Abstract
Amalgamation SNP (ASNP) is a fragment of existential second-order logic that strictly contains binary connected MMSNP of Feder and Vardi and binary connected guarded monotone SNP of Bienvenu, ten Cate, Lutz, and Wolter; it is a promising candidate for an expressive subclass of NP that exhibits a complexity dichotomy. We show that ASNP has a complexity dichotomy if and only if the infinite-domain dichotomy conjecture holds for constraint satisfaction problems for first-order reducts of binary finitely bounded homogeneous structures. For such CSPs, powerful universal-algebraic hardness conditions are known that are conjectured to describe the border between NP-hard and polynomial-time tractable CSPs. The connection to CSPs also implies that every ASNP sentence can be evaluated in polynomial time on classes of finite structures of bounded treewidth. We show that the syntax of ASNP is decidable. The proof relies on the fact that for classes of finite binary structures given by finitely many forbidden substructures, the amalgamation property is decidable.
Introduction
Feder and Vardi in their groundbreaking work [15] formulated the famous dichotomy conjecture for finite-domain constraint satisfaction problems, which has recently been resolved [11, 26]. Their motivation to study finite-domain CSPs was the question which fragments of existential second-order logic might exhibit a complexity dichotomy in the sense that every problem that can be expressed in the fragment is either in P or NP-complete. Existential second-order logic without any restriction is known to capture NP [14] and hence does not have a complexity dichotomy by an old result of Ladner [24]. Feder and Vardi proved that even the fragments of monadic SNP and monotone SNP do not have a complexity dichotomy since every problem in NP is polynomial-time equivalent to a problem that can be expressed in these fragments. However, the dichotomy for finite-domain CSPs implies that monotone monadic SNP (MMSNP) has a dichotomy, too [15, 23].
MMSNP is also known to have a tight connection to a certain class of infinite-domain CSPs [7]: an MMSNP sentence is equivalent to a connected MMSNP sentence if and only if it describes an infinite-domain CSP. Moreover, every problem in MMSNP is equivalent to a finite disjunction of connected MMSNP sentences. The infinite structures that appear in this connection are tame from a model-theoretic perspective: they are reducts of finitely bounded homogeneous structures (see Sect. 4.1). CSPs for such structures are believed to have a complexity dichotomy, too; there is even a known hardness condition such that all other CSPs in the class are conjectured to be in P [8]. The hardness condition can be expressed in several equivalent forms [1, 2].
In this paper we investigate another candidate for an expressive logic that has a complexity dichotomy. Our minimum requirement for what constitutes a logic is relatively liberal: we require that the syntax of the logic should be decidable. The same requirement has been made for the question whether there exists a logic that captures the class of polynomial-time solvable decision problems (see, e.g., [19, 20]). The idea of our logic is to modify monotone SNP so that only CSPs for model-theoretically tame structures can be expressed in the logic; the challenge is to come up with a definition of such a logic which has a decidable syntax. We would like to require that the (universal) first-order part of a monotone SNP sentence describes an amalgamation class. We mention that the Joint Embedding Property (JEP), which follows from the Amalgamation Property (AP), has recently been shown to be undecidable [10]. In contrast, we use the fact that the AP for binary signatures is decidable (Sect. 5). We call our new logic Amalgamation SNP (ASNP). This logic contains binary connected MMSNP; it also contains the more expressive logic of binary connected guarded monotone SNP. Guarded monotone SNP (GMSNP) has been introduced in the context of knowledge representation [3] (see Sect. 6). We show that ASNP has a complexity dichotomy if and only if the infinite-domain dichotomy conjecture holds for constraint satisfaction problems for first-order reducts of binary finitely bounded homogeneous structures. In particular, every problem that can be expressed in ASNP is a CSP for some countably infinite 
-categorical structure 
. In Sect. 7 we present an example application of this fact: every problem that can be expressed in one of these logics can be solved in polynomial time on instances of bounded treewidth.
Constraint Satisfaction Problems
Let 
be structures with a finite relational signature 
; each symbol 
is equipped with an arity
. A function 
is called a homomorphism from
to
if for every 
and 
we have 
; in this case we write 
. We write 
for the class of all finite 
-structures 
such that 
.
Example 1
If 
is the 3-clique, i.e., the complete undirected graph with three vertices, then 
is the graph 3-colouring problem, which is NP-complete [18].
Example 2
If 
then 
is the digraph acyclicity problem, which is in P.
Example 3
If 
for 
then 
is the Betweenness problem, which is NP-complete [18].
A homomorphism h from 
to 
is called an embedding of
into
if h is injective and for every 
and 
we have 
if and only if 
; in this case we write 
. The union of two 
-structures 
is the 
-structure 
with domain 
and the relation 
for every 
. The intersection
is defined analogously. A disjoint union of 
and 
is the union of isomorphic copies of 
and 
with disjoint domains. As disjoint unions are unique up to isomorphism, we usually speak of the disjoint union of 
and 
, and denote it by 
. A structure is connected if it cannot be written as a disjoint union of at least two structures with non-empty domain. A class of structures 
is closed under inverse homomorphisms if whenever 
and 
homomorphically maps to 
we have 
. If 
is a finite relational signature, then it is well-known and easy to see [5] that 
for a countably infinite 
-structure 
if and only if 
is closed under inverse homomorphisms and disjoint unions.
Monotone SNP
Let 
be a finite relational signature, i.e., 
is a set of relation symbols R, each equipped with an arity
. An SNP (
-) sentence is an existential second-order (
-) sentence with a universal first-order part, i.e., a sentence of the form
 |
where 
is a quantifier-free formula over the signature 
. We make the additional convention that the equality symbol, which is usually allowed in first-order logic, is not allowed in 
(see [15]). We write 
for the class of all finite models of 
.
Example 4
for the SNP 
-sentence 
given below.
 |
A class 
of finite 
-structures is said to be in SNP if there exists an SNP 
-sentence 
such that 
; we use analogous definitions for all logics considered in this paper. We may assume that the quantifier-free part of SNP sentences is written in conjunctive normal form, and then use the usual terminology (clauses, literals, etc).
Definition 1
An SNP 
-sentence 
with quantifier-free part 
and existentially quantified relation symbols 
is called
monotone if each literal of
with a symbol from 
is negative, i.e., of the form 
for 
.monadic if all the existentially quantified relations are unary.
connected if each clause of
is connected, i.e., the following 
-structure 
is connected: the domain of 
is the set of variables of the clause, and 
if and only if 
is a disjunct of the clause.
The SNP sentence from Example 4 is monotone, but not monadic, and it can be shown that there does not exist an equivalent MMSNP sentence [4].
Theorem 1
([5]). Every sentence in connected monotone SNP describes a problem of the form 
for some relational structure 
. Conversely, for every structure 
, if 
is in SNP then it is also in connected monotone SNP.
Amalgamation SNP
In this section we define the new logic Amalgamation SNP (ASNP). We first revisit some basic concepts from model theory.
The Amalgamation Property
Let 
be a finite relational signature and let 
be a class of 
-structures. We say that 
is finitely bounded if there exists a finite set of finite 
-structures 
such that 
if and only if no structure in 
embeds into 
; in this case we also write 
. Note that 
is finitely bounded if and only if there exists a universal 
-sentence 
(which might involve the equality symbol) such that for every finite 
-structure 
we have 
if and only if 
. We say that 
has
the Joint Embedding Property (JEP) if for all structures
there exists a structure 
that embeds both 
and 
.the Amalgamation Property (AP) if for any two structures
such that 
induce the same substructure in 
and in 
(a so-called amalgamation diagram) there exists a structure 
and embeddings 
and 
such that 
for all 
.
Note that since 
is relational, the AP implies the JEP. A class of finite 
-structures which has the AP and is closed under induced substructures and isomorphisms is called an amalgamation class.
The age of 
is the class of all finite 
-structures that embed into 
. We say that 
is finitely bounded if 
is finitely bounded. A relational 
-structure 
is called homogeneous if every isomorphism between finite substructures of 
can be extended to an automorphism of 
. Fraïssé’s theorem implies that for every amalgamation class 
there exists a countable homogeneous 
-structure 
with 
; the structure 
is unique up to isomorphism, also called the Fraïssé-limit of 
. Conversely, it is easy to see that the age of a homogeneous 
-structure is an amalgamation class. A structure 
is called a reduct of a structure 
if 
is obtained from 
by restricting the signature. It is called a first-order reduct of 
if 
is obtained from 
by first expanding by all first-order definable relations, and then restricting the signature. An example of a first-order reduct of 
is the structure 
from Example 3.
Defining Amalgamation SNP
As we have mentioned in the introduction, the idea of our logic is to require that a certain class of finite structures associated to the first-order part of an SNP sentence is an amalgamation class. We then use the fact that for binary signatures, the amalgamation property is decidable (Sect. 5).
Definition 2
Let 
be a finite relational signature. An Amalgamation SNP
-sentence is an SNP sentence 
of the form 
where
are binary;
is a conjunction of 
-formulas and of conjuncts of the form 
where 
and 
is a 
-formula;the class of
-reducts of the finite models of 
is an amalgamation class.
Note that ASNP inherits from SNP the restriction that equality symbols are not allowed. Also note that Amalgamation SNP sentences are necessarily monotone. This implies in particular that the class of 
-reducts of the finite models of 
is precisely the class of finite 
-structures that satisfy the conjuncts of 
that are 
-formulas (i.e., that do not contain any symbol from 
).
Example 5
The monotone SNP sentence from Example 4 describing 
is in ASNP. The problem 
from Example 3 can be expressed by the ASNP sentence
 |
Note that every finite-domain CSP can be expressed in ASNP; this can be seen similarly as in the argument of Feder and Vardi that finite-domain CSPs can be expressed in MMSNP [15].
Then the class of finite models of the first-order part of 
has the JEP, and since equality is not allowed in SNP the class is even closed under disjoint unions; it follows that also 
is closed under disjoint unions. It can be shown as in the proof of Theorem 1 that every Amalgamation SNP sentence can be rewritten into an equivalent connected Amalgamation SNP sentence.
ASNP and CSPs
We present the link between ASNP and infinite-domain CSPs.
Theorem 2
For every ASNP 
-sentence 
there exists a first-order reduct 
of a binary finitely bounded homogeneous structure such that 
.
Proof
Let 
be the set of existentially quantified relation symbols of 
. Let 
, for a quantifier-free formula 
in conjunctive normal form, be the first-order part of 
. Let 
be the class of 
-reducts of the finite models of 
; by assumption, 
is an amalgamation class. Moreover, 
is finitely bounded because it is the class of models of a universal 
-sentence. Let 
be the Fraïssé-limit of 
; then 
is a finitely bounded homogeneous structure. Let 
be the 
-structure which is the first-order reduct of the structure 
where the relation 
for 
is defined as follows: if 
are all the 
-formulas such that 
contains the conjunct 
for all 
, then the first-order definition of S is given by 
.
Claim 1
If 
is a finite 
-structure such that 
, then 
.
Let 
be a homomorphism. Let 
be the 
-expansion of 
where 
of arity l denotes 
. Then 
satisfies 
: to see this, let 
and let 
be a conjunct of 
. Since 
we have in particular that 
and so there must be a disjunct 
of 
such that 
. Then one of the following cases applies.
is a 
-literal and hence must be negative since 
is a monotone SNP sentence. In this case 
implies 
since h is a homomorphism.
is a 
-literal. Then by the definition of 
we have that 
if and only if 
.
Hence, 
. Since the conjunct 
of 
and 
were arbitrarily chosen, we have that 
. Hence, 
satisfies 
.
Claim 2
If 
is a finite 
-structure such that 
, then 
.
If 
has a 
-expansion 
that satisfies 
, then there exists an embedding from the 
-reduct 
of 
into 
by the definition of 
. This embedding is in particular a homomorphism from 
to 
. 
Theorem 3
Let 
be a first-order reduct of a binary finitely bounded homogeneous structure 
. Then 
can be expressed in ASNP.
Proof
Let 
be the signature of 
and 
the signature of 
. We may assume without loss of generality that 
contains a binary relation E that denotes the equality relation; it is easy to see that an expansion by the equality relation preserves finite boundedness. Consider the structure 
with the domain 
where
 |
To show that 
is homogeneous, let h be an isomorphism between finite substructures of 
. Let 
be the set of all first entries of elements of the first structure. Define 
by picking for 
an element of the form 
and defining by 
. This is well-defined: if h is defined on 
and on 
, then 
, and hence 
. The same consideration for 
shows that g is a bijection, and in fact an isomorphism between finite substructures of 
. By the homogeneity of 
there exists an extension 
of g. For each 
pick a permutation 
of 
that extends the bijection given by 
. Then the map 
given by 
is an automorphism of 
that extends h. Since 
is finitely bounded, there exists a universal 
-formula 
such that 
. Note that 
might contain the equality symbol (which we do not allow in SNP sentences). Let 
be the formula obtained from 
by
replacing each occurrence of the equality symbol by the symbol
;joining conjuncts that imply that E denotes an equivalence relation;
- joining for every
of arity n the conjunct
(implementing indiscernibility of identicals for the relation E).
We claim that 
. To see this, let 
be a finite 
-structure. If 
satisfies 
, then every induced substructure 
of 
with the property that 
implies that at most one of x and y is an element of A, satisfies 
, and hence is a substructure of 
. This in turn means that 
is in 
. The implications in this statement can be reversed which shows the claim.
Let 
be the formula obtained from 
as follows. For each 
let 
be the first-order definition of 
in 
; since 
is homogeneous we may assume that 
is quantifier-free [21]. Furthermore, we may assume that 
is given in conjunctive normal form. Let k be the arity of S. We then add for each conjunct 
of 
the conjunct
 |
By construction, the sentence 
obtained from 
by quantifying all relation symbols of 
is an ASNP 
-sentence. 
Corollary 1
ASNP has a complexity dichotomy if and only if the infinite-domain dichotomy conjecture is true for first-order reducts of binary finitely bounded homogeneous structures.
Deciding Amalgamation
In this section we show how to algorithmically decide whether a given existential second-order sentence is in ASNP. The following is a known fact in the model theory of homogeneous structures (the first author has learned the fact from Gregory Cherlin), but we are not aware of any published proof in the literature.
Theorem 4
Let 
be a finite set of finite binary relational 
-structures. There is an algorithm that decides whether 
has the amalgamation property.
Proof
Let m be the maximal size of a structure in 
, and let 
be the number of isomorphism types of two-element structures in 
. It is well-known and easy to prove that 
has the amalgamation property if and only if it has the so-called 1-point amalgamation property, i.e., the amalgamation property restricted to diagrams 
where 
. Suppose that 
is such an amalgamation diagram without amalgam. Let 
. Let 
and 
. Let 
be a 
-structure 
with domain 
such that 
and 
are substructures of 
. Since 
by assumption is not an amalgam for 
, there must exist 
such that the substructure of 
induced by 
embeds a structure from 
.
Note that the number of such 
-structures 
is bounded by 
since they only differ by the substructure induced by p and q. So let 
be a list of sets witnessing that all of these structures 
embed a structure from 
. Let 
be the substructure of 
induced by 
and 
be the substructure of 
induced by 
. Suppose for contradiction that 
has an amalgam 
; we may assume that this amalgam is of size at most 
. Depending on the two-element structure induced by 
in 
, there exists an 
such that the structure induced by 
in 
embeds a structure from 
, a contradiction. 
Corollary 2
There is an algorithm that decides for a given existential second-order sentence 
whether it is in ASNP.
Proof
Let k be the maximal number of variables per clause in the first-order part 
of 
, and let 
be the set of all structures at most the elements 
that do not satisfy 
. Then 
and the result follows from Theorem 2. 
Guarded Monotone SNP
In this section we revisit an expressive generalisation of MMSNP introduced by Bienvenu, ten Cate, Lutz, and Wolter [3] in the context of ontology-based data access, called guarded monotone SNP (GMSNP). It is equally expressive as the logic MMSNP
introduced by Madelaine [25]1. We will see that every GMSNP sentence is equivalent to a finite disjunction of connected GMSNP sentences (Proposition 1), each of which lies in ASNP if the signature is binary (Theorem 5).
Definition 3
A monotone SNP 
-sentence 
with existentially quantified relations 
is called guarded if each conjunct of 
can be written in the form
 |
are atomic 
-formulas, called body atoms,
are atomic 
-formulas, called head atoms,for every head atom
there is a body atom 
such that 
contains all variables from 
(such clauses are called guarded).
We do allow the case that 
, i.e., the case where the head consists of the empty disjunction, which is equivalent to 
(false).
The next proposition extends a well-known fact for MMSNP to guarded SNP.
Proposition 1
Every GMSNP sentence 
is equivalent to a finite disjunction 
of connected GMSNP sentences.
Proof
We prove Proposition 1. Let 
be a guarded SNP sentence. Suppose that the quantifier-free part of 
has a disconnected clause 
(Definition 1). By definition the variable set can be partitioned into non-empty variable sets 
and 
such that for every negative literal 
of the clause either 
or 
. The same is true for every positive literal, since otherwise the definition of guarded clauses would imply a negative literal on a set that contains 
, contradicting the property above. Hence, 
can be written as 
for non-empty disjoint tuples of variables 
and 
. Let 
be the formula obtained from 
by replacing 
by 
, and let 
be the formula obtained from 
by replacing 
by 
.
Let 
be the existential predicates in 
, and let 
be the input signature of 
. It suffices to show that for every 
-expansion 
of 
we have that 
satisfies 
if and only if 
satisfies 
or 
. If 
falsifies a clause of 
, there is nothing to show since then 
satisfies neither 
nor 
. If 
satisfies all clauses of 
, it in particular satisfies a literal from 
; depending on whether this literal lies in 
or in 
, we obtain that 
satisfies 
or 
, and hence 
or 
. Iterating this process for each disconnected clause of 
, we eventually arrive at a finite disjunction of connected guarded SNP sentences. 
It is well-known and easy to see [17] that each of 
can be reduced to 
in polynomial time. Conversely, if each of 
is in P, then 
is in P, too. It follows in particular that if connected GMSNP has a complexity dichotomy into P and NP-complete, then so has GMSNP.
Theorem 5
For every sentence 
in connected GMSNP there exists a reduct 
of a finitely bounded homogeneous structures such that 
. If all existentially quantified relation symbols in 
are binary then it is equivalent to an ASNP sentence.
In the proof of Theorem 5 we use a result of Cherlin, Shelah, and Shi [12] in a strengthened form due to Hubička and Nešetřil [22], namely that for every finite set 
of finite 
-structures, for some finite relational signature 
, there exists a finitely bounded homogeneous 
-structure 
such that a finite 
-structure 
homomorphically maps to 
if none of the structures in 
homomorphically maps to 
. We now prove Theorem 5.
Proof
Let 
be a 
-sentence in connected guarded monotone SNP with existentially quantified relation symbols 
. Let 
be the signature which contains for every relation symbol 
two new relation symbols 
and 
of the same arity and for every relation symbol 
a new relation symbol 
. Let 
be the first-order part of 
, written in conjunctive normal form, and let n be the number of variables in the largest clause of 
. Let 
be the sentence obtained from 
by replacing each occurrence of 
by 
and each occurrence of 
by 
, and finally each occurrence of 
by 
. Let 
be the (finite) class of all finite 
-structures with at most n elements that do not satisfy 
. We apply the mentioned theorem of Hubička and Nešetřil to 
, and obtain a finitely bounded homogeneous 
-structure 
such that the age of the 
-reduct 
of 
equals 
. We say that 
is correctly labelled if for every 
of arity m and 
we have 
if and only if 
. Let 
the 
-expansion of 
where 
of arity m denotes
 |
Since 
is finitely bounded homogeneous, 
is finitely bounded homogeneous, too. Let 
be the 
-reduct of 
. We claim that 
. First suppose that 
is a finite 
-structure that satisfies 
. Then it has an 
-expansion 
that satisfies 
. Let 
be the 
-structure with the same domain as 
where
denotes 
for each 
;
denotes 
for each 
;
denotes 
for each 
.
Then 
satisfies 
, and hence embeds into 
. This embedding is a homomorphism from 
to 
since the image of the embedding is correctly labelled by the construction of 
.
Conversely, suppose that 
has a homomorphism h to 
. Let 
be the 
-expansion of 
by defining 
if and only if 
, for every n-ary 
. Then each clause of 
is satisfied, because each clause of 
is guarded: let 
be the variables of some clause of 
. If 
satisfy the body of this clause, and 
is a head atom of such a clause, then the set 
is correctly labelled. This implies that some of the head atoms of the clause must be true in 
because 
satisfies 
. The second statement follows from Theorem 3. 
The following example shows that GMSNP does not contain ASNP.
Example 6
is in ASNP (see Example 5) but not in GMSNP. Indeed, suppose that 
is a GMSNP sentence which is true on all finite directed paths. We assume that the quantifier-free part 
of 
is in conjunctive normal form. Let 
be the existentially quantified relation symbols of 
, let 
, and let l be the number of variables in 
. Every directed path, viewed as a 
-structure, satisfies 
, and therefore has an 
-expansion 
that satisfies 
. Note that there are finitely many different 
-expansions of a path of length 
; let 
be this number. Hence, for a path of length 
, there must be 
with 
such that the substructures of 
induced by 
and by 
are isomorphic. We then claim that the directed cycle 
satisfies 
: this is witnessed by the 
-expansion inherited from 
which satisfies 
since each clause in 
is guarded. Hence, 
does not express digraph acyclicity.
Application: Instances of Bounded Treewidth
If a computational problem can be formulated in ASNP or in GMSNP, then this has remarkable consequences besides a potential complexity dichotomy. In this section we show that every problem that can be formulated in ASNP or in GMSNP is in P when restricted to instances of bounded treewidth. The corresponding result for Monadic Second-Order Logic (MSO) instead of ASNP is a famous theorem of Courcelle [13]. We strongly believe that ASNP is not contained in MSO (consider for instance the Betweenness Problem from Example 3), so our result appears to be incomparable to Courcelle’s.
In the proof of our result, we need the following concepts from model theory. A first-order theory T is called 
-categorical if all countable models of T are isomorphic [21]. A structure 
is called 
-categorical if its first-order theory (i.e., the set of first-order sentences that hold in 
) is 
-categorical. Note that with this definition, finite structures are 
-categorical. Another classic example is the structure 
. The definition of treewidth can be treated as a black box in our proof, and we refer the reader to [6].
Theorem 6
Let 
be an ASNP or a connected GMSNP 
-sentence and let 
. Then the problem to decide whether a given finite 
-structure 
of treewidth at most k satisfies 
can be decided in polynomial time with a Datalog program (of width k).
Proof
Since structures that are homogeneous in a finite relational language are 
-categorical [21] and first-order reducts of 
-categorical structures are 
-categorical [21], Theorem 2 and Theorem 5 imply that the problem to decide whether a finite 
-structure satisfies 
can be formulated as CSP
for an 
-categorical structure 
. Then the statement follows from Corollary 1 in [6]. 
Remark 1
In Theorem 6 it actually suffices to assume that the core of 
has treewidth at most k.
Corollary 3
Let 
be a GMSNP 
-sentence and let 
. Then there is a polynomial-time algorithm that decides whether a given 
-structure of treewidth at most k satisfies 
.
Proof
Conclusion and Open Problems
ASNP is a candidate for an expressive logic with a complexity dichotomy: every problem in ASNP is NP-complete or in P if and only if the infinite-domain dichotomy conjecture for first-order reducts of binary finitely bounded homogeneous structures holds. See Fig. 1 for the relation to other candidate logics that are known to have a dichotomy, might have a complexity, or provably do not have a dichotomy.
Fig. 1.
Fragments of existential second-order logic and complexity dichotomies.
We presented an application of ASNP concerning the evaluation of computational problems on classes of structures of bounded treewidth. We also proved that the syntax of ASNP is algorithmically decidable. The following problems concerning ASNP are open.
Is the Amalgamation Property decidable for (not necessarily binary) classes given by finitely many forbidden substructures?
Is every binary CSP in Monadic Second-Order Logic (MSO) also in ASNP?
Is every problem in NP polynomial-time equivalent to a problem in Amalgamation SNP if we drop the monotonicity assumption?
Is there a natural logic (which in particular has an effective syntax) that contains both ASNP and connected GMSNP and which describes CSPs for reducts of finitely bounded homogeneous structures?
Footnotes
MMSNP
relates to MMSNP as Courcelle’s MSO
relates to MSO [13].
This work has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 681988, CSP-Infinity) and by DFG Graduiertenkolleg 1763 (QuantLA).
Contributor Information
Marcella Anselmo, Email: manselmo@unisa.it.
Gianluca Della Vedova, Email: gianluca.dellavedova@unimib.it.
Florin Manea, Email: flmanea@gmail.com.
Arno Pauly, Email: arno.m.pauly@gmail.com.
Manuel Bodirsky, Email: manuel.bodirsky@tu-dresden.de.
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