Miguel Couceiro
email: miguel.couceiro@inria.fr

Nicolas Hug
email: nh.nicolas.hug@gmail.com

Henri Prade
email: prade@irit.fr

Gilles Richard
email: richard@irit.fr

Behavior of Analogical Inference w.r.t. Boolean Functions

It has been observed that a particular form of analogical inference, based on analogical proportions, yields competitive results in classification tasks. Using the algebraic normal form of Boolean functions, it has been shown that analogical prediction is always exact iff the labeling function is affine. We point out that affine functions are also meaningful when using another view of analogy. We address the accuracy of analogical inference for arbitrary Boolean functions and show that if a function is ε-close to an affine function, then the probability of making a wrong prediction is upper bounded by 4ε. This result is confirmed by an empirical study showing that the upper bound is tight. It highlights the specificity of analogical inference, also characterized in terms of the Hamming distance.

Introduction

There have been several proposals for formalizing analogical inference in logical settings [START_REF] Davies | [END_REF]Russell, 1989;[START_REF] Gust | [END_REF]. More recently, a propositional logic modeling of analogical proportions has been developed [Miclet and Prade, 2009;Prade and Richard, 2013]. This latter view is at the basis of a successful classification method for which empirical evidence was provided in classification problems [START_REF] Miclet | [END_REF]Bounhas et al., 2017].

Analogical proportions are statements of the form "a is to b as c is d", usually denoted a : b :: c : d. Such a proportion expresses that the ways a and b differ are the same as the ways c and d differ. In this paper, one should think of a, b, c and d as being represented by tuples of m Boolean feature values, i.e., as elements of the m-dimensional Boolean space B m = {0, 1} m . How does analogy-based inference work?

Let f be a Boolean function from B m to B and x ∈ B m such that f (x) is unknown (f (x) can be viewed as a class label). The inference amounts to say that if there exist three items a, b, c with known labels such that a : b :: c : x holds, then f (x) may be the solution (if any) of the analogical proportion f (a) : f (b) :: f (c) : f (x) [Stroppa and Yvon, 2005].

The notion of analogical proportion and the associated inference mechanism may appear a bit uncommon at first glance. First, as far as we know, the relationship between analogical proportion and Hamming distance has never been investigated despite the fact that the latter is a key notion for appreciating differences in a Boolean framework. Second, analogical proportion seems a quite sophisticated relation between 4 items and one may wonder whether it occurs rarely in practice.

In this paper, we shed some light into these concerns. Another important issue is to understand why it works in practice. Beyond the general good reputation of analogy, no explanation for its good results has been provided until now, with the exception of the recent contribution [Couceiro et al., 2017] where a remarkable behavior w.r.t. affine Boolean functions is established. Namely, given a sample S of elements where a function f is known, it is shown that we can guess with certainty the value of f (x) for x / ∈ S for some classes of functions f , using analogical inference. The key result is that analogical inference does not make any error iff f is an affine Boolean function. Nevertheless, no theoretical or experimental results were provided regarding the possible link between the accuracy of the inference process and the distance between the target labeling function f and the set of affine functions. Here, we address the latter concern.

The paper is organized as follows. Section 2 gives the necessary background on two models (M and K) of Boolean analogical proportions. In Section 3, we link analogical proportions with Hamming distance. We show that, in general, it is possible to find non trivial intermediary items b and c given any a and d such that a : b :: c : d holds. Section 4 surveys Boolean functions and their algebraic normal forms. In Section 5, we compare the class of model-preserving functions, based on M and K definitions, and show that both coincide with affine functions. In Section 6, we consider the case where the target function is not affine, and show that if a function is ε-close to an affine function, then the probability of making wrong predictions based on analogical inference is upper bounded linearly w.r.t. ε.

Analogical Proportions

An analogy A (or analogical proportion 1 ) over a nonempty set X is a quaternary relation over X that satisfies four axioms [START_REF] Miclet | [END_REF] (all formulas are universally quantified):

1. Reflexivity: A (a, b, a, b).

1
In the paper, the term analogy is short for analogical proportion 2. Symmetry: A (a, b, c, d) =⇒ A(c, d, a, b).

Central permutation:

A(a, b, c, d) =⇒ A(a, c, b, d). 4. Unicity: A(a, b, a, x) =⇒ x = b.
In this paper, we restrict the interpretation domain X to be the Boolean set B = {0, 1} and A(a, b, c, d) will be denoted a : b :: c : d. It leads to 2 remarkable Boolean models: M = {0000, 1111, 1010, 0101, 1100, 0011}, K = {0000, 1111, 1010, 0101, 1100, 0011, 0110, 1001}.

The K model was proposed, although very informally, by the American researcher S. Klein [Klein, 1982]

c : d ≜ m ∧ i=1 a i : b i :: c i : d i .
When an analogy is defined over a set X, given 3 elements a, b, c of X, the requirement that the relation A (a, b, c, x) holds defines an equation where x is the unknown. In the Boolean case, with M or K as models, there is at most one solution. When equation a : b :: c : x is solvable, we abbreviate it with solvable(a, b, c). In this case, sol(a, b, c) denotes the solution. K-analogies extend to B m as well, and they are always solvable.

Throughout this paper, we will adopt model M .

A New View on Analogies

We first show that an analogy is valid only if certain constraints (in terms of Hamming distances) on their 4 items hold. Then we highlight that analogies emerge as soon as 2 items are compared. We focus on M -analogies here.

Links with the Hamming Distance

A standard distance over B m is the Hamming distance:

H m (x, y) = |{i ∈ [1, m] : x i ̸ = y i }|, which satisfies: H m+1 (x, y) = H m (x/m, y/m) + H 1 (x m+1 , y m+1 )
where x/m denotes the projection of x over B m . We write H when there is no ambiguity about the dimension. A first link between Boolean analogy and Hamming distance is given by: Adding these two equations leads to the expected result. The same reasoning applies to the two remaining equalities. □ However the reverse property does not hold for 0 : 1 :: 1 : 0 and for 1 : 0 :: 0 : 1. A stronger relation can be established: And we are done by adding these two equations. The second equality is deduced from the first one using Property 1. □ As Prop. 2 rules out 0 : 1 :: 1 : 0 and 1 : 0 :: 0 : 1, we may wonder if it characterizes analogy.

Property 3 When m = 1, H(a, b) + H(a, c) = H(a, d) (or H(d, c) + H(d, b) = H(d, a)) implies a : b :: c : d. Proof: If H(a, d) = 0, then H(a, b) = H(a, c) = 0 and a = b = c = d. If H(a, d) = 1, then either H(a, b) = 0 or H(a, b) = 1.
if ∀k = 1, . . . , m, H(a k , b k ) + H(a k , c k ) = H(a k , d k ).
If we consider B m as a subset of R m , a : b :: c : d means that the 4 vectors are the vertices of a parallelogram. Note that this parallelogram can be flat (a : b :: a : b), or even reduce to a single vertex (a : a :: a : a). With this interpretation, it is quite clear that given 3 vectors a, b, c ∈ B m , it is not always possible to find a fourth vector d ∈ B m such that a : b :: c : d (although this 4th vector always exists in R m ).

Analogy and Comparison of Two Items

Analogy is implicitly present when we compare two objects. Indeed, for any two distinct objects represented as Boolean vectors a and d, there is a way to fill in the gap between them in a "smooth" way. More precisely, it is possible to find two other vectors b and c such that a : b :: c : d. To formalize this idea, let Agr(a, d) be the set of indices where a and d agree and Dis(a, d) the set of indices where the two vectors differ. Then we have the following easy-to-check result. Property 5 Let a, d ∈ B m and take b, c ∈ B m such that:

1. ∀i ∈ Agr(a, d), a i = b i = c i = d i 2. ∀i ∈ Dis(a, d), either (b i = a i and c i = d i ) or (b i = ¬a i and c i = ¬d i ).
. We can check that H(a, b) = H(c, d) = 1, H(a, c) = H(b, d) = 2, while H(a, d) = H(b, c) = 3 = |Dis(a, d)|.
Analogy allows us to compare a to d via some transitional steps involving intermediary values b to c, which are closer to a (or to d) than a is to d. In a visual context, analogies were recently used for expressing constraints in multi-class categorization tasks and attribute transfer [START_REF] Hwang | [END_REF][START_REF] Jing | [END_REF]. The latter reference illustrates the creation of "intermediate" images b and c building an analogy with two given images a and d (in an approximative way).

Boolean Functions: Brief Overview

We now survey some useful results on Boolean functions.

B as a Field and B m as a Vector Space

When it comes to represent a Boolean function, an algebraic view may be more relevant than sticking to the standard logical operators. In that case, multiplication in B is the usual one (corresponding to conjunction), but addition is addition modulo 2 (still denoted + and corresponding to XOR), making B a 2-element field such that x + x = 0. In this way, B m becomes a vector space over the 2-element field B: scalar multiplication distributes componentwise and addition of vectors is likewise defined componentwise as addition modulo 2. The Hamming weight wt(x) is |{x i |x i = 1}| so that the Hamming distance H(x, y) is just wt(x + y). By endowing B with the total order 0 < 1, B m becomes totally ordered via the lexicographic order. It is sometimes useful to represent a Boolean function f ∈ F m by the unique vector of B 2 m of its values: wt(f ) is the number of vectors x such that f (x) = 1.

A linear form over B m is a Boolean function g ∈ F m such that for every x, y ∈ B m and α ∈ B we have g(x + y) = g(x) + g(y) and g(αx) = αg(x). Note that the second equality reduces to g(0) = 0 in this case. The set of linear forms over B m is usually denoted (B m ) * and called the dual space of B m . This is also a vector space over B. B m is naturally equipped with a dot product given by < x, y >= Σ m i=1 x i y i . This dot product induces the following well known properties

• B m and (B m ) * are isomorphic.

• x → g x where g x (y) =< x, y > is an isomorphism. Analogy over B (thought of as a field) can then be defined by: a : b ::

c : d ≜ (a + d = b + c) ∧ (ad = bc).
Extension to B m is made componentwise. These definitions are strictly equivalent to the ones given in Section 2. We have to note that a : b :: c : d implies d = a + b + c and, with the notations of Section 2, if solvable(a, b, c) then sol(a, b, c) = a + b + c. This is also true with vectors in B m . The reverse property does not hold as can be seen with 0110 which is not an analogy but where the last element is still the sum of the 3 first elements.

Polynomial Representation of Boolean Functions

A monomial is a term of the form x I ≜ ∏ i∈I x i , for some (possibly empty) finite set of positive integers I with the convention that 1 is the empty monomial x ∅ . The size | I | is called the degree of x I denoted d(x I ). A polynomial is a sum of monomials ∑ I⊆{1,...,m} ω I • x I where each ω I belongs to B (addition is understood as addition modulo 2). Equipped with scalar multiplication over B and addition, the set B[x 1 , . . . , x m ] of polynomials is a ring over B. The degree of a polynomial is the largest degree of its monomials. The monomial 0 is just 0 • x ∅ . Note that 0 and 1 are the only monomials of degree 0. A projection (corresponding to the selection of one feature) is obviously reduced to a monomial of degree 1.

Each Boolean function f ∈ F m is uniquely represented by a polynomial called ANF for Algebraic Normal Form [Stone, 1936;Zhegalkin, 1927] i.e.,

f (x 1 , . . . , x m ) = ∑ I⊆{1,...,m} ω I • x I .
The degree d(f ) of a Boolean function f is the degree of its ANF. With no danger of ambiguity, we confuse f with its ANF. The negation of a function

f is just f + 1. When d(f ) ≤ 1, we say that f is affine. In other words, a function f ∈ F m is affine iff f (x 1 , . . . , x m ) = ∑ i∈[1,m] ω i • x i + ω 0 =< ω, x > +ω 0 ,
where ω = (ω 1 , . . . , ω m ). The set of affine functions of arity m is denoted by L m , so that L = ∪ m L m is the set of all affine functions. If the term ω 0 is 0, the affine function is just a linear form. Thus, affine functions are either linear forms (including the constant function 0) or their complements.

5 Analogy and Affine Functions

Predicting by Analogy

For m ≥ 1, let F m = F(B m , B), denote the set of all Boolean functions of arity m, and let F = ∪ m∈N F m , i.e., the set of all Boolean functions. Given a sample set S ⊆ B m and a function f ∈ F m , the analogical root of a given element x ∈ B m , denoted by R S (x, f ), can be defined as follows [Hug et al., 2016]: Definition 1 For f ∈ F m , define R S (x, f ) to be the set In other words, R S (x, f ) is the set of triples over S that are analogically linked to x or, equivalently, the set of vertices a, b, c ∈ S 3 such that (a, b, c, x) is a parallelogram in the Boolean cube B m . The set R S (x, f ) can be used to provide a prediction for the label f (x) of x using analogical inference. For any element x ∈ B m \ S such that R S (x, f ) ̸ = ∅, the analogical label of x is defined as follows: Definition 2

x S,f ≜ M ode({sol(f (a), f (b), f (c)) : (a, b, c) ∈ RS(x, f )})
where M ode(Σ) returns the most frequent element of a multiset Σ (i.e. x S,f is the most common label among all candidate labels). In case of a tie, the returned element is randomly chosen among the most frequent elements. It is clear that, for these elements, we do not necessarily have

x S,f = f (x). Definition 3 A function f ∈ F m is said to be analogy pre- serving (AP ) if for every x ∈ B m \S such that R S (x, f ) ̸ = ∅, we have x S,f = f (x).
Note that, thanks to code independency property, f is AP iff ¬f is AP . Proposition 1, which has been proved in [Couceiro et al., 2017], gives an alternative description of AP functions.

(f (a), f (b), f (c)) =⇒ sol(f (a), f (b), f (c)) = f (d).
Analogical prediction may look quite similar, at least in principle, to a k-NN technique where a final vote among k candidates is performed. Nevertheless, we can ensure the accuracy of the prediction: this is not the case with k-NN algorithms.

From AP Functions to KP Functions

In this section we recall the explicit description (Proposition 3 below) of the class of AP functions, which was recently presented in [Couceiro et al., 2017] by making use of tools in clone theory [START_REF] Denecke | [END_REF]Lau, 2006]. The intuition comes from the following result in [Szendrei, 1980].

Proposition 2 A function f ∈ F m is affine if and only if for every a, b, c ∈ B m , we have f (a + b + c) = f (a) + f (b) + f (c).
Inspired by the latter result, the following explicit description of AP functions has been obtained:

Proposition 3 A function f is AP if and only if it is affine.
This result provides a clear picture of what can be expected from a prediction outside a given Boolean sample: only affine functions can ensure sound predictions. If the number of Boolean functions of arity m is 22 m , the number of affine functions of arity m is just 2 m+1 . This shows that affine functions constitute a rather small subclass of functions in this landscape. This fact asks for an extension of our results to non affine functions.

In fact, everything which has been done in the previous sections, based on the smallest model M of analogy, can be done with the model K.

Approximately Affine Functions

Applying analogical inference without caution may lead to incorrect labelling of instances x / ∈ S: if f is not AP , we may have x S,f ̸ = f (x) for some x. It is natural to ask whether the use of analogical inference to approximately affine functions can ensure labels which are also approximately correct. Let us formalize what we mean by "approximation".

The Meaning of Approximately Affine

The word approximation is understood in various ways within the Boolean algebra community. In this paper we will take a probabilistic approach. Given two functions f, g ∈ F m , the Hamming distance H(f, g) between f and g is just wt(f + g) when f and g are represented as vectors (see Section 4.1). In other words,

H(f, g) = |dis(f, g)| = |{x ∈ B m : f (x) ̸ = g(x)}|.
Let P denote the uniform distribution 2 over B m . Then

P (x ∈ dis(f, g)) = H(f,g) 2 m
is a distance over the set F m of all Boolean functions, and we will denote it by d(f, g). Now, we say that a function g approximates f at level ε or that it is ε-close to f if d(f, g) ≤ ε. This can also be read as "the proportion of x ∈ B m such that f (x) ̸ = g(x) does not exceed ε". Given a set C ⊆ F m , d(f, C) is defined as usual as min g∈C d(f, g). Obviously, when it comes to computing the distance d(f, L) of f to the set L of affine functions, we are sure that d(f, L) ≤ 1 2 . Indeed, if there is an affine function g such that d(f, g) > 1 2 , then g + 1 is still affine and d(f, g + 1) < 1 2 , by definition.

Definition 5 A function f is ε-close to C if d(f, C) ≤ ε. If f is ε-close to the set L of affine functions for a threshold ε, then f is said ε-approximately affine (necessarily ε ∈ [0, 1 2 ]
). As we will see, this notion of ε-closeness allow us to assess the quality of the analogical inference.

Sound Analogical Inference

In this section, we mainly recall concepts introduced in [Couceiro et al., 2017]. Given a subset S ⊆ B m , a function f ∈ F m , and an element x ∈ B m , if R S (x, f ) ̸ = ∅, then an analogical label x S,f can be associated to x. Let us then define the analogical extension E S (f ) of S w.r.t. f as the set of x ∈ B m such that R S (x, f ) ̸ = ∅. Formally:

Definition 6 E S (f ) ≜ {x ∈ B m : ∃(a, b, c) ∈ S 3 , a : b :: c : x and solvable(f (a), f (b), f (c))}.
Intuitively, E S (f ) can be seen as the set of all x ∈ B m that are the fourth vertex of a parallelogram built with 3 vertices from the sample set S, provided that the equation related to the associated labels is solvable. is clear that S ⊆ E S (f ) because a : a :: a : a always holds. Define

err S,f = P ({x ∈ E S (f ) \ S | x S,f ̸ = f (x)}).
Hence, we are interested in the elements x ∈ E S (f ) \ S whose predicted label x S,f is f (x). Definition 7 We say that E S (f ) is sound if err S,f = 0.

Due to the definition of the probability P , err S,f = 0 means the proportion of wrong analogical labels is zero. In this case, E S (f ) can be used as an extended sample set for classification tasks. In the opposite case, E S (f ) is said to be unsound (i.e. noisy) which is equivalent to err S,f > 0. Thus Proposition 1 can then be restated as follows:

Proposition 5 A function f ∈ F m is AP if and only if for every S ⊆ B m , E S (f ) is sound.

Approximately Sound Extension

By Proposition 5, if f is not affine (and thus not AP ), then there exists a sample S such that E S (f ) is unsound. It means that 3 P (err S,f > 0) > 0 where P relates to the probability over all possible choices of the sample S.

This probability provides a measure of the imperfection of f w.r.t. analogical inference. Given a function f ε-close to L, can we determine an upper bound of this probability? The following proposition gives a positive answer.

3 From a rigorous viewpoint, we should denote P ′ (err S,f > 0) where P is the probability distribution over the universe B m , and P ′ is the induced distribution over the powerset 2 B m . In our case, err S,f > 0 simply means {x ∈ ES(f Now, for the sake of contradiction, suppose that

) : |x ̸ = f (x)} ̸ = ∅. Proposition 6 Let ε ∈ [0, 1 2 ]. If f is ε-approximately affine, then P (err S,f > 0) ≤ 4ε. Proof: Let be f ε-close to L and g ∈ L such that d(f, g) ≤ ε, i.e., P ({x ∈ B m |f (x) ̸ = g(x)}) ≤ ε.
f (a) = g(a), f (b) = g(b), f (c) = g(c) and f (x) = g(x).
In this case, solvable(g(a), g(b), g(c)), and thus sol(g(a), g(b), g(c)) = g(x) since g is affine. Furthermore, f (x) = g(x) implies that

f (x) ̸ = sol(f (a), f (b), f (c)) = sol(g(a), g(b), g(c)) = g(x) = f (x),
which yields the desired contradiction. As a consenquence, one of the following conditions must hold:

f (a) ̸ = g(a) or f (b) ̸ = g(b) or f (c) ̸ = g(c) or f (x) ̸ = g(x).
But the probability of this event is upper bounded by

P (f (a) ̸ = g(a)) + P (f (b) ̸ = g(b)) + P (f (c) ̸ = g(c)) + P (f (x) ̸ = g(x)) ≤ 4ε.
Hence, the occurrence of the event (E S (f ) is unsound) is upper bounded by 4ε, which completes the proof. □

This result tells us that if we can upper bound the distance between f and L by a small ε, then the probability that no noise will be introduced by using E S (f ) instead of S in our classifier4 , will be at least 1 -4ε.

From a practical point of view, testing that a function is approximately linear (i.e. affine where the final constant in its ANF is 0) can be checked via a simple test due to [START_REF] Blum | Self-testing/correcting with applications to numerical problems[END_REF] also known as BLR test. It can be proved that if BLR test accepts f with a probability at least 1 -4ε, then f is ε-close to L. The test can be performed with O( 1 ε ) queries. See [START_REF] Kaufman | [END_REF] for a comprehensive analysis of the test.

Relaxing the Bounds

Our definition of approximately sound extension is a rather strict. For a sample S, having a probability err S,f strictly positive does not mean that the classifier will performs modestly starting from S. For instance, starting with any S, if this probability is in ]0, 0.001], then the probability of having P (err S,f ) > 0 is 1 despite the algorithm being highly satisfactory. This fact asks for a relaxation of our definition. A bad performance would be more accurately described by err S,f > δ, for some δ ∈

[0, 1[. Definition 8 We say that E S (f ) is δ-sound if err S,f ≤ δ. Similarly, f is δ-sound if ∀S ⊂ B m , E S (f ) is δ-sound 5 .
If f is δ-sound (for a small δ), then we can consider the analogical inference principle as a valuable inference principle for f since the error rate is quite low. With this view, we are interested in the probability of f not being δ-sound, which then becomes: η f (δ) = P (err S,f > δ). In the case when δ = 0 and f is ε-approximately affine, we have that η f (0) ≤ 4ε. In fact, we can prove the following:

Proposition 7 Let ε ∈ [0, 1 2 ], δ ∈ [0, 1]. If f is ε- approximately affine, then P (err S,f > δ) ≤ 4ε • (1 -δ).
Proof: Given a sample set S and x ∈ E S (f ) \ S, it follows from Prop. 6 that P (x S,f ̸ = f (x)) ≤ 4ε. For err S,f > δ to occur, such an x should lie outside an area of radius δ and the probability of picking such an x is 1 -δ. Obviously if δ = 1, then ∀f, P (err S,f > 1) = 0 since ∀S, err S,f ≤ 1. □

Experiments

In order to investigate whether the bounds in Propositions 6 and 7 are tight, we implemented the following protocol.

For X = B 8 , we started from 3 affine functions: 1. g 1 (x) = x 1 which is the first projection.

2. g 2 (x) = x 1 + x 2 (all variables but x 1 , x 2 are irrelevant) 3. g 3 (x) = Σ m i=1 x i where all variables are relevant. To get an ε-close function, we just flip a random fraction ε of the values of g i on the universe X. Doing so, we are sure to get a function f i,ε such that d(f i,ε , g i ) ≤ ε and then d(f i,ε , L) ≤ ε. There is no need to go beyond 0.5 since d(f, g) > 0.5 =⇒ d(f, g + 1) < 0.5,

where g + 1 ∈ L if g ∈ L.
For each approximately affine function f i,ε , we performed 100 experiments with a random S of size |X| 2 = 2 7 : 1. For each S, we estimated the probability err S,fi,ε by

P ({x ∈ E S (f i,ε ) \ S : x S,fi,ε ̸ = f i,ε (x)}),
i.e., the proportion of elements x in E S (f i,ε ) \ S whose analogical predicted label is not equal to f i,ε (x). 2. The probability of interest η fi,ε (δ) = P (err S,fi,ε > δ) is taken as the proportion of errors over the 100 experiments.

ηf 3 (δ) 4ε ⋅ (1 -δ) Figure 1: Values of η f i ,ε (δ)for δ = 0 0.0 0.1 0.2 0.3 0.4 0.5 ε 0.0 0.2 0.4 0.6 0.8 1.0 ηf 1 (δ) ηf 2 (δ) ηf 3 (δ) 4ε ⋅ (1 -δ) Figure 2: Values of η f i ,ε (δ)for δ = 0.3
In figures 1, 2 and 3, we provide η(δ) as a function of ε for δ = 0, 0.3, 0.5 respectively. The upper theoretical bound 4ε • (1 -δ) from Proposition 7 is shown as a dashed line. We observe that for all approximately affine functions f 1 , f 2 , f 3 , the computed probability remains below the theoretical bound. Figures 1 and2 show that the bounds are tight.

To give an idea of the behavior of 1-NN in the same context, we show (Fig. 4) a comparison between the evolution of ω = 1err and the proportion τ of properly classified elements in E S (f i,ε ) \ S by 1-NN wrt ε. As expected, 1-NN becomes better when we move away from affine functions, i.e., the opposite behavior to analogy-based classifier.

Conclusion

Analogical inference allows to predict the exact labels of elements outside a given sample set S, provided that the labelling function f is affine. In the case when f is only approximately affine, we provided a probability of null error: it is less than 4 times the distance between the target function f and the set L of affine functions. As this distance can be evaluated via the BLR test, we have a way to upper bound the error associated with any Boolean function. Our experiments agree with this upper bound, and indicate that the bound may be tight. When relaxing the error requirement to a non null error bounded by δ, we get another theoretical upper bound.

We note that the upper bound does not rely on the size of S, just because the external probability P is considered over the whole set of potential samples S. Introducing the size of S as a parameter could further refine this upper bound. These results provide clues suggesting why (and when) analogical inference gives good prediction results. Still, there exist known cases of functions f that are far from being affine, but for which the analogical inference works well [Bayoudh et al., 2007]. This is yet to be explained.

Also, in real life applications, we rarely deal with purely Boolean datasets: often they mix Boolean with symbolic attributes taking values in finite sets. A typical example of such an attribute is color. Obviously, on the set of values of color, there is no structure, no order and no known operator. Binarizing the attribute values, leads us to considering partial Boolean functions, which is a topic of ongoing research.

  more than 25 years before the introduction of the Boolean modeling of analogical proportion. It obeys the 4 postulates of analogical proportion but it has what may be considered as an undesirable behavior: 'a is to b as c is to d is the same as 'b is to a as c is to d. Still it is of interest to keep it for comparison with genuine analogical proportion. It is noteworthy that K is such that any added element of B 4 would contradict unicity. M is the smallest Boolean model obeying the 4 axioms above. We denote by a : b :: c : d an analogical proportion w.r.t. M . It can be defined by the logical expression: a : b :: c : d ≜(a ∧ ¬b ↔ c ∧ ¬d) ∧ (¬a ∧ b ↔ ¬c ∧ d) where ∧, ¬, ↔ are respectively the logical connectives for conjunction, negation and equivalence. It expresses the fact that a differs from b as c differs from d (and vice-versa: b differs from a as d from c). Considering B as a subset of R, another equivalent formulation is given by: a : b :: c : d if and only if a -b = c -d. Note however that the latter reformulation is not suitable in the Boolean setting since a -b may not belong to B when a, b ∈ B. It can be checked that M (and K) satisfies the code independence property: a : b :: c : d if and only if ¬a : ¬b :: ¬c : ¬d, which guarantees that 0 and 1 play symmetric roles in the encoding of the Boolean features. An M -analogy over B m is defined componentwise, i.e., ∀a, b, c, d ∈ B m , a : b ::

Property 1

 1 ∀a, b, c, d ∈ B m s.t. a : b :: c : d, we have: H(a, b) = H(c, d), H(a, c) = H(b, d), H(a, d) = H(b, c). Proof: We prove the first equality H(a, b) = H(c, d) by induction on m. Clearly, it holds for m = 1. By definition of analogy on B m+1 , given a, b, c, d ∈ B m+1 , a : b :: c : d iff a/m : b/m :: c/m : d/m and a m+1 : b m+1 :: c m+1 : d m+1 The induction hypothesis leads to H(a/m, b/m) = H(c/m, d/m) and H(a m+1 , b m+1 ) = H(c m+1 , d m+1 ).

Property 2

 2 ∀a, b, c, d, such that a : b :: c : d, we have H(a, b) + H(a, c) = H(a, d). Similarly, we have H(d, c) + H(d, b) = H(d, a). Proof: The first equality is easily proved by induction on m. It is true for m = 1. Since we have a : b :: c : d iff : a/m : b/m :: c/m : d/m and a m+1 : b m+1 :: c m+1 : d m+1 we can apply the induction hypothesis to get H(a/m, b/m) + H(a/m, c/m) = H(a/m, d/m) and H(a m+1 , b m+1 ) + H(a m+1 , c m+1 ) = H(a m+1 , d m+1 ).

  In the first case, a = b and H(a, c) = 1, i.e. c = d since m = 1 and variables can only take values 0 or 1. Similarly if H(a, b) = 1, b = d and H(a, c) = 0, i.e. a = c. The other relation follows by permutation. □ When m > 1, H(a, b)+H(a, c) = H(a, d) is not enough to ensure a : b :: c : d as we can see by taking a = 000, b = 100, c = 110, d = 111. Indeed a i : b i :: c i : d i holds only for i = 2, and not for i = 1 or 3, while both H(a, b) + H(a, c) = H(a, d) and H(d, c) + H(d, b) = H(d, a) hold. So, we should require that the relation holds componentwise for characterizing analogy: Property 4 ∀a, b, c, d ∈ B m , a : b :: c : d if and only

  Then a : b :: c : d. Moreover if |Dis(a, d)| ≥ 2, then we can find a, b, c, d that are pairwise distinct. Example: Let a = 01100 and d = 00111 in B 5 . Then Agr(a, d) = {1, 3} and Dis(a, d) = {2, 4, 5}. By 1., we have b = 0b 2 1b 4 b 5 and c = 0c 2 1c 4 c 5 , while 2. gives us more freedom, for instance, b = 01110 and c = 00101 are solutions. Indeed, 01100 : 01110 :: 00101 : 00111

{

  (a, b, c) ∈ S 3 : a : b :: c : x and solvable(f (a), f (b), f (c))}.

Proposition 1 A

 1 function f : B m → B is AP if and only if for every a, b, c, d ∈ B m , the following implication holds: a : b :: c : d and solvable

  Let us denote K(a, b, c, d) the fact that a, b, c, d belong to model K. Obviously a : b : c : d =⇒ K(a, b, c, d) and the main difference with analogy is that every triple in B m is uniquely K-solvable i.e., ∀a, b, c, ∃!d such that K(a, b, c, d). In fact, d = a + b + c is the unique solution of the equation K(a, b, c, x). The definitions we have used for defining AP functions still apply and lead to the following definition for an KP (for K-preserving) function: Definition 4 f is KP if and only if ∀a, b, c, d ∈ B m : K(a, b, c, d) =⇒ K(f (a), f (b), f (c), f (d)) (1) Using Proposition 2, we get the following result: Proposition 4 The class of KP functions is the class of affine functions. Proof: Observe that K(a, b, c, d) means d = a + b + c, and thus if f is KP , then f (a + b + c) = f (a) + f (b) + f (c) by (1). It then follows from Proposition 2 that f is affine. □ So considering analogical inference, using model M or K does not make any difference w.r.t. preserving functions.

  Given a sample set S, an element x ∈ E S (f ) \ S, the occurrence of x S,f ̸ = f (x) implies that there are a, b, c ∈ S: a : b :: c : x and solvable(f (a), f (b), f (c)) and sol(f (a), f (b), f (c)) ̸ = f (x). This implies that there are a, b, c ∈ B m such that a : b :: c : x and solvable(f (a), f (b), f (c)) and sol(f (a), f (b), f (c)) ̸ = f (x).

Figure 3 :Figure 4 :

 34 Figure 3: Values of η f i ,ε (δ)for δ = 0.5

Using the uniform probability will allow to use relative frequencies as probability approximation.

The factor 4 has no link with the VC-dim of an analogical classifier, which is infinite[Hug et al., 2016]. This multiplicative scalar is due to the fact that analogical proportion involves 4 items.

Note that f is 0-sound iff f is AP.

Acknowledgements

This work is partially supported by ANR-11-LABX-0040-CIMI (Centre International de Mathématiques et d'Informatique) within the program ANR-11-IDEX-0002-02, project ISIPA.


H Hernández-Moreno

F Collombet
email: francis.collombet@iut-tlse3.fr

B Douchin

D Choqueuse

P Davies

Entire Life Time Monitoring of Filament Wound Composite Cylinders Using Bragg Grating Sensors: III. In-Service External Pressure Loading

Keywords:

This article is the third of three papers describing a study of the monitoring of filament wound composite cylinders for underwater applications. Part I described the technological issues and the development of specimens instrumented with embedded gratings and thermocouples, with the aim of monitoring the temperature and strain changes during the cylinder manufacturing presented in Part II. Residual strains are not negligible, over 1,000 axial micro-strain at the end of the curing cycle. Part III describes the response of these cylinders to hydrostatic pressure loading. The same embedded fiber optical Bragg gratings (FBGs) used for parts I and II of the study are here used as strain gauges. Their response is compared to that of classical resistive strain gages bonded to the inner surface of the tube. Results from these initial tests demonstrate the embedded FBG sensor's capability to monitor structural health of an underwater structure from fabrication throughout its service life. Embedded instrumentation records strains during pressure cycles up to final failure, without affecting the cylinder response.

Introduction

Composite materials are well suited for underwater applications, where their light weight and excellent corrosion resistance make them an attractive alternative to metallic materials. Several authors have shown that composite cylinders can be used to resist the hydrostatic pressures which accompany deep sea applications [START_REF] Stachiw | Graphite fiber reinforced plastic pressure hull mod 2 for the advanced unmanned search system vehicle[END_REF][START_REF] Smith | Design of Marine Structures in Composite Materials[END_REF][START_REF] Stevenson | The mechanical design and implementation of an autonomous submersible[END_REF][START_REF] Graham | Composite pressure hulls for deep ocean submersibles[END_REF][START_REF] Gruber | Buckling performance of hydrostatic compression loaded 7-inch diameter thermoplastic composite monocoque cylinders[END_REF][START_REF] Davies | Thermoplastic composite cylinders for underwater applications[END_REF]. If such applications are to become more widespread, particularly for the offshore oil industry, it is essential to be able to monitor the material condition during manufacturing and then throughout the service life. Two previous papers in this series [START_REF] Hernández-Moreno | Entire life time monitoring of filament wound composite cylinders using Bragg grating sensors: I. Adapted tooling and instrumented specimen[END_REF] [START_REF] Hernández-Moreno | Entire life time monitoring of filament wound composite cylinders using Bragg grating sensors: II. Process monitoring[END_REF] have shown how a filament wound composite cylinder can be instrumented with Bragg gratings, which enable its condition during and after manufacture to be evaluated. In the present paper one of those cylinders, which was previously monitored during fabrication, is subjected to increasingly high external pressure cycles up to failure, in order to demonstrate how monitoring can be performed in service. The test was carried out at the hyperbaric testing facility at IFREMER, the French Ocean Research Institute. The pressure which acts on an immersed structure depends on the depth according to the expression [START_REF]AFNOR standard XP X 10-812, Marine environment, oceanographic equipment[END_REF]: P = C 1 *H + C 2 * H 2 (1) where P = pressure (MPa), H = immersion depth (m), C 1 is equal to 0.01 MPa/m and C 2 equal to 0.05 x 10 -6 MPa/m². For deep sea applications this loading is often the main design criterion. In recent years optical fiber based sensors, such as Bragg gratings, have been increasingly used for in situ strain and temperature monitoring, references can be found in the previous papers [START_REF] Hernández-Moreno | Entire life time monitoring of filament wound composite cylinders using Bragg grating sensors: I. Adapted tooling and instrumented specimen[END_REF] [START_REF] Hernández-Moreno | Entire life time monitoring of filament wound composite cylinders using Bragg grating sensors: II. Process monitoring[END_REF]. One example developed specifically for underwater applications was described by Brower [START_REF] Brower | Structural properties measurements in deepwater oil and gas fields using an advanced fiber optic sensor monitoring system[END_REF], who presented a "smart pipe system" composed of a "smart layer" (composite material) with embedded Bragg gratings, placed between the pipe outer skin and the thermal insulation layer, this system allows pipe monitoring during service. Several recent publications have described the response of glass/epoxy composite cylinders to external pressure [START_REF] Hinton | Strength of composite laminates under biaxial loads[END_REF][START_REF] Davies | Influence of delamination on strength of externally pressurized glass/epoxy cylinders[END_REF][START_REF] Soden | Biaxial test results for strength and deformation of a range of E-glass and carbon fibre reinforced composite laminates[END_REF][START_REF] Gning | Damage development in thick composite tubes under impact loading and influence on implosion pressure: experimental observations[END_REF]. Two failure modes can occur; buckling, usually associated with thin walled tubes, and material failure in compression observed in thicker cylinders. Figure 1 shows the two failure regions, together with some IFREMER data for 55° glass/epoxy cylinders. The limit between the two modes depends on the choice of failure criteria, both for buckling and material failure, and there is still considerable debate over the appropriate choice for this type of loading [START_REF] Soden | Recommendations for designers and researchers resulting from the world-wide failure exercise[END_REF]. An analytical expression for buckling proposed by Mistry et al. [START_REF] Mistry | Failure of composite cylinders under combined external pressure and axial loading[END_REF] is plotted in Figure 1, together with a simple maximum hoop stress criterion for which the critical value is taken as 500 MPa for material failure. Given the uncertainty in predicting implosion pressures most applications rely on qualification testing. This is performed in pressure vessels, on tubes which are closed by metallic end plates. The design of these end closures, which determine the test boundary conditions, is particularly important when thick wall tubes are tested if premature failure is to be avoided [START_REF] Blake | A shear deflection theory for analysis of end plugs for external pressure tests of composite cylinders[END_REF] [START_REF] Davies | Long term behaviour of fibre reinforced structures for deep sea applications[END_REF]. In a test cylinder, strain sensors (strain gages) are placed on the structure at the moment when the specimen is being prepared for testing, so the initial strain within the material is unknown, though sometimes it can be estimated. One aim of the present study is to include the specimen residual strain, which is measured during fabrication, in the design of the pressure vessel. For this it was necessary to demonstrate that the same Bragg gratings could be used to follow both manufacturing and pressure testing.

Testing

Cylinder characteristics

A first series of five cylinders without instrumentation was pressure tested to failure in order to establish the reference baseline. These were nominally identical to the instrumented specimens, filament wound on the same machine with the same winding parameters, and have been reported previously [START_REF] Hernández-Moreno | Influence of winding pattern on the mechanical behavior of filament wound composite cylinders under external pressure[END_REF]. All were 4.4 mm thick, 125 mm internal diameter, reinforced with 3.5 mm width glass fibre rovings (1200 tex). The epoxy resin system used is a mixture of Araldite LY 5052 and hardener HY 5052. Winding angle is ±55° with respect to the cylinder axis. The number of layers is 7 (one filament wound layer is composed of the equivalent of two unidirectional layers, at +55° and -55°). The specimen has zones of reinforcing fibers at both ends with a winding angle of 90°. These allow plane surfaces to be obtained by a machining operation in order to form the interfaces with metallic closures (Figure 2), [START_REF] Hernandez-Moreno | Monitoring de la fabrication de tubes composites réalisés par enroulement filamentaire et comportement mécanique sous pression externe[END_REF]. The instrumented specimen used for pressure testing here was cylinder 37 (see Part II [START_REF] Hernández-Moreno | Entire life time monitoring of filament wound composite cylinders using Bragg grating sensors: II. Process monitoring[END_REF]), which has one rhomboid pattern in the circumferential direction. The Bragg gratings are located at mid-length of the cylinder, between the first and second reinforcement layers near the inner surface. Based on the results from previous tests (Figure 1) these are thin wall cylinders with a t/d ratio of 0.035, and would be expected to fail during pressure testing by buckling.

Sensor characteristics

Bragg grating sensors have a central wavelength  B (Bragg wavelength). Temperature variation T, and longitudinal strain , both produce a wavelength shift  B (Kang [START_REF] Kang | Strain monitoring of a filament wound composite tank using fiber Bragg grating sensors[END_REF]); grating behavior can be written as equation 1 with a B , and b B being the sensitivities to temperature and strain respectively:

     b T a B B   (1)
The technique used in this research for decoupling strain and temperature responses is to perform an initial characterization of the thermo-optical sensor response before embedding. Once the thermal response is known, the strain can be obtained using equation 2. A complete description of calibration procedures is presented in Part II [START_REF] Hernández-Moreno | Entire life time monitoring of filament wound composite cylinders using Bragg grating sensors: II. Process monitoring[END_REF].

  ) T ( b reading B       1 (2)
The Bragg grating wave length response to temperature excitation is linear, so with calibration data a linear regression can be obtained for each grating, having the form of equation 3:

   b T m ) T (   (3)
where, m  is the slope in the wavelength vs. temperature graph, and b  is the ordinate to origin. Substituting equation 3 into equation 2 leads to equation 4:

       b T m b reading B    1 ( 4
)
where reading is obtained from the Bragg grating interrogator system and T is the actual temperature at the place where Bragg grating is located. This temperature is obtained by embedding, beside each grating, a small diameter (250m) K type thermocouple, to provide a temperature reference. If "b B " (strain sensitivity) has a value of 1.21 pm/°C, for a Bragg wavelength of 1550 nm, the constant "b" has a value of 7.80.10 -7 / [START_REF] Mulle | Thermal expansion of carbon-epoxy laminates measured with embedded FBGS -Comparison with other experimental techniques and numerical simulation[END_REF] .

The details of the two Bragg grating sensors embedded in the cylinder tested under external pressure are given in Table 1. As indicated in Part I, [START_REF] Hernández-Moreno | Entire life time monitoring of filament wound composite cylinders using Bragg grating sensors: I. Adapted tooling and instrumented specimen[END_REF] the length of the Bragg gratings is 10 mm and they are UV uniformly inscribed with a phase mask and apodization in a 125 m diameter optical fiber having a polyacrylate coating (SMF-28e ® ). Each grating measures a mean strain value over the sensor length.

Test characteristics and cylinder conditioning

Cylinder 37 was tested under external pressure to validate embedded instrumentation during service and to continue the monitoring of a composite filament wound structure until failure.

Taking into account the failure mode of cylinders under pressure (buckling) the sensor must be placed on the structure at the adequate location in order to be sensitive to the resulting deformation. The sensor was therefore placed in the mid-section of the cylinder, between the first and the second ply of the structure. In plane deformations of the central part of the cylinder correspond to the sum of the global compression deformation of the structure and the flexural deformation induced by the buckling mode of the structure. The flexural deformations of the ring induced by the buckling of the structure for the more common modeshapes (modes 2 and 3) are shown in figure 3a and3b. This indicates that the flexural deformation depends on: the localization of the sensor along the circumference of the ring, the distance of the sensor from the neutral fibre of the beam. The response to flexural deformation of a sensor will be zero if it is placed at a node point of the mode-shape and maximum at the anti-node point, Figure 3c. In the same way the sensor placed on the surface of the material (outer or inner) will have the maximum sensitivity to the flexural deformation while a sensor placed at the mid thickness of the material will not measure any bending strain. In this initial study to validate the use of embedded optical sensors the cylinder was equipped with two Bragg gratings, one placed in the axial direction and the second lying along the circumference, as shown in Figure 4a. Results from fabrication monitoring indicated the presence of residual strains. Residual strains at 21°C for this cylinder at the end of manufacturing were: in the axial direction -1025 , in the circumferential direction 70. For this test, one end closure was provided with a sealed passage tube, in order to connect the optical fibers to the chamber cover. In order to monitor strains during the entire service life of the structure (up to 10 years for some underwater applications) the optical fiber must be carefully protected from the external environment. The main objectives are: To protect the fiber from hydrolytic attack. Glass fiber is very sensitive to water contact, which may induce stress corrosion cracking; To leave the non embedded part of the optical fiber free of pressure in order to avoid parasite effects induced by pressure acting on the fiber. Based on experience from previous projects at Ifremer these two points were solved by placing the optical fiber in a small diameter stainless steel tube capable of withstanding the hydrostatic pressure. This cylinder was also instrumented with strain gages to compare with the Bragg grating results. These were bonded on the inner surface of the cylinder, three in the circumferential direction and three in the axial direction, but only one of the latter functioned correctly during the test. These are also shown on Figure 4a. The conditioned cylinder before testing is shown in figure 4b. The test was performed using the 1000 bar capacity hyperbaric testing chamber shown in figure 5.

Loading and unloading pressure cycles were applied, Figures 6 and7, with pressure increments of 10 bar, starting at 10 bar. Once the pressure had stabilized at 10 bars, the first loading -unloading cycle was composed of a pressure ramp until 20 bar, followed by a constant pressure segment lasting 5 minutes and an unloading ramp to 10 bars. Subsequent cycles have the same shape, but each time the final value is higher by 10 bar. The loading and unloading rate was 1 MPa/min for the first 4 cycles (up to 5 MPa) then increased to 3 MPa/min for the last 3 cycles (up to collapse). During the entire test, strain responses from gages and pressure signals are recorded with a data acquisition system, with a sampling frequency of 4 readings per minute. This sampling frequency is limited by the data acquisition program available, which was developed to interrogate a large number of strain gages inside the pressure vessel via a single connecting cable. Central Bragg grating wavelengths were recorded continuously using a different acquisition system with a higher frequency of 5 Hz. Bragg grating wavelengths are transformed into strains following the procedure described in section 2.2. Spectra of both gratings were also recorded at different times during the test, in order to check that there was no influence of transverse loading on the optical fiber strain.

Test results

Results from the five initial tests to failure indicated a mean value of implosion pressure of 6.5 MPa, with a standard deviation of 0.4 MPa (Table 2), as described previously [START_REF] Hernández-Moreno | Influence of winding pattern on the mechanical behavior of filament wound composite cylinders under external pressure[END_REF]. In all but one case the cylinders buckled with a 3-lobe circumferential failure mode. The implosion pressure for the test on the cylinder with the Bragg grating was 6.7 MPa, which is within one standard deviation of the mean value obtained on non-instrumented specimens, and showed the same 3-lobe buckling failure mode, Figure 8. This result indicates that the embedded instrumentation has no significant influence on mechanical strength at implosion.

Bragg grating strain response to pressure

Spectral response of both gratings (axial and circumferential) were recorded at specific points during the test, specifically before and after pressure vessel closure and at the different pressure steps [START_REF] Brower | Structural properties measurements in deepwater oil and gas fields using an advanced fiber optic sensor monitoring system[END_REF][START_REF] Hernandez-Moreno | Monitoring de la fabrication de tubes composites réalisés par enroulement filamentaire et comportement mécanique sous pression externe[END_REF][START_REF] Brower | Structural properties measurements in deepwater oil and gas fields using an advanced fiber optic sensor monitoring system[END_REF]30,[START_REF] Brower | Structural properties measurements in deepwater oil and gas fields using an advanced fiber optic sensor monitoring system[END_REF]40,[START_REF] Brower | Structural properties measurements in deepwater oil and gas fields using an advanced fiber optic sensor monitoring system[END_REF]50,55,60,10 Bar), in order to check for any spectrum shape variation. Examples of circumferential grating spectra are shown in Figure 9, the axial spectra are very similar. Analyzing spectral response it can be seen that throughout the test, the shape of spectra does not vary, this means that loads transverse to the optical fiber axis do not influence the grating response significantly, so the grating response is solely due to membrane strains and temperature (in this case the latter is almost constant). The Bragg grating strain response is initially proportional to pressure (Figures 6 and7), but for the last three cycles there is a slight deviation, which may indicate a change in shape due to buckling or damage initiation. Bragg grating data is recorded for slightly longer than the pressure data at rupture due to the difference in sampling periods, 15 s for the pressure gage and 0.2 s for Bragg gratings, indicating the functionality of the latter until the last moments of the specimen life.

Bragg grating strain response and gage strains

The Bragg grating and strain gage responses show similar behavior (see Figures 10 and11).

As noted with respect to pressure, the Bragg grating strain during the last three cycles shows a small deviation compared with the strain gage values. As previously noted the absolute values measured will depend on the location of the measurement points with respect to the buckled shape. As the lobes develop some regions are more highly strained while others are unloaded, due to the induced local flexure. Only three strain gages were bonded round the circumference in this test, and this is not sufficient to represent the evolution of the buckled shape accurately. In order to be sure to capture the buckling mode correctly a larger number of strain gages are needed, and in some large cylinder tests over 100 can be used. In the axial direction the strain gage begins with an almost zero value while the Bragg grating already has 1128 Figure 8). This is due to the fact that the Bragg grating followed the strain evolution from fabrication including residual strain. In the circumferential direction (see Figure 7), this difference is not obvious because the residual strain indicated by the Bragg grating at the beginning is only 21. It is important to note that these residual strains evolved very little (by around 100 micro-strain) in the time between the end of manufacture and the start of the pressure test (6 months). This indicates that these are indeed permanent residual strains, which must be taken into account in cylinder design.

Pressure vs. strain

When pressure-strain diagrams are analyzed (Figures 12 and13), a critical pressure can be observed around 5.5 MPa (the pressure at which the pressure versus strain plot becomes non-linear). Beyond this point, a Bragg grating hysteresis response is observed, which is not detected by the strain gages. The reasons for this hysteresis are not clear, more tests are required to investigate its origins. Observing each diagram between 0 and 5 MPa (before pressure reaches a critical value), the Bragg grating strain response follows a very similar slope to that of the strain gages, within the variations between the strain gages (due to different flexural strains at different points on the mode-shape as indicated in Figure 3c).

Future instrumentation strategy

One of the major advantages of Bragg grating strain measurements for in-situ measurements is the possibility to place several strain measurement points on the same fibre, significantly reducing the number of cables and connectors. For example, multi sensor fibres can be implanted during cylinder manufacture with six sensors equally distributed along the sensitive zone in order to detect either a 2-nodes mode-shape (over half the circumference of the central part) and a 3 node mode-shape (over one third of the circumference). In addition, if embedded fibre optic sensors can also be implanted at the neutral fibre the comparison with the results from sensors placed close to the surface would enable compression strains to be distinguished from flexural (buckling) deformation. This comparison may also provide information on the distribution of residual stresses through the thickness of the structure.

It should be noted however, that in this study Bragg gratings have only been tested up to 7 MPa. For higher pressures (deeper marine applications) the connectors and FBG response must be verified.

Conclusion

The results from an external pressure test in a hyperbaric chamber indicate that embedded optical fiber instrumentation allows real time material monitoring within the structure. The presence of the instrumentation has no influence on failure mode nor on implosion pressure, at least for the cylinder conditions studied here. In-situ fiber optic instrumentation is a valuable tool to provide a complete record of the evolution of cylinder response, in terms of strain and temperature, from fabrication throughout the service life. The residual strains which appear during manufacture are not negligible (over 1100 axial micro-strain for the tube tested here) and may significantly affect subsequent behaviour during service. If reliable predictive models are to be developed for deep sea pressure vessel applications these strains must be included in failure criteria, and Bragg grating instrumentation provide the means to quantify them. They may also enable manufacturers to evaluate strategies to reduce these residual strains, by optimizing cure cycles.

Figures

Cylinder failure modes

Figure 1 .Figure 2 .Figure 3

 123 Figure 1. Implosion pressure versus thickness to mean diameter (t/d) ratio, showing transition between buckling and material failure modes.

Figure 4 .

 4 Figure 4. Instrumented cylinder with a) Strain and temperature measurement points, b) Complete cylinder and closure system before external pressure testing in hyperbaric chamber.

Figure 5 .

 5 Figure 5. 1000 bar hyperbaric pressure testing chamber.

Figure 6 .

 6 Figure 6. Axial strain from Bragg grating and pressure vs. time.

Figure 7 .

 7 Figure 7. Circumferential strain from Bragg grating and pressure vs. time.

Figure 9 .

 9 Figure 9. Circumferential FBG spectra during pressure test cycles.

Figure 10 .

 10 Figure 10. Bragg grating strain response and strain gage placed in axial direction.

Figure 11 .

 11 Figure 11. Bragg grating and strain gage response in circumferential direction.

Figure 12 .

 12 Figure 12. Axial pressure-strain plots.

Figure 13 .

 13 Figure 13. Circumferential pressure-strain plots.

Table 1 .

 1 Bragg grating temperature response and metrological characteristics.

	Tables
						-6 /°C)	sensitivity (nm/°C)
	axial	1535.155	0.0103	1534.974	6.7	0.0103
	circumferential	1550.055	0.0105	1549.900	6.7	0.0105
	Table 2. Results from external pressure tests
	Cylinder		Implosion pressure (MPa)	Buckling mode
	Non-instrumented [19]
			6.0		2
			6.8		3
			6.5		3
			6.4		3
			6.7		3
	FBG Instrumented (No. 37)	6.7		n=3

Cylinder directions for OFBG placement

Nominal wavelength (nm) m  (nm/°C)  b  (nm)  a (10

Acknowledgments

H. Hernández Moreno wishes to thank the National Council of Science and Technology of Mexico (CONACYT) and the National Polytechnic Institute of Mexico (IPN) for their scholarship sponsoring. The authors thank I. Fernandez Hernandez, J. Bauw, F. Afonso, and E. Vargas Rojas for their collaboration during their internship at LGMT. Also thanks to M. Mulle at LGMT, for his collaboration during the instrumented implosion test, and to IFREMER for financial and technical support, with special thanks to P. Warnier and A. Duff.