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Concurrency Theorems for Non-linear Rewriting Theories

May 2021

DOI:10.48550/arXiv.2105.02842

Authors:

Nicolas Behr

Université de Paris CNRS

Russ Harmer

École Normale Supérieure de Lyon

Jean Krivine

French National Centre for Scientific Research

Preprints and early-stage research may not have been peer reviewed yet.

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.

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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:A→Band

g:B→Dis the largest, i.e. least general, Ctogether with arrows g0:A→C

and f0:C→Dfor 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:C→Cand a natural transformation η:IDC

−→ Tsuch that

1. for all X∈obj(C),ηX:X→T(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=f◦efor m∈rm(C)and e∈epi(C), then e∈iso(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=m◦e, with morphisms Ae

−→ ¯

Bin epi(C)and ¯

−→ Ain M

(uniquely up to isomorphism in ¯

B).

–It possesses a strict initial object ∅∈obj(C)[32, A1.4], i.e., for every

object X∈obj(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

X∈obj(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 X∅Y

is XX+YY(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:EG→VG. Morphisms of Graph between G, H ∈obj(Graph)

are of the form ϕ= (ϕV, ϕE), with ϕV:VG→VHand ϕE:EG→EHsuch

that ϕV◦sG=sH◦ϕEand ϕV◦tG=tH◦ϕE.

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:E→V×Vis an injective function. A morphism

f= (fV, fE)between objects Sand S0is a pair of functions fV:V→V0and

fE:E→E0such that ι0◦fE= (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 G∈obj(Graph)[17, Sec. 2.1], ηG:G→T(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 :vn→vpfor each pair of vertices (vn, vp)∈V0

G×V0

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 E⊆V×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:ST(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=mV◦eVlifts 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 eV∈epi(Set)).

E E0E˜

EE0

V×V V 0×V0V×Vim(fV)×im(fV)V0×V0

V V 0Vim(fV)V0

eVmV

∆∆∆

eV×eVmV×mV

∃!f0

Ef00

ι0

˜ι

fV×fV

∆∆

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

−→ 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

FCB

T(A)T(B)B

∃!n

¯n

T(f)

¯m

ηA

ηb

(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=m0◦b

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 δ∈iso(C)such that δ◦a=a0

and d0◦δ=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 a◦p=n,

take the pushout (1), which by universal property of pushouts induces an

arrow De

−→ C; if e∈iso(C),(a, d)is a contribution to the M-multi-POC

of (f, b).

A B A B

PDPD

Q E FC

(i) (ii)

aPO

b∃!a

∃!d

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 p◦a=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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<latexit sha1_base64="+9t7lfzG4m39xKwQx+QGWNhDGvE=">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</latexit>

⌘B

<latexit sha1_base64="npH7tXZjA5DYXKBTMyajVTP4FXM=">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</latexit>

<latexit sha1_base64="/B1XSKVJvrHoLKjblLBbR7G85Ss=">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</latexit>

<latexit sha1_base64="XOesLas+EHE8gkjtGzbnI6d2z7s=">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</latexit>

⌘B

<latexit sha1_base64="I1J3TGcWyTjKV+ZTjuBZP3iTum0=">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</latexit>

<latexit sha1_base64="w/HImxGYYAKFAC6Lzl5taJSiDHw=">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</latexit>

¯m

<latexit sha1_base64="9GACVUnEyIndndtYcl1Nzl5VkoE=">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</latexit>

⌘A

<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>

PBPB

FPC

POC

<latexit sha1_base64="kEXi/4JbMwFIWRtZUhOAfb1thUw=">AAACxHicjVHLSsNAFD2Nr1pfVZdugkXoqiQi6rJQEJct2AfUIsl0WkPzYmYilKI/4Fa/TfwD/QvvjCmoRXRCkjPn3nNm7r1+GgZSOc5rwVpaXlldK66XNja3tnfKu3sdmWSC8TZLwkT0fE/yMIh5WwUq5L1UcC/yQ971Jw0d795xIYMkvlLTlA8ibxwHo4B5iqhW46ZccWqOWfYicHNQQb6aSfkF1xgiAUOGCBwxFOEQHiQ9fbhwkBI3wIw4QSgwcY57lEibURanDI/YCX3HtOvnbEx77SmNmtEpIb2ClDaOSJNQniCsT7NNPDPOmv3Ne2Y89d2m9Pdzr4hYhVti/9LNM/+r07UojHBuagioptQwujqWu2SmK/rm9peqFDmkxGk8pLggzIxy3mfbaKSpXffWM/E3k6lZvWd5boZ3fUsasPtznIugc1xzT2tO66RSr+ajLuIAh6jSPM9QxyWaaBvvRzzh2bqwQkta2WeqVcg1+/i2rIcP5wGPMw==</latexit>

(1)(2)

⌘B

¯m

⌘A

¯n

T(f)

PBPB

FPC

POC

(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

¯a

¯α

e◦¯α

n◦α

(2)

(1)

(7)

We deﬁne an M-FPC augmentation (FPA) of the pushout square (1) as a di-

agram formed from an epimorphism e∈epi(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 (n◦α, 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)|e∈epi(C)∧e◦¯α, n ∈ M∧(f, n◦α) = 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

−→ 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=y◦fand b=y◦g, and moreover

(f, g)as well as yare unique up to universal isomorphisms.

∅

A B A B

A+B A +B

Y Y P

Z Z

(i) (ii)

inAinB

yAyB

[a,b]

inAinB

ιAιB

xAxB

ιX

∃!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=e◦inA, yB=e◦inB), 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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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 S0→Sjfor 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= (O←K→I)∈

span(C)into an object X∈obj(C)is deﬁned as

MSqP O

r(X) := {Im

−→ X|m∈rm(C)}.(10)

ASqPO-type direct derivation7of X∈obj(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)¯

m∗

¯m

¯o¯

(2) (1)

(11)

–The set of SqPO-type admissible matches of rules r2, r1∈span(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, r2∈span(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

j2j1

j21

j∗

¯o2

ι∗

21 ¯

¯o2

¯o1

j∗

¯o1¯

j1ι21

ι1

(12)

(22)(11)(21)

(3)

(4)(5)

(6)

(13)

We then deﬁne the composite rule via span composition:

^r1:= (O21 ←K2→J21)◦(J21 ←K1→I21 ) (14)

Deﬁnition 10. General DPO-rewriting semantics over an rm-adhesive category

–The set of DPO-admissible matches of a rule rule r= (O←K→I)∈

span(C)into an object X∈obj(C)is deﬁned as

MDP O

r(X) := {(m, ¯m,¯

i)|m∈rm(C)∧( ¯m,¯

i)∈ P(i, m)}.(15)

ADPO-type direct derivation of X∈obj(C)with rule ralong m∈MDPO

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, r1∈span(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, r2∈span(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

j2j1

j∗

¯o2¯o1

j∗

(12)

(22)(11)(21)

(17)

We then deﬁne the composite rule via span composition:

Jr1:= (O21 ←K2→J21)◦(J21 ←K1→I21 ) (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 X0∈obj(C)be an object, and let

r2, r1∈span(C)be two (generic) rewriting rules.

Concurrency Theorems for Non-linear Rewriting Theories 15

1. Synthesis: For every pair of admissible matches m1∈MSqP 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 m1∈MSqP O

r1(X0)and m2∈MSqP 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 X0∈obj(C)be an object,

and let r2, r1∈span(C)be (generic) spans in C.

–Synthesis: For every pair of admissible matches m1∈MDPO

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 m1∈MDP O

r1(X0)and m2∈MSqP 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.

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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

∃!¯x

d b

FPC

a◦x

∃!x0

Then the following properties hold:

1. Universal property of pushouts (POs): Given a commutative diagram as in

(i), there exists a morphism D−¯e→Ethat is unique up to isomorphisms.

2. Universal property of pullbacks (PBs): Given a commutative diagram as in

(ii), there exists a morphism X−¯x→Athat is unique up to isomorphisms.

3. Universal property of ﬁnal pullback complements (FPCs): Given a commu-

tative diagram as in (iii)where (a◦x.y)is a PB of (d, c0), there exists a

morphism Y−¯

x0→Cthat 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

D C

(∗)

(†)

(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 a∈mono(C)or b∈mono(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 p∈rm(C)or h∈rm(C), then the induced morphism x:D→Eis a

monomorphism [26, Prop. 2.4]:

A B

C D

∃!x

(21)

A.3 Single-square lemmata speciﬁc to M-adhesive categories

Lemma 5. Let Cbe an M-adhesive category.

A B

C D

γ(∗)β

(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 g◦f∈ 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

a0b0

βχ

(1) (2) ϕ

ϕ0

γ0

(3)

(4)

(†)

(∗)b0◦a0γ0◦ϕ0

(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., (γ◦ϕ, h)is an FPC of (f, γ 0◦ϕ0))

(b) if (3) + (4) is an FPC (i.e., (γ◦ϕ, h)is an FPC of (f, γ0◦ϕ0)) 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., (γ◦ϕ, h)is an FPC of (f, γ 0◦ϕ0)), 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¯

A0B0C0

a0b0

β0χ

β00

βχ

(2)

(1)

(24)

–The existence of the morphisms A−a0→¯

Band B−β00 →¯

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

−−→ 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 eQ∈epi(C) and mQ∈ M) the

epi-M-factorization of the morphism P→Zthat 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+BeQ◦eP

−−−−→ 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 eQ∈mono(C)∩epi(C).

∅

A B X00

A+BA B

X0Q

A0B0X

P0A B

A B (ii)

(i)

ιAιB

xAxB

a0b0

a00 b00

pApB

∃!x

ιX0

inAinB

∃!f

PB PB

(2)

qAqB

xAxB

(3A) (3B)

∃!

(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 X0∈obj(C) be an object, rj= (Oj←Kj→Ij)∈span(C) (j= 1,2)

generic rules, and let (m1:I1X0)∈ MSqP O

r1(X0) and (m2:I2X1)∈

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 (I2J21 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

PO FPC

m∗

¯m1m1

PO FPC

j2j1

∃!

(26)

Take the pullback (J21 ←¯

K1→¯

X0) of (J21 →X1←¯

X0), and the pullback

(¯

X1←¯

K2→J21) of ( ¯

X1→X1←J21), resulting in the following diagram:

O2K2I2O1K1I1

K2J21 K1

X2X1X1X0X0

m∗

¯m2m2m∗

1¯m1

j2j1

j21 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→¯

and K2→¯

K2.

–By the decomposition property of M-morphisms, (K1→¯

K1),(K2→¯

K2)∈

–Since by assumption (O1X1←¯

X0) is the pushout of (O1←K1¯

X0),

and since pushouts along M-morphisms are pullbacks, invoking pullback-

pullback decomposition yields that (O1←K1¯

K1) is a pullback of (O1

J21 ←¯

K1). A completely analogous argument reveals that ( ¯

K2K2→I2)

is a pullback of ( ¯

K2→J21 I2).

–Since moreover (O1J21)∈ 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 (I2J21)∈ 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∗

¯m2m2m∗

1¯m1

j2j1

j21 PB

(12)

(22)

(11)

(21)

∃!

(32)

(42)(41)

(31)

(28)

–By stability of M-morphisms under pushouts, (I1→I21),(O2→O21)∈ 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∗

¯m2

m2m∗

¯m1

j2j1

j21

(12)(11)PO

(32)(31)

e21

m21

∃!(40

∃!

(400

(20

(200

(20

(40

(400

(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)

e21

¯m21

j21

¯m0

¯m∗

¯m2¯m∗

1¯m1

FPC

(400

(20

(200

FPC

(200

(20

2)(40

(400

(32)(12)

FPC

(31)

(40

(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 (O1←K1→I1) with match ¯m1into ¯

J21, and then by

rule (O2←K2→I2) 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

j2j1

(11)

e21

¯m21

j21

¯m0

¯m∗

¯m2¯m∗

1¯m1

FPC

∃!

(5)

(6)

(72)(71)

(400

(20

(200

FPC

(200

(20

2)(40

(400

(32)(12)

FPC

(31)

(40

(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

m∗

2m1

j2¯

m∗

21 ¯m21

k21

(¯

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

m∗