Jérôme Mengin
email: jerome.mengin@lri.fr

A theorem prover for default logic based on prioritized conflict resolution and an extended resolution principle

This paper presents a theorem prover for Reiter's default logic, one of the most studied nonmonotonic logics. Our theorem prover is based on a decomposition of default logic into two main elements: we describe an extension of the resolution principle, that handles the "monotonic" aspect of the defaults, and we provide a generalization of Reiter's and Levy's algorithms for the computation of hitting sets, that takes care of the nonmonotonic part of default logic. Lastly, we describe how these two components can be separately modified in order to obtain theorem provers for other variants of default logic, notably prioritized default logic.

Introduction

Default reasoning occurs whenever an agent must complete the certain information he has about a domain, with some knowledge which is plausible but not infallible. Pieces of such knowledge, also called pieces of default knowledge or defaults for short, enable the agent to draw reliable conclusions, which may have to be retracted in presence of new information.

In many logical approaches to default reasoning, the certain information is completed with defaults that are coherent with one another, to generate an extension of the certain information. As there are, usually, several possible combinations of defaults that are coherent with one another, several extensions can be built. This notion of coherence can be the consistency in the sense of classical logic [START_REF] David | A logical framework for default reasoning[END_REF][START_REF] Gärdenfors | Nonmonotonic inference based on expectations[END_REF] or of some modal logics [START_REF] Schwind | Modal semantics for hypothesis theory[END_REF][START_REF] Truszczyński | Modal interpretation of default logic[END_REF]. Other notions of coherence have been shown [START_REF] Levy | Computing extensions of default theories[END_REF][START_REF] Froidevaux | Default logics: a unified view[END_REF] to underly Reiter's default logic [START_REF] Reiter | A logic for default reasoning[END_REF] and its variants.

In order to restrict the number of extensions, it is often necessary to use priorities among the defaults, like the so-called specificity principle, widely used in taxonomy. Other kinds of priorities, studied in the context of default reasoning, include ordering relations among the defaults, representing their relative reliabilities [START_REF] Brewka | Preferred subtheories: An extended logical framework for default reasoning[END_REF][START_REF] Junker | Handling partially ordered defaults in TMS[END_REF][START_REF] Geffner | Conditional entailment: bridging two approaches to default reasoning[END_REF][START_REF] Cayrol | Un modèle logique général pour le raisonnement révisable[END_REF], or some notions implicitly embedded in fixed-point constructs, as it is the case in Reiter's default logic [START_REF] Reiter | On interacting defaults[END_REF][START_REF] Brewka | Cumulative default logic: In defense of nonmonotonic inference rules[END_REF][START_REF] Delgrande | Default logic revisited[END_REF].

A growing number of studies deal with algorithmic problems related to the computation of extensions. Some of these studies are based on TMS like algorithms [START_REF] Junker | Computing the extensions of autoepistemic and default logics with a truth maintenance system[END_REF][START_REF] Junker | Handling partially ordered defaults in TMS[END_REF]. Other methods rely on more ad hoc approaches [START_REF] Niemelä | Decision procedures for autoepistemic logics[END_REF][START_REF] Schwind | A tableau-based theorem prover for a decidable subset of default logic[END_REF][START_REF] Schwind | A tableau-based characterization for default logic[END_REF][START_REF] Hopkins | Default logic: Orderings and extensions[END_REF][START_REF] Satoh | A top down proof procedure for default logic by using abduction[END_REF]. In this paper, following an approach by Levy [START_REF] Levy | Computing extensions of default theories[END_REF], we present two methods that separately handle the notion of coherence between the defaults, and that of priorities among them.

More precisely, we propose a generalization of the resolution principle to defaults. That is, we define so-called "extended clauses", that are some kind of clausal representations of defaults, and we then define some new resolution rules for these extended clauses. We can then check the coherence of a set of defaults using these resolution rules, by trying to generate an empty clause.

We then propose a generalization of Levy's [START_REF] Levy | Reason maintenance systems and default theories[END_REF] algorithm for the computation of sets of defaults that generate extensions. This algorithm works by "eliminating", from an initial set of defaults, as many defaults as possible in order to restore some coherence of this set. The particularity of our algorithm is that it is not specially designed for default logic. It can handle a wide class of priorities among pieces of default knowledge.

In the next section, we present Reiter's original definition of default logic, and we recall the characterization that is used in our theorem prover. In Sect. 3, we describe our extended resolution principle, together with the new type of clauses that we introduce. Sect. 4 presents our elimination algorithm, and Sect. 5 describe some modifications that can be done to these two methods in order to obtain other variants of default logic.

Default logic

A default theory is a pair (W, D), where W is a set of closed formulas of first order predicate calculus, representing the certain information, and D is a set of defaults, of the.form f :g h , where f, g, h are closed formulas of the language of first order predicate calculus. The default f :g h is intended to mean: "if f is true, and if nothing proves that g is false, then conclude h". We suppose, in the sequel, that W is consistent. Reiter [START_REF] Reiter | A logic for default reasoning[END_REF] defines sets of theorems that are consequences of a default theory: Definition 1 ( [START_REF] Reiter | A logic for default reasoning[END_REF]). A set of formulas E is an extension of a default theory (W, D) iff E = Γ (E), where Γ (E) is the smallest set that is deductively closed, contains W , and such that for all f :g h ∈ D, if f ∈ Γ (E) and ¬g / ∈ E, then h ∈ Γ (E).

In [START_REF] Mengin | Prioritized conflict resolution for default reasoning[END_REF], we give another characterization of the extensions of a default theory, based on the following definitions: Definition 2 ( [START_REF] Mengin | Prioritized conflict resolution for default reasoning[END_REF]). Let (W, D) be a closed default theory. The set of formulas generated by a subset U of D, denoted by Th def (W ∪ U ), is the smallest set of formulas that contains W , is deductively closed (in the sense of predicate calculus), and such that for all f :g h ∈ U , if f ∈ Th def (W ∪U ) then h ∈ Th def (W ∪ U ). We will denote by Th the deduction operator of the predicate calculus.

Definition 3 ([19]

). A conflict of a default theory (W, D) is a subset C of D minimal that contains a default f :g h such that f ∧ ¬g ∈ Th def (W ∪ U ). In the sequel, Ξ denotes the set of conflicts of a theory.

Definition 4 ([19]

). An elimination function on a default theory (W, D) is a function φ that associates, to each conflict C ∈ Ξ, a set of pairs (d, V ) ∈ C ×2 D . A subset U of D is φ-preferred if it does not contain any conflict of the theory and verifies:

∀d ∈ D \ U, ∃C ∈ Ξ, C ⊆ U ∪ {d} and (d, U ) ∈ φ(C)
The intended meaning of (d, V ) ∈ φ(C) is that, in order to resolve the conflict C, one can eliminate d, as soon as the elements of V are not themselves eliminated. See [START_REF] Mengin | Prioritized conflict resolution for default reasoning[END_REF] for a more detailled presentation of these notions.

Theorem 1 ([19]

). A set of formulas E is an extension of a default theory (W, D) iff E is generated by an φ DL -preferred subset of D, where φ DL is the elimination function defined by: φ

DL (C) = {( f :g h , V ) ∈ C × 2 D | f ∧ ¬g ∈ Th def (W ∪ V )}.
The next proposition shows that the extended deduction operator Th def is strongly related to the notion of conflict. It can be compared to a general result of classical logic: a formula f is classicaly entailed by a consistent set of formulas

W if and only if W ∪ {f → ⊥} is inconsistent. Proposition 1 ([20]). Given a default theory (W, D), whose set of conflicts is empty, then f ∈ Th def (W ∪ U ) iff f :⊥ ⊥ is in a conflict of the theory W, U ∪ { f :⊥ ⊥ }.
Let us now outline the major steps taken by our theorem prover. Given a default theory (W, D) and a formula f , we want to decide whether f is in at least one extension of the theory:

1. Compute the conflicts of (W, D ∪ { f :⊥ ⊥ }) (including those of (W, D)). 2. Compute φ DL on the conflicts of (W, D). 3. Compute the φ DL -preferred subsets of D. 4. f is in the extension generated by an φ DL -preferred subset U of D iff there is a conflict V ∪ { f :⊥ ⊥ } of (W, D ∪ { f :⊥ ⊥ }) such that V ⊆ U . c1 ¬c 2 :c c d ¬c 1 :c c d ¬c 2 :c c d ¬2:c c d ¬2:c c d c1 ¬2:c 2 d ¬2:c 1 d ¬2:c 2 d ¬c:c c d ¬c:c c d c ¬2:c d ¬2:c d ¬2:c d (1) (2) (3) (4) (5) (6)
Fig. 1. Extended resolution rules (c1 and c2 denote two clauses whose resolvent is c)

An extended resolution principle

In order to compute the conflicts of a default theory, we extend the classical resolution principle to defaults. Figure 1 presents the "extended resolution rules", that we will use together with the classical one.

Rules ( 1) and ( 2) are used to prove the prerequisite of a default d from the clauses of W and the consequents of other defaults obtained from rule (3) (see below): we make resolutions against the negation of the prerequisite of the default only. Therefore, when the extended clause " ¬2:c c d " is produced, the prerequisite of this default is proved.

Once the prerequisite of d has been proved, its consequent can be used like any formula of W (rule 3). However, the consistency of its justification is checked by making resolutions on it (rules 5-6). The production, using these rules, of an extended clause of the form ¬2:2 d proves a conflict. Rule ( 4) is there to avoid producing again the consequent of d using rule (3) everytime a resolution is made on its justification.

The production of an empty clause, using these extended resolution rules and the classical one, shows that the falsity is a consequence, in the sense of Th def , of the set of defaults. In this case, the negation of the justification of any default can also be proved, therefore it also proves a conflict.

Notice how the indexes of the extended defaults have to be taken into account in rules ( 2) and ( 6): we can only make a resolution between the prerequisites (respectively the justifications) of two extended clauses if they have the same indexes, that is, if they have been produced from the same default. We will see the effect of relaxing these conditions in Sect 5.

Theorem 2 ([20]

). Let (W, D) be a default theory, such that W is a finite set of clauses and D is finite and can be put under clausal form. A subset C of D is a conflict of the theory iff it is minimal such that the empty clause or an extended clause of the form ¬2:2 d can be produced from W ∪ C using rules (1) to [START_REF] Cayrol | Un modèle logique général pour le raisonnement révisable[END_REF] and the classical resolution rule. Example 1. Let W = {p ∨ q, ¬r ∨ ¬s} and D = {d, d }, where d = p∨q:r s and d = q:t t . Figure 2 shows the extended resolutions produced to prove that {d} is a conflict of the theory.

p ∨ q ¬(¬p):r s d ¬(¬q):r s d ¬r ∨ ¬s ¬(¬q):t t d X X X ¬(q):r s d X X X ¬(2):r s d ¬(2):r d s X X X X X X X X X ¬r X X X ¬ (
In order to compute all the conflicts of a theory, we use [START_REF] Mengin | Raisonnement par défaut : résolutions de conflits et priorités[END_REF] Besnard et al.'s saturation by set [START_REF] Besnard | A theorem prover for a decidable subset of default logic[END_REF], which returns all conflicts of a set of (extended) clauses, and which is incremental: given two sets of clauses, it is possible to saturate E ∪ F using the result of the saturation of F . Thus, given a default theory (W, D), we will first saturate W ∪ D in order to compute the conflicts of the theory. Given some formula f , this result significantly simplify the saturation of

W ∪ D ∪ { f :⊥ ⊥ }.
Our implementation uses a particular resolution strategy, the head-literal resolution, which was proved in [START_REF] Bossu | Saturation, nonmonotonic reasoning and the closed world assumption[END_REF] to be complete and decidable for the class of so-called groundable clauses.

Lastly, let us briefly describe the computation of φ DL : a default d can be eliminated in a context V iff its prerequisite and the negation of its justification can be proved from V . This corresponds to the production of extended clauses of the form ¬(2):c c d and ¬(2):2 d during the saturation process. Thus, the elimination function φ DL can be computed from the result of the saturation.

Prioritized conflict resolution

The computation of the φ-preferred subsets of D, given a default theory (W, D), is done using an adaptation of Levy's algorithm for the computation "augmented HS-trees" [START_REF] Levy | Reason maintenance systems and default theories[END_REF]. Levy's has shown how his algorithm can be used for the computation of Reiter's default logic. We generalize below this result, by extending his notion of exception to a wider class of elimination functions. This result will be used in the next section, when we describe how to adapt our theorem prover to other variants of default logic. Definition 6. An elimination function φ on the set of conflicts Ξ of a default theory (W, D) is 2-monotonic if for all C ∈ Ξ and (d,

V ∪ V ) ∈ D × 2 D , if (d, V ) ∈ φ(C) then (d, V ∪ V ) ∈ φ(C). An exceptions is a pair (d, V ∪ (C \ {d})) where C ∈ Ξ and (d, V ) ∈ φ(C)}. (d, V ) is minimal if for no exception of the form (d, V ), V ⊂ V .
We denote by Σ the set of minimal exceptions of a theory. Given a subset E of Σ, we denote

E 1 = {d | ∃V ⊆ D, (d, V ) ∈ E} and E 2 = ∪ V ∈{V |∃d∈D,(d,V )∈E} V .
Theorem 3. Let (W, D) be a default theory, and φ a 2-monotonic elimination function on Ξ. A subset U of D is φ-preferred iff there exists E ⊆ Σ such that

U = E 1 , E 1 ∩ E 2 = ∅ and ∀C ∈ Ξ, C ∩ E 1 = ∅. Proof (sketch) 1. Suppose U is φ-preferred, then ∀d ∈ D \ U, ∃C d ∈ Ξ, C d \ {d} ⊆ U and (d, U ) ∈ φ(C d ). Let V d ⊆ U ∪ (C d \ {d}) be minimal such that (d, V d ) ∈ Σ, and let E = {(d, V d ) | d ∈ D \ U }. Clearly U = D \ E1. One can check that E1 ∩ E2 = ∅ and ∀C ∈ Ξ, C ∩ E1 = ∅. For the converse, let E ⊆ Σ s.t. E1 ∩ E2 = ∅ and ∀C ∈ Ξ, C ∩ E1 = ∅, and let U = D \ E1. For all C ∈ Ξ, since E1 ∩ C = ∅, C ⊆ U . Moreover let d ∈ D \ U = E1, and let V ⊆ D and C ∈ Ξ s.t. (d, V ∪ (C \ {d})) ∈ E: since E1 ∩ E2 = ∅, V ⊆ U and C \ {d} ⊆ U . By definition of Σ, (d, V ) ∈ φ(C), hence (d, U ) ∈ φ(C).
We compute the φ-preferred subsets of D by constructing a binary tree, whose nodes are labelled by pairs of the form (χ, Υ ), where χ ⊆ 2 D and Υ ⊆ D: χ is a set of conflicts that remain to be solved, and Υ is a set of defaults that cannot be eliminated any more, because they justify previous eliminations. The root is labelled by (Ξ, ∅), and we consider an enumeration of Σ. Consider a new exception (d, V ) in this enumeration. For each leaf of the tree labeled by (χ,

Υ ), if d / ∈ Υ , and if ∅ / ∈ {C \ V | C ∈ χ, d / ∈ C} = χ , build two edges.
The first one, labelled by d, leads to a new leaf labelled by (χ , Υ ∪ V ). The second one is not labelled, and leads to a leaf labelled by (χ, Υ ). Then U is an φ-preferred subsets of D iff U is the complementary in D of the set of labels of the edges on a path from the root to a leaf labelled by a pair of the form (∅, Υ ).

This tree can be pruned by a depth-first search. Let U be the φ-preferred subset of D corresponding to a leaf labelled by (∅, Υ ). Consider a node labelled by (χ , Υ ), such that the set L of labels of the edges from the root to this node, joined with ∪ C∈χ C, is included in D \ U : if this branch leads to a new solution U , it will be the complementary of a set included in L ∪ (∪ C∈χ C). Since φpreferred subsets of D are maximal such that they do not contain any conflict of the theory, we cannot obtain a new solution in the branches below this node.

Other variants of default logic

A major interest of this decomposition of the theorem prover into two completely independent tasks is that it enables us to adapt it to several other variants of default logic. In this section, we describe three such modifications.

Strong regularity Whereas Reiter's default logic requires that the justifications of the defaults used to generate an extension are separately consistent with the extension, several variants put a stronger condition: they require that the justifications of all the defaults together are consistent with the extensions [START_REF] Brewka | Cumulative default logic: In defense of nonmonotonic inference rules[END_REF][START_REF] Schaub | Assertional default theories: a semantical view[END_REF]. Froidevaux and Mengin [START_REF] Froidevaux | Default logics: a unified view[END_REF] formalize it with a definition similar to the following one:

¬2:c 1 d 1 ¬2:c 2 d 2 ¬c 1 :c 1 c 1 d 1 ¬c 2 :c 2 c 2 d 2 ¬c 1 :c 1 c 1 d 1 ¬c 2 :c 2 c 2 d 2 ¬2:c d 1 ¬c:c 1 c 1 ∨c 2 d 1 ¬c:c 2 c 1 ∨c 2 d 2 (6 ) (2 ) (2 )
W if Th def (W ∪ D) is consistent with {g | f :g h ∈ D}.
We can modify the definitions of conflict and of irregularity accordingly. In order for our extended resolution procedure to compute these irregularities, we simply have to replace rule [START_REF] Cayrol | Un modèle logique général pour le raisonnement révisable[END_REF] with rule (6') above,where c is the resolvent of c 1 and c 2 . So we are now allowed to make resolutions between clauses that have been produced with the justifications of several defaults. If this leads to some extended clause ¬2:2 , then it means that the set of defaults is strongly irregular.

Reasoning by cases It is well-known that Reiter's default logic does not enable one to reason by cases using defaults: from the theory defined by W = {a ∨ b} and D = { a:c c , b:d d } it is not possible to conclude c ∨ d. Several authors have propose to modify Reiter's definition of the extension in order to allow such reasoning by cases. The idea is that one must be able to fire defaults "by sets", that is, to conclude the disjunction of the consequences of some defaults once the disjuction of their prerequisite has been proved. In order to adapt our theorem prover to reasoning by cases, we simply have to replace rule (2) with rules (2') and (2") abobe, where c is the resolvent of c 1 and c 2 . As an example, it is possible to deduce the empty clause, using rules (1), (3) to [START_REF] Cayrol | Un modèle logique général pour le raisonnement révisable[END_REF], the above rule and the classical resolution rule, from W ∪ D ∪ { c:2 2 , d:2 2 }, where the latter set contains the extended clausal form of c∨d:2 2 . Moinard [START_REF] Moinard | Reasoning by cases without contraposition in default logic[END_REF] describes how this modification of default logic leads to some unwanted contrapositions of the defaults. He proposes to remedy to this problem by strenghtening the consistency condition on the justifications of the defaults, and the condition that allows one to eliminate a default. It would be interesting to specify the notion of irregularity as well as the elimination function that correspond to Moinard's definitions.

Prioritized default logic Several authors have proposed to add to default logic the possibility to define an ordering among the defaults, such that d 1 ≤ d 2 means that the default d 1 is less reliable than the default d 2 . The idea is that if d 1 and d 2 are conflicting, then one must use d 2 to generate an extension rather that d 1 . In [START_REF] Mengin | Prioritized conflict resolution for default reasoning[END_REF], we have proposed to associate, to such an ordering, the elimination function min ≤ , such that min

≤ (C) = {(d, V ) ∈ C ×2 D | ∀d ∈ C, d < d}.
The meaning of this elimination function is that in order to resolve a conflict, one must eliminate one of its minimal (for ≤) elements. In order to adapt our theorem prover to the use of such a relation among the defaults, we simply have to apply the elimination algorithm to the set of conflicts of a theory, considering the elimination function min

≤ •φ DL : min ≤ •φ DL (C) = {(d, V ) ∈ φ DL (C) | ∀d ∈ C, d < d}.

Conclusion

We have described here the two main elements of a theorem prover for default logic. The first one generalizes the resolution principle to the (monotonic) deductions that can be made using defaults. The main interest of our presentation, by means of the introduction of special resolution rules, is that it is completely independent from any resolution strategy. We have implemented a theorem prover for default logic that uses theses rules combined to a resolution strategy by Bossu and Siegel [START_REF] Bossu | Saturation, nonmonotonic reasoning and the closed world assumption[END_REF], the resolution on head literals. But other resolution strategies could have been used. It is important to note that the resolution steps obtained using rules (1) to ( 6) correspond to resolution steps that would be obtained using the classical rule only on a clausal form of W ∪ {¬f, g, h | f :g h ∈ D}. Consequently, any resolution strategy that is complete and that terminates on some particular class of clauses can be generalized to an extended resolution strategy that terminates and is complete on the corresponding class of extended clauses.

In comparison to many other theorem provers for default logic, we do not simply use some classical theorem prover to compute proofs for the prerequisites of the defaults or counter-arguments against their justifications. We provide, together with the extension of the resolution principle, a kind of "deduction theorem" that permits to make the link between the notion of conflicting defaults and that of monotonic deduction with defaults. Notice also that we do not need the introduction of any new propositional variable, as it is the case for the theorem provers that use some kind of traduction of the defaults into classical logic. This results in a greater clarity of the resolution steps, and simpler implementation. We have also described how simple modifications of some of the extended resolution rules can lead to other variants of default logic, that strengthen the consistency condition on the justification of the defaults or allow one to reason by cases.

Whereas the first part of our theorem prover is specific to default logic, the second one, that is, the elimination algorithm, is completely independent from any particular nonmonotonic logic. Levy [START_REF] Levy | Reason maintenance systems and default theories[END_REF] gives a detailed comparison between his algorithm and TMS-based algorithms used to compute extensions of default logic. As the algorithm that we have presented in this paper is not fundamentally different from Levy's one, this comparison still holds for our algorithm.

The main difference between Levy's algorithm and ours is that we do not need to check that the leaves of our tree correspond to valid sets of defaults generating extensions.

We have described in [START_REF] Mengin | Raisonnement par défaut : résolutions de conflits et priorités[END_REF] how the notion of prioritized conflict resolution underlies some other famous nonmonotonic logic, like McDermott and Doyle's one [START_REF] Macdermott | Non-monotonic logic 1[END_REF]. In [START_REF] Mengin | Prioritized conflict resolution for default reasoning[END_REF], we have also shown how elimination functions are interesting from the knowledge representation point of view.

The elements presented above are part of a system for prioritized default reasoning that we have implemented in Caml Light, a language of the ML family.

Definition 5 .

 5 The clausal form of a default d = ¬CN :CN CN , where CN , CN and CN are conjunctions of clauses, is the set of defaults d = { ¬c:c c d | c ∈ CN, c ∈ CN , c ∈ CN }. We will denote by D the union of the clausal forms of the set of defaults D, and call extended clauses the elements of D.

2 ): 2 dFig. 2 .

 222 Fig. 2. Extended resolutions on Example 1

Fig. 3 .

 3 Fig. 3. Modified extended resolution rules

Acknowledgements

The author is greatly indebted to Christine Froidevaux, who helped a lot during this work, and to Philippe Chatalic and Viorica Ciorba for their sound criticisms on the algorithms. The author also acknwledges sound criticisms by two anonymous referees.


Philippe Camail

Bruno Allard

Maxime Darnon

Charles Joubert

Christian Martin

Joao Pedro Trovao

A Study of the Superposition of a Hybrid PV/CPV Utility-Scale Power Plant

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Objectives & Issues

o

  Cloud passages suppress CPV energy production. o Inverters enter standby mode when input voltage <200V o It takes about 1min to start inverters è What is the impact of inverter start-up time on CPV systems energy production? è Can we reduce this impact by hybridizing CPV systems with flat panel PV? A Study of the Superposition of a Hybrid PV/CPV Utility-Scale Power Plant P. Camail 1,2 , B. Allard 1 , M. Darnon 2 , C. Joubert 1 , C. Martin 1 and J. Trovao 3(1) Laboratoire Ampère -CNRS UMR 5005 \ (2) LN2 / 3iT -CNRS UMI 3463 \ (3) Université de SherbrookeResults[1] Yamada, Noboru, Hirai, Daiki, "Maximization of conversion efficiency based on global normal irradiance using hybrid concentrator photovoltaic architecture", Progress in Photovoltaics: Research and Applications, 2006.[2] Algora, Carlos, Rey-Stolle, Ignacio, Handbook of Concentrator Photovoltaic Technology, 2016.

Fig. 1 è

 1 Fig.1 GNI and DNI (top), power of trackers #5 and #6 (middle) and voltage across one input of the corresponding inverters (bottom), the red line corresponds to the trigger voltage of the inverter

Acknowledgements


Léo Dutriaux
email: leodutriaux@gmail.com

Valérie Gyselinck

The postural effect on the memory of manipulable objects: interference or facilitation?

Keywords: Grounded cognition, Long Term Memory, Affordance, Concepts, Language

The grounded cognition approach posits the involvement of sensory-motor processes in the representation of knowledge. However, the functional impact of these processes on cognition has been questioned, and some authors have explored the effect of motor interference on memory in order to test causally this hypothesis. In a seminal study, Dutriaux and Gyselinck (2016) showed that keeping the hands behind the back during learning decreases the memory of manipulable objects, but not the memory of nonmanipulable objects.

The aim of this paper is to shed light on the mechanism behind the effect of posture in memory observed by Dutriaux and Gyselinck. The present experiment replicated the posture manipulation during learning, but asked participants to keep their hands behind the back during recall. Results showed a similar detrimental effect of the hands behind the back specific to manipulable objects. This shows that the mechanism behind this effect arises from postural interference rather than from a compatibility between the posture during learning and the posture during recall, and adds new evidence in favor of the sensory-motor grounding of knowledge.

Introduction

From a situated and grounded cognition perspective, memory is grounded on the sensorymotor system. As a result, memory retrieval is conceived as a reenactment of the sensorymotor activity at encoding, namely, a sensory-motor simulation [START_REF] Barsalou | Perceptual symbol systems[END_REF][START_REF] Versace | Act-In: An integrated view of memory mechanisms[END_REF]. This means that, for instance, the activation of the concept HAMMER involves the reinstatement of the perceptual patterns of activation related to past experiences with hammers, as well as the reinstatement of the motor patterns related to its use, that is, their affordances. The main objective of this report is to account for the mechanisms through which the motor system contributes to the long-term memory of manipulable objects.

Several studies tried to assess whether concepts of manipulable objects involve activation of the related motor patterns. For instance, [START_REF] Tucker | Action priming by briefly presented objects[END_REF] showed their participants pictures or names of manipulable objects. The manipulable objects could be grasped either with a precision grip or a whole-hand grip. Participants were instructed to judge whether the object was an artifact or not. To do so, they used either a response device requiring a precision grip or a whole-hand grip. Results showed that responses were faster when the grip evoked by the objects and the grip required to answer were compatible. Importantly, a similar effect is also observed with pictures and with the names of manipulable objects. It suggests that the conceptual processing of manipulable objects activates the related motor pattern which subsequently facilitates the related action (for an alternative account of these effects, see [START_REF] Guerineau | Anticipating the magnitude of response outcomes can induce a potentiation effect for manipulable objects[END_REF][START_REF] Heurley | Size coding of alternative responses is sufficient to induce a potentiation effect with manipulable objects[END_REF]. Consistently, reading the names of manipulable objects activates the motor system (e.g., [START_REF] Carota | Body-part-specific representations of semantic noun categories[END_REF][START_REF] Rueschemeyer | Body schematics: on the role of the body schema in embodied lexical-semantic representations[END_REF]). However, it has been objected that this kind of activation might result from a mere cascade of activation instead of being causally involved in the representation of the object [START_REF] Mahon | A critical look at the embodied cognition hypothesis and a new proposal for grounding conceptual content[END_REF]. As recently acknowledged in the literature [START_REF] Ostarek | Towards Strong Inference in Research on Embodiment -Possibilities and Limitations of Causal Paradigms[END_REF][START_REF] Ostarek | Six Challenges for Embodiment Research[END_REF], a challenge for the grounded view is to provide paradigms that demonstrate that motor information has a causal role in the representation of knowledge.

To address this issue, an increasing number of studies have tried to interfere with motor simulation to see whether it impacts the memory of manipulable objects. Most of them used a short-term memory paradigm, and show conflicting results. Guérard and colleagues found that performing a motor task during the encoding of lists of manipulable objects decreases the immediate recall of the objects [START_REF] Downing-Doucet | A motor similarity effect in object memory[END_REF][START_REF] Guérard | The role of motor affordances in immediate and long-term retention of objects[END_REF][START_REF] Lagacé | Norms for grip agreement for 296 photographs of objects[END_REF], while Pecher and colleagues failed to find such a specific interfering effect of a motor task with manipulable objects [START_REF] Pecher | No Role for Motor Affordances in Visual Working Memory[END_REF][START_REF] Pecher | The role of affordances for working memory for objects[END_REF][START_REF] Quak | Effects of motor congruence on visual working memory[END_REF]. To date, only a few studies have considered such effects on long-term memory. Canits, Pecher, and Zeelenberg (2018) replicated a stimulus-response grasp compatibility effect as in [START_REF] Tucker | Action priming by briefly presented objects[END_REF], but they failed to find any effect on the long-term recall of names or pictures of the objects. With a paradigm imposing a conflicting action during the retention interval, van Dam, [START_REF] Van Dam | Embodied grounding of memory: toward the effects of motor execution on memory consolidation[END_REF] asked their participants to learn words denoting objects that were associated with either a pressing or a twisting action (e.g., "piano" or "screwdriver") and words that were not associated to actions (e.g., "house"). Half of the participants then had to perform a distractive task that required the execution of pressing responses, and the other half had to perform a similar task but with twisting responses instead. They were finally asked to carry out a recognition test of the objects.

Results of three experiments revealed a better memory for words that denoted objects for which the functional use was congruent with the action performed during the retention interval, than for words that denoted objects for which the functional use was incongruent.

These findings suggest that a reactivation of motor codes affects the process of memory consolidation of the related concepts. [START_REF] Dutriaux | Learning is better with the hands free: The role of posture in the memory of manipulable objects[END_REF] considered the role of motor interference during the encoding process. Participants were presented with lists of pictures or names of manipulable and nonmanipulable objects. They were instructed to learn the lists taking two different postures. For half the lists, they had to put their hands in front of them, on the desk; and for the other half, they had to keep their hands behind their back. They then performed a short distractive task, during which they all had their hands in front of them. Still with their hands on the desk, participants were instructed to recall orally the objects while having their hands in front of them. Results showed that keeping their hands behind their back during the study phase decreased the subsequent number of names of manipulable objects recalled, but not of nonmanipulable objects. Similar results were observed with pictures of objects and words (see also [START_REF] Dutriaux | Aging and posture in the memory of manipulable objects[END_REF]. These effects were in addition extended to short phrases including a verb and a noun in further experiments [START_REF] De Vega | Crossing hands behind your back reduces recall of manual action sentences and alters brain dynamics[END_REF][START_REF] Dutriaux | How to change your memory of an object with a posture and a verb[END_REF]. The results obtained in these experiments suggest that motor simulation has a functional role in the encoding process (see also [START_REF] Villatte | Short-term upper limb immobilization and the embodied view of memory: A pilot study[END_REF][START_REF] Vitale | Enhancing Motor Brain Activity Improves Memory for Action Language: A tDCS Study[END_REF].

The rationale behind this posture manipulation comes from several studies suggesting that motor simulation depends whether the context or the task favors action (e.g., [START_REF] Bub | Evocation of functional and volumetric gestural knowledge by objects and words[END_REF]. For instance, motor simulation seems to be negatively impacted when our hands are behind our back [START_REF] Sirigu | Motor and visual imagery as two complementary but neurally dissociable mental processes[END_REF], when the object is outside of our peripersonal space (e.g., [START_REF] Cardellicchio | The space of affordances: a TMS study[END_REF][START_REF] Costantini | Where does an object trigger an action? An investigation about affordances in space[END_REF], or when the linguistic context does not favor action, as when the object name is included in a negative sentence [START_REF] De Vega | Sentential Negation Might Share Neurophysiological Mechanisms with Action Inhibition. Evidence from Frontal Theta Rhythm[END_REF] or when combined with a non-action verb [START_REF] Ambrosini | Handles lost in non-reachable space[END_REF][START_REF] Masson | Kicking calculators: Contribution of embodied representations to sentence comprehension[END_REF]. To account for these results, we hypothesized that the hands behind the back posture induces a context that does not favor action. This context would interfere with the motor simulation at encoding, which may then affect the memory of manipulable objects. For this reason, we called this effect the Postural Interference effect (PI effect). An alternative interpretation is that this effect is due to a facilitation of the hands in front of the body posture during retrieval, resulting from a mechanism close to the encoding specificity principle [START_REF] Tulving | Elements of episodic memory[END_REF]. The encoding specificity principle states that memory is improved when the information available at encoding is similar to the information available at retrieval. Importantly, in the case of the PI effect described in [START_REF] Dutriaux | Learning is better with the hands free: The role of posture in the memory of manipulable objects[END_REF], the posture with the hands behind the back had a detrimental effect only on manipulable objects, and not on the other materials learned in the same postural context. This then rules out a general compatibility effect as accounting for the PI effect. An explanation which is nevertheless in line with a form of encoding specificity principle is that motor cues created by the posture are relevant only for the items that hold similar motor information. Given that an object affordance depends on the organism's posture, this same posture at retrieval could then become a cue to memories of manipulable objects that facilitate their motor reinstatement. This hypothesis of motor state-dependency effect [START_REF] Matheson | Investigating grounded conceptualization: motor system state-dependence facilitates familiarity judgments of novel tools[END_REF] will be called for convenience the "compatibility hypothesis" here, which will be considered an alternative to the previous "interference hypothesis".

The primary aim of this study is thus to disentangle the interference and compatibility hypotheses. To do so, the present experiment is an adaptation of the Experiment 2 reported in [START_REF] Dutriaux | Learning is better with the hands free: The role of posture in the memory of manipulable objects[END_REF]. The material is exactly the same and the procedure is identical except for the posture during the recall phase. Participants had to learn lists of names of manipulable and nonmanipulable objects, either with their hands behind their back or in front of them. After a short distraction task, they had to recall the objects orally. Contrary to [START_REF] Dutriaux | Learning is better with the hands free: The role of posture in the memory of manipulable objects[END_REF], participants had to keep their hands behind their back at recall, instead of having them in front of them. The interference hypothesis predicts that keeping the hands behind the back during the study phase will produce a PI effect, decreasing the recall of manipulable objects, but not of nonmanipulable objects. Conversely, the compatibility hypothesis predicts a motor state-dependency effect, that is, the recall of manipulable objects will be higher with the hands behind the back during learning because the posture will be similar to the recall posture, allowing participants to benefit from the postural cues encoded at learning. The data and material from the present experiments are publicly available on the Open Science Framework website (https://osf.io/be6yc/).

Method

Participants

Twenty-nine right-handed undergraduates studying psychology at Paris Descartes University took part in the experiment for course credits (five males, mean age = 21.9 years). All participants gave their informed consent to the experimental procedure. All procedures performed in the present studies involving human participants were in accordance with the ethical standards of the institutional and national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Material

The material used in this experiment is the same as in [START_REF] Dutriaux | Learning is better with the hands free: The role of posture in the memory of manipulable objects[END_REF].

The French names of 36 manipulable objects (i.e., tools, e.g., phone, hammer), and 36 nonmanipulable objects (e.g., carpet, bench) were distributed in six lists of 12 words (for wordlists, see SM-1). They were similar in terms of word length (t < 1), objective (t(70) = 1.68, p = .097) and subjective frequencies (t(70) = 1.56, p = .123), or imageability (t(70) = 1.30, p = .198). Objective frequency was drawn from the LEXIQUE database [START_REF] New | Une base de données lexicales du français contemporain sur internet: LEXIQUE[END_REF], and the subjective frequency and imageability means were drawn from a combination of three norms [START_REF] Bonin | L'imageabilité : normes et relations avec d'autres variables psycholinguistiques[END_REF][START_REF] Desrochers | Valeurs de fréquence subjective et d'imagerie pour un échantillon de 1916 substantifs de la langue frangaise[END_REF][START_REF] Desrochers | Subjective frequency and imageability ratings for 3600 French nouns[END_REF]. Each list included six manipulable objects and six nonmanipulable ones. The words were presented on a grey background (arial 100) in the center of the screen.

Procedure and design

Participants were tested individually in a quiet room with the experimenter present.

The setting of the room was such as only the screen was in the visual field of participants, without any objects on the table, nor keyboard or mouse, avoiding the presence of any strong motor cues in the direct environment. For each list, participants had to perform a study phase, a distractive phase, and a test phase. During the study phase, participants were shown each word one at a time during 1 s with an interval of 500 ms, in a random order. They were instructed to learn as many words as possible while adopting two different postures. For three lists, they had to put their hands in front of them in a relaxed manner, and were told that they could move their hands if necessary (e.g., to scratch their head). They were then asked to hold their hands behind their back for the distractive and the test phase. For the other three lists, they had to keep their hands behind their back, holding one wrist with the other hand both for the learning and for the test phase. The order of the two blocks of three lists was counterbalanced between participants, and the order of the lists was randomized. All the participants had then to keep their hands behind their back for the distractive task. For each of the 50 trials of the distractive task, participants were presented with a lowercase letter on the left, and a 3x3 grid with nine uppercase letters on the right. They had to say orally whether the lowercase letter matched one of the nine uppercase letters on the right. This task lasted for 2 minutes. Finally, they had to recall the names of the objects without a time limit, still with their hands behind their back.

Results

Figure 1 (left) shows the mean percentages of objects correctly recalled for each condition in the present study, together with the percentage obtained in [START_REF] Dutriaux | Learning is better with the hands free: The role of posture in the memory of manipulable objects[END_REF] on the right. The present data were analyzed using a 2 × 2 repeated measure analysis of variance (ANOVA), with the proportion of objects recalled as a dependent variable, and with manipulability and posture as within-subject variables 1 . Concerning the main effects, none of them were significant (Fs < 1). As expected though, the interaction between manipulability and posture was significant (F(1, 28) = 6.50, p = .015, η²p = 0.19).

Consistent with the PI effect, planned comparisons revealed that the posture with the hands behind the back during learning decreased the memory of manipulable objects (F(1, 28) = 5.95, p = .021, η²p = 0.18), but not of nonmanipulable objects (F < 1). Post-hoc t-test applying a Bonferroni correction for multiple comparisons did not show any other significant differences between our conditions (All corrected ps > .05).

1 Shapiro-Wilks normality tests showed that all our repeated measures are normally distributed (all ps > .05) except the "manipulable-on the front" condition (p = .023). However, there is now very strong evidence that ANOVA is strongly robust to non-normality (e.g., [START_REF] Blanca | Datos no normales: ¿es el ANOVA una opción válida?[END_REF]. Furthermore, a Mauchly test showed that the assumption of sphericity cannot be rejected (p > .05). The compatibility hypothesis states that the recall of manipulable objects should increase when postures at learning and recall match. According to the present results, this hypothesis can be discarded. Keeping one's hands behind the back affects the memory of manipulable objects even when the posture at recall is the same. The effect of posture should thus rather be interpreted as resulting from an interference effect, not a facilitative one. Furthermore, to further explore the effects of posture upon memory, a complementary analysis has been conducted including both the present data and those from [START_REF] Dutriaux | Learning is better with the hands free: The role of posture in the memory of manipulable objects[END_REF] shown in Figure 1, right. To do so, the posture at recall was added as a between-participant variable to the ANOVA. Results showed that the interaction between manipulability and posture at learning isas expectedstill significant (F(1, 61) = 10.05, p < .001, η²p = 0.14), with a detrimental effect of posture on manipulable objects (F(1, 61) = 13.52, p < .001, η²p = 0.18) but not on nonmanipulable objects (F < 1). However, no main effect of posture at learning nor interaction with this factor were statistically reliable (all Fs < 1). Critically, planned comparisons did not show a reliable effect of the posture at recall for either manipulable or nonmanipulable objects, whatever was the posture at learning whatever was the posture at learning (all Fs < 1). This complementary analysis is again consistent with the interference hypothesis and not with the compatibility hypothesis, as only the posture at learning seems to have an effect on recall.

Discussion

This experiment was conducted to investigate the mechanism behind the PI effect observed on the long-term memory of manipulable objects. Two hypotheses were put forward to account for the effect of postural interference on the retrieval of manipulable objects: the interference hypothesis, and the compatibility hypothesis. The interference hypothesis predicts that recall of manipulable objects will be worse with the hands behind the back than in front during learning, regardless of the retrieval posture; while the compatibility hypothesis predicts a motor state-dependency effect, that is, keeping the hands behind the back would facilitate the recall of manipulable objects only if the posture at encoding was similar. In the present experiment, participants had to learn lists of manipulable and nonmanipulable objects, and to recall them after a distractive task. While the present study is similar to [START_REF] Dutriaux | Learning is better with the hands free: The role of posture in the memory of manipulable objects[END_REF] at learning, requiring participants to put their hands in front of them or behind their back during learning, it differed from it in asking them to always keep their hands behind the back during recall. Results showed a classical PI effect, meaning that the recall of manipulable object was worse with the hands behind the back than with the hands in front of them, with no effect of posture on nonmanipulable objects.

First of all, the replication of the PI effect confirms the view that motor information can be part of the memory trace and that motor simulation has a functional role in the representation of knowledge. Our results also cast a new light on the previous work that observed a PI effect (Manuel de Vega et al., 2021;[START_REF] Dutriaux | How to change your memory of an object with a posture and a verb[END_REF][START_REF] Dutriaux | Aging and posture in the memory of manipulable objects[END_REF][START_REF] Dutriaux | Learning is better with the hands free: The role of posture in the memory of manipulable objects[END_REF]. As emphasized in the introduction, this pattern of results could have been interpreted as an interference of the hands behind the back posture, but also as a facilitation of the hands with the hands in front. Facilitation may have resulted from an encoding specificity effect specific to the manipulable objects, which has been called motor state-dependency effect by [START_REF] Matheson | Investigating grounded conceptualization: motor system state-dependence facilitates familiarity judgments of novel tools[END_REF]. Namely, the reinstatement of the motor information related to the manipulable object might have been facilitated by the similarity between the motor state at learning and recall, while this would not be the case for non-motoric information.

However, the present results invalidated this hypothesis by showing that the hands behind the back posture during the study phase decreased the recall of manipulable objects, even though participants also had their hands behind their back during the test phase. This is consistent with recent results showing that maintaining the hands behind the back during the learning phase decreases motor related activity. Further, these results are in accordance with the idea that a context that does not favor action has a negative impact on motor simulation.

Still, the question of the exact mechanism behind the effect of posture at recall remains open. To provide elements of response to this question, we analyzed the present data combined with those of the 2016 study. The compatibility hypothesis would have predicted a three-way interaction, where participants would have shown a better memory for manipulable objects with the hands behind the back at recall when the hands were also behind the back at learning; and better memory performance with the hands in front of them at recall when the hands are also in front of them at learning. Instead, our complementary analysis showed no effect of posture at recall whatsoever, suggesting that posture mainly impacts the encoding phase. However, in part because the distracting task was not exactly the same from one experiment to the other, future work should replicate these results with the posture that would change both at leaning and recall.

These results raise also several questions that require further investigation. First, participants may have used motor imagery to learn the list. As this strategy is more likely to be used with manipulable objects, it is possible that the posture manipulation had an impact on the motor imagery process rather than on an automatic motor simulation. Further research could disentangle these two hypotheses using for instance incidental learning rather than explicit learning to minimize the contribution of conscious strategies. Nonetheless, this interpretation is still consistent with an embodied account of memory, as posture would still interfere with a form of motor information (here motor imagery), which results in a negative impact on memory. Finally, one can wonder which properties of the posture used in this experiment caused the PI effect. For instance, would an active motoric task or putting only the dominant hand behind the back produce similar results?

To sum up, this experiment shed light on the mechanism behind the PI effect, showing that the hands behind the back posture has interfering effect on the encoding of names of manipulable objects. It also provides further evidence for the involvement of the sensorymotor system in the representation of knowledge.

Figure 1 .

 1 Figure 1. Means for the percentage of items recalled as a function of manipulation and

Acknowledgments

The authors wish to thank Yingying Ge for assistance with testing. The work was funded by the French National Research Agency (Grant ANR-13-APPR-0009 to Valérie Gyselinck).