Equational weighted tree transformations

Acta Informatica (2012) 49:29–52DOI 10.1007/s00236-011-0148-5

ORIGINAL ARTICLE

Equational weighted tree transformations

Symeon Bozapalidis · Zoltán Fülöp · George Rahonis

Received: 31 March 2011 / Accepted: 22 November 2011 / Published online: 6 December 2011© Springer-Verlag 2011

Abstract We consider systems of equations of weighted tree transformations with finitesupport over continuous and commutative semirings. We define a weighted relation to beequational, if it is a component of the least solution of such a system of equations in a pair ofalgebras. In particular, we focus on equational weighted tree transformations which are equa-tional relations obtained by considering the least solutions of such systems in pairs of termalgebras. We characterize equational weighted tree transformations in terms of weighted treetransformations defined by different weighted bimorphisms. To demonstrate the robustnessof equational weighted tree transformations, we give an equational definition of the classof linear and nondeleting weighted top-down tree transformations and of the class of linearand nondeleting weighted extended top-down tree transformations. Finally, we prove that aweighted relation is equational if and only if it is, roughly speaking, the morphic image of aweighted equational tree transformation.

1 Introduction

The classical theory of recognizable tree languages and tree transformations [12,28,29] hasbeen generalized to weighted recognizable tree languages and weighted tree transformations(also called tree series and tree series transformations, respectively). Roughly speaking, in

Research of the second author was supported by the TÁMOP-4.2.1/B-09/1/KONV-2010-0005 program ofthe Hungarian National Development Agency and the third author by the RISC-Linz Transnational AccessProgram, project SCIEnce (Contract No. 026133) of the European Commission FP6 for IntegratedInfrastructures Initiatives.

S. Bozapalidis · G. RahonisDepartment of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greecee-mail: [email protected]

G. Rahonise-mail: [email protected]

Z. Fülöp (B)Department of Computer Science, University of Szeged, Árpád tér 2., 6720 Szeged, Hungarye-mail: [email protected]

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30 S. Bozapalidis et al.

the generalization a weight is associated to each tree of a tree language, as to each pair oftrees of a tree transformation. The weights are taken from a semiring [30] or other appropri-ate algebraic structures. The weights make it possible to investigate tree languages and treetransformations from a qualitative but also from a quantitative point of view. For instance, inthe classical theory we are interested in whether an output tree is a translation of a given inputtree, but using the semiring of real numbers we can ask for the probability of this event. Theclassical theory, which we call the unweighted case sometimes, is reobtained as the particular“weighted theory” using the Boolean semiring.

Recognizable weighted tree languages over semirings were defined and considered in [1],over fields in [4], and then investigated in several works, cf. e.g. [6–8,23,26]. The structureof weights is a complete distributive lattice in [24], a strong bimonoid in [17], and a multiop-erator monoid in [34]. In this paper, we will mainly be interested in recognizable weightedtree languages over continuous semirings, which were considered in [9,23,31,32].

The main automaton model for recognizable weighted tree languages is the K -�-treeautomaton introduced in [1], where K is the underlying semiring and � is the ranked alpha-bet of input tree symbols. Recently this model is rather called a weighted tree automaton.However, in the works [9,23,31,32] a recognizable weighted tree language is defined as acomponent of the least fixpoint of a system of equations of weighted tree languages overa continuous and commutative semiring. Since the underlying semiring is continuous, thespace of the potential solutions of the system becomes a complete poset and the system canbe realized as a continuous mapping over that space. Hence, the classical fixpoint theoremassures that the minimal solution of the system exists. This approach is the generalizationof the equational definition of recognizable tree languages given in [28,29]. The idea comesfrom [39], in which the equational definition of a recognizable subset of an arbitrary algebrawas given.

Weighted tree transformations are defined as the semantics of machines called weightedtree transducers. There are two kinds of semantics, the initial algebra one [22,33–35], whichallows statements to be proved by induction easily, and the rewriting one [25,27], which hasimportance in practical applications like natural language processing. An equational defini-tion of (unweighted) tree transformations was given only recently in [10]. The aim of thispaper is to give an equational definition for weighted tree transformations over a continuousand commutative semiring and hereby generalize the results of [10]. Let us mention that,before [10] and our present work, several other papers in the literature dealt with differentinterpretations of the equational approach of [39], see [14,15,20,21] for instance.

We introduce the concept of an equational weighted relation and, as a particular case of it,of an equational weighted tree transformation in the following way. We generalize [IO]-sub-stitution of weighted tree languages (cf. [11]) to [IO]-substitution of weighted relations overthe direct product of two algebras in weighted tree transformations with variables. Moreover,we introduce the concept of a system of equations of weighted tree transformations. Such asystem (E) consists of n ≥ 1 equations of the form xi = ρi , whereρi ∈ K 〈T�(Xn)×T�(Xn)〉is a weighted tree transformation of finite support over the ranked alphabets�,�; the variableset Xn = {x1, . . . , xn}; and the continuous and commutative semiring K for every 1 ≤ i ≤ n.For all algebras A = (A, �A) and B = (B,�B), the system (E) under the [IO]-substitutionmode induces a continuous mapping �A,B

E over the complete poset K 〈〈A × B〉〉n , whereK 〈〈A × B〉〉 is the class of weighted relations from A to B. Then the classical fixpoint theo-rem guarantees that the least fixpoint fix�A,B

E of that mapping exists, which we call the least[IO]-solution of (E) in (A,B, K ). A weighted relation in K 〈〈A × B〉〉 is [IO]-equational if itappears as a component of the least [IO]-solution of a system (E) of equations of weightedtree transformations in (A,B, K ). Finally, a weighted tree transformation τ ∈ K 〈〈T� × T�〉〉

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Equational weighted tree transformations 31

is [IO]-equational, if it is an [IO]-equational relation with A = T� and B = T�, which arethe corresponding term algebras over � and �, respectively.

In our paper, we focus on equational weighted tree transformations. We give a sufficientcondition for the existence and uniqueness of the [IO]-solution of a system of equations ofweighted tree transformations in the corresponding term algebras. We characterize weightedtree transformations in terms of weighted bimorphisms of [26,36]. In fact, we show that theclass of [IO]-equational weighted tree transformations coincides with the class of weightedtree transformations defined by jointly complete bimorphisms.

Then we consider two well-known weighted tree transformation classes which are charac-terized in terms of weighted bimorphisms in [25,37], and give a characterization for them interms of equational weighted tree transformations. These classes are ln-T O P and ln-XT O P ,i.e., the classes of weighted tree transformations computed by linear and nondeleting top-down weighted tree transducers, and by linear and nondeleting top-down weighted extendedtree transducers, respectively.

Finally, we establish a Mezei–Wright like relationship between [IO]-equational weightedtree transformations and [IO]-equational weighted relations. Namely we show that a weightedrelation is [IO]-equational if and only if it is, roughly speaking, the morphic image of an[IO]-equational weighted tree transformation.

The paper is organized as follows. In Sect. 2, we introduce the necessary notions andnotation. In Sect. 3, we define [IO]-substitution of weighted relations in weighted tree trans-formations with variables, and in Sect. 4, we prove some technical results concerning seriesand substitutions in weighted tree transformations. In Sect. 5, we define the concept of asystem of equations of weighted tree transformations and of [IO]-equational weighted rela-tions and tree transformations. In Sect. 6, we give the characterization of [IO]-equationalweighted tree transformations in terms of weighted bimorphisms, and we give equationalcharacterizations for the mentioned fundamental classes of weighted tree transformations.In Sect. 7, we prove the Mezei–Wright like characterization of [IO]-equational weightedrelations.

2 Definitions

2.1 General notation

We denote by N the set of nonnegative integers.Let V be a set, n ≥ 1, 1 ≤ i1 < · · · < ik ≤ n, and a1, . . . , ak ∈ V . We introduce a

notation for the set of those elements of V n , each of which has a j as its i j th component forj = 1, . . . , k. Namely, we set

V n |(i1,a1)...(ik ,ak ) = {(b1, . . . , bn) ∈ V n | bi1 = a1, . . . , bik = ak}.

For n = 0, we define V n = {( )} (even if V = ∅), where ( ) is the empty vector.

2.2 Fixpoint theorem

A partially ordered set (for short: poset) is a pair (V,≤), where V is a set and ≤ is a partialorder, i.e., a reflexive, antisymmetric, and transitive relation on V . We will write just Vfor (V,≤) and, for every V ′ ⊆ V , we denote by sup V ′ the supremum of V ′ in V pro-vided it exists. A poset V is called ω-complete if it has a least element ⊥ and every ω-chaina0 ≤ a1 ≤ . . . in V has a supremum in V , denoted by supi≥0 ai .

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Let f : V → V be a mapping. A fixpoint of f is an element a ∈ V such that f (a) = a.A fixpoint a of f is the least fixpoint if a ≤ a′ for every fixpoint a′ of f . Moreover, f iscalled ω-continuous if for every ω-chain a0 ≤ a1 ≤ . . . in V which has a supremum, thesupremum of { f (ai ) | i ≥ 0} exists and f (supi≥0 ai ) = sup{ f (ai ) | i ≥ 0}. It is obviousthat if f is ω-continuous, then it is monotonic, meaning that f (a) ≤ f (a′) whenever a ≤ a′for all a, a′ ∈ V . We will use the following result, called fixpoint theorem, cf. e.g. [40, Sect.1.5, Thm. 7].

Proposition 1 Let (V,≤) be an ω-complete poset and f : V → V an ω-continuous map-ping. Then f has a least fixpoint fix f , and fix f = sup{ f (i)(⊥) | i ≥ 0}, where f (i) denotesthe i-fold application of f .

2.3 Semirings and �-algebras

Let (M,+, 0), denoted also by M , be a commutative monoid. Then M is called naturallyordered if the relation defined by

m1 ≤ m2 ⇐⇒ there exists m ∈ M such that m2 = m1 + m for all m1,m2 ∈ M

is a partial order on M .A sum operation

∑over M is a family (

∑I | I is an arbitrary set), where

∑I associates

with every family (mi | i ∈ I ) of elements of M a further element of M which we denote by∑i∈I mi . Then M equipped with a sum operation

∑is called

∑-complete if the following

conditions hold

(i)∑

i∈{ j} mi = m j ,∑

i∈{ j,k} mi = m j + mk, for j �= k,

(ii)∑

j∈J

(∑i∈I j

mi

)= ∑i∈I mi , for all index sets I, J , and I j for every j ∈ J and

elements mi ∈ M with i ∈ I such that⋃

j∈J I j = I and I j ∩ I j ′ = ∅ for j �= j ′.A naturally ordered and

∑-complete monoid M is

∑-continuous if, for every set I , family

(mi | i ∈ I ) of elements of M , and m ∈ M the following implication holds:

if∑

i∈F mi ≤ m for every finite subset F of I , then∑

i∈I mi ≤ m.

Finally, M is complete (resp. continuous), if there is a sum operation∑

over M such that M is∑-complete (resp.

∑-continuous). We recall a well-known result, which we will need later.

Proposition 2 (cf. [38, p. 195, Thm. 8], [31, Thm. 2.3]) Let M be a continuous monoid andm0 ≤ m1 ≤ . . . be an ω-chain in M with mk = ∑0≤i≤kni for every k ≥ 0, where ni ∈ Mfor 0 ≤ i ≤ k. Then supk≥0 mk exists and supk≥0 mk =∑i≥0 ni .

A commutative semiring (K ,+, ·, 0, 1) is an algebraic structure such that (K ,+, 0) and(K , ·, 1) are commutative monoids, 0 �= 1, the multiplication · distributes over addition +,and m · 0 = 0 for every m ∈ K . If no confusion arises, then we denote the commutative se-miring simply by K . Then K is called naturally ordered if the monoid (K ,+, 0) is naturallyordered. In a naturally ordered K , multiplication preserves the order ≤.

Moreover, K is called complete if there is a sum operation∑

over the monoid (K ,+, 0)such that this latter is

∑-complete and the generalized distributivity law

∑

i∈I

(m · mi ) = m ·(∑

i∈I

mi

)

is satisfied for each family (mi | i ∈ I ) of elements of K and m ∈ K (cf. [5,13,16,18,31]).

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Finally, K is called continuous if the monoid (K ,+, 0) is continuous. It follows fromAxiom (ii) and Proposition 2 that in a continuous K the multiplication is continuous, mean-ing that

m · sup{mi | i ≥ 0} = sup{m · mi | i ≥ 0}for every ω-chain m0 ≤ m1 ≤ . . . in K and m ∈ K .

Example 3 Examples of commutative semirings are the Boolean semiring (B,∨,∧, 0, 1),where B = {0, 1}, the semiring Nat∞ = (N ∪ {∞},+, ·, 0, 1) of natural numbers with infin-ity, the arctic semiring Arct = (N ∪ {∞,−∞},max,+,−∞, 0), and the tropical semiringTrop = (N ∪ {∞},min,+,∞, 0). Each of these semirings is continuous [16].

A commutative monoid (M,+, 0) is called a (left) K -semimodule if K acts on the left ofM , i.e., there is a mapping of type K × M → M such that

n(m1 + m2) = nm1 + nm2 n1 (n2m) = (n1 · n2)m

(n1 + n2)m = n1m + n2m 1m = m 0m = 0 = n0

for every n, n1, n2 ∈ K ,m,m1,m2 ∈ M . The K -semimodule (M,+, 0) is called completeif M is a complete monoid and the scalar product preserves (infinite) sums. Moreover, it iscontinuous if the monoid M is continuous.

A ranked alphabet is a pair (�, rk) (simply denoted by �) where � is a finite set andrk : � → N is the rank function. As usual, we set �k = {σ ∈ � | rk(σ ) = k} for everyk ≥ 0.

A �-algebra is a pair A = (A, �A) where A is a nonempty set, called the domain setof A, and �A is a family (σA | σ ∈ �) of operations on A such that for every k ≥ 0 andσ ∈ �k , we have σA : Ak → A. In particular, σA ∈ A for every σ ∈ �0. If no confusionarises, then sometimes we drop A from�A and σA in what follows. Given a further�-alge-bra B = (B, �), a �-algebra morphism from A to B is a mapping H : A → B such thatH(σA(a1, . . . , ak)

) = σB (H(a1), . . . , H(ak)) for σ ∈ �k, k ≥ 0, and a1, . . . , ak ∈ A.Hence H

(σA) = σB for every σ ∈ �0.

2.4 Series and weighted relations

In the rest of the paper K will denote a continuous and commutative semiring.

Let A be a set. A series over A and K (or (A, K )) is a mapping η : A → K . For everya ∈ A, we write (η, a) for the value η(a) ∈ K and refer to it as the coefficient (or: weight)of a in η. The support of η is the set supp(η) = {a ∈ A | (η, a) �= 0}. If supp(η) is finite,then η is called a polynomial over (A, K ). We denote by K 〈〈A〉〉 (resp. K 〈A〉) the set of allseries (resp. polynomials) over (A, K ).

Let η, η′ ∈ K 〈〈A〉〉 and m ∈ K . The sum η + η′ and the scalar product mη are seriesin K 〈〈A〉〉 which are defined element-wise, i.e., for every a ∈ A we have

(η + η′, a

) =(η, a)+ (η′, a

)and (mη, a) = m · (η, a). The following statement is straightforward.

Proposition 4(

K 〈〈A〉〉,+, 0̃)

is a continuous and commutative monoid and, with the scalar

product defined above, it is a continuous K -semimodule. In particular, K 〈〈A〉〉 is a poset withrespect to the natural order.

For every a ∈ A and m ∈ K , we denote by m.a the series with value m at a and 0 for allother b ∈ A. Then we may write a series η as the sum

∑a∈A(η, a).a. In particular, if η is

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a polynomial, then we may write it as m1.a1 + · · · + mk .ak , where supp(η) = {a1, . . . , ak}and mi = (η, ai ) for every 1 ≤ i ≤ k. For every m ∈ K , we denote by m̃ the constant seriesdefined by (m̃, a) = m for every a ∈ A.

Now let B be a further set, f : A → B a mapping, and consider its extension f :K 〈〈A〉〉 → K 〈〈B〉〉 defined by

f (η) =∑

a∈A

(η, a). f (a)

for every η ∈ K 〈〈A〉〉. The proof of the following statement is easy. It can be obtained e.g. byadapting the proof of Lemma 6 of [11].

Proposition 5 The mapping f : K 〈〈A〉〉 → K 〈〈B〉〉 is a continuous mapping with respect tothe natural order.

Finally, a weighted relation over A, B, and K (or (A, B, K )) is a series over (A × B, K ).

2.5 Weighted tree languages and tree transformations

In the rest of the paper �,�, and will denote ranked alphabets such that eachcontains at least one nullary symbol.

Let V be a finite set with V ∩ � = ∅. The set T�(V ) of finite trees (or: terms) over� and V is defined by induction to be the smallest set T such that (i) V ⊆ T and (ii) ifk ≥ 0, σ ∈ �k, and t1, . . . , tk ∈ T , then σ(t1, . . . , tk) ∈ T . If σ ∈ �0, then we write just σfor σ( ) and we write T� for T�(∅). Note that T� �= ∅ since �0 �= ∅. For every V ′ ⊆ V , wedefine �(V ′) = {σ (v1, . . . , vk) | k ≥ 0, σ ∈ �k, and v1, . . . , vk ∈ V ′}.

A particular�-algebra is the term algebra T�(V ) = (T�(V ),�) of all trees over� and V ,where σT�(V )(t1, . . . , tk) = σ(t1, . . . , tk) for every k ≥ 0, σ ∈ �k , and t1, . . . , tk ∈ T�(V ).In fact, it is the free �-algebra generated by V in the class of all �-algebras, i.e., forevery �-algebra A, any mapping h : V → A extends uniquely to a �-algebra morphismh : T�(V ) → A. If V = ∅, then we denote the unique morphism from T� to A by HA.

Any subset of T�(V ) is called a tree language and any relation of the form S ⊆ T�(V )×T�(V ) is called a tree transformation.

Let X = {x1, x2, . . .} be a countably infinite set of variables, which is disjoint fromany ranked alphabet considered in the paper. We set Xn = {x1, . . . , xn} for n ≥ 0, henceX0 = ∅. Let t ∈ T�(Xn) be a tree. The set sub(t) ⊆ T�(Xn) of subtrees of t and the setvar(t) ⊆ Xn of variables in t are defined such that sub(t) = var(t) = {t} if t ∈ Xn ,and sub(t) = {t} ∪⋃k

i=1 sub(ti ) and var(t) = ⋃ki=1 var(ti ) if t = σ(t1, . . . , tk) for some

k ≥ 0, σ ∈ �k , and t1, . . . , tk ∈ T�(Xn). We denote by |t |xi the number of occurrences ofxi in t for all 1 ≤ i ≤ n. Then t is called linear (resp. nondeleting) in Xn if |t |xi ≤ 1 (resp.|t |xi ≥ 1) for every 1 ≤ i ≤ n. A subset L ⊆ T�(Xn) is linear (resp. nondeleting), if eacht ∈ L is linear (resp. nondeleting) in Xn . A pair (s, t) ∈ T�(Xn) × T�(Xn) is linear (resp.nondeleting) if both s and t are linear (resp. nondeleting) in Xn . Furthermore, it is calledweakly variable symmetric if var(s) = var(t) for all 1 ≤ i ≤ n. We lift these concepts toan arbitrary tree transformation R ⊆ T�(Xn)× T�(Xn) in the obvious way.

Let now � = {ξ1, ξ2, . . .} be another set of variables, which is disjoint from X and anyranked alphabet considered in the paper, and let �n = {ξ1, . . . , ξn} for every n ≥ 0. Wedefine tree substitution. For this, let V ⊆ X or V ⊆ �, let t, t1, . . . , tn ∈ T�(V ) and letv1, . . . , vn be pairwise different elements of V . We denote by t (t1/v1, . . . , tn/vn) the tree

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which we obtain by substituting simultaneously ti for every occurrence of vi in t for every1 ≤ i ≤ n. In particular, we abbreviate t (t1/x1, . . . , tn/xn) by t (t1, . . . , tn).

A tree homomorphism from � to � is a family of mappings (hk | k ≥ 0) such that forevery k ≥ 0, hk : �k → T� (�k). Such a tree homomorphism is called linear (for short l)(resp. nondeleting or complete, for short c) if for every k ≥ 1 and σ ∈ �k the tree hk(σ ) islinear (resp. nondeleting) in �k . It is called a relabeling if for every k ≥ 0 and σ ∈ �k wehave hk(σ ) = δ(ξ1, . . . , ξk) for some δ ∈ �k . We note that our relabeling is a restricted (infact deterministic) version of the one introduced in [19, Def. 3.1].

For every finite set V , the tree homomorphism (hk | k ≥ 0) induces a mapping h :T� (V ) → T� (V ) defined inductively in the following way. For every t ∈ T� (V ) welet

• h(t) = t if t ∈ V , and• h(t) = hk(σ )(h (t1) /ξ1, . . . , h (tk) /ξk) if t = σ (t1, . . . , tk) with k ≥ 0, σ ∈ �k , and

t1, . . . , tk ∈ T� (V ).

As usual, we also call the induced mapping h tree homomorphism. We will use the factwithout reference that the class of all tree homomorphisms is closed under composition [19].We denote by H the class of all tree homomorphisms and, for any combination w of l andc we denote by w-H the class of w-tree homomorphisms, respectively. The class of all rel-abelings is denoted by RE L . Finally, a pair (h, h′) of tree homomorphisms h : T (Xn) →T� (Xn) and h′ : T (Xn) → T� (Xn) is called jointly nondeleting (or: jointly complete) ifvar(hk(γ )) ∪ var(h′

k(γ )) = {ξ1, . . . , ξk} for every γ ∈ k . We denote by jc(H, H) theclass of all jointly complete pairs of tree homomorphisms.

Next we introduce weighted tree languages and tree transformations. A series ϕ ∈K 〈〈T� (Xn)〉〉 is called a weighted tree language over �, Xn, and K (or: over (�, Xn, K )).In case n = 0 we call ϕ a weighted tree language over (�, K ). We say that ϕ is linear ifsupp(ϕ) is linear.

We will freely use the concept of a recognizable weighted tree language. The reader whois unfamiliar with this concept may consult [1,9,26].

A weighted relation ρ ∈ K 〈〈T� (Xn) × T� (Xn)〉〉 is called a weighted tree transforma-tion over �,�, Xn, and K (or: (�,�, Xn, K )). In case n = 0 we call ρ a weighted treetransformation over (�,�, K ). We say that ρ is linear (resp. weakly variable symmetric) ifsupp(ρ) is linear (resp. weakly variable symmetric).

3 Substitutions in weighted tree transformations

In the rest of the paper A = (A, �) and B = (B,�) will denote an arbitrary �- anda �-algebra, respectively.

In this section we introduce the [IO]-substitution of weighted relations of K 〈〈A × B〉〉 inweighted tree transformations. We begin with some elementary concepts.

Let h : Xn → A be any mapping with h(xi ) = ai , 1 ≤ i ≤ n. For every s ∈ T�(Xn),we denote h(s) by s(a1, . . . , an)A and call it the evaluation of s at (a1, . . . , an) in A. Henceσ( )A = σA for every σ ∈ �0. If no confusion arises, then we drop the index A froms(a1, . . . , an)A.

Remark 6 If xi �∈ var(s), then s(a1, . . . , ai , . . . , an) = s(a1, . . . , a, . . . , an) for every ele-ment a ∈ A.

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Now let s ∈ T� (Xn) with var(s) = {xi1 , . . . , xik } and η1, . . . , ηn ∈ K 〈〈A〉〉. The [IO]-substitution of η1, . . . , ηn in s is the series

s [η1, . . . , ηn] =∑

a1,...,ak∈A

(ηi1 , a1

) · . . . · (ηik , ak).s(b1, . . . , bn),

where, for every a1, . . . , ak ∈ A, the sequence b1, . . . , bn is an arbitrary element ofAn |(i1,a1)...(ik ,ak ). Note that, by Remark 6, the value in the right-hand side of the aboveequation is independent of the choice of the sequences b1, . . . , bn , hence [IO]-substitutionof series is well-defined.

For every ϕ ∈ K 〈〈T�(Xn)〉〉 the [IO]-substitution of η1, . . . , ηn in ϕ is the series

ϕ [η1, . . . , ηn] =∑

s∈T�(Xn)

(ϕ, s)s [η1, . . . , ηn] .

Next we define [IO]-substitutions of weighted relations in pairs of terms using evaluationsof pairs of terms. Let (s, t) ∈ T� (Xn) × T� (Xn) with var(s) ∪ var(t) = {xi1 , . . . , xik }and θ1, . . . , θn ∈ K 〈〈A × B〉〉. The [IO]-substitution of θ1, . . . , θn in (s, t) is the weightedrelation

(s, t) [θ1, . . . , θn] =∑

(a1,b1),...,(ak ,bk )∈A×B

(θi1 , (a1, b1)

) · . . . · (θik , (ak, bk)).(s(c1, . . . , cn), t (d1, . . . , dn)

),

where, for every (a1, b1), . . . , (ak, bk) ∈ A × B, the sequence (c1, d1), . . . , (cn, dn) is anarbitrary element of (A × B)n |(i1,(a1,b1))...(ik ,(ak ,bk )). It is easy to see that [IO]-substitution ofweighted relations is well-defined.

Remark 7 If xi �∈ var(s) ∪ var(t), then

(s, t) [θ1, . . . , θi , . . . , θn] = (s, t) [θ1, . . . , θ, . . . , θn]

for every θ ∈ K 〈〈A × B〉〉.Proof By definition, neither θi nor θ contributes to the values on the corresponding side ofthe equation. ��

In this paper we will mainly be interested in [IO]-substitutions of weighted tree transfor-mations over (�,�, K ) in pairs (s, t) of terms in T�(Xn)× T�(Xn). In this particular casewe evaluate (s, t) in (T�, T�) and thus the evaluation becomes the usual tree substitution.Now we give an example of such substitutions.

Example 8 Let σ ∈ �3, δ ∈ �2, (s, t) = (σ (x1, x1, x3), δ(x3, x1)), and (s′, t) =(σ (x1, x1, x2), δ(x3, x1)). Moreover, let θ1 = min{1.(s1, t1), 2.(s′

1, t ′1)}, θ2 = ∞̃, θ ′2 =

2.(s2, t2), and θ3 = 1.(s3, t3) be weighted tree transformations over (�,�,Trop). Then

(s, t) [θ1, θ2, θ3] = (s, t)[θ1, θ

′2, θ3]

= min{2.(σ(s1, s1, s3), δ(t3, t1)

), 3.(σ(s′

1, s′1, s3), δ(t3, t ′1)

)},

(s′, t) [θ1, θ2, θ3] = ∞̃, and,

(s′, t)[θ1, θ

′2, θ3] = min

{4.(σ(s1, s1, s2), δ(t3, t1)

), 5.(σ(s′

1, s′1, s2), δ(t3, t ′1)

)}.

We note that we deviate from the notation which we introduced for polynomials in Sect. 2.4and write ‘min’ in the usual form.

123


Finally, we define the [IO]-substitution of weighted relations in weighted tree transforma-tions. For every τ ∈ K 〈〈T� (Xn)× T� (Xn)〉〉, we define the [IO]-substitution of θ1, . . . , θn

in τ by

τ [θ1, . . . , θn] =∑

(s,t)∈T�(Xn)×T�(Xn)

(τ, (s, t))(s, t) [θ1, . . . , θn] .

In the same way, we can define OI-evaluation of trees (and pairs of trees), see Section3 of [10]. Moreover, we can generalize OI-substitution of relations in tree transformationsof [10] to OI-substitution of weighted relations in weighted tree transformations. However,in this paper we do not introduce this OI-substitution in detail, because we refer to it onlyimplicitly in Propositions 13 and 16.

4 Preliminary results

In this section, we prove some technical results concerning series and substitutions inweighted tree transformations which we will use in proving our main results.

Lemma 9 Let (s, t) ∈ T� (Xn)×T� (Xn) and 1 ≤ i ≤ n be such that xi ∈ var(s)∪var(t).Moreover, let θ1, . . . , θi−1, θi+1, . . . , θn ∈ K 〈〈A × B〉〉, J a set, and (η j | j ∈ J ) be a familyof weighted relations in K 〈〈A × B〉〉. Then

(s, t)

⎡

⎣θ1, . . . ,∑

j∈J

η j , . . . , θn

⎤

⎦ =∑

j∈J

(s, t)[θ1, . . . , η j , . . . , θn

],

where the sum occurs in the i th argument.

Proof Let var(s) ∪ var(t) = {xi1 , . . . , xik } and assume that i = im . In the following com-putation, for every (a1, b1) . . . , (ak, bk) ∈ A × B, the sequence (c1, d1), . . . , (cn, dn) is anarbitrary element of (A × B)n |(i1,(a1,b1))...(ik ,(ak ,bk )). Then

(s, t)

⎡

⎣θ1, . . . ,∑

j∈J

η j , . . . , θn

⎤

⎦

=∑

(a1,b1),...,(ak ,bk )∈A×B

(θi1 , (a1, b1)) · . . . ·⎛

⎝

⎛

⎝∑

j∈J

η j

⎞

⎠ , (am, bm)

⎞

⎠ · . . . · (θik , (ak, bk)).

(s(c1, . . . , cn), t (d1, . . . , dn)

)

=∑

(a1,b1),...,(ak ,bk )∈A×B

(θi1 , (a1, b1)) · . . . ·⎛

⎝∑

j∈J

(η j , (am, bm)

)⎞

⎠ · . . . · (θik , (ak, bk)).

(s(c1, . . . , cn), t (d1, . . . , dn)

)

=∑

(a1,b1),...,(ak ,bk )∈A×B

∑

j∈J

(θi1 , (a1, b1)) · . . . · (η j , (am, bm)) · . . . · (θik , (ak, bk)).

(s(c1, . . . , cn), t (d1, . . . , dn)

)

123


=∑

j∈J

∑

(a1,b1),...,(ak ,bk )∈A×B

(θi1 , (a1, b1)) · . . . · (η j , (am, bm)) · . . . · (θik , (ak, bk)).

(s(c1, . . . , cn), t (d1, . . . , dn)

)

=∑

j∈J

(s, t)[θ1, . . . , η j , . . . , θn

].

��Next we investigate weighted tree transformations defined by means of weighted tree

languages and pairs of tree homomorphisms. For this, let h : T(Xn) → T�(Xn) andh′ : T(Xn) → T�(Xn) be a pair of tree homomorphisms, and consider the mapping⟨h, h′⟩ : T(Xn) → T�(Xn)×T�(Xn) defined by

⟨h, h′⟩ (u) = (h(u), h′(u)) for u ∈ T(Xn).

Then extend it to⟨h, h′⟩ : K 〈〈T(Xn)〉〉 → K 〈〈T�(Xn)× T�(Xn)〉〉

by letting⟨h, h′⟩ (ϕ) =

∑

u∈T(Xn)

(ϕ, u).⟨h, h′⟩ (u)

for every weighted tree language ϕ ∈ K 〈〈T(Xn)〉〉. By Proposition 5, the last extension⟨h, h′⟩ is ω-continuous. Moreover, it is easy to see that it is a continuous K -semimodule

morphism from (K 〈〈T(Xn)〉〉,+, 0̃) to (K 〈〈T�(Xn)× T�(Xn)〉〉,+, 0̃). (The statement alsofollows in a more general setting from Corollary 2.5 of [23]). This proves the following.

Proposition 10 The mapping⟨h, h′⟩ is ω-continuous. Moreover, for every set I , family (ϕi ∈

K 〈〈T(Xn)〉〉 | i ∈ I ),m ∈ K , ϕ ∈ K 〈〈T(Xn)〉〉, and pairs of tree homomorphisms h :T(Xn) → T�(Xn) and h′ : T(Xn) → T�(Xn), we have

(1)⟨h, h′⟩ (

∑i∈I ϕi ) =∑i∈I

⟨h, h′⟩ (ϕi ) and

(2)⟨h, h′⟩ (mϕ) = m

⟨h, h′⟩ (ϕ).

Lemma 11 Let ψ ∈ K 〈(Xn) ∪ Xn〉, ϕ1, . . . , ϕn ∈ K 〈〈T(Xn)〉〉 be weighted tree lan-guages and h : T(Xn) → T�(Xn) and h′ : T(Xn) → T�(Xn) be a jointly complete pairof tree homomorphisms. Then

⟨h, h′⟩ (ψ[ϕ1, . . . , ϕn]) = ⟨h, h′⟩ (ψ)

[⟨h, h′⟩ (ϕ1) , . . . ,

⟨h, h′⟩ (ϕn)

].

Proof Let us abbreviate (Xn) ∪ Xn by (Xn)∪. First we show that for every u ∈ (Xn)

∪⟨h, h′⟩ (u[ϕ1, . . . , ϕn]) = ⟨h, h′⟩ (u)

[⟨h, h′⟩ (ϕ1) , . . . ,

⟨h, h′⟩ (ϕn)

].

Let var(u) = {xi1 , . . . , xik }. In the next computation, for every u1, . . . , uk ∈ T(Xn), thesequence v1, . . . , vn is an arbitrary element of T(Xn)

n |(i1,u1)...(ik ,uk ). Moreover, for every(s1, t1), . . . , (sk, tk) ∈ T�(Xn)× T�(Xn), the sequence (s1, t1), . . . , (sn, tn) is defined in ananalogous way. Then we have⟨h, h′⟩ (u[ϕ1, . . . , ϕn])

= ⟨h, h′⟩⎛

⎝∑

u1,...,uk∈T(Xn)

(ϕi1 , u1

) · . . . · (ϕik , uk).u(v1, . . . , vn)

⎞

⎠

=∑

u1,...,uk∈T(Xn)

(ϕi1 , u1

) · . . . · (ϕik , uk).⟨h, h′⟩ (u(v1, . . . , vn))

123


=∑

u1,...,uk∈T(Xn)

(ϕi1 , u1

) · . . . · (ϕik , uk).⟨h, h′⟩ (u)

( ⟨h, h′⟩ (v1), . . . ,

⟨h, h′⟩ (vn)

)

=∑

u1,...,uk∈T(Xn)〈h,h′〉(u1)=(s1,t1),...,〈h,h′〉(uk )=(sk ,tk )

(ϕi1 , u1) · . . . · (ϕik , uk). 〈h, h′〉(u)[(s1, t1), . . . , (sn, tn)]

=∑

(s1,t1),...,(sk ,tk )∈T�(Xn)×T�(Xn)⎛

⎜⎜⎝

∑

u1∈T(Xn)〈h,h′〉(u1)=(s1,t1)

(ϕi1 , u1)

⎞

⎟⎟⎠ · . . . ·

⎛

⎜⎜⎝

∑

uk∈T(Xn)〈h,h′〉(uk )=(sk ,tk )

(ϕik , uk)

⎞

⎟⎟⎠ .

〈h, h′〉(u)[(s1, t1), . . . , (sn, tn)]=

∑

(s1,t1),...,(sk ,tk )∈T�(Xn)×T�(Xn)⎛

⎝∑

u1∈T(Xn)

(ϕi1 , u1).〈h, h′〉(u1), (s1, t1)

⎞

⎠ · . . . ·⎛

⎝∑

uk∈T(Xn)

(ϕik , uk).〈h, h′〉(uk), (sk , tk)

⎞

⎠ .

〈h, h′〉(u)[(s1, t1), . . . , (sn, tn)]=

∑

(s1,t1),...,(sk ,tk )∈T�(Xn)×T�(Xn)

(〈h, h′〉(ϕi1), (s1, t1)) · . . . · (〈h, h′〉(ϕik ), (sk , tk)

).

〈h, h′〉(u)[(s1, t1), . . . , (sn, tn)]= ⟨h, h′⟩ (u) [⟨h, h′⟩ (ϕ1) , . . . ,

⟨h, h′⟩ (ϕn)]

where at the second equality we use Proposition 10 (1) and (2), and the last equalityholds because (h, h′) is a jointly complete pair of tree homomorphisms, hence var(h(u)) ∪var(h′(u)) = {xi1 , . . . , xik }. Moreover, we have

⟨h, h′⟩ (ψ[ϕ1, . . . , ϕn])

= ⟨h, h′⟩⎛

⎝∑

u∈(Xn)∪(ψ, u)u[ϕ1, . . . , ϕn]

⎞

⎠

=∑

u∈(Xn)∪(ψ, u)

⟨h, h′⟩ (u[ϕ1, . . . , ϕn])

=∑

u∈(Xn)∪(ψ, u)

⟨h, h′⟩ (u)

[⟨h, h′⟩ (ϕ1), . . . ,

⟨h, h′⟩ (ϕn)

]

=∑


⎛

⎜⎜⎜⎝

∑

u∈(Xn)∪〈h,h′〉(u)=(s,t)

(ψ, u)

⎞

⎟⎟⎟⎠(s, t)

[⟨h, h′⟩ (ϕ1) , . . . ,

⟨h, h′⟩ (ϕn)

]

=∑


(〈h, h′〉(ψ), (s, t))(s, t)

[⟨h, h′⟩ (ϕ1) , . . . ,

⟨h, h′⟩ (ϕn)

]

= ⟨h, h′⟩ (ψ)[⟨

h, h′⟩ (ϕ1), . . . ,⟨h, h′⟩ (ϕn)

],

where at the second equality we use Proposition 10 (1) and (2), and the third equality isjustified above. ��

123


5 Systems of equations of weighted tree transformations

In this section we introduce systems of equations of weighted tree transformations withfinite support and define [IO]-equational weighted relations, as well as weighted tree trans-formations as the least [IO]-solutions of such systems. Then we give sufficient conditionsfor the existence and uniqueness of the [IO]-solution of a system of equations of weightedtree transformations. Finally, we recall systems of equations of weighted tree languages, andassociate a system of equations of weighted tree transformations with a system of equationsof weighted tree languages and a pair of tree homomorphisms.

We begin with some preparation. Let ρ ∈ K 〈T�(Xn)× T�(Xn)〉. We define the mapping

�(A,B)ρ : K 〈〈A × B〉〉n → K 〈〈A × B〉〉, (θ1, . . . , θn) �−→ ρ [θ1, . . . , θn]

for all (θ1, . . . , θn) ∈ K 〈〈A × B〉〉n . Then we can prove the following.

Lemma 12 The mapping �(A,B)ρ is ω-continuous.

Proof Let(θ1,k, . . . , θn,k

)k≥0 be an ω-chain in K 〈〈A × B〉〉n . Then

supk≥0

(ρ[θ1,k, . . . , θn,k

])

= supk≥0

⎛

⎝∑


(ρ, (s, t))(s, t)[θ1,k, . . . , θn,k

]⎞

⎠

=∑


(ρ, (s, t)) supk≥0

((s, t)

[θ1,k, . . . , θn,k

])

=∑


(ρ, (s, t))(s, t)[

supk≥0

((θ1,k, . . . , θn,k)

) ]

= ρ[

supk≥0

((θ1,k, . . . , θn,k)

) ],

where at the second equality we used the fact that K 〈〈A× B〉〉 is a continuous K -semimodule,cf. Proposition 4. Next we verify the third equality by showing that the two underlined expres-sions are equal. Assume that θi,k =∑0≤ ji ≤k ηi, ji for every 1 ≤ i ≤ n and k ≥ 0. Moreover,let var(s)∪ var(t) = {xi1 , . . . , xil } and define the sequence (c1, d1), . . . , (cn, dn) for every(a1, b1), . . . , (al , bl) ∈ A × B in the same way as in the definition of the [IO]-substitution.Then

supk≥0

((s, t)

[θ1,k , . . . , θn,k

])

= supk≥0

⎛

⎝(s, t)

⎡

⎣∑

0≤ j1≤k

η1, j1 , . . . ,∑

0≤ jn≤k

ηn, jn

⎤

⎦

⎞

⎠

=supk≥0

⎛

⎝∑

(a1,b1),...,(al ,bl )∈A×B

⎛

⎝

⎛

⎝∑

0≤ ji1 ≤k

ηi1, ji1

⎞

⎠ , (a1, b1)

⎞

⎠ · . . . ·⎛

⎝

⎛

⎝∑

0≤ jil ≤k

ηil , jil

⎞

⎠ , (al , bl )

⎞

⎠ .

(s(c1, . . . , cn), t (d1, . . . , dn)

))

= supk≥0

⎛

⎝∑

(a1,b1),...,(al ,bl )∈A×B

⎛

⎝∑

0≤ ji1 ≤k

(ηi1, ji1

, (a1, b1))⎞

⎠ · . . . ·⎛

⎝∑

0≤ jil ≤k

(ηil , jil

, (al , bl ))⎞

⎠ .

123


(s(c1, . . . , cn), t (d1, . . . , dn)

))

= supk≥0

⎛

⎝∑

(a1,b1),...,(al ,bl )∈A×B

∑

0≤ ji1 ,..., jil ≤k

(ηi1, ji1

, (a1, b1))

· . . . ·(ηil , jil

, (al , bl )).

(s(c1, . . . , cn), t (d1, . . . , dn)

))

= supk≥0

⎛

⎝∑

0≤ ji1 ,..., jil ≤k

∑

(a1,b1),...,(al ,bl )∈A×B

(ηi1, ji1

, (a1, b1))

· . . . ·(ηil , jil

, (al , bl )).

(s(c1, . . . , cn), t (d1, . . . , dn)

))

= (by Remark 7)

supk≥0

⎛

⎝∑

0≤ ji1 ,..., jil ≤k

(s, t)[τ1, . . . , τn]⎞

⎠ ,

where τm = ηiκ , jiκ if m = iκ for some 1 ≤ κ ≤ l and τm =∑

jm≥0

ηm, jm otherwise

= (by Proposition 2)∑

0≤ ji1 ,...,0≤ jil

(s, t)[τ1, . . . , τn],

where τm = ηiκ , jiκ if m = iκ for some 1 ≤ κ ≤ l and τm =∑

jm≥0

ηm, jm otherwise

= (by applying Lemma 9 l times)

(s, t)

⎡

⎣∑

j1≥0

η1, j1 , . . . ,∑

jn≥0

ηn, jn

⎤

⎦

= (s, t)

[

supk≥0

((θ1,k , . . . , θn,k)

)]

.

Hence we obtain that

�(A,B)ρ

(

supk≥0

((θ1,k, . . . , θn,k

)))

= supk≥0

(�(A,B)ρ

((θ1,k, . . . , θn,k

))).

��

Now we define the main concept of this paper. A system of equations of weighted treetransformations over �,�, Xn , and K (or: over (�,�, Xn, K )) is a system

(E) x1 = ρ1, . . . , xn = ρn,

where ρi ∈ K 〈T� (Xn) × T� (Xn)〉, i.e., ρi is a polynomial over (�,�, Xn, K ) for every1 ≤ i ≤ n. The system (E) is called linear (resp. weakly variable symmetric) if ρi is linear(resp. weakly variable symmetric) for every 1 ≤ i ≤ n.

We associate with (E) the mapping

�(A,B)E : K 〈〈A × B〉〉n → K 〈〈A × B〉〉n

123


defined by �(A,B)E (θ1, . . . , θn) =(�(A,B)ρ1 (θ1, . . . , θn), . . . , �

(A,B)ρn (θ1, . . . , θn)

)for every

(θ1, . . . , θn) ∈ K 〈〈A × B〉〉n . By Lemma 12, the mapping �(A,B)ρi (1 ≤ i ≤ n) is ω-continu-

ous, hence�(A,B)E is also ω-continuous. Then, by Proposition 1, the least fixpoint fix�(A,B)E

exists. In fact, fix�(A,B)E = supk≥0((θ1,k, . . . , θn,k

)), where

θi,0 = 0̃, for 1 ≤ i ≤ n, and θi,k+1 = ρi[θ1,k, . . . , θn,k

], for 1 ≤ i ≤ n and k ≥ 0.

In the following we also call a fixpoint of �(A,B)E an [IO]-solution of (E) in (A,B, K ), and

fix�(A,B)E the least [IO]-solution of (E) in (A,B, K ).A weighted relation θ ∈ K 〈〈A × B〉〉 is called [IO]-equational if it is a component of the

least [IO]-solution in (A,B, K ) of a system of equations of weighted tree transformationsover (�,�, Xn, K ). Moreover, we prefix [IO]-equational by ‘l’, ‘ws’, (or ‘l-ws’) if we wantto refer to an [IO]-equational weighted relation which is defined in the above way in termsof a system of equations of weighted tree transformations which is linear, weakly variablesymmetric (or has both properties), respectively.

Based on the OI-substitution of weighted relations into weighted tree transformations (cf.the note at the end of Sect. 3) we can define OI-equational weighted relations in an analogousway. Then we can generalize Corollary 13 of [10], which says that OI-equational relations andl-[IO]-equational relations coincide, to weighted relations and obtain the following result.

Proposition 13 A weighted relation is OI-equational if and only if it is l-[IO]-equational.

This means that all results obtained for l-[IO]-equational weighted relations also hold forOI-equational weighted relations. We note that in the unweighted case all such results arestated and proved explicitly in terms of OI-substitution in [10].

In the rest of the paper we will focus on [IO]-equational weighted tree transforma-tions over (�,�, K ), i.e., [IO]-equational weighted relations in K 〈〈T� × T�〉〉. They areobtained by considering the least [IO]-solution of systems (E) of equations of weightedtree transformations over (�,�, Xn, K ) in (T�, T�, K ). For the sake of simplicity, wecall an [IO]-solution (resp. the least [IO]-solution) of such an (E) in (T�, T�, K ) just an[IO]-solution (resp. the least [IO]-solution) of (E). We define [IO]-equational (resp. l-[IO]-equational, ws-[IO]-equational, and l-ws-[IO]-equational) weighted tree transformations aswe defined the corresponding concepts for weighted relations. We will denote by EQUT(resp. l-EQUT ,ws-EQUT , and l-ws-EQUT ) the corresponding classes of weighted treetransformations.

Example 14 Let � = {σ, γ, α, β} and � = {δ, ω, γ, α} with rk(σ ) = rk(δ) = rk(ω) =2, rk(γ ) = rk(β) = 1, and rk(α) = 0. For the sake of better readability, we write γ n(α) forthe tree γ (. . . γ (α) . . .) with n occurrences of γ .

(a) Consider the following system (E) of equations of weighted tree transformations over(�,�, X3,Nat∞):

x1 = 1.(γ (x1), γ (x1))+ 1.(γ (x1), x1)+ 1.(α, α)

x2 = 1.(β(x1), ω(x1, x1))

x3 = 1.(σ (x3, x2), δ(x2, x3))+ 1.(α, α).

We note that (E) is weakly variable symmetric. Let us abbreviate �(T�,T�)E by � anddetermine fix�. Its first approximations are as follows, where ⊥ = (̃0, 0̃, 0̃).

123


�(0)(⊥) = ⊥�(1)(⊥) = (1.(α, α), 0̃, 1.(α, α))

(where, e.g., the first component is obtained as the result of the substitution

(1.(γ (x1), γ (x1))+ 1.(γ (x1), x1)+ 1.(α, α))[⊥])�(2)(⊥) = (1.(γ (α), γ (α))+ 1.(γ (α), α)+ 1.(α, α), 1.(β(α), ω(α, α)), 1.(α, α))

(where now the first component is the result of the substitution

(1.(γ (x1), γ (x1))+ 1.(γ (x1), x1)+ 1.(α, α))[1.(α, α), 0̃, 1.(α, α)])�(3)(⊥) = (1.(γ 2(α), γ 2(α))+ 2.(γ 2(α), γ (α))+ 1.(γ 2(α), α)+ 1.(γ (α), γ (α))

+1.(γ (α), α)+ 1.(α, α),

1.(β(γ (α)), ω(γ (α), γ (α)))+ 1.(β(γ (α)), ω(α, α))+ 1.(β(α), ω(α, α)),

1.(σ (α, β(α)), δ(ω(α, α), α))+ 1.(α, α)....

Let us denote the i th component of fix� by τi for i = 1, 2, 3. We can see easily that

τ1 =∑

0≤m≤n

(n

m

).(γ n(α), γm(α)

)

and thus

τ2 =∑

0≤m≤n

(n

m

).(β(γ n(α)), ω(γm(α), γm(α))

).

Finally, τ3 is the weighted tree transformation such that every pair in its support has theform(σ(σ (. . . (α, sk) . . . , s2), s1), δ(ω(t1, t1), δ(ω(t2, t2), . . . , δ(ω(tk, tk), α) . . .))

), (1)

where k ≥ 1, and for every 1 ≤ i ≤ k we have si = β(γ ni (α)) and ti = γmi (α) forsome 0 ≤ mi ≤ ni . The weight of such a pair is

∏ki=1

(nimi

), see Fig. 1. The weighted

tree transformation τi is ws-[IO]-equational for i = 1, 2, 3.(b) Now consider the system (G)

x1 = min{1.(γ (x1), γ (x1)), 1.(γ (x1), x1), 0.(α, α)}x2 = 0.(β(x1), ω(x1, x1))

x3 = min{0.(σ (x3, x2), δ(x2, x3)), 0.(α, α)}.of equations of weighted tree transformations over (�,�, X3,Trop). Using the sameabbreviations and notation as in (a), we obtain that

τ1 = min0≤m≤n

{n.(γ n(α), γm(α)

)}

and thus

τ2 = min0≤m≤n

{n.(β(γ n(α)), ω(γm(α), γm(α))

)}.

Moreover, the support of τ3 consists of pairs of the form (1) and the weight of such a pair ismin1≤i≤k{ni }. The weighted tree transformation τi is ws-[IO]-equational for i = 1, 2, 3.

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Fig. 1 The pair (1) of trees visualized

In the next theorem we give a sufficient condition for the unicity of the [IO]-solution of asystem of equations of weighted tree transformations. Hereby, we generalize the correspond-ing result for systems of equations of weighted tree languages obtained in [4, Prop. 6.1]. Forthis, we define a system xi = ρi , 1 ≤ i ≤ n of equations of weighted tree transformations tobe proper if supp(ρi ) ⊆ (T� (Xn) \ Xn)× (T� (Xn) \ Xn) for every 1 ≤ i ≤ n.

Theorem 15 Any proper and weakly variable symmetric system of equations of weightedtree transformations has a unique [IO]-solution.

Proof Let

(E) x1 = ρ1, . . . , xn = ρn,

be a proper and weakly variable symmetric system of equations of weighted tree transfor-mations over (�,�, Xn, K ), and (τ1, . . . , τn) an [IO]-solution of (E). For every (u, v) ∈T� × T�, we have

(τi , (u, v)) = (ρi [τ1, . . . , τn] , (u, v))

=∑

(s,t)∈supp(ρi )var(s)=var(t)={xi1 ,...,xik }(s1,t1)...,(sk ,tk )∈T�×T�

(s[s1,...,sn ],t[t1,...,tn ])=(u,v)

(ρi , (s, t)) · (τi1 , (s1, t1)) · . . . · (τik , (sk , tk)

)

=∑

(s,t)∈supp(ρi )var(s)=var(t)={xi1 ,...,xik }

(s1,t1)...,(sk ,tk )∈(

sub(u)\{u})×(sub(v)\{v})

(s[s1,...,sn ],t[t1,...,tn ])=(u,v)

(ρi , (s, t)) · (τi1 , (s1, t1)) · . . . · (τik , (sk , tk)

),

where for every (s1, t1), . . . , (sk, tk) ∈ T� × T�, the sequence (s1, t1), . . . , (sn, tn) is anarbitrary element of (T� × T�)n |(i1,(s1,t1))...(ik ,(sk ,tk )). In the first summation, we can writevar(s) = var(t) = {xi1 , . . . , xik } because (E) is weakly variable symmetric, and the thirdequality is justified by the fact that (E) is proper. Hence (τi , (u, v)) is uniquely determinedby ρi and by the values of the τ j ’s on pairs, of which the components are proper subtrees ofu and v, respectively. It follows that (E) has a unique [IO]-solution. ��

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Finally, we recall equational weighted tree languages. A system of equations of weightedtree languages over , Xn , and K (or (, Xn, K )) is a system

(E) x1 = ψ1, . . . , xn = ψn,

whereψi ∈ K 〈T(Xn)〉 for every 1 ≤ i ≤ n. The system (E) is called linear (resp. simple) ifψi is linear (resp. ψi ∈ K 〈(Xn)〉) for every 1 ≤ i ≤ n. The concept of the least [IO]-solu-tion of (E) can be defined as for systems of equations of weighted tree transformations using[IO]-substitution of weighted tree languages rather than weighted tree transformations. Obvi-ously, the least [IO]-solution of (E) is an n-tuple (ϕ1, . . . , ϕn) ∈ K 〈〈T〉〉n of weighted treelanguages. The same holds for OI-substitution. Similarly to weighted tree transformations, aweighted tree language is called l-[IO]-equational (resp. OI-equational) if it is a componentof the least [IO]-solution of a linear system (resp. OI-solution of a system) of equations ofweighted tree languages. It is well-known (see, e.g., [9,26]), that a weighted tree languageis recognizable if and only if it is OI-equational. We will use the following variant of thisresult.

Proposition 16 A weighted tree language ϕ ∈ K 〈〈T〉〉 is recognizable if and only if it is acomponent of the least [IO]-solution of a linear system

(E) x1 = ψ1, . . . , xn = ψn,

of equations of weighted tree languages over (, Xn, K ) with ψi ∈ K 〈(Xn) ∪ Xn〉,1 ≤ i ≤ n.

Proof By [4, Lm. 6.3] or [23, Cor. 3.6], there is a simple system (G) of equations of weightedtree languages over (, Xn, K ) such that ϕ is a component of the least OI-solution of (G).Then by Proposition 13 we obtain the system (E) with the desired properties. We note thatthe transformation of (G) to (E) may yield equations of the form xi = 1.(x j , x j ) in (E), cf.the (“weighted version” of) the linearization algorithm appearing in the proof of Lemma 12in [10]. ��

Now, let

(E) x1 = ψ1, . . . , xn = ψn,

be a system of weighted tree languages over (, Xn, K ), and h : T → T� and h′ : T → T�be tree homomorphisms. The system of equations of tree transformations over (�,�, Xn, K )associated with (E), h and h′ is the system

⟨h, h′⟩ (E) x1 = ⟨h, h′⟩ (ψ1) , . . . , xn = ⟨h, h′⟩ (ψn) .

We will need the following statement.

Lemma 17 Let

(E) x1 = ψ1, . . . , xn = ψn,

be a linear system of equations of weighted tree languages over (, Xn, K ) such that ψi ∈K 〈(Xn) ∪ Xn〉 for every 1 ≤ i ≤ n, and let h : T → T� and h′ : T → T� be a jointlycomplete pair of tree homomorphisms. Moreover, let (ϕ1, . . . , ϕn) be the least [IO]-solutionof (E). Then the least [IO]-solution of

⟨h, h′⟩ (E) is

(⟨h, h′⟩ (ϕ1) , . . . ,

⟨h, h′⟩ (ϕn)

).

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Proof We have⟨h, h′⟩ (ϕi ) = ⟨h, h′⟩ (ψi [ϕ1, . . . , ϕn])

= ⟨h, h′⟩ (ψi )[⟨

h, h′⟩ (ϕ1) , . . . ,⟨h, h′⟩ (ϕn)

],

for every 1 ≤ i ≤ n, where the second equality follows from Lemma 11. Hence(⟨h, h′⟩ (ϕ1) , . . . ,

⟨h, h′⟩ (ϕn)

)is an [IO]-solution of

⟨h, h′⟩ (E).

Now assume that (ζ1, . . . , ζn) is another [IO]-solution of⟨h, h′⟩ (E) and that (ϕ1, . . . , ϕn)=

supk≥0((ϕ1,k, . . . , ϕn,k

)), where

– ϕi,0 = 0̃, for 1 ≤ i ≤ n, and ϕi,k+1 = ψi[ϕ1,k, . . . , ϕn,k

], for 1 ≤ i ≤ n and k ≥ 0.

We show by induction that, for every 1 ≤ i ≤ n and k ≥ 0, we have⟨h, h′⟩ (ϕi,k

) ≤ ζi . Fork = 0 this is true by definition. Then, for every k ≥ 0, we have

⟨h, h′⟩ (ϕi,k+1

) = ⟨h, h′⟩ (ψi [ϕ1,k, . . . , ϕn,k])

= ⟨h, h′⟩ (ψi )[⟨

h, h′⟩ (ϕ1,k), . . . ,

⟨h, h′⟩ (ϕn,k

)]

≤ ⟨h, h′⟩ (ψi ) [ζ1, . . . , ζn] = ζi ,

where the second equality follows from Lemma 11, and the inequality holds by the induc-tion hypothesis and by Lemma 12 (proving that the [IO]-substitution is ω-continuous andthus monotonic). Hence, by the fact that

⟨h, h′⟩ is ω-continuous (Proposition 10), we get⟨

h, h′⟩ (ϕi ) ≤ ζi for every 1 ≤ i ≤ n. This proves that(⟨

h, h′⟩ (ϕ1) , . . . ,⟨h, h′⟩ (ϕn)

)is the

least [IO]-solution of⟨h, h′⟩ (E). ��

6 Characterizing equational weighted tree transformations in terms of weightedbimorphisms

In this section we give a characterization of the class EQUT and some of its subclasses interms of certain weighted bimorphisms. The concept of an (unweighted) bimorphism wasintroduced in [2,3], that of a weighted bimorphism was introduced in [26,36]. Moreover,we show that in fact the class ws-EQUT is the closure of ws-l-EQUT under complete treehomomorphisms. Then we consider two weighted tree transformation classes which are char-acterized in terms of weighted bimorphisms in [25,37]. These are ln-T O P and ln-XT O P ,i.e., the classes of weighted tree transformations computed by linear and nondeleting top-down weighted tree transducers, and by linear and nondeleting top-down weighted extendedtree transducers, respectively. We give a characterization for these two classes in terms ofequational weighted tree transformations.

First we introduce some necessary concepts. A weighted bimorphism (over ,�,�, andK ) is a triple (h, ϕ, h′), where ϕ ∈ K 〈〈T〉〉 is a recognizable weighted tree language, andh : T → T� and h′ : T → T� are the input and the output tree homomorphism,respectively. The weighted tree transformation computed by (h, ϕ, h′) is 〈h, h′〉(ϕ). For anycombination w1 and w2 of l, c, we denote by B(w1-H, w2-H) the class of all weightedtree transformations computed by bimorphisms with input homomorphism of type w1

and output homomorphism of type w2. Furthermore, we denote by B( jc(H, H)) (resp.B( jc(l-H, l-H))) the class of all weighted tree transformations computed by bimorphismswhose input and output homomorphism constitute a jointly complete pair of tree homo-morphisms (resp. linear tree homomorphisms). We note that B(c-H, c-H) ⊆ B( jc(H, H))(resp. B(lc-H, lc-H) ⊆ B( jc(l-H, l-H))).

123


A system xi = ρi , 1 ≤ i ≤ n, of equations of weighted tree transformations over(�,�, Xn, K ) is called rule-like if supp(ρi ) is rule-like for every 1 ≤ i ≤ n, i.e., if eachpair (s, t) ∈ supp(ρi ) has the form (x j , x j ) or the form (σ (xi1 , . . . , xik ), t) for some k ≥ 0and σ ∈ �k , such that σ(xi1 , . . . , xik ) is linear, and t ∈ T�({xi1 , . . . , xik }). For instance, theequation systems (E) and (G) appearing in Example 14 are rule-like (and weakly variablesymmetric). A weighted tree transformation τ ∈ K 〈〈T�×T�〉〉 is called rl-[IO]-equational ifit is a component of the least [IO]-solution of a rule-like system of equations of weighted treetransformations over (�,�, Xn, K ). We will denote by rl-EQUT the corresponding class ofweighted tree transformations and we will combine the restrictions ‘l’, ‘ws’, and ‘rl’ in theusual way.

Now we are ready to give the mentioned bimorphism characterizations of certain classesof equational weighted tree transformations. In our statements we will apply the restriction‘l’ optionally by writing it in the form [l]. We obtain an instance of such a statement byeither replacing all occurrences of [l] by ‘l’ consistently or by dropping all occurrencesof [l].

Theorem 18

(a) [l]-EQUT = B( jc([l]-H, [l]-H))(b) ws-[l]-EQUT = B([l]c-H, [l]c-H)(c) rl-ws-[l]-EQUT = B(RE L , [l]c-H)

Proof (a) B( jc([l]-H, [l]-H)) ⊆ [l]-EQUT : Assume that τ = 〈h, h′〉(ϕ), where ϕ ∈K 〈〈T〉〉 is a recognizable weighted tree language and h : T → T�, h′ : T → T� isa jointly complete pair of tree homomorphisms. If ϕ = 0̃, then obviously τ = 0̃ ∈ EQUT .

Otherwise, by Proposition 16, we can assume that ϕ is a component of the least [IO]-solution (ϕ1, . . . , ϕn) of a linear system

(E) x1 = ψ1, . . . , xn = ψn,

of weighted tree languages over (, Xn, K ) which satisfies ψi ∈ K 〈(Xn) ∪ Xn〉 forevery 1 ≤ i ≤ n. Since (h, h′) is jointly complete, by Lemma 17 we obtain that(⟨

h, h′⟩ (ϕ1) , . . . ,⟨h, h′⟩ (ϕn)

)is the least [IO]-solution of

⟨h, h′⟩ (E). This means that τ is a

component of the least [IO]-solution of⟨h, h′⟩ (E), hence τ ∈ EQUT . If the tree homomor-

phisms h and h′ are linear, then the system⟨h, h′⟩ (E) is linear.

[l]-EQUT ⊆ B( jc([l]-H, [l]-H)): Let τ ∈ K 〈〈T� × T�〉〉 be a component of the least[IO]-solution of a system

(E) x1 = ρ1, . . . , xn = ρn,

of equations of weighted tree transformations over (�,�, Xn, K ).For every 1 ≤ i ≤ n, we set Ri = supp(ρi ) ∩ {(x j , x j ) | 1 ≤ j ≤ n}, and for every

(s, t) ∈ (supp(ρi ) \ Ri ) we specify a new symbol σs,t with rank m = |var(s)∪ var(t)|. Let be the ranked alphabet consisting of all such symbols. Consider the system of equations

(E′) x1 = ψ1, . . . , xn = ψn,

of weighted tree languages over (, Xn, K ), where for every 1 ≤ i ≤ n we let

supp(ψi )={σs,t (xi1 , . . . , xim ) | (s, t) ∈ (supp(ρi ) \ Ri ), var(s) ∪ var(t)={xi1 , . . . , xim },

1 ≤ i1 < . . . < im ≤ n} ∪ {x j | (x j , x j ) ∈ Ri

},

123


and define the weights(ψi , σs,t (xi1 , . . . , xim )

) = (ρi , (s, t)) and(ψi , x j

) = (ρi , (x j , x j )).

Clearly, the system (E′) is linear.Now consider the tree homomorphisms h : T → T� and h′ : T → T� determined

by hm(σs,t) = s(ξ1/xi1 , . . . , ξm/xim ) and h′

m

(σs,t) = t (ξ1/xi1 , . . . , ξm/xim ) for every

m ≥ 0 and σs,t ∈ m . Obviously,⟨h, h′⟩ (σs,t (xi1 , . . . , xim )) = (s, t), which means that⟨

h, h′⟩ (ψi ) = ρi for every 1 ≤ i ≤ n, i.e., that (E) = ⟨h, h′⟩ (E′). Moreover,⟨h, h′⟩ is jointly

complete, and thus by Lemma 17 we obtain that τ = ⟨h, h′⟩ (ϕ), where ϕ is a component ofthe least [IO]-solution of (E′). If the system (E) is linear, then both h and h′ are linear. Sinceϕ is a recognizable series (Proposition 16), our proof is completed.

To prove (b) and (c) we follow the proof of (a) and make the following additional remarks.(b) For the proof of the first inclusion we note that the tree homomorphisms h and h′

are nondeleting (and thus the pair (h, h′) is jointly complete), which implies that the system⟨h, h′⟩ (E′) is weakly variable symmetric. The second inclusion also holds because the sys-

tem (E) is now weakly variable symmetric and thus the tree homomorphisms h and h′ arenondeleting.

(c) The first inclusion holds because the assumption that h is a relabeling and h′ is non-deleting implies that the system

⟨h, h′⟩ (E) is rule-like and weakly variable symmetric. For

the proof of the second one, we note that the system (E) is rule-like and weakly variablesymmetric, hence by construction, h is a relabeling and h′ is nondeleting. ��

Next we show that the class of ws-[IO]-equational weighted tree transformations is theclosure of the class of ws-l-[IO]-equational weighted tree transformations under nondeletingtree homomorphisms. In this way we generalize a corresponding result concerning equationalweighted tree languages (cf. [11, Thm. 3]). We will need the subsequent notation.

Let (s, t) ∈ T� × T� and g : T� → T�′ and g′ : T� → T�′ two tree homomorphisms.We set

⟨g, g′⟩ ((s, t)) = (g(s), g′(t)). Furthermore, for every weighted tree transformation

τ ∈ K 〈〈T� × T�〉〉, we define the weighted tree transformation⟨g, g′⟩ (τ ) over (�′,�′, K ),

such that⟨g, g′⟩ (τ ) =

∑

(s,t)∈T�×T�

(τ, (s, t)).⟨g, g′⟩ ((s, t)).

For a class C of weighted tree transformations over K , we let 〈c-H, c-H〉 (C) = {⟨g, g′⟩ (τ )| τ ∈ C, τ ∈ K 〈〈T� × T�〉〉, and g : T� → T�′ and g′ : T� → T�′ are nondeleting treehomomorphisms}.Proposition 19 〈c-H, c-H〉 (ws-l-EQUT) = ws-EQUT.

Proof We observe that 〈c-H, c-H〉 (B(lc-H, lc-H)) = B(c-H, c-H). The proof followsfrom the following obvious facts. The composition of a complete homomorphism and a lin-ear and complete homomorphism is a complete homomorphism. Moreover, any completehomomorphism appears as the composition of a complete homomorphism and a linear andcomplete homomorphism (which may be, e.g, the complete homomorphism itself and theidentity mapping, respectively). Then the statement follows from Theorem 18. ��

Now we are ready to establish equational characterizations of the classes ln-T O P (resp.ln-X T O P) of weighted tree transformations computed by linear and nondeleting top-downweighted tree transducers (resp. linear and nondeleting top-down weighted extended treetransducers). For this, we do not recall the original definition of all these classes, but we just

123


recall the following weighted bimorphism characterization of them given in [25, Obs. 3.5,Thm. 4.3, Cor. 4.4], which will be enough for our purpose.

Proposition 20

(a) ln-T O P = B(RE L , lc-H)(b) ln-X T O P = B(lc-H, lc-H).

Thus we obtain the following equational characterizations of the mentioned weighted treetransformation classes.

Theorem 21

(a) rl-ws-l-EQUT = ln-T O P(b) ws-l-EQUT = ln-XT O P.

Proof To prove (a), we combine Theorem 18(c) with Proposition 20(a). Moreover, we get(b) by Theorem 18(b) and Proposition 20(b). ��

7 A Mezei–Wright like relationship

In this section, we give a Mezei–Wright type result which relates [IO]-equational weightedtree transformations and [IO]-equational weighted relations. Namely we show that a weightedrelation is [IO]-equational if and only if it is, roughly speaking, the morphic image of an[IO]-equational weighted tree transformation. First we recall a preparatory result from [10].

Proposition 22 (cf. [10, Lm. 29]) For every n ≥ 0, s ∈ T�(Xn), and s1, . . . , sn ∈ T� , wehave

HA (s (s1, . . . , sn)) = s (HA(s1), . . . , HA(sn)) .

Next we define the mapping H(A,B) : K 〈〈T� × T�〉〉 → K 〈〈A × B〉〉 such that for anyτ ∈ K 〈〈T� × T�〉〉 we have

H(A,B)(τ ) =∑

(s,t)∈T�×T�

(τ, (s, t)).H(A,B)((s, t)),

where H(A,B)((s, t)) = (HA(s), HB(t)) for all (s, t) ∈ T� × T�. By Proposition 5, weimmediately obtain the following.

Proposition 23 The mapping H(A,B) is ω-continuous.

We will also need the following technical result.

Lemma 24 For every n ≥ 0, ρ ∈ K 〈〈T�(Xn)× T�(Xn)〉〉, τ1, . . . , τn ∈ K 〈〈T� × T�〉〉, wehave

H(A,B) (ρ [τ1, . . . , τn]) = ρ[H(A,B)(τ1), . . . , H(A,B)(τn)

].

Proof First we prove that

H(A,B) ((s, t) [τ1, . . . , τn]) = (s, t)[H(A,B)(τ1), . . . , H(A,B)(τn)

]

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for every (s, t) ∈ T�(Xn)× T�(Xn). Let var(s) ∪ var(t) = {xi1 , . . . , xik }. Then

H(A,B) ((s, t) [τ1, . . . , τn])

= H(A,B)

⎛

⎝∑

(si ,ti )∈T�×T�,1≤i≤k

(τi1 , (s1, t1)

) · . . . · (τik , (sk , tk)).(s(u1, . . . , un), t (v1, . . . , vn)

)⎞

⎠

=∑

(si ,ti )∈T�×T�,1≤i≤k

(τi1 , (s1, t1)

) · . . . · (τik , (sk , tk)).H(A,B)

((s(u1, . . . , un), t (v1, . . . , vn))

)

=∑

(si ,ti )∈T�×T�,1≤i≤k

(τi1 , (s1, t1)

) · . . . · (τik , (sk , tk)).(HA (s(u1, . . . , un)) , HB (t (v1, . . . , vn))

)

=∑

(si ,ti )∈T�×T�,1≤i≤k

(τi1 , (s1, t1)

) · . . . · (τik , (sk , tk)).

(s(HA(u1), . . . , HA(un)), t (HB(v1), . . . , HB(vn))

)

=∑

(si ,ti )∈T�×T�,1≤i≤k

(τi1 , (s1, t1)

) · . . . · (τik , (sk , tk)).

(s, t)(H(A,B)((u1, v1)), . . . , H(A,B)((un, vn))

)

= (s, t)[H(A,B)(τ1), . . . , H(A,B)(τn)

],

where for every (s1, t1), . . . , (sk, tk) ∈ T� × T�, the sequence (u1, v1), . . . , (un, vn) is anarbitrary element of (T� × T�)n |(i1,(s1,t1))...(ik ,(sk ,tk )). Moreover, at the fourth equality we useProposition 22. Finally, we have

H(A,B) (ρ [τ1, . . . , τn])

= H(A,B)

⎛

⎝∑


(ρ, (s, t)) (s, t) [τ1, . . . , τn]

⎞

⎠

=∑


(ρ, (s, t)) H(A,B) ((s, t) [τ1, . . . , τn])

=∑


(ρ, (s, t)) (s, t)[H(A,B)(τ1), . . . , H(A,B)(τn)

]

= ρ[H(A,B)(τ1), . . . , H(A,B)(τn)

].

��Now we are ready to state and prove the mentioned Mezei–Wright like correspondence (cf.

[39, Thm. 5.5]) between [IO]-equational weighted relations and [IO]-equational weightedtree transformations.

Theorem 25 A weighted relation θ ∈ K 〈〈A × B〉〉 is [l]-[IO]-equational if and only if thereexists an [l]-[IO]-equational weighted tree transformation τ ∈ K 〈〈T� × T�〉〉 such thatH(A,B)(τ ) = θ .

Proof Assume first that θ is [IO]-equational. Then there is a system

(E) x1 = ρ1, . . . , xn = ρn,

of equations of weighted tree transformations over (�,�, Xn, K ) such that θ is acomponent of its least [IO]-solution (θ1, . . . , θn) in (A,B, K ). Let (τ1, . . . , τn) be the least

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[IO]-solution of (E). We show that H(A,B)(τi ) = θi for every 1 ≤ i ≤ n. By Lemma 24 wehave

H(A,B)(τi ) = H(A,B) (ρi [τ1, . . . , τn]) = ρi[H(A,B) (τ1) , . . . , H(A,B) (τn)

],

i.e.,(H(A,B) (τ1) , . . . , H(A,B) (τn)

)is an [IO]-solution of (E) in (A,B, K ). We show that

in fact it is the least [IO]-solution of (E). For this let

(τ1, . . . , τn) = supk≥0

((τ1,k, . . . , τn,k

)),

where τi,0 = 0̃, for 1 ≤ i ≤ n, and τi,k+1 = ρi[τ1,k, . . . , τn,k

], for 1 ≤ i ≤ n and k ≥ 0.

We show by induction that, for every 1 ≤ i ≤ n and k ≥ 0, we have H(A,B)(τi,k) ≤ θi . Fork = 0 this is true by definition. Then, for every k ≥ 0, we have

H(A,B)(τi,k+1) = H(A,B)(ρi[τ1,k, . . . , τn,k

])

= ρi[H(A,B)

(τ1,k), . . . , H(A,B)

(τn,k)]

≤ ρi [θ1, . . . , θn] = θi ,

where the second equality holds by Lemma 24 and the inequality holds by the inductionhypothesis and that the [IO]-substitution is monotonic. By Proposition 23 we concludeH(A,B)(τi ) = θi for every 1 ≤ i ≤ n. If θ is l-[IO]-equational, then the system (E) islinear and the proof can be finished in the same way. This proves one half of our theorem.The other direction can be proved similarly. ��Acknowledgments The authors are indebted to the anonymous referees for their valuable suggestions.

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Equational weighted tree transformations