Chenavier, Cyrille

Lucas, Maxime

English

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The Diamond Lemma for non-terminating rewriting
systems using deterministic reduction strategies
Cyrille Chenavier, Maxime Lucas
To cite this version:
Cyrille Chenavier, Maxime Lucas. The Diamond Lemma for non-terminating rewriting systems using
deterministic reduction strategies. IWC 2019 - 8th International Workshop on Confluence, Jun 2019,
Dortmund, Germany. pp.1-5. ￿hal-02385139￿
The Diamond Lemma for non-terminating rewriting
systems using deterministic reduction strategies
Cyrille Chenavier1 and Maxime Lucas2
1 Inria Lille - Nord Europe, Équipe Valse
cyrille.chenavier@inria.fr
2 Inria Rennes - Bretagne Atlantique, Équipe Gallinette
maxime.lucas@inria.fr
Abstract
We study the confluence property for rewriting systems whose underlying set of terms
admits a vector space structure. For that, we use deterministic reduction strategies. These
strategies are based on the choice of standard reductions applied to basis elements. We
provide a sufficient condition of confluence in terms of the kernel of the operator which
computes standard normal forms. We present a local criterion which enables us to check
the confluence property in this framework. We show how this criterion is related to the
Diamond Lemma for terminating rewriting systems.
1 Introduction
The fact that local confluence together with termination implies confluence has been known
for abstract rewriting systems since Newman’s work [8]. For rewriting on noncommutative
polynomials, a similar result known as the Diamond lemma was introduced by Bergman [2]
nearly 30 years later, in order to compute normal forms in noncommutative algebras using
rewriting theory. It asserts that for terminating rewriting systems, the local confluence property
can be checked on monomials.
One difficulty of rewriting polynomials is that the naive notion of rewriting path (obtained as
the closure of the generating rewriting relations under reflexivity, transitivity, sum and product
by a scalar) does not terminate. Instead, one needs to first consider well-formed rewriting steps
before forming the reflexive transitive closure.
Nevertheless the Diamond lemma has proved to be very useful : together with the works of
Bokut [3], it has given birth the theory of noncommutative Gröbner bases [7]. The latter have
provided applications to various areas of noncommutative algebra such as the study of embed-
ding problems (which appear in the works of Bokut and Bergman), homological algebra [4, 5]
or Koszul duality [1, 9].
Computation of normal forms in noncommutative algebra is also used to provide formal
solutions to partial differential equations. In this framework, a confluence criterion analogous
to the Diamond Lemma is given by Janet bases [10], which specify a deterministic way to reduce
each polynomial into normal form using standard reductions [6]. The confluence criterion may
then be asserted as follows: for each monomial m and each non-standard reduction m → f , f
is reducible into m̂, where the latter is obtained from m using only standard reductions.
In the presented paper, we propose an extension of the Diamond Lemma which offers two
improvements over the one from Bergman: first it allows the treatment of non-terminating
rewriting relations, and second it does not presuppose a notion of well-formed rewriting steps.
This last property seems particularly promising in order to extend the Diamond Lemma to
other structures.
Instead of supposing that the rewriting relation studied is terminating, we suppose given an
ordering on the monomials, independent of the rewriting relation. We then use methods based
on standard reductions: for every monomial m, we select exactly one reduction with left-hand
side m, which is decreasing for the ordering chosen. Such choices induce a deterministic way
to reduce each polynomial, obtained by applying simultaneously standard reductions on every
monomial appearing in its decomposition. When these deterministic reductions terminate, one
defines an operator which maps every polynomial to its unique standard normal form.
From this operator, we define a suitable notion of confluence in our setting, and show in
Proposition 3.4 that it implies the usual notion of confluence for the rewriting system studied.
We then provide an effective method for checking this criterion in Theorem 3.7. This method is
based on a local analysis corresponding to checking local confluence on monomials. In particular,
when the rewriting system is terminating, we show (Theorem 3.9) that we recover the Diamond
Lemma as a particular case of Theorem 3.7.
2 Local strategies and h-normal forms
We fix a commutative field K as well as a well-founded partially ordered set (X,<X). We
denote by KX the vector space spanned by X: an element v ∈ KX is a finite formal linear
combination of elements of X with coefficients in K. The sum of u =
∑
λxx and v =
∑
µxx
equals
∑
(λx + µx)x and the product of λ ∈ K by v equals
∑
(λλx)x. For every v ∈ KX, there
exists a unique finite set supp(v) ⊂ X, called the support of v, such that
v =
∑
x∈supp(v)
λxx and x ∈ supp(v)⇒ λx 6= 0. (1)
We extend the order <X into the multiset order on KX, denoted <KX : for any u, v ∈ KX,
u <KX v if supp(u) 6= supp(v) and for any x ∈ supp(u) \ supp(v), there exists y ∈ supp(v) \
supp(u) such that y >X x. Note that <KX and <X coincide when restricted to X, so we simply
denote this order by < in the rest of this paper.
We fix a set R ⊆ X × KX which represents rewrite rules of the form x −→
R
r. The set R
induces a rewriting relation on KX which reduces many x’s at once and defined as follows:∑
x
λxx+ v −→
R
∑
x
λxrx + v, (2)
where v is any element of KX, and for any x ∈ X appearing in the sum, λx 6= 0, x −→
R
rx ∈ R
and x /∈ supp(v). Finally we denote by ∗←→
R
the closure of −→
R
under transitivity, symmetry
and sum.
Definition 2.1. A local strategy h for R is the choice, for every x ∈ X not minimal for <, of
a rewriting rule hx = x −→
R
rx such that rx < x.
In the rest of this paper, we suppose chosen such a local strategy h (note that such an h may
not exist). Any vector v can be decomposed in a unique way as
∑
λxx+v
′, where y ∈ supp(v′)
implies that y is minimal for <, and x ∈ supp(v)\supp(v′) is not. We define a rewriting relation
−→
h
as follows:
hv =
∑
λxx+ v
′ −→
h
∑
λxrx + v
′, (3)
where for every x, hx = x −→
R
rx. Note in particular that if x is minimal for <, then hx = x→ x
is the identity on x.
Definition 2.2. A vector v is said to be an h-normal form if it is a normal form for −→
h
.
Example 2.3. Let X = {x, y, z, t}, x −→
R
y, y −→
R
z + t, z −→
R
y − t. Note that this is not
terminating since we have the infinite loop y −→
R
z+ t −→
R
(y− t)+ t = y. We choose the order
x > y > z, t, and the following distinguished rewrite rules: hx = x −→
h
y and hy = y −→
h
z + t.
Then the R-normal forms are the λtt, while the h-normal forms are all the expressions of the
form λtt+ λzz.
Lemma 2.4. Let v be a vector in KX. Either v is minimal for <, or there exists v′ < v such
that v −→
h
v′. In particular, h-normal forms are precisely the minimal elements of KX for <.
For each v ∈ KX and strategy local strategy h, there exists at most one v′ such that v −→
h
v′,
and −→
h
is compatible with the termination order <. As a consequence, any v ∈ KX is sent by
multiple applications of −→
h
to a unique h-normal form that we denote by H(v). This defines
a map H : KX → KX.
Proposition 2.5. The map H is a linear projector, in the sense that for all u, v ∈ KX and
λ ∈ K, H(u+ v) = H(u) +H(v), H(λu) = λH(u) and H(H(u)) = H(u).
Proof. The h-normal forms are closed under sums, so that H(H(v)) = H(v) for every v, that
is H is a projector. Moreover, if u −→
h
u′ and v −→
h
v′, then we have u + v −→
h
u′ + v′.
Iterating −→
h
, we get H(u+ v) = H(H(u) +H(v)) = H(u) +H(v).
3 A confluence criterion
In this section we investigate the confluence properties of R. The main idea behind this section
is that under suitable hypothesis −→
h
should form a terminating, confluent subrelation of −→
R
.
We start in Definition 3.1 and the following propositions by relating the confluence of −→
R
to properties on h. Then Theorem 3.7, we prove a confluence criterion to check whether R
satisfies Definition 3.1.
Definition 3.1. We say that R is h-confluent if for every rewrite rule x −→
R
v ∈ R, we have
H(x− v) = 0.
Example 3.2. Let us take the same example as in Example 2.3. We have three equations to
check:
H(x) = z + t = H(y), H(y) = z + t = H(z + t), H(z) = z = H(y − t),
and so R is h-confluent. Replacing the rule z −→
R
y − t by z −→
R
y, we get H(z) = z and
H(y) = z + t, so R is not h-confluent anymore.
Proposition 3.3. If R is h-confluent, then u ∗←→
R
v if and only if H(u− v) = 0.
Proof. The relation ∗←→
R
is the closure of −→
R
under transitivity, symmetry and sum. Since the
relation H(u− v) = 0 is closed under these operations, we get one implication.
Reciprocally, if H(u − v) = 0 then by definition of H we have u ∗←→
h
v, and in particular
u
∗←→
R
v.
Proposition 3.4. If R is h-confluent then −→
R
is confluent.
Proof. Let v, v1, v2 ∈ KX be such that v ∗−→
R
vi, for i = 1, 2. From Proposition 3.3, v1 − v2
belongs to ker(H), that is H(v1) = H(v2). Denoting by u the common value, we get vi
∗−→
R
u,
which proves the proposition.
Note that the previous proposition is a sufficient but not a necessary condition: taking X
to be the integers, with the relations n −→
R
n+ 1 is confluent, but there exist no local strategy
h making R h-confluent.
We now introduce our criterion to show that R is h-confluent. For that, we assume that the
set of relations R is equipped with a well-founded order ≺ satisfying the following decreasingness
property:
Definition 3.5. We say that R is locally h-confluent if for every x ∈ X and f = x −→
R
v, then
letting hx = x −→
h
rx, we have the confluence diagram:
x
f //
hx

vOO
∗

rx oo
∗ // v′,
where each rewriting step occurring in the dotted arrows is strictly smaller than f with respect
to ≺.
Example 3.6. Continuing with Example 2.3, let us define an order ≺ on R by the following
ordering: (x −→
R
y), (y −→
R
z + t) ≺ (z −→
R
y − t). This is guided by the heuristic that rules
advancing towards an h-normal form should be favored over rules that do not: here z is an
h-normal form so the rule rewriting it is large for ≺. The following diagrams show that R is
locally h-confluent:
x
R
//
hx

y
y y
y
R
//
hy

z + t
z + t z + t
z
R
//
hz
y − t
R

z z
Our main result is the following.
Theorem 3.7. If R is locally h-confluent, then R is h-confluent. In particular, −→
R
is confluent.
Proof. We reason by induction on r according to the order ≺. Looking at the square corre-
sponding to r:
x
r //
hx

vOO
∗

rx oo
∗ // v′,
we have H(x) = H(rx) by definition of H, and H(rx) = H(v′) = H(v) by induction hypothesis,
which concludes the proof.
Remark 3.8. Local h-confluence implies that the pair of rewriting relations (−→
h
,−→
R
) is
decreasing with respect to conversions (see [11, Definition 3]), using the order ≺ on R and the
discrete ordering on −→
h
. By [11, Theorem 3], this implies that (−→
h
,−→
R
) commute. Using the
fact that −→
h
⊆−→
R
, one can then recover that −→
R
is confluent.
Let us show how the Diamond Lemma fits as a particular case of our setup.
Theorem 3.9 ([2]). Assume that −→
R
is terminating and that for every x ∈ X, x −→
R
r and
x −→
R
r′ ∈ R, r and r′ are joinable. Then, −→
R
is confluent.
Proof. We define an ordering x > y on X as the transitive closure of the relation “there exists
v ∈ KX such that x −→
R
v and y ∈ supp(v)”. This is well-founded since by hypothesis −→
R
is
terminating. By definition, if x ∈ X is not minimal for >, then x is not an R-normal form.
Let us fix an arbitrary rewriting step hx = x −→
h
rx. By definition of >, for any y ∈ supp(rx)
we have y < x and so rx < x, which shows that h is a local strategy. Ordering the rewrite
rules by their left hand sides makes R locally h-confluent. Theorem 3.7 finally shows that R is
confluent.
Conclusion. We introduced a sufficient condition, based on deterministic reduction strate-
gies, of confluence for rewriting systems on vector spaces. As a particular case, we recover the
Diamond Lemma. This work maybe extended in particular into two main directions. The first
one consists in weakening our assumption on the set K of coefficients, by allowing non invertible
coefficients. A second extension consists in characterising Janet bases in this framework, with
the objective to develop constructive methods in the analysis and formal resolution of PDE’s.
References
[1] Roland Berger. Koszulity for nonquadratic algebras. J. Algebra, 239(2):705–734, 2001.
[2] George M. Bergman. The diamond lemma for ring theory. Adv. in Math., 29(2):178–218, 1978.
[3] Leonid A. Bokut′. Imbeddings into simple associative algebras. Algebra i Logika, 15(2):117–142,
245, 1976.
[4] Yuji Kobayashi. Complete rewriting systems and homology of monoid algebras. J. Pure Appl.
Algebra, 65(3):263–275, 1990.
[5] Yuji Kobayashi. Gröbner bases of associative algebras and the Hochschild cohomology. Trans.
Amer. Math. Soc., 357(3):1095–1124, 2005.
[6] Paul-André Melliès. Axiomatic Rewriting Theory I: A Diagrammatic Standardization Theorem,
pages 554–638. Springer Berlin Heidelberg, Berlin, Heidelberg, 2005.
[7] Teo Mora. An introduction to commutative and noncommutative Gröbner bases. Theoret. Comput.
Sci., 134(1):131–173, 1994.
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Math. (2), 43:223–243, 1942.
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[10] Werner M. Seiler. Spencer cohomology, differential equations, and Pommaret bases. In Gröbner
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The Diamond Lemma for non-terminating rewriting systems using deterministic reduction strategies

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HAL Id: hal-02385139https://hal.archives-ouvertes.fr/hal-02385139Submitted on 28 Nov 2019HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.The Diamond Lemma for non-terminating rewritingsystems using deterministic reduction strategiesCyrille Chenavier, Maxime LucasTo cite this version:Cyrille Chenavier, Maxime Lucas. The Diamond Lemma for non-terminating rewriting systems usingdeterministic reduction strategies. IWC 2019 - 8th International Workshop on Confluence, Jun 2019,Dortmund, Germany. pp.1-5. ￿hal-02385139￿The Diamond Lemma for non-terminating rewritingsystems using deterministic reduction strategiesCyrille Chenavier1 and Maxime Lucas21 Inria Lille - Nord Europe, Équipe Valsecyrille.chenavier@inria.fr2 Inria Rennes - Bretagne Atlantique, Équipe Gallinettemaxime.lucas@inria.frAbstractWe study the confluence property for rewriting systems whose underlying set of termsadmits a vector space structure. For that, we use deterministic reduction strategies. Thesestrategies are based on the choice of standard reductions applied to basis elements. Weprovide a sufficient condition of confluence in terms of the kernel of the operator whichcomputes standard normal forms. We present a local criterion which enables us to checkthe confluence property in this framework. We show how this criterion is related to theDiamond Lemma for terminating rewriting systems.1 IntroductionThe fact that local confluence together with termination implies confluence has been knownfor abstract rewriting systems since Newman’s work [8]. For rewriting on noncommutativepolynomials, a similar result known as the Diamond lemma was introduced by Bergman [2]nearly 30 years later, in order to compute normal forms in noncommutative algebras usingrewriting theory. It asserts that for terminating rewriting systems, the local confluence propertycan be checked on monomials.One difficulty of rewriting polynomials is that the naive notion of rewriting path (obtained asthe closure of the generating rewriting relations under reflexivity, transitivity, sum and productby a scalar) does not terminate. Instead, one needs to first consider well-formed rewriting stepsbefore forming the reflexive transitive closure.Nevertheless the Diamond lemma has proved to be very useful : together with the works ofBokut [3], it has given birth the theory of noncommutative Gröbner bases [7]. The latter haveprovided applications to various areas of noncommutative algebra such as the study of embed-ding problems (which appear in the works of Bokut and Bergman), homological algebra [4, 5]or Koszul duality [1, 9].Computation of normal forms in noncommutative algebra is also used to provide formalsolutions to partial differential equations. In this framework, a confluence criterion analogousto the Diamond Lemma is given by Janet bases [10], which specify a deterministic way to reduceeach polynomial into normal form using standard reductions [6]. The confluence criterion maythen be asserted as follows: for each monomial m and each non-standard reduction m → f , fis reducible into m̂, where the latter is obtained from m using only standard reductions.In the presented paper, we propose an extension of the Diamond Lemma which offers twoimprovements over the one from Bergman: first it allows the treatment of non-terminatingrewriting relations, and second it does not presuppose a notion of well-formed rewriting steps.This last property seems particularly promising in order to extend the Diamond Lemma toother structures.Instead of supposing that the rewriting relation studied is terminating, we suppose given anordering on the monomials, independent of the rewriting relation. We then use methods basedon standard reductions: for every monomial m, we select exactly one reduction with left-handside m, which is decreasing for the ordering chosen. Such choices induce a deterministic wayto reduce each polynomial, obtained by applying simultaneously standard reductions on everymonomial appearing in its decomposition. When these deterministic reductions terminate, onedefines an operator which maps every polynomial to its unique standard normal form.From this operator, we define a suitable notion of confluence in our setting, and show inProposition 3.4 that it implies the usual notion of confluence for the rewriting system studied.We then provide an effective method for checking this criterion in Theorem 3.7. This method isbased on a local analysis corresponding to checking local confluence on monomials. In particular,when the rewriting system is terminating, we show (Theorem 3.9) that we recover the DiamondLemma as a particular case of Theorem 3.7.2 Local strategies and h-normal formsWe fix a commutative field K as well as a well-founded partially ordered set (X,<X). Wedenote by KX the vector space spanned by X: an element v ∈ KX is a finite formal linearcombination of elements of X with coefficients in K. The sum of u =∑λxx and v =∑µxxequals∑(λx + µx)x and the product of λ ∈ K by v equals∑(λλx)x. For every v ∈ KX, thereexists a unique finite set supp(v) ⊂ X, called the support of v, such thatv =∑x∈supp(v)λxx and x ∈ supp(v)⇒ λx 6= 0. (1)We extend the order <X into the multiset order on KX, denoted <KX : for any u, v ∈ KX,u <KX v if supp(u) 6= supp(v) and for any x ∈ supp(u) \ supp(v), there exists y ∈ supp(v) \supp(u) such that y >X x. Note that <KX and <X coincide when restricted to X, so we simplydenote this order by < in the rest of this paper.We fix a set R ⊆ X × KX which represents rewrite rules of the form x −→Rr. The set Rinduces a rewriting relation on KX which reduces many x’s at once and defined as follows:∑xλxx+ v −→R∑xλxrx + v, (2)where v is any element of KX, and for any x ∈ X appearing in the sum, λx 6= 0, x −→Rrx ∈ Rand x /∈ supp(v). Finally we denote by ∗←→Rthe closure of −→Runder transitivity, symmetryand sum.Definition 2.1. A local strategy h for R is the choice, for every x ∈ X not minimal for <, ofa rewriting rule hx = x −→Rrx such that rx < x.In the rest of this paper, we suppose chosen such a local strategy h (note that such an h maynot exist). Any vector v can be decomposed in a unique way as∑λxx+v′, where y ∈ supp(v′)implies that y is minimal for <, and x ∈ supp(v)\supp(v′) is not. We define a rewriting relation−→has follows:hv =∑λxx+ v′ −→h∑λxrx + v′, (3)where for every x, hx = x −→Rrx. Note in particular that if x is minimal for <, then hx = x→ xis the identity on x.Definition 2.2. A vector v is said to be an h-normal form if it is a normal form for −→h.Example 2.3. Let X = {x, y, z, t}, x −→Ry, y −→Rz + t, z −→Ry − t. Note that this is notterminating since we have the infinite loop y −→Rz+ t −→R(y− t)+ t = y. We choose the orderx > y > z, t, and the following distinguished rewrite rules: hx = x −→hy and hy = y −→hz + t.Then the R-normal forms are the λtt, while the h-normal forms are all the expressions of theform λtt+ λzz.Lemma 2.4. Let v be a vector in KX. Either v is minimal for <, or there exists v′ < v suchthat v −→hv′. In particular, h-normal forms are precisely the minimal elements of KX for <.For each v ∈ KX and strategy local strategy h, there exists at most one v′ such that v −→hv′,and −→his compatible with the termination order <. As a consequence, any v ∈ KX is sent bymultiple applications of −→hto a unique h-normal form that we denote by H(v). This definesa map H : KX → KX.Proposition 2.5. The map H is a linear projector, in the sense that for all u, v ∈ KX andλ ∈ K, H(u+ v) = H(u) +H(v), H(λu) = λH(u) and H(H(u)) = H(u).Proof. The h-normal forms are closed under sums, so that H(H(v)) = H(v) for every v, thatis H is a projector. Moreover, if u −→hu′ and v −→hv′, then we have u + v −→hu′ + v′.Iterating −→h, we get H(u+ v) = H(H(u) +H(v)) = H(u) +H(v).3 A confluence criterionIn this section we investigate the confluence properties of R. The main idea behind this sectionis that under suitable hypothesis −→hshould form a terminating, confluent subrelation of −→R.We start in Definition 3.1 and the following propositions by relating the confluence of −→Rto properties on h. Then Theorem 3.7, we prove a confluence criterion to check whether Rsatisfies Definition 3.1.Definition 3.1. We say that R is h-confluent if for every rewrite rule x −→Rv ∈ R, we haveH(x− v) = 0.Example 3.2. Let us take the same example as in Example 2.3. We have three equations tocheck:H(x) = z + t = H(y), H(y) = z + t = H(z + t), H(z) = z = H(y − t),and so R is h-confluent. Replacing the rule z −→Ry − t by z −→Ry, we get H(z) = z andH(y) = z + t, so R is not h-confluent anymore.Proposition 3.3. If R is h-confluent, then u ∗←→Rv if and only if H(u− v) = 0.Proof. The relation ∗←→Ris the closure of −→Runder transitivity, symmetry and sum. Since therelation H(u− v) = 0 is closed under these operations, we get one implication.Reciprocally, if H(u − v) = 0 then by definition of H we have u ∗←→hv, and in particularu∗←→Rv.Proposition 3.4. If R is h-confluent then −→Ris confluent.Proof. Let v, v1, v2 ∈ KX be such that v∗−→Rvi, for i = 1, 2. From Proposition 3.3, v1 − v2belongs to ker(H), that is H(v1) = H(v2). Denoting by u the common value, we get vi∗−→Ru,which proves the proposition.Note that the previous proposition is a sufficient but not a necessary condition: taking Xto be the integers, with the relations n −→Rn+ 1 is confluent, but there exist no local strategyh making R h-confluent.We now introduce our criterion to show that R is h-confluent. For that, we assume that theset of relations R is equipped with a well-founded order ≺ satisfying the following decreasingnessproperty:Definition 3.5. We say that R is locally h-confluent if for every x ∈ X and f = x −→Rv, thenletting hx = x −→hrx, we have the confluence diagram:xf //hxvOO∗rx oo∗ // v′,where each rewriting step occurring in the dotted arrows is strictly smaller than f with respectto ≺.Example 3.6. Continuing with Example 2.3, let us define an order ≺ on R by the followingordering: (x −→Ry), (y −→Rz + t) ≺ (z −→Ry − t). This is guided by the heuristic that rulesadvancing towards an h-normal form should be favored over rules that do not: here z is anh-normal form so the rule rewriting it is large for ≺. The following diagrams show that R islocally h-confluent:xR//hxyy yyR//hyz + tz + t z + tzR//hzy − tRz zOur main result is the following.Theorem 3.7. If R is locally h-confluent, then R is h-confluent. In particular, −→Ris confluent.Proof. We reason by induction on r according to the order ≺. Looking at the square corre-sponding to r:xr //hxvOO∗rx oo∗ // v′,we have H(x) = H(rx) by definition of H, and H(rx) = H(v′) = H(v) by induction hypothesis,which concludes the proof.Remark 3.8. Local h-confluence implies that the pair of rewriting relations (−→h,−→R) isdecreasing with respect to conversions (see [11, Definition 3]), using the order ≺ on R and thediscrete ordering on −→h. By [11, Theorem 3], this implies that (−→h,−→R) commute. Using thefact that −→h⊆−→R, one can then recover that −→Ris confluent.Let us show how the Diamond Lemma fits as a particular case of our setup.Theorem 3.9 ([2]). Assume that −→Ris terminating and that for every x ∈ X, x −→Rr andx −→Rr′ ∈ R, r and r′ are joinable. Then, −→Ris confluent.Proof. We define an ordering x > y on X as the transitive closure of the relation “there existsv ∈ KX such that x −→Rv and y ∈ supp(v)”. This is well-founded since by hypothesis −→Risterminating. By definition, if x ∈ X is not minimal for >, then x is not an R-normal form.Let us fix an arbitrary rewriting step hx = x −→hrx. By definition of >, for any y ∈ supp(rx)we have y < x and so rx < x, which shows that h is a local strategy. Ordering the rewriterules by their left hand sides makes R locally h-confluent. Theorem 3.7 finally shows that R isconfluent.Conclusion. We introduced a sufficient condition, based on deterministic reduction strate-gies, of confluence for rewriting systems on vector spaces. As a particular case, we recover theDiamond Lemma. This work maybe extended in particular into two main directions. The firstone consists in weakening our assumption on the set K of coefficients, by allowing non invertiblecoefficients. A second extension consists in characterising Janet bases in this framework, withthe objective to develop constructive methods in the analysis and formal resolution of PDE’s.References[1] Roland Berger. Koszulity for nonquadratic algebras. J. Algebra, 239(2):705–734, 2001.[2] George M. Bergman. The diamond lemma for ring theory. Adv. in Math., 29(2):178–218, 1978.[3] Leonid A. Bokut′. Imbeddings into simple associative algebras. Algebra i Logika, 15(2):117–142,245, 1976.[4] Yuji Kobayashi. Complete rewriting systems and homology of monoid algebras. J. Pure Appl.Algebra, 65(3):263–275, 1990.[5] Yuji Kobayashi. Gröbner bases of associative algebras and the Hochschild cohomology. Trans.Amer. Math. Soc., 357(3):1095–1124, 2005.[6] Paul-André Melliès. Axiomatic Rewriting Theory I: A Diagrammatic Standardization Theorem,pages 554–638. Springer Berlin Heidelberg, Berlin, Heidelberg, 2005.[7] Teo Mora. An introduction to commutative and noncommutative Gröbner bases. Theoret. Comput.Sci., 134(1):131–173, 1994.[8] Maxwell H. A. Newman. On theories with a combinatorial definition of “equivalence.”. Ann. ofMath. (2), 43:223–243, 1942.[9] Stewart B. Priddy. Koszul resolutions. Trans. Amer. Math. Soc., 152:39–60, 1970.[10] Werner M. Seiler. Spencer cohomology, differential equations, and Pommaret bases. In Gröbnerbases in symbolic analysis, volume 2 of Radon Ser. Comput. Appl. Math., pages 169–216. Walterde Gruyter, Berlin, 2007.[11] Vincent Van Oostrom. Confluence by decreasing diagrams. In International Conference on Rewrit-ing Techniques and Applications, pages 306–320. Springer, 2008.

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The Diamond Lemma for non-terminating rewriting systems using deterministic reduction strategies

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