphs and
simple graphs. In particular, we have established the existence of appropriate
notions of M-multi-sums, M-multi-POCs and M-FPC-PO-augmentations in
this setting that permit a proof of the concurrency theorem for general non-
linear rules.
Our immediate next goal is to prove associativity of our notion of rule com-
position in order to enable the use of rule algebra constructions [5,9,7] and
tracelets [3] for static analysis [6,4] of systems generated by non-linear SqPO
or DPO transformations. Intuitively, associativity is necessary in order to guar-
antee that one may consistently analyze and classify derivation traces based
upon nested applications of the concurrency theorem, in the sense that recursive
rule composition operations should yield a “catalogue” of all possible ways in
which rules can interact in derivation sequences. The latter is formalized as the
so-called tracelet characterization theorem in [3], whereby any derivation trace
is characterized as an underlying tracelet and a match of the tracelet into the
initial state of the trace. As illustrated in the worked example presented in (19),
which highlighted the intriguing effect that comparatively complicated interme-
diate state in derivation traces involving cloning and fusing of graph structures
are consistently abstracted away via performing rule compositions, one might
hope that this type of effect persists also in n-step derivation traces for arbitrary
n, for which however associativity is a prerequisite. Concretely, without the asso-
ciativity property, the tentative “summaries” of the overall effects of derivation
traces via their underlying tracelets would not be mathematically consistent, as
they would only encode the causality of the nesting order in which they were
Concurrency Theorems for Non-linear Rewriting Theories 17
calculated via pairwise rule composition operations. Preliminary results indicate
however that indeed our generalized SqPO- and DPO-type semantics both sat-
isfy the requisite associativity property, which will be presented in future work.
Beyond known applications to rule-based descriptions of complex systems,
such as in Kappa [12] and related formalisms, we hope to exploit this frame-
work in graph combinatorics and structural graph theory [11]—which frequently
employ operations such as edge contraction, which requires input-linear but
output-non-linear rules, and node expansion, which further requires input-non-
linear rules—to provide stronger tools for reasoning about graph reconfigurations
as used, for example, in the study of coloring problems. We moreover expect
this framework to be useful in strengthening existing approaches to graph-based
knowledge representation [28], particularly for the extraction and manipulation
of audit trails [30] that provide a semantic notion of version control in these
settings.
References
1. Adamek, J., Herrlich, H., Strecker, G.: Abstract and concrete categories: The joy
of cats. Reprints in Theory and Applications of Categories (17), 1–507 (2006),
http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf
2. Behr, N.: Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra
Framework. In: Proceedings of GCM 2019. EPTCS, vol. 309, pp. 23–52 (2019).
https://doi.org/10.4204/eptcs.309.2
3. Behr, N.: Tracelets and tracelet analysis of compositional rewriting sys-
tems. In: Proceedings of ACT 2019. EPTCS, vol. 323, pp. 44–71 (2020).
https://doi.org/10.4204/EPTCS.323.4
4. Behr, N.: On Stochastic Rewriting and Combinatorics via Rule-Algebraic Meth-
ods. In: Proceedings of TERMGRAPH 2020. vol. 334, pp. 11–28 (2021).
https://doi.org/10.4204/eptcs.334.2
5. Behr, N., Danos, V., Garnier, I.: Stochastic mechanics of graph rewriting. In: Pro-
ceedings of LiCS ’16. ACM Press (2016). https://doi.org/10.1145/2933575.2934537
6. Behr, N., Danos, V., Garnier, I.: Combinatorial Conversion and Moment Bisim-
ulation for Stochastic Rewriting Systems. LMCS Volume 16, Issue 3 (2020),
https://lmcs.episciences.org/6628
7. Behr, N., Krivine, J.: Rewriting theory for the life sciences: A unifying framework
for CTMC semantics. In: Graph Transformation (ICGT 2020). Theoretical Com-
puter Science and General Issues, vol. 12150 (2020). https://doi.org/10.1007/978-
3-030-51372-6
8. Behr, N., Krivine, J.: Compositionality of Rewriting Rules with Conditions. Com-
positionality 3(2021). https://doi.org/10.32408/compositionality-3-2
9. Behr, N., Sobocinski, P.: Rule Algebras for Adhesive Categories (extended journal
version). LMCS Volume 16, Issue 3 (2020), https://lmcs.episciences.org/6615
10. Bonifati, A., Furniss, P., Green, A., Harmer, R., Oshurko, E., Voigt, H.:
Schema Validation and Evolution for Graph Databases. In: Conceptual Model-
ing, LNCS, vol. 11788, pp. 448–456. Springer International Publishing (2019).
https://doi.org/10.1007/978-3-030-33223-5 37
11. Bousquet-M´elou, M.: Counting planar maps, coloured or uncoloured, p. 1–50. Lon-
don Mathematical Society Lecture Note Series, Cambridge University Press (2011).
https://doi.org/10.1017/CBO9781139004114.002
18 N. Behr, R. Harmer and J. Krivine
12. Boutillier, P., et al.: The Kappa platform for rule-based modeling. Bioinformatics
34(13), i583–i592 (2018). https://doi.org/10.1093/bioinformatics/bty272
13. Braatz, B., Golas, U., Soboll, T.: How to delete categorically — Two pushout com-
plement constructions. Journal of Symbolic Computation 46(3), 246–271 (2011).
https://doi.org/https://doi.org/10.1016/j.jsc.2010.09.007
14. Cockett, J., Lack, S.: Restriction categories I: categories of partial maps. Theo-
retical Computer Science 270(1), 223–259 (2002). https://doi.org/10.1016/S0304-
3975(00)00382-0
15. Cockett, J., Lack, S.: Restriction categories II: partial map classification. Theo-
retical Computer Science 294(1), 61–102 (2003). https://doi.org/10.1016/S0304-
3975(01)00245-6
16. Corradini, A., et al.: Sesqui-Pushout Rewriting. In: Graph Transformations. LNCS,
vol. 4178, pp. 30–45. Springer Berlin Heidelberg (2006)
17. Corradini, A., et al.: AGREE – Algebraic Graph Rewriting with Controlled Em-
bedding. In: Graph Transformation (ICGT 2015). LNCS, vol. 9151, pp. 35–51.
Cham (2015). https://doi.org/10.1007/978-3-319-21145-9 3
18. Danos, V., et al.: Reversible Sesqui-Pushout Rewriting. In: Graph Trans-
formation (ICGT 2014). LNCS, vol. 8571, pp. 161–176. Cham (2014).
https://doi.org/10.1007/978-3-319-09108-2 11
19. Diers, Y.: Familles universelles de morphismes, Publications de l’U.E.R.
math´ematiques pures et appliqu´ees, vol. 145. Universit´e des sciences et techniques
de Lille I (1978)
20. Dyckhoff, R., Tholen, W.: Exponentiable morphisms, partial products and pullback
complements. Journal of Pure and Applied Algebra 49(1-2), 103–116 (1987)
21. Ehrig, H., et al.: Fundamentals of Algebraic Graph Transformation. Monographs
in Theoretical Computer Science (2006). https://doi.org/10.1007/3-540-31188-2
22. Ehrig, H., Golas, U., Hermann, F.: Categorical frameworks for graph transforma-
tion and HLR systems based on the DPO approach. Bulletin of the EATCS (102),
111–121 (2010)
23. Ehrig, H., et al.: Adhesive High-Level Replacement Categories and Systems. In:
LNCS, vol. 3256, pp. 144–160 (2004). https://doi.org/10.1007/978-3-540-30203-2 -
12
24. Ehrig, H., et al.: M-adhesive transformation systems with nested application con-
ditions. Part 1: parallelism, concurrency and amalgamation. MSCS 24(04) (2014).
https://doi.org/10.1017/s0960129512000357
25. Gabriel, K., et al.: Finitary M-adhesive categories. MSCS 24(04) (2014).
https://doi.org/10.1017/S0960129512000321
26. Garner, R., Lack, S.: On the axioms for adhesive and quasiadhesive categories.
TAC 27(3), 27–46 (2012)
27. Golas, U., Habel, A., Ehrig, H.: Multi-amalgamation of rules with ap-
plication conditions in M-adhesive categories. MSCS 24(04) (2014).
https://doi.org/10.1017/s0960129512000345
28. Harmer, R., Le Cornec, Y.S., L´egar´e, S., Oshurko, E.: Bio-curation
for cellular signalling: The kami project. IEEE/ACM Transactions on
Computational Biology and Bioinformatics 16(5), 1562–1573 (2019).
https://doi.org/10.1109/TCBB.2019.2906164
29. Harmer, R., Oshurko, E.: Knowledge representation and update in hierarchies of
graphs. JLAMP 114, 100559 (2020)
30. Harmer, R., Oshurko, E.: Reversibility and composition of rewriting in hierarchies.
EPTCS 330, 145–162 (2020). https://doi.org/10.4204/eptcs.330.9
Concurrency Theorems for Non-linear Rewriting Theories 19
31. Heindel, T.: Hereditary Pushouts Reconsidered. In: Ehrig, H., Rensink, A., Rozen-
berg, G., Sch¨urr, A. (eds.) Graph Transformations (ICGT 2010). LNCS, vol. 6372,
pp. 250–265. Springer Berlin Heidelberg, Berlin, Heidelberg (2010)
32. Johnstone, P.T.: Sketches of an Elephant – A Topos Theory Compendium, vol. 1.
Oxford University Press (2002)
33. Johnstone, P.T., Lack, S., Soboci´nski, P.: Quasitoposes, Quasiadhesive Categories
and Artin Glueing. In: Algebra and Coalgebra in Computer Science. LNCS,
vol. 4624, pp. 312–326 (2007). https://doi.org/10.1007/978-3-540-73859-6 21
34. Lack, S., Soboci´nski, P.: Adhesive Categories. In: FoSSaCS 2004. LNCS, vol. 2987,
pp. 273–288 (2004). https://doi.org/10.1007/978-3-540-24727-2 20
35. Lack, S., Soboci´nski, P.: Adhesive and quasiadhesive categories. RAIRO
- Theoretical Informatics and Applications 39(3), 511–545 (2005).
https://doi.org/10.1051/ita:2005028
36. owe, M.: Polymorphic Sesqui-Pushout Graph Rewriting. In: Graph Transforma-
tion. LNCS, vol. 9151, pp. 3–18 (2015). https://doi.org/10.1007/978-3-319-21145-
9 1
37. Monro, G.: Quasitopoi, logic and heyting-valued models. Jour-
nal of Pure and Applied Algebra 42(2), 141–164 (1986).
https://doi.org/https://doi.org/10.1016/0022-4049(86)90077-0
20 N. Behr, R. Harmer and J. Krivine
A Collection of definitions and auxiliary properties
A.1 Universal properties
Lemma 2. Let Cbe a category.
X X
A B A B A B
D C D C D C
E Y
(i) (ii) (iii)
e
e0
c
d
a
b
PO
e
a
d
c
b
x
x
x0
PB
a
d b
c
FPC
x
ax
c0
y
!x0
Then the following properties hold:
1. Universal property of pushouts (POs): Given a commutative diagram as in
(i), there exists a morphism D¯eEthat is unique up to isomorphisms.
2. Universal property of pullbacks (PBs): Given a commutative diagram as in
(ii), there exists a morphism X¯xAthat is unique up to isomorphisms.
3. Universal property of final pullback complements (FPCs): Given a commu-
tative diagram as in (iii)where (ax.y)is a PB of (d, c0), there exists a
morphism Y¯
x0Cthat is unique up to isomorphisms, and which satis-
fies that (x, y)is the PB of (b, x0).
A.2 Stability properties
Definition 11. Let Cbe a category.
A0B0
D0C0
AB
D C
c
d
a
b
()
δ
χ
α
β
d0
c0
a0
b0
y
y
y
()
y
(20)
Apushout () in Cis said to be stable under pullbacks iff for every com-
mutative cube over the pushout ()such as in the diagram above where all
vertical squares are pullbacks, the top square ()is a pushout.
Afinal pullback complement (FPC) () in Cis said to be stable under
pullbacks iff for every commutative cube over the FPC ()such as in the
diagram above where all vertical squares are pullbacks, the top square ()is
an FPC.
Lemma 3. Two important examples of categories for which suitable stability
properties for pushouts hold are given as follows:
Concurrency Theorems for Non-linear Rewriting Theories 21
1. In every adhesive category C, pushouts along monomorphisms (i.e., pushouts
such as ()in (20) with amono(C)or bmono(C)) are stable under
pullback [34]. This property is indeed the “if” direction of the so-called van
Kampen property of adhesive categories [26].
2. In a regular mono (rm)-quasiadhesive category [26, Def. 1.1 and Cor. 4.7],
all pushouts along regular monomorphisms exist, these pushouts are also pull-
backs, and in particular pushouts along regular monomorphisms are stable
under pullbacks. A useful characterization of rm-quasiadhesive categories is
the following: a small category Cwith all pullbacks and with pushouts along
regular monomorphisms is rm-quasiadhesive iff it has a full embedding into
a quasi-topos (preserving the aforementioned two properties).
Lemma 4 ([18], Lem. 1). Let Cbe a category with all pullbacks. Then FPCs
are stable under pullbacks.
Proposition 3. In a quasi-topos C,unions of regular subobjects are effec-
tive [33, Prop. 10], i.e., the union of two subobjects is computed as the pushout of
their intersection, and moreover the following property holds: in a commutative
diagram such as below, where (c, a)is the pullback of (h, p),(d, b)the pushout of
(c, a), where all morphisms (except possibly x) are monomorphisms, and where
either prm(C)or hrm(C), then the induced morphism x:DEis a
monomorphism [26, Prop. 2.4]:
A B
C D
E
a
cb
d
h
!x
p
PO
(21)
A.3 Single-square lemmata specific to M-adhesive categories
Lemma 5. Let Cbe an M-adhesive category.
A B
C D
a
γ()β
d
(22)
1. Pushouts along M-morphisms are pullbacks: if ()is a pushout and β∈ M,
then ()is also a pullback.
2. Stability of M-morphisms under pushouts: if ()is a pushout and β∈ M,
then γ∈ M.
3. Stability of M-morphisms under pullbacks: if ()is a pullback and γ∈ M,
then β∈ M.
4. If ()is a pullback, γ=idAand a, β ∈ M, then a∈ M.
Since ()for γ=idAand β∈ M ⊂ mono(C)is always a pullback, 4. may be
reformulated as follows:
22 N. Behr, R. Harmer and J. Krivine
4.’ Decomposition property of M-morphisms: if gf∈ M and g∈ M, then
f∈ M.
A.4 Double-square lemmata
Lemma 6. Let Cbe a category.
A B C F F 0
A0B0C0G G0F0
A0H H0
ab
α
a0b0
βχ
(1) (2) ϕ
γ
ϕ0
γ0
h
g
f
(3)
(4)
()
()b0a0γ0ϕ0
(23)
Given commutative diagrams as above, the following statements hold:
1. Pushout-pushout-(de-)composition: if (1) is a pushout, (1) + (2) is a pushout
iff (2) is a pushout.
2. Pullback-pullback-(de-)composition: if (2) is a pullback, (1)+ (2) is a pullback
iff (1) is a pullback.
3. Pushout-pullback-decomposition [18, Lem. 4]: if (1)+(2) is a stable pushout9
and (1),(2),()are pullbacks, then (1) and (2) are both pushouts. (Note: If
a0and b0are monomorphisms, the condition on ()is always satisfied.)
4. Pullback-pushout-decomposition (variant of [27, Lem. B.2]): if χis in M,
(1) + (2) is a pullback and (1) is stable pushout, then (1) and (2) are both
pullbacks.
5. Horizontal FPC-FPC-(de-)composition: if (2) is an FPC (i.e., (β, b0)is an
FPC of (b, χ)), (1) + (2) is an FPC iff (1) is an FPC.
6. Vertical FPC-FPC-(de-)composition [36, Prop. 36]: if (3) is an FPC (i.e.,
(ϕ, g)is an FPC of (f, ϕ0)),
(a) if (4) is an FPC (i.e., (γ, h)is an FPC of (g, γ0)), then (3) + (4) is an
FPC (i.e., (γϕ, h)is an FPC of (f, γ 0ϕ0))
(b) if (3) + (4) is an FPC (i.e., (γϕ, h)is an FPC of (f, γ0ϕ0)) and if
(4) is a pullback, then (3) is an FPC (i.e., (ϕ, g)is an FPC of (f, ϕ0)).
7. Vertical FPC-pullback decomposition [18, Lem. 3]: if (3) + (4) is an FPC
(i.e., (γϕ, h)is an FPC of (f, γ 0ϕ0)), both (4) and ()are pullbacks,
and if the diagram commutes, then (3) is an FPC (i.e., (ϕ, g)is an FPC of
(f, ϕ0)) and (4) is an FPC (i.e., (γ , h)is an FPC of (g, γ0)). (Note: If γ0
and ϕ0are monomorphisms, the condition on ()is always satisfied.)
Proof. Most of the above results are cited from previous works (with references
provided), yet the pullback-pushout decomposition statement is a slight gener-
alization of the variant provided in [27, Lem. B.2] and thus requires a proof.
9Here, “stable” refers to stability under pullbacks.
Concurrency Theorems for Non-linear Rewriting Theories 23
Construct the diagram below via taking a pullback:
A B C
A¯
BC
A0B0C0
b
α
a0b0
β0χ
ab
PB
a0
β00
βχ
(2)
(1)
(24)
The existence of the morphisms Aa0¯
Band Bβ00 ¯
Bfollows from
the universal property of pullbacks.
By stability of M-morphisms under pullbacks,β0is in M, and thus by the
decomposition property of M-morphisms,β00 is also in M. The latter entails
that (1) is a pullback.
By pullback-pullback decomposition, (2) is a pullback.
Since (1) and (2) are pullbacks, (1)+(2) a pushout along a M-morphism (i.e.,
a stable pushout), and since β0, β00 ∈ M,bypushout-pullback decomposition,
both (1) and (2) are pushouts. Therefore, B
=B0, and the claim follows.
B Proof of Lemma 1
The only non-trivial part about the existence statement concerns the fact that
yAand yBare in M, which follows from the decomposition property of M-
morphisms. As for the construction of multi-sum elements, let us first prove
that A+BeP
Pis indeed an epimorphism. To this end, for a cospan of M-
morphisms ApA
PpB
Bobtained via pushout of some span of M-morphisms
AxA
XxB
B, let A+Bf
Pfor f= [pA, pB] denote the induced morphism,
and let A+Bef
P0mF
Pdenote the epi-M-factorization of f. Taking
pullbacks along mfto obtain the squares marked (1A) and (1B) in (25)(i), which
by the universal property of pullbacks entails the existence of the morphisms
marked a00 and b00, by stability of M-morphisms under pullbacks,a0and b0are in
M. Thus (a00, a0, idA, idA) and (b00, b0, idA, idA) are pullbacks, whence by stability
of isomorphisms under pullbacks, a0, a00, b0, b00 are isomorphisms. Form the square
marked (2) as a pullback, which by the universal property of pullbacks also yields
morphisms ι0
X
X0and X0x
X. By pullback-pullback decomposition, all
squares of the bottom commutative cube are pullbacks, so that by stability of
isomorphisms under pullbacks, X0
=X. The bottom-most square is a pushout
along M-morphisms and thus a stable pushout, whence by stability of stable
pushouts under pullbacks, the square marked (2) is a pushout. Thus by the
universal property of pushouts,P0
=P, and we have proved that A+Bf
Pis
indeed an epimorphism (henceforth referred to as eP).
To proceed, denote by PeQ
QmQ
(with eQepi(C) and mQ∈ M) the
epi-M-factorization of the morphism PZthat exists by the universal prop-
erty of pushouts. By uniqueness of epi-M-factorizations up to isomorphisms [1,
24 N. Behr, R. Harmer and J. Krivine
Prop. 14.4], since A+BeQeP
PmQ
Zand A+Be
Ym
Zare epi-M-
factorizations of A+B[a,b]
Z, we find that P
=Y. Finally, since mQ∈ M,
the squares marked (3A) and (3B) in (25)(ii) are pullbacks. Forming the square
marked (4) as a pullback, by pullback-pullback decomposition also the two back
vertical squares in (25)(ii) are pullbacks, whence by stability of isomorphisms
under pullbacks, X0
=X. Since Pis a pushout of M-morphisms and since (4)
is a pullback along M-morphisms, it follows from effectiveness of binary unions
of regular subobjects that Pq
Qis a monomorphism, which proves the claim
that eQmono(C)epi(C).
A B X00
A+BA B
X0Q
A0B0X
P0A B
XZ
A B (ii)
P
(i)
ιAιB
xAxB
a0b0
a00 b00
pApB
PO
mf
!x
ιX0
inAinB
ef
!f
PB
PB PB
(2)
qAqB
an
xAxB
PB
mQ
(3A) (3B)
!
PB
(4)
(1A)(1B)
(25)
C Proof of the SqPO-type concurrency theorem
Recall that we assume Cis a quasi-topos, and for the compatibility part of the
theorem in addition that Cis finitary.
C.1 “Synthesis” part
Let X0obj(C) be an object, rj= (OjKjIj)span(C) (j= 1,2)
generic rules, and let (m1:I1X0)∈ MSqP O
r1(X0) and (m2:I2X1)
MSqP O
r2(X1) be S qP O-admissible matches, where X1:= r1m1(X0). Consider
then a sequence of Sq P O-type direct derivations, which yields a diagram as
presented in (26), and identify the multi-sum element (I2J21 O1), which
is in particular a cospan of M-morphisms, and unique up to isomorphisms. By
Concurrency Theorems for Non-linear Rewriting Theories 25
the universal property of multi-sums, there exists an M-morphism J21 X1:
O2K2I2O1K1I1
J21
X2X1X1X0X0
m
2¯m2
m2
PO FPC
m
1
¯m1m1
PO FPC
j2j1
!
(26)
Take the pullback (J21 ¯
K1¯
X0) of (J21 X1¯
X0), and the pullback
(¯
X1¯
K2J21) of ( ¯
X1X1J21), resulting in the following diagram:
O2K2I2O1K1I1
K2J21 K1
X2X1X1X0X0
m
2
¯m2m2m
1¯m1
m1
j2j1
j21 PB
PB
(12)
(22)
(11)
(21)
(27)
By stability of M-morphisms under pullbacks, ( ¯
K1¯
X0),(¯
K2¯
X0)∈ M.
By the universal property of pullbacks, there exist the morphisms K1¯
K1
and K2¯
K2.
By the decomposition property of M-morphisms, (K1¯
K1),(K2¯
K2)
M.
Since by assumption (O1X1¯
X0) is the pushout of (O1K1¯
X0),
and since pushouts along M-morphisms are pullbacks, invoking pullback-
pullback decomposition yields that (O1K1¯
K1) is a pullback of (O1
J21 ¯
K1). A completely analogous argument reveals that ( ¯
K2K2I2)
is a pullback of ( ¯
K2J21 I2).
Since moreover (O1J21)∈ M and (J21 X1)∈ M, so that in particular
the square (11)+(21) is a pushout that is stable under pullbacks, by pushout-
pullback decomposition the squares (11) and (21) are also pushouts.
Since (I2J21)∈ M and (J21 X1)∈ M, since the square (12) + (22)
is an FPC, and since (12) and (22) are pullbacks, by vertical FPC-pullback
decomposition the squares (12) and (22) are FPCs.
Next, form the squares marked (31) and (32) in the diagram below by taking
pushouts:
O2K2I2O1K1I1
O21 K2J21 K1I21
X2X1X1X0X0
m
2
¯m2m2m
1¯m1
m1
j2j1
j21 PB
PB
(12)
(22)
(11)
(21)
PO
!
PO
!
(32)
(42)(41)
(31)
(28)
By stability of M-morphisms under pushouts, (I1I21),(O2O21)∈ M.
26 N. Behr, R. Harmer and J. Krivine
Since (32) + (42) and (32) are pushouts, by pushout-pushout decomposition
(42) is a pushout, and since moreover ¯
K2¯
X1is in M,bystability of
M-morphisms under pushouts we find that (O21 X2)∈ M.
In order to analyze the structure of the induced squares (31) and (41) in
further detail, let us invoke the epi-M-factorization of the morphism I21 X0,
and construct the following diagram:
O2K2I2O1K1I1
O21 K2J21 K1I21
¯
O21 ¯
¯
K2¯
J21 ¯
¯
K1¯
I21
X2X1X1X0X0
m
2
¯m2
m2m
1
¯m1
m1
j2j1
j21
(12)(11)PO
PO
(32)(31)
e21
m21
!(40
1)
!
!
!
(400
1)
PB
PO
(20
1)
(200
1)
PB
(200
2)
(20
2)
(40
2)
PO
(400
2)
(29)
Since (400
1) is constructed as a pullback and m21 ∈ M, by stability of M-
morphisms under pullbacks ¯
¯
K1¯
X0is in M.
By the decomposition property of M-morphisms, the morphisms I1¯
I21,
¯
K1¯
¯
K1and K1¯
¯
K1are all in M.
By vertical FPC-pullback decomposition, both (400
1) and (41)+(40
1) are FPCs.
By pushout-pushout decomposition, since (20
1) is constructed as a pushout
and (20
1) + (200
1) is a pushout, (200
1) is also a pushout.
By stability of M-morphisms under pushouts, the morphisms J21 ¯
J21 and
¯
J21 X1are in M.
Since (200
2) is constructed as a pullback and since (20
2) + (200
2) is an FPC (and
thus a pullback), by pullback-pullback decomposition (20
2) is a pullback.
By stability of M-morphisms under pullbacks, the morphisms ¯
K2¯
¯
K2and
¯
¯
K2¯
X1are in M.
By vertical FPC-pullback decomposition, (20
2) and (200
2) are both FPCs.
Since (40
2) is constructed as a pushout and since (40
2) + (400
2) is a pushout, by
pushout-pushout decomposition (400
2) is a pushout.
By stability of M-morphisms under pushouts, the morphisms O21 ¯
O21
and ¯
O21 X2are in M.
As an intermediate summary, we have thus derived the following information:
O2K2I2O1K1I1
O21 K2J21 K1I21
¯
O21 ¯
¯
K2¯
J21 ¯
¯
K1¯
I21
X2X1X1X0X0
j2j1
(11)
PO
PO
e21
¯m21
¯
j21
¯m0
21
¯m
21
¯m
2
¯m2¯m
1¯m1
FPC
PO
PO
FPC
FPC
PO
(400
1)
PB
PO
(20
1)
(200
1)
FPC
(200
2)
(20
2)(40
2)
PO
(400
2)
(32)(12)
PO
FPC
(31)
PO
(40
1)
(30)
Concurrency Theorems for Non-linear Rewriting Theories 27
As indicated via the dotted lines, the vertical composition of the top two rows
of the diagram yields a two-step sequence of SqPO-type direct derivations
from ¯
I21 along rule (O1K1I1) with match ¯m1into ¯
J21, and then by
rule (O2K2I2) with match ¯m1to ¯
O21.
The data of squares (31) and (40
1) furnishes an M-FPC augmentation (i.e., of
the pushout square (30
1) via morphisms I1¯
I21,I21 ¯
I21 and ¯
K1¯
¯
K1).
To proceed, form the squares (5) and (6) in the diagram below by taking
pullbacks:
O2K2I2O1K1I1
O21 K2J21 K1I21
¯
O21 ¯
¯
K2¯
J21 ¯
¯
K1¯
I21
X2X1X1X0X0
K21
¯
K21
j2j1
(11)
PO
PO
e21
¯m21
¯
j21
¯m0
21
¯m
21
¯m
2
¯m2¯m
1¯m1
FPC
PO
PO
FPC
FPC
PO
!
PB
PB
(5)
(6)
(72)(71)
(400
1)
PB
PO
(20
1)
(200
1)
FPC
(200
2)
(20
2)(40
2)
PO
(400
2)
(32)(12)
PO
FPC
(31)
PO
(40
1)
(31)
By the universal property of pullbacks, there exists a morphism K21 ¯
K21.
By pullback-pullback decomposition, since (5) + (200
1) and (6) are pullbacks,
(72) is a pullback, and analogously since (5) + (200
2) is a pullback and (6) is
a pullback, (71) is a pullback.
By stability of M-morphisms under pullbacks, the morphism K21 ¯
K21 is
in M.
Since (200
1) is a pushout along an M-morphism and thus stable under pull-
backs, (72) is a pushout. Thus by pushout composition, (72) + (400
2) is a
pushout.
Since (200
2) is an FPC and FPCs are stable under pullbacks, (71) is an FPC.
Thus by horizontal FPC composition, (71) + (400
1) is an FPC.
This concludes the proof of the “synthesis” part of the concurrency theorem,
since the latter two points exhibit the data of a single-step SqPO-type direct
derivation (of X0along ( ¯
O21 K21 ¯
I21) along match ¯m21).
28 N. Behr, R. Harmer and J. Krivine
C.2 “Analysis” part
Suppose we were given an SqPO-type composite rule as defined via the data in
the diagram below:
O2K2I2O1K1I1
¯
O21 K2J21 K1¯
I21
X2X0
K21
K21
m
2m1
¯
j2¯
j1
m
21 ¯m21
¯
k21
PB
(¯
12)
(¯
32)(¯
11)(¯
31)
(81)
(5)
(82)
(32)
Here, compared to the diagram in (31), we have for brevity only explicitly de-
picted the vertical compositions of the top two rows in (31) (i.e., (¯
32) and (¯
11) are
pushouts, while (¯
12) and (¯
31) are FPCs). According to the definition of SqPO-
type direct derivations, we furthermore are given that (81) is an FPC and (82)
a pushout.
Extend this diagram by forming FPC (9), pushout (10) and FPC (11):
O2K2I2O1K1I1
¯
O21 K2J21 K1¯
I21
X2X1X1X0X0
K21
K21
m
2m1
¯
j2¯
j1
m
21 ¯m21
¯
k21
!
PB
FPC
FPC
!
(¯
12)
(¯
32)(¯
11)(¯
31)
(81)
(121)
(5)
(122)
(13)
(82)
(9)
(11) (10)
PO
(33)
By the universal property of FPCs, there exist morphisms K21 ¯
X0and
K21 ¯
X1.
By stability of M-morphisms under pullbacks and under pushouts, respec-
tively, the morphisms K1¯
X0,J21 X1and K2¯
X1are in M.
Since (81) = (121)+ (9) and (9) are FPCs, by horizontal FPC decomposition,
(121) is an FPC.
To proceed, we have to construct the following auxiliary diagram, where the
square (14) on the bottom of the diagram is formed by taking the pullback
Concurrency Theorems for Non-linear Rewriting Theories 29