Meghyn Bienvenu
email: meghyn.bienvenu@u-bordeaux.fr

Camille Bourgaux
email: camille.bourgaux@ens.fr

Querying and Repairing Inconsistent Prioritized Knowledge Bases: Complexity Analysis and Links with Abstract Argumentation

In this paper, we explore the issue of inconsistency handling over prioritized knowledge bases (KBs), which consist of an ontology, a set of facts, and a priority relation between conflicting facts. In the database setting, a closely related scenario has been studied and led to the definition of three different notions of optimal repairs (global, Pareto, and completion) of a prioritized inconsistent database. After transferring the notions of globally-, Pareto-and completion-optimal repairs to our setting, we study the data complexity of the core reasoning tasks: query entailment under inconsistencytolerant semantics based upon optimal repairs, existence of a unique optimal repair, and enumeration of all optimal repairs. Our results provide a nearly complete picture of the data complexity of these tasks for ontologies formulated in common DL-Lite dialects. The second contribution of our work is to clarify the relationship between optimal repairs and different notions of extensions for (set-based) argumentation frameworks. Among our results, we show that Pareto-optimal repairs correspond precisely to stable extensions (and often also to preferred extensions), and we propose a novel semantics for prioritized KBs which is inspired by grounded extensions and enjoys favourable computational properties. Our study also yields some results of independent interest concerning preference-based argumentation frameworks.

Introduction

Ontology-mediated query answering (OMQA) improves data access through the use of an ontology, which provides a convenient vocabulary for query formulation and captures domain knowledge that is exploited during query evaluation [START_REF] Poggi | Linking data to ontologies[END_REF][START_REF] Bienvenu | Ontology-mediated query answering with data-tractable description logics[END_REF][START_REF] Xiao | Ontologybased data access: A survey[END_REF]. There is now a large literature on OMQA, with much of the work adopting description logics (DLs) [START_REF] Baader | An Introduction to Description Logic[END_REF], or the closely related OWL (2) standard [START_REF] Owl Working Group | OWL 2 Web Ontology Language: Document overview. W3C Recommendation[END_REF], as the ontology specification language. In particular, the DL-Lite family of lightweight DLs [START_REF] Calvanese | Tractable reasoning and efficient query answering in description logics: The DL-Lite family[END_REF][START_REF] Artale | The DL-Lite family and relations[END_REF]) (which underpin the OWL 2 QL profile [START_REF] Motik | OWL 2 Web Ontology Language profiles[END_REF])) has been a main focus of both theoretical and practical research due to its favourable computational properties.

While it is often reasonable to assume that the ontology has been debugged and contains only trusted knowledge, real-world datasets are plagued by data quality issues, which may render the KB inconsistent. This has led to a large body of work on how to handle data inconsistencies in OMQA, with the proposal of several inconsistency-tolerant semantics to provide meaningful answers to queries posed over inconsistent KBs (see e.g. the surveys [START_REF] Bienvenu | Inconsistencytolerant querying of description logic knowledge bases[END_REF]) and [START_REF] Bienvenu | Inconsistency handling in ontologymediated query answering: A progress report[END_REF]). Many of these semantics are based upon some notion of repair, defined as an inclusion-maximal subset of the data that is consistent with the ontology. This is in particular the case for the AR, brave, and IAR semantics, which correspond respectively to a query answer holding w.r.t. all repairs, at least one repair, or the intersection of all repairs [START_REF] Lembo | Inconsistency-tolerant semantics for description logics[END_REF][START_REF] Bienvenu | Tractable approximations of consistent query answering for robust ontologybased data access[END_REF]. The computational properties of these and other semantics are now quite well understood, and some practical algorithms and implementations have begun to be developed, see e.g. [START_REF] Lembo | Inconsistency-tolerant query answering in ontology-based data access[END_REF][START_REF] Bienvenu | Computing and explaining query answers over inconsistent DL-Lite knowledge bases[END_REF][START_REF] Tsalapati | Efficient query answering over expressive inconsistent description logics[END_REF][START_REF] Trivela | A framework and positive results for IAR-answering[END_REF].

In many scenarios, there is some information about the dataset that can be used to select the most relevant repairs (e.g. we might know which relations or sources are most reliable, or the likelihood of certain kinds of facts being correct). To exploit such information, variants of the preceding semantics have been considered in which we restrict attention to the most preferred repairs based upon cardinality, weight, or a stratification of the dataset into priority levels [START_REF] Bienvenu | Querying inconsistent description logic knowledge bases under preferred repair semantics[END_REF]. While the latter forms of preferences are quite natural, there are other relevant ways of defining preferred repairs that are worth exploring. In particular, in the database area, the seminal work of [START_REF] Staworko | Prioritized repairing and consistent query answering in relational databases[END_REF] supposes that preferences are given in terms of a priority relation (i.e. acyclic binary relation) between conflicting facts. Such 'fact-level' preferences have been shown to naturally arise in applications like information extraction and can e.g. be declaratively specified using rules [START_REF] Fagin | Declarative cleaning of inconsistencies in information extraction[END_REF]. To lift a priority relation between facts to the level of repairs, three different methods were proposed by Staworko et al. Pareto-optimal and globally-optimal repairs are defined as those for which there is no possible substitution of facts that leads to a (Pareto / global) improvement, while completion-optimal repairs correspond to greedily constructing a repair based upon some compatible total order (see Section 3 for formal definitions). The complexity of reasoning with these three kinds of optimal repair has been investigated, primarily focusing on constraints given as functionality dependencies [START_REF] Fagin | Dichotomies in the complexity of preferred repairs[END_REF][START_REF] Kimelfeld | Detecting ambiguity in prioritized database repairing[END_REF][START_REF] Livshits | Counting and enumerating (preferred) database repairs[END_REF].

In this paper, we explore the use of fact-level preferences and optimal repairs for inconsistency handling in OMQA. Our first contribution is a data complexity analysis of the central reasoning problems related to optimal repairs: conjunctive query entailment under inconsistency-tolerant semantics based upon optimal repairs, uniqueness of optimal repairs, and enumeration of all optimal repairs. Our results provide a nearly complete picture of the data complexity for the three types of optimal repair and for ontologies formulated in DL-Lite core and DL-Lite H horn ; however, as we make precise later, many of our results transfer to other datatractable ontology languages. Not surprisingly, we find that reasoning with optimal repairs is generally more challenging than for standard repairs. In particular, query entailment under variants of the AR, IAR, and brave semantics based upon any of the three notions of optimal repair is intractable in data complexity, whereas with standard repairs, the IAR and brave semantics allow for tractable query answering.

Our second contribution is to establish connections with abstract argumentation frameworks (AFs). More precisely, we show that every prioritized KB with only binary conflicts naturally corresponds to a preference-based AF [START_REF] Kaci | Preference in abstract argumentation[END_REF]. To handle general prioritized KBs, we need preference-based set-based AFs (SETAFs), which we introduce as a natural extension of SETAFs [START_REF] Flouris | A comprehensive study of argumentation frameworks with sets of attacking arguments[END_REF]. This correspondence enables us to compare repairs and extensions. We determine that Pareto-optimal repairs are precisely the stable extensions of the corresponding preference-based (SET)AF, and under reasonable assumptions, also coincide with the preferred extensions. To establish the latter result, we prove a technically challenging result about symmetric preferencebased SETAFs that we believe is of independent interest. Globally-optimal and completion-optimal repairs correspond to proper subsets of the stable extensions, but do not at present have any analog in the argumentation setting.

The argumentation connection situates optimal repairs within a broader context and lays the foundations for importing ideas and results from the argumentation literature. Indeed, our third contribution is to propose a new notion of grounded repair, directly inspired by grounded extensions from argumentation. We show that the (unique) grounded repair is contained in the intersection of Pareto-optimal repairs. As the grounded repair can be computed in polynomial time from the dataset and conflicts, it yields a tractable approximation of the Pareto variant of the IAR semantics for the considered DLs. Moreover, we show that it is more productive than the recently proposed Elect semantics (Belabbes, [START_REF] Belabbes | Elect: An inconsistency handling approach for partially preordered lightweight ontologies[END_REF]Belabbes and Benferhat 2019). These advantages motivate us to take a closer look at the computational properties of the grounded semantics. We prove in particular a matching P lower bound and show how the semantics can be computed via the wellfounded semantics of logic programs.

Proofs are omitted due to lack of space but can be found in the long version [START_REF] Bienvenu | Querying and repairing inconsistent prioritized knowledge bases: Complexity analysis and links with abstract argumentation[END_REF].

Preliminaries

Even if many of our results apply to more general settings, our focus is on description logic (DL) knowledge bases, and we will in particular consider the DL-Lite core and DL-Lite H horn dialects of the DL-Lite family.

Syntax. A DL knowledge base (KB) K = T , A consists of an ABox A and a TBox T , both constructed from a set N C of concept names (unary predicates), a set of N R of role names (binary predicates), and a set N I of individuals (constants).

The ABox (dataset) consists of a finite number of concept assertions of the form A(a) and role assertions of the form R(a, b), where

A ∈ N C , R ∈ N R , a, b ∈ N I .
The TBox (ontology) consists of a set of axioms whose form depends on the DL in question. In DL-Lite core , TBox axioms are concept inclusions B C built according to the following grammar, where A ∈ N C and R ∈ N R :

B := A | ∃S, C := B | ¬B, S := R | R -.

DL-Lite H

horn extends DL-Lite core with role inclusions of the form S Q where Q := S | ¬S and concept inclusions of the form

B 1 • • • B n C.
Semantics. An interpretation has the form I = (∆ I , • I ), where ∆ I is a non-empty set and

• I maps each A ∈ N C to A I ⊆ ∆ I , each R ∈ N R to R I ⊆ ∆ I ×∆ I ,
= ∆ I \ B I 1 , (B 1 B 2 ) I = B I 1 ∩ B I 2 , (R -) I = {(c, d) | (d, c) ∈ R I } and (∃Q) I = {c | ∃d : (c, d) ∈ Q I }. An interpretation I satisfies G H if G I ⊆ H I ; it satisfies A(a) (resp. R(a, b)) if a I ∈ A I (resp. (a I , b I ) ∈ R I ). We call I a model of K = T , A if I satisfies all axioms in T and assertions in A. A KB K is consistent if it has a model; otherwise it is inconsistent. We say that an ABox A is T - consistent if the KB K = T , A is consistent.
Queries. A first-order (FO) query is a first-order logic formula whose atoms are built using the predicate symbols in N C ∪ N R and constants in N I . We will focus on the subclass of conjunctive queries (CQs) which take the form q( x) = ∃ y ψ( x, y), where ψ is a conjunction of atoms of the forms A(t) or R(t, t ) whose terms are either variables from x ∪ y or individuals. When a CQ consists of a single atom, we called it an instance query (IQ). A query without free variables is called Boolean, and we refer to Boolean CQs as BCQs. A Boolean query q is satisfied by an interpretation I, written I |= q, iff q is satisfied in I according to standard first-order logic semantics; q is entailed by a KB K, written K |= q, iff I |= q for every model I of K. Henceforth, when we speak of a (Boolean) query, without specifying the type of query, we mean a (Boolean) CQ.

An inconsistent KB entails every Boolean query under the standard semantics. Several inconsistency-tolerant semantics have been defined to obtain meaningful answers in this context (see [START_REF] Bienvenu | Inconsistencytolerant querying of description logic knowledge bases[END_REF] for a survey). Most of these semantics are based on the notion of repairs, which correspond to the different ways of restoring consistency by minimally removing some assertions. Definition 1 (Repair). An ABox repair of a KB K = T , A , or repair for short, is an inclusion-maximal T -consistent subset of A. The set of repairs of K is denoted by SRep(K).

Another central notion is that of conflicts, which are the minimal subsets of assertions that contradict the TBox. Definition 2 (Conflict). A conflict of a KB K = T , A , is a minimal T -inconsistent subset of A. The set of all conflicts of K is denoted by Conf (K) and can be represented as a conflict hypergraph whose vertices are assertions and whose hyperedges are the conflicts.

To simplify the presentation, we assume throughout the paper that the ABox does not contain any assertion that is inconsistent with the TBox, i.e., every conflict contains at least two assertions. Note that self-contradictory assertions do not occur in any repair, and such assertions can be readily identified and removed using existing reasoning algorithms.

In the database setting, Staworko, Chomicki, and Marcinkowski (2012) defined three notions of optimal repairs based on a priority relation over facts, which expresses preferences between conflicting facts. Such a relation could e.g. represent expert knowledge of how to resolve certain kinds of conflicts or be learned from manual conflict resolution [START_REF] Martinez | Policy-based inconsistency management in relational databases[END_REF][START_REF] Tanon | Learning how to correct a knowledge base from the edit history[END_REF]. We recast these notions in the KB setting. Definition 3 (Priority relation, completion). Given a KB K = T , A , a priority relation for K is an acyclic binary relation over the assertions of A such that if α β, then there exists C ∈ Conf (K) such that {α, β} ⊆ C. A priority relation is total iff for all α = β, if there exists C ∈ Conf (K) such that {α, β} ⊆ C, then α β or β α. A completion of is a total priority relation ⊇ . Definition 4 (Prioritized KB). A prioritized KB is a pair (K, ) consisting of a KB K and priority relation for K. The notation K will be used as shorthand for (K, ). Definition 5 (Optimal repairs). Consider a prioritized KB K with K = T , A , and let A ∈ SRep(K).

• A Pareto improvement of A (w.r.t. ) is a T -consistent A ⊆ A such that there exists β ∈ A \ A such that β α for every α ∈ A \ A . • A global improvement of A (w.r.t. ) is a T -consistent
A ⊆ A such that A = A and for every α ∈ A \ A , there exists β ∈ A \ A such that β α. The repair A is a:

• Pareto-optimal repair of K if there is no Pareto improvement of A w.r.t. . • Globally-optimal repair of K if there is no global improvement of A w.r.t. .

• Completion-optimal repair of K if A is a globallyoptimal repair of K , for some completion of . We denote by GRep(K ), PRep(K ) and CRep(K ) the sets of globally-, Pareto-, and completion-optimal repairs. [START_REF] Staworko | Prioritized repairing and consistent query answering in relational databases[END_REF] showed the following relation between optimal repairs:

CRep(K ) ⊆ GRep(K ) ⊆ PRep(K ) ⊆ SRep(K).
They also show that A is a completion-optimal repair iff it can be obtained by the following greedy procedure: while some assertion has not been considered, pick an assertion that is maximal w.r. We next introduce the three natural inconsistency-tolerant semantics based upon repairs. The AR semantics is arguably the most natural and well-known semantics and requires that a query be entailed from every repair of the KB. The IAR semantics is a more cautious semantics that evaluates queries over the intersection of all repairs. Finally, the brave semantics is the most adventurous semantics, returning yes to queries that are entailed by some repair. Definition 7 (AR, IAR and brave semantics). For X ∈ {C, G, P, S}, a prioritized KB K entails a query q under

• X-AR semantics, written K |= X AR q, iff T , A |= q for every A ∈ XRep(K ); • X-IAR semantics, written K |= X IAR q, iff T , A ∩ |= q where A ∩ = A ∈XRep(K ) A ; • X-brave semantics, written K |= X brave q, iff T , A |= q for some A ∈ XRep(K ).
We remark that these semantics are related as follows:

K |= X IAR q ⇒ K |= X AR q ⇒ K |= X brave q. Example 8 (Ex. 6 cont'd).
Considering the different semantics based upon completion-optimal repairs, we observe that If we consider now AR semantics for the different kinds of optimal repairs, we find that

• K |= C IAR Carnivorous(a), • K |= C AR Snake(a) but K |= C IAR Snake(a), • K |= C brave Boa(a) but K |= C AR Boa(a).
• K |= C AR Snake(a) but K |= G AR Snake(a), • K |= G AR Carnivorous(a) but K |= P AR Carnivorous(a), • K |= P AR Eat(a, b) but K |= S AR Eat(a, b).

Complexity Analysis

In this section, we analyze the data complexity of the central reasoning tasks related to optimal repairs.

Overview

We define the problems and complexity classes we consider as well as some relevant properties of DL languages.

Problems. Given a prioritized KB K and a type of repairs X ∈ {C, G, P, S}, repair checking (ISREP) is deciding whether a set of assertions belongs to XRep(K ); AR (resp. IAR, brave) entailment (AR, IAR, BRAVE) is deciding whether K entails a given BCQ under X-AR (resp. X-IAR, X-brave) semantics; uniqueness (UNIQUE) is deciding whether |XRep(K )| = 1; and enumeration (ENUM) is enumerating all elements of XRep(K ).

Complexity classes. All of the complexity results stated in this paper are for the data complexity measure, i.e., the complexity is with respect to the size of A, with T and q treated as fixed (hence of constant size).

For decision problems, we will refer to the following complexity classes: AC 0 is the class of problems that can be solved by a uniform family of cicuits of constant depth and polynomial-size, with unbounded-fanin AND and OR gates; P is the class of problems solvable in polynomial time; NP is the class of problems solvable in non-deterministic polynomial time; Σ p 2 is the class of problems solvable in nondeterministic polynomial time with access to an NP oracle; coNP (resp. Π p 2 ) is the class of problems whose complement is in NP (resp. Σ p 2 ). These classes are related as follow:

AC 0 ⊂ P ⊆ NP ⊆ Σ p 2 and AC 0 ⊂ P ⊆ coNP ⊆ Π p 2 .
It is widely believed that all of the inclusions are proper.

For enumeration problems, we will use the following classes: TOTALP is the class of enumeration problems that can be solved in polynomial time in n + m where n is the input size and m the output size; DELAYP is the class of enumeration problems that can be solved by an algorithm such that the delay between the k th and the k + 1 th solution is bounded by a polynomial in the input size; INCP is the class of enumeration problems that can be solved by an algorithm such that the delay between the k th and the k + 1 th solution is bounded by a polynomial in the size of the input and k. These classes are related as follow: DELAYP ⊆ INCP ⊆ TOTALP. If the decision problem associated to an enumeration problem is NP-hard, then the enumeration problem is not in TOTALP.

SRep(K) PRep(K ) GRep(K ) CRep(K ) ISREP in P in P coNP-c in P AR coNP-c coNP-c Π p 2 -c coNP-c IAR in AC 0 coNP-c Π p 2 -c coNP-c BRAVE in AC 0 NP-c Σ p 2 -c NP-c UNIQUE in P coNP-c Π p 2 -c ‡ in P ENUM DELAYP * not TOTALP † not TOTALP † DELAYP *

Results

. Table 1 presents an overview of the complexity results for DL-Lite. However, since most of the results hold for some DLs outside the DL-Lite family, we will state them for DLs that satisfy some of the following properties:

• BOUNDCONF: the size of conflicts is bounded;

• BINCONF: conflicts have size at most two;

• POLYBCQ: polynomial time BCQ entailment;

• POLYCONS: polynomial time consistency checking.

The most commonly considered DL-Lite dialects, such as DL-Lite core and DL-Lite R (the basis for OWL 2 QL), satisfy all of these properties, while DL-Lite H horn satisfies all except BINCONF (the size of conflicts is bounded by the number of concept and role names occurring in the TBox).

Query Entailment

We start by establishing the complexity of repair checking and BCQ entailment under the various semantics.

The upper bounds are stated in the following theorem and can be proven by adapting existing results from databases and inconsistency-tolerant OMQA (recall that proofs of all results are provided in [START_REF] Bienvenu | Querying and repairing inconsistent prioritized knowledge bases: Complexity analysis and links with abstract argumentation[END_REF]). Observe that this result applies to all DLs satisfying POLY-CONS and POLYBCQ, which includes prominent Horn DLs like EL and Horn-SHIQ. Theorem 9. For DLs satisfying POLYCONS, repair checking is in coNP for globally-optimal repairs, in P for Paretoor completion-optimal repairs.

For DLs satisfying POLYCONS and POLYBCQ, G-AR and G-IAR entailment are in Π p 2 , G-brave entailment in Σ p 2 , P-AR, P-IAR, C-AR and C-IAR entailment are in coNP, and P-brave and C-brave entailment in NP.

For Pareto-optimal and completion-optimal repairs, coNP-hardness (resp. NP-hardness) of AR and IAR (resp. brave) IQ entailment in DL-Lite core follow from the special case where the priority relation is given by priority levels, for which it is known the three families of optimal repairs we consider coincide [START_REF] Bourgaux | Inconsistency Handling in Ontology-Mediated Query Answering[END_REF]. The following theorem establishes the remaining lower bounds. Theorem 10. In DL-Lite core , repair checking is coNP-hard for globally-optimal repairs, G-AR and G-IAR entailment are Π p 2 -hard, and G-brave entailment Σ p 2 -hard, even for IQs.

Uniqueness and Enumeration

We now turn our attention to the problems of deciding uniqueness and enumerating repairs. Note that in the case of classical repairs, deciding whether |SRep(K)| = 1 amounts to checking whether K is consistent (recall our assumption that all self-contradictory assertions have been removed).

As observed by [START_REF] Livshits | Counting and enumerating (preferred) database repairs[END_REF] in the database setting, classical repairs correspond to the maximal independent sets (MISs) of the conflict hypergraph. It is known that MIS enumeration for graphs is in DELAYP [START_REF] Johnson | On generating all maximal independent sets[END_REF], while for hypergraphs with bounded-size hyperedges, the problem is in INCP [START_REF] Eiter | Identifying the minimal transversals of a hypergraph and related problems[END_REF][START_REF] Boros | An efficient incremental algorithm for generating all maximal independent sets in hypergraphs of bounded dimension[END_REF]. Since the conflict hypergraph can be tractably computed for DLs with bounded-size conflicts, enumeration of SRep(K) is in INCP for DLs with bounded conflicts, and in DELAYP for DLs with binary conflicts. [START_REF] Kimelfeld | Detecting ambiguity in prioritized database repairing[END_REF] and Livshits and Kimelfeld (2017) provide intractability results for UNIQUE and ENUM in the case of globally-and Paretooptimal repairs of databases with functional dependencies. Their proofs do not transfer to DL-Lite KBs, but via different proofs, we can establish analogous results in our context. Theorem 11. Uniqueness is coNP-complete for Paretooptimal repairs, Π p 2 -complete for globally-optimal repairs. Upper bounds hold for DLs satisfying POLYCONS. Lower bounds hold for DL-Lite core . Corollary 12. Enumeration of Pareto-optimal or globallyoptimal repairs is not in TOTALP for DL-Lite core . [START_REF] Kimelfeld | Detecting ambiguity in prioritized database repairing[END_REF] provide an algorithm which, given an arbitrary conflict hypergraph and a priority relation, decides uniqueness for completionoptimal repairs in polynomial time. Hence deciding whether |CRep(K )| = 1 is in P for DLs with bounded conflicts, since in this case the conflict hypergraph can be computed in polynomial time. [START_REF] Livshits | Counting and enumerating (preferred) database repairs[END_REF] describe an algorithm that enumerates completion-optimal repairs with polynomial delay for conflict graphs. It follows that for DLs with binary conflicts, enumeration of completion-optimal repairs is in DELAYP. The case of completion-optimal repairs with non-binary conflicts remains open.

Case of Transitive Priority Relations

A natural case to consider is when the priority relation is transitive. This occurs in particular when the priority relation captures the relative reliability of the facts. [START_REF] Kimelfeld | Detecting ambiguity in prioritized database repairing[END_REF] proved that when the priority relation is transitive, deciding uniqueness for globally-optimal repairs can be done in polynomial time when the conflict hypergraph is given. This result is applicable to all DLs for which the conflict hypergraph is computable in polynomial time. A natural question is whether any of our other problems become easier if we assume is transitive. We found that this is not the case, although the transitivity requirement makes the lower bounds proofs more involved. In particular, we can no longer use the coNPhardness proof for uniqueness to derive that the enumeration of globally-optimal repairs is not in TOTALP, but instead show this via the coNP-hardness of deciding whether a given set of ABoxes is exactly GRep(K ). Theorem 14. When is transitive, uniqueness is in P for globally-optimal repairs and DLs satisfying BOUNDCONF. All other lower bounds in Table 1 still hold for DL-Lite core .

Definition 13. A priority relation is transitive if α 1 • • • α n and {α 1 , α n } ⊆ C ∈ Conf (K) implies α 1 α n .
Score-structured priority relations. An interesting special case of transitive priority relation arises when every assertion α is assigned a natural number called its score (or priority level) s(α), which induces a priority relation as follows: for every pair of assertions α and β that belongs to a conflict, α β if and only if s(α) > s(β). Livshits and Kimelfeld (2017) call such a priority relation score-structured and show that deciding whether a priority is score-structured is in P and that if is score-structured, CRep(K ) = GRep(K ) = PRep(K ). This relation was also given in [START_REF] Bienvenu | Querying inconsistent description logic knowledge bases under preferred repair semantics[END_REF][START_REF] Bourgaux | Inconsistency Handling in Ontology-Mediated Query Answering[END_REF] where it is shown that they further coincide with another notion of preferred repairs directly defined from ABoxes partitioned into priority levels.

For score-structured priorities, the complexity of all decision problems is the same as in the case of completionoptimal repairs, with the lower bounds shown in [START_REF] Bourgaux | Inconsistency Handling in Ontology-Mediated Query Answering[END_REF]. By contrast, for enumeration, it follows from the algorithm for enumerating score-structured priority repairs of Livshits and Kimelfeld that enumeration is in INCP for DLs with bounded conflicts and consistency checking in P.

Preference-Based Argumentation

In preparation for the following section, we recall the basics of argumentation frameworks and previously studied extensions with preferences and collective attacks. We also prove some new results (Theorems 22, 23, 28, and 32).

Argumentation Frameworks

We consider finite argumentation frameworks. Argumentation frameworks over an infinite set of arguments have also been studied in the literature but finiteness is an ordinary assumption and is made for all results we recall or extend here.

Definition 15. An argumentation framework (AF) is a pair (Args, ) where Args is a finite set of arguments and ⊆ Args × Args is the attack relation. When (α, β) ∈ , we say that α attacks β, alternatively denoted by α β.

We recall some commonly used notation and terminology, letting (Args, ) be the considered AF and A ⊆ Args. We use A + = {β | α β for some α ∈ A} to denote the set of arguments attacked by arguments from A. We say that A defends γ ∈ Args (or, γ is defended by

A) iff {β | β γ} ⊆ A + . A set A ⊆ Args is conflict-free if A ∩ A + = ∅. The characteristic function Γ F : 2 Args → 2 Args of F = (Args, ) is defined as follows: Γ F (A) = {α | α is defended by A}. A set A ⊆ Args is admissible if it is conflict-free and A ⊆ Γ F (A).
Argumentation semantics are usually based upon extensions, intended to capture coherent sets of arguments. We recall some prominent notions of extension: Definition 16. Let F = (Args, ). Then E ⊆ Args is a:

• complete extension iff E is conflict-free and E = Γ F (E); • grounded extension iff E is the ⊆-minimal complete extension, or equivalently, the least fixpoint of Γ F ; • preferred extension iff E is a ⊆-maximal admissible set;

• stable extension iff E + = Args \ E.
While every stable extension is a preferred extension, the converse does not hold in general [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. The term coherent designates AFs for which these two notions coincide (i.e. every preferred extension is stable). Coherence is viewed as a desirable property, and several sufficient conditions for coherence have been identified. In particular: Definition 17. An AF (Args, ) is symmetric iff is symmetric and irreflexive.

Theorem 18. (Coste-Marquis, Devred, and Marquis 2005) Every symmetric AF is coherent.

Enriching AFs with Preferences

There has been significant interest in extending AFs to allow for preferences between arguments (Amgoud and Cayrol 2002; Kaci, van der Torre, and Villata 2018).

Definition 19. A preference-based argumentation framework (PAF) is a triple (Args, , ), where Args and are as before, and is an acyclic binary relation over Args (called the preference relation). Symmetric PAFs are obtained by requiring (Args, ) to be symmetric.

It is typical to assume that the preference relation is transitive. This is a reasonable assumption, but for the sake of generality, the preceding definition does not impose this.

The standard way of defining the semantics of PAFs is via reduction to (plain) AFs: Definition 20. Given a PAF (Args, , ), the corresponding AF is (Args,

), where α β iff α β and β α. A subset E ⊆ Args is a stable (resp. preferred, grounded, complete) extension of a PAF (Args, , ) iff it is a stable (resp. preferred, grounded, complete) extension of the corresponding AF.

While alternative reductions have been proposed in (Amgoud and Vesic 2014) and [START_REF] Kaci | Preference in abstract argumentation[END_REF], all of these reductions coincide for symmetric PAFs, which is the case that will interest us here. Theorem 21. (Amgoud and Vesic 2014) Every symmetric PAF with a transitive preference relation is coherent.

We generalize the preceding result by showing that transitivity is not required: Theorem 22. Every symmetric PAF is coherent.

Theorem 23 gives a characterization of AFs that correspond to symmetric PAFs. It generalizes the strict acyclicity condition characterizing the so-called conflict+preference AFs [START_REF] Kaci | Acyclic argumentation: Attack = conflict + preference[END_REF] that requires for any cycle α 1 α 2 . . .

α n α 1 that α 1 α n . . . α 2 α 1 .
Theorem 23. For every AF F = (Args, ), the following statements are equivalent.

1. F is the corresponding AF of some symmetric PAF. 2. For any cycle α 1 α 2 . . . α n α 1 , there exists (j, i) ∈ {(1, 2), . . . , (n -1, n), (n, 1)} with α i α j .

Set-Based Argumentation Frameworks

Another well-studied extension of AFs is to allow collective attacks, in which a set of arguments together form an attack [START_REF] Nielsen | A generalization of Dung's abstract framework for argumentation: Arguing with sets of attacking arguments[END_REF][START_REF] Flouris | A comprehensive study of argumentation frameworks with sets of attacking arguments[END_REF]. Definition 24. A set-based argumentation framework (SETAF) is a pair (Args, ) where Args is a finite set of arguments, and ⊆ (2 Args \ {∅}) × Args is the attack relation. We'll write S α to mean (S, α) ∈ .

When working with SETAFs, we define A + as {β | S β for some S ⊆ A} and say A defends β iff A + ∩ S = ∅ whenever S β. The definitions of characteristic function, conflict-free, admissibility, different types of extensions, and coherence for SETAFs are the same as for AFs but using these modified notions of defeated and defended arguments.

Symmetric SETAFs were recently defined as follows: Definition 25. [START_REF] Diller | Investigating subclasses of abstract dialectical frameworks[END_REF]) A SETAF (Args, ) is symmetric if the following two conditions hold: (Symm-1) if S β and α ∈ S, then S ∪ {β} α for some S , and (Irr) there is no S α with α ∈ S.

Unfortunately, Diller et al. show their definition does not preserve the nice properties of symmetric AFs. In particular: Theorem 26. Some symmetric SETAFs are not coherent.

We propose an alternative, and we believe quite natural, notion of symmetric SETAF, which we term 'strongly symmetric' to distinguish it from the preceding notion: Definition 27. A SETAF (Args, ) is strongly symmetric if it satisfies (Irr) and (Symm-2): for every attack S β and every α ∈ S, we have S \ {α} ∪ {β} α. With our definition, symmetry implies coherence: Theorem 28. Strongly symmetric SETAFs are coherent.

Adding Preferences to SETAFs

It is natural to combine the preceding two extensions, but to the best of our knowledge, this combination has not yet been considered. We propose the following definition: Definition 29. A preference-based set-based argumentation framework (PSETAF) is a triple (Args, , ), where (Args, ) is a SETAF, and is an acyclic binary relation over Args, called the preference relation.

To define the semantics of PSETAFs, we give a reduction to SETAFs, which generalizes the one for PAFs: Definition 30. Given a PSETAF (Args, , ), its corresponding SETAF is (Args,

), where the relation ⊆ (2 Args \ {∅}) × Args is defined as follows: S α iff S α and α β for every β ∈ S. A subset E ⊆ Args is a stable (resp. preferred, grounded, complete) extension of a PSETAF (Args, , ) iff it is a stable (resp. preferred, grounded, complete) extension of the corresponding SETAF.

We will focus on strongly symmetric PSETAFs: Definition 31. A PSETAF (Args, , ) is strongly symmetric if the SETAF (Args, ) is strongly symmetric.

The following result lifts Theorem 21 to PSETAFs. The proof is quite intricate, and we leave open whether the same result holds without the transitivity assumption: Theorem 32. Every strongly symmetric PSETAF with a transitive preference relation is coherent.

Linking Prioritized KBs and PSETAFs

This section places optimal repairs into a broader context by exhibiting a tight relationship between optimal repairs and extensions of argumentation frameworks. The argumentation connection is then exploited to define a new kind of prioritized repair with favourable computational properties.

Each prioritized KB naturally gives rise to a PSETAF in which the ABox assertions play the role of arguments, and the attack relation captures the conflicts. Definition 33. Given a prioritized KB K with K = T , A , the associated PSETAF is

F K, = (A, K , ), where K = {(C \ {α}, α) | C ∈ Conf (K), α ∈ C}. Remark 34.
Recall that we assume the ABoxes do not contain self-conflicting assertions, hence C \ {α} = ∅. If we choose not to make this assumption, we should omit such assertions when constructing the PSETAF, i.e., using A \ {α | T , {α} |= ⊥} for the set of arguments and only considering C ∈ Conf (K) with |C| ≥ 2 to construct K .

In this section, results are formulated for arbitrary KBs, except where otherwise noted.

Optimal Repairs vs. PSETAF Extensions

Now that we have translated prioritized KBs into argumentation frameworks, it is possible to compare preferred repairs and extensions. The following theorem shows that Paretooptimal repairs correspond precisely to stable extensions.

Theorem 35. A ⊆ A is a Pareto-optimal repair of K = ( T , A , ) iff A is a stable extension of F K, .
Observe that our translation always produces strongly symmetric PSETAFs, which makes it possible to transfer Theorem 32 when the priority relation is transitive.

Theorem 36. If is a transitive priority relation, then the PSETAF F K, is coherent.

Theorems 35 and 36 together show that for transitive , Pareto-repairs coincide also with preferred extensions.

For KBs which contain only binary conflicts (e.g. core DL-Lite dialects), the resulting PSETAF is actually a PAF, so we can drop the transitivity requirement:

Theorem 37. For every prioritized KB K such that C ∈ Conf (K) implies |C| = 2, F K, is coherent. Thus, A ∈ PRep(K ) iff A is a preferred extension of F K, .
Note that it follows from Theorem 35 that globallyoptimal and completion-optimal repairs correspond to proper subsets of the stable extensions of F K, , but they do not at present have any analog in the argumentation setting. For a (SET)AF F that corresponds to a (strongly) symmetric P(SET)AF P , it is possible to define new types of extensions that will correspond to globally-and completionoptimal repairs of a KB having P for associated P(SET)AF and to import results from the prioritized KB setting. In particular, when F is an AF that respects the conditions of Theorem 23, we can adapt the algorithms for completionoptimal repairs to tractably enumerate a non-empty subset of the stable/preferred extensions of F , while enumerating its stable/preferred extensions is not in TOTALP in general.

Grounded Semantics for Prioritized KBs

We next turn to the relationship between grounded extensions and optimal repair semantics. We start by clarifying the situation for the simplest case, in which there are no preferences (cf. [START_REF] Croitoru | What can argumentation do for inconsistent ontology query answering?[END_REF] for a similar result in a different but related setting).

Theorem 38. Let K = T , A be a KB, and let F K be the SETAF corresponding to F K, ∅ (with ∅ the empty relation). Then the grounded extension of F K coincides with the intersection of the repairs of K.

For prioritized KBs, we can use Theorem 35 and the fact that every stable extension is a complete extension to relate grounded extensions with Pareto-optimal repairs: Theorem 39. If G is the grounded extension of F K, , then G ⊆ B for every B ∈ PRep(K ).

Thus, the grounded extension contains only assertions common to all Pareto-optimal repairs. It may not contain all such assertions, as the following example illustrates:

Example 40. Consider a prioritized KB K in which Conf (K) = {{α, β}, {α, γ}, {β, γ}, {γ, δ}} and where contains α γ, β γ, and γ δ. Then δ appears in the two Pareto-optimal repairs of K , which are {α, δ} and {β, δ}, while the grounded extension of F K, is ∅.

We propose to use the grounded extension to define a new inconsistency-tolerant semantics for prioritized KBs: Definition 41. A query q is entailed from K under grounded semantics, denoted K |= GR q, iff T , G |= q, where G is the grounded extension of F K, .

As follows from Theorem 39, our new semantics provides an under-approximation of the P-IAR semantics:

Theorem 42. For every prioritized KB K and query q: K |= GR q implies K |= P IAR q. Importantly, since the grounded extension of a (SET)AF can be computed in polynomial time, this semantics allows for tractable query answering in some relevant settings.

Theorem 43. Let L be a DL satisfying POLYBCQ and POLYCONS. Then the problem of BCQ entailment over L-KBs under grounded semantics is in P w.r.t. data complexity.

Corollary 44. For DL-Lite H horn KBs, BCQ entailment under grounded semantics is in P w.r.t. data complexity.

The grounded semantics can be compared to another recently proposed semantics for prioritized KBs, the Elect semantics, defined for KBs with binary conflicts by [START_REF] Belabbes | Elect: An inconsistency handling approach for partially preordered lightweight ontologies[END_REF] then generalized for nonbinary conflicts (Belabbes and Benferhat 2019). Elect evaluates queries over a set of elected assertions and is tractable for DLs for which computing the conflicts and BCQ entailment can be done in polynomial time w.r.t. data complexity.

Definition 45. (Belabbes and Benferhat 2019) An assertion α ∈ A is elected iff for every C ∈ Conf (K), if α ∈ C, then there exists β ∈ C such that α β. The set of elected assertions is denoted by Elect(K ).

The grounded semantics is more productive than Elect, i.e., is a more precise under-approximation of P-IAR.

Theorem 46. Elect(K ) ⊆ G where G is the grounded extension of F K, .

Example 47. Consider a prioritized KB K in which Conf (K) = {{α, β}, {β, γ}} and where contains α β and β γ. Then Elect(K ) = {α} while G = {α, γ}. Indeed, α ∈ G and γ is defended by {α} in the AF corresponding to F K, , so γ ∈ G.

Our next result establishes that reasoning under grounded semantics is P-complete in the settings of Theorem 43.

Theorem 48. IQ entailment under grounded semantics is P-hard w.r.t. data complexity for DL-Lite core . [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] observed that the grounded extension can be computed via the well-founded semantics of a simple logic program. The following lemma generalizes this result to k-SETAFs, i.e., SETAFs for which S β implies |S| ≤ k.

Theorem 49. Let F = (Args, ) be a k-SETAF. Then α is in the grounded extension of F iff acc(α) belongs to the well-founded model of the following normal logic program:

{def(x) ← att i (y 1 , .., y i , x),acc(y 1 ),...,acc(y

i ) | 1 ≤ i ≤ k} ∪ {acc(x) ← arg(x), ¬def(x)} ∪ {arg(α) | α ∈ Args} ∪ {att i (α 1 , . . . , α i , β) ← | {α 1 , . . . , α i } β}
Intuitively, the preceding logic program computes the sets of defeated (def) and accepted (acc) arguments starting from the original sets of arguments (arg) and attacks (att i encodes i-ary attacks). By adding rules to compute conflicts and populate the attack relations att i , we can show that the grounded extension of F K, can be computed via logic programming. While stated for DL-Lite H horn , the next theorem holds for any ontology (or constraint) language for which inconsistency can be characterized by a finite set of BCQs. Theorem 50. For every DL-Lite H horn TBox T , there exists a normal logic program Π T such that for every ABox A, priority relation for K = T , A , and assertion α, the following are equivalent:

• α belongs to the grounded extension of the PSETAF F K,

• acc(id(α)) belongs to the well-founded model of

Π T ∪ {γ ← | γ ∈ A id } ∪ {pref(id(α), id(β)) ← | α β},
where A id is obtained from A by adding an extra argument to every assertion β containing a unique id, denoted id(β).

As the well-founded semantics is implemented in logic programming systems like XSB [START_REF] Rao | XSB: A system for effciently computing WFS[END_REF], the preceding theorem presents a method for implementing grounded semantics. Alternatively, the grounded extension can be naturally under-approximated by fixing d > 0 and considering Γ d F (∅) rather than the least fixpoint of Γ F . The logic program from Theorem 50 can be modified to yield a non-recursive stratified program that computes Γ d F K, (∅), which in turn can be expressed as a first-order (∼ SQL) query and evaluated using a relational database system.

Related Work

Our complexity results are similar to those obtained for prioritized databases with functional dependencies [START_REF] Staworko | Prioritized repairing and consistent query answering in relational databases[END_REF][START_REF] Kimelfeld | Detecting ambiguity in prioritized database repairing[END_REF][START_REF] Livshits | Counting and enumerating (preferred) database repairs[END_REF]. Compared to this work, we additionally consider IAR and brave semantics, study the impact of transitivity on all reasoning tasks, and establish connections with argumentation.

Inconsistency-tolerant semantics based on other kinds of preferred repairs have been investigated both in the database and DL or Datalog ± contexts, often with a focus on repairs that have maximal cardinality or weight (Lopatenko and Bertossi 2007;[START_REF] Du | Weight-based consistent query answering over inconsistent SHIQ knowledge bases[END_REF][START_REF] Baget | Inconsistency-tolerant query answering: Rationality properties and computational complexity analysis[END_REF][START_REF] Lukasiewicz | Complexity of inconsistency-tolerant query answering in Datalog+/-under cardinality-based repairs[END_REF]. However, such global optimality criteria lead to a higher computational complexity (typically ∆ P 2 [O(log n)]-hard). By contrast, [START_REF] Bienvenu | Querying inconsistent description logic knowledge bases under preferred repair semantics[END_REF] show that for preferred repairs based on priority levels and setinclusion (corresponding to optimal repairs in the scorestructured case), BCQ entailment remains in the first level of the polynomial hierarchy for DL-Lite. Our results that even if the priority relation over facts is not score-structured, and not even transitive, these computational properties are retained for Pareto-and completion-optimal repairs. When is score-structured, we can compare the grounded extension and the intersection of optimal repairs to the proposals of [START_REF] Benferhat | How to select one preferred assertional-based repair from inconsistent and prioritized DL-Lite knowledge bases?[END_REF] for selecting a single preferred T -consistent subset of the ABox. In particular, the non-defeated repair is the union of the intersections of the repairs of S 1 , S 1 ∪ S 2 , . . . , S 1 ∪ • • • ∪ S n , where S 1 , . . . , S n is a partition of A into priority levels. Since Elect coincides with non-defeated semantics for scorestructured priorities (Belabbes and Benferhat 2019), the non-defeated repair is included in the grounded extension by Theorem 46; as the priority relation in Example 47 is scorestructured, the inclusion may be strict. Another proposal is the prioritized inclusion-based non-defeated repair, which coincides with the intersection of the optimal repairs. The grounded extension thus lies between the non-defeated and prioritized inclusion-based non-defeated repairs. Other proposals are either included in non-defeated, or are not sound approximations of the intersection of the optimal repairs. For partially preordered ABoxes, Belabbes and Benferhat (2019) propose to go beyond Elect by intersecting the optimal repairs w.r.t. the score-structured priorities obtained by extending the original partial preorder into a total preorder. Interestingly, the obtained set of assertions corresponds exactly to the intersection of the completion-optimal repairs of the prioritized KB underpinned by the partially preordered KB (cf. appendix of [START_REF] Bienvenu | Querying and repairing inconsistent prioritized knowledge bases: Complexity analysis and links with abstract argumentation[END_REF]).

Argumentation frameworks derived from inconsistent KBs have been considered before. Arguments are generally defined as pairs of a support (subset of the KB) and a conclusion (consequence of the support), with various attack relations (see e.g., [START_REF] Amgoud | Inferring from inconsistency in preference-based argumentation frameworks[END_REF][START_REF] Gorogiannis | Instantiating abstract argumentation with classical logic arguments: Postulates and properties[END_REF]. A series of papers starting with (Croitoru and Vesic 2013) links argumentation and inconsistency-tolerant querying of KBs, where the support of the argument is a subset of the ABox, the conclusion is a conjunction of facts entailed from the support and T , and α attacks β iff the conclusion of α is T -inconsistent with the support of β (in some papers, the argument consists of a whole derivation sequence, not just the support and conclusion). Stable and preferred extensions of such AFs correspond to repairs in the following sense: every such extension contains all those arguments whose supports are included in some particular repair. The work by [START_REF] Croitoru | Introducing preference-based argumentation to inconsistent ontological knowledge bases[END_REF] shows that it is possible to use a preference relation over facts to define notions of optimal extensions for AFs induced by inconsistent KBs in this fashion, and observe that such extensions correspond to optimal repairs. While touching on similar topics, an essential difference between this line of work and our own lies in the definition of the AF. Indeed, in the AFs of Croitoru et al., even if we group together arguments with the same supports, we would still have an exponential number of arguments. By contrast, our translation can be carried out in polynomial time under reasonable assumptions, enabling us to import tractability results from ar-gumentation to OMQA. Moreover, since we use assertions as arguments, each priority relation yields a preference relation over arguments, enabling a transparent reduction to preference-based (SET)AFs. Croitoru, Thomopoulos, and Vesic do not discuss any connections to preference-based, nor set-based AFs, and our results on the equivalence between Pareto-optimal repairs and stable and preferred extensions of P(SET)AFs do not follow from their results.

Conclusion and Future Work

We have explored the problem of how to repair and query inconsistent KBs while taking into account a priority relation over facts. By leveraging connections to database repairs and abstract argumentation, we obtained a number of novel results for our setting, while at the same time contributing new results and research questions to these two areas.

After importing the notions of Pareto-, globally-, and completion-optimal repairs into the OMQA setting, our first contribution was a data complexity study that showed, unsurprisingly, that reasoning with optimal repairs is typically intractable and more challenging than for classical repairs. Nevertheless, there are several cases which are 'only' NP/ coNP, which suggests that it may be interesting to devise practical SAT-based procedures, as has been successfully done for some other forms of repair [START_REF] Bienvenu | Querying inconsistent description logic knowledge bases under preferred repair semantics[END_REF]. It would also be relevant to implement and experiment our proposed grounded semantics as well as the first-order approximations we suggested. Our complexity study could be expanded to include further ontology languages as well as the combined complexity measure.

In order to provide an elegant translation of prioritized KBs to argumentation frameworks, we were naturally led to consider preference-based (SET)AFs. We believe that the new results we established in Section 5, in particular, Theorem 32, should be of interest to the argumentation community. Moreover, the connections we established between extensions of PSETAFs and repairs of prioritized KBs motivate a more detailed study of PSETAFs (a concrete open question is whether Theorem 32 holds in the absence of transitivity). Furthermore, this correspondence could be leveraged to explore new notions of extension for (strongly) symmetric P(SET)AFs inspired by completion-or globally-optimal repairs, or to design benchmarks for (P)(SET)AFs via a translation from inconsistent KBs.

An important question that has not yet been satisfactorily answered in the database literature is which of the three forms of optimal repair is most natural, independently of their computational costs. While we do not claim to provide a definitive answer to this question, we believe that our result showing that Pareto-optimal repairs coincide with stable extensions (and often also preferred extensions) speaks to the interest of adopting Pareto-optimal repairs. We should emphasize that while phrased for KBs, the connection between Pareto-optimal repairs and stable / preferred extensions holds equally well for databases with denial constraints (such as functional dependencies). Moreover, the tractable grounded semantics we proposed can be applied not just to KBs but also to prioritized databases.

  and each a ∈ N I to a I ∈ ∆ I . The function • I is straightforwardly extended to general concepts and roles, e.g. (¬B 1 ) I

  t. among those not yet considered, and add it to the current set if it does not lead to a contradiction. Example 6. Consider the prioritized KB K given on Figure 1. The sets of optimal repairs of K are as follows. CRep(K ) ={{Rattler(a), Carnivorous(a), EatMeat(a, b)}, {Boa(a), Carnivorous(a), EatMeat(a, b)}} GRep(K ) =CRep(K ) ∪ {{Mammal(a), Oviparous(a), Carnivorous(a), EatMeat(a, b)}} PRep(K ) =GRep(K ) ∪ { {Rattler(a), Herbivorous(a), EatPlant(a, b)}, {Boa(a), Herbivorous(a), EatPlant(a, b)}, {Mammal(a), Oviparous(a), Herbivorous(a), EatPlant(a, b)}} SRep(K) additionally contains repairs with Stone(b) instead of the EatPlant(a, b) or EatMeat(a, b) assertions.

TFigure 1 :

 1 Figure 1: Prioritized KB example. T formalizes knowledge about animal groups and A gives information about individuals a and b. An arrow from α to β indicates that α β while a dotted line indicates a conflict without priority between assertions.

Table 1 :

 1 Data complexity overview. Lower bounds hold for DL-Litecore. Upper bounds hold for DL-Lite H horn except those noted with * which hold only for DL-Litecore. Results for the SRep(K) case are stated (or are straightforward consequences of results) in (Bienvenu and Bourgaux 2016; Bienvenu and Rosati 2013; Livshits and Kimelfeld 2017). ‡ in P if is transitive † INCP if is score-structured

* INCP for DL-Lite H horn KBs with a scorestructured priority relation (see Section 4.4 for the definition)

Acknowledgements

This work was supported by Camille Bourgaux's CNRS PEPS grant and the ANR Chair INTENDED.


Professor, Jacques Le Cacheux

How to herd cats: economic policy coordination in the Euro zone in tough times i

This paper addresses the issue of fiscal policy coordination in the context of the current crisis. It first aims at clarifying the economic rationale for fiscal policy coordination in a monetary union with decentralized fiscal authorities, and at exploring the foundations of the kind of coordination devices chosen, as well as the incentives and constraints on member states' governments arising from the fiscal rules in the Euro Zone, both in tranquil and in stormy economic times. We then proceed with an analysis of the difficulties arising from the heterogeneous nature of the Euro Zone. In Section 3, we explore some of the possible causes of heterogeneity, with an emphasis on the issue of collective action and country size, with the coexistence of large and small countries, facing different incentives and constraints, hence tending to adopt divergent strategies in the occurrence of common macroeconomic shocks. Section 4 addresses the possible evolution in automatic fiscal stabilizers and Section 5 documents the size and structure of national fiscal stimulus packages. The concluding section advocates a better mix of rules and discretionary coordination for fiscal policies in the Euro Zone.

Introduction

The economic and financial crisis that has been spreading over the world economy since the beginning of 2008 has elicited strong policy reactions from national governments in almost all countries. In the immediate aftermath of the banking crisis and stock market collapse of September-October 2008, OECD countries have, for the first time in decades, announced large fiscal stimulus packages of Keynesian inspiration, while central banks have been pursuing extremely expansionary monetary policies. Yet, in spite of mounting evidence that the recession will, in 2009-2010 at least, be much more severe in the European Union (EU) than in the US, there have been no signs that additional policy measures will be implemented in European economies. Quite the contrary, on the eve of the London G20 summit of April 2, 2009, one of the major items of contention between the US government and Europeans was the size and significance of the various national fiscal stimulus plans: indeed, comparing the figures of the announced discretionary measures on both sides of the Atlantic, it appears that discretionary fiscal policies in EU member states are considerably less ambitious than what was enacted by the US Congress in February 2009.

One major difference between the US and the EU is, of course, that the latter is a collection of sovereign states, with only limited common institutions and procedures to deal with common fiscal policy. Indeed, the 2008-2009 deep recession may be regarded as the first common macroeconomic shock of large magnitude hitting the infant European monetary union -the tenth anniversary of the euro was celebrated in January 2009. In this large group of relatively small countries, macroeconomic policy coordination resorts to the logic of collective action [START_REF] Olson | The Logic of collective action[END_REF]. Factors explaining differences in incentives facing national governments should therefore be expected to play an important role in the choice of national strategies. One prominent parameter is country size, with smaller states being more tempted than larger ones by free rider strategies.

The apparent difficulties with fiscal policy coordination within the EU and, more specifically, within the Euro zone, are the topic of this paper. Hence the image of herding cats. Several questions are in effect raised: first what are the circumstances under which cats would gain from being herded; then what are the possible reasons why they are so hard to herd; and finally how could institutions be modified to make it easier to herd these cats. This paper tries to clarify the economic rationale for fiscal policy coordination in a monetary union and to shed light on the sources of current difficulties and tensions in the field of fiscal policy in the Euro zone. In the first section, we survey the major arguments for policy coordination and discuss the possible objectives of coordination, as well as the major devices to coordinate fiscal policies. Section 2 emphasizes one important source of difficulties in the current EU configuration, namely the heterogeneity of national macroeconomic performances and initial situations, at the outbreak of the recession. Section 3 then analyzes the constraints and incentives that arise from these various tools of policy coordination, in order to better understand the choices facing national governments in such settings; we stress that country size is probably a relevant dimension to consider when analyzing the incentives facing national governments in a monetary union. In Section 4, we raise the issue of automatic fiscal stabilizers and of whether differences in their magnitude might explain the difference in discretionary fiscal stimuli. Section 5 briefly discusses the contents of existing fiscal stimulus plans of a sample of EU countries. Finally Section 6 offers some tentative conclusions about the design of rules and institutions for a better macroeconomic policy mix in the Euro Zone.

1. Economic interdependencies, the nature of shocks and the rationale for coordination What emerges from the abundant economic literature on policy coordination is the generic idea that the need for coordination arises in contexts characterized by interdependencies, due either to collective goods or to externalities iii : in such contexts, decentralized decision-making in the absence of coordination devices will lead to sub-optimal, non cooperative, Nash equilibria. In a monetary union with decentralized fiscal authorities, economic interdependencies may arise from different channels. They result from the existence of collective goods, such as monetary stability or reputation on financial markets and vis-à-vis private agents in general, of public goods and common policies (defence, infrastructure building with network effects, etc.), or from spillovers, i.e. unintended consequences of national macroeconomic policies on other member states economies. Such spillover effects may be positive or negative.

Sources of interdependencies

When a group of countries participate in a regional economic integration process such as the European Union's Single market, they decide to abolish barriers to private transactions in goods, capital and, in the European case, obstacles to labour mobility, all measures fostering market integration, which in turn is likely to increase the magnitude of spillover effects resulting from the use of national economic policy tools, such as fiscal policies iv . Whether or not this process then spontaneously generates economic and financial convergence, intended as similarity in market conditions -prices and real incomes -as well as synchronicity and similar magnitude of business cycle fluctuations is still an open issue. Similarly, it is still unclear whether market integration, which generates a lowering of transaction costs, hence a mixture of specialization in production and agglomeration of productive activities, will make macroeconomic shocks more uniformly distributed across the whole area, or conversely more asymmetric and idiosyncratic v .

When countries form a monetary union, they share in common a single currency, and therefore uniform nominal variables vi -mostly the short-term nominal interest rate, which is the main instrument in the hands of modern central banks --, which may in turn accentuate the reduction in transaction costs, hence goods and financial market integration. Monetary and financial stability of the currency area is then properly regarded as a "collective good", insofar as it primarily depends on the joint use of their single monetary instrument, event though it may also be influenced by individual countries' actions or circumstances. But members of the monetary union may also decide that other aspects of their joint venture are to be considered "collective goods", be they of the traditional, public good kind, such as defence, or simply goods for which members collectively care, such as economic growth, or even some concept of distributional justice. In all such cases, the "collective good", once it has been recognized such, is indeed a common concern and henceforth subject to all the difficulties arising in a context of interdependencies.

The theory of "fiscal federalism" (Oates, 1972 and[START_REF]Le gouvernement économique de la zone euro[END_REF] suggests that in all cases where spillovers are of significant magnitude, the simplest way of ensuring the emergence of collective action is to deprive local, decentralized authorities from the competence over the corresponding policy instrument and to transfer it to the central government. But in the case of the EU, in contrast to what has progressively emerged in federations such as the US, there is no such thing as a significant central budget. Or rather, the EU budget is the common budget of all 27 member states, not just of the Euro zone; and it is too small and constrained by a ceiling on its overall size, with a balanced-budget rule preventing any countercyclical management at the central level. Hence, in cases when collective action is required in the field of fiscal policy, it can only emerge as the outcome of a coordination of decentralized governments.

The nature and transmission of macroeconomic shocks

But when is collective action really beneficial, and do the circumstances prevailing in 2008 correspond to such a context? One of the numerous contributions of Optimum Currency Area (OCA) theory has been the celebrated distinction between two types of macroeconomic shocks hitting a currency union: common, or symmetric shocks, i.e. those that affect all members of the union in a similar fashion; and specific, idiosyncratic, or asymmetric shocks, i.e. those that hit only one, or a subgroup of members of the union. The conventional wisdom derived from this theory was that common shocks could be dealt with by using the only common instrument in the currency union, namely monetary policy, whereas asymmetric shocks would give rise to more trouble, because countries would no longer enjoy the benefit of independent currencies and because the channels of adjustments to such shocks might not be very effective in the case of the Euro Zone. This seems to be the reason why much energy was dedicated to designing rules, such as the Stability and Growth Pact (SGP), that essentially aim at preventing national governments of individual countries from adopting divergent, "harmful" fiscal policies vii . And conversely, not much had been put in place for the occurrence of common shocks, under the assumption that monetary policy alone could handle them. It is true that Broad Economic Policy Guidelines (BEPG) were instituted to make sure that at least some degree of coherence in national macroeconomic policies -fiscal policies and supply-side measures-is maintained. But BEPG are mostly concerned with medium-term strategies, and cannot be relied upon as a coordination device in the face of a sudden common shock calling for immediate action; it also appears that BEPG have been only moderately successful as an instrument of coordination viii . Not much attention was paid to the possibility of having to face a common macroeconomic shock: in the first version of the SGP, special circumstances that could excuse exceeding the 3%-of-GDP ceiling for budget deficits were defined as a more than 2% recession; but it was widely held most unlikely.

Conventional wisdom also implicitly assumed that monetary policy would suffice to deal with common macroeconomic shock in the unlikely occurrence in which they might materialize. But the experience of Japan in the 1990s should have taught policy makers that there are circumstances in which monetary policy can no longer be relied upon, because it becomes ineffective due to the "zero barrier" -or "liquidity trap" in the language of old Keynesian economics-that prevents central banks from lowering real interest rates when inflation rates are close to zero or indeed negative. And for the first time since at least the Second World War, such a situation occurred in the Euro zone, as well as in the US and the UK. Whereas monetary policy alone can deal with common inflationary shocks, it cannot fight deflationary shocks of a large magnitude. In such cases, an appropriate policy mix is required, whence aggregate fiscal policy plays a major role.

At first sight, the 2008-2009 crisis should be regarded as a -large--common negative shock, in the terminology of the OCA theory. But is it really so? Or at least, was it perceived as such by national governments? Looking at the timing of quarterly macroeconomic evolutions, there are signs that the impact of the global crisis was not synchronized amongst EU members, and that the time lags may help understand some significant differences in national fiscal policy reactions. Table 1 indicates the quarterly changes in real GDP, as published by Eurostat in early 2009. Focusing on larger EU members, it shows that, for instance, Germany enjoyed a very buoyant growth in the first quarter on 2008, while France, as well as most other European countries, was experiencing a marked slowdown in economic growth. In the last quarter of 2008, the opposite happened, the slowdown in Germany being much steeper than in France, which may explain why the French government decided not to add to the already announced fiscal stimulus plan, whereas the German government went for additional spending in February 2009.

[Table 1, approximately here]

Macroeconomic spillovers from fiscal policies

Because the European monetary union was conceived at a time when monetary stability was widely held to be the single, most desirable objective, and with the aim of minimizing centralization, i.e. transferring only monetary powers at the supranational level, while leaving most attributes of economic sovereignty, especially fiscal and tax policies, in the hands of national governments, interdependencies stemming from the use of these instruments were given most attention in the debate over economic policy coordination. But due to the belief that large common negative shocks were not likely, the emphasis was put on fiscal rules, rather than on instruments to promote positive coordination.

Macroeconomic spillovers from national fiscal policies in a monetary union arise as a consequence either of market integration or, due to monetary integration, of the interactions between the aggregate outcome of decentralized fiscal policies and the central bank's decision process. Let us focus on the first category, such spillovers may be either positive -the so-called traditional Keynesian spillovers, that result from the multiplier effects of fiscal policies and their "locomotive effects" through trade-or negative, mostly through the consequences of deficit financing on financial variables, be they interest rates or exchange rates.

A major indicator of the strength of traditional, Keynesian interdependencies is the degree of trade openness ix . As is apparent from Table 2, the situation of EU members on the eve of the current crisis was, in this regard, contrasted, with most larger countries only moderately open, while smaller ones were, on average, much more open to trade, albeit with a few exceptions (Greece and Portugal, in particular).

[Tables 2 and3, and Chart 1, approximately here] An additional piece of evidence is provided by the change over time in the trade openness ratio, shown in Chart 1 for a selection of EU member states. In most smaller countries, not only was this ratio already high in the mid-90s, but it has been steadily rising since then, while the ratio has been mostly flat, and much lower, for all but one larger EU members, the exception being Germany.

The rationale for the SGP, the rule imposed in the Amsterdam Treaty on national fiscal policies x , is directly inspired by the assumption that such trade spillovers are relatively negligible, compared to other sources of, this time negative, spillovers arising in a monetary union whose priority in monetary stability. The latter tend to result from the weakening of discipline devices bearing on national fiscal policies for countries forming a currency union and from the nuisances that are inflicted on partners in the union by over-expansionary, and possibly unsustainable, national fiscal policies xi .

Economic theory suggests that positive spillovers or collective good situations should be dealt with by a centralized authority or other devices that foster collective action, either by common policy instruments or by incentives for decentralized decision-makers to act in the common interest, whereas negative spillovers can in principle be fought either by negative incentives or by the adoption of rules imposing limits and constraints on national fiscal policies, in order to prevent the nuisances arising from "bad behaviour" and "excessive public deficits". But these rules have a cost, in terms of lost opportunities or reduced flexibility, and they should be properly directed at making national fiscal policies sustainable, not necessarily through an upper limit on current deficits xii .

Heterogeneity in the Euro zone

Part of the difficulty in agreeing on common macroeconomic orientations may arise from differences in initial conditions, and in perceived margins of manoeuvre. Indeed, in spite of efforts during the transition phase to economic and monetary union, especially through the imposition of convergence criteria -the so-called Maastricht criteria--, the economies of the Euro zone have proved much more heterogeneous than had been expected. This is clearly shown in the standard macroeconomic performance indicators, especially measures of economic growth, long term real interest rates and public finance indicators, as well as in the current account positions of the various member states (Tables 4 to 7). How to explain such heterogeneity? That it may result from asymmetric shocks seems hard to believe, although there have undoubtedly been some divergences in domestic policy orientations. But whatever the sources of such divergences, they have, in the Euro zone, yielded different monetary conditions in the member states: in the last decade, under monetary union, with identical nominal short-term interest rates and, until recently, almost equal long-term ones, differences in domestic inflation rates were reflected in different real interest rates; and lower real rates not only tend to boost economic growth by making investment and private indebtedness less costly, but also reduce the real burden of existing public debt, thus easing the task of fiscal policy making.

[Tables 4 to 7, approximately here] 3. Possible causes of macroeconomic heterogeneity: Small and large states coping with differences in incentives Although the equal treatment principle is deeply entrenched in the democratic ideals and widely regarded as the only fair organization rule in a democracy, there seems to be a problem with the "one-size-fits-all" principle in the field of fiscal policy rules in the EU. In particular, the implementation of the SGP has shown weaknesses that have been clearly correlated with country size: thus, in 2004, all four large countries of the EU had budget deficits above the 3%-of-GDP ceiling; and will again in 2009. The 2004 episode, as well as the conflict over qualified majority rules that forestalled the adoption of the constitutional treaty project, have revealed a profound cleavage between small and large countries in the EU, a distinction that had never been apparent before. For scholars of the history of federal states and institutions, especially of the United States of America, this should not come as a surprise (see [START_REF] Laurent | L'Union européenne en quête d'un « Grand compromis[END_REF].

Nor indeed for economists, as it has long been emphasized that in contexts of "collective action", size matters and smaller players are more likely to free-ride and exploit larger ones [START_REF] Olson | The Logic of collective action[END_REF], and can easily be shown, the incentives -costs and benefits of various courses of action--facing a small open economy are not at all the same as the ones facing a medium-sized one, such as Germany or France, which may help explain the differences in performances and strategies that seem so systematic in the recent history of the euro zone.

For a small open economy, traditional fiscal policy of the Keynesian kind will usually be of limited effectiveness, whereas all policies that improve the competitiveness of the national economy by lowering production costs of firms located in the domestic economy are relatively more powerful: this may explain why fiscal consolidations in small countries have been found to have "non-Keynesian" effects (see [START_REF] Giavazzi | Non-Keynesian Effects of Fiscal Policy Changes: International Evidence and the Swedish Experience[END_REF]; it also suggests that tax competition, "structural reforms" and wage moderation policies will all have very powerful, positive effects for a small open economy, both because exports represent a major fraction of demand to domestic firms and because the elasticity of the supply of external capital -in particular foreign direct investments -is higher, the smaller and the more open the economy is. In addition, policies that lower production costs in a small economy do not harm domestic demand very much, and they have little incidence on domestic inflation, so that they do not raise real interest rates, as nominal rates in a monetary union tend to be uniform across countries and to be relatively little influenced by the policies of a single, small country xiii .

For large countries, on the contrary, free riding is almost impossible, and the various policy choices reviewed above tend to be more costly, or even counterproductive. Traditional, Keynesian-style demand-management policies, especially fiscal policies are more efficient than for a small open economy, because demand spillovers are relatively lower. On the other hand, all policies tending to lower production costs are less effective, and they all tend to lead to a lower domestic inflation, which then results in a higher real interest rate, so that they tend to be costly in terms of economic activity and growth. The contrasted fates of Germany and Ireland over the past few years seem to be a perfect illustration of this difficulty of large countries in an economic and monetary union [START_REF] Laurent | The Irish Tiger and the German Frog. A Tale of Size and Growth in the Euro Area[END_REF].

Automatic fiscal stabilizers

But aren't cats herded enough after all? In the weeks preceding the G20 summit in London, the controversy between the US government and EU national governments over the required size of national fiscal stimulus packages revolved around the magnitude and potential effectiveness of automatic fiscal stabilizers: whereas the US discretionary fiscal stimulus was, indeed, much larger than the announced EU plans. Europeans pointed to the allegedly much larger automatic fiscal stabilizers existing in Europe, with the implication that they needed not do as much as the US federal government in terms of discretionary fiscal stimulus. This dispute over automatic fiscal stabilizers is reminiscent of some of the European controversies over the SGP: in the minds of defenders of the SGP, with national budgets at or close to balance on average over the economic cycle, 3% of GDP was regarded as a sufficient margin of manoeuvre for national fiscal policies faced with the usual economic downturns; and this magnitude was deemed to correspond, at least roughly, to the size of automatic fiscal stabilizers incorporated in European national public sectors.

In theory, in the context of monetary unions and according the standard OCA theory arguments, one should distinguish between automatic stabilizers embedded in the functioning of the central budget [START_REF] Sachs | Fiscal transfers in currency areas[END_REF][START_REF] Zumer | Stabilisation et redistribution budgétaires entre régions : Etat centralisé, Etat fédéral[END_REF][START_REF] Melitz | International and interregional risk-sharing and lessons for the EMU[END_REF], and those arising from the functioning of national public sectors. With regard to the first category, it is well known that the EU budget is both too small and little sensitive to cyclical fluctuations to play a significant role, whereas the US federal budget has been shown to act as an automatic cushioning mechanism in case of asymmetric shocks occurring within the US economy. But it is nonetheless the case that the state and local public sectors pursue highly pro-cyclical fiscal policies in the US, for various reasons, including the institutional setting in which state and local fiscal policies are decided: in this respect, it may be argued that the magnitude of the discretionary federal fiscal stimulus is, at least in part, aimed at counteracting the pro-cyclicality of state and local fiscal policies [START_REF] Le Cacheux | Les finances publiques locales américaines entre la récession et la nouvelle politique fédérale, Note de la Mission Scientifique[END_REF][START_REF] Schelkle | Good governance in the crisis: A comparison of the EU and the US[END_REF].

What about the magnitude of automatic fiscal stabilizers embedded in European national public sectors? At first sight, they may look larger, insofar as the size of public sectors and the extent of social welfare devices are much larger on average in EU countries than in the US. However, it is also the case in Europe that local public sectors tend to behave pro-cyclically, even though this is probably less documented than in the US. And there is evidence that, over the past decades, general governments' budgets have evolved toward lesser automatic stabilizers (Solow, 2004;[START_REF] Creel | Automatic stabilisation, discretionary policy and the Stability Pact[END_REF]: (personal and corporate) income tax schedules have become less progressive, while in many countries the generosity and length of various counter-cyclical, welfare devices -and most prominently unemployment benefits-have been significantly reduced. How important is this reduction in the magnitude of automatic stabilizers is hard to assess precisely.

The heterogeneity of national fiscal stimulus packages

Table 8 gives the composition of fiscal stimulus plans in a selection of EU countries xiv . Whereas it provides only mixed evidence on the effect of country size on national fiscal policy choices, it clearly supports the view that national discretionary measures are very heterogeneous, both in size and in the precise targeting of sectors and types of expenditures. In particular, the UK appears to be the only country to have resorted to an across-the-board measure to support private domestic consumption, a VAT temporary cut, which, if effective at all, will benefit foreign as well as domestic producers; by contrast, all other countries featuring in the table -all members of the Euro zone, incidentally-have tended to favour stimulus measures targeted at home producers, such as public investment. But, as shown in Table 9, the sizes and composition of national fiscal stimulus plans are not such that any defined pattern emerges, except for the fact that they definitely are smaller than the one enacted in the US.

[Tables 8 and9, approximately here] 6. Concluding remarks: fiscal policy coordination without a gouvernement économique As abundantly illustrated by the vicissitudes of the Stability and Growth Pact and by the poor performance of the economies of Euro zone over the past few years, fiscal policy coordination is still embryonic and not very satisfactory in the European monetary union, after more than ten years of existence of the European currency. Though not really surprising, when considered in a historical perspective -it took much more than ten years for the US Federal Reserve System, created in 1913, and the Federal government to set up the subtle mechanisms of macroeconomic management that are now seen to operate in the US economy -this difficulty is clearly hampering the achievement of other goals and negatively affecting the whole process of institution-building and integration in the EU. This lack of coordination appears especially costly in the current crisis, with the prospects of a very deep and protracted recession, followed by a very shallow recovery in the Euro zone.

Because of the still very large divergences in preferences and perceptions of costs and benefits of the various possible strategies for fiscal policies, and more generally for economic policy-making amongst member governments of the Euro zone, the prospects for a genuine economic government of the Euro zone are rather gloomy. It is therefore not surprising that in the face of the massive recession that has hit the EU economies, coordinated policy responses have not easily been forthcoming. This is all the more problematic as an appropriate aggregate fiscal orientation, as well as an appropriate macroeconomic policy mix, would probably prove more effective in fighting the recession than disorderly national policy responses that seem to be inspired, at least in part, by free-riding considerations. And although all EU governments are now facing very similar fiscal circumstances -high public deficits and sharply increasing public debt ratios -it seems unlikely that cats will be more easily herded when it comes to public finance consolidation plans.

In the absence of a genuine gouvernement économique xv that would in some way aggregate preferences over common rules and collective goods and make use of policy instruments to ensure their provision, there may be ways to improve on the current performance within existing institutions. In 2005, in response to difficulties in the implementation of the SGP, the latter was reformed and notably improved to face small, idiosyncratic shocks. But this is clearly insufficient to elicit coordinated fiscal policies in much stormier economic times. A more ambitious reflection would try to explore the possibilities of developing federalism, either by embarking on a reinforcement of the central EU budget along the lines once suggested by the MacDougall Report (EC Commission, 1977) or, more realistically perhaps, to devise an original brand of highly decentralized federalism, which may be labelled "Pigouvian federalism" [START_REF] Laurent | L'Union européenne en quête d'un « Grand compromis[END_REF], in which the central budget would remain relatively small, but where an appropriate mix of ruleswith desirable features, such as an amended "golden rule" of public finance, whereby expenditures to be encouraged are not counted in the agreed deficit ceiling or target --and financial incentives, in the European budget, is set up in order to induce national governments to undertake actions that are collectively considered to be in the common interest. ii Correspondance: Jacques Le Cacheux, CATT, Faculté de droit, de sciences économiques et de gestion, avenue du Doyen Poplawski -64000 PAU -FRANCE.

e-mail : jacques.lecacheux@sciences-po.fr.

iii Rather than a clear-cut distinction between these two generic situations of interdependencies, they may be thought of as pertaining in a continuum, as convincingly advocated by [START_REF] Buchanan | The demand and supply of public goods[END_REF], who also points to the similarities between public goods and "club goods", here aggregated under the label "collective goods", and to the subjective nature of such economic interdependencies, insofar as they may not be purely "technological", but may also be generated by perceived interdependencies of individuals' (actors') utility functions.

iv This is of course reminiscent of the well-established tradition of functionalist analysis of regional integration processes, which does not have to be teleological, as the rest of this chapter endeavours to show. v As emphasized in the economic literature of "optimal currency areas" (OCA), initiated by [START_REF] Mundell | A Theory of Optimal Currency Areas[END_REF], the nature of macroeconomic shocks, and most importantly the distinction between common, or symmetric, and idiosyncratic, or asymmetric shocks, is a major dimension of the analysis of stabilization policies in a monetary union. See below.

vi But not necessarily monetary conditions and all nominal variables: in particular, inflation rates will usually differ amongst countries in a monetary union, as will be made clearer in the following.

vii On OCA implications for the Euro Zone, one may refer to [START_REF] Obstfeld | Regional non adjustment and fiscal policy[END_REF]. On the Commission view, see [START_REF] Buti | Economic Policy in EMU. A Study by the European Commission Services[END_REF]. On the rationale for the Stability Pact, see, in particular, [START_REF] Buiter | Excessive deficits: sense and nonsense in the Treaty of Maastricht[END_REF][START_REF] Le Cacheux | To co-ordinate or not to co-ordinate: An economist perspective on the rationale for fiscal policy coordination in the euro zone[END_REF].

viii On the functioning of the BEPG, see the recent paper by [START_REF] Deroose | The Broad Economic Policy Guidelines: Before and after the Re-Launch of the Lisbon Strategy[END_REF] and the numerous references therein. Their overall judgement is moderately positive, but they don't raise the issue of coordination in the face of large common shocks. ix Of course, rather than measuring a country's total trade with the rest of the world, it would be preferable to focus on intra-European trade. But because the average share of intra-European trade is approximately 2/3, the rough indication given with total trade is sufficient. An indication is given for the case of Germany in Table 3.

x The Stability and Growth Pact was conceived as a sequel of the so-called "Maastricht criteria", asset of conditions that had meant to secure convergence of public finances amongst the EU countries before the creation of the monetary union. The Pact, instead, is a permanent fiscal rule adopted in June 1997 in a Protocol of the Amsterdam treaty.

xi On the rationale for "rules rather than coordination", a more detailed analysis in offered in Le Cacheux (2007).

xii There has been a very abundant literature on the merits and inconveniences of a ceiling on public deficits, such as included in the Stability and Growth Pact, following the critical piece on the "Maastricht criteria" by [START_REF] Buiter | Excessive deficits: sense and nonsense in the Treaty of Maastricht[END_REF]. See, for instance, [START_REF] Eichengreen | The Stability Pact: more than a minor nuisance?[END_REF]. For recent assessments, see the collection of papers in Farina and Tamborini, eds., 2008. xiii The notion of "small, open economy" also refers to the idea that it has no influence on its environment, therefore is a "price taker" and, in game theory terms, chooses strategies without caring about possible reactions or retaliation from partners. Our arguments are also somewhat reminiscent of, though different from the wellknown result of trade theory, namely that small countries gain from a unilateral tariff reduction, whereas larger countries are likely to lose. Another way of putting the argument would be to say that a small country faces a 2 1 much more elastic supply of foreign capital and/or firms that a large one, or else that the former faces a much more favourable Laffer curve than the latter.

xiv Additional evidence may be found in Saha and von Weizsäker (2009). They refer to those measures of national fiscal stimulus packages taking effect in 2009, and their figures are not fully consistent with those in the table.

xv On this notion, see various other contributions in Linsenmann, Meyer andWessels, eds., 2007, as well as Commissariat général du Plan, 1999.

Table 1 : GDP growth rates in EU countries, quarterly profiles Percentage change compared with the previous quarter

 1

	1
	0
	Percentage
	change
	compared with
	the same quarter
	of the previous
	year

Table 2 : Trade to GDP in 2005 for OECD countries

 2

	United States	13,4	Finland	39
	Japan	13,6	Korea	41,2
	Australia	21	Switzerland	44,5
	Greece	22	Sweden	44,9
	Italy	26,3	OECD average	45
	France	26,6	Denmark	46,2
	Spain	28,2	EU15 average	50,7
	United Kingdom	28,3	Austria	51,9
	New Zealand	29,1	Netherlands	66,1
	Mexico	30,7	Hungary	67,1
	Turkey	30,7	Czech Republic	70,8
	Portugal	32,9	Ireland	74,9
	Canada	36	Slova k Republic	79,8
	Norway	36,7	Belgium	86
	Poland	37,2	Slovenia	129,7
	Germany	38,1	Luxembourg	148,6
	Iceland	38,3
	Source : OECD.

Trade to GDP ratio for selected EU member states, 1995-2005

											1
											3
	55
	50
	45										Austria Denmark
	40										Finland
											France
	35										Germany
											Italy
	30										Portugal
	25										Sweden UK
	20
	15
	1995	1996	1997	1998	1999	2000	2001	2002	2003	2004	2005
	Source: OECD, and Laurent and Le Cacheux, 2007.

Table 3 : First 15 trade partners of Germany in 2005

 3

	Exports	912,2	%	Imports	796,2	%
	France	79	8,7	France	53,7	6,7
	US	69,3	7,6	Netherlands	51,8	6,5
	UK	60,4	6,6	US	41,8	5,2
	Italy	53,9	5,9	China	40,8	5,1
	Netherlands	49	5,4	UK	39,1	4,9
	Belgium	43,6	4,8	Italy	36,3	4,6
	Austria	43,3	4,7	Belgium	28,8	3,6
	Spain	40	4,4	Austria	26	3,3
	Swiss	29,6	3,2	Swiss	22,6	2,8
	Poland	22,3	2,4	Russia	22,3	2,8
	China	21,2	2,3	Japan	21,8	2,7
	Russia	17,3	1,9	Spain	18,1	2,3
	Tch. Rep.	19,2	2,1	Tch.Rep.	17,7	2,2
	Sweden	17,2	1,9	Poland	16,8	2,1
	Hungary	13,6	1,5	Norway	15,1	1,9
	Source: Destatis, and Laurent and Le Cacheux, 2007.

Table 4 : GDP growth rates in Euro Zone countries, ten-year averages

 4

	Average		Average	Average
	1979-1988	1989-1998	1999-2008
	AUT	2,21	2,66	2,27
	BEL	2,04	2,09	2,27
	DEU	1,97	2,52	1,56
	ESP	2,22	2,69	3,54
	FIN	3,70	1,74	3,36
	FRA	2,24	1,96	2,16
	GRC	0,73	1,95	4,15
	IRL	2,84	6,58	5,85
	ITA	2,78	1,58	1,36
	LUX	3,81	4,93	4,88
	NLD	1,72	3,17	2,35
	PRT	3,27	3,17	1,70
	EURO	2,23	2,25	2,12
	Ecart-type	0,87	1,47	1,41

Table 5 : Current accounts in Euro Zone countries, (% of GDP, ten-year averages)

 5

		Average	Average	Average
		1979-1988	1989-1998	1999-2008
	AUT	-0,43	-1,20	0,57
	BEL	0,52	4,68	3,26
	DEU	1,43	0,12	3,23
	ESP	-0,57	-1,73	-5,90
	FIN	-1,42	-0,07	5,93
	FRA	-1,10	0,48	0,42
	GRC	-4,58	-3,28	-8,75
	IRL	-6,19	1,42	-2,00
	ITA	-0,74	0,49	-1,26
	LUX
	NLD	2,17	4,11	5,17
	PRT	-3,41	-2,01	-9,13
	EURO	0,05	0,26	0,31
	Ecart-type	2,52	2,44	5,27

Table 6 : Real long-term interest rates in Euro Zone countries, ten-year averages

 6

		Average	Average	Average
		1979-1988	1989-1998	1999-2008
	AUT	4,32	4,40	2,60
	BEL	5,81	5,20	2,27
	DEU	4,80	4,15	2,81
	ESP		5,03	1,16
	FIN	2,91	6,62	2,65
	FRA	4,32	5,27	2,61
	GRC	-1,75	5,58	0,66
	IRL	3,53	5,47	0,70
	ITA	2,79	6,18	2,22
	LUX	5,83	4,20	0,97
	NLD	5,22	4,61	2,15
	PRT	4,04	6,76	1,55
	EURO			2,21
	Ecart-type	2,11	0,89	0,81

Table 7 : Public deficits in Euro Zone countries, (% of GDP, ten-year averages)

 7

		Average	Average	Average
		1979-1988	1989-1998	1999-2008
	AUT	-3,26	-3,41	-1,51
	BEL	-10,61	-5,26	-0,27
	DEU	-2,39	-2,40	-1,99
	ESP	-4,72	-4,78	0,15
	FIN	3,49	-1,84	3,87
	FRA	-2,15	-3,90	-2,68
	GRC	-7,09	-9,26	-4,25
	IRL	-10,27	-1,43	1,35
	ITA	-10,46	-8,39	-2,80
	LUX		2,15	2,27
	NLD	-4,62	-3,18	-0,32
	PRT	-6,62	-5,12	-3,41
	EURO	-4,54	-4,22	-1,74
	Ecart-type	4,30	3,06	2,44
	Sources of Tables 4 to 7: OECD, calculations by Jérôme Creel., and Fitoussi and Le Cacheux, eds.,
	2009.

Table 8 : National fiscal stimulus plans in a selection of EU countries, as of January 2009

 8 Source: Natixis, compiled from national sources.

	Bn €	UK	Germany France Italy	Spain
	Housing	2.1	1.8	1.8	0.3	0.5
	Public infrastructures	1.2	4.6	10.5	1.4	8.0
	Private investment incentives 0.5	19.5	10.7	1.2	7.3
	Incentives to buy durable goods		3.5	1.0	2.6	1.8
	Private consumption support 15.0	0.5	1.5		0.6
	Public consumption	1.2	2.0			0.9
	Total (bn €)	20.0	31.9	25.5	5.5	19.1
	Total (% of GDP)	1.2	1.3	1.3	0.4	1.7

Table 9 : Automatic and discretionary variations in fiscal balances in the 11 original Euro zone members*

 9 Comments on a previous version by Iain Begg, Reinhard Felke, Eric De Souza, and participants in the EU-CONSENT Workshop, Brugge, April 7-8, 2009, as well as two referees from this Journal, are gratefully acknowledged. Research collaboration with economists of OFCE, especially Jérôme Creel and Eloi Laurent, have also been important in feeding my reflection. Of course, I remain the sole responsible for any remaining error.

				(% of GDP)
	Observed		Impact of	Discretionary	Total fiscal
				recession	package	balance**
	2007	2008	2009	2010

i Third draft.