Karam Abdulahhad
email: karam.abdulahhad@imag.fr

Jean-Pierre Chevallet
email: jean-pierre.chevallet@imag.fr

Catherine Berrut
email: catherine.berrut@imag.fr

A New Lattice-Based Information Retrieval Theory

Logic-based Information Retrieval (IR) models represent the retrieval decision as an implication d → q between a document d and a query q, where d and q are logical sentences. However, d → q is a binary decision, we thus need a measure to estimate the degree to which d implies q, noted P (d → q). The main problems in the logic-based IR models are the difficulties to implement the decision algorithms and to define the uncertainty measure P as a part of the logic. In this study, we chose the Propositional Logic (PL) as the underlying framework. We propose to replace the implication d → q by the material implication d ⊃ q. However, we know that there is a mapping between PL and the lattice theory. In addition, Knuth [13] introduced the notion of degree of inclusion to quantify the ordering relations defined on lattices. Therefore, we position documents and queries on a lattice, where the ordering relation is equivalent to the material implication. In this case, the implication d → q is replaced by an ordering relation between documents and queries, and the uncertainty P (d → q) is redefined using the degree of inclusion measure. This new IR model is: 1-general where it is possible to instantiate most of classical IR models depending on our lattice-based model, 2-capable to formally prove the intuition of Rijsbergen about replacing P (d → q) by P (q|d), and 3-easy to implement.

Idempotency

Definition 2.4 (Distributive Lattice). Lattices that respect the following two conditions are called distributive lattices.

Introduction

Many studies [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF][START_REF] Nie | An outline of a general model for information retrieval systems[END_REF][START_REF] Nie | An information retrieval model based on modal logic[END_REF][START_REF] Nie | Towards a probabilistic modal logic for semantic-based information retrieval[END_REF][START_REF] Chiaramella | About retrieval models and logic[END_REF][START_REF] Meghini | A model of information retrieval based on a terminological logic[END_REF][START_REF] Sebastiani | A probabilistic terminological logic for modelling information retrieval[END_REF][START_REF] Chevallet | Experiences in information retrieval modelling using structured formalisms and modal logic[END_REF][START_REF] Crestani | Logic and uncertainty in information retrieval[END_REF][START_REF] Losada | A logical model for information retrieval based on propositional logic and belief revision[END_REF] showed that Information Retrieval (IR) could be represented as a logical implication d → q, where d represents a document and it is a set of logical sentences in a certain logic, and q represents a query and it is also a set of logical sentences in the same logic of d.

Using the logical implication d → q for representing the retrieval decision is quiet limited, because d → q is a binary decision, i.e. either d implies q or not. In addition, it is almost impossible to represent the exact semantic content of d and q [START_REF] Chiaramella | About retrieval models and logic[END_REF]. We thus need a more flexible notion of implication between d and q for reflecting this loss of information. We need to estimate the degree of implication or the uncertainty of implication, noted P (d → q).

All studies that talked about representing the retrieval decision by logical implication d → q, also presented some methods to estimate the uncertainty P (d → q). Most of proposals for estimating the value of P (d → q) were very complex and costly algorithms. Even that there were a few studies, e.g. [START_REF] Losada | A logical model for information retrieval based on propositional logic and belief revision[END_REF], presented practical and simple algorithms.

Concerning the type of logic, a wide range of logics have been used to represent d, q, and consequently d → q.

Chevallet [START_REF] Chevallet | Experiences in information retrieval modelling using structured formalisms and modal logic[END_REF] uses the First-Order Logic, represented by Conceptual Graph, for representing d, q, and d → q. He also uses the notion of graph projection for estimating P (d → q). Losada et al. [START_REF] Losada | A logical model for information retrieval based on propositional logic and belief revision[END_REF] and Abdulahhad et al. [START_REF] Abdulahhad | The effective relevance link between a document and a query[END_REF] use the Propositional Logic, and they use the notion of model intersection for estimating P (d → q). Nie [START_REF] Nie | An information retrieval model based on modal logic[END_REF][START_REF] Nie | Towards a probabilistic modal logic for semantic-based information retrieval[END_REF] uses the Modal Logic, and he uses the notion of Possible Worlds and the relations between them for estimating P (d → q). In [START_REF] Nie | Towards a probabilistic modal logic for semantic-based information retrieval[END_REF], he also uses the probability besides possible worlds for estimating P (d → q). Meghini et al. [START_REF] Meghini | A model of information retrieval based on a terminological logic[END_REF] and Sebastiani [START_REF] Sebastiani | A probabilistic terminological logic for modelling information retrieval[END_REF] use the Description Logic, and they use probability to estimate P (d → q). The choice of the appropriate logic depends on its expressive power and the complexity of its deduction algorithms, where there is a trade-off between the expressive power and deduction algorithm complexity.

In this study, we still represent d and q by logical sentences, but we depend on Lattices as an algebraic structure to position d and q and to exploit the degree of inclusion (or implication) metric defined on lattices [START_REF] Knuth | Deriving laws from ordering relations[END_REF][START_REF] Knuth | Lattice duality: The origin of probability and entropy[END_REF] for estimating P (d → q) in a simple, general, and practical way.

On the one hand, we choose the Propositional Logic (PL), because it is simple logic and in our point of view has a sufficient expressive power to represent documents, queries, and the retrieval decision.

On the other hand, it is known that PL corresponds to a lattice [START_REF] Dominich | The Modern Algebra of Information Retrieval[END_REF]. Moreover, Knuth [START_REF] Knuth | Deriving laws from ordering relations[END_REF][START_REF] Knuth | Lattice duality: The origin of probability and entropy[END_REF] defined a degree of implication or inclusion on lattices. It is then possible to exploit the lattice structure and Knuth's notion of the degree of implication for estimating P (d → q). This study is structured as follow: In section 2, we give a brief mathematical introduction about the lattice theory, the degree of inclusion measure, and about the propositional logic and its model-based interpretation. We claim, in section 3, that the implication d → q could be replaced by the material implication d ⊃ q. In section 4, we introduce a new mapping between the propositional logic and the lattice theory, more precisely, we introduce a new lattice where its nodes are logical clauses and its ordering relation is the material implication. In section 5, we redefine the basic IR notions depending on the lattice that defined in section 4, or in other words, we build a new IR model depending on the defined lattice and using the degree of inclusion measure for ranking. We discuss, in section 6, some important properties of our model, like its generality. We conclude in section 7.

Mathematical Preliminaries

Lattice

A lattice is an algebraic structure, or in other words, it is a set of elements satisfying certain properties. 1. If a unique "least upper bound" or "the supremum" of a and b exists, it is called the join, noted a ∨ b, and it satisfies the following conditions:

-a ∨ b ∈ L -a ≤ (a ∨ b) -b ≤ (a ∨ b)
-∃c ∈ L where a ≤ c and b ≤ c and c ≤ (a ∨ b) 2. If a unique "greatest lower bound" or "the infimum" of a and b exists, it is called the meet, noted a ∧ b, and it satisfies the following conditions:

-a ∧ b ∈ L -(a ∧ b) ≤ a -(a ∧ b) ≤ b -∃c ∈ L where c ≤ a and c ≤ b and (a ∧ b) ≤ c ⊓ ⊔ Definition 2.3 (Lattice). A lattice (L, ∧, ∨) is defined either as a poset (L, ≤)
where the join ∨ and the meet ∧ exist for each pair of elements in L or as an algebraic structure consisting of a set of elements L and two binary operations meet ∧ and join ∨ satisfying: In any lattice (L, ∧, ∨), the consistency relations explicitly express the relationship between the order relation ≤ and the meet ∧ and the join ∨ binary operations, as follow:

1. Distributivity of ∧ over ∨: ∀a, b, c ∈ L, a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) 2. Distributivity of ∨ over ∧: ∀a, b, c ∈ L, a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) ⊓ ⊔ Definition
∀a, b ∈ L, a ≤ b ⇔ a ∧ b = a a ∨ b = b ⊓ ⊔ Definition 2.9 (Sublattice). Assume that (L, ∧, ∨) is a lattice. (L ′ , ∧, ∨) is a sublattice of (L, ∧, ∨) iff, 1. L ′ ⊆ L and 2. ∀a, b ∈ L ′ , a ∧ b ∈ L ′ and a ∨ b ∈ L ′ . ⊓ ⊔

The Degree of Inclusion

In any poset (O, ≤), it is possible and helpful to quantify the notion of inclusion by introducing the zeta function (2.1).

∀x, y ∈ O, ζ(x, y) = 1 if x ≤ y 0 if x ≤ y (2.1)
The zeta function describes whether y includes x or not. It is possible to define its dual function dual of the zeta function

(2.2). ∀x, y ∈ O, ζ ∂ (x, y) = 1 if x ≥ y 0 if x ≥ y (2.
2)

The dual of the zeta function describes whether x includes y or not. However, knowing that y includes x, x ≤ y, for any two distinct elements x any y of a poset (O, ≤), then clearly x does not include y, x ≥ y. Even x does not include y, it is possible to describe the degree to which x includes y. Knuth [START_REF] Knuth | Lattice duality: The origin of probability and entropy[END_REF] generalizes the inclusion (2.2) to the degree of inclusion represented by real numbers. He introduced the z function

(2.3). ∀x, y ∈ O, z(x, y) =    1 if x ≥ y 0 if x ∧ y =⊥ z otherwise, where 0 < z < 1 (2.3)
z(x, y) quantifies the degree to which x includes y. Knuth [START_REF] Knuth | Lattice duality: The origin of probability and entropy[END_REF] says: "The motivation here is that, if we are certain that x includes y then we want to indicate this knowledge. However, if we know that x does not include y, then we can quantify the degree to which x includes y".

Assume that instead of working with elements of poset x, y ∈ O, we work with elements of a Distributive Lattice x, y ∈ L where (L, ∧, ∨) is a distributive lattice. In this case, the function z should be consistent with the structure of distributive lattices. The function z should therefore satisfy the following rules by which the degree of inclusion should be manipulated [START_REF] Knuth | Lattice duality: The origin of probability and entropy[END_REF]: for any distributive lattice (L, ∧, ∨), ∀x, y, t ∈ L:

1. Sum rule z(x ∨ y, t) = z(x, t) + z(y, t) -z(x ∧ y, t) (2.4) 2. First Product rule z(x ∧ y, t) = z(x, t) + z(y, t) -z(x ∨ y, t) (2.5) 3. Second Product rule z(x ∧ y, t) = C • z(x, t) • z(y, x ∧ t) (2.6)
where the constant C acts as a normalization factor. 4. Bay's Theorem rule

z(y, x ∧ t) = z(y, t) • z(x, y ∧ t) z(x, t) (2.7)
The z function is simply the probability when it is defined on a Boolean algebra or lattice (B, ∧, ∨, ¬, ⊤, ⊥) (2.8) [START_REF] Knuth | Deriving laws from ordering relations[END_REF][START_REF] Knuth | Lattice duality: The origin of probability and entropy[END_REF].

∀x, y ∈ B, P (x|y) = z(x, y) (2.8)
where P is a probability function.

Propositional Logic PL

This study depends on Propositional Logic PL as a theoretical and mathematical basis. Therefore, when saying that s is a logical sentence, we implicitly mean that s is a logical sentence under PL.

Definition 2.10 (Alphabet A). We define A = {a 1 , . . . , a n } as a set of all atomic propositions. The set A forms our alphabet, and it is a finite set |A| = n. Any proposition a i ∈ A can take only one of two possible values: True T , or False F .

⊓ ⊔

A formal logic has a syntax but can also have a semantic. This semantic translates the formal sentences of that logic into another mathematical world. For example, we get the semantic of a logical sentence in PL by assigning a truth value (T or F ) to each proposition in that sentence.

Any set of atomic propositions A corresponds to 2 |A| possible translations or interpretations ∆ A depending on the truth values assignations. Definition 2.11 (Interpretations ∆ A ). The set of interpretations ∆ A of a set of atomic propositions A is defined as follow: andδ ′ i contains the propositions that have true as a truth value in δ i . For a specific interpretation δ ′ i , all propositions that do not belong to δ ′ i are implicitly false. In the rest of this study we will use this simple notation for interpretations.

∆ A = {δ i |δ i = {(a 1 , x i 1 ), . . . , (a n , x i n )}, a j ∈ A, x i j ∈ {T,
∆ ′ A = {δ ′ i |δ ′ i = {a j |(a j , T ) ∈ δ i }} where |∆ ′ A | = |∆ A |,

⊓ ⊔

In other words, for any alphabet A, the set of interpretations actually correspond to the different rows of the truth table that is built in terms of A. For example, suppose that A = {a, b, c} contains three propositions, the truth table, the set of interpretations ∆ A , and the set ∆ ′ A are depicted in Figure 2.1.

Definition 2.12 (Models M ). For any logical sentence s, the subset

M s ⊆ ∆ ′ A that make s true is called the set of models of s, noted M s |= s ( M s models s),
where: for any model m ∈ M s , if we substitute each atomic proposition in s by its truth value in m then the truth value of s will be true. The notation |= s means that s is a tautology, or in other words, s is true under any interpretation (M s = ∆ ′ A ). The notation |= s means that s is false under all interpretations (M s = φ).

We have

M s ⊆ ∆ ′ A is a set of models and each model m ∈ M s is a set of atomic propositions m ⊆ A. ⊓ ⊔ Fig. 2.1.
The truth table of A = {a, b, c}, the corresponding set of interpretations ∆A, and the set ∆ ′ A , where ∆A = {δ1, δ2, δ3, δ4, δ5, δ6, δ7, δ8} and

∆ ′ A = {δ ′ 1 , δ ′ 2 , δ ′ 3 , δ ′ 4 , δ ′ 5 , δ ′ 6 , δ ′ 7 , δ ′ 8 } a b c δi δ ′ i F F F δ1 = {(a, F ), (b, F ), (c, F )} δ ′ 1 = {} F F T δ2 = {(a, F ), (b, F ), (c, T )} δ ′ 2 = {c} F T F δ3 = {(a, F ), (b, T ), (c, F )} δ ′ 3 = {b} F T T δ4 = {(a, F ), (b, T ), (c, T )} δ ′ 4 = {b, c} T F F δ5 = {(a, T ), (b, F ), (c, F )} δ ′ 5 = {a} T F T δ6 = {(a, T ), (b, F ), (c, T )} δ ′ 6 = {a, c} T T F δ7 = {(a, T ), (b, T ), (c, F )} δ ′ 7 = {a, b} T T T δ8 = {(a, T ), (b, T ), (c, T )} δ ′ 8 = {a, b, c} Fig. 2.2.
The truth table of the material implication ⊃, the set of interpretations ∆ ′ A , and the set of models M |= a ⊃ b, where

M = {δ ′ 1 , δ ′ 2 , δ ′ 4 } a b δ ′ i a ⊃ b F F δ ′ 1 = {} T {δ ′ 1 } |= a ⊃ b F T δ ′ 2 = {b} T {δ ′ 2 } |= a ⊃ b T F δ ′ 3 = {a} F {δ ′ 3 } |= a ⊃ b T T δ ′ 4 = {a, b} T {δ ′ 4 } |= a ⊃ b
For example, assume that A = {a, b} and assume that the logical sentence s is the material implication a ⊃ b. The set of models

M |= a ⊃ b is depicted in Figure 2.2, where M = {δ ′ 1 , δ ′ 2 , δ ′ 4 }.
Theorem 2.1 (Sentences vs. Models). By moving from the syntax space to the semantic space, for any two logical sentences s 1 and s 2 we have:

[|= s 1 ⊃ s 2 ] ⇔ [M s1 ⊆ M s2 ]
where

M s1 |= s 1 is the set of models of s 1 , M s2 |= s 2 is the set of models of s 2 . Proof. |= s 1 ⊃ s 2 means that any model of s 1 should also be a model of s 2 , noted M s1 |= s 2 or simply s 1 |= s 2 . From the truth table in Figure 2.2, s 1 ⊃ s 2 is true iff there is no model of s 1 is not a model of s 2 , which means M s1 ⊆ M s2 . ⊓ ⊔ Theorem 2.2 (Models-Based Boolean Algebra B M ). The algebraic struc- ture B M = (2 ∆ ′ A , ∧, ∨, ¬, ⊤, ⊥
) is a Boolean algebra and we refer to it as Models-Based Boolean algebra, where:

1. 2 ∆ ′ A is the power set of ∆ ′ A 2. ∀M 1 , M 2 ∈ 2 ∆ ′ A , M 1 ∧ M 2 = M 1 ∩ M 2 3. ∀M 1 , M 2 ∈ 2 ∆ ′ A , M 1 ∨ M 2 = M 1 ∪ M 2 4. ∀M ∈ 2 ∆ ′ A , ¬M = M where M = ∆ ′ A \ M 5. ⊤ = ∆ ′ A 6. ⊥ = φ
The ordering relation ≤ defined on B M is:

∀M 1 , M 2 ∈ 2 ∆ ′ A , [M 1 ≤ M 2 ] ⇔ [M 1 ⊆ M 2 ] ⊓ ⊔
From Theorems 2.1 and 2.2, the order relation on B M is equivalent to the material implication. Therefore, the implication of the Boolean lattice B M can be generalized to the degree of implication represented by real numbers (2.9).

∀M x , M y ∈ Ω A , P (M y |M x ) =    1 if M x ⊆ M y 0 if M x ∩ M y = φ p otherwise, where 0 < p < 1 (2.9)
where x and y are two logical sentences, M x |= x is the set of models of x, and M y |= y is the set of models of y. We know that [|= x ⊃ y] ⇔ [M x ⊆ M y ] (Theorem 2.1). We also know that each set of models M x correspond to a set of logically-equivalent sentences (it is possible to choose x as a representative of this equivalent class). Therefore, for any two logical sentences x and y, P (M y |M x ) (2.9) could be rewritten as (2.10).

P (y|x) =    1 if x ⊃ y 0 if x ∧ y = F = ⊥ p otherwise, where 0 < p < 1 (2.10)
Here also, Knuth [START_REF] Knuth | Lattice duality: The origin of probability and entropy[END_REF] says: "Probabilities are functions of pairs of logical statements and quantify the degree to which one logical statement implies another".

Revisiting IR in terms of PL

Many studies [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF][START_REF] Nie | An outline of a general model for information retrieval systems[END_REF][START_REF] Nie | An information retrieval model based on modal logic[END_REF][START_REF] Nie | Towards a probabilistic modal logic for semantic-based information retrieval[END_REF][START_REF] Chiaramella | About retrieval models and logic[END_REF][START_REF] Meghini | A model of information retrieval based on a terminological logic[END_REF][START_REF] Sebastiani | A probabilistic terminological logic for modelling information retrieval[END_REF][START_REF] Chevallet | Experiences in information retrieval modelling using structured formalisms and modal logic[END_REF][START_REF] Crestani | Logic and uncertainty in information retrieval[END_REF][START_REF] Losada | A logical model for information retrieval based on propositional logic and belief revision[END_REF] argue that the retrieval process could be represented as a logical implication between a document d and a query q. They say: if d and q are sets of logical sentences in a specific logic then d should be retrieved iff it logically implies q, noted d → q. In this study, we use the Propositional Logic PL as underlying logic.

Rijsbergen [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF] formalized the implication →, as follows: for any two sentences or sets of sentences X and Y , X → Y means that 'if X is true then Y '. In other words, if both X and Y are true under an interpretation then X → Y is also true under that interpretation. The truth of X → Y does not simply depend on the evaluation of X and Y in one interpretation.

Chiaramella et al. [START_REF] Chiaramella | About retrieval models and logic[END_REF], by their turn, formalized the implication →, as follow:

D → Q is true iff Q is true given that D is true.
However, all studies [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF][START_REF] Nie | An outline of a general model for information retrieval systems[END_REF][START_REF] Nie | An information retrieval model based on modal logic[END_REF][START_REF] Nie | Towards a probabilistic modal logic for semantic-based information retrieval[END_REF][START_REF] Chiaramella | About retrieval models and logic[END_REF][START_REF] Meghini | A model of information retrieval based on a terminological logic[END_REF][START_REF] Sebastiani | A probabilistic terminological logic for modelling information retrieval[END_REF][START_REF] Chevallet | Experiences in information retrieval modelling using structured formalisms and modal logic[END_REF][START_REF] Crestani | Logic and uncertainty in information retrieval[END_REF][START_REF] Losada | A logical model for information retrieval based on propositional logic and belief revision[END_REF] claim that the implication → is different from the classic material implication ⊃. In addition, → is more appropriate than ⊃ for IR. In this paper, we claim that the two implications → and ⊃ are equivalent.

All definitions of the implication X → Y , depicted in [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF][START_REF] Nie | An outline of a general model for information retrieval systems[END_REF][START_REF] Nie | An information retrieval model based on modal logic[END_REF][START_REF] Nie | Towards a probabilistic modal logic for semantic-based information retrieval[END_REF][START_REF] Chiaramella | About retrieval models and logic[END_REF][START_REF] Meghini | A model of information retrieval based on a terminological logic[END_REF][START_REF] Sebastiani | A probabilistic terminological logic for modelling information retrieval[END_REF][START_REF] Chevallet | Experiences in information retrieval modelling using structured formalisms and modal logic[END_REF][START_REF] Crestani | Logic and uncertainty in information retrieval[END_REF][START_REF] Losada | A logical model for information retrieval based on propositional logic and belief revision[END_REF], agree that X → Y can only be evaluated in the cases where the antecedent X is true and in those cases the consequent Y should also be true. In other words, the evaluation space for X → Y is restricted to the cases or interpretations that make X true, whereas the evaluation space of X ⊃ Y contains all the possible cases or interpretations.

We think that the main problem in those studies [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF][START_REF] Nie | An outline of a general model for information retrieval systems[END_REF][START_REF] Nie | An information retrieval model based on modal logic[END_REF][START_REF] Nie | Towards a probabilistic modal logic for semantic-based information retrieval[END_REF][START_REF] Chiaramella | About retrieval models and logic[END_REF][START_REF] Meghini | A model of information retrieval based on a terminological logic[END_REF][START_REF] Sebastiani | A probabilistic terminological logic for modelling information retrieval[END_REF][START_REF] Chevallet | Experiences in information retrieval modelling using structured formalisms and modal logic[END_REF][START_REF] Crestani | Logic and uncertainty in information retrieval[END_REF][START_REF] Losada | A logical model for information retrieval based on propositional logic and belief revision[END_REF] was the inability to imagine the meaning of d → q when d is false, where d represents a document and q represents a query. However, the impossibility of imagining some thing is not a sufficient reason for do not modeling that thing. In other words, d → q when d is false is a part of the model even if this case is not realistic.

Before we go forward, we should clarify the potential map between the logical notions and the IR notions:

1. Each proposition a i ∈ A corresponds to only one unique indexing term.

Therefore, A is also the set of indexing terms. 2. Any document d and query q are logical sentences in PL, where: d is equivalent to a set of models M d |= d, and q is equivalent to a set of models M q |= q. 3. We claim that the retrieval decision d → q is equivalent to the material implication d ⊃ q between the two sentences d and q. Theorems 2.1 and 3.1 show that:

[s 1 → s 2 ] ⇔ [s 1 ⊃ s 2 ] (3.1)
Theorem 3.1. The IR implication →, which is defined in [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF] and [START_REF] Chiaramella | About retrieval models and logic[END_REF], is equivalent to the set inclusion between models. For any two logical sentences s 1 , s 2

[|= s 1 → s 2 ] ⇔ [M s1 ⊆ M s2 ]
where M s1 |= s 1 is the set of models of s 1 , and M s2 |= s 2 is the set of models of s 2 .

Proof. 1. s 1 → s 2 is true means that 'if s 1 is true then s 2 ', or every model of s 1 should also be a model of s 2 . 2. M s1 ⊆ M s2 means that when s 1 is true then s 2 is also true.

⊓ ⊔ Equation 3.1 does not contradict with previous studies, which say that queries are only evaluable in the cases (interpretations) where d is true. From the truth table in Figure 2.2 and after replacing a by d and b by q, we have:

M d⊃q = (M d ∩ M q ) ∪ M d (3.2)
Although that M d are possible models of d ⊃ q, the implementations of IR models do not, in general, take M d into account, and they neglect them, because M d corresponds to the cases where d is false. Therefore, we think that modeling the cases where d is false will not pose any problem, as long as, these cases will be neglected at the implementation time.

After the previous discussion our main hypothesis is: the material implication d ⊃ q is the appropriate implication for modeling the retrieval decision in the IR field. Henceforth, we will use the two implications d → q and d ⊃ q interchangeably.

In general, IR is an uncertain process [START_REF] Chiaramella | About retrieval models and logic[END_REF], so we need to evaluate the uncertainty of d → q. Rijsbergen [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF] and Nie [START_REF] Nie | An outline of a general model for information retrieval systems[END_REF] depend on the Possible Worlds semantic to evaluate the uncertainty of the implication P (d → q). More precisely, according to Rijsbergen [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF] the P (d → q) could be replaced by the conditional probability P (q|d), whereas, Nie [START_REF] Nie | An outline of a general model for information retrieval systems[END_REF] depends on the notion of distance between the possible worlds over the path from d to q.

In this section, we also introduce, in an intuitive manner, a way of estimating the uncertainty of an implication. We could simply claim that:

P (d → q) = |M d ∩ M q | (3.3)
The intuitive meaning of this formula could be that the limit to which d and q are compatible, or how many system (people) could assign the same interpretation for both d and q. In next sections, we will introduce a more formal measure for estimating the implication uncertainty.

A New Logic-Based Lattice

The application field of this study will be the Information Retrieval (IR) field.

IR could be formalized depending on the lattice of models B M (Theorem 2.2). Any document or query could be represented as a node in B M . However, B M is a very general and huge lattice (if

|A| = 3 then |2 ∆ ′ A | = 2 2 3 = 2 8
). Moreover, IR notions like documents and queries are generally modelized in a very simple way e.g. set of terms. Therefore, we here propose a new lattice B C depending on rewriting the sentences that represent documents and queries in their DNF form. This new lattice will simplify the B M lattice, without loss of generality. Definition 4.1 (Clauses C A ). We define the set of clauses C A on the alphabet A as follow:

C A = ({a 1 , ¬a 1 , T } × • • • × {a n , ¬a n , T }) \ {T }
where |C A | = 3 |A| -1 and {a i , ¬a i , T }×{a j , ¬a j , T } = {a i ∧a j , a i ∧¬a j , a i , ¬a i ∧ a j , ¬a i ∧ ¬a j , ¬a i , a j , ¬a j , T }.

Note that {a i , ¬a i , T } × {a j , ¬a j , T } = {a j , ¬a j , T } × {a i , ¬a i , T } because the conjunction ∧ is commutative. Any clause s ∈ C A can be defined as follow:

∀s ∈ C A , ∃A s ⊆ A, s = ai∈As b i
where A s = φ and b i is a literal. Any literal b i is an atomic proposition a i or its negation ¬a i .

⊓ ⊔

It is possible to split the set A s into two disjoint sets: A + s that contains the propositions occurring in their non-negative form, and A - s that contains the propositions occurring in their negative form, where A + s ∪ A - s = A s and A + s ∩A - s = φ. We also define the set A ± s = A\A s , which contains the propositions that do not occur in s. 1. If a i ∈ A + s then a i should be mapped into true in any model of s, so a i ∈ m.

If a i ∈ A -

s then a i should be mapped into false in any model of s, so a i ∈ m.

If a i ∈ A ±

s then a i could be mapped either into true or false in any model of s, so whether a i ∈ m or not. Definition 4.3 (Full Clause Model). Depending on the set of models M s , we define one unique model m s , where {m s } |= s, as follow:

∀s ∈ C A , m s = m∈Ms (m)
where M s |= s is the set of models of s. m s is the full clause model of s.

⊓ ⊔

The model m s is equivalent to the clause s after completing it by the negation form of all propositions that do not occur in it a i ∈ A ± s . For example, suppose A = {a, b, c} and s = a ∧ b then the model m s is equivalent to the following clause a ∧ b ∧ ¬c. In other words, m s will only contain the propositions that occur in s in their non-negative form. Theorem 4.1 (Full Clause Models Boolean Algebra B C ). The algebraic structure B C = (Ψ A , ∧, ∨, ¬, ⊤, ⊥) is a Boolean algebra and we refer to it as the Full Clause Models Boolean algebra, where: The ordering relation ≤ defined on B C is:

1. Ψ A = {m s |s ∈ C A } 2. ∀m 1 , m 2 ∈ Ψ A , m 1 ∧ m 2 = m 1 ∩ m 2 3. ∀m 1 , m 2 ∈ Ψ A , m 1 ∨ m 2 = m 1 ∪ m 2 4. ∀m ∈ Ψ A ,
∀m 1 , m 2 ∈ Ψ A , [m 1 ≤ m 2 ] ⇔ [m 1 ⊆ m 2 ] ⊓ ⊔ Theorem 4.

(Relationship between Material Implication → and B C ).

The potential relationship between the material implication → and the ordering relations ⊆ defined on B C is:

∀s 1 , s 2 ∈ C A , [|= s 1 → s 2 ] ⇔ [(m s2 ⊆ m s1 ) ∧ (M s1 ∩ M s2 = φ)]
where M s1 |= s 1 is the set of all models of s 1 , M s2 |= s 2 is the set of all models of s 2 , and m s1 , m s2 are the full clause models of s 1 and s 2 , respectively.

Proof. From the following points, it is possible to prove this theorem.

1. From Theorem 2.1, we have:

[|= s 1 → s 2 ] ⇔ [M s1 ⊆ M s2 ]. 2. [M s1 ⊆ M s2 ] ⇒ [µ(M s2 ) ⊆ µ(M s1 )] ⇒ [m s2 ⊆ m s1 ]. 3. [M s1 ⊆ M s2 ] ⇒ [M s1 ∩ M s2 = φ], this is correct knowing that M s1 = φ be- cause A s1 = φ. 4. [M s1 ∩ M s2 = φ] ⇒ A + s1 ∩ A - s2 = φ ∧ A - s1 ∩ A + s2 = φ .
In other words, there is no proposition occurring in two different forms in s 1 and s 2 . 5. [m s2 ⊆ m s1 ] ⇒ A + s2 ⊆ A + s1 . According to the two sets A - s1 and A - s2 , we have: (a) If A - s2 ⊆ A - s1 then (M s1 ⊆ M s2 ) because s 1 has more propositions and hence has less number of models. We have

A ± s1 ⊆ A ± s2 , so |M s1 | ≤ |M s2 |. (b) Otherwise A - s1 ⊂ A - s2 , we define the set A - s2\s1 = A - s2 \A - s1 = A ± s1 ∩A - s2
, which contains the propositions that are mapped into false in s 2 and could be mapped into either true or false in s 1 . As all propositions A ± s1 that do not occur in s 1 could be either true or false then it is possible to assume, without loss of generality, that: 1-all propositions A - s2\s1 are false in s 1 , or in other words, we move the set A - s2\s1 of propositions from A ± s1 to A - s1 . In this case, we have A - s2 = A - s1 , and we back to the case (a), 2-all propositions A - s2\s1 are true in s 1 , or in other words, we move the set A - s2\s1 of propositions from A ± s1 to A + s1 . In this case, we have a contradiction with point (4), and hence, this assumption is not possible. Actually, this operation of moving some propositions from one set to another will not change any thing concerning the models m s1 and m s2 (Definition 4.3).

From points 1, 2 and 3, we prove that:

[|= s 1 → s 2 ] ⇒ [(m s2 ⊆ m s1 ) ∧ (M s1 ∩ M s2 = φ)]
From points 4 and 5, we prove that:

[|= s 1 → s 2 ] ⇐ [(m s2 ⊆ m s1 ) ∧ (M s1 ∩ M s2 = φ)] ⊓ ⊔
Theorem 4.2 shows that the ordering relation ≤ defined on the lattice B C is equivalent to the materiel implication → defined on the clauses of C A (Definition 4.1).

Revisiting IR in terms of B C and the z Function

As we said in previous sections, the alphabet A forms the set of terms because each term corresponds to one unique proposition, both documents and queries are logical sentences in PL, the retrieval decision is modeled by the implication d → q, and finally the degree of implication P (d → q) forms the ranking mechanism.

Documents and Queries

In IR, documents and queries could be written as a clause of C A (Definition 4.1) or a set of clauses of C A connected by disjunction. In other words, documents and queries are DNF logical sentences. Any logical sentence could be rewritten in the DNF form. Therefore, there are no restrictions on the logical sentences that could be used to represent documents and queries, and hence there is no loss of generality. More formally, suppose we have a set of documents D and a query q:

M d = si∈S d (M si ) ⊓ ⊔
Definition 5.2 (Query). The query q is a DNF logical sentence, and it corresponds to one unique non-empty set of clauses S q ⊆ C A connected by disjunction, or equivalently: q = si∈Sq s i

where S q = φ. The set of models M q of a query q is defined as follow:

M q = si∈Sq (M si ) ⊓ ⊔

Relevance

Most of studies in the logical IR models defined the relevance between a document d and a query q by an implication between them [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF][START_REF] Nie | An outline of a general model for information retrieval systems[END_REF][START_REF] Nie | An information retrieval model based on modal logic[END_REF][START_REF] Nie | Towards a probabilistic modal logic for semantic-based information retrieval[END_REF][START_REF] Chiaramella | About retrieval models and logic[END_REF][START_REF] Meghini | A model of information retrieval based on a terminological logic[END_REF][START_REF] Sebastiani | A probabilistic terminological logic for modelling information retrieval[END_REF][START_REF] Chevallet | Experiences in information retrieval modelling using structured formalisms and modal logic[END_REF][START_REF] Crestani | Logic and uncertainty in information retrieval[END_REF][START_REF] Losada | A logical model for information retrieval based on propositional logic and belief revision[END_REF]. d is relevant concerning q iff d implies q, noted d → q. We claim that the implication d → q could be replaced by the material implication d ⊃ q (Equation 3.1). Of course, that is right if we use the propositional logic PL as an underling logic. Assume that d is a document (Definition 5.1) and q is a query (Definition 5.2). We have M d |= d the set of document models and M q |= q the set of query models. Theorem 4.2 shows that |= d → q is equivalent to two conditions:

-C.1: (M d ∩ M q = φ) which means that there is no contradiction between d and q, or in other words, d ∧ q = F . The verification of this condition depends on the actual implementation of d and q. For example, if a user asks for documents that do not contain "information retrieval" then the system should not return documents talking about "information retrieval", because in that case d ∧ q will be false F . More formally,

[M d ∩ M q = φ] ⇔ ∃s i ∈ S d , ∃s j ∈ S q , M si ∩ M sj = φ and M si ∩ M sj = φ ⇔ A + si ∩ A - sj = φ ∧ A - si ∩ A + sj = φ (5.1)
In the rest of this study,

[C.1 = T ] ⇔ [M d ∩ M q = φ] [C.1 = F ] ⇔ [M d ∩ M q = φ] (5.2)
-C.2: (m q ⊆ m d ) on the one hand, this condition corresponds to the ordering relation ≤ defined on B C (Theorem 4.1). On the other hand, the verification of this condition depends on the number of clauses |S d | and |S q | in d and q, respectively. According to that we have the following cases, where we suppose that C.1 = T (5.2) as a pre-condition:

• case 1 (|S d | = |S q | = 1): suppose that S d = {s d } and S q = {s q } then [|= d → q] ⇔ m sq ⊆ m s d (5.3) see Theorem 4.2. • case 2 (|S d | = 1 and |S q | > 1): suppose that S d = {s d } then [|= d → q] ⇔ [∃s i ∈ S q , m si ⊆ m s d ] (5.4)
because for any logical sentences s 1 , s 2 , s 3 , we have:

[s 1 → (s 2 ∨ s 3 )] ⇔ [(s 1 → s 2 ) ∨ (s 1 → s 3 )] • case 3 (|S d | > 1 and |S q | = 1): suppose that S q = {s q } then [|= d → q] ⇔ ∀s i ∈ S d , m sq ⊆ m si (5.5)
because for any logical sentences s 1 , s 2 , s 3 , we have:

[(s 1 ∨ s 2 ) → s 3 ] ⇔ [(s 1 → s 3 ) ∧ (s 2 → s 3 )] • case 4 (|S d | > 1 and |S q | > 1): [|= d → q] ⇔ ∀s i ∈ S d , ∃s j ∈ S q , m sj ⊆ m si (5.6)
because for any logical sentences s 1 , s 2 , s 3 , s 4 , we have:

[(s 1 ∨ s 2 ) → (s 3 ∨ s 4 )] ⇔   [(s 1 → s 3 ) ∨ (s 1 → s 4 )] ∧ [(s 2 → s 3 ) ∨ (s 2 → s 4 )]  

Uncertainty

IR is an uncertain process [START_REF] Chiaramella | About retrieval models and logic[END_REF]. Therefore, it is mandatory to define a measure for quantifying the implication d → q, written P (d → q). It is rarely the case where d directly implies q, so we need a measure to estimate the degree to which d implies q, and then ranking documents according to the decreasing value of this measure.

According to Definitions 5.1 and 5.2, documents and queries are sets of clauses, and then they correspond to sets of nodes in B C . Moreover, B C is a Boolean algebra (Theorem 4.1). Knuth [START_REF] Knuth | Lattice duality: The origin of probability and entropy[END_REF] defines the z function (2.3) on distributive lattices. The z(x, y) function measures the degree to which x includes or implies y for any two distinct elements x and y of a distributive lattice. Therefore, it is possible to replace the uncertainty function P by the degree of implication function z. First we should define the z function on our two lattices B M , and B C .

-The lattice B M : assume s 1 and s 2 are two logical sentences, M s1 |= s 1 is the set of models of s 1 , and

M s2 |= s 2 is the set of models of s 2 , z M (M s1 , M s2 ) =    1 if M s2 ⊆ M s1 0 if C.1 = F z M
otherwise, where 0 < z M < 1

(5.7)

C.1 = F (5.2) means M s1 ∩ M s2 = φ.
-The lattice B C : assume s 1 and s 2 are two clauses in C A , where m s1 is the full clause model of s 1 and m s2 is the full clause model of s 2 ,

z C (m s1 , m s2 ) =    1 if m s2 ⊆ m s1 0 if C.1 = F z C
otherwise, where 0 < z C < 1

(5.8)

C.1 = F (5.2) means (A + s1 ∩ A - s2 = φ) ∨ (A - s1 ∩ A + s2 = φ).
The condition C.1 can not be directly verified using the lattice B C . It needs external notions (A + and A -) to be verified.

Let us now come back to our initial uncertain implication P (d → q). d and q are, in general, logical sentences. By rewriting d and q in their DNF form, each of them corresponds to one or several clauses in C A . Therefore, it is possible to rewrite P in terms of z M (5.7) or z C (5.8), as follow:

-P (d → q) corresponds to z M (M q , M d ) because z M (M q , M d ) = 1 when M d ⊆ M q which is equivalent to |= d → q. -if (M d ∩ M q = φ) then P (d → q) corresponds to z C (m d , m q ) because z C (m d , m q ) = 1 when d → q is true (Theorem 4.
2), where d, q ∈ C A , m d is the full clause model of d, m q is the full clause model, M d |= d is the set of models of d, and M q |= q is the set of models of q. This is true when d and q correspond to two distinct clauses in B C , but it could be easily generalized in the other cases as we will see in the next section.

The Relevance Status Value RSV (d, q)

In this section, we will depend on the full clause models Boolean algebra B C (Theorem 4.1), and we will study the different cases of d and q. For simplifying the notation, we will refer to z C by z.

Nie [START_REF] Nie | An outline of a general model for information retrieval systems[END_REF] differentiates between the two implications Exhaustivity d → q and Specificity q → d. He supposes that the matching score RSV (d, q) is written as follow:

RSV (d, q) = F [P (d → q), P (q → d)] (5.9)

We take this general form of matching score (5.9), and we build our discussion on it. According to the case where d and q correspond to only one clause

(|S d | = |S q | = 1) or a set of clauses (|S d | > 1 and |S q | > 1) in B C , we have: in this discussion, we suppose that C.1 = T , 1. case 1 (|S d | = |S q | = 1): suppose that S d = {s d } and S q = {s q } then -P (d → q) = z(m s d , m sq ) -P (q → d) = z(m sq , m s d ) RSV (d, q) = F [P (d → q), P (q → d)] = F z(m s d , m sq ), z(m sq , m s d ) (5.10) 2. case 2 (|S d | = 1 and |S q | > 1): suppose that S d = {s d } then -P (d → q) = G si∈Sq (z(m s d , m si ))
, where G : R n → R and G respects the condition (5.4). For example, G could the normal sum + or the max function.

-P (q → d) = G ′ si∈Sq (z(m si , m s d ))
, where G ′ : R n → R and G ′ respects the condition (5.5). For example, G ′ could the normal product × or the min function.

RSV (d, q) = F [P (d → q), P (q → d)] = F G si∈Sq (z(m s d , m si )) , G ′ si∈Sq (z(m si , m s d )) (5.11) 3. case 3 (|S d | > 1 and |S q | = 1): suppose that S q = {s q } then -P (d → q) = G ′ si∈S d z(m si , m sq ) -P (q → d) = G si∈S d z(m sq , m si ) RSV (d, q) = F [P (d → q), P (q → d)] = F G ′ si∈S d z(m si , m sq ) , G si∈S d z(m sq , m si ) (5.12) 4. case 4 (|S d | > 1 and |S q | > 1): -we define G ′′ (S d ; S q ) = G ′ si∈S d • G sj ∈Sq
where

G ′′ (S d ; S q ) = G ′ si∈S d G sj ∈Sq z(m si , m sj ) -P (d → q) = G ′′ (S d ; S q ) -P (q → d) = G ′′ (S q ; S d )
RSV (d, q) = F [P (d → q), P (q → d)] = F [G ′′ (S d ; S q ), G ′′ (S q ; S d )] (5.13) Equation 5.13 is the most general form of the matching score RSV (d, q) between a document d and a query q.

Discussion

After presenting our general IR model (Equation 5.13), it is now possible to discuss some results.

Result 1

If d and q are two logical sentences then they correspond to two distinct sets of models M d , M q ∈ 2 ∆ ′ A , or equivalently two distinct nodes in B M . We also know that B M is a Boolean lattice, so the z M (x, y) function is the probability P M (x|y).

We already said that P (d → q) corresponds to z M (M q , M d ) (Equation 5.7), so P (d → q) corresponds to P M (M q |M d ). However, we know that each node M ∈ 2 ∆ ′ A in the lattice B M is a set of models of a set of equivalent logical sentences. By this way, M q is a set of models of a set of logical sentences equivalent to q. We choose q as a representative to this equivalent class. We do the same thing for d. Therefore, P (d → q) = P (q|d) (6.1) Equation ( 6.1) justifies the definition of P (d → q) that is presented by Rijsbergen [START_REF] Van Rijsbergen | A non-classical logic for information retrieval[END_REF]. To our knowledge, this is the first study that present a formal justification of the Rijsbergen's intuition.

Result 2

Assume that d and q correspond to two distinct full clause models m d , m q ∈ Ψ A , or equivalently two distinct nodes m d and m q in B C . The results that we will obtain could be easily generalized to the cases where d and q are two sets of nodes.

Exhaustivity P (d → q) corresponds to z(m d , m q ) in B C (Equation 5.8). Using the first product rule (2.5), we obtain:

z(m d , m q ) = z(m q ∧ m d , m q ) (6.
2)

The same for Specificity P (q → d):

z(m q , m d ) = z(m d ∧ m q , m d ) (6.3)
Those two results (6.2) and (6.3) correspond to our intuition presented in our previous study [START_REF] Abdulahhad | The effective relevance link between a document and a query[END_REF]. Depending on the Bay's Theorem rule (2.7), it is possible to deduce that:

z(m d , m q ) = z(m d , ⊤) z(m q , ⊤) × z(m q , m d ) (6.4)
where z(x, ⊤) represent prior probabilities, which can be arbitrary assigned [START_REF] Knuth | Lattice duality: The origin of probability and entropy[END_REF].

Using the Sum rule (2.4), we obtain:

z(m d , m q ) = 1 -z(¬m d , m q ) (6.5)
Actually, there are many interesting properties could be deduced from playing on the rules of the z function (2.4, 2.5, 2.6, and 2.7).

Result 3

We think that (5.13) forms a general IR framework and some well-known IR models could be derived from (5.13).

Language Models (LM). Assume that each proposition a i ∈ A corresponds to one term in the document collection D. A document d ∈ D is written as follow:

d = ai∈A d a i (6.6)
where A d = φ and a i is the proposition that corresponds to the term t i that occurs in d. In other words, d is a conjunction of the terms that occur in it. For any document d, we have: A - d = φ, or in other words, the negation of terms is not modelized. The query q is represented in the same way, q = ai∈Aq a i (6.7)

Hence, d, q ∈ C A are two clauses, and they correspond to two distinct nodes m d and m q in B C , respectively. Therefore, G ′′ ({d}; {q}) = z(m d , m q ) and also G ′′ ({q}; {d}) = z(m q , m d ).

We choose F as the Weighted-Sum of two values, so (5.13) could be rewritten as follows:

RSV (d, q) = α × z(m d , m q ) + β × z(m q , m d )
Now, assume that α = 0 and β = 1 (Specificity without Exhaustivity) then

RSV (d, q) = z(m q , m d )
As B C is a Boolean lattice, it is possible to replace z by the probability P C then RSV (d, q) = P C (m q |m d )

Now, suppose that the elements of m q are conditionally independent, and let us define a probability distribution θ d on the set m d then

RSV (d, q) = ai∈mq P C (a i |θ d )
which is the general form of language models. Therefore, language models are instances of our general framework.

Probabilistic Models (PM). Probabilistic Models (PMs) depend on the Probability Ranking Principle [START_REF] Robertson | The Probability Ranking Principle in IR[END_REF], according to which: documents are ranked according to the decreasing value of the probability P (R|d, q). More precisely, PMs use the notion of odds (6.8).

RSV (d, q) ∝ P (R|d, q) P (NR|d, q) (6.8) R means document is relevant whereas NR means document is non-relevant.

The main problem in PMs is that the relevance information is not available in advance. Therefore, it is hard to estimate the two probabilities P (R|d, q) and P (NR|d, q). However, using Bay's rule and with some simplifications, the ranking formula (6.8) becomes:

RSV (d, q) ∝ P (d|R, q) P (d|NR, q) (6.9)

To estimate the two probabilities, we should have samples of the relevant and non-relevant documents of each query.

Some studies [START_REF] Hiemstra | Relating the new language models of information retrieval to the traditional retrieval models[END_REF][START_REF] Lavrenko | Relevance based language models[END_REF] claim that R and NR could be seen as two sets of relevant and non-relevant documents for a specific query, respectively. Robertson et al. [START_REF] Robertson | Relevance weighting of search terms[END_REF] also assume that a set of relevance judgements for each request should be available to estimate the relevance weights. If we take this viewpoint, where R and NR are two sets of documents, and project it on the lattice B C , we obtain:

assume that each proposition a The previous formula is the general form of probabilistic models. Therefore, probabilistic models are instances of our general framework. Theorem 6.1. A document d is relevant to a query q if:

|= (d → R) and |= (d → NR) Proof. Assume R = d R 1 ∨ • • • ∨ d R k , where ∀1 ≤ i ≤ k, |= d R i → q. Assume NR = d NR 1 ∨ • • • ∨ d NR l , where ∀1 ≤ i ≤ l, |= d NR i ∧ q. |= d → R means that ∃1 ≤ i ≤ k, |= d → d R
i , which in its turn means |= d → q because |= R → q. Suppose that |= d → NR then:

-|= (d → d NR 1 ) ∨ • • • ∨ (d → d NR 1 ) -∃1 ≤ i ≤ l, |= d → d NR i -|= ¬d ∨ d NR i -we know that |= d NR i ∧ q then: |= ¬(¬d ∨ d NR i ) ∨ (d NR i ∧ q) -|= d ∧ ¬d NR i ∨ d NR i ∧ d ∧ ¬d NR i ∨ q -|= (d ∨ q) ∧ d ∨ d NR i ) ∧ (¬d NR i ∨ q -|= (d ∨ q) ∧ (d ∧ q) -|= (d ∧ q)
We should thus change our hypothesis from (|= d → NR) to ( |= d → NR).

⊓ ⊔

Unlike the previous implementations of PMs, lattices allow us to define the two sets R and NR in advance. We first define the up-set of a node m in our lattice B C , noted ↑ m:

∀m ∈ Ψ A , ↑ m = {m ′ |m ′ ∈ Ψ A , m ≤ m ′ }
We define the non-relevant nodes N NR in B C :

N NR = {m|m ∈ Ψ A , z(m, m q ) = 0}
Now, it is possible to define the set of nodes N R , N R = (↑ m q ) \ N NR The lattice B C allows us to define the family of probabilistic models. Moreover, it allows us to determine the relevant and non-relevant documents in advance, which are very important to estimate the two probabilities P (d|R, q) and P (d|NR, q). Vector Space Model (VSM). Assume that each proposition a i ∈ A corresponds to one term in the document collection D. Suppose that any document d ∈ D has the same definition presented in LM (6.6), so d corresponds to only one node m d in B C . Suppose that any query q has the same definition presented in LM (6.7), so q corresponds to only one node m q in B C .

Definition 2 . 1 (

 21 Partially Ordered Set (poset)). A partial order over a set of elements L is a binary relation ≤ L , simply ≤, satisfying the following conditions: 1. Reflexivity: ∀a ∈ L, a ≤ a 2. Antisymmetry: ∀a, b ∈ L, if a ≤ b and b ≤ a then a = b 3. Transitivity: ∀a, b, c ∈ L, if a ≤ b and b ≤ c then a ≤ c The set L with the order relation (L, ≤) is called a partially ordered set or poset. ⊓ ⊔ Definition 2.2 (Meet ∧ and Join ∨). Assume (L, ≤) is a poset. For any two elements a, b ∈ L, we have:

2 . 5 (Definition 2 . 6 (Definition 2 . 7 (Definition 2 . 8 (

 25262728 Bounded Lattice). The algebraic structure (L, ∧, ∨, ⊤, ⊥) is called bounded lattice iff (L, ∧, ∨) is a lattice, and ⊤ ∈ L and ⊥ ∈ L are the top and the bottom of (L, ∧, ∨), respectively, where:1. ∀a ∈ L, a ≤ ⊤ and a ∧ ⊤ = a and a ∨ ⊤ = ⊤ 2. ∀a ∈ L, ⊥ ≤ a and a ∨ ⊥ = a and a ∧ ⊥ = ⊥ ⊓ ⊔ Complemented Lattice).If for any element a ∈ L in the bounded lattice (L, ∧, ∨, ⊤, ⊥), there exists a unique element b ∈ L, noted b = ¬a, satisfying:1. a ∧ ¬a = ⊥ 2. a ∨ ¬a = ⊤The algebraic structure (L, ∧, ∨, ¬, ⊤, ⊥) is called a complemented lattice.⊓ ⊔ Boolean Algebra). Any distributive and complemented lattice (L, ∧, ∨, ¬, ⊤, ⊥) is a Boolean algebra. ⊓ ⊔ Consistency Relations).

Definition 4 . 2 (

 42 Alphabet Splitting). Each clause s ∈ C A splits the alphabet A into three mutually disjoint sets of atomic propositions:1. A + s contains the propositions a i ∈ A s where b i = a i . 2. A - s contains the propositions a i ∈ A s where b i = ¬a i . 3. A ± s = A \ A s contains the propositions that do not occur in s. ⊓ ⊔ Example 4.1. Suppose A = {a, b} then C A = ({a, ¬a, T } × {b, ¬b, T }) \ {T } = {a ∧ b, a ∧ ¬b, a, ¬a ∧ b, ¬a ∧ ¬b,¬a, b, ¬b}. ⊓ ⊔ Example 4.2. Suppose A = {a, b, c}, for the clause s = ¬a ∧ b ∈ C A we have: A s = {a, b}, A + s = {b}, A - s = {a}, and A ± s = {c}. ⊓ ⊔ Any clause s ∈ C A corresponds to a set of models M s (Definition 2.12), where M s |= s. In any model m ∈ M s of the clause s, the propositions A + s should be mapped into true, the propositions A - s should be mapped into false, and the propositions A ± s could be mapped into true or false. Hence, the number of models |M s | of any clause s is |M s | = 2 |A ± s | . For any model m ∈ M s and any proposition a i ∈ A:

  ¬m = m where m = A \ m 5. ⊤ = m ⊤ where m ⊤ = A which is the full clause model of the clause a 1 ∧ • • • ∧ a n . 6. ⊥ = m ⊥ where m ⊥ = φ which is the full clause model of clauses that do not contain any proposition in non-negative form, e.g. ¬a i .

Definition 5 . 1 (

 51 Document). Any document d ∈ D is a DNF logical sentence, and it corresponds to one unique non-empty set of clauses S d ⊆ C A connected by disjunction, or equivalently: ∀d ∈ D, d = si∈S d s i where S d = φ. The set of models M d of a document d is defined as follow:

  i ∈ A corresponds to one term in the document collection D.any document d ∈ D has the same definition presented in LM(6.6), so it corresponds to only one node m d in B C . -any query q has the same definition presented in LM (6.7), so it corresponds to only one nodem q in B C . -R is a disjunction of a set of documents R = d 1 ∨• • •∨d k where any document d i satisfies |= d i → q. That is correct because |= R → q means that for any d i we have |= d i → q. Therefore, R corresponds to a set of nodes N R in B C . N R = {m di |1 ≤ i ≤ k, |= d i → q} -NR is also a disjunction of a set of documents NR = d 1 ∨ • • • ∨ d l where any document d i satisfies |= d i ∧ q.That is correct because |= NR ∧ q means that for any d i we have |= d i ∧ q. Therefore, NR corresponds to a set of nodesN NR in B C . N NR = {m di |1 ≤ i ≤ l, |= d i ∧ q}-The retrieval decision could be reformulated as follow: d is relevant to q if |= d → R and |= d → NR (Theorem 6.1). By taking the degree of implication z into account, we have:RSV (d, q) ∝ G ′′ ({d}; R) G ′′ ({d}; NR) G ′′ ({d}; R) could be simplified to G r∈R (z(m d , m r )), and we choose the max function to replace G (5.11). Moreover, as B C is a Boolean algebra then z(x, y) could be replaced by the probability P C (x|y). We also suppose that the elements of m d are conditionally independent. Finally, we obtain the following ranking formula:RSV (d, q) ∝ ai∈m d P C (a i |m R ) P C (a i |m NR )where m R is the node that maximises G r∈R (z(m d , m r )), and m NR is the node that maximises G mnr∈NR (z(m d , m nr )).

  F }} where |∆ A | = 2 |A| because each proposition a i ∈ A could have one of two possible values (T or F ), and ∀δ i ∈ ∆ A , |δ i | = |A|. It is possible to build a new notation ∆ ′A simpler than but equivalent to ∆ A , as follow:

We present in this study a new theoretical framework for representing documents, queries, and the retrieval decision including a ranking mechanism. We use the Propositional Logic for representing documents and queries, and then we claim that the retrieval decision corresponds to the material implication between a document and a query. After that, we position documents and queries on a lattice, more precisely on a Boolean algebra. We then exploit the degree of implication metric z, defined on the lattice, for representing the ranking mechanism.

This study, on the one hand, presents a new vision of logic-based IR models through exploiting the implicit link between lattices and Propositional Logic. On the other hand, it presents a general IR framework capable of representing the classical IR models, like Language Models, Probabilistic Models, and Vector Space Models.

Our model also provides a theoretical proof for the definition of P (d → q) that is presented by Rijsbergen [1]. In addition, it provides a theoretical proof for the choices that are made in [11].

The most important point in this study, in our point of view, is the simplicity and flexibility of the framework that it provides. We discussed a few capabilities of our model, but there still exists so many potential capabilities waiting to be discovered, especially through working on the rules of the z function (2.4, 2.5, 2.6, and 2.7), and through exploiting the partial order relation defined on the Boolean algebra B C .

follow:

where | -→ m| = ti w i . However, the z function (6.10) satisfies the sum rule of the z function (2.4), where:

We choose the classical product (×) instead of F , so (5.13) could be rewritten as follows:

According to (6.2) and (6.3), it is possible to rewrite the previous equation as follow:

which is compatible with our previous publication [START_REF] Abdulahhad | The effective relevance link between a document and a query[END_REF].


F Hille

Subspace-based damage detection on steel frame structure under changing excitation

Keywords: Damage detection, Ambient vibration, Changing excitation, Subspace methods, Robust tests

Damage detection can be performed by detecting changes in the modal parameters between a reference state and the current (possibly damaged) state of a structure from measured output-only vibration data. Alternatively, a subspace-based damage detection test has been proposed and applied successfully, where changes in the modal parameters are detected, but the estimation of the modal parameters themselves is avoided. Like this, the test can run in an automated way directly on the vibration measurements. However, it was assumed that the unmeasured ambient excitation properties during measurements of the structure in the reference and possibly damaged condition stay constant, which is hardly satisfied by any application. A new version of the test has been derived recently that is robust to such changes in the ambient excitation. In this paper, the robust test is recalled and its performance is evaluated both on numerical simulations and a real application, where a steel frame structure is artificially damaged in the lab.

Introduction

In the last twenty years, monitoring the integrity of the civil infrastructure has been an active research topic, including in connected areas such as automatic control, for mastering the aging of bridges, or the resistance to seismic events and the protection of the cultural heritage. The structural response to unknown ambient excitation is measured, and vibration-based monitoring can be performed under the premise that changes in the structural properties due to damage lead to changes in the eigenstructure of a system. Since the ambient excitation is unknown, it can change during the monitoring of a structure, e.g. in changing wind or traffic conditions. The focus of this paper is damage detection under possible changes in the ambient excitation.

A standard approach for damage detection is modal parameter identification and their comparison to a reference state of the monitored structure, e.g. as in [START_REF] Magalh Ães | Dynamic monitoring of a long span arch bridge[END_REF][START_REF] Ramos | Monitoring historical masonry structures with operational modal analysis: Two case studies[END_REF]. The modal parameters are not afflicted by different ambient excitations, but their automatic estimation and matching from measurements of different states of the structure might require an extensive preprocessing step.

In this paper we consider the subspace-based damage detection approach developed in [START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF][START_REF] Basseville | Statistical model-based damage detection and localization: subspace-based residuals and damage-to-noise sensitivity ratios[END_REF][START_REF] Mevel | Subspace-based fault detection robust to changes in the noise covariances[END_REF][START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF]. With this approach, a data-driven model obtained in the reference state is compared to data from the possibly damaged state using a subspacebased residual function and a χ 2 -test built on it for a hypothesis test, without actually estimating the modal parameters in the tested, possibly damaged states. This has the advantage that the computations in the tested (possibly damaged) state are fully automated. The computed χ 2 -value is compared to a threshold to decide if damage occurred or not.

Since this kind of methods process the measurement data in a much more direct way, the original algorithms from [START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF][START_REF] Basseville | Statistical model-based damage detection and localization: subspace-based residuals and damage-to-noise sensitivity ratios[END_REF] have the drawback of being influenced by changes in the unknown ambient excitation, which may lead to false alarms. Recently, the original algorithms have been extended and modified to be robust under such changes in the excitation in [START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF]. This approach has been shown to be promising under changing excitation in numerical simulations [START_REF] Mevel | Statistical subspace-based damage detection under changing ambient excitation[END_REF][START_REF] Strout | Changes in the statistics of ambient excitations in the performance of two damage detection schemes[END_REF][START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF]. It is the purpose of this paper to validate the new approach on a case study in the lab under different excitation levels.

This paper is organized as follows. In Section 2, the basics of the considered damage detection approach are presented and in Section 3 the extension from [START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF] that is robust to changing ambient excitation. In Section 4, a numerical application is presented on a simulated truss, as well as a case study on a steel frame structure in the lab that was artificially damaged. Finally, concluding remarks are given in Section 5.

Subspace-based damage detection

Models and parameters

The behavior of a structure is assumed to be described by a linear time-invariant (LTI) dynamical system

M Ẍ (t) + C Ẋ (t) + K X (t) = υ(t) (1)
where t denotes continuous time, M , C , K ∈ R d×d are the mass, damping and stiffness matrices respectively and X ∈ R d collects the displacements of the d degrees of freedom (DOF) of the structure. The external and non-measured force υ(t) is modeled as white noise.

Let the system (1) be observed at r coordinates, e.g. with accelerometers. Discretizing system (1) in time and transformation to a first order system leads to the discrete time state-space model

x k+1 = Ax k + v k y k = Cx k + w k (2)
with the states x k ∈ R n , the outputs y k ∈ R r , the state transition matrix A ∈ R n×n and the observation matrix C ∈ R r×n , where r is the number of sensors and n is the system order. The excitation v k is an unmeasured Gaussian white noise sequence with zero mean and (during a measurement) constant covariance matrix Q:

E(v k v T k ′ ) def = Q δ(k -k ′ )
, where E(•) denotes the expectation operator, and w k is the measurement noise.

Properties from subspace-based system identification

For the damage detection method in [START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF][START_REF] Basseville | Statistical model-based damage detection and localization: subspace-based residuals and damage-to-noise sensitivity ratios[END_REF] a residual function is constructed based on properties from covariance-driven outputonly subspace-based system identification [START_REF] Van Overschee | Subspace Identification for Linear Systems: Theory, Implementation, Applications[END_REF][START_REF] Peeters | Reference-based stochastic subspace identification for output-only modal analysis[END_REF].

Let G = E(x k+1 y T k ) be the cross-covariance between the states and the outputs, let R i = E(y k y T k-i ) = CA i-1 G be the theoretic output covariances and

H p+1,q def =      R 1 R 2 . . . R q R 2 R 3 . . . R q+1 . . . . . . . . . . . . R p+1 R p+2 . . . R p+q      def = Hank (R i ) (3)
be the theoretic block Hankel matrix of size (p + 1)r × qr, where parameters p and q are chosen such that min{pr, qr} ≥ n with often p + 1 = q. Matrix H p+1,q possesses the well-known factorization property

H p+1,q = O p+1 C q (4)
into the matrices of observability and controllability

O p+1 =      C CA . . . CA p      , C q = G AG . . . A q-1 G . ( 5
)
From the observability matrix O p+1 , the matrices C and A could be recovered [START_REF] Van Overschee | Subspace Identification for Linear Systems: Theory, Implementation, Applications[END_REF][START_REF] Peeters | Reference-based stochastic subspace identification for output-only modal analysis[END_REF] and subsequently the modal parameters. However, the fact is used that damages lead to changes in A and C and subsequently in H p+1,q through properties 4 and 5, which will be directly checked in a statistical test, instead of doing system identification.

The damage detection test

In the following, the non-parametric damage detection test [START_REF] Balm Ès | Merging sensor data from multiple temperature scenarios for vibration-based monitoring of civil structures[END_REF] based on [START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF][START_REF] Basseville | Statistical model-based damage detection and localization: subspace-based residuals and damage-to-noise sensitivity ratios[END_REF] is described. Using measured data (y k ) k=1,...,N , a consistent estimate H p+1,q of the Hankel matrix is obtained from the empirical output covariances

R i = 1 N N ∑ k=1 y k y T k-i , H p+1,q = Hank R i . (6)
Let H ref p+1,q be the Hankel matrix in the reference state. Compute its left null space S from the singular value decomposition (SVD)

H ref p+1,q = U 1 U 0 ∆ 1 0 0 ∆ 0 V T 1 V T 0 (7)
as S = U 0 , where ∆ 1 is of size n × n and where ∆ 0 ≈ 0.

The characteristic property of the reference state then writes as

S T H p+1,q ≈ 0, (8)
while the product deviates from 0 in the damaged state. To decide whether measured data corresponds to the reference state or not, the residual vector

ζ with ζ = √ N vec S T H p+1,q (9)
is defined [START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF][START_REF] Basseville | Statistical model-based damage detection and localization: subspace-based residuals and damage-to-noise sensitivity ratios[END_REF]. It is tested if this residual is significantly different from zero or not, using the χ 2 test

χ 2 ζ = ζ T Σ -1 ζ ζ, (10)
where the empirical residual covariance Σ ζ = cov(ζ) is computed on several datasets from the reference state. To decide if damage occurred or not, the test value χ 2 is compared to a threshold, which can also obtained from χ 2 test values on several datasets from the reference state.

Robust damage detection test under changing excitation

A change in the covariance Q of the unmeasured ambient excitation v k of system (2) provokes a change in the the crosscovariance G between the states and the outputs and thus in the Hankel matrix estimate H p+1,q (see ( 3)), even if no structural change occurs. Hence, the residual ζ and the corresponding test value χ 2 are influenced by changes in the ambient excitation, which may lead to false alarms. Due to these reasons, a robust variant of the presented damage detection test was derived in [START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF]. It is based on the property that the Hankel matrix H p+1,q and the matrix of its principal left singular vectors U 1 (see SVD in [START_REF] Mevel | Statistical subspace-based damage detection under changing ambient excitation[END_REF]) share the same left null space S -another characteristic property of the reference state can be written as

S T U 1 ≈ 0,
analogous to [START_REF] Magalh Ães | Dynamic monitoring of a long span arch bridge[END_REF]. While the Hankel matrix H p+1,q is dependent on the ambient excitation properties, matrix U 1 is a matrix with orthonormal columns and thus can be regarded as independent of the excitation properties. Note that matrix U 1 is defined by a unique SVD (e.g. by forcing the first entry in each column to be positive) to ensure no changing modal basis.

Since U 1 is independent of the excitation, a robust residual vector is defined as

ξ = √ N vec S T U 1 (11)
is defined [START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF]. It is tested if this residual is significantly different from zero or not, using the χ 2 test

χ 2 ξ = ξ T Σ -1 ξ ξ, (12)
where the empirical residual covariance Σ ξ = cov(ξ) is computed on several datasets from the reference state as detailed in [START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF].

Applications

Numerical application: simulated truss structure

A truss model with 25 DOF (see Figure 1) was considered in a first numerical application as in [START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF]. The output data was generated at six sensor positions in vertical direction at the lower chord by exciting the structure at the same positions with white noise.

The excitation noise at these six positions was generated with a diagonal covariance matrix Q, whose diagonal entries were randomly chosen from a uniform distribution in the interval [START_REF] Balm Ès | Merging sensor data from multiple temperature scenarios for vibration-based monitoring of civil structures[END_REF]36]. 5% white noise were added on the generated outputs. Damage was simulated by decreasing the stiffness of element 16 in a first step by 10% and in a second step by 20%, leading to a decrease of the structure's natural frequencies of up to 1.0% and 2.2% compared to the reference state, respectively. In the reference state, the null space S and the residual covariance Σ ζ (for the conventional test [START_REF] Peeters | Reference-based stochastic subspace identification for output-only modal analysis[END_REF]) and Σ ξ (for the robust test ( 12)) were computed on 10 data sets to set up the parameters of the damage detection tests.

In each structural state, data sets of length N = 25 000 at a sampling frequency of 50 Hz were generated. To compare both damage detection tests under changing excitation, 100 new data samples were generated with random excitation covariance in the reference and both damaged states, on which the χ 2 -tests were computed in Figure 2. For each of the compared tests, an empirical threshold was computed from the χ 2 -values of the reference state allowing a 5% type I error (horizontal dashed line). As can be seen in Figure 2(a), the damage detection fails with the conventional test under changing excitation properties, while the new robust damage detection test in Figure 2(b) manages to separate the reference state from the damaged states very well.

Laboratory application: Steel frame structure

A test series has been performed at a jacket-like steel structures within the lab facilities of BAM to verify the usability, reliability and robustness of the introduced damage detection algorithm under changing excitation properties. The test structure provides the ability to gradually and reversibly increase artificially induced structural damage. The test stand can be used to analyze vibration based damage identification methods or just the sensitivity of single parameters influencing the identification of structural damage.

The lab structure and damage modeling

For the lack of options to test damage identification methods on real size civil engineering structures it is common to experimentally analyze newly developed procedures on scaled medium size laboratory structures. This is especially true for new structural types as e.g. offshore wind energy converters. As part of the research work on subspace-based damage detection on offshore structures at BAM a particular steel frame structure has been constructed and is provided as laboratory model.

The model structure is made of steel pipe components representing a scaled two-dimensional section of a jacket-type support structure for an offshore wind energy converter. For this purpose one and a half diagonal bracings are designed between the two legs. The upper completion is formed by an I-sectional steel beam. At the lower end the legs end in steel foot plates. With exception of the damage sections all structural parts of the model are welded. The foot plates are screwed via a transition piece to the floor. In the z-direction the structure is held on its head girder by a complex construction, whose two fork-like support poles allow a movement of the structure in the x-direction. The lab structure is constructed such that a defined local loss of stiffness can be modeled, which complies with the requirements of a realistic projection of an individual fatigue crack as a damage, and the reversibility of the induced damages is assured for the repeatability of test series. Though, cracks as damage can be induced artificially at one K-type gusset of the model structure by loosening bolts of a flange connection. Four of such flange connections were designed, each at the four ends of the K-type gusset. The progress of a crack-like damage is then simulated by the number of bolts that are loosened. Each flange joint was equipped with a maximal possible number of bolts to provide a high resolution of the induced stiffness loss.

Measurement setup

The excitation for the tests was provided by an electrodynamic shaker. A broadband random acceleration signal with a frequency content between 10 Hz and 1000 Hz was produced by a shaker control unit and was induced via the shaker at the top of the structure after amplification. The shaker was equipped with a 5 kg excitation mass and the excitation direction was horizontal and approximately 30 • rotated out of the in-plane-direction.

Nine piezoelectric accelerometers were applied at the structure for the vibration measurements. The sensors were connected with the structure by permanent magnets and measured in the direction perpendicular to the surface they were applied to. The signals were amplified, recorded and stored with a 20 channel DIFA measurement unit.

Damage detection test

A test series was conducted to validate the reliability and sensitivity of the described robust damage detection procedure under changing excitation. In the tests, the time signals were measured for 16.4 s with a sampling rate of 2500 Hz, which results in 40,960 data points per signal.

The damage location was chosen at the lower brace of the flange. In the reference state all bolts were screwed tight. For inducing damage, successively 1, 2, 3, 5 or 7 adjacent bolts were unscrewed, each representing a fatigue crack of increasing length. Thereby, the loosening of 3 bolts is comparable with a reduction of the moment of inertia by 3%, whereas the unscrewing of 7 bolts means a loss of already 30% of the bending stiffness.

To validate the damage detection algorithm under changing excitation, three excitation (power) levels were used for each damage state. Besides the full scale level, a reduction of 5 dB and a reduction of 10 dB with respect to the full scale level were performed and are denoted as "Excitation 1", "Excitation 2" and "Excitation 3" in the tests, respectively. A reduction of 5 dB accounts for a power ratio of ≈0.31 and an amplitude ratio of ≈0.56 and a reduction of 10 dB for a power ratio of 0.1 and an amplitude ratio of ≈0.31. The full scale excitation power level was chosen preliminary in dependence of the shaker performance.

For each damage level and each excitation level, four signals of 16.4 s length was measured.

Results

Both the conventional damage detection test presented in Section 2.3 and the robust damage detection test under changing excitation in Section 3 are applied to the measured data with exactly the same parameters.

Two data sets from Excitation 1 and two data sets from Excitation 3 in the reference state are used to compute the null space S and the covariance Σ for both test variants. The tests were set up with the parameters of the block Hankel matrix p+1 = q = 15 and model order n = 120, resulting in the dimension of the null space S of qrn = 15. The mentioned data sets were separated into altogether 1000 pieces for the computation of the covariance matrix. See also reference [START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF] for the numerical details and an efficient implementation of these computations. Then, both tests (see Equations ( 10) and ( 12), respectively) are applied to all the measured data sets in the reference and damaged states at the three different excitation levels. The computed χ 2 test values are presented in Figures 4(a) and 4(b) for both tests, respectively, together with an empirical threshold that was set up using the χ 2 test values of the reference state.

In Figure 4(a) it can be seen that the conventional test reacts strongly to different excitation levels. A decrease in the excitation level results also in a decrease in the χ 2 test value, independently of the damage. While the χ 2 test values increase with the damage size for each excitation level, this increase is not distinguishable for the different excitation levels. Only the χ 2 test values of a significant damage (7 loosened bolts) all lie over the threshold that was established in the reference state. Thus, only damages can be clearly detected with the conventional test, whose influence on the χ 2 test is stronger than the changes in excitation.

On the other side, the results of the new test in Figure 4(b) clearly show a stronger robustness to changing excitation properties. While there are still small fluctuations of the test values in the reference state and in the states with small damage, these fluctuations are less significant. For the damages from 3 loosened bolts on, these variations seem to be independent from the excitation level, which validates the robustness of the new test to changing excitation. Furthermore, the relatively small damage of only 3 loosened bolts can be clearly detected under changing excitation with the new test thanks to its robustness.

Conclusions

In this paper, the recently developed statistical subspace-based damage detection test from [START_REF] Mevel | Subspace-based damage detection under changes in the ambient excitation statistics[END_REF] that is robust to changing ambient excitation properties was successfully tested on real data from a lab experiment for the first time. In this experiment a steel frame structure was artificially damaged by loosening bolts of a flange connection and output-only data was recorded under different excitation levels. The new test that was designed to be robust to changing excitation properties showed much better damage detection results in this setting with an earlier detection of damage than a previous conventional damage detection test. Since the applied damage detection framework already proved to be feasible for real civil structures in the field (see e.g. case studies [START_REF] Mevel | Stochastic subspace-based structural identification and damage detection and localization -Application to the Z24 Bridge benchmark[END_REF][START_REF] Zhou | Vibration-based damage detection to the composite tank filled with fluid[END_REF][START_REF] Öhler | Structural health monitoring with statistical methods during progressive damage test of S101 Bridge[END_REF]), the presented test robust to excitation changes should be of value for such applications, where the excitation is ambient and uncontrolled, and where the new test is thus expected to detect damage earlier and more reliably. The application of the new test to structures in the field is part of future work.

Figure 1 :

 1 Figure 1: Truss structure with six sensors.

Figure 2 :

 2 Figure 2: Comparison of damage detection tests on simulated truss structure (log-scale).

Figure 3 :

 3 Figure 3: Test stand for proof testing of damage identification algorithms; left: laboratory structure, middle: flange connection for damage modeling, right: electrodynamic shaker for excitation

Figure 4 :

 4 Figure 4: Comparison of damage detection tests on steel frame structure in a lab experiment (log-scale).

Acknowledgments

The partial support from the European project FP7-PEOPLE-2009-IAPP 251515 ISMS is gratefully acknowledged.