Stefano Bosi
email: stefano.bosi@univ-evry.fr

Epee

Cuong Le Van
email: cuong.le-van@univ-paris1.fr

Ngoc-Sang Pham
email: pns.pham@gmail.com

Intertemporal equilibrium with production: bubbles and efficiency

Keywords: Intertemporal equilibrium, physical capital bubble, efficiency, infinite horizon. JEL Classification : C62, D31, D91, G10

We consider a general equilibrium model with heterogeneous agents, borrowing constraints, and exogenous labor supply. First, the existence of intertemporal equilibrium is proved even if the aggregate capitals are not uniformly bounded above and the production functions are not time invariant. Second, (i) we call by physical capital bubble a situation in which the fundamental value of physical capital is lower than its price, (ii) we say that the interest rates are low if the sum of interest rates is finite. We show that physical capital bubble is equivalent to a situation with low interest rates. Last, we prove that with linear technologies, every intertemporal equilibrium is efficient. Moreover, there is a room for both efficiency and bubble.

Introduction

Following Becker, Bosi, Le [START_REF] Becker | On existence and bubbles of Ramsey equilibrium with borrowing constraints[END_REF], we consider a dynamic general equilibrium model with heterogeneous agents. However, our framework is different from their model in three points: (i) for simplicity, we consider exogenous labor suply, (ii) our technology is not stationary, (iii) aggregate capital stock is not necessarily uniformly bounded from above. Heterogeneous agents decide to invest and consume. If they invest in physical capital, this asset will not only give them return in term of consumption good at the next period but also give back a fraction of the same asset (after being depreciated). Agents cannot borrow.

Our first contribution is to prove the existence of intertemporal general equilibrium. To do it, we firstly prove the existence of equilibrium for each T -truncated economy. Hence, we have a sequence of equilibria which depend on T . We then prove that this sequence has a limit (for the product topology) which is an equilibrium for the infinite horizon economy.

We say that physical capital bubble occurs at equilibrium (for short, bubble) if the price of the physical asset is greater than its fundamental value. We say interest rates are low at equilibrium (for short, low interest rates) if the sum of returns on capital is finite. Our second contribution to prove that bubble is equivalent to low interest rates. The no-bubble result in Becker, Bosi, Le [START_REF] Becker | On existence and bubbles of Ramsey equilibrium with borrowing constraints[END_REF] can be viewed as a particular case of our result. Indeed, in Becker, Bosi, Le [START_REF] Becker | On existence and bubbles of Ramsey equilibrium with borrowing constraints[END_REF], thanks to the concavity of a stationary technology, the aggregate capital stock is uniformly bounded, and then real return of the physical capital is uniformly bounded from below. Therefore, the sum of returns equals infinity. According to our result, the physical capital bubble is ruled out. However, when we allow non-stationary production functions, there may be a bubble at equilibrium. To see the point, take linear production functions whose productivity at date t is denoted by a t . At equilibrium, real return of physical capital at date t must be a t . As mentioned above, there is a bubble if and only if ∞ t=0 a t < ∞. We can now see clearly that there is a bubble if productivities decrease with sufficiently high speed.

Our third contribution is about the efficiency of intertemporal equilibrium. An intertemporal equilibrium is called to be efficient if its aggregate capital path is efficient in sense of [START_REF] Malinvaud | Capital accumulation and efficient allocation of resources[END_REF]. We prove that with linear production functions, every intertemporal equilibrium is efficient. However, as we mentioned above, this efficient intertemporal equilibrium may have bubble if productivities decrease with sufficiently high speed. Therefore, we have both efficient and bubble at equilibrium with such technologies. Note that our result does not require any conditions about the convergence or boundedness of the capital path as in previous literature.

Related literature

(1) On rational bubbles. [START_REF] Tirole | On the possibility of speculation under rational expectations[END_REF] proved that there is no financial asset bubble in a rational expectation model without endowment. A survey on bubble in models with asymmetric information, overlapping generation, heterogeneous-beliefs can be found in [START_REF] Brunnermeier | Bubbles, financial crises, and systemic risk[END_REF]. Doblas-Madrid Doblas-Madrid (2012) presented a model of speculative bubbles where rational agents buy an overvalued asset because given their private information, they believe they have a good chance of reselling at a profit to a greater fool. Martin and Ventura (1953), [START_REF] Ventura | Economic growth with bubbles[END_REF] did not define bubble as we do. Instead, they defined bubble as a short-lived asset.

(2) On the efficiency of a capital path. [START_REF] Malinvaud | Capital accumulation and efficient allocation of resources[END_REF] introduced the concept of efficiency of a capital path and give a sufficient condition of the efficiency: lim t→∞ p t K t = 0, where (p t ) is a sequence of competitive prices, (K t ) is the capital path.1 Following Malinvaud, [START_REF] Cass | On capital overaccumulation in the aggregative, neoclassical model of economic growth: A complete characterization[END_REF] considered capital path which is uniformly bounded from below. Under the concavity of a stationary production function and some mild conditions, he proved that a capital path is inefficient if and only if the sum (over time) of future values of a unit of physical capital is finite. [START_REF] Cass | Present values playing the role of efficiency prices in the one-good growth model[END_REF] given a necessary and sufficient condition for a consumption plan (C) to be efficient, which can be stated that the limit inferior of differences between the present value of any consumption plan and the plan (C) is negative. Our paper is also related to [START_REF] Becker | Efficient Ramsey equilibria[END_REF] where they proved that a Ramsey equilibrium is efficient if the most patient household is not credit constrained from some date. However, their result is based on the fact that consumption of each household is uniformly bounded from below. In our paper, we do not need this condition. Instead, the efficient capital path in our model may converge to zero. Mitra and [START_REF] Ray | On the Phelps-Koopmans theorem[END_REF] studied the efficiency of a capital path with nonconvex production technologies and examined whether the Phelps-Koopmans theorem is valid. However, their results are no longer valid without the convergence or the boundedness of capital paths.

(3) Another concept of efficiency is constrained efficiency. About the constrained efficiency in general equilibrium models with financial asset, see [START_REF] Kehoe | Debt Constrained Asset Markets[END_REF], Alvarez and Jermann (2000), Bloise and Pietro (2011). About the constrained efficiency in the neoclassical growth model, see Davila, Hong, Krusell and Rios-Rull (2012).

The remainder of the paper is organized as follows. Section 2 describes the model. In section 3, existence of equilibrium is proved. Section 4 studies physical capital bubble. Section 5 explores our results on the efficiency of equilibria. Conclusion will be presented in Section 6. Technical details are gathered in Appendix.

Model

We follow Becker, Bosi, Le [START_REF] Becker | On existence and bubbles of Ramsey equilibrium with borrowing constraints[END_REF], but we consider: (i) exogenous labor supply, (ii) non-stationary production functions.

Consumption good: at each period t, price of consumption is denoted by p t and agent i consumes c i,t units of consumption good.

Physical capital: at time t, if agent i decides to buy k i,t+1 ≥ 0 units of new capital, then at period t + 1, after begin depreciated, agent i will receive (1δ)k i,t+1 units of old capital and a return on capital k i,t+1 at the rate r t+1 . Here, δ is the rate of capital depreciation rate.

Each household i takes sequences of prices (p, r) = (p t , r t ) ∞ t=0 as given and solves

(P i (p, r)) : max (c i,t ,k i,t+1 ) m i=1 +∞ t=0 +∞ t=0 β t i u i (c i,t ) (1)
subject to :

k i,t+1 ≥ 0 (2) p t (c i,t + k i,t+1 -(1 -δ)k i,t ) ≤ r t k i,t + θ i π t (p t , r t ), (3)
where (θ i ) m i=1 is the share of profit, θ i ≥ 0 for all i and

m i=1 θ i = 1.
Firm: For each period, there is a representative firm who takes prices (p t , r t ) as given and maximizes its profit.

(P (r t )) :

π t (p t , r t ) := max Kt≥0 p t F t (K t ) -r t K t
We write π t instead of π t (p t , r t ) if there is no confusion.

Definition 1. A sequence of prices and quantities pt , rt ,

(c i,t , ki,t+1 ) m i=1 , Kt +∞ t=0 is an equilibrium of the economy E = (u i , β i , k i,0 , θ i ) m i=1
, F, if the following holds.

(i) Price positivity: pt , rt > 0 for t ≥ 0.

(ii) Market clearing: at each t ≥ 0, good :

m i=1 [c i,t + ki,t+1 -(1 -δ) ki,t ] = F t ( Kt ) (4) capital : Kt = m i=1 ki,t . (5)
(iii) Optimal consumption plans: for each i,

(c i,t , ki,t+1 ) m i=1 ∞ t=0
is a solution to problem

(P i (p, r)).
(iv) Optimal production plan: for each t ≥ 0, ( Kt ) is a solution to problem (P (r t )).

The existence of equilibrium

The following result proves that the feasible aggregate capital and the feasible consumption are bounded for the product topology.

Lemma 1. Feasible individual and aggregate capitals and feasible consumptions are in a compact set for the product topology. Moreover, they are uniformly bounded if there exists t 0 and an increasing, concave function G such that: (i) for every t ≥ t 0 we have F t (K) ≤ G(K) for every K, (ii) there exists x > 0 such that G(y) + (1δ)y ≤ y for every y ≥ x.

Proof. Denote

D 0 := D 0 (F 0 , δ, K 0 ) := F 0 (K 0 ) + (1 -δ)K 0 , D t := D t ((F s ) t s=0 , δ, K 0 ) := F t (D t-1 ((F s ) t-1 s=0 , δ, K 0 )) + (1 -δ)D t-1 ((F s ) t-1 s=0 , δ, K 0 ), ∀t ≥ 0. Then m i=1 c i,t + K t+1 ≤ D t for every t ≥ 0.
We now assume t 0 and the function G (as in Lemma 1) exist. We are going to prove that 0

≤ K t ≤ max{D 0 , ..., D t 0 -1 , x} =: M . Indeed, K t ≤ D t-1 ≤ M for every t ≤ t 0 . For t ≥ t 0 , we have K t+1 = m i=1 k i,t+1 ≤ G(K t ) + (1 -δ)K t . Then K t 0 +1 ≤ G(K t 0 ) + (1 -δ)K t 0 ≤ G(M ) + (1 -δ)M ≤ M . Iterating the argument, we obtain K t ≤ M for each t ≥ 0. Feasible consumptions are bounded because m i=1 c i,t ≤ G(K t ) + (1 -δ)K t .
We need the following assumptions.

Assumption (H1): For each i, the utility function u i of agent i is strictly increasing, strictly concave, continuous differentiable, and u(0

) = 0, u ′ (0) = ∞.
Assumption (H2): F t (•) is continuously differentiable, strictly increasing, concave, the input is essential (F t (0) = 0) and F t (∞) = ∞.

Assumption (H3): δ ∈ (0, 1) and k i,0 > 0 for every i.2 Assumption (H4): For each i, utility of agent i is finite

∞ t=0 β t i u i (D t ) < ∞.

Existence of equilibrium in E T

We define Ttruncated economy E T as E but there are no activities from period T + 1 to the infinity, i.e., c i,t = k i,t = 0 for every i = 1, . . . , m, and for every t ≥ T + 1.

In the economy E T , agent i takes sequences of prices (p, r) = (p t , r t ) T t=0 as given and maximizes his intertemporal utility by choosing consumption and investment levels.

(P i (p, r)) : max (c i,t ,k i,t+1 ) T t=0 T t=0 β t i u i (c i,t ) (6)
subject to:

k i,t+1 ≥ 0, (7)
(budget constraints) p t (c i,t + k i,t+1 -(1 -δ)k i,t ) ≤ r t k i,t + θ i π t , (8)
where k i,T = 0. Then we define the bounded economy E T b as E T but all variables are bounded in the following bounded sets:

(c i,t ) T t=0 ∈ C i := [0, B c ] T +1 (k i,t ) T +1 t=1 ∈ K i := [0, B k ] T K := (K t ) T +1 t=1 ∈ K := [0, B] T , where B c > max t F t (B) + (1 -δ)B, B > mB k .
Proposition 1. Under Assumptions (H1) -(H3), there exists an equilibrium for E T b .

Proof. See Appendix.

Proposition 2. An equilibrium of the economy E T b is also an equilibrium of the unbounded economy E T .

Proof. Similar to the one in Becker, Bosi, Le [START_REF] Becker | On existence and bubbles of Ramsey equilibrium with borrowing constraints[END_REF].

Existence of equilibrium in E

Theorem 1. Under Assumptions (H1)-(H4), there exists an equilibrium.

Proof of Theorem 1. We have shown that for each T ≥ 1, there exists an equilibrium for the economy E T . We denote by (p T , rT , (c T i , kT i ) m i=1 , KT ) an equilibrium of Ttruncated economy E T . We can normalize by setting pT t + rT t = 1 for every t ≤ T . We see that

0 < cT i,t , KT t ≤ D t .
Without loss of generality, we can assume that

(p T , rT , (c T i , kT i ) m i=1 , KT )
T →∞ ----→ (p, r, (c i , ki ) m i=1 , K) (for the product topology ).

We are going to prove that: (i) all markets clear, (ii) at each date t, Kt is a solution to the firm's maximization problem, (iii) rt > 0 for each t ≥ 0, (iv) (c i , ki ) is a solution to the maximization problem of agent i for each i = 1, . . . , m, (v) pt > 0 for each t. Consequently, we obtain that (p, r, (c i , ki ) m i=1 , K) is an equilibrium for the economy E.

(i) By taking the limit of market clearing conditions for the truncated economy, we obtain the market clearing conditions for the economy E.

(ii) Take K ≥ 0 arbitrary. We have pT

t F t (K) -rT t K ≤ pT t F t ( KT t ) -rT t KT t .
Let T tend to infinity, we obtain that pt F t (K)rt K ≤ pt F t ( Kt )rt Kt . Therefore, the optimality of Kt is proved.

(iii) If rt = 0 then pt = 1 (since rT t + pT t = 1). The optimality of Kt implies that Kt = ∞. This is a contradiction, because we have Kt = lim (iv) First, we give some notations. For each i and t, we define B T i (p, r) and C T i (p, r) as follows

B T i (p, r) := (c i,t , k i,t+1 ) T t=0 ∈ R T +1 + × R T +1 + : (a) k i,T +1 = 0, (b) ∀t = 0, . . . , T, k i,t+1 > 0, pt [c i,t + k i,t+1 -(1 -δ)k i,t ] < rt k i,t + θ i π t (p t , rt ) C T i (p, r) := (c i,t , k i,t+1 ) T t=0 ∈ R T +1 + × R T +1 + : (a) k i,T +1 = 0, (b) ∀t = 0, . . . , T, k i,t+1 ≥ 0, pt [c i,t + k i,t+1 -(1 -δ)k i,t ] ≤ rt k i,t + θ i π t (p t , rt ) .
Since rt > 0 for every t, it is easy to prove that B T i (p, r) = ∅.

Let (c i , k i ) be an feasible allocation of the problem P i (p, r). We have to prove that

∞ t=0 β t i u i (c i,t ) ≤ ∞ t=0 β t i u i (c i,t ). We define (c ′ i,t , k ′ i,t+1 ) T t=0 as follows: c ′ i,t = c i,t for every t ≤ T , = 0 if t > T ; k ′ i,t+1 = k i,t+1 for every t ≤ T -1, = 0 if t ≥ 0. We see that (c ′ i,t , k ′ i,t+1 ) T t=0 belongs to C T i (p, r). Since B T i (p, r) = ∅, there exists a sequence (c n i,t , k n i,t+1 ) T t=0 ∞ n=0 ∈ B T i (p, r) with k n
i,T +1 = 0, and this sequence converges to (c ′ i,t , k ′ i,t+1 ) T t=0 when n tends to infinity. We have

pt (c n i,t + k n i,t+1 -(1 -δ)k n i,t ) < rt k n i,t + θ i π t (p t , rt )
We can chose s 0 > T , high enough, such that: for every s ≥ s 0 , we have

ps t (c n i,t + k n i,t+1 -(1 -δ)k n i,t ) < rs t k n i,t + θ i π t (p s t , rs t ).
It means that (c n i,t , k n i,t+1 ) T t=0 ∈ C T i (p s , rs ). Therefore, we get T t=0

β t i u i (c n i,t ) ≤ s t=0 β t i u i (c s i,t ).
Let s tend to infinity, we obtain T t=0

β t i u i (c n i,t ) ≤ ∞ t=0 β t i u i (c i,t ).
Let n tends to infinity, we have T t=0

β t i u i (c i,t ) ≤ ∞ t=0 β t i u i (c i,t
) for every T .

Let T tend to infinity, we obtain

∞ t=0 β t i u i (c i,t ) ≤ ∞ t=0 β t i u i (c i,t ).
(v) p t is strictly positive thanks to the strict increasingness of the utility functions.

Physical asset bubble

Let p t , r t , (c i,t , k i,t ) m i=1 , K t +∞ t=0 be an equilibrium. Without loss of generality, we assume that p t = 1 for every t.

Lemma 2. For each t, we have

1 = (1 -δ + r t+1 p t+1 )γ t+1 (9)
where γ t+1 := max i∈{1,...,m}

β i u ′ i (c i,t+1 ) u ′ i (c i,t )
.

Proof. Firstly, we write all FOCs for the economy E. Denote by λ i,t the multiplier with respect to the budget constraint of agent i and by µ t+1 the multiplier with respect to the borrowing constraint (i.e., k T i,t+1 ≥ 0) of agent i.

β t i u ′ i (c i,t ) = λ i,t p t λ i,t p t = λ i,t+1 (r t+1 + p t+1 (1 -δ)) + µ i,t+1 µ i,t+1 k i,t+1 = 0.
Therefore, we have

p t+1 r t+1 + p t+1 (1 -δ) ≥ β i u ′ i (c i,t+1 ) u ′ i (c i,t )
for every i.

Since K t > 0 at equilibrium, there exists i such that k i,t+1 > 0. For such agent, we have µ i,t+1 = 0. Thus, λ i,t p t = λ i,t+1 (r t+1 + p t+1 (1δ)). Consequently, we get (9)

Definition 2. We define the discount factor of the economy from initial date to date t as follows

Q 0 := 1, Q t := t s=1 γ s , t ≥ 1. ( 10
)
According to Lemma 2, we have

Q t = (1 -δ + r t+1 p t+1 )Q t+1 for every t ≥ 0. As a consequence, we can write 1 = (1 -δ + r 1 p 1 )Q 1 = (1 -δ)Q 1 + r 1 p 1 Q 1 = (1 -δ)(1 -δ + r 2 p 2 )Q 2 + r 1 p 1 Q 1 = (1 -δ) 2 Q 2 + (1 -δ) r 2 p 2 Q 2 + r 1 p 1 Q 1 = • • • = (1 -δ) T Q T + T t=1 (1 -δ) t-1 r t p t Q t .
Interpretation. In this model, physical capital is viewed as a long-lived asset.

1. At date 1, one unit (from date 0) of this asset will give (1-δ) units of physical capital and r 1 p 1 units of consumption good as its dividend.

2. At date 2, (1δ) units of physical capital will give (1δ) 2 units of physical capital and (1δ) r 2 p 2 units of consumption good ... Therefore, the fundamental value of physical capital at date 0 can be defined by

F V 0 = ∞ t=1 (1 -δ) t-1 r t p t Q t .
Definition 3. We say that there is a capital asset bubble if physical capital's price is greater that its fundamental value, i.e., 1

> ∞ t=1 (1 -δ) t-1 rt pt Q t .
We can see that there is a bubble on capital asset if and only if lim

t→∞ (1 -δ) t Q t > 0.
8 Documents de travail du Centre d'Economie de la Sorbonne -2014.43

Definition 4. We say that interest rates are low at equilibrium if

∞ t=1 r t p t < ∞. ( 11
)
Otherwise, we say that interest rates are high.

We now state our main result in this section.

Proposition 3. There is a bubble if and only if interest rates are low Proof. According ( 9), we see that

Q t = (1 -δ + r t+1 p t+1 )Q t+1 . Hence, we have 1 = (1 -δ + r 1 p 1 )Q 1 = (1 -δ + r 1 p 1 )(1 -δ + r 2 p 2 )Q 2 = . . . = Q T T t=1 (1 -δ + r t p t ) = Q T (1 -δ) T T t=1 [1 + r t (1 -δ)p t ].
Consequently, lim

t→∞ (1 -δ) t Q t > 0 if and only if ∞ t=1 [1 + rt (1-δ)pt ] < +∞. This condition is equivalent to ∞ t=1 r t p t < ∞. (12)
It means that interest rates are low.

We point out some consequences of Proposition 3.

Corollary 1. [START_REF] Becker | On existence and bubbles of Ramsey equilibrium with borrowing constraints[END_REF]) Assume that F t = F for every t, F is strictly increasing, strictly concave. Then there is no bubble at equilibrium.

Proof. Since F is strictly increasing and strictly concave, aggregate capital stock is uniformly bounded, i.e., there exists 0 < K < ∞ such that K t ≤ K. Consequently,

rt pt = F ′ (K t ) > F ′ (K) > 0 for every t. This implies that ∞ t=1 rt pt = ∞.
According to Proposition 3, there is no bubble.

Corollary 2. Assume that F t (K) = a t K for each t. Then there is a bubble at equilibrium if and only if

∞ t=1 a t < ∞.
Proof. This is a direct consequence of Proposition 3 .

This result shows that if the productivity decrease to zero with high speed, a bubble in physical capital will appear.

In this section, we study the efficiency of intertemporal equilibrium. Following [START_REF] Malinvaud | Capital accumulation and efficient allocation of resources[END_REF], we define the efficiency of a capital path as follows.

Definition 5. Let F t be a production function, δ be capital depreciation rate. A feasible path of capital is a positive sequence

(K t ) ∞ t=0 such that 0 ≤ K t+1 ≤ F t (K t ) + (1 -δ)K t for every t ≥ 0 and K 0 is given. A feasible path is efficient if there is no other feasible path (K ′ t ) such that F t (K ′ t ) + (1 -δ)K ′ t -K ′ t+1 ≥ F t (K t ) + (1 -δ)K t -K t+1
for every t with strict inequality for some t.

Here, aggregate feasible consumption at date t is defined by

C t := F t (K t ) + (1 -δ)K t - K t+1 .
Definition 6. We say that an intertemporal equilibrium is efficient if its aggregate feasible capital path (K t ) is efficient.

Our main result in this section requires some intermediate steps. First, we have as in [START_REF] Malinvaud | Capital accumulation and efficient allocation of resources[END_REF]).

Lemma 3. An equilibrium is efficient if lim t→∞ Q t K t+1 = 0.
Proof. Let (K ′ t , C ′ t ) be a feasible sequence. We have just to show that lim inf

T →+∞ T t=0 Q t C t -C ′ t ≥ 0. ( 13
)
It is enough to prove that feasibility and first-order conditions imply

T t=0 Q t C t -C ′ t ≥ -Q T K T +1 (14)
Let us prove inequality (14). We have By noticing that K 0 = K ′ 0 and Q t+1 1δ + r t+1 p t+1 -Q t = 0, we then get:

∆ T ≡ T t=0 Q t C t -C ′ t = T t=0 Q t F t (K t ) -F t K ′ t + (1 -δ) K t -K ′ t -K t+1 -K ′ t+1 ≥ T t=0 Q t F ′ t (K t ) K t -K ′ t + (1 -δ) K t -K ′ t ] - T t=0 Q t K t+1 -K ′ t+1 = T t=0 Q t 1 -δ + r t p t K t -K ′ t - T t=0 Q t K t+1 -K ′
∆ T ≥ T t=1 Q t 1 -δ + r t p t K t -K ′ t - T t=0 Q t K t+1 -K ′ t+1 = T -1 t=0 Q t+1 1 -δ + r t+1 p t+1 -Q t K t+1 -K ′ t+1 -Q T K T +1 -K ′ T +1 ≥ T -1 t=0 Q t+1 1 -δ + r t+1 p t+1 -Q t K t+1 -K ′ t+1 -Q T K T +1 = -Q T K T +1 .
We also have the transversality condition of each agent.

Lemma 4. At any equilibrium, we have lim t→∞

β t i u ′ i (c i,t
)k i,t+1 = 0 for every i.

Proof. See [START_REF] Kamihigashi | A simple proof of the necessity of the transversality condition[END_REF].

The following result shows the impact of borrowing constraints on the efficiency of an intertemporal equilibrium.

Lemma 5. Consider an equilibrium. If there exists a date such that, from this date on, the borrowing constraints of agents are not binding at this equilibrium, then it is efficient.

Proof. Assume that there exists t 0 such that k i,t > 0 for every i and for every t ≥ t 0 . Then we have: for every

t ≥ t 0 Q t Q t 0 = β t-t 0 i u ′ i (c i,t ) u ′ i (c i,t 0 )
.

According to Lemma 4, we have lim t→∞

β t i u ′ i (c i,t )k i,t+1 = 0. Then lim t→∞ Q t k i,t+1
= 0 for every i. This implies that lim t→∞ Q t K t+1 = 0. Therefore, this equilibrium is efficient.

We now state our main finding in this section.

Proposition 4. Assume that the production functions are linear. Then every equilibrium path is efficient.

Proof. Since production functions are linear, profit equals to zero. Recall that we have c i,t > 0 for every i and every t. This implies that k i,t > 0 at equilibrium. According to Lemma 5, every equilibrium path is efficient.

Corollary 2 and Proposition 4 indicate that with linear production functions, there exists an equilibrium the capital path of which is efficient and a bubble may arise at this equilibrium.

We build infinite-horizon dynamic deterministic general equilibrium models in which heterogenous agents invest in physical capital and consume. We proved existence of equilibrium in this model, even if technologies are not stationary and aggregate capital is not uniformly bounded.

We say there is a bubble of physical capital at equilibrium if the physical capital's price is greater than its fundamental value. We point out that bubble is equivalent to low interest rates. In particular, there is a bubble if productivities decrease with high speed.

With linear technologies, every intertemporal equilibrium is efficient. Interestingly, it is possible to have both bubble and efficient at equilibrium.

A Appendix: Existence of equilibrium for the truncated economy

Proof of Proposition 1. Denote ∆ :

= {z 0 = (p, r) : 0 ≤ p t , r t ≤ 1, p t + r t = 1 ∀t = 0, . . . , T }, B i (p, r) := (c i , k i ) ∈ C i × K i such that : ∀t = 0, . . . , T p t (c i,t + k i,t+1 -(1 -δ)k i,t ) < r t k i,t + θ i π t , and
C i (p, r) := (c i , k i ) ∈ C i × K i such that : ∀t = 0, . . . , T p t (c i,t + k i,t+1 -(1 -δ)k i,t ) ≤ r t k i,t + θ i π t ,
Denote by Bi (z 0 ) the closure of B i (z 0 ). Lemma 6. For every (p, r) ∈ P, we have B i (p, q) = ∅ and Bi (p, q) = C i (p, q). Proof. We rewrite B i (p, r) as follows

B i (p, r) := (c i , a i ) ∈ C i × A i such that : ∀t = 0, . . . , T 0 < p t ((1 -δ)k i,t -c i,t -k i,t+1 ) + r t k i,t + θ i π t . Since (1 -δ)k i,0 > 0, we can choose c i,0 ∈ (0, B c ) and k i,1 ∈ (0, B k ) such that 0 < p 0 ((1 -δ)k i,0 -c i,0 -k i,1 ) + r 0 k i,0 + θ i π 0 .
By induction, we see that B i (p, r) is not empty.

Lemma 7. B i (p, r) is a lower semi-continuous correspondence on P := ∆ T +1 . And C i (p, r) is upper semi-continuous on P with compact convex values.

Proof. Clearly, since B i (p, r) is empty and has an open graph.

We define Φ :

= ∆ × m i=1 (C i × K i ) × K. An element z ∈ Φ is in the form z = (z i ) m+1 i=0
where z 0 := (p, r), z i := (c i , k i ) for each i = 1, . . . , m, and z m+1 = K.

We now define correspondences. First, we define ϕ 0 (for additional agent 0)

ϕ 0 : m i=1 (C i × K i ) × K → 2 ∆ ϕ 0 ((z i ) m+1 i=1 ) := arg max (p,r)∈∆ T t=0 p t m i=1 [c i,t + k i,t+1 -(1 -δ)k i,t ] -F t (K t ) + T t=0 r t K t - m i=1 k i,t .
For each i = 1, . . . , m, we define

ϕ i : ∆ → 2 C i ×K i ϕ i (p, r) := arg max (c i ,k i )∈C i (p,r) T t=0
β t i u i (c i,t ) .

For each i = m + 1, we define

ϕ m+1 : ∆ → 2 K ϕ i (p, r) := arg max K∈K T t=0 p t F t (K t ) -r t K t .
Lemma 8. ϕ i is upper semi-continuous convex-valued correspondence for each i = 0, 1, . . . , m+ 1.

Proof. This is a direct consequence of the Maximum Theorem.

According to the Kakutani Theorem, there exists (p, r, (c i , ki ) We know that Xt , Ȳt ≤ 0 and pt , rt > 0. Then, Xt = Ȳt = 0. The optimality of (c i , ki ) and K comes from ( 16) and ( 17).

6

  t→∞KTt ≤ D t < ∞. Documents de travail du Centre d'Economie de la Sorbonne -2014.43

  du Centre d'Economie de laSorbonne -2014.43

  ,t + ki,t+1 -(1δ) ki,t ] = F ( Kt )Proof. Since prices are strictly positive and the utility functions are strictly increasing, all the budget constraints are binding and, summing them across the individuals, we get pt Xt + rt Ȳt = 0.(23)

  + ki,t+1 -(1δ) ki,t ] -F t ( Kt ) and Ȳt = Ktand capital respectively. For every (p, r) ∈ ∆ T +1 , we have Therefore, we have Xt , Ȳt ≤ 0, which implies that Lemma 9. pt , rt > 0 for t = 0, . . . , T .Proof. If pt = 0 then ci,t = B c > (1δ)B + F t (B). Therefore, we get ci,t + ki,t+1 > F t ( Kt ) which is a contradiction. Hence, pt > 0.If rt = 0, then the optimality of K implies that K t = B. However, we have ki,t ≤ B k for every i, t. Consequently, ≤ mB k < B = K t , contradiction to (22). Therefore, we get rt > 0.

			m	m
			ci,t + ki,t+1 ≤ (1 -δ)	ki,t + F t ( Kt )	(21)
			i=1	i=1
				m
			Kt ≤	ki,t .	(22)
				i=1
	(1 -δ) ki,t Lemma 10. m i=1 ki,t + m i=1
					m i=1 , K) such that
			(p, r) ∈ ϕ 0 ((c i , ki ) m i=1 , K)	(15)
			(c i , ki ) ∈ ϕ i (p, r)	(16)
			K ∈ ϕ m+1 (p, r).	(17)
	Denote by Xt :=	m	[c i,t m	ki,t the excess
		i=1			i=1
	demands for goods T	T
			(p t -pt ) Xt +	(r t -rt ) Ȳt ≤ 0.	(18)
			t=0	t=0
	By summing the budget constraints, for each t, we get
			pt Xt + rt Ȳt ≤ 0.	(19)

Hence, we have: for every (p t , r t ) ∈ ∆

p t Xt + q t Ȳt ≤ pt Xt + rt Ȳt ≤ 0. (

20

)

13 Documents de travail du Centre d'Economie de la Sorbonne -2014.43

Documents de travail du Centre d'Economie de laSorbonne -2014.43

See Malinvaud (1953), Lemma 5, page

Documents de travail du Centre d'Economie de laSorbonne -2014.43

Becker, Bosi, Le Van and Seegmuller (3) weekens H3 by assuming m i=1 ki,0 > 0 because of positive labor income.

J. Ventura, Bubbles and capital flows, Journal of Economic Theory 147,738-758 (2012).


Julio Cesar

Dos Reis

Duy Dinh

Cédric Pruski

Marcos Da Silveira

Chantal Reynaud-Delaître

The influence of similarity between concepts in evolving biomedical ontologies for mapping adaptation

Keywords: mapping adaptation, mapping evolution, mapping maintenance, similarity metrics, ontology alignment, ontology evolution, ontology versions

Biomedical ontologies continuously evolve and impact associated mappings. This demands to adapt ontology mappings to maintain them up-to-date. This article studies whether similarity calculated between values of concept attributes issued from successive ontology versions plays a role in deciding mapping adaptation actions. We empirically analyse the evolution of official mappings established between large biomedical ontologies. The results point out the relevance of this factor for mapping adaptation.

Introduction

The needs of exchanging and retrieving data between biomedical systems have increasingly become relevant. Ontology mappings play a key role in enabling semantic interoperability in this context [START_REF] Burgun | Accessing and integrating data and knowledge for biomedical research[END_REF]. They interconnect concepts of domain-related ontologies allowing systems to interpret data annotated with different ontologies. The huge size and existing intersections between ontologies force linking them through mappings. However, to remain useful and reflect the most up-to-date knowledge of the domain, these ontologies evolve and new versions are periodically released. This potentially impacts existing mappings demanding methods to ensure, as automatic as possible, their semantic consistency over time.

We have designed the DyKOSMap framework to adapt ontology mappings through a set of Mapping Adaptation Actions (MAAs) [START_REF] Reis | Mapping adaptation actions for the automatic reconciliation of dynamic ontologies[END_REF]. Our approach aims at deciding which action to apply when some ontology change affects the source concept of a correspondence. Existing tools enable to calculate simple and complex ontology changes given two successive ontology versions [START_REF] Hartung | COnto-Diff: Generation of Complex Evolution Mappings for Life Science Ontologies[END_REF]. Indeed, various factors may influence the action decision for each correspondence individually [START_REF] Reis | Mapping adaptation actions for the automatic reconciliation of dynamic ontologies[END_REF]. We have studied how different types of ontology changes correlate with the adaptation actions [START_REF] Gross | How do computed ontology mappings evolve ? A case study for life science ontologies[END_REF][5] [START_REF] Reis | Characterizing Semantic Mappings Adaptation via Biomedical KOS Evolution: A Case Study Investigating SNOMED CT and ICD[END_REF]. Nevertheless, the complete understanding of this phenomenon demands further studies inquiring other factors to take into account in the mapping adaptation process.

This article reports on the influence of the similarity relatedness for mapping adaptation. We calculate the similarity between textual statements from source concepts of mappings with textual statements of parents, children and sibling concepts issued from a new ontology version. These statements are attribute values characterizing the concepts. For instance, an attribute a i , of type name, contains the value "cardio vascular disease". We investigate whether the MAAs correlate with the behaviour of similarity values observed. We hypothesize that the similarity aspect stands for an element which may help deciding the adequate actions to adapt mappings.

Methods

An ontology O consists of a set of concepts interrelated by directed relationships. Each concept c i j at time j has a unique identifier, and a set of attributes that characterizes concepts where ATT(c i )={a 1 , a 2 ,..., a n } (e.g., name, definition, synonym, etc.). We define a set of concepts of an ontology O X at time j as CNC(O X j ) = {c 1 j , c 2 j ,..., c n j }. A relationship r interconnects two concepts and has a specific type, e.g., "subsumption", "part-of", etc.

The context CT of a concept c i in the ontology stands for the union of the sets of sup(c i ), sub(c i ) and sib(c i ) concepts of c i , as following:

CT(c i ) = sup(c i ) ∪ sub(c i ) ∪ sib(c i ) where :sup(c i ) ={c k |c k ∈ CNC(O), c i ⊏c k ∧ c i c k }, sub(c i ) ={c k |c k ∈ CNC(O), c k ⊏c i ∧ c i c k }, sib(c i ) ={c k |c k ∈ CNC(O), sup(c k )∩sup(c i )  } to which c i ⊏c k stands for "ci is narrower
or more specific than cj", e.g., "hypotension" is more specific than"vascular disease".

An ontology mapping M S,T We have expressed behaviours of mapping adaptation as Mapping Adaptation Actions [START_REF] Reis | Mapping adaptation actions for the automatic reconciliation of dynamic ontologies[END_REF] (cf. Figure 1). In MoveM the source concept c s of the correspondence is replaced by another concept c k . Similarly, in DeriveM the original correspondence remains and a new correspondence appears connecting a concept c k with c t . In both actions, c k 1 ∈ CT(c s 1 ). In RemoveM the correspondence is deleted and the action ModifySemTypeM consists in modifying the type of semantic relation. The no-action refers to the cases where correspondences remain unchanged.

We conduct our experiments using various releases of official mappings established between biomedical ontologies including: SNOMED-CT (SCT), MeSH, ICD-9-CM (ICD9) and ICD-10-CM (ICD10). In particular, mappings interconnect SCT(2010)-ICD9(2009) and SCT(2012)-ICD9(2011); MeSH(2012)-ICD10(2011) and MeSH(2013)-ICD10(2011). We observe the evolution of the biomedical ontologies and mappings through the following procedure:

 Calculate MAAs: Given all correspondences between two ontologies, for each correspondence impacted by some ontology change, we determine a list of MAAs. We remove correspondences where the source concept remains unchanged from one ontology version to another, or where ontology changes simultaneously affect both source and target concepts. We use the COnto-Diff tool to calculate ontology changes [START_REF] Hartung | COnto-Diff: Generation of Complex Evolution Mappings for Life Science Ontologies[END_REF]. Having successive releases of mappings allows us to calculate the MAAs for each correspondence. We compare the elements composing a correspondence (identifier of source and target concept, and relation type). For instance, given a correspondence at time j, we search its elements in the mapping at time j+1. If we fail to find, we sign the RemoveM action for such correspondence. We use a similar approach to determine the other MAAs.  Identify relevant attributes: For each source concept impacted by ontology changes, we identify a minimal set of relevant attributes (we assign 3, but it can be parameterized) representing the most relevant attributes for a given correspondence, since this is established on partial information (e.g., some attributes) defining concepts. These source concept attributes consist of the most similar to the ones in the target concept. We pre-selected groups of attributes to calculate the similarity value. For instance, we do not compare the ICD9 attribute "Exclude" with the SCT attribute "Name", if a correspondence exists.  Calculate similarity: For each relevant attribute of a source concept c s in an impacted correspondences at time j 0 , we calculate the similarity relatedness with attributes issued from concepts in the context of c s at time j 1 . To this end, we explore string-based similarity metrics, especially the bi-gram because this measure performs well on ontology matching tasks [START_REF] Cheatham | String similarity metrics for ontology alignment[END_REF]. The similarity function receives two attribute values and returns a value ranging from 0 to 1. The higher the result is, the more similar these attributes are. We analyze the density of calculated similarity values ranging from 0 to 1 for each MAAs by using the kernel density estimation method [START_REF] Sheather | A Reliable Data-Based Bandwidth Selection Method for Kernel Density Estimation[END_REF]. The density stands for a smoothing distribution of frequencies of the similarity values similar to a histogram. We observe the similarity's influence on the MAAs by studying the following scenarios: 1. REL_ATTS_changedCT: For each impacted correspondence, we search the highest similarity value calculated among the relevant attributes identified with all attributes issued of changed concepts in the context of c s at time j 1. We register such maximum similarity value for the MAAs applied. 2. Best_REL_ATT_changedCT: This is similar to scenario 1, but we only consider the best relevant attribute for calculating the similarity (i.e., the most significant attribute for c s regarding the c t ). We aim to examine the influence of this particular attribute with respect to all relevant attributes. 3. REL_ATTS_unchangedCT: This scenario also performs similar to scenario 1, but examines the unchanged part of the context of c s (i.e., all unchanged concepts in the context). replaces the concept c s . We calculate the density of the highest similarity values among the relevant attributes and the attributes characterizing c k 1 . 6. Best_REL_ATT_conceptMAA: This is similar to scenario 5, but we only consider the best relevant attribute. We repeated this procedure for both set of mappings (SCT-ICD9 and MeSH-ICD10). The total number for each type of MAAs analyzed remained: MoveM (635); DeriveM (768); ModSemTypeM (58), RemoveM (193) and no-action (9280).

Results

Figure 2 presents the obtained kernel estimation based density of the similarity values for the DeriveM and MoveM actions. For DeriveM, we observe a similar behaviour of the density of similarity values in scenarios 1 (REL_ATTS_changedCT) and 2 (Best_ REL_ATT__changedCT), revealing a high frequency of low similarity values. We explain this by the fact that most frequently the changed CT set is empty so similarity values remain zero. In scenarios 3 (REL_ATTS_unchangedCT) and 4 (Best_ REL_ATT_unchangedCT), we observe the contrary behaviour with a bigger density of high similarity values. This shows that the original source concepts appear similar with unchanged concepts in the context. Scenario 4 presents a lower density of high similarity values compared with scenario 3. We point out a higher density of medium similarity in scenario 5 (REL_ATTS_conceptMAA) compared with scenario 6. This reveals that the best relevant attributes frequently are not the most similar ones to conceptMAA. Results for scenarios in MoveM indicate a similar behaviour of the DeriveM except for the scenario 1 (REL_ATTS_changedCT). This underscores that when finding similarity with the changed CT a MoveM action occurs.

Figure 3 presents the achieved results for the ModSemTypeM, RemoveM and noaction. For the ModSemTypeM, scenarios 1 (REL_ATTS_changedCT) and 2 (Best REL_ATT_changedCT) behave very similar, with a high density of low similarity values. On the contrary, scenario 3 (REL_ATTS_unchangedCT) shows a higher density of high similarity values which remains lower for the scenario 4. All scenarios for RemoveM behave very similar presenting a high density of low similarity values. This may rely on the fact that when correspondences evolve by RemoveM action, the whole context contained unchanged or not similar concepts that could be candidates for replacing the source concept. When no-action is applied, the scenarios with the changed context (1 and 2) show a high density of low similarity values also because of the inexistence of changed concepts. However, we observe a high density of high similarity values in the scenarios with unchanged context (3 and 4) despite that correspondences remained unchanged (i.e. any mapping adaptation action was applied).

Discussion

Our results pointed out well-defined behaviours with respect to the density of similarity values for the MAAs. We found the similarity between ontology versions a relevant factor for mapping adaptation. This might help deciding which action applying, but seems insufficient for a completely automatic mapping adaptation. For example, for DeriveM and MoveM actions (cf. Figure 2), the density of the highest similarity values in scenarios 3 and 4 (unchanged context) is bigger than the density observed in scenarios 5 and 6 (conceptMAA). We expected that the similarity value with the concept where a DeriveM or MoveM action happens would remain higher than the similarity found with unchanged concepts. On the other hand, the similarity aspect can help deciding for a RemoveM action (cf. Figure 3). In addition, we expected to observe a more determinant influence of the best relevant attributes in the studied scenarios. We conclude that a system for mapping adaptation must combine the similarity with other aspects to achieve enough facts enabling to trigger different MAAs. Our future work involves studying these further factors.

j,

  established at time j, interrelates a set of given concepts by semantic correspondences. A correspondence cor st = (c s , c t , conf, semType) links two concepts c s ∈CNC(O S j ) and c t ∈CNC(O T j ) through the semType relation. The conf denotes the similarity value between c s and c t indicating the confidence of their relation. We consider the following types of semantic relation: unmappable [⊥], equivalent [=], narrow-to-broad [≤], broad-to-narrow [≥] and overlapped [≈].

Figure 1 .

 1 Figure 1. Mapping Adaptation Actions

4 .

 4 Best_REL_ATT_unchangedCT: Similar to scenario 2, we observe the similarity density only considering the best relevant attribute with the unchanged context. 5. REL_ATTS_conceptMAA: This scenario applies only for MoveM and DeriveM since they involve a concept c k 1 (denoted conceptMAA) which

Figure 2 .Figure 3 .

 23 Figure 2. Similarity values density between concept attributes for DeriveM and MoveM actions

Corresponding Author.


Julio Cesar

Dos Reis

Marcos Da Silveira
email: [marcos.dasilveira@tudor.lu

Duy Dinh
email: duy.dinh@tudor.lu

Cédric Pruski
email: cedric.pruski]@tudor.lu

Chantal Reynaud-Delaître

Requirements for Implementing Mapping Adaptation Systems

Keywords: Ontology Alignment, Mapping Maintenance, Mapping Adaptation, Ontology Evolution, Semantic Interoperability, Knowledge Engineering I

Ontologies, or more generally speaking, Knowledge Organization Systems (KOS) have been developed to support the correct interpretation of shared data in collaborative applications. The quantity and the heterogeneity of domain knowledge often require several KOS to describe their content. In order to assure unambiguous interpretation, overlapped concepts of different, but domain-related KOS are semantically connected via mappings. However, in various domains, KOS periodically evolve creating the necessity of reviewing the validity of associated mappings. The size of KOS remains a barrier for a manual review of mappings, and rather requires the support of (semi-) automatic solutions. This article describes our experiences in understanding how KOS evolution affects mappings. We present our lessons learned from various empirical experiments, and we derive primary elements and requirements for improving the automation of mapping maintenance.

INTRODUCTION

The growing number of integrated and collaborative Web environments demands knowledge-intensive software applications and semantic technologies to improve information retrieval, management, reasoning and sharing. Knowledge Organization Systems (KOS) [START_REF] Hodge | Systems of Knowledge Organization for Digital Libraries: Beyond Traditional Authority Files[END_REF] encompass all types of conceptual models for organizing knowledge (e.g., taxonomies, thesauri and ontologies). They play a key role for Web-based collaborative applications, making the semantic of information explicit at different degrees of expressivity. However, the knowledge described by one KOS is often limited to specific sub-domains. Thus, Web-based collaborative systems need to rely on different KOS to cover the scope of their applications, resulting in semantic interoperability problems. This occurs for example, when the knowledge described by two or more KOS overlaps.

The necessity of semantically correlating overlapped computer-interpretable knowledge through mappings have been largely discussed in the literature [START_REF] Euzenat | Ontology Matching[END_REF]. However, few approaches consider the dynamic aspects of knowledge and their impact on mappings. Considering the increasing size of KOS (e.g., SNOMED-CT, ICD, etc. for the biomedical domain) and the existing efforts to correlate data published on the Web (e.g., Linked Open Data), automatic mapping adaptation has become a necessity to assure a certain level of reliability in dynamic environments. Mapping adaptation addresses the maintenance of mappings referring to the task of modifying existing mappings according to changes affecting the KOS, to keep them semantically valid and complete over time [START_REF] Reis | Mapping adaptation actions for the automatic reconciliation of dynamic ontologies[END_REF].

This article presents the lessons learned from the DynaMO project that investigates the impact of KOS evolution on the adaptation of associated mappings. We recall the outcomes of various experiments conducted to (i) identify KOS changes [START_REF] Reis | Understanding Semantic Mapping Evolution by Observing Changes in Biomedical Ontologies[END_REF][START_REF] Reis | Characterizing Semantic Mappings Adaptation via Biomedical KOS Evolution: A Case Study Investigating SNOMED CT and ICD[END_REF], (ii) mapping changes [START_REF] Reis | Mapping adaptation actions for the automatic reconciliation of dynamic ontologies[END_REF], and (iii) the potential correlation between them [START_REF] Gross | Semi-Automatic Adaptation of Mappings between Life Science Ontologies[END_REF]. On this ground, our contributions are twofold. First, we derive the principal requirements to develop an automatic system for mapping adaptation. Second, we describe and model the processes for managing the evolution of dynamic KOS and mappings between them.

We structure the remainder of this article as follows: Section II presents the related work; Section III reports on the methods adopted in our experiments; Section IV presents the achieved results while Section V gives a discussion with the lessons learned and some requirements for handling the mapping adaptation problem; Section VI wraps up the conclusions and provides an outlook on future work.

II. RELATED WORK

Mappings express the semantic interrelations of concepts issued from different KOS. We consider a mapping m as a triple (s, t, r) [START_REF] Euzenat | Ontology Matching[END_REF], where s refers to a concept from the source KOS S , t is a concept from the target KOS T , (KOS S ≠ KOS T ), and r stands for the type of semantic relation between s and t (i.e., equivalence (≡), more general than (≥), less general than (≤), and partially matched ()).

Mappings can be manually or automatically created by knowledge engineers or through alignment tools respectively. However, if KOS evolve, modifications affecting one of the concepts (s or t) may invalidate the relation r. Examples of Mapping Maintenance tasks include the identification of invalid mappings, the interpretation of existing mappings and their adaptation.

Mapping maintenance refers to the process aiming to keep existing mappings in an updated state, reflecting changes affecting KOS entities at evolution time [START_REF] Reis | Understanding Semantic Mapping Evolution by Observing Changes in Biomedical Ontologies[END_REF].

In this article, we focus on a part of the mapping maintenance which deals with the mapping adaptation problem. Some research work related to mapping adaptation has been previously proposed. Yu & Popa [START_REF] Yu | Semantic Adaptation of Schema Mappings when Schemas Evolve[END_REF] studied the evolution of database schemas and they proposed to isolate the KOS entities that were modified between two versions of the same KOS, and to identify high level types of changes (e.g., deletion of a table column, etc.). According to the change types, they applied a strategy of mapping adaptation. However, the accurate identification of these change types can be as complex as creating new mappings between KOS. Groß et al. [START_REF] Gross | Semi-Automatic Adaptation of Mappings between Life Science Ontologies[END_REF] have extended this technique to the context of ontologies. Additionally, they proposed to complete the resulting set of mappings by using matching techniques over newly added ontology concepts. This work improves the mapping adaptation strategies using ontology complex changes. Groß et al. [START_REF] Groß | How do computed ontology mappings evolve? -A case study for life science ontologies[END_REF] empirically investigate the correlation between mapping evolution and ontology evolution in the domain of life sciences. They proposed a measure of impact ratio for the impact of ontology changes on mapping evolution with respect to three general ontology change types. Their work gives a good overview of the mapping adaptation needs, and motivates deeper investigations to refine the change types to automatically adapt mappings.

An et al. [START_REF] An | Maintaining Mappings between Conceptual Models and Relational Schemas[END_REF] consider KOS with two different models (database schemas and ontologies). They define mappings as links between columns of relational tables and properties of concepts in ontology. They represent semantic mappings as formulas relating tables in a schema with a subset of conjunctive formulas encoding a sub-tree in the ontology graph (s-tree). They characterize the validity of mappings through these formulas. The approach requests domain experts to declare the similarities between the old version and the new version of KOS, and to select the appropriate adaptation strategies.

Martins & Silva [START_REF] Martins | A User-driven and a Semantic-based Ontology Mapping Evolution Approach[END_REF] proposed to adapt mappings impacted by changes in ontologies according to a pre-selected adaptation strategy. The authors assume that all mappings are instances of a pre-defined ontology (SBO) and changes should occur in mappings only if concepts of one of the connected ontologies are removed. Although promising, such approaches lack flexibility in terms of KOS changes considered, an interpretation of the definition of established mappings and the availability of complex behaviours for adapting mapping.

The findings of the aforementioned studies highlight the relevance of gaining a more in-depth understanding of how an advanced and fine-grained classification of KOS changes would affect mapping evolution. Evaluating the accuracy of detection methods for KOS changes and the correlations with mapping changes in real-world datasets may significantly contribute to the knowledge engineering research field. This article highlights findings from our experiments to understand KOS changes and evaluate their effects on mappings.

III. METHODS

In the experiments, we consider a KOS a set of Concepts, Attributes and Relationships. Relationships represent the ways in which concepts of the same KOS are related to one another while Attributes characterize Concepts' attributes. We use two different notations namely relation and relationship to distinguish a semantic link (e.g., 'equivalent', 'less specific', 'more specific', etc.) between two concepts in a mapping from a semantic link (e.g., 'is-a', 'part-of', 'related-to', etc.) between them within a KOS, respectively. Our experiments only consider the relationship "IS-A" in KOS.

We conducted the experiments with data collected from the biomedical domain. We made this choice because of various aspects: (1) the dynamic characteristics of the biomedical domain and (2)the availability of KOS (and mappings) created and validated by domain experts. We have analysed six different KOS: SNOMED CT (SCT), ICD-9-CM (ICD9), NCI Thesaurus (NCI), MedDRA, MeSH, and ICD-10-CM (ICD10) and their related mappings during a period of three years (from 2009 to 2012). These KOS are characterized by the absence of both concept's instances and by attributes' values containing only textual statements [START_REF] Bodenreider | Comparing SNOMED CT and the NCI Thesaurus through Semantic Web Technologies[END_REF].

We defined the experiments to investigate real-world cases of the evolution of KOS and mappings. To this end, we organized the experiments aiming to (1) understand KOS evolution (cf. Section A), (2) understand mappings evolution (cf. Section B) and ( 3) understand the impact of KOS evolution on mappings (cf. Section C).

A. Understanding KOS evolution

Historically, users have paid minimal attention to lists of KOS changes and knowledge engineers have often not provided comprehensive and detailed documentation of changes in computer-interpretable formats [START_REF] Oliver | Representation of change in controlled medical terminologies[END_REF]. Therefore, we defined a set of experiments that aim to observe the evolution of KOS and search for possible KOS "change operation patterns" and supplementary elements that can contribute to adequately adapt mappings. Change operations stand for sequences of changes semi-automatically implemented with the help of knowledge engineers to transform the current KOS version into a new one [START_REF] Klein | Ontology versioning and change detection on the web[END_REF]. For this purpose, we compare two versions of the same KOS to identify the differences between them, and have characterized them as KOS change operations (KCOs). We used the COnto-Diff tool [START_REF] Hartung | COnto-Diff: generation of complex evolution mappings for life science ontologies[END_REF] to support this activity, which returns a set of KCOs. By observing these results, we can determine the most common and some rare evolution cases. We use this output to thoroughly investigate cases that result in complex mappings adaptation processes (e.g., split of concepts) and to define further methods to identify KCOs [START_REF] Reis | Characterizing Semantic Mappings Adaptation via Biomedical KOS Evolution: A Case Study Investigating SNOMED CT and ICD[END_REF].

The set of experiments implemented in this phase includes:  KOS Overview [START_REF] Reis | Analyzing and supporting the mapping maintenance problem in biomedical knowledge organization systems[END_REF]. A quantitative and qualitative analysis of KOS was performed, targeting the identification of basic correlations with mappings evolution. Our study consisted in observing all KOS changes that occurred in NCI from version 10.01 (March 2010) to version 11.09 (October 2011) and in identifying the amount of mappings between NCI and ICD9 (v. 2010) and between NCI and MedDRA (v. 12.0) affected by these changes.  KOS Change Pattern Operations [START_REF] Reis | Characterizing Semantic Mappings Adaptation via Biomedical KOS Evolution: A Case Study Investigating SNOMED CT and ICD[END_REF]. To refine the KOS overview experiments, we searched for specific and recurring KOS changes that we considered as change patterns. We described them according to KCOs performed in KOS. KCOs include the revision, deletion and addition of KOS entities (cf. Table III), which directly influence the reliability of associated mappings. These experiments analysed three KOS (NCI, SCT, ICD9) during the period of 2009 to 2012.

B. Understanding Mapping evolution

As observed for KOS changes, a better insight into the concrete reasons for mapping changes would give more accurate understanding for implementing mapping adaptation. However, tracking of mapping changes was unavailable for the analysed sets of mappings. Hence, we implemented several experiments aiming to observe the evolution of mappings and to search for well-defined actions expressing mappings change and/or supplementary elements that might describe the way mappings evolve.

We performed the following set of experiments:  Mapping Evolution [START_REF] Reis | Understanding Semantic Mapping Evolution by Observing Changes in Biomedical Ontologies[END_REF][START_REF] Reis | Characterizing Semantic Mappings Adaptation via Biomedical KOS Evolution: A Case Study Investigating SNOMED CT and ICD[END_REF]. The goal is to observe and analyse the most recurrent behaviours of modifications in mappings.  Mapping Analysis [START_REF] Reis | The influence of similarity between concepts in biomedical ontology evolution for mapping adaptation[END_REF]. These experiments support the identification of a "sufficient" subset of attributes from a concept that is relevant for explaining existing mappings.

In this investigation, we conducted a set of experiments to assess the quality of the identified attributes using two versions of two biomedical ontologies (SCT 2010 and 2012, and ICD9 2009 and 2011) and their mappings.  Mapping Adaptation Actions [START_REF] Reis | Mapping adaptation actions for the automatic reconciliation of dynamic ontologies[END_REF].According to the outcomes of our observations on mapping evolution, we formally describe specific changes to apply in mappings as a set of mapping adaptation actions.

C. Understanding the impact of KOS evolution on mappings

This set of experiments aim to detect potential correlations between changes in KOS and changes in mappings. To this end, we assume that only modified concepts of KOS associated to a mapping can impact on the evolution of mappings (the associated one or other ones).

We performed the following experiments:  Map-KOS Correlation [START_REF] Reis | Understanding Semantic Mapping Evolution by Observing Changes in Biomedical Ontologies[END_REF]. Given the occurrence of a mapping adaptation action, we searched in KOS for any change involving the source concept (of the mapping) or at least one of its close neighbourhoods (i.e., concepts that are directly linked to the source concept). For each type of mapping adaptation actions (cf. Table IV), we selected and analysed the most frequent set of KOS changes to understand and classify the conditions that lead (or not) to the observed changes in mapping. These experiments aim to identify changes in the KOS that may potentially lead to a change in mappings.  KOS-Map Correlation [START_REF] Reis | Understanding Semantic Mapping Evolution by Observing Changes in Biomedical Ontologies[END_REF]. To complement the first analysis (Map-KOS Correlation), we applied the same approach in the opposite sense. Given a KOS change operation, where one or several concepts are involved, we searched for changes in mappings that have one of the modified concepts as its source concept in mapping. For each KOS change operation, we selected a subset of mappings to manually perform a qualitative analysis.  Selection of Mapping Adaptation Strategy [START_REF] Reis | Mapping adaptation actions for the automatic reconciliation of dynamic ontologies[END_REF]. The outcome from these two previous empirical analyses led to the specification of criteria (based on the KOS change operations) that can "activate" one mapping adaptation action or a sequence of these. Human intervention is required in cases the system cannot decide.

IV. RESULTS

Table I shows an overview of the datasets with the number of changed and unchanged mappings. D i corresponds to the date of mapping releases (every six months, selected from January 2010). In total, we analysed more than 300.000 mappings. . The results of this analysis showed some expected situations. For example, the addition (removal) of mappings is mainly associated with addition (removal) of concepts. However, some cases do not follow this correlation. For instance, in D 3 (cf. A in Table II), only 66% of the added mappings result from the addition of newly added concepts. Another interesting observation indicates that changes in KOS are not always associated with changes in mappings. For instance, unchanged mappings (U) are associated with 0.06% of changed concepts. This indicates that changes in concepts do not always trigger mapping adaptation actions. This raises the following questions: What types of KOS changes can impact the mapping evolution? Which attributes of interrelated concepts impact the underlying mappings and why? How to identify these attributes? To address these questions, we investigated the refinement of KOS change operations (KCOs). By analysing two versions of the same KOS, we identified a set of operations used to perform the KOS changes. Table III [START_REF] Hartung | COnto-Diff: generation of complex evolution mappings for life science ontologies[END_REF] shows a nonexhaustive set of KCOs. Column "Nr. of changes" presents the absolute number of each change operation observed in our experiments with SCT and ICD9 [START_REF] Hartung | COnto-Diff: generation of complex evolution mappings for life science ontologies[END_REF]. A KCO with a frequency equal to zero was added based on our assumption that all KCOs observed have an inverse KCO. For instance, merge stands for the inverse KCO of split. We also designed mapping adaptation actions (MAA) composed of actions used to perform changes in mappings. Table IV presents the formalization of MAA and Fig. 1 defines the notations used in Table IV. This also implies knowing that CT(c) refers to the context of a concept c composed of all super, sub and siblings concepts.

Fig. 1 Mapping adaptation based on KOS changes

The relation r st between interrelated concepts is among the following types: unmappable(), equivalent(), less specific than (), more specific than (), and partially matched().The function sim(c 0 s , c 1 k ) represents the similarity measure adopted, where the outcome refers to a value between 0 and 1 (higher is the value, more similar are the concepts). The symbol σ defines the threshold for considering two concepts as semantically similar. Details about the studied similarity measures can be found in [START_REF] Reis | The influence of similarity between concepts in biomedical ontology evolution for mapping adaptation[END_REF].The set of concepts of a given KOS at time t 0 is given by the function Concepts(K 0 x ).

Move

This is a composed action for which an existing mapping from M 0 ST is re-allocated in M 1 ST, thus the source concept is different.

moveM(mst,c 1 k)  m 0 stM 0 ST m 1 stM 1 ST c 1 kConcepts(K 1 S) (c 1 kCT(c 1 s), m 1 kt M 1 STsim(c 0 s, c 1 k)σ)
Derivation This is a composed action for which an existing mapping in M 0 ST is copied in M 1 ST with a different source concept.

DeriveM(mst,c 1 k)  m 0 stM 0 ST  m 1 stM 1 STc 1 k Concepts(K 1 S) (c 1 kCT(c 1 s), m 1 kt M 1 ST sim(c 0 s, c 1 k)σ)

Change Relation

This is a composed action in which the type of the semantic relation of a given mapping is modified.

chgR(mst,new_rst)  m 1 st M 1 ST new_r 1 st   (m 1 st M 1 ST, new_r 1 st r 0 st)
No-Action In this case, any modification is observed in the mapping

no-action(mst)  m 0 stM 0 ST m 1 stM 1 ST
At this stage, we aim to understand the correlations between the list of KCOs and the list of MAAs. We observe that for the studied KOS where concepts are described by textual attributes. We investigated how these attributes can be correlated to established mappings. To this end, we developed the TopA algorithm [START_REF] Reis | The influence of similarity between concepts in biomedical ontology evolution for mapping adaptation[END_REF], which gives the N most relevant attributes for a given mapping. TopA relies on the adaptation of different semantic similarity measures targeting the lexical level [START_REF] Levenshtein | Binary codes capable of correcting deletions, insertions and reversals[END_REF], the syntactic level [START_REF] Maedche | Measuring similarity between ontologies[END_REF] and the semantic level [START_REF] Jiang | Semantic similarity based on corpus statistics and lexical taxonomy[END_REF].

We use the detected attributes during our correlation analysis, to verify, for example, whether changing these attributes to a different concept in another KOS version relates to moving the associated mapping. We defined four specific types of change patterns at the level of attributes: Total transfer (TT)when an attribute is deleted from one concept and entirely moved to another one; Partial Transfer (PT)when an attribute is deleted from one concept and a modified version of this attribute is added into another one), Total Copy (TC)when an entire copy of the attribute is added into another concept; Partial Copy (PC) -a modified copy of the attribute is added into another concept. The partially copied PC or transferred attributes PT are identified by the degree of similarity with the original attribute (over a threshold σ). We assume that if the similarity sim is close to 1 (i.e., sim ≥ σ), we consider the attribute as totally copied TC or transferred TT.

Table V presents our findings regarding correlations between each MAA and the defined change patterns. We do not consider mapping addition in this analysis because MAA are only applied to modify established mappings. Our achieved results point out the correlation between MAA and the proposed change patterns. However, this demands further studies to explain why some correlations lack in some cases. While KOS maintenance belongs to the lifecycle of KOS, efforts to track changes about their evolution in the biomedical domain to automatically maintain their associated mappings up-to-date remain insufficient. Consequently, to define methods aiming to understand and classify KOS changes over time requires a complex and time consuming work. To overcome this problem, we had mostly to deal with: -Related to KOS evolution:  Data quality. Data sources are often published in a proprietary format or are not fully available in a computer-interpretable format. For instance, ICD9 is published as a MS-Word document that requires a parser to extract the KOS content. This parser is unavailable, which forces each research team to develop their own one (potentially leading to different datasets and errors).  Types of change operations. We have defined a nonexhaustive list of change operations according to the conducted experiments and available tools. This may lead to unknown situations regarding how KOS evolve.  Accuracy of change operations identification between KOS versions. Since we mainly ground our approach to mapping adaptation on changes in concept attributes (we have defined change patterns), and since these attributes stand for textual statements, it remains a very complex task to correctly these operations and interpret their consequences on concept attributes values. -Related to mappings:  Identification of relevant KOS entities for interpreting mappings. KOS's statements to justify why and how mappings were created and KOS's entities that impact the alignment of the concepts are unavailable in an explicit way for mappings considered in our studies, which may influence the accuracy of maintenance tasks.  Mappings' modification reasons. Reasons explaining changes affecting mappings from one release to another are unavailable in datasets. We identified at least two main reasons for mapping adaptation in our work: (1) the propagation of KOS changes; and (2) the correction of erroneous mappings (when the validation process fails).  Mapping adaptation techniques. The lessons learned from our conducted experiments have guided us to establish a set of useful heuristics to support automatic mapping maintenance. Inspired by the investigation of Stojanovic [START_REF] Stojanovic | User-driven ontology evolution management[END_REF] and Noy et al. [START_REF] Noy | A framework for ontology evolution in collaborative environments[END_REF] and based on how we managed the difficulties previously listed for the evolution of KOS and mappings, we propose a model for the process of KOS evolution (cf. Fig. 2B). We partially transpose this model to inherently describe a mapping evolution process (cf. Fig. 2A).

Following the phases proposed by Stojanovic, the KOS evolution process demands at least the list of KOS's entities requiring modification and of their respective types. We separate this information within two concepts Elements and Goals. Additionally, we deem useful to find the specific reasons for mapping changes. For instance, the correction of spelling mistakes can indicate a reason. Note that for clarity reasons, Fig. 2 hides some relationships and attributes, e.g., the relationship between goals and change patterns.

Considering the types of KOS changes, we assume that atomic changes are implemented alone and that we can associate them to achieve more complex changes [START_REF] Klein | Ontology versioning and change detection on the web[END_REF]. We suggest three types of complex changes. The Lexical Complex Changes include modifications in attributes' values of concepts (we only consider textual attributes) (cf. Section Methods). These attributes can be partially or totally copied, or transferred to another concept, in another KOS version. This information might help identifying Syntactic Complex Changes. For instance, totally transferring an attribute to another concept can represent one of the evidences necessary to characterize a split [START_REF] Reis | Characterizing Semantic Mappings Adaptation via Biomedical KOS Evolution: A Case Study Investigating SNOMED CT and ICD[END_REF]. The semantic consequences of a Lexical Complex Change are described in the Semantic Complex Change. For instance, an attribute transferred from one concept to another can have semantic consequences as follows: the former concept becomes more general and the later concept becomes more specific.

The implementation of KOS changes applies one or more change operations according to the pre-conditions and in a temporal order (Sequence). The change Validation consists of an important and laborious task where KOS engineers can detect and eliminate potential inconsistencies. Finally, the set of changes are propagated to associated artefacts (e.g., mappings, annotations, queries, etc.).

The mapping evolution process (cf. Fig. 2A) mostly differs from the KOS evolution process in the capturing phase, describing complex changes and representing the changes' state. In the capturing phase, we thoroughly detail the elements that best describe how mappings were established (Relevant Information), the similarity measures used (Similarity metadata), their values (Similarity value), and the level of relevance of a specific element for mapping creation (Relevance level). Thus, if KOS evolution affects any relevant attribute (an identified KOS entity), we can measure the impact on mappings with more accuracy.

Complex changes applied to mappings include Derivation (a copy of a mapping with a new source concept), Move (transferring a mapping into another source concept) and Change relation (changing the semantic relation between source and target concepts). The process also tracks the state mappings change, and the status of each change includes valid, under validation, or invalid.

VI. CONCLUSION

This article performed a meta-analysis on a series of previously conducted empirical experiments that allowed on this basis a discussion of lessons learned. We highlighted the complexity of analysing KOS evolution when failing to report KOS changes in a computer-interpretable format. We focused our analyses on issues related to the propagation of KOS changes on mappings. Empirical experiments analysed were performed using a set of biomedical KOS and their associated mappings to observe and classify correlations between their changes. We originally introduced models for representing the KOS evolution process and the mapping evolution process. We designed these models to support mapping maintenance tasks.

Having methods that semi-automatically update KOS and maintain the validity of mappings might enable interoperable systems to follow logical reasoning and explain its implications to authoring groups in charge of the maintenance of KOS and mappings. Developers whose software depends on these semantic artefacts will allocate less time and costs for maintenance tasks if they can reduce the manual work required to handle different techniques used by KOS constructors for representing the domain and to report its changes over time.

In our future work, we will particularly improve the proposed methods and tools to automatically identify semantic complex changes between KOS versions. We also plan to generalize our approach evaluating in other domains, and to study new similarity measures to apply for KOS with instances and non-textual attributes.

Fig. 2 .

 2 Fig.2. Modeling KOS and Mapping evolution process

TABLE I .

 I ANALYSIS OF SCT-ICD9 MAPPINGS FROM JAN/2010 TO JUL/2011Table II presents a deeper overview, where we consider four KOS entities (concept, attribute, relationship, and the neighbours) as well as the following mapping changes: Unchanged (U), Added (A), Removed (R), Modification of target concept (M t ), and Modification of the relation (M r ) or both Modifications in target concept and relation (M t_r

	Dataset	Total	Unchanged Changed
	D 1	100.875 100.394	481
	D 2	101.254 100.281	973
	D 3	102.601 101.076	1.525

TABLE II .

 II CHANGES IN SCT ENTITIES CORRELATED WITH MAPPING CHANGES.

	MCO	DS	Concept	Attribute (%)	Relationship	Neighbour
			(%)		(%)	(%)
	U	D1	0.06	0.25	6.75	1.71
		D2	0.02	0.12	3.00	1.17
		D3	0.03	0.11	3.34	1.38
	A	D1	100	100	100	12.93
		D2	100	100	100	20.87
		D3	66.18	66.18	66.5	28
	R	D1	98.22	98.18	100	39.2
		D2	100	100	100	43.9
		D3	83.64	83.64	100	47.27
	M t	D2	1.4	1.4	4.7	0.47
		D3	0	0	10	5
	M r	D2	0.5	0.5	10	3.51
		D3	0	0	0	0
	M t_r D2 1.2	1.2	4.7	2.38
		D3	0	0	3.45	3.45

TABLE III .

 III KOS CHANGE OPERATIONS

	Change Operation	Description	Nr. of changes
				SCT	ICD9
		addC(c)	Add a new concept	-	-
		delC(c)	Delete an existing concept -	-
		addA(a,c)	Add a new attribute	7.720	79
	Atomic	delA(a,c) addR(r,c 1 ,c 2 )	Delete an existing attribute Add a relationship	4.003 4.327	25 110
			between c 1 and c 2
		delR(r,c 1 ,c 2 )	Delete a relationship	7.210	39
			between c 1 and c 2
		chgA(c,a,v)	Change the value of an	950	155
			attribute
		moveC(c,p 1 ,p 2 ) Move a concept (and sub-	7.140	31
			tree) from the parent p 1 to
			p 2
		subst(c 1 ,c 2
	Complex

TABLE IV .

 IV MAPPING ADAPTATION ACTIONS. SEE FIG 1 FOR NOTATIONS

	MAA	Description		Definition
	Remove	This is an atomic action through which a mapping m 0	st	rmvM(mst)  m 0 stM m 1 stM 1 ST	0 ST
		is deleted from M 0 ST
	Addition	This is an atomic action through which a new		addM(mst)  m 0 stM m 1 stM 1 ST	0 ST
		mapping m 1 st is added to M 1 ST

TABLE V .

 V MAP-KOS CORRELATIONS

	MAA	Nr.	TT PT TC PC No-change
	Move	362	68	1	190 14	223
	Derive	583	2	0	133 54	419
	Remove	167	3	0	16	7	156
	Change Relation 55	0	0	2	1	45
	No-action	9024 16	2	176 70	8073
	V. LESSONS LEARNED

ACKNOWLEDGMENT

The National Research Fund (FNR) of Luxembourg (Grant #C10/IS/786147) entirely supports this work under the DynaMO research project.