CONTINUOUS FIRST ORDER LOGIC FOR UNBOUNDED METRIC STRUCTURES ITA Ï BEN YAACOV

Keywords: 2000 Mathematics Subject Classification. 03C35, 03C90, 03C95 unbounded metric structure, continuous logic, emboundment

We present an adaptation of continuous first order logic to unbounded metric structures. This has the advantage of being closer in spirit to C. Ward Henson's logic for Banach space structures than the unit ball approach (which has been the common approach so far to Banach space structures in continuous logic), as well as of applying in situations where the unit ball approach does not apply (i.e., when the unit ball is not a definable set).

We also introduce the process of single point emboundment (closely related to the topological single point compactification), allowing to bring unbounded structures back into the setting of bounded continuous first order logic.

Together with results from [Benc] regarding perturbations of bounded metric structures, we prove a Ryll-Nardzewski style characterisation of theories of Banach spaces which are separably categorical up to small perturbation of the norm. This last result is motivated by an unpublished result of Henson.

Introduction

Continuous first order logic is an extension of classical first order logic, introduced in [BU] as a model theoretic formalism for metric structures. It is convenient to consider that continuous logic also extends C. Ward Henson's logic for Banach space structures (see for example [START_REF] Henson | Ultraproducts in analysis, Analysis and Logic[END_REF]), even though this statement is obviously false: continuous first order logic deals exclusively with bounded metric structures, immediately excluding Banach spaces from the picture. This is a technical hurdle which is relatively easy to overcome. What one usually does (e.g., in [START_REF] Ben | Continuous first order logic and local stability[END_REF]Example 4.5] and the discussion that follows it) is decompose a Banach space into a multi-sorted structure, with one sort for, say, each closed ball of radius n ∈ N. One may further rescale all such sorts into the sort of the unit ball, which therefore suffices as a single sorted structure. The passage between Banach space structures in Henson's logic and unit ball structures in continuous logic preserves such notions as elementary classes, elementary extensions, type-definability of subsets of the unit ball, etc. This approach has allowed so far to translate almost every model theoretic question regarding Banach space structures to continuous logic.

The unit ball approach suffers nonetheless from several drawbacks. One drawback, which served as our original motivation, comes to light in the context of perturbations of metric structures introduced in [Benc]. Specifically, we wish to consider the notion of perturbation of the norm of a Banach space arising from the Banach-Mazur distance. However, any linear isomorphism of Banach spaces which respects the unit ball is necessarily isometric, precluding any possibility of a non trivial Banach-Mazur perturbation. Another drawback of the unit ball approach, also remedied by the tools introduced in the present paper, is that in some unbounded metric structures the unit ball is not a definable set (even though it is always type-definable), so naming it as a sort (and quantifying over it) adds undesired structure. For example, this is the case with complete normed fields (i.e., of fields equipped with a complete non trivial multiplicative valuation in R), considered in detail in [Benb].

In the present paper we replace the unit ball approach with the formalism of unbounded continuous first order logic, directly applicable to unbounded metric structures and in particular to Banach space structures. Using some technical definitions introduced in Section 1, the syntax and semantics of unbounded logic are defined in Section 2. In Section 3 we prove Loś's Theorem for unbounded logic, and deduce from it a Compactness Theorem inside bounded sets. It follows that the type space of an unbounded theory is locally compact. In Section 4 we show that unbounded continuous first order logic has the same expressive power as Henson's logic of positive bounded formulae.

In order to be able to apply to unbounded structures tools which are already developed in the context of standard (i.e., bounded) continuous logic, we introduce in Section 5 the process of emboundment. Trough the addition of a single point at infinity, to each unbounded metric structure we associate a bounded one, to which established tools apply. This method is used in Section 6 to adapt the framework of perturbations, developed in [Benc] for bounded structures, to unbounded ones. In particular, Theorem 6.9 asserts that the Ryll-Nardzewski style characterisation of ℵ 0 -categoricity up to perturbation [START_REF]On perturbations of continuous structures[END_REF]Theorem 3.5] holds for unbounded metric structures as well.

As an application, we prove in Section 7 a Ryll-Nardzewski style characterisation of theories of Banach spaces which are ℵ 0 -categorical up to arbitrarily small perturbation of the norm. This result is motivated by an unpublished result of Henson, whom we thank for the permission to include it in the present paper.

Notation is mostly standard. We use a, b, c, . . . to denote members of structures, and use x, y, z, . . . to denote variables. Bar notation is used for (usually finite) tuples, and uppercase letters are used for sets. We also write ā ∈ A to say that ā is a tuple consisting of members of A, i.e., ā ∈ A n where n = |ā|. When T is an L-theory (whether bounded or unbounded) we always assume that T is closed under logical consequences. In particular, |T | = |L| + ℵ 0 and T is countable if and only if L is. We shall assume familiarity with (bounded) continuous first order logic, as developed in [BU]. For the parts dealing with perturbations, familiarity with [Benc] is assumed as well. For a general survey of the model theory of metric structures we refer the reader to [START_REF] Ben Yaacov | Model theory for metric structures, Model theory with Applications to Algebra and Analysis[END_REF].

Gauged spaces

We would like to allow unbounded structures, while at the same time keeping some control over the behaviour of bounded parts thereof. The "bounded parts" of a structure are given by means of a gauge.

Definition 1.1. Let (X, d) be a metric space, ν : X → R any function. We define X ν≤r = {x ∈ X : ν(x) ≤ r} and similarly X ν≥r , X ν<r , etc.

(i) We call X ν≤r and X ν<r the closed and open ν-balls of radius r in X, respectively.

(ii) We say that ν is a gauge on (X, d), and call the triplet (X, d, ν) a (ν-)gauged space if ν is 1-Lipschitz in d and every ν-ball (of finite radius) is bounded in d. Note that this implies that the bounded subsets of (X, d) are precisely those contained in some ν-ball.

Remark 1.2. We could have given a somewhat more general definition, replacing the 1-Lipschitz condition with the weaker condition that the gauge ν should be bounded and uniformly continuous on every bounded set. This does not cause any real loss of generality, since in that case we could define

d ′ (x, y) = d(x, y) + |ν(x) -ν(y)|.
Then ν is 1-Lipschitz with respect to d ′ , and the two metrics d and d ′ are uniformly equivalent and induce the same notion of a bounded set.

Definition 1.3. Recall that a (uniform) continuity modulus is a left-continuous increasing function δ

: (0, ∞) → (0, ∞) (i.e., δ(ε) = sup ε<ε ′ δ(ε ′ )).
We say that a mapping f : (X, d X , ν X ) → (Y, d Y , ν Y ) between two gauged spaces respects δ under ν if for all ε > 0:

ν X (x), ν X (y) < 1 ε , d X (x, y) < δ(ε) =⇒ d Y (f (x), f (y)) ≤ ε, ν Y (x) ≤ 1 δ(ε) . (UC ν )
We say that f is uniformly continuous under ν if it respects some δ under ν.

While respecting a given δ under ν depends on the choice of ν, the fact that some δ is respected under ν does not.

Lemma 1.4. Let X and Y be gauged spaces, f : X → Y a mapping. Then (i) Let δ : (0, ∞) → (0, ∞) be any mapping, and assume that f respects δ under ν in the sense of (UC ν ). Define δ ′ (ε) = ε ∧ sup 0<ε ′ <ε δ(ε ′ ). Then δ ′ ≤ id is a continuity modulus and f respects δ ′ under ν as well. (If we used sup alone we could obtain infinite values, whence the need for truncation at ε.)

(ii) A mapping f : X → Y between gauged spaces is uniformly continuous under ν if and only if it restriction to every bounded set is uniformly continuous and bounded.

Proof. Easy.

1.4

Definition 1.5. A Cartesian product of gauged metric spaces X = i<n X i is equipped with a gauged metric structure as follows:

d(x, ȳ) = i<n d(x i , y i ), ν(x) = i<n ν(x i ). (1)
In particular, if n = 0 then X = { * } and d( * , * ) = ν( * ) = 0.

We also identify R + with the gauged space (R + , |x -y|, |x|).

Lemma 1.6. Let X, Y , and so on, denote gauged spaces.

(i) The projection mapping X × Y → X respects the identity uniformly under ν.

(ii) Let f i : X → Y i , i < n, be mappings between gauged spaces, each respecting δ f i under ν. Then f : X → Y i respects the continuity modulus δ f = i<n δ f i under ν.
In addition, if δ f i ≤ id for all (indeed, for some) i < n then δ f ≤ id as well. (iii) Let X, Y and Z be gauged spaces. Assume that f : X → Y and g : Y → Z respect continuity moduli δ f and δ g , respectively, under ν. Assume moreover that δ f , δ g ≤ id. Then h = g • f : X → Z respects the continuity modulus

δ h = δ f • δ g • δ f under ν.
In particular, δ h ≤ id is a continuity modulus. (iv) Let X and Y be gauged spaces, and let f : X × Y → R + and g : Y → R + mappings which respect δ f and δ g under ν, respectively. Assume also that f is eventually equal to g, namely that there exists a constant C such that f (x, y) = g(y) whenever ν(x) ≥ C. Define

h 1 (y) = sup x∈X f (x, y), h ′ 1 (y) = g(y) ∨ sup x∈X f (x, y), h 2 (y) = inf x∈X f (x, y), h ′ 2 (y) = g(y) ∧ inf x∈X f (x, y), δ h (ε) = δ g (ε) ∧ δ f (ε ∧ 1 C ).
Then h i , h ′ i : Y → R + are well defined (i.e., the supremum is always finite) and respect δ h under ν. Moreover, if either δ f ≤ id or δ g ≤ id then δ h ≤ id.

Proof. The first two items are easy.

For the third item we only prove that δ h is respected under ν. Indeed, let ε > 0, x, y ∈ X, and assume that ν(x), ν(y) < 1 ε and d(x, y) < δ h (ε). By the left continuity assumption there are s, t such that: d(x, y) < δ f (s), s < δ g (t), t < δ f (ε). In particular s < t < ε. Using our hypotheses we obtain from top to bottom: ε) , as desired. For the fourth item, the existence of h 1 and h ′ 1 follows from the fact that for a fixed y, the function x → f (x, y) is bounded on bounded sets and eventually constant. We show that h 1 respects δ h under ν, a similar argument applies to the other functions. Let y 1 , y 2 ∈ Y , and assume that ν(y

d(x, y) < δ f (s), ν(x), ν(y) < 1 ε < 1 s , d f (x), f (y) ≤ s < δ g (t), ν • f (x), ν • f (y) ≤ 1 δ f (ε) < 1 t , d h(x), h(y) ≤ t < ε, ν • h(x), ν • h(y) ≤ 1 δg(t) < 1 s . In addition, s could have been chosen arbitrarily close to δ g • δ f (ε) whereby ν • h(x) ≤ 1 δg•δ f (ε) ≤ 1 δ f •δg •δ f (ε) = 1 δ h (
i ) < 1 ε , d(y 1 , y 2 ) < δ h (ε). Let r = ε ∧ 1 C , so ν(y i ) < 1 r and d(y 1 , y 2 ) < δ g (ε) ∧ δ f (r).
We may choose a point x ∈ X such that f (x, y 1 ) is arbitrarily close to h 1 (y 1 ). There are two cases to consider:

I. ν(x) ≥ C f (x, y 1 ) = g(y 1 ) ≤ 1 δg(ε) ≤ 1 δ h (ε) , |f (x, y 1 ) -f (x, y 2 )| = |g(y 1 ) -g(y 2 )| ≤ ε, II. ν(x) < C ≤ 1 r f (x, y 1 ) ≤ 1 δ f (r) ≤ 1 δ h (ε) , |f (x, y 1 ) -f (x, y 2 )| ≤ r ≤ ε.
Either way we obtain that h 1 (y 1 ) ≤ 1 δ h (ε) and that h 1 (y 1 ) ≤ h 1 (y 2 ) + ε, which is enough. If there exists x ∈ X such that ν(x) ≥ C then h 1 = h ′ 1 . If not then when dealing with h ′ 1 we need to consider the possibility that h ′ 1 (y 1 ) = g(y 1 ), which is treated identically to case I. The functions h 2 and h ′ 2 are treated analogously.

1.6

Unbounded continuous logic

We turn to define a R + -valued variant of continuous logic which can accommodate unbounded metric structures. We shall refer to this logic as unbounded continuous logic. The [0, 1]-valued (or, more generally, bounded) continuous logic defined in [BU] will be referred to here as standard or bounded.

Definition 2.1. An unbounded continuous signature L consists of the following data:

(i) A set of relation (or predicate) symbols and of function symbols, each equipped with its arity (zero-ary function symbols are also called constant symbols). (ii) For each n-ary symbol s, a continuity modulus δ s : (0, ∞) → (0, ∞). (iii) For each sort S, a distinguished binary predicate symbol d S called the distance symbol, as well as a distinguished unary predicate symbol ν S called the gauge symbol. The subscript S is usually omitted. We usually write down a signature merely by listing its non distinguished symbols.

Definition 2.2. Let L be an unbounded signature, and for the sake of simplicity let us assume it is single-sorted. An (unbounded) L-structure is a complete metric gauged space (M, d, ν) = (M, d M , ν M ), possibly empty, equipped with interpretation of the symbols:

(i) The interpretation of an n-ary function symbol f is a mapping f M : M n → M which respects δ f under ν. (ii) The interpretation of an n-ary predicate symbol P is a mapping P M : M n → R + which respects δ P under ν. For this purpose we view M n with a gauged space (M n , d, ν) as per Definition 1.5. Similarly, R + admits a standard gauge structure (R + , d, id).

Thus, restricted to a ν-ball, everything is bounded and uniformly continuous as in bounded continuous logic, and closed ν-balls are metrically closed and therefore complete.

Remark 2.3. If the language contains a constant symbol 0 then the formula ν ′ (x) = d(x, 0) can act as an alternative gauge. Indeed, if r ∈ R + then M ν ′ ≤r ⊆ M ν≤r+ν(0) , since ν is 1-Lipschitz, and conversely M ν≤r ⊆ M ν ′ ≤β d (r,ν(0)) by definition of an unbounded structure. Thus we can pass between ν-balls and ν ′ -balls in a way which depends only on L.

In most cases, ν will indeed be equal to d(x, 0).

A standard continuity modulus for an n-ary symbol, when n > 0, is the function x → x n . If a symbol s is 1-Lipschitz is each argument and ν(x) = d(x, 0) then s indeed respects the standard continuity modulus under ν. For a zero-ary symbol the standard continuity modulus is the identity.

Example 2.4. Let L be a standard (i.e., [0, 1]-valued) continuous signature as defined in [BU]. In that case we chose to equip each n-ary symbol s with individual continuity moduli δ s,i , i < n, one for each argument. Let L ′ be the unbounded signature obtained from L by adding a gauge symbol ν S for each sort S, and by setting δ s (x) = 1 ∧ i δ s,i ( x n ) (for zero-ary s let δ s = 1). Then every L-structure M can be naturally viewed as an unbounded L ′ -structure by interpreting all gauges as the constant 0. If L admits a constant symbol 0 then interpreting ν(x) = d(x, 0) works as well.

Example 2.5 (Banach spaces). We would like to view Banach spaces as unbounded structures. Let L = {0, +, m r : r ∈ Q}, where m r is unary scalar multiplication by r. We view x as shorthand for d(x, 0), and take it to be the gauge. Let δ mr (x) = |r|x and let all other continuity moduli be standard. Then every real Banach space is naturally an (unbounded) L-structure.

This can be extended to additional structure on the Banach space. For example a complex Banach space also has a function symbol for multiplication by i, while a Banach lattice is given by binary function symbols ∨, ∧ (again with standard continuity moduli).

Example 2.6 (Naming constants). Let L be an unbounded signature, M and L-structure. Let A ⊆ M. We define L(A) as L ∪ A, where each a ∈ A is viewed as a new constant symbol. We equip each symbol a with the uniform continuity modulus δ a = id ∧ 1 ν(a) .

Then for every ε > 0 we have ν(a) ≤ 1 δa(ε) , and we may render M an L(A)-structure by interpreting a M = a.

We now define the syntax of continuous logic. A term is defined, as usual, as either being a variable or a composition of a function symbol with simpler terms. Similarly, an atomic formula is a composition of a predicate symbol with terms. Connectives are continuous functions from (R + ) n to R + , or any convenient family of such functions which is dense in the compact-open topology, i.e., in the topology of uniform convergence on every compact set. We shall use the system {1, x -. y, x + y, x/2} which generates such a dense set through composition. While alternative systems may be legitimate, we shall always require the presence of 1 and -. in what follows. We point out that as functions from (R + ) n → R + , all the basic connectives we chose respect their respective standard continuity moduli (according to their arity). As one may expect, every combination of formulae by connectives is a formula.

On the other hand, care is needed when defining quantified formulae. First, sup x ϕ could be infinite. Second, even if ϕ is bounded, we still need a uniform rate of convergence for sup ν(x)<C ϕ → sup x ϕ as C → ∞, or else we may run into trouble with compactness as well as with uniform continuity under ν. In Henson's logic of positive bounded formulae [START_REF] Henson | Ultraproducts in analysis, Analysis and Logic[END_REF], where the truth values are True/False, one gets around this by restricting quantifiers to bounded balls (and then again, one needs to play around with the radii of the balls when considering approximations). If we tried to do the same thing with continuous quantifiers we could again run into trouble if, say, sup ν(x)<r ϕ < sup ν(x)≤r ϕ. We shall follow a different path, looking for the simplest syntactic conditions on a formula ϕ that ensure that inf x ϕ and sup x ϕ are semantically legitimate. This approach will allow us nonetheless to recover approximate versions of bounded quantifiers later on.

Definition 2.7. We define formulae by induction, and at the same time we define whether a formula is syntactically eventually constant in a variable x and/or bounded.

• Atomic formulae are defined as above.

-If ϕ is atomic and x does not appear in ϕ, then ϕ is eventually constant in x.

-No atomic formula is bounded. • A combination of formulae by connectives is a formula.

-If all the components are bounded (respectively, eventually constant in x) then so is the compound formula.

-If ϕ is bounded then ϕ -. ψ is bounded for any ψ and ϕ -. ν(x) is eventually constant in x. • If ϕ is eventually constant in x then inf x ϕ and sup x ϕ are formulae (but not otherwise).

-If ϕ is bounded (respectively, eventually constant in a variable y) then so are sup x ϕ and inf x ϕ. In particular, sup x ϕ and inf x ϕ are eventually constant in x.

Notice that the formula 1, being a combination of no formulae, is bounded and eventually constant in every variable. Similarly, every dyadic number r

= k 2 m = ( 1 2 ) m (1+• • •+1)
can be viewed as a formula, and is syntactically bounded and constant as such. It follows that for every formula ϕ, the formula ϕ ∧ r = r -. (r -. ϕ) is syntactically bounded.

The qualitative syntactic properties of boundedness and eventual constancy can be translated to quantitative information.

Definition 2.8. For every syntactically bounded formula ϕ we extract a syntactic bound B ϕ as follows:

ϕ = θ( ψ), ψ i bounded: B ϕ = sup x∈ Q [0,B ψ i ] θ(x), ϕ = ψ -. χ, or sup x ψ, or inf x ψ, ψ bounded: B ϕ = B ψ .
Notice that no ambiguity arises for ψ -. χ when both ψ and χ are syntactically bounded.

Similarly, for a formula ϕ(x, ȳ) which is syntactically eventually constant in x we extract a syntactic constancy threshold C ϕ,x ∈ R + and a formula ϕ(∞, ȳ), whose free variables lie among ȳ, and which is intended to agree with ϕ(x, ȳ) once ν(x) ≥ C ϕ,x .

x not free in ϕ:

ϕ(∞, ȳ) = ϕ, C ϕ,x = 0, ϕ = θ( ψ), ψ i e.c. in x: ϕ(∞, ȳ) = θ ψ(∞, ȳ) , C ϕ,x = C ψ i ,x , ϕ = ψ -. ν(x), ψ bounded: ϕ(∞, ȳ) = 0, C ϕ,x = B ψ , ϕ = sup z ψ, ψ e.c. in x = z: ϕ(∞, ȳ) = sup z ψ(∞, z, ȳ), C ϕ,x = C ψ,x .
Again, when cases overlap the definitions agree.

The definition of the semantics can be somewhat delicate. The model for the definition is an unbounded structure M in which elements of arbitrarily high gauge exist (e.g., a non trivial Banach space). In this case the definition is entirely straightforward, namely

(f τ ) M (ā) = f M • τ M (ā) ∈ M, (P τ ) M (ā) = P M • τ M (ā) ∈ R + , θ( φ) M (ā) = θ φM (ā) (where θ is a connective), inf x ϕ(x, ā) M = inf b∈M ϕ M (b, ā), idem for sup . (Q)
Let us state some properties of this model situation, for the time being without proof. First, the interpretation of every term and formula is uniformly continuous under ν (essentially by Lemma 1.6). Second, if ϕ is syntactically bounded then it is bounded by

B ϕ . Third, if ϕ(x, ȳ) is syntactically eventually constant in x then ϕ(x, ȳ) = ϕ(∞, ȳ) whenever ν(x) ≥ C ϕ,x .
In this case, ϕ(x, ā) is bounded for every ā, so the interpretation of the quantifiers makes sense and the following holds:

inf x ϕ(x, ā) M = inf b∈M ∪{∞} ϕ M (b, ā), idem for sup . (Q ∞ )
However, we must also take into account structures in which elements of arbitrarily high gauge need not exist. In order for ultra-products to behave reasonably, i.e., in order for Loś's Theorem to hold, the definition of quantifier semantics in the general case must follow (Q ∞ ) and not (Q). This is illustrated in Remark 3.2 below. Definition 2.9. Let M be an L structure. Then terms, atomic formulae and connectives are interpreted naturally, by composition. Quantifiers are interpreted according to (Q ∞ ), where ϕ(∞, ā) is understood as per Definition 2.8.

Theorem 2.10. Let M be an L-structure. Then:

(i) All formulae are interpreted as R + -valued functions on Cartesian powers of M.

In particular, in the interpretation of quantified formulae in M all the suprema are finite. (ii) Every term τ and every formula ϕ are uniformly continuous under ν.

(iii) If a formula ϕ is syntactically constant then ϕ M (ā) ≤ B ϕ for all ā ∈ M. (iv) If a formula ϕ(x, ȳ) is syntactically eventually constant in x then ϕ(b, ā) = ϕ(∞, ā) whenever ν(b) ≥ C ϕ,x .
Proof. We prove this by induction on the complexity of terms and formulae. We observe that if ϕ(x, ȳ) is syntactically eventually constant in x then ϕ(∞, ȳ) is of lesser or equal complexity. Thus, when treating sup x ϕ and inf x ϕ, we may use the induction hypotheses both for ϕ(x, ȳ) and for ϕ(∞, ȳ). We may assume that all the continuity moduli of symbols lie below the identity, and construct as we go continuity moduli below the identity for each term and formula. The induction step itself now follows immediately from the definitions, the induction hypotheses and Lemma 1.6.

2.10

We leave it as an exercise to the reader to check that with our choice of connectives, every formula is equivalent to one in prenex normal form (one needs to make sure in particular that the natural transformations towards a prenex form do not violate the restrictions on quantification imposed in Definition 2.7).

It will be convenient later on to have some analogue of the restricted quantifier sup ν(x)≤r ϕ (which is not part of our language). Let us assume that ϕ is syntactically bounded and let k = ⌈B ϕ ⌉, namely the least integer syntactic bound for ϕ. We observe that for a dyadic r, the formula ϕ -. (ν(x) -. r) is equivalent to (ϕ + (ν(x) ∧ r)) -. ν(x) which is syntactically bounded and eventually constant in x. It follows that for every natural m > 0 the formula ϕ -. m(ν(x) -. r) is equivalent to one which is syntactically bounded and eventually constant in x. Let 0 < r < r ′ , and find the least m such that we can write r ≤ s = ℓ2 -m < (ℓ + 1)2 -m ≤ r ′ , and choose the least possible s. Define:

ϕ↓ x≤r,r ′ = ϕ -. k2 m (ν(x) -. s), sup x r,r ′ ϕ = sup x ϕ↓ x≤r,r ′ , ϕ↑ x≤r,r ′ = k -. (k -. ϕ)↓ x≤r,r ′ , inf x r,r ′ ϕ = inf x ϕ↑ x≤r,r ′ .
Both formulae on the left are syntactically bounded and eventually constant in x, so the expressions on the right are indeed formulae. By construction we always have ϕ↓ x≤r,r ′ ≤ ϕ, and in addition ϕ↓ x≤r,r ′ = ϕ when ν(x) ≤ r and ϕ↓ x≤r,r ′ = 0 when ν(x) ≥ r ′ . Thus sup ν(x)≤r ϕ ≤ sup r,r ′ x ϕ ≤ sup ν(x)<r ′ ϕ. Similarly, inf ν(x)≤r ϕ ≥ inf r,r ′ x ϕ ≥ inf ν(x)<r ′ ϕ. We may further extend these abbreviations to the case where ϕ is not syntactically bounded by truncating it at 1, defining ϕ↓ x≤r,r ′ = (ϕ∧1)↓ x≤r,r ′ (and proceeding as above). This will only be used in conditions of the form sup r,r ′

x inf s,s ′ y . . . ϕ = 0, whose satisfaction does not depend on our particular choice of constant at which we truncate.

Loś's Theorem, compactness and theories

Let L be an unbounded signature, {M i : i ∈ I} a family of L-structures and U an ultra-filter on I.

Let I 0 = {i : M i = ∅}. If I 0 ∈ U define N 0 = (a i ) ∈ i∈I 0 M i : lim U ν M i (a i ) < ∞ ,
otherwise N 0 = ∅. Alternatively, one may introduce a new formal element ∞ with ν(∞) = +∞, and define

N 0 = (a i ) ∈ i∈I M i ∪ {∞} : lim U ν M i (a i ) < ∞ .
Under this definition a member (a i ) ∈ N 0 can have few (according to U ) coordinates which are equal to ∞ and which may be ignored in the definitions that follow. Either approach leads to the same construction.

For a function symbol f or predicate symbol P , and arguments (a i ), (b i ), . . . ∈ N 0 , define:

f N 0 (a i ), (b i ), . . . = (f M i (a i , b i , . . .)), P N 0 (a i ), (b i ), . . . = lim U P M i (a i , b i , . . .).
Note that by definition of N 0 , the values of P M i (a i , b i , . . .) are bounded on a large set of indexes, so lim U P M i (a i , b i , . . .) ∈ R + . It is now straightforward verification that N 0 is an L-pre-structure, i.e., that it verifies all the properties of a structure with the exception that d N 0 might be a pseudo-metric and needs not be complete. Let N = N 0 be the associated L-structure, obtained by dividing by the zero distance equivalence relation and passing to the metric completion. We call N the ultra-product of

{M i : i ∈ I} modulo U , denoted M i /U . The image in N of (a i ) ∈ N 0 will be denoted [a i ].
(Compare with the construction of ultra-products of Banach spaces in [START_REF] Henson | Ultraproducts in analysis, Analysis and Logic[END_REF] and of bounded continuous structures in [BU].) Theorem 3.1 ( Loś's Theorem). For every formula ϕ(x) and

[a i ], [b i ], . . . ∈ M i /U : ϕ([a i ], [b i ], . . .) Q M i /U = lim U ϕ(a i , b i , . . .) M i .
Proof. Mostly as for bounded logic. The only significant difference is in the treatment of quantifiers, which we sketch below.

If lim U inf x ϕ(x, a i , . . .) M i < r then there is a large set on which inf x ϕ(x, a i , . . .) M i < r and we can find witnesses b i there (possibly the formal infinity) such that ϕ(b i , a i , . . .

) M i < r. If ν(b i ) ≤ C ϕ,x on a large set then ν([b i ]) ≤ C ϕ,x , so in particular [b i ] belongs to the ultra-product and inf x ϕ(x, [a i ], . . .) ≤ ϕ([b i ], [a i ], . . .) = lim U ϕ(b i , a i , . . .) ≤ r. If, on the other hand, b i = ∞ or ν(b i ) ≥ C ϕ,x on a large set then inf x ϕ(x, [a i ], . . .) ≤ ϕ(∞, [a i ], . . .) = lim U ϕ(∞, a i , . . .) = lim U ϕ(b i , a i , . . .) ≤ r.
Conversely, assume that inf x ϕ(x, [a i ], . . .) < r. Then again, either there is

[b i ] such that ϕ([b i ], [a i ], . . .) < r or ϕ(∞, [a i ],
. . .) < r, and in either case lim U inf x ϕ(x, a i , . . .) ≤ r.

3.1

Remark 3.2. Loś's Theorem might fail if our semantic interpretation did not take the value at infinity into account. For example, consider the sentence ϕ = inf x (1 -. ν(x)). Let M n be the structure consisting of two points, ν(a n,0 ) = 0, ν(a n,1 ) = n. Then the ultra-product contains a single point a 0 = [a n , 0], ν(a 0 ) = 0, and we would have ϕ Mn = 0 for all n ≥ 1 and yet ϕ Q Mn/U = 1. Worse still, if M n consisted only of a n,1 then M i /U would be empty, making the naïve interpretation of quantifiers meaningless. An empty ultra-product can also be obtained with unbounded structures, for example M n = E B(n) where E is a Banach space and B(n) is its open ball of radius n. (These and other pathological examples were pointed out to the originally over-optimistic author by C. Ward Henson.) Definition 3.3. Say that a family of conditions Σ = {ϕ i ≤ r i : i ∈ λ} is approximately finitely satisfiable if for every finite w ⊆ λ and ε > 0, the family Σ 0 = {ϕ i ≤ r i +ε : i ∈ w} is satisfiable.

Corollary 3.4. If a set of sentential conditions (i.e., conditions without free variables) is approximately finitely satisfied in a family of structures, then it is satisfied in some ultra-product of these structures.

Proof. Standard.

3.4

Corollary 3.5 (Bounded compactness for unbounded continuous logic). Let L be an unbounded signature, r ∈ R + , and let Σ be a family of conditions in the free variables x <n . Then Σ ∪ {ν(x i ) ≤ r : i < n} is satisfiable of and only if it is approximately finitely satisfiable.

As usual, a theory is a set of sentential conditions. The complete theory of a structure M, elementary equivalence and elementary embeddings are defined as usual.

Corollary 3.6. Two structures M and N are elementarily equivalent if and only if M embeds elementarily into an ultra-power of N.

Proof. One direction is clear. For the other we observe that if M and N are elementarily equivalent, then the elementary diagram of M is approximately finitely satisfiable in N. Indeed, let ā ∈ M and say that ϕ(ā) = 0. Let also ε > 0 and r = ν(ā). Then N inf r,r+ε x ϕ(x) = 0, so there are b ∈ N such that ν( b) < r + ε and ϕ( b) < ε.

3.6

We could prove an analogue of the Shelah-Keisler theorem that if N and M are elementarily equivalent then they have isomorphic ultra-powers. We give a more elementary proof of a lesser result, which will suffice just as well later on.

Lemma 3.7.

(i) Two models M and N are elementarily equivalent if and only if there are sequences

M = M 0 M 1 . . . and N = N 0 N 1 . . . where each M n+1 (N n+1
) is an ultra-power of M n (N n ) and n∈N M n ≃ n∈N N n (so their completions are isomorphic as well).

(ii) A class of structures K is elementary if and only if it is closed under elementary equivalence and ultra-products.

Proof. For the first item, right to left by the elementary chain lemma, which is proved as usual. For left to right, assume that M ≡ N. Then there is an ultra-power N 1 = N U and an elementary embedding

f 0 : M → N 1 . Then (M, M) ≡ (N 1 , f 0 (M)
) (in a language with all elements of M named) so there exists an ultra-power M 1 = M U ′ and an elementary embedding g 0 : N 1 → M 1 such that g 0 • f 0 = id M . Proceed in this manner to obtain the sequences.

The second item is standard.

3.7

It is easily verified that any theory is logically equivalent to one which only consists of conditions of the form ϕ = 0. A universal theory is one which only consists of conditions of the form sup x ϕ(x) = 0 where ϕ is quantifier-free (and syntactically bounded and eventually constant in each x i ). Observe that:

• For any formula ϕ we can express ∀x ϕ(x) = 0 by the universal axiom scheme sup n,n+1

x ϕ(x) = 0. • If t and s are terms we can express ∀x t = s by ∀x d(t, s) = 0.

• If ϕ and ψ are formulae we can express ∀x ϕ ≥ ψ by ∀x ψ -. ϕ = 0.

Example 3.8. We can continue Example 2.5 and give the (universal) theory of the class of Banach spaces:

Universal equational axioms of a vector space.

∀x s x ≤ m r (x) ≤ s ′ x s, s ′ dyadic, s ≤ |r| ≤ s ′ ∀xy x + y ≤ x + y ∀xy d(x, y) = x + m -1 (y) .
More generally, it will be convenient to write Notice that in the ∀∃ notation, the universal quantifier holds literally, while the existential quantifiers holds in an approximate sense, with respect to the quantification radius as well as with respect to the value of ϕ (which may both be slightly bigger than s or 0, respectively.)

Example 3.9 (Measure algebras). Let L = {0, ∨, ∧, }, where 0 is a constant symbol, ∨, ∧, are binary function symbols. We use µ(x) as shorthand for d(x, 0), and take it to be the gauge. All the continuity moduli are standard.

The universal theory of measure algebras (which are the topic of [START_REF] Fremlin | Measure Theory Volume 3: Measure Algebras, Torres Fremlin[END_REF]) consists of:

Universal equational axioms of relatively complemented distributive lattices ,

∀xy µ(x) + µ(y) = µ(x ∧ y) + µ(x ∨ y), µ(0) = 0, ∀xy d(x, y) = µ(x y) + µ(y x).
We can further say that a measure algebra is atomless by the axiom scheme:

∀ <n x ∃ ≤n y |µ(x ∧ y) -µ(x)/2| = 0, n ∈ N.
Example 3.10 (Replacing a function with its graph). Let L be an unbounded signature, f ∈ L an n-ary function symbol. We define its graph to be the (n + 1)-ary predicate G f (x, y) = d(f (x), y). Since it is defined by a formula it respects a continuity modulus under ν uniformly in all L-structures, and we may add it to the language. The axiom scheme ∀xy G f (x, y) = d(f (x), y) is universal. We may further drop f from the language. Indeed, we observe that a predicate G f is the graph of a function f with continuity modulus δ f if and only if the following theory holds. The second axiom ensures that in the third axiom there actually exists a unique y = f (x) such that G f (x, y) = 0. Then the first two axioms imply that G f is the graph of f , and the two last axioms together ensure that f respects δ f under ν.

∀x, y, z G f (x, y) ≤ G f (x, z) + d(y, z) ∀x, y, z d(y, z) ≤ G f (x, y) + G f (x, z) ∀ <ε -1 x ∃ ≤δ f (ε) -1 y G f (x, y) = 0 ε > 0 ∀ <ε -1 xȳ ∀ <δ f (ε) -1 +1 z δ f (ε) -. d(x, ȳ) ∧ G f (x, z) -. G f (ȳ, z) -. ε = 0 ε > 0
Types and type spaces are defined more or less as usual:

Definition 3.11. Fix an unbounded signature L.

(i) Given an n-tuple ā, we define its type p(x) = tp(ā) as usual as the set of all L-conditions in the variables x <n satisfied by ā. The type p(x) determines the value of ϕ(ā) for every formula ϕ, and we may write ϕ p = ϕ(x) p(x) = ϕ(ā). (ii) A complete n-type (in L) is the type of some n-tuple. By Corollary 3.5, this is the same as a maximal finitely consistent set of conditions p(x <n ) such that for some r ≥ 0 we have ν(x i ) ≤ r ∈ p for all i < n. (iii) The set of all n-types is denoted S n . The set of all n-types containing a theory T (equivalently: realised in models of T ) is denoted S n (T ). (iv) For every condition s in the free variables x <n , [s] Sn(T ) (or just [s], if the ambient type space is clear from the context) denotes the set of types {p ∈ S n (T ) : s ∈ p}. (v) The family of all sets of the form [s] S n (T ) forms a base of closed sets for the logic topology on S n (T ). It is easily verified to be Hausdorff.

For each n ∈ N, we can define ν : S n (T ) → R by ν(p) = i<n ν(x i ) p . With this definition, (S n (T ), d, ν) is a gauged space. Applying previous definitions we have:

S ν≤r n (T ) = i<n ν(x i ) ≤ r = i<n ν(x i ) ≤ r .
By Corollary 3.5, S ν≤r n (T ) is compact. If S n (T ) = S ν≤r n (T ) for some r, then S n (T ) is compact. Conversely, if S n (T ) is compact for n ≥ 1, then ν is necessarily bounded on models of T , so there is some r such that T ⊢ sup x ν(x)∧(r+1) ≤ r and S m (T ) = S ν≤r m (T ) for all m ∈ N. In this case all the other symbols are also bounded in models of T , so up to re-scaling everything into [0, 1] we are in the case of standard continuous first order logic.

In the non compact case we still have S n (T ) = r S ν≤r n (T ). Thus each p ∈ S n (T ) there is r such that p ∈ S ν≤r n (T ), and S ν≤r+1 n (T ) is a compact neighbourhood of p (since it contains the open set [( ν(x i )) < r + 1]). Therefore S n (T ) is locally compact.

On the relation with Henson's positive bounded logic

We sketch out here how unbounded continuous logic generalises, in an appropriate sense, Henson's logic of approximate satisfaction of positive bounded formulae in Banach space structures. For this purpose we assume familiarity with the syntax and semantics of Henson's logic (see for example [START_REF] Henson | Ultraproducts in analysis, Analysis and Logic[END_REF]).

The classical presentation of Henson's logic involves a purely functional signature L H with a distinguished sort for R. There is no harm in assuming that the distinguished sort only appears as the target sort of some function symbols (otherwise we can add a second copy and a single function symbol for the identity mapping into the copy, and treat the copy as the distinguished sort). Also, there is no harm in replacing R with R + .

We can therefore define an unbounded continuous signature L by dropping the distinguished sort and replacing all function symbols into it with R + -valued predicate symbols.

As every sort is assumed to be normed, we identify ν with • . While a signature in Henson's logic does not specify continuity moduli, in every class under consideration each symbol satisfies some continuity modulus uniformly under • which we may use (or else the logic would fail to describe the class). It is a known fact that there exists a (universal) L H -theory, call it T 0 , whose models are precisely the structures respecting these continuity moduli under • .

From now on by "structure" we mean a model of T 0 , or equivalently a L-structure (as these can be identified). The ambiguity concerning whether a structure is a Henson or unbounded continuous structure is further justified by the fact that the definitions of isomorphism and ultra-products in either logic coincide. As we can moreover prove Lemma 3.7 for Henson's logic just as well, we conclude: Theorem 4.1. A class of structures K is elementary in Henson's logic if and only if it is elementary in unbounded continuous logic.

Recall:

Fact 4.2. Let X = n∈N X n be a topological space where each X n is closed and

X n+1 is a neighbourhood of X n . Then a subset F ⊆ X is closed if and only if F ∩ X n is for all n.
An n-type is the same thing as a complete theory with n new constant symbols (more precisely, a type p with ν(p) ≤ r corresponds to a complete theory with new constants symbols with continuity moduli δ a ≤ 1 r ). is open in both topologies to conclude that this is a global homeomorphism.

4.3

This can be restated as:

Corollary 4.4. For every set Σ(x) of L H -formulae there exists a set Γ(x) of L-conditions, and for every set Γ(x) of L-conditions there exists a set Σ(x) of L H -formulae, such that for every structure M and ā ∈ M:

M A Σ(ā) ⇐⇒ M Γ(ā).
Remark 4.5. In Henson's logic, the bounded quantifier ∀ ≤r x (∃ ≤r x) mean "for all (there exists) x such that x ≤ r". Thus Henson's logic coincides with unbounded continuous logic of normed structures where ν = • . One may generalise Henson's logic to allow an arbitrary ν and obtain full equivalence of the two logics.

For the benefit of the reader who finds this proof a little too obscure, let us give one direction explicitly. We know that every formula in Henson's logic is equivalent to one in prenex form

∀ ≤r 0 x 0 ∃ ≤r 1 x 1 . . . ϕ(x, ȳ),
where ϕ is a positive Boolean combination of atomic formulae of the form t i (x, ȳ) ≥ r i or t i ≤ r i . Every term t i can be identified with an atomic L-formula, and replacing t i with t i -. r i or with r i -. t i , we may assume all these atomic formulae are of the form t i ≤ 0. Since (t i ≤ 0) ∧ (t j ≤ 0) ⇐⇒ (t i ∨ t j ) ≤ 0 and (t i ≤ 0) ∨ (t j ≤ 0) ⇐⇒ (t i ∧ t j ) ≤ 0, we can find a single t such that ϕ(x, ȳ) is equivalent to t ≤ 0. We thus reduced to:

∀ ≤r 0 x 0 ∃ ≤r 1 x 1 . . . t(x, ȳ) ≤ 0 .
We can view t as a quantifier-free L-formula, in which case the above holds approximately if and only if the following holds (with the notation preceding Example 3.9):

∀ <r 0 x 0 ∃ ≤r 1 x 1 . . . t(x, ȳ) = 0.
Thus the approximate satisfaction of a L H -formula, and therefore of a partial type, are equivalent to the satisfaction of a partial type in L.

Emboundment

As we mentioned earlier, the multi-sorted approach to unbounded structures allows us to reduce many issues concerning unbounded structures to their well-established analogues in bounded continuous logic, but this does not work well for perturbations when we wish to perturb ν itself. In addition, if the bounded balls are not definable in the unbounded structure then their introduction as sorts adds unexpected structure -this may happen, for example, when considering a field equipped with a valuation in R as an unbounded metric structure.

We could of course generalise everything we did to the unbounded case, but that would be extremely tedious to author and reader alike. Instead, we seek a universal reduction of unbounded logic to the more familiar (and easier to manipulate) bounded one. This reduction goes through a construction which we call emboundment. Thus, for example, a bounded set X ⊆ M n in an unbounded structure is said to be definable (a term we knowingly used above without a definition) if it is definable in the embounded structure M ∞ . An easy verification yields that this is equivalent to the predicate d(x, X) being definable in M, i.e., a uniform limit of formulae on every bounded set. (See [Bena] for definable sets in bounded structures.)

One naïve approach would be to choose a continuous function mapping

R + into [0, 1], say θ(x) = x
x+1 , and apply it to all the predicate symbols: for every L-structure M we define M θ as having the same underlying set, and for every predicate symbol P we define P M θ (ā) = θ(P M (ā)). It can be verified that θ(x + y) ≤ θ(x) + θ(y) for all x, y ≥ 0 (this is true when x = 0, and the partial derivative with respect to x of the left hand side is smaller). It follows that d M θ is a metric:

d M θ (a, b) = θ(d M (a, b)) ≤ θ(d M (a, c) + d M (c, b)) ≤ θ(d M (a, c)) + θ(d M (c, b)) = d M θ (a, b) + d M θ (a, b).
Of course d M θ needs not be a complete metric, so we obtain new elements when passing to the completion. Similarly, if T θ = Th{M θ : M T }, then we have a natural embdding of S n (T ) in S n (T θ ), and it can be verified that the latter is the Stone-Čech compactification of the former. This is essentially the same thing as allowing ∞ as a legitimate truth value (as θ extends to a homeomorphism [0, ∞] → [0, 1]). As usual with the Stone-Čech compactification, this adds too many new types to be manageable. In short, this naïve construction does yield bounded structures but it is not at all clear that the structures (or theories) thus obtained are meaningful. For example, even the following is not clear (to the author), and one would expect it to be false: Question 5.1. Is every model of T θ of the form M θ , where M T ? For a better approach, we take a second look on the construction of unbounded logic and its semantics, as well as on the construction of unbounded ultra-products. Throughout these constructions appeared a formal infinity element ∞, which, while not a member of the structures, was treated for many intents and purposes as if it were. Indeed, the quantifier semantics included ∞ in the set over which quantification takes place, and the ultra-product construction could be restated informally as "add ∞, take a usual ultraproduct, then take ∞ out". In particular, unbounded structures may be formally empty since, from a practical point of view, they still always contain the ideal point at infinity.

With this motivation in mind, we seek to equip each unbounded structure M with a new metric, denoted d M ∞ such that every sequence (a n ) in M which goes to infinity in the sense that ν(a n ) → ∞, is Cauchy in d M ∞ , converging to a new element representing the formal infinity. Such a metric is naturally bounded. Moreover, every predicate on M which is uniformly continuous under ν can be modified to yield a bounded predicate which is in uniformly continuous in the usual sense with respect to d M ∞ . On the other hand, this does not work well for function symbols (for example, we cannot give a sense to ∞ + ∞ in the emboundment of a Banach space). We shall therefore replace every function symbol in the language with its graph G f (x, y) = d(f (x), y) as in Example 3.10, and assume that the signature L is purely relational. We then define

L ∞ = L ∪ {∞}
where ∞ is a new constant symbol. We may consider L to consist, as a set, of its non distinguished symbols alone, in which case ν gets dropped (or more precisely, both d and ν are dropped, and then L ∞ is equipped with its own distinguished distance symbol d).

Whether or not ν is kept will be of no essential difference to the construction. We do not specify at this point the uniform continuity moduli, but we shall show below that such moduli can be chosen that do fit our purpose.

For every L-structure M we define an L ∞ -structure M ∞ . Its domain is the set M ∪ {∞}. For elements coming from M we interpret the symbols as follows (we recall that

d(a, ∞) = ν(∞) = ∞, θ(∞) = 1, and ν(x) = ν(x i )): d M ∞ (a, b) = θ • d M (a, b) 1 + ν M (a) ∧ ν M (b) , P M ∞ (ā) = θ • P M (ā) 1 + ν M (ā) , (P = d).
So in particular:

d M ∞ (a, ∞) = 1 1 + ν M (a) , P M ∞ (. . . , ∞, . . .) = 0, (P = d).
Notice that if we interpreted d M ∞ as with other symbols we would have d M ∞ (a, ∞) = 0 for all a, and thus not obtain a metric. Conversely, we can reconstruct M from M ∞ , first recovering ν M from d M ∞ (x, ∞) and then recovering d M and P M from d M ∞ and P M ∞ , respectively, using the fact that θ -1 (y) = y 1-y . Let us show that d M ∞ is a metric. The only non trivial property to verify is the triangle inequality, namely

θ • d M (a, c) 1 + ν M (a) ∧ ν M (c) ≤ θ • d M (a, b) 1 + ν M (a) ∧ ν M (b) + θ • d M (b, c) 1 + ν M (b) ∧ ν M (c)
.

If b has the smallest gauge among the three then this follows from the fact that θ • d M is a metric, which we verified earlier. Otherwise we may assume without loss of generality that a has the smallest gauge, say r. Let t = d M (a, b), s = d M (b, c). Then ν M (b) ≤ r + t and d M (a, c) ≤ t + s, and it is enough to verify that

θ(t + s) 1 + r ≤ θ(t) 1 + r + θ(s) 1 + r + t .
Moving the second term to the left and developing we obtain:

θ(t + s) 1 + r - θ(t) 1 + r = s (1 + r)(1 + t)(1 + t + s) = s (1 + r)(1 + t + s) + (1 + r)t(1 + t + s) ≤ s (1 + r)(1 + s) + t(1 + s) = s (1 + r + t)(1 + s) = θ(s) 1 + r + t , as desired. Once we know that d ∞ is a metric it is clear that a n → ∞ in d M ∞ if and only if ν M (a n ) → ∞.
Example 5.2. Let M be a bounded structure, and turn it into an unbounded structure M ′ as in Example 2.4. Then M ∼ = (M ′ ) ∞ {∞}, so all we did was add a single isolated point with distance 1 to the original structure.

Lemma 5.3. The gauged space (M, d M , ν M ) and the bounded metric space (M, d M ∞ ) are related as follows:

(i) We have d M ≥ d M ∞ on all of M, and the two metrics are uniformly equivalent on every bounded subset of M (bounded in the sense of M). (ii) For every r ′ > r the ν-ball M ν<r ′ contains a uniform d M ∞ -neighbourhood of M ν≤r (of radius θ(r ′ -r) 1+r ).

Proof. The inequality a,b) 1+r

d M ≥ d M ∞ is immediate. Let us fix r ≥ 0 and let a ∈ M ν≤r , b ∈ M. Then by definition d M ∞ (a, b) ≥ θ•d M (
. Thus, for all ε > 0

d M ∞ (a, b) < θ(ε) 1 + r =⇒ d M (a, b) < ε,
concluding the proof of the first item. This also proves the third item, since

B d M ∞ M ν≤r , θ(r ′ -r) 1 + r ⊆ B d (M ν≤r , r ′ -r) ⊆ M ν<r ′ .

5.3

Proposition 5.4. For every L-structure M, M ∞ as defined above is an L ∞ -structure, called the emboundment of M. That is to say that M ∞ is complete, and that we can complete the definition of L ∞ choosing uniform continuity moduli for its symbols which are satisfied in every M ∞ .

Proof. For completeness, let (a n ) n∈N be a Cauchy sequence in

d M ∞ . If ν(a n ) → ∞ (where again, ν(∞) = ∞) then a n → ∞ in d ∞ .
Otherwise, there is r such that a n ∈ M ν<r infinitely often. Passing to a sub-sequence, we may assume that the entire sequence fits inside M ν<r . By Lemma 5.3(i) the sequence is Cauchy in d M and therefore admits a limit in M, which is necessarily also its limit in d M ∞ . For uniform continuity, let P ∈ L be an (n + 1)-ary predicate symbol. Let ε > 0 be given, and we wish to find δ > 0 such that for all a, b ∈ ε , so is P . By Lemma 5.3(i), P is uniformly continuous with respect to d M ∞ on M ν< 1 ε , whence the existence of δ as desired.

M ∞ d M ∞ (a, b) ≤ δ =⇒ sup x | P (a, x) -P (b, x)| M ∞ ≤ ε. First, if ν(a), ν(b) ≥ 1 ε -1 (where ν(∞) = ∞)

5.4

It is straightforward to verify that the emboundment construction commutes with the ultra-product construction, since everything is continuous:

M i /U ∞ = M ∞ i /U .
In particular, all the tuples (a i ) such that lim U ν M i (a i ) = ∞, which were dropped during the construction of

M i /U , satisfy [a i ] = [∞ M i ] = ∞ in M ∞ i /U .
Similarly, emboundment commutes with unions of increasing chains, and by Lemma 3.7 we have M ≡ N ⇐⇒ M ∞ ≡ N ∞ for any two L-structures M and N.

If N ⊆ M then working with L(N) we get N M ⇐⇒ N ∞ M ∞ . Similarly, if N ′ M ∞
where N ′ is an L ∞ -structure then we can recover an L-structure on N = N ′ {∞} ⊆ M, so N ′ = N ∞ and N M.

Proposition 5.5. Let K be a class of L-structures, and let

K ∞ = {M ∞ : M ∈ K}.
Then K is elementary if and only if K ∞ is.

Proof. Assume K is elementary. Then, by the arguments above, K ∞ is closed under ultra-products, isomorphism and elementary substructures. It is therefore elementary. Similarly for the converse.

5.5

By Proposition 5.5 we may replace every L-theory T (in unbounded logic) with its emboundment T ∞ = Th L ∞ (Mod(T ) ∞ ), which is a theory in standard bounded logic. By naming constants we further see that L-types of tuples in M are in bijection with L ∞ -types of tuples in M ∞ {∞} (i.e., in M again, but this time viewed as a subset of a L ∞ -structure).

Given a tuple ā ∈ M ∞ , let w = w(ā) = {i < n : a i = ∞}. Then we may identify tp M ∞ (ā) with the pair (w, tp M (a ∈w )). We can therefore express the set of types S n (T ∞ ) as w⊆n {w} × S |w| (T ). For w ⊆ n, r ∈ R + and ϕ(x ∈w ) ∈ L, define:

V n,w,r,ϕ = (v, q(x ∈v )) ∈ S n (T ∞ ) : w ⊆ v ⊆ n, ϕ q < 1, i∈v w ν(x i ) q > r .
Given a type (w, p) ∈ S n (T ∞ ), one can verify that the family of all sets of the form V n,w,r,ϕ where ϕ p = 0 forms a base of neighbourhoods for (w, p). In particular, the natural inclusion S n (T ) ֒→ S n (T ∞ ), consisting of sending p → (n, p), is an open topological embedding. In case T is complete (so | S 0 (T )| = 1), this embedding for n = 1 is a single point compactification of S 1 (T ) obtained by adding the type at infinity. We may therefore also refer to T ∞ as the compactification of T .

Once we understand types we know what saturation means. Among other things we have:

Lemma 5.6. An L-structure M is approximately ℵ 0 -saturated if and only if M ∞ is. Definition 6.4. Let ρ a perturbation pre-radius for T .

(i) We say that ρ respects infinity if for all r ∈ R + there exists

r ′ ∈ R + such that [ν(x) ≥ r ′ ] ρ ⊆ [ν(x) ≥ r] and [ν(x) ≤ r] ρ ⊆ [ν(x) ≤ r ′ ].
(ii) We define when ρ respects equality, respects ∃, or is permutation-invariant as in the bounded case.

Proposition 6.5. Let ρ be a perturbation pre-radius for T . The the following are equivalent:

(i) ρ is a perturbation radius.

(ii) ρ respects infinity, and for every n, m ∈ N and mapping σ : n → m, the induced mapping σ * : S m (T ) → S n (T ) satisfies that for all p ∈ S m (T ):

σ * (p ρ ) = σ * (p) ρ .
(I.e., σ * • ρ m = ρ n • σ * as multi-valued functions). (iii) ρ respects ∞, =, ∃, and is permutation-invariant. (iv) ρ ∞ separates ∞, respects = and ∃ and is permutation-invariant.

Proof.

(i) =⇒ (ii). Assume ρ is a perturbation radius, so ρ ∞ is a perturbation radius respecting infinity. If ρ does not respect infinity, then by definition of ρ ∞ we have in ρ ∞ 1 a pair (p, q) where p is the type of a finite elements and q = tp(∞) or vice versa, contradicting the assumption on ρ ∞ .

Since ρ ∞ is a perturbation radius, for all σ : n → m we have in S(T ∞ ):

σ * •ρ ∞ m = ρ ∞ n •σ * .
As ρ ∞ also separates infinity we can restrict this to S(T ) and obtain σ * • ρ m = ρ n • σ * .

(ii) =⇒ (iii). By restricting to the case where σ is the mapping 2 → 1, n ֒→ n + 1, or a permutation of n ∈ N.

(iii) =⇒ (iv). By a mirror-image to the argument above, if ρ respects ∞ then ρ ∞ must separate ∞.

We claim that since ρ respects ∞ and ∃ and is permutation-invariant, we have for all n ∈ N: ρ ∞ n = ((w, p), (w, q)) ∈ S n (T ∞ ) : w ⊆ n, (p, q) ∈ ρ |w| (i.e., the right hand side is a closed set). Indeed, assume we have pairs ((w i , p i ), (w i , q i )) for i ∈ I and U is an ultra-filter on I, and let ((v, p), (u, q)) = lim U ((w i , p i ), (w i , q i )). We need to show that v = u and (p, q) ∈ ρ |v| . First, as there are finitely many possibilities for w i ⊆ n we may assume that w i = w ⊆ n for all i. Then we might as well assume w = n throughout.

For s ⊆ n, let p s i and q s i be the restrictions of p i and q i , respectively, to x ∈s . As ρ respect ∃ and is permutation-invariant, (p s i , q s i ) ∈ ρ |s| . As ρ respects infinity we have:

k / ∈ v ⇐⇒ p {k} i → U tp(∞) ⇐⇒ q {k} i → U tp(∞) ⇐⇒ k / ∈ u.
Therefore v = u, and as ρ |v| is closed (p, q) = lim U (p v i , q v i ) ∈ ρ |v| . This proves our claim.

It is now immediate that as ρ respects = and ∃ and is permutation-invariant, the same holds of ρ ∞ .

(iv) =⇒ (i). Since then ρ ∞ is a perturbation radius.

6.5

Corollary 6.6. Perturbation systems p for T are in a natural one-to-one correspondence with families {d p,n : n ∈ N}, in which each d p,n is a [0, ∞]-valued metric on S n (T ), and such that: (i) For every n, the set (p, q, ε) ∈ S n (T ) 2 × R + : d p,n (p, q) ≤ ε is closed.

(ii) For every n, m ∈ N and mapping σ : n → m, the induced mapping σ * : S m (T ) → S n (T ) satisfies for all p ∈ S m (T ) and q ∈ S n (T ):

d p,m (p, (f * ) -1 (q)) = d p,n (f * (p), q).
(Here we follow the convention that d

p,m (p, ∅) = inf ∅ = ∞.) (iii) For every r ∈ R + there is r ′ ∈ R + such that if p, q ∈ S 1 (T ) and d p,1 (p, q) ≤ r, then ν(x) p ≥ r ′ =⇒ ν(x) q ≥ r
Similarly, perturbation pre-systems are in one-to-one correspondence with families of metrics satisfying the first condition alone.

Proof. Same as [START_REF]On perturbations of continuous structures[END_REF]Lemma 1.24], where condition (iii) corresponds to the requirement that every p(ε) respect infinity.

6.6

Let us fix a perturbation system p for T , and let p ∞ be the corresponding perturbation system for T ∞ . As for plain approximate ℵ 0 -saturation, we have Lemma

6.7. A model M T is p-approximately ℵ 0 -saturated if and only if M ∞ is p ∞ -approximately ℵ 0 -saturated.
Proof. As for Lemma 5.6.

6.7

In particular, and two separable p-approximately ℵ 0 -saturated models of T must be p-isomorphic.

Similarly:

Lemma 6.8. Two models M, N T are p-isomorphic if and only if M ∞ and N ∞ are p ∞ -isomorphic. The theory T is p-ℵ 0 -categorical if and only if T ∞ is p ∞ -ℵ 0 -categorical.
We conclude that [START_REF]On perturbations of continuous structures[END_REF]Theorem 3.5] holds as stated for unbounded structures: Theorem 6.9. Let T be a complete countable unbounded theory, p a perturbation system for T . Then the following are equivalent: (i) The theory T is p-ℵ 0 -categorical.

(ii) For every n ∈ N, finite ā, p ∈ S n (ā) and ε > 0, the set [p p(ε) (x ε , āε )] has non empty interior in S n (ā).

(iii) Same restricted to n = 1.

Proof. The idea is to reduce to [START_REF]On perturbations of continuous structures[END_REF]Theorem 3.5]. Most of the reduction is in the preceding results: T is complete if and only if T ∞ is, T is p-ℵ 0 -categorical if and only if T ∞ is p ∞ -ℵ 0 -categorical, etc. The last thing to check is that the property

p(x, ā) ∈ S n (ā), ε > 0 =⇒ [p p(ε) (x ε , āε )] • = ∅ ( * )
holds for T, p if and only it holds for T ∞ , p ∞ . Indeed, assume first ( * ) holds for T ∞ , p ∞ . Let ā ∈ M T , p(x, ā) ∈ S n (ā). Then ā can be viewed also as a tuple in M ∞ T ∞ , and we can identify p(x, ā) with a type p ∞ (x, ā) ∈ S L ∞ n (ā). Then p p(ε) (x, ȳ) and p ∞p ∞ (ε) (x, ȳ) coincide more or less by definition, and fit in S ν≤r (T ) for some r ∈ R + . It is not true that p p(ε) (x ε , ȳε ) and p ∞p ∞ (ε) (x ε , ȳε ) coincide since in the metrics on models of T and T ∞ differ. But as everything fits inside some ν-ball, and the two metrics are uniformly equivalent on every ν-ball, we can still find ε ′ > 0 such that

[p(ε)(x ε , ȳε )] • ⊇ [p ∞p ∞ (ε ′ ) (x ε ′ , ȳε ′ )] • = ∅.
For the converse, consider a finite tuple ā ∈ M ∞ T ∞ , and a type p(x, ā) ∈ S L ∞ n (ā). As ∞ is definable in T ∞ (it is the unique element satisfying P 1 (x) = 0, for example) we never need it as a parameter, so we may assume that ā ∈ M. Assume first that p(x, ā) says that all x i are finite as well. Then in fact p(x, ā) ∈ S L n (ā), and we conclude as above by the uniform equivalence of the metric. In the general case we may need to write p(x, ȳ) as (w, q) where w ⊆ |x, ȳ|, and q ∈ S |w| (T ). Then q is a type of finite elements and is taken care of by the previous case, while the infinite coordinates are taken care of by the fact that ∞ is definable, so [d L ∞ (x, ∞) < ε] defines an open set in S L ∞ (ā).

6.9

The discussion at the end of [START_REF]On perturbations of continuous structures[END_REF]Section 3], and in particular the characterisation of p-ℵ 0 -categoricity for an open perturbation system p by coincidence of topologies ( [START_REF]On perturbations of continuous structures[END_REF]Theorem 3.15]), can be transferred to an unbounded theory T via reduction to T ∞ in precisely the same way.

7. An example: Henson's categoricity theorem Let T 0 be the (unbounded) theory of pure Banach spaces as given in Example 3.8. Definition 7.1. Let E and F be Banach spaces (i.e., models of T 0 ). Say that a mapping f : E → F is an ε-isomorphism if it is an isomorphism of the underlying vector spaces, and satisfies in addition:

∀v ∈ E e -ε v ≤ f (v) ≤ e ε v .
Definition 7.2. Let ā ∈ E 0 T 0 . Define the Banach-Mazur distance between two types p, q ∈ S n (ā), denoted d BM,n (p, q), as the minimal ε > 0 such that there exist models (E, ā), (F, ā) Th(E 0 , ā), and tuples b ∈ E, c ∈ F realising p and q, respectively, and an ε-isomorphism f : E → F fixing ā and sending b to c. If no such ε > 0 exists then d BM,n (p, q) = ∞.

The following result is very similar to an unpublished result communicated to the author orally by C. Ward Henson. It is one of the original motivations for the present paper as well as for [Benc].

Corollary 7.3. Let T be a complete theory of Banach spaces with no additional structure (i.e., a completion of T 0 ). Then the following are equivalent:

(i) If E and F are two separable models of T , then for every ε > 0 there exists an ε-isomorphism (i.e., a bijective ε-embedding) from E to F . (ii) For n ∈ N and finite tuple ā ∈ E T , let S * n (ā) be the space of types of n-tuples which are linearly independent over ā. Then every Banach-Mazur ball in S * n (ā) has non empty interior in the logic topology on S * n (ā). Proof. First we observe that the Banach-Mazur distance defines a perturbation system BM by Corollary 6.6. Therefore, by Theorem 6.9, the first condition is equivalent to the one saying that for all ε > 0 and p(x, ā) ∈ S n (ā): [p BM (ε) (x ε , āε )] • = ∅ in S n (ā). We need to show that this is equivalent to the second condition. Since the Banach-Mazur perturbation preserves linear dependencies we may drop superfluous parameters and always assume that the tuple ā is linearly independent. Thus, if p(x, ā) ∈ S * (ā) then p(x, ȳ) ∈ S * (T ).

Observe also that S * n (ā) is a dense open subset of S n (ā) (indeed, it is metrically dense there in the usual metric on types). It follows that a subset X ⊆ S * n (ā) has the same interior in S n (ā) and in S * n (ā), so we may simply speak of its interior. Moreover, a subset X ⊆ S n (ā) has non empty interior if and only if X ∩ S * n (ā) has. For left to right, let us show that if p ∈ S * n (T ) and ε > 0 then there exists δ > 0 such that [p(x δ )] ⊆ [p BM (ε) ]. So let Λ = {λ ∈ F n :

|λ i | = 1}, i.e., the (compact) space of all formal linear combinations of n variables of • 1 -norm 1, and let s = min{ λ(x) p(x) : λ ∈ Λ} > 0. We claim that δ = sε 2n > 0 will do. Indeed, let q ∈ [p(x δ )]. Let E be a model, b, c ∈ E such that b p, c q and b i -c i ≤ δ for all i < n. For i < n define a linear functional η i : Span( b) → F by η i ( λ j b j ) = λ i . Then η i ≤ s -1 , and by the Hahn-Banach Theorem we may extend them to ηi : E → F such that ηi ≤ s -1 . Define a linear operator S : E → E by S(x) = i ηi (x)(b i -c i ). Then a simple calculation shows that S(b i ) = b i -c i and S ≤ ε/2. Assuming ε was small enough to begin with (which we may), I -S is invertible, its inverse being I + S + S 2 + . . .. Finally, for all v ∈ E:

e -ε v ≤ (1 -ε/2) v ≤ v -S(v) ≤ (1 + ε/2) v ≤ e ε v .
We conclude that I -S is an ε-automorphism sending b to c, so q ∈ p BM (ε) .

Re-choosing our numbers we find ε/2 > δ > 0 such that that [p(x δ )] ⊆ [p BM (ε/2) (x)], so [p(x δ )] BM (δ) ⊆ [p BM (ε) (x)]. As the former has non empty interior so does the latter (in S n (T ) as well as when restricted to S * n (T )). When considering parameters we have

  Corollary 4.3. Two n-tuples in a structure have the same type in one logic if and only if they have the same type in the other, and this identification induces a homeomorphism S L H n (T 0 ) ≃ S L n . Proof. The first statement is by Theorem 4.1. Also, a set X ⊆ S • ≤r n is closed if and only if the class {(M, ā) : tp(ā) ∈ X} is elementary: the bounds on the norm are needed since we need to impose bounds on the norms of constant symbols. It follows from Theorem 4.1 that the bijection S L H n (T 0 ) ≃ S L n is a homeomorphism when restricted to S • ≤n n . Now use Fact 4.2 and the fact that S • ≤r n is compact and S • <r n

  then the above is satisfied regardless of d M ∞ (a, b). Otherwise, without loss of generality we have ν(a) < 1 ε -1, and if d M ∞ (a, b) < ε 2 then ν(b) < 1 ε by Lemma 5.3(ii). Since P and ν are uniformly continuous with respect to d M on M ν< 1

Research partially supported by NSF grant DMS-0500172, ANR chaire d'excellence junior THEMOD-MET (ANR-06-CEXC-007) and by Marie Curie research network ModNet. The author would like to thank C. Ward Henson for many helpful discussions and comments. March 2009.

Proof. Follows from the facts that there is a unique point at infinity, which belongs to M ∞ , and that in the neighbourhood of every other point d M and d M ∞ are equivalent.

5.6

Finally, we point out that the theory T is bounded to begin with if and only if the point at infinity in models of T ∞ is isolated, in analogy with what happens when one attempts to add a point at infinity to a space which is already compact.

Perturbations of unbounded structures

We now adapt the framework of perturbation of bounded metric structures to unbounded structures, essentially by reducing the unbounded case to the bounded one through emboundment. For this purpose we assume close familiarity with the original development in [Benc]. We fix an unbounded theory T and its emboundment T ∞ . Definition 6.1. A perturbation pre-radius for T is defined as for a bounded theory, i.e., as a family ρ = {ρ n ⊆ S n (T ) : n ∈ N} containing the diagonals. We define X ρ , Pert ρ (M, N), BiPert ρ (M, N), ρ , ρ as in [Benc].

Let ρ be a perturbation pre-radius for T . We can always extend it to a perturbation radius ρ * for T ∞ by:

Clearly, this is a perturbation pre-radius for T ∞ . Conversely, if ρ ′ is a perturbation preradius for T ∞ then its restriction to S(T ), denoted ρ ′ ↾ S(T ) , is a perturbation pre-radius for T , and as the inclusion S n (T ) ⊆ S n (T ∞ ) is open we have the identity:

We define perturbation radii for T directly by reduction to T ∞ : Definition 6.2.

(i) Let ρ ′ be a perturbation pre-radius for T ∞ . We say that ρ

(ii) A perturbation pre-radius ρ for T is a perturbation radius if ρ ∞ is a perturbation radius for T ∞ which separates infinity.

Definition 6.3. A perturbation pre-system for T is a decreasing family p of perturbation pre-radii satisfying downward continuity, symmetry, triangle inequality and strictness as in [START_REF]On perturbations of continuous structures[END_REF]Definition 1.23]. It is a perturbation system if p(ε) is a perturbation radius for all ε, i.e., if p ∞ is a perturbation system separating infinity for T ∞ .

We turn to characterise perturbation radii as in [Benc], and establish more precisely the relation between perturbations of T and of T ∞ . p(x, ā) ∈ S * n (ā) such that p(x, ȳ) ∈ S * n+m (T ), so we find δ > 0 such that [p(x δ , ȳδ )] BM (δ) ⊆ [p BM (ε) (x, ȳ)], and thus [p(x δ , āδ )] BM (δ) ⊆ [p BM (ε) (x, ā)], concluding as above.

For the other direction, let us show that for all p ∈ S n (T ) and

, and therefore in S n (T ), as S * n (T ) is open in S n (T ). In case p / ∈ S * n (T ) we need to be more delicate. Up to a permutation of the variables we may assume that p is of the form p(x <m , y <k ), where m + k = n, q(x) = p↾ x ∈ S * m (T ), and p ⊢ i<k (y i = λ i (x)) for some linear combinations λ i .

Then we know there is a formula ϕ(x) such that ∅ = [ϕ < 1/2] ⊆ q BM (ε) . Then in S n (T ) we have:

We by variable-invariance we may find p ′′′ ∈ (p ′′ ) BM (ε) such that p ′′′ ↾ x = q. As the linear structure is left untouched by the Banach-Mazur perturbation we must have p ′′′ (x, ȳ) ⊢ i<k (y i = λ i (x)), so in fact p ′′′ = p, as required. The case with parameters is proved identically (with each y i being equal to a linear combination of x and ā).


This paper addresses the scheduling maintenance operations for a set of geographically dispersed production equipment. We consider various equipment subject to uncertain failures that need to be monitored and repaired by a central maintenance workshop (CMW). We focus on the associated preventive maintenance actions, which require optimal scheduling. A mobile maintenance workshop (MMW) aims to transport all the resources to systematically replace the equipment with spare parts provided by the CMW following the predefined schedule. We propose a general framework and a model to minimize the maintenance and routing costs while optimizing equipment reliability. The novelty of the proposed model is the reduction of computation time through the proposition of heuristic methods while considering capacity constraints and various types of equipment. We perform several instances of problems in the oil and gas field. The studied approach optimizes the location of the CMW and the capacity of the MMW, solving the NP-Hard Problem.

I. INTRODUCTION

Manufacturing systems reliability and availability depend mainly on the quality of maintenance management [START_REF] Antosz | Spare parts' criticality assessment and prioritization for enhancing manufacturing systems' availability and reliability[END_REF]. In general, the aim is to monitor production assets, schedule maintenance actions and conduct various preventive or corrective tasks. Fortunately, industry 4.0 technologies allow anticipating the failures of production equipment. Therefore, operations are becoming more and more smart and remote [START_REF] Gopalakrishnan | Data-driven machine criticality assessmentmaintenance decision support for increased productivity[END_REF]. But it is well known that the major constraint faced by managers remains the limited budget dedicated to the maintenance [START_REF] Garg | Maintenance management: literature review and directions[END_REF]. One of the solutions to reduce costs consists of sharing the same resources (spare parts, operators and tools) for different facilities. However, an allocation problem arises since the schedule of several maintenance tasks depends strongly on the availability of resources [START_REF] Zhang | Optimal maintenance planning and resource allocation for wind farms based on non-dominated sorting genetic algorithm-ΙΙ[END_REF]. It is even more accentuated when the facilities are geographically distributed [START_REF] Manco | Maintenance management for geographically distributed assets: a criticality-based approach[END_REF].

In this paper, we study the implementation of Distributed Maintenance [START_REF] Simeu-Abazi | Optimisation of distributed maintenance: Modelling and application to the multi-factory production[END_REF]. The concept consists of gathering all the resources in a central maintenance workshop (CMW), also known as repair shop [START_REF] Djeunang Mezafack | Systematic literature review of repair shops: focus on sustainability[END_REF]. A mobile maintenance workshop (MMW) is in charge of the routing between the CMW and geographically spread production sites (PS), where pieces of manufacturing equipment are subject to failures. The objective is to schedule preventive actions for each PS, optimize the MMW preventive visits' and restore defective equipment in the CMW as soon as possible.

Industrial applications of Distributed Maintenance are emerging in industries where the distance between geographically dispersed equipment is not too great [START_REF] Djeunang Mezafack | Systematic literature review of repair shops: focus on sustainability[END_REF]. Centralization could reduce maintenance costs while increasing equipment availability [START_REF] Abbou | Les réseaux de Petri pour la modélisation et l'analyse des performances d'un atelier de maintenance[END_REF]. For example, several locomotives share the same maintenance workshop for preventive actions in the railway sector [START_REF] Hani | Ant colony optimization for solving an industrial layout problem[END_REF]. In the oil and gas industry, the location of oil platforms (onshore or offshore) sharing the same central control depends on the sources of raw materials. The centralized entity manages and carries out maintenance operations on geographically dispersed equipment. As well as in the aviation industry, defective aircraft parts are replaced directly without transporting the aircraft. A centralized workshop is needed to diagnose the origin of faults and repair them [START_REF] Sanchez | An optimisation framework for airline fleet maintenance scheduling with tail assignment considerations[END_REF]. In other applications, a third party maintains distributed installations of different companies. This paper aims to optimize maintenance costs while reducing downtime of geographically dispersed equipment throughout Distributed Maintenance. The literature provides several studies dealing with this objective. But, one of the main difficulties faced by the authors is the combination of maintenance scheduling of PS and routing optimization of MMW. Indeed, López-Santana et al. (2016) proposed one of the first modular models called CMR (Combined Maintenance and Routing) [START_REF] López-Santana | On the combined maintenance and routing optimization problem[END_REF]. It is a two-step iterative process that makes a balance between maintenance and routing costs. But this model does not consider the capacity constraint of transportation between the PS with equipment to be maintained. Hence, Vega-Figueroa et al. (2022) proposed a hybrid algorithm [START_REF] Vega-Figueroa | Hybrid algorithm for the solution of the periodic vehicle routing problem with variable service frequency[END_REF] and Allaham et al. (2022) defined a MILP (Mixed Integer Linear Programming) [START_REF] Allaham | MILP of multitask scheduling of geographically distributed maintenance tasks[END_REF] to allocate teams and tasks for each maintenance operation considering the capacity constraint of spare parts. Djeunang Mezafack et al. (2022) propose a MILP and a general framework called CMCR (Combined Maintenance and Capacitated Routing) [START_REF] Djeunang Mezafack | Optimization of Distributed Maintenance: Design, Scheduling and Capacitated Routing Problem[END_REF] to choose the capacity of the MMW and the location of the CMW. However, all these approaches require exponential computation time since they tackle an NP-Hard problem. Thus, the existing models only allow solving small instances of the problem. From this gap, this paper provides an improved model called OMCR (Optimized Maintenance and Capacitated Routing). The novelty of the approach is to reduce the complexity of the optimization algorithm while considering the capacity constraints of the MMW and various types of equipment under Distributed Maintenance. Thus, we state two research questions as follow:

• How to implement Distributed Maintenance with diverse types of equipment?

• How to optimize the capacitated routing of the MMW with a low computation time?

The paper is structured into five sections. Following an introduction about the context of this work in section 1, section We consider a system with a set of N Production Sites (PS) geographically distributed over an area. Each PS has one piece of equipment subject to uncertain failures. With a schedule, a Mobile Maintenance Workshop (MMW) is responsible for transporting preventive maintenance resources (spare parts and tools) to visit the set of PS within a horizon of time, as presented in Figure 1. A Centralized Maintenance Workshops (CMW) monitors the state of each piece of equipment and stores the spare parts. The MMW starts in the CMW with a limited capacity of spare parts and visits all the PS following the optimal scheduling. When the MMW reaches a PS, the piece of equipment is replaced systematically by a spare part. The objective is to find the optimal routing of the MMW through the PS while reducing maintenance and routing costs. The main assumptions of the problem are summarized as follows:

1. The MMW is a fleet of m homogeneous vehicles, each with a limited capacity 𝑄.

2.

A vehicle transports only one operator in charge of replacing a piece of equipment with a brand-new spare part.

3. The travel times are deterministic and do not change over the scheduling horizon .

4. The CMW is a depot with an unlimited capacity.

5. The PS are N geographically-spread customers, each with a single piece of equipment.

6. A piece of equipment starts in "As Good As New" condition and, after the replacement, it returns to " As Good As New " condition.

7. All the pieces of equipment are mutually independent according to their failures' behaviour. It means that the state of one piece of equipment does not disrupt those of another. Thus, we consider it is possible to optimize the frequency of preventive maintenance (PM) operations for each customer separately.

8. Each customer is subject to a hard time window constraint out of which a PM operation cannot be conducted.

9. In the case of a failure, the customer waits and the following PM operation replaces the defective equipment.

In the next section, we will present, first, the general framework used to optimize Distributed Maintenance and, second, the OMCR model in detail.

III. OPTIMIZATION OF DISTRIBUTED MAINTENANCE

A. General framework A Distributed Maintenance works with three main entities: MMW/vehicles, a CMW/depot and PS/customers. Figure 2 illustrates the framework, which gives an overview of the optimization process. It allows choosing the capacity of the vehicles, the position of the depot and the scheduling of the operations. Indeed, generally, two loops allow to compare the optimal scheduling, first for different depot locations and second for various vehicles capacity [START_REF] Djeunang Mezafack | Optimization of Distributed Maintenance: Design, Scheduling and Capacitated Routing Problem[END_REF]. The goal is to choose the best parameters satisfying the optimal maintenance and routing costs. The position first loop considers the routing cost is low when the depot 𝐶𝑀𝑊 is close to a customer 𝑃𝑆 𝑖 , 𝑖 = 1, … , 𝑁 [START_REF] Simeu-Abazi | Implementation of a cost optimization algorithm in a context of distributed maintenance[END_REF]. The second loop of capacity tests different values of Q from Q min to Q max . But, this general process requires excessive computation time due to these two loops. However, after integrating the new OMCR model into the framework, the loop concerning the position of the depot is no more necessary.

It could therefore reduce the computation time. Hence, the core model, OMCR, is an extended version of the CMCR [START_REF] Djeunang Mezafack | Optimization of Distributed Maintenance: Design, Scheduling and Capacitated Routing Problem[END_REF]. The main contribution of our model is the integration of the depot position choice in the optimization model reducing computation time. Indeed, as in the CMCR, the proposed model considers as input the capacity of vehicles and data of customers. It concerns their geographical locations and historical statistics on their equipment failures. As an innovation, OMCR enables a suitable position for the depot to be found automatically without going through the loop of different locations.

B. Optimized Maintenance and Capacitated Routing

The OMCR model consists in solving two different sub-models iteratively as presented in Figure 3: ). This sub-model does not change from the CMR to OMCR and allows to find the optimal period of PM for each piece of equipment separately. The output is the time windows for each customer at which the PM must be carried-out within the scheduling horizon. We focus now on the HRM sub-model. We consider a complete directed graph G=(V,A), where V={0,1,2,…,N} is a set of nodes with the depot 0, and Vc=V\{0} a subset of customers. A={(i,j):i,jV} represents the set of links between all pairs of nodes. The vehicles set is defined by K={1,2,…,m}, each with a 1 wi is obtained exactly as in the CMR. capacity Q. Each customer iVc is associated with a certain positive demand di < Q and an on-site service time TPMi. Nonnegative travel time tij and distance dij are associated to each arc (i,j)A. auxiliary directed graph G'=(V',A'), where A'={(i',j'):i',j'V'} represents the set of arcs. For each node i'V' we can find the equivalent iV such as di'=di. And, for each arc (i',j')A' we can find the equivalent arc (i,j)A such as ti'j'=tij and di'j'=dij.

The problem therefore consists in solving a classical CVRPTW considering the graph G' such that:

• Each PM operation i'V'\{0} is performed exactly once.

• A vehicle cannot transport spare parts over its capacity Q.

• Each time windows [ej,lj]i ; iV\{0} is equivalent to a time window [ei',li'] ; i'V'\{0}.

For the rest of the paper, we use the index i instead of i' to denote each PM operation. CVRPTW is an NP-hard problem whose only small instances can be solved. The most recent and relevant model is proposed by [START_REF] Borcinova | Two models of the capacitated vehicle routing problem[END_REF]. Based on this classical model, we can define the Mixed-Integer Linear Programming model (MILP) of the HRM. However, for large instances of the problem, it is necessary to combine the MILP with a heuristic. We choose to adapt and implement a divide to conquer algorithm [START_REF] Mariescu-Istodor | VRPDiv: A Divide and Conquer Framework for Large Vehicle Routing Problems[END_REF] as illustrated in Figure 4. Firstly, the large list of PM operations is sorted from earliest to latest and divided into small sub-lists. Secondly, each sub-list is solved through the MILP and afterwards the results are combined to obtain PM operation scheduling We can now focus on the MILP formulation. A binary decision variable xij is defined to indicate if a vehicle crosses an arc (i,j) in the optimal solution. A vehicle arrives for a service i at a time denoted by si and with a load yi. [START_REF] Gopalakrishnan | Data-driven machine criticality assessmentmaintenance decision support for increased productivity[END_REF] represents the minimum number of vehicles needed to serve all the operations. The constraint (3) imposes that exactly m vehicles leave the depot. The classical flow constraints (4), ( 5) and ( 6) guarantee that each vehicle can leave the depot exactly once, and each PM operation is performed only once. In the constraint [START_REF] Djeunang Mezafack | Systematic literature review of repair shops: focus on sustainability[END_REF], the capacity of vehicles is stated such that the difference of a vehicle's load between two successive services i and j do not exceed the demand of j. The constraint [START_REF] Abbou | Les réseaux de Petri pour la modélisation et l'analyse des performances d'un atelier de maintenance[END_REF] ensures that the time between two successive services i and j do not exceed TPMi +Tij. The constraints ( 9), ( 10) and ( 11) restrict the upper and lower bounds of decision variables. In the next paragraph, we will implement the proposed model in an academic case study.

IV. NUMERICAL STUDY

A. Test instances

In this section, we run the OMCR in the oil and gas field to highlight the relevance of the proposed model. We apply the general framework proposed in Section 3.1.

(1) Equipment: we study the case of onshore pumps.

(2) Customers: we consider ten distributed pumping stations in a radius of 300 km. We assume that each customer has exactly one pump subject to uncertain failures. A pump is systematically replaced within a PM operation and transported to the depot.

The objective is to find the optimal location of the depot, the capacity of the vehicles and the scheduling of PM operations. We first get data on pump failures and operational costs [START_REF] López-Santana | On the combined maintenance and routing optimization problem[END_REF], as presented in Table 1. Secondly, we generate nine instances of the problem by randomly locating the ten pumping stations in a Cartesian plane (0, x, y) (Appendix 1 for details). Then, we consider the distance between each pair of pumping stations Euclidian. The MMW has a fleet of homogeneous vehicles whose capacity needs to be optimized. The unit costs related to each vehicle are:

CD=0.476$/km, CT=30$/h and CP=5000$. We consider three types of vehicles, each with a nominal speed of 80 km/h:

(1) Medium: Q = 4 pumps, Coef = 1;

(2) Heavy: Q = 6 pumps, Coef = 1.5;

(3) Extra-heavy: Q = 8 pumps, Coef = 2;

We choose the software Scilab 5.5.2 to implement the case study. All the tests have been run using the library "FOSSEE Optimization Toolbox" adapted for MILP. We perform the experiments on Windows 8, 64 bits machine, with an Intel(R) Core (TM) i7-10850H, CPU 2.70 GHz and 32 Go of RAM.

B. Results

The OMCR allows optimizing the PM operations of geographically dispersed equipment considering the capacity constraint of vehicles while reducing computation time. With the case study, we have conducted experiments to highlight the relevance of the proposed model. We run the iterative process until reaching 10 successive iterations where the value of E[DMC] do not change to more than 1%. We run all nine instances using the general framework, first without removing the loop of depot position and second after removing it. Figure 5 presents the results obtained for the first instance with the loop. It can be seen that going through the old optimization framework with the OMCR model takes 125 minutes. The best choice for vehicles is the medium size and, position 1 is the best for the depot. The optimal E[DMC] is $92.5/h. Then the optimal number of PM operations to perform within a month is 77, as presented in figure 6. Table 2 and Figure 7 allow comparing the results obtained with and without the loop of depot position. First, the proposed framework allows dividing the computation time by more than ten, as presented in Table 3. And, even though it does not produce an optimal E[DMC], the result remains in an acceptable interval for all nine instances. Indeed, to better interpret the results, we have defined an interval as presented in Figure 7.

The lower bound is the optimal results obtained by situating the depot in the customers' locations. The upper bound is the results obtained if the depot's position is chosen randomly and once among customers. Finally, by implementing the proposed framework in the nine instances, the computation time is deeply reduced, the gap between the results are less than 8%. Furthermore, in a realworld application, the number of customers is higher than ten. Managers may decide to locate the depot randomly among the clients' locations to tackle the increase in computing time. In this case, even if the computation time becomes small, the difference between the result obtained may be more than 21%, as illustrated in instance 2 of Figure 7. With the proposed framework, the results could be less. Furthermore, Table 3 presents the selection of CMW location and vehicles capacity for the 9 instances. It can be observed that from one instance to another, the location differs much more with the loop process than without it. The Average distance between each depot is 116km with a standard deviation of 68km with the loop rather than 43km and 29km without it. It could be interesting in the case where the company decides to add or change the position of a PS after constructing the CMW. V. CONCLUSION This paper deals with the combination of maintenance scheduling and routing optimization. We focus on a Distributed Maintenance context where a set of geographically dispersed production sites need to be visited frequently for preventive maintenance operations. We proposed an optimized framework and a novel model to consider the capacity constraints of the vehicles transporting spare parts and operator while reducing computation time. We made some assumptions to fit the problem with the known approach of Operational Research and Maintenance Management. The main contribution of this work is the reduction of maintenance and routing costs while optimizing the scheduling of preventive maintenance operations. We implemented the proposed model in some academic instances concerning the oil and gas industry. The study allows dividing the computation time by more than ten. And, the best costs obtained have a gap from the optimal to less than 8% Future research based on this work could extend experiments to a more extensive case study with a higher number of customers to test the scalability of the proposed model. Furthermore, it could be interesting to remove some assumptions in the modelling of the distributed maintenance system. We assumed that defective equipment in a production site must always wait for the following preventive maintenance operation for the replacement. However, this equipment may require an emergency repair. Thus, re-scheduling could be studied to reflect the real-world process. In addition, it is necessary to consider the uncertainty of the travel time between two production sites.

Figure 1 .

 1 Figure 1. Distributed Maintenance context [18].

Figure 2 .

 2 Figure 2. General framework with the part to be removed after integrating the OMCR model.

( 1 )

 1 Maintenance Model (MM): The objective is to minimize the Expected Maintenance Costs (E[MC]

( 2 )

 2 Heuristic Routing Model (HRM): The objective is to minimize the Expected Routing Costs (E[RC]); this model is the first main change from CMR to OMCR. It solves the well-known CVRPTW (Capacitated Vehicle Routing Problem with Time Windows) in Operational Research.In the CVRPTW, a fleet of homogeneous vehicles must service customers with known demands subject to opening hours[START_REF] Konstantakopoulos | Vehicle routing problem and related algorithms for logistics distribution: a literature review and classification[END_REF]. Based on the mathematical model used to solve this problem, the HRM proposed the optimal schedule of PM operations.The MM and HRM sub-models are linked first by the position of the depot, which is the second and last main change in the model. Indeed, we consider that the depot is located at the barycentre of the customers weighted by each number of PM operations i obtained from the MM sub-model. In addition, the two sub-models are linked by the expected waiting time wi 1 which is the period elapsing between an expected failure and the beginning of the next PM operation. These two models loop and the sum of 𝐸[𝑀𝐶] and 𝐸[𝑅𝐶] converge 2 to an optimal solution. The Expected Distributed Maintenance Cost is represented by: 𝐸[𝐷𝑀𝐶] = 𝐸[𝑀𝐶] + 𝐸[𝑅𝐶]

Figure 3 .

 3 Figure 3. Principle of construction of Optimized Maintenance and Capacitated Routing Model (OMCR) Each customer iVc has i times a PM operation over the scheduling horizon . The MM sub-model allows to obtain time windows for each operation [ej,lj]i ; j=1,2,…, i. We thus consider an auxiliary grouped set of nodes V'={0,1,2,…,n} where 0 is the depot and n=i represents the cardinal of the set of all the PM operations over the horizon . Then, we define an

Figure 4 .

 4 Figure 4. Divide to Conquer algorithm, adapted to the HRM sub-model The MILP of the HRM sub-model can be stated as follows: 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 ∑ ∑ (𝐶𝑜𝑒𝑓. 𝐶𝐾. 𝑑 𝑖𝑗 + 𝐶𝐻. 𝑡 𝑖𝑗 ). 𝑥 𝑖𝑗 + 𝐶𝑜𝑒𝑓. 𝐶𝑈. 𝑚 𝑛 𝑗=0 𝑛 𝑖=0 (1) 𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 -𝑄. 𝑚 ≤ -∑ 𝑑 𝑖 𝑛 𝑖=1 ; (2) ∑ 𝑥 0𝑗 𝑛 𝑗=1 -𝑚 = 0 ; (3) ∑ 𝑥 𝑖𝑖 𝑛 𝑖=0 = 0 ; (4) ∑ 𝑥 𝑖𝑗 𝑛 𝑖=0,𝑖≠𝑗 = 1, ∀𝑗 ∈ 𝑉 ′ -{0} ; (5) ∑ 𝑥 𝑖𝑗

Figure 5 .

 5 Figure 5. General framework without removing the loop of position: instance 1

Figure 6 .

 6 Figure 6. Number of PM operations within a month: instance 1

Distributed Maintenance: Design, Scheduling and Capacitated Routing Problem*

	2 is dedicated to the problem definition. The optimization of
	Distributed Maintenance is presented in section 3. Section 4
	conducts experiments to highlight the relevance of the proposed
	approach. The last section contains our conclusion and future
	research perspectives.
	Rony A. DJEUNANG MEZAFACK, Zineb SIMEU-ABAZI, Maria DI MASCOLO II. PROBLEM DEFINITION

TABLE I

 I

	.	DATA ON PUMP'S FAILURES

i CPMi a ($) TPMi a (hour) Cwi a ($/hour) fi(t) b

	1	183	9	15	N(45,4)
	2	121	5	14	N(54,4.5)
	3	193	6	19	W(66,3.5)
	4	156	8	14	W(100,3.5)
	5	138	8	13	W(63,3.5)
	6	194	10	17	N(44,4.4)
	7	163	9	15	W(84,3.5)
	8	100	9	12	N(78,7)
	9	193	6	13	N(96,8.728)
	10	105	10	18	N(75,6.819)

a. CPMi: service cost of replacing a pump; TPMi: time necessary to replace a pump; Cwi: downtime cost per time unit b. N(, ) denotes the Normal probability density function with mean  and standard deviation ; and W(, k) denotes the Weibull probability density function with scale parameter  and shape parameter k. [hours]

TABLE II .

 II COMPARISON OF THE PROPOSED FRAMEWORK

			E[DMC] ($/h)		Computation Time (min)
	Instances	With Loop	Without Loop	Gap	With Loop	Without Loop	Gap
	1	92.5	93.1	1%	125.4	9.2	-1263%
	2	87.5	90.0	3%	128.5	11.4	-1032%
	3	86.1	93.6	8%	118.6	10.7	-1005%
	4	93.8	99.3	6%	129.2	11.0	-1080%
	5	93.7	95.2	2%	124.6	12.3	-911%
	6	92.9	94.7	2%	122.3	12.2	-901%
	7	89.3	92.2	3%	148.2	12.4	-1098%
	8	92.3	96.3	4%	156.9	9.5	-1546%
	9	88.1	95.9	8%	163.9	14.2	-1052%
	Figure 7. Interval of the results using the proposed model

It is necessary to define a tolerance interval for convergence. The optimization stops if E[DMC] doesn't exceed the bound after several iterations.