Zhanhao Xiao
email: zhanhao.xiao@ut-capitole.fr

Andreas Herzig

Laurent Perrussel

Hai Wan

Xiaoheng Su

Hierarchical Task Network Planning with Task Insertion and State Constraints

We extend hierarchical task network planning with task insertion (TIHTN) by introducing state constraints, called TIHTNS. We show that just as for TIHTN planning, all solutions of the TIHTNS planning problem can be obtained by acyclic decomposition and task insertion, entailing that its planexistence problem is decidable without any restriction on decomposition methods. We also prove that the extension by state constraints does not increase the complexity of the plan-existence problem, which stays 2-NEXPTIME-complete, based on an acyclic progression operator. In addition, we show that TIHTNS planning covers not only the original TIHTN planning but also hierarchyrelaxed hierarchical goal network planning.

Introduction

Hierarchical task network (HTN) planning [Erol et al., 1994] is an approach for building plans via step-wise refinement of high-level tasks into lower-level tasks in a top-down manner. A task network may contain both compound (high-level) and primitive tasks. Primitive tasks correspond to STRIPS-like actions that can be applied in states where their preconditions are met, while compound tasks are abstractions: for every compound task, the domain features a set of decomposition methods, each mapping the task to a task network. The complexity of the plan-existence problem for HTN planning ranges up to undecidability even for propositional HTN planning [Erol et al., 1994]. Even so, hierarchical planning approaches are often chosen for real world application scenarios [START_REF]Web service composition with user preferences[END_REF][START_REF] Biundo | [END_REF]. On the other hand, it is usually a challenge to provide a complete domain which includes all possible methods for all compound tasks, while defining only a partially hierarchical domain is not sufficient to produce all desired solutions. Several HTN researchers have investigated how partially hierarchical domain knowledge can be exploited during planning without relying on the standard HTN formalism [START_REF] Kambhampati | [END_REF][START_REF] Biundo | [END_REF][START_REF] Geier | [END_REF][START_REF] Shivashankar | [END_REF]. Among them, hierarchical task network with task insertion (TIHTN) planning [START_REF] Geier | [END_REF] relaxes the restriction on solutions and allows solutions generated not only by the decomposition of compound tasks, but also by the insertion of primitive tasks from outside the decomposition hierarchy.

Unfortunately, only ordering constraints are considered in propositional TIHTN planning [START_REF] Geier | [END_REF] and lifted TIHTN planning [Alford et al., 2015b]. In contrast, state constraints are taken into account in the conventional HTN planning [Erol et al., 1994], but there is few work on considering them in TIHTN planning.

In this paper, we investigate the extension of TIHTN with state constraints, noted TIHTNS for short. We first prove that, just as for the TIHTN planning problem, all solutions of the TIHTNS problems can be obtained by acyclic decomposition and task insertion, entailing that it is decidable without any restrictions on decomposition methods. We then show that the extension by state constraints does not cause an increase in complexity of the plan-existence problem, which stays 2-NEXPTIME-complete, based on an acyclic progression operator. We also show that TIHTNS planning includes lifted TIHTN planning. As under task insertion semantics, hierarchical goal network (HGN) planning [START_REF] Shivashankar | [END_REF] and goal-task network (GTN) planning [START_REF][END_REF] can be translated to lifted TIHTN, our framework, TI-HTNS, actually covers the two kinds of planning approaches. In addition, we give an alternative embedding of hierarchyrelaxed HGN (HR-HGN) [START_REF] Shivashankar | [END_REF] in TI-HTNS without introducing fresh operators. State constraints State constraints can capture the pre-and postcondition of compound tasks, though in the standard HTN planning there is no notion of pre-and postcondition of compound tasks. A compound task is considered as accomplished if its subtasks are accomplished. With a state constraint, a formula, as a postcondition, can be required to hold after accomplishing a compound task.

Ordering constraints cannot fully represent state constraints. The 'immediate' state constraint, which requires a formula holds immediately before or after a compound task, can be simulated via introducing a virtual subtask to check whether the formula holds. However, the 'maintenance' state constraints (also called trajectory constraints in the Planning Domain Definition Language 3 (PDDL3) [START_REF] Gerevini | Alfonso Gerevini and Derek Long. Plan constraints and preferences in pddl3[END_REF]) cannot be represented easily. For instance, suppose a robot is required to always keep 10% battery for emergency. Its initial compound task is to "clean a room", de-composed into "clean the ground" (t 1 ) and "clean the table" (t 2 ). Suppose that "clean the ground" requires the full battery. Then there is no solution for the TIHTNS planning problem with such a state constraint. 1Furthermore, state constraints are introduced into PDDL3 [START_REF] Gerevini | Alfonso Gerevini and Derek Long. Plan constraints and preferences in pddl3[END_REF] to support hard constraints over state properties of a trajectory and the specification of preferences. Later [Sohrabi et al., 2009] extends PDDL3 into HTN planning where state constraints are used to capture user preferences. State constraints are also necessary in real-world applications, such as in web service composition where state constraints are used to describe user preferences [START_REF]Web service composition with user preferences[END_REF] and to additionally capture the enforcement of regulations [START_REF] Sohrabi | [END_REF].

The rest of the paper is structured as follows. Section 2 presents the definition of TIHTNS. Section 3 presents how to embed lifted TIHTN and HR-HGN into TIHTNS. Section 4 presents the complexity results of TIHTNS. Section 5 concludes and discusses the future work.

Extended TIHTN Planning

In this section we adapt the original TIHTN planning formalism of [START_REF] Geier | [END_REF]Alford et al., 2015b]. First, we define a function-free first order language L from a set of variables and a finite set L 0 of predicates and constants. Next we take parts of variables in L as task symbols to identify tasks. Every task is associated with an action (task name), which is syntactically a first-order atom in L. That is, every action typically is associated with an arity and contains variables that can be eliminated via grounding. For example, by grounding the action "openDoor(X)" where X is quantified as a door, we can obtain a set of ground actions "openDoor(a)", "openDoor(b)", etc. Those actions which can be executed directly are called operators, noted O, while others are called compound actions,2 noted C. Each operator o ∈ O is represented as a triple of formulas: precondition pre(o), positive effect add(o), and negative effect del(o) where add(o), del(o) are finite sets of atomic formulas. The variables in add(o) and del(o) should be associated with the variables in the operator so that they are instantiated simultaneously. For example, the positive effect of "openDoor(X)" being "Opened(X)", after grounding "openDoor(a)" has positive effect "Opened(a)" but not "Opened(b)". The positive and negative effect of an operator should not conflict, i.e., add(o) ∩ del(o) = ∅ for every o ∈ O. We suppose that O contains the 'empty' primitive operator skip where pre(skip) = ⊤ and add(skip) = del(skip) = ∅.

For a function f : R → S, its restriction to a subset X of its domain is

f | X = {(r, s) ∈ f | r ∈ X}. Given a binary relation Q ⊆ R × R, we define its restriction to X ⊆ R by Q| X = Q ∩ (X × X)
; similarly for ternary relations. We extend the set union operator ∪ to tuples:

(Q 1 , Q 2 ) ∪ (Q ′ 1 , Q ′ 2 ) = (Q 1 ∪ Q ′ 1 , Q 2 ∪ Q ′ 2 )
and extend functions to sequences: f ( t 1 , . . . , t n ) = f (t 1 ), . . . , f (t n ) .

HTN Problems

Task networks A task network tn = (T, ∆, α) is a tuple, where

• T is a finite and non-empty set of task symbols;

• ∆ ⊆ (T ∪{nil}) × L × (T ∪{nil}) is a set of constraints over T • α : T → C ∪ O labels every task with an action.

With Function α, we allow multiple instances of an action in a task network. Compared to the ordering constraints in [START_REF] Geier | [END_REF] which are in form of task-task pairs, we use a triple (t i , ϕ, t j ) to denote a state constraint which intuitively means that formula ϕ must be true in all states between t i and t j . Specially, we introduce an idle task symbol nil which designates a task that is accomplished immediately: (nil, ϕ, t j ) and (t i , ϕ, nil) mean formula ϕ holds immediately before t j and after t i , respectively. We suppose nil only occurs in constraints. When ϕ is the truth constant ⊤ then the state constraint (t i , ϕ, t j ) becomes an ordering constraint that just requires that t i is before t j . A task is primitive if it is associated to an operator, otherwise compound. A task network is primitive if it only contains primitive task.

We say that two task networks tn = (T, ∆, α) and tn ′ = (T ′ , ∆ ′ , α ′ ) are isomorphic, noted tn ∼ = tn ′ , if there exists a bijection δ : T → T ′ where for all t, t ′ ∈ T it holds that

α(t) = α ′ (δ(t)) and (t, ϕ, t ′ ) ∈ ∆ iff (δ(t), ϕ, δ(t ′ )) ∈ ∆ ′ .
Methods Non-primitive task networks contain compound tasks which cannot be executed directly by the agent, and decomposition methods tell us how to decompose these hierarchically. Each decomposition method m is a tuple (c, tn m ), where c is a compound action, called the method's head, and tn m is a task network, whose inner tasks are called the method's subtasks. The intuition is that compound action c can be reduced by the subtask network tn m . HTN problems TIHTNS problems only differ HTN problems in the solution criterion and share the syntactical problem description. An HTN domain is a tuple D = (L, C, O, M) where M is a set of methods and C ∩ O = ∅. An HTN problem is a tuple P = (D, s I , tn I ) where s I is the ground initial state and tn I is the initial task network.

The semantics of HTN planning is given through grounding. According to [Alford et al., 2015a], it is easy to translate a lifted HTN problem into a ground (or propositional) HTN problem as the set of relations and constants L 0 is finite. For an HTN problem (L, C, O, M, s I , tn I ), we use P = (L, C, O, M, s I , tn ′ I ) to denote the ground (or propositional) problem obtain from it.

A ground state is a subset of the ground atoms in L 0 . A set of ground operators O determines a state-transition function γ : 2 L0 × O -→ 2 L0 , where:

• γ(s, o) is defined iff s |= pre(o); • γ(s, o) = (s \ del(o)) ∪ add(o) if γ(s, o) is defined.
A sequence of operators o 1 , . . . , o n is executable in a state s 0 iff there exists a sequence of states s 1 , . . . , s n such that

∀ 1≤i≤n γ(s i-1 , o i ) = s i .
Example 1. Suppose we have an initial compound action goM C for "go to Melbourne center", operator o 1 for "fly to Melbourne", and operator o 2 for "take a taxi to the center". We use At(Mc)and At(Ma) to denote "being at Melbourne center" and "being at Melbourne airport". The primitive operators are:

• o 1 = (⊤, {At(Ma)}, ∅) • o 2 = (At(Ma), {At(Mc)}, {At(Ma)})
and the method is:

• m = (goM C, tn m ), where tn m = (t 2 , ∅, (t 2 , o 1 )) Then the HTN problem P is (L, goM C, {o 1 , o 2 }, m, s I , tn I ) with s I = ∅ and tn I = (t 1 , (t 1 , At(Mc), nil), (t 1 , goM C)). 3

Task Decomposition

Next we borrow the notion of decomposition in [START_REF] Geier | [END_REF] and define how to decompose a compound task into a task network. In order to indicate the starting point and the end point of a compound task t, we introduce a pair of 'dummy' primitive tasks, noted * t and t * . As nil only occurs in constraints, we suppose the restriction of constraint set ∆ to a set of tasks

T is ∆| T = ∆∩((T ∪{nil})×L×(T ∪{nil})). Definition 1 (Decomposition). Given an HTN domain D = (L, C, O, M), let tn = (T, ∆, α) be a task network and t ∈ T be a compound task. Let m = (α(t), (T m , ∆ m , α m )) and T m ∩ T = ∅. The decomposition of task t by method m is tn ′ = (T ′ , ∆ ′ , α ′ )
where

T ′ = (T \ {t}) ∪ T m ∪ { * t, t * } ∆ ′ = ∆| T \{t} ∪ ∆ m ∪ {( * t, ⊤, t j ), (t j , ⊤, t * ) | t j ∈ T m } ∪ {(t 1 , ϕ, * t) | (t 1 , ϕ, t) ∈ ∆} ∪ {(t * , ϕ, t 2 ) | (t, ϕ, t 2 ) ∈ ∆} α ′ = α| T \{t} ∪ α m ∪ {( * t, skip), (t * , skip)} We write tn --→ t,m
tn ′ when tn ′ is the decomposition of t by m.

In the resulting task network, the decomposed compound task is replaced with subtasks defined by the method applied and its corresponding starting and terminating tasks. The latter two are dummy tasks and are mapped to the action skip.

All state constraints about the decomposed task t are propagated by * t and t * in ∆ ′ . More precisely, if ϕ holds before t then it also holds before * t and if ϕ ′ holds after t then it also holds after t * . Subtasks should satisfy the inner constraints introduced by the decomposition method and should be performed between * t and t * . Example 2 (Example 1 continued). We apply the method m in tn I to decompose t 1 , i.e., tn I ---→ t1,m tn ′ , where tn ′ is:

T ′ ={t 2 , * t 1 , t 1 * } ∆ ′ ={(t 2 , ⊤, t 1 * ), ( * t 1 , ⊤, t 2 ), (t 1 * , At(Mc), nil)} α ′ ={(t 2 , o 1 ), ( * t 1 , skip), (t 1 * , skip)}
3 We will sometimes omit the braces of singleton sets.

Decomposition tree

The hierarchical procedure of task decomposition can be viewed as a tree. Given an HTN domain D = (L, C, O, M ), a decomposition tree is a five-tuple Tr = (T, E, ∆, α, β) where (T, E) is a tree, rooted in t 0 which is a fresh task symbol, with nodes T and with directed edges E pointing towards the leaves; ∆ is a set of constraints over T ; function α : T → C∪O links tasks and actions where α(t 0 ) = c 0 such that c 0 is a fresh compound action with an intuition "to achieve the initial task network". β : T → M ′ labels inner nodes with methods where M ′ = M ∪{(c 0 , tn 0 )} and tn 0 is some task network.

We write τ (Tr) for the nodes (tasks) of the decomposition tree Tr and ch(Tr, t) for the direct children of t ∈ τ (Tr) in Tr. Hereafter sub(t) denotes the set of subtasks of t. The leaf nodes of Tr together with the constraints about these nodes define a task network, denoted by ϑ(Tr). Definition 2 (valid decomposition tree). A decomposition tree Tr is valid w.r.t.

an HTN problem P = (L, C, O, M, s I , tn I ) iff the root node of Tr is t 0 where β(t 0 ) = (c 0 , tn I ) and for any inner node t where β(t) = (c, tn m ), the followings hold:

1. α(t) = c 2. if t = t 0 then ch(Tr, t) = sub(t), otherwise ch(Tr, t) = sub(t) ∪ { * t, t * } such that • (sub(t), ∆| sub(t) , α| sub(t) ) ∼ = tn m • for every t ′ ∈ sub(t), (t ′ , ⊤, t * ), ( * t, ⊤, t ′ ) ∈ ∆ 3. for every t ′ ∈ τ (Tr) ∪ {nil}: • if (t, ϕ, t ′ ) ∈ ∆ then (t * , ϕ, t ′ ) ∈ ∆ • if (t ′ , ϕ, t) ∈ ∆ then (t ′ , ϕ, * t) ∈ ∆
4. there are no constraints in ∆ except for those demanded by criterion 2. and 3. When tn ′ is reachable from tn by a finite sequence of decompositions then we write tn -→ * D tn ′ . The decomposition tree focuses on the whole procedure of decomposition and records all intermediate tasks and constraints during the procedure. In contrast, in a task network the application of a decomposition method abandons the decomposed task. However, they are compatible as the following proposition states. Proposition 1. Given an HTN problem P, tn I -→ * D tn iff there exists a valid decomposition tree Tr with respect to P where ϑ(Tr) = tn.

Solutions

A solution of an HTN problem is a sequence of primitive tasks which is also called a plan of the problem. Consistency with constraints Given a primitive task network tn = (T, ∆, α) where |T | = n, let σ : T → {1, . . . , n} be a bijection. We use σ to form a total ordering, noted σ(tn), of tasks in T as: σ -(1), . . . , σ -(n) where σ -is the inverse function of σ, i.e., σ -(σ(t)) = t. Suppose α(σ(tn)) is executable in s 0 , i.e., there exists a sequence s 1 , . . . , s n such that γ(s i-1 , α(t i )) = s i for every i such that 1 ≤ i ≤ n. We say that σ(tn) is consistent with ∆ in s 0 if for every σ -(i), σ -(j) ∈ T the following hold:

• for every (nil, ϕ, σ -(j)) ∈ ∆, s j-1 |= ϕ;

• for every (σ -(i), ϕ, nil) ∈ ∆, s i |= ϕ;

• for every (σ -(i), ϕ, σ -(j)) ∈ ∆, i < j and s k |= ϕ for every i ≤ k<j. Intuitively, a 'maintenance' state constraint (t, ϕ, t ′ ) is satisfied if all states between t and t ′ satisfy ϕ. An 'immediate' state constraint (nil, ϕ, t) is satisfied if ϕ holds in the state right before * t occurs (or t if t is a primitive task), in other words right before all subtasks of t. Finally, an 'immediate' state constraint (t, ϕ, nil) is satisfied if ϕ holds right after the state where * t (or t if t primitive task) occurs, in other words right after all subtasks of t. Note that it is impossible to satisfy (t, ⊥, t ′ ) because there is no state s such that s |= ⊥. Executability A task network tn is primitive iff it contains only primitive tasks. A primitive task network tn is executable in a state s iff there exists a total ordering σ(tn) of the tasks in tn that is consistent with ∆ in s. We called such a σ(tn) a plan of tn in s, noted σ tn,s .

It is possible that the task network is not executable in a state. For instance, the primitive task network tn ′ in Example 2 is not executable in s I because apart from skip it only involves the operator o 1 and it is impossible to satisfy At(Mc). However, if we extend the task network by inserting some tasks, then we can make it executable. Insertion Let tn = (T, ∆, α) and tn ′ = (T ′ , ∆ ′ , α ′ ) be two task networks where tn ′ is primitive. Inserting tn ′ into tn results in the task network tn 1 = tn ∪ tn ′ .

Note that it is not required that T ′ ∩ T = ∅ because the insertion can involve some constraints about tasks in tn. Solutions With respect to some initial tn, if tn ′ is reachable by a finite sequence of decompositions and an insertion we write tn -→ * DI tn ′ . The plan obtained only by decomposition is called an HTN solution while the plan obtained additionally by insertion is called a TIHTNS solution:

Let tn be a primitive task network such that there exists a plan of tn in s I . Given an HTN problem P, we call σ tn,sI an HTN solution of P if tn I → * D tn; we call σ tn,sI a TIHTNS solution of P if tn I → * DI tn. Example 3 (Example 2 continued). The plan * t 1 , t 2 , t 3 , t 1 * where α(t 3 ) = o 2 is a TIHTNS solution of P.

When considering whether an HTN problem P has a TI-HTNS solution, we call P a TIHTNS problem.

The next proposition states that the state constraints about compound tasks are satisfied in the solutions of the problem. Proposition 2. Given a TIHTNS problem P, suppose Tr is a valid decomposition tree with respect to P and tn ′ is the final primitive task network obtained from ϑ(Tr) by insertion and σ tn ′ ,sI is a solution of P. Then all constraints in Tr are satisfied by σ tn ′ ,sI . Sketch of proof. Suppose there are two compound tasks t i and t j in Tr and tn ′ = (T ′ , ∆ ′ , α ′ ). For every (t i , ϕ, t j ) ∈ ∆, we have (t i * , ϕ, * t j ) ∈ ∆ ′ and then all states s k such that σ(t i * ) ≤ k < σ( * t j ), s k |= ϕ. For every (t i , ϕ, nil) ∈ ∆, we have (t i * , ϕ, nil), (st, ⊤, t i * ) ∈ ∆ ′ for all subtasks st ∈ sub(t i ) and then s σ(ti * ) |= ϕ and σ(st) < σ(t i * ) for all st ∈ sub(t i ). The case of (nil, ϕ, t j ) is similar. If one or both of t i , t j are primitive then just consider * t j and t i * as t j and t i , respectively. Then all constraints in Tr are satisfied.

Remark 1. The state constraint here we define is weaker than that in the original HTN planning [START_REF] Erol | [END_REF]. Their semantics considers every compound task starts by its first 'real' subtask and terminates by its last 'real' subtask instead of the virtual starting and terminating tasks. The distinction between two semantics stands on whether it is allowed to insert tasks between * t and the first real subtask of t (between the last real subtask and t * ). Our weaker semantics can capture the pre-and postcondition of compound actions better. In Example 1, after the subtask t 2 of goM C is accomplished, the agent's desirable goal "being in the center" does not hold while by allowing the insertion of t 3 , the goal is achieved.

Embedding TIHTN and HGN

Recently HTN researchers work on enhancing the semantics of HTN planning and have proposed variants of HTN planning. Propositional TIHTN was first proposed in [START_REF] Geier | [END_REF] and later was extended into lifted TIHTN in [Alford et al., 2015b]. Hierarchy goal network (HGN) planning [START_REF] Shivashankar | [END_REF] operates over a hierarchy of goals with methods that decompose goals with further subgoals. [START_REF] Shivashankar | [END_REF] relaxes the hierarchy of HGN planning and translates the variant which is called HR-HGN planning into classical planning. Hereafter, we detail that how lifted TIHTN and HR-HGN planning can be easily encoded in our framework in polynomial time.

Embedding TIHTN Let us now translate lifted TIHTN into our extended TIHTNs. By replacing all occurrences of t i ≺ t j in lifted TIHTN problem P with (t i , ⊤, t j ), we obtain a TIHTNS problem, noted Γ T (P). The following proposition shows that the two problems are equivalent. Proposition 3. For a lifted TIHTN problem P, P has a solution iff Γ T (P) has a solution. Sketch of proof. Under the lifted TIHTN semantics, consider the decomposition of t by m, the new ordering constraint set is obtained as: (1) keep the constraints not involving t; (2) introduce the constraints about the subtasks; (3) if t is before t ′ then all subtasks are before t ′ ; (4) if t is after t ′ then all subtasks are after t ′ . Under the extended semantics, according to Definition 1, the change of state constraints includes (1) and ( 2) and is analogous to (3) and ( 4). If t is before t ′ , i.e., (t, ⊤, t ′ ), there will be (t * , ⊤, t ′ ) introduced. As (t m , ⊤, t * ) for all subtasks t m are introduced, it entails that (t m , ⊤, t ′ ) which means t m is before t ′ . Then (3) is simulated and the case of ( 4) is similar. The proposition follows. [START_REF][END_REF] combine HTN and HGN planning into goal-task network (GTN) planning where an element of the network consists of a goal and a task. They also show that allowing task insertion, HGN and GTN planning problems can be translated polynomially into lifted TIHTN problems. Therefore, our extension TIHTNS actually also covers them. Embedding HR-HGN Hierarchical goal network planning is a formalism which extends classical planning to include hierarchical decomposition using methods.

HGN planning talks about goal network gn = (G, ≺) where G is a set of formulas in disjunctive normal form over ground literals, called goal formulas, and ≺⊆ G × G is a strict partial order on G. Similar with HTN problems, an HGN problem is a tuple P = (L, O, M, s I , gn I ) where M is a set of HGN methods and gn I is an initial goal network. For an HR-HGN problem, M is a empty set and omitted. The solutions of HR-HGN are defined as the set of all operators sequences that are executable in the initial state s I and that achieve all initial goals according to the order. In [START_REF] Shivashankar | [END_REF], by introducing fresh operators with the number of |gn I |, an HR-HGN problem can be translated into a classical planning problem. Now, we translate HR-HGN planning into our framework without introducing any fresh operators. Allowing task insertion, state constraints simulate a goal formula by requiring that the formula holds immediately before an empty task. Formally, given an HR-HGN problem P = (L, O, s I , gn I ), we define a TIHTNS problem Γ G (P) = (L, ∅, O, ∅, s I , tn) as:

• for every g ∈ G I , λ g ∈ tn and (nil, g, λ g ) ∈ ∆ and α(λ g ) = skip • for every g i ≺ g j , (λ gi , ⊤, λ gj )

Next we show that the translation Γ G is correct. Proposition 4. For an HGN problem P, P has a solution iff Γ G (P) has a solution. Sketch of proof. Suppose σ tn ′ ,sI is a solution of Γ G (P). Then tn ′ is obtained from tn by an insertion. It forms a sequence s 0 , . . . , s n where s 0 = s I . For every g ∈ G I , we have λ g ∈ T ′ and (nil, g, λ g ) ∈ ∆ ′ then s i |= g where σ(λ g ) = i; for every g i ≺ g j , we have (λ gi , ⊤, λ gj )∆ ′ then i ′ < j ′ and s i ′ |= g i , s j ′ |= g j ,where σ(λ gi ) and σ(λ gj ).

Complexity

In this section, we show that the solutions of TIHTNS can be obtained by acyclic decomposition and it does not increase the complexity, staying 2-NEXPTIME-complete. Substitution Now we adapt the notion of subtree substitution initially proposed in [START_REF] Geier | [END_REF] which replaces a subtree with another subtree. Given a decomposition tree (T, E, ∆, α, β) and a node t ∈ T , we define the subtree of Tr induced by t, as

Tr[t] = (T ′ , E ′ , ∆| T ′ , α| T ′ , β| T ′ ) where (T ′ , E ′ ) is the subtree in (T, E) which is rooted in t.
Let Tr = (T, E, ∆, α, β) be a decomposition tree and t i , t j ∈ T be two nodes of Tr where t i is an ancestor of t j . We define the result of the subtree substitution on Tr that substitutes t i by t j , written

Tr[t i ← t j ] = (T ′ , E ′ , ∆ ′ , α| T ′ , β| T ′ ), where
T ′ = (T \ τ (Tr[t i ])) ∪ τ (Tr[t j ]) E ′ = E| T ′ ∪ {(p, t j ) | (p, t i ) ∈ E} ∆ ′ = (∆| T ′ \ {(nil, ϕ 1 , * t j ), (t j * , ϕ 2 , nil) ∈ ∆}) ∪ {(t j , ϕ 1 , t 1 ), (t j * , ϕ 1 , t 1 ) | (t i , ϕ 1 , t 1 ) ∈ ∆} ∪ {(t 2 , ϕ 2 , t j ), (t 2 , ϕ 2 , * t j ) | (t 2 , ϕ 2 , t i ) ∈ ∆}
Compared to [START_REF] Geier | [END_REF], the substitution operator here requires to remove t j corresponding starting and terminating tasks before replacement because they are generated from t j 's parent which will be dropped. The following proposition states that the resulting tree is still valid.

Proposition 5. Let Tr = (T, E, ∆, α, β) be a valid decomposition tree with respect to TIHTNS problem P and two nodes t i ∈ T, t j ∈ τ (Tr[t i ]) with α(t i ) = α(t j ). Then Tr[t i ← t j ] is also a valid decomposition tree w.r.t. P. Acyclic decomposition The sequence of decompositions is acyclic if for every node t in its corresponding tree Tr, the ancestors of t have different actions.

The insertion of tasks allow us to break the loop of generating same compound tasks during the decomposition procedure and find a shortcut to generate the solution. The following theorem states that we only need to consider the acyclic sequences of decompositions to compute the solution. Theorem 1. A TIHTNS problem P has a solution iff P has a solution which is generated by an acyclic sequence of decompositions and an insertion.

Proof. The right-to-left direction is straightforward. Now we prove the left-to-right direction.

Suppose σ tn,sI is a TIHTNS solution of P where tn I → * D tn 1 and tn is obtained from tn 1 by insertion. Then tn 1 is a primitive task network. If the sequence of decompositions from tn I to tn 1 is acyclic, then it is proved. Now suppose the sequence is not acyclic. Then in its corresponding tree Tr, there exist two ancestors t i , t j of some t such that α(t i ) = α(t j ). By Proposition 5, the tree Tr[t i ← t j ] is valid with respect to P. By Proposition 1, there exists a primitive task network tn 2 such that tn 2 = ϑ(Tr[t i ← t j ]) and tn I → * D tn 2 . As the substitution does not introduce any new node, the leaf nodes of Tr[t i ← t j ] are a subset of the leaf nodes of Tr. The only state constraints about leaf nodes which are introduced by the substitution are (t j * , ϕ 1 , t 1 ) and ( * t j , ϕ 2 , t 2 ). Now we replace all occurrences of t j * with t i * , * t j with * t i in Tr[t i ← t j ]. Then except for those constraints between subtasks of t j and either t j * or * t j , in form of ( * t j , ⊤, t) and (t, ⊤, t j * ), the constraints in tn 2 are a subset of the constraints of tn 1 . Because t i is an ancestor of t j in Tr, there must be some tasks t ′ , t ′′ , . . . , t (k) such that ( * t i , ⊤, t ′ ), (t ′ , ⊤, t ′′ ), . . . , (t (k) , ⊤, * t j ). Thus, in the plan σ tn,sI , * t i is before * t j and then before all subtasks of t j . Then those constraints ( * t j , ⊤, t) are satisfied; the case of t j * is similar. So σ tn,sI includes all tasks in tn 2 and is consistent with all constraints in tn 2 and then tn can be obtained from tn 2 by inserting an appropriate task network.

The next lemma allows us to only check whether there is a solution with an upper bound on length. Lemma 1. If TIHTNS problem P = (L, C, O, M, s I , tn I ) has a solution then P has a solution with a length of at most

|T I | × (k + 2) |C|×c m × 2 p×c n ,
where k is the maximal number of subtasks in one method, c and p are the number of constants and predicates respectively, m and n are the maximal arity of compound actions and predicates respectively. Sketch of proof. A valid decomposition tree generated by acyclic decomposition has at most |T I |×(k+2) |C| leaf nodes where |C| is the number of ground compound actions that is |C| × c m . As the number of states is bounded by 2 |L0| where |L 0 | is the number of ground atoms that is p × c n , at most 2 |L0| tasks can be inserted between two neighbor tasks.

As the state-transition function γ satisfies the frame axiom, before performing an operator we can check whether a formula is true in the next state. Given a formula ϕ, we use ϕ - o to denote the formula obtained from ϕ by replacing p with ⊤ and q with ⊥ where p ∈ add(o) and q ∈ del(o). As all variables outside the effect of operator o keep their truth value, we can check whether a formula ϕ holds in the next state before operator o is performed, as the following lemma states: Lemma 2. s |= ϕ - o iff γ(s, o) |= ϕ. Acyclic progression We adapt the acyclic progression operator proposed in [Alford et al., 2015b] to TIHTNS. To capture the 'maintenance' state constraints we introduce a set Σ of pairs (ϕ, t) of formula and task which means ϕ should be satisfied until performing task t. We use Fml(Σ) to denote the conjunction of all formulas in Σ. To forbid the recursion of tasks, when decomposing a compound task t by method m, its subtasks cannot contain t's ancestors denoted by h(t), where h : T -→ 2 C . Formally, we use a tuple (s, tn, Σ, h) to represent that task network tn still needs to be accomplished at the current state s. Given a tuple (s, tn, Σ, h), we define its acyclic progression is (s ′ , tn ′ , Σ ′ , h ′ ) with tn ′ = (T ′ , ∆ ′ , α ′ ), which is obtained from three possible options:

• Task insertion: if s |= pre(o) ∧ Fml(Σ) - o then s ′ = γ(s, o) and tn ′ = tn, Σ ′ = Σ, h ′ = h
The task inserted should satisfy the state constraints.

• Task performing: for a primitive task t

∈ T , if s |= pre(α(t)) ∧ Fml(Σ -t ) - α(t)
where Σ -t = Σ ∩ (L × (T \ {t})) and the followings hold:

there is no t ′ such that t ′ = nil and (t ′ , ϕ, t) ∈ ∆ for every (nil, ϕ, t) ∈ ∆, s |= ϕ for every (t, ϕ, t

′ ) ∈ ∆, s |= ϕ - α(t) then -T ′ = T \ {t}, ∆ ′ = ∆| T ′ , α ′ = α| T ′ -Σ ′ = Σ -t ∪ {(ϕ, t ′ ) | (t, ϕ, t ′ ) ∈ ∆ and t ′ = nil} -h ′ = h| T ′ and s ′ = γ(s, α(t))
A primitive task is chosen only if its all predecessors have been accomplished, its precondition is satisfied and the state constraints in Σ, except for those constraints ending with the primitive task, will hold after performing it. The 'maintenance' state constraints starting from the primitive task will be added into Σ. • Task decomposition: for a compound task t ∈ T and a method (α(t), (T m , ∆ m , α m )) ∈ M , if h(t) ∩ T m = ∅, then tn --→ t,m tn ′ and

-h ′ = h| T ′ ∪ {(t m , h(t) ∪ {α(t)}) | t m ∈ T m } -Σ ′ = Σ| T ′ ∪ {(ϕ, * t) | (ϕ, t) ∈ Σ}
For a compound task, the method chosen to decomposed cannot contain an action which is its ancestor in order to avoid the recursion of tasks. Note that the cardinality of Σ may be exponential on size but Fml(Σ) is polynomial which is restricted by the constraints in M . It entails that applying a step of acyclic progression is in P.

If all tasks can be eliminated by the acyclic progression, then the TIHTNS problem has a solution: Lemma 3. Given a TIHTNS problem P, P has a solution iff there is a sequence of acyclic progressions from (s I , tn I , ∅, h I ) to (s, tn ∅ , ∅, ∅) where h I = α I and tn ∅ is the empty task network.

As TIHTNS can cover lifted TIHTN whose plan-existence problem is 2-NEXPTIME-complete [Alford et al., 2015b], the plan-existence problem for TIHTNS is 2-NEXPTIMEhard. The next theorem states its completeness by proving the upper bound.

Theorem 2. Deciding whether a TIHTNS problem has a solution is 2-NEXPTIME-complete.

Proof. Theorem 1 reduces deciding the problem into checking all solutions generated by acyclic decomposition and Lemma 1 reduces further deciding the problem into checking solutions with an upper bound on length. By introducing a counter on operators according to [Alford et al., 2015b], the length of a solution can be restricted with a polynomial translation on the problem. As the number of decomposing compound tasks is bounded double exponentially, it can ensure that every sequence of acyclic progression terminates after a double exponential number of steps of insertion, performing and decomposition. Therefore, by applying nondeterministically acyclic decomposition, either it can reach a solution or it cannot progress any more which entails that the problem has no solution.

Conclusion

In this paper, we extended TIHTN so that state constraints can be captured. We have shown that this extension does not increase the complexity while it can represent cover planning frameworks: lifted TIHTN and HR-HGN.

State constraints are written as linear temporal logic (LTL) formulas in PDDL3. It seems not difficult to adapt the notion of consistency with constraint to LTL formulas.

The experimental results in [START_REF] Shivashankar | [END_REF] show that the completeness of methods can contribute to the performance of the HGN planners. It is also interesting to extend TIHTNS again with allowing method insertion. A promising reference is [START_REF] Herzig | [END_REF] which provides a dynamic view of HTN planning and models decomposition in propositional dynamic logic.

For the original TIHTN planning, there is an intuitive attempt to simulate that constraint: introducing a virtual primitive task pt whose precondition is "more than 10% battery" and introducing two ordering constraints t1 ≺ pt and pt ≺ t2. Assume "charge" means to charge the battery, then the plan t1;"charge";pt;t2 is a solution of the original TIHTN planning problem. But it is counter-intuitive and the state constraint is still violated.

In the original HTN, "non-primitive or compound task name" is used to distinguish from STRIPS-like actions, i.e., operators.

Acknowledgments

We wants to appreciate the reviewers for their insightful comments. Zhanhao Xiao is partially supported by CSC (Chinese Scholarship Council) and Hai Wan is partially supported by National Natural Science Foundation of China under grants 61573386, Natural Science Foundation of Guangdong Province under grant 2016A030313292 and Guangdong Province Science and Technology Plan projects under grant 2016B030305007. Laurent Perrussel is partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02.


Cédric Foy

Cerema, Ifsttar Umrae

Estimation of the impact noise level in wood constructions: choice of a global acoustic indice

Keywords:

This work concerns the characterization of the impact sound level in wood constructions from in situ measurements. Two global acoustic indices have been estimated and compared with each other, one associated with the excitation source that is the tapping machine, the other related to the excitation source that is the impact ball, likely to better characterize footsteps. These global acoustic indices were estimated over the frequency range [100 Hz: 3150 Hz] and over the frequency range [50 Hz: 3150 Hz], A-weighted or A-unweighted. A statistical study was then carried out on the difference between the global acoustic indice related to the tapping machine and that related to the impact ball. According to the choice of the frequency range and the use of the A-weighting, this statistical study made it possible to conclude on the sensitivity of these global acoustic indices with respect to the correction term normalizing the influence of the reception room and, with respect to the low frequencies [50 Hz: 100 Hz]. Finally, this statistical study concludes on the relevance of the use of a global acoustic indice compared to the other in order to characterize the impact sound level in these wood constructions.

INTRODUCTION

This work focuses on building acoustics. More specifically, it aims to focus on acoustic indicators of the level of impact noise for light wooden constructions. To date in France, the impact noise level is estimated from measurements using a tapping machine as excitation source. This is the same excitation source used for heavy concrete construction. However, several works seem to question this excitation source in favor of the impact ball. The sound signal of this excitation source is here supposed to be closer to that resulting from the noise of step, often troublesome in the light constructions. This work compares an acoustic indicator associated with the "tapping machine" excitation source with an acoustic indicator linked to the "impact ball" excitation source. A series of measurements have been made on wood constructions. The impact noise levels associated with each of these excitation sources were then measured. Standardized and non-standardized acoustic indicators, A-weighted and A-unweighted acoustic indicators, integrating or not the low frequencies [50 Hz : 100 Hz], were then estimated, as well as the deviations between these different acoustic indicators. The present work is based on the study of these deviations. At first, the influence of the A-weighting and the low frequencies on these deviations were studied by directly observing the value of these deviations, but also by carrying out a statistical analysis. The latter is based on a non-parametric test which, by definition, doesn't assume that these deviations obey a particular distribution. In a second step, this work ends with the study of the relevance of the choice of the acoustic indicator for these wood constructions. As before, a statistical analysis is performed.

MEASUREMENT CAMPAIGNS AND ACOUSTIC INDICES

The sites selected for this study are individual houses with wood floors with different types of floor coverings (Figure 1). The emission rooms and the reception rooms are superimposed on each other or diagonally. A measurement campaign is associated with a emission room and a reception room. Nine measurement campaigns were carried out. For measurement campaign number 1, five emission positions and five reception positions were considered. For all other measurement campaigns, three emission positions and three reception positions were only considered. For each pair "emission position / receive position", the impact noise level was estimated for the two commonly used excitation sources: the impact machine and the impact ball [START_REF] Schultz | Alternative test method for evaluating impact noise[END_REF][START_REF] Ljunggren | Correlation between sound insulation and occupants' perception -proposal of alternative single number rating of impact sound[END_REF][START_REF] Schoenwald | Influence of receive room properties on impact sound pressure level measured with heavy impact sources[END_REF][START_REF] Jeon | Use of the standard rubber ball as an impact source with heavyweight concrete floors[END_REF]. The emission positions and the reception positions chosen are the same for the case where the excitation source is the tapping machine and for the case where the excitation source is the impact ball.

For the tapping machine, the recording time is fifteen seconds. For the impact ball, the recording time is adapted to the case of a single drop from a height of one meter, the height corresponding to the distance between the underside of the ball and the ground surface considered. Regarding the measurement of the reverberation time in the reception rooms, it is made from a impulse noise. Three reception positions were systematically considered. Finally, the reverberation time introduced within the expressions of the impact noise level is arithmetically averaged over these three measurements.

The acoustic indicators chosen are based on international standards [START_REF]Acoustic -Rating of Sound Insulation in Buildings and of Building Elements -Part 2: Impact Sound Insulation[END_REF][START_REF]Acoustic -Laboratory Measurements of Sound Insulation of Building Elements -Part 3: Measurement of Impact Sound Insulation[END_REF]. The acoustic indicator associated with the "tapping machine" excitation source is the standard sound pressure level L nT (see Table 1). The acoustic indicator associated with the "impact ball" excitation source is the maximum sound pressure level L F Max (see Table 2). The frequency ranges studied are [100 Hz : 3150 Hz] and [50 Hz : 3150 Hz]. The frequency range [50 Hz : 100 Hz] corresponds in our study at low frequencies. This choice is common [START_REF] Rasmussen | Sound insulation between dwellings -descriptors applied in building regulations in europe[END_REF][START_REF] Antonio | Influence of low frequency bands on airborne and impact sound insulation single nnumber for typical portuguese buildings[END_REF] even if it can be questioned in the context of the noise annoyance in wood constructions [START_REF] Ljunggren | Correlation between sound insulation and occupants' perception -proposal of alternative single number rating of impact sound[END_REF][START_REF] Blazier | Investigation of low-frequency footfall noise in wood-frame, multifamily building construction[END_REF][START_REF] Olynyk | Subjective judgments of footstep-noise transmission through floors[END_REF]. Finally, these acoustic indicators were estimated with or without A-weighting. (A-weighted) sound pressure level for the i thirdoctave band and for the j measure L (A)eq, j = 10 log n i=1 10 L (A)eq,i, j /10

(A-weighted) global sound pressure level for the j measure L n(A)T,i, j = L (A)eq,i, j -10 log( T R i T R re f ) (A-weighted) standardized sound pressure level for the i third-octave band and for the j measure L n(A)T, j = 10 log n i=1 10 L n(A)T,i, j /10 (A-weighted) global standardized sound pressure level for the j measure Table 1: Acoustic indicator related to the impact noise level measured from the tapping machine, with T R i the average reverberation time in the reception room and T R re f = 0.5s.

L (A)F Max,i, j (A-weighted) maximum sound pressure level for the third octave band i and for the j measure L (A)F Max, j = 10 log n i=1 10 L (A)F Max,i, j /10 (A-weighted) maximum global sound pressure level for the j measure

L (A)F MaxV,T,i, j = L (A)F Max,i, j + 10 log( V V re f ) - 10 log         1-C -1 re f 1-C -1         C (1-C) -1 -C -(1-C -1 ) -1 C (1-C re f ) -1 re f -C -(1-C -1 re f ) -1 re f                
(A-weighted) maximum standardized sound pressure level for the i third-octave band and for the j measure

L (A)F MaxV,T, j = 10 log n i=1 10 L (A)F MaxV,T,i, j /10
(A-weighted) maximum global standardized sound pressure level for the j measure Table 2: Acoustic indicator related to the impact noise level measured from the impact ball, with C re f = T R re f /1.7275, C = T R i /1.7275 and V re f = 50m 3 .

RESULTS AND ANALYSIS

Influence of the correction term -Comparison of standardized and nonstandardized levels

The first study concerns the influence of the correction term on the value of acoustic indicators. For each indicator, the deviation ∆ corr between the standardized noise level and the non-standard noise level is estimated:

∆ corr = L standardized -L non-standardized . (1)
The non-standard noise level corresponds to the standardized noise level (see Tables 1 and2) for which the logarithmic terms related to the reception room are not taken into account. These deviations were estimated for the cases where the indicators are calculated over the frequency ranges [50 Hz : 3150 Hz] and [100 Hz : 3150 Hz], A-weighted or Aunweighted.

Concerning the acoustic indicator associated with the "tapping machine" excitation source (see Table 1), the figure 2 shows, for example, the deviations calculated from the measured impact noise levels on the range of frequencies [50 Hz : 3150 Hz], A-weighted and A-unweighted. The mean deviation was also estimated. Finally, in all cases ([50 Hz : 3150 Hz] or [100 Hz : 3150 Hz], A-weighted and A-unweighted), the mean deviation is negative and of the order of -1.5 dB (or dB(A)). The standardized impact noise level is lower than the non-standardized impact noise level. The influence of the correction term therefore seems significant. This means that the expression of the proposed correction term makes it possible to take into account a certain influence of the reception room. Moreover, the fact that these deviations are of the same order of magnitude whether the Aweighting is taken or not int account, implies that the correction terme semms independant of the A weigthing. This result is in good agreement with the fact that the correction term is theoretically only related to the influence of the reception room.

Concerning the acoustic indicator associated with the "impact ball" excitation source, in all cases ([50 Hz : 3150 Hz] or [100 Hz : 3150 Hz], A-weighted and A-unweighted), the mean deviation is less than one decibel (or dB(A)). The standardized impact noise level is therefore of the same order of magnitude as the non-standardized impact noise level. Thus, the influence of the term of correction seems here not very significant. This suggests that the proposed expression of the correction term is inadequate since it doesn't seem to reflect the influence of the receiving room. On the other hand, as previously, A-weighting doesn't affect the value of the correction term.

As a result, the statistical analysis performed and based on these ∆ corr deviations uses the Wilcoxon test. This is a non-parametric test which aims here to check the validity of the hypothesis "the mean deviation is zero" (hypothesis H0) from the sample of measurements. So if the hypothesis H0 is verified then the deviation ∆ corr can be considered as zero. As a result, the standardized level L standardized is equal to the unstandardized level L non-standardized . It follows that the associated correction term has no influence on the value of the acoustic indicator. If the hypothesis H0 is not verified (hypothesis H1), the conclusion is reversed. Classically, accepting or rejecting the H0 assumption is based on the value of the estimated p-value (noted p) by the Wilcoxon test:

                  
p ≤ 0.01 : very strong presumption against the hypothesis H0 0.01 < p ≤ 0.05 : strong presumption against the hypothesis H0 0.05 < p ≤ 0.1 : weak presumption against the hypothesis H0 p > 0.1 : very weak presumption against the hypothesis H0

(2)

The table 3 summarizes the results obtained. First, these results show that, for the two acoustic indicators, the A-weighting doesn't seem to play a role on the value of this correction term which was expected. Secondly, if the frequency range considered is [100 Hz : 3150 Hz], then the correction term have an impact on the value of the acoustic indicator, whether it is the "tapping machine" noise level or the "impact ball" noise level. Thirdly, if the low frequencies [50 Hz : 100 Hz] are integrated, the acoustic indicator associated with the "tapping machine" excitation source remains sensitive to the correction term but not the acoustic indicator associated with the "impact ball" excitation source. This suggests that the expression of the associated correction term must be reviewed at least in this frequency range [50 Hz : 100 Hz]. This is however not clearly established with regard to the values of p-value.

Influence of low frequencies on the value of acoustic indices

The second study concerns the influence of low frequencies [50 Hz : 100 Hz] on the value of acoustic indicators. For each acoustic indicator, the deviation ∆ low between the estimated impact noise level over the frequency range [50 Hz : 3150 Hz] and the estimated impact noise level over the range Frequency [100 Hz : 3150 Hz] was estimated, A-weighted and A-unweighted: (

∆ low = L [
Concerning the acoustic indicator associated with the "tapping machine" excitation source (see Table 1), the figure 3 shows the calculated deviations from the impact noise levels calculated, weighted A and unweighted A. In the case where the A-weighting is not considered, the mean deviation is positive and equal to 1.1 dB. The estimated impact noise level over the frequency range [50 Hz: 3150 Hz] is therefore higher than the estimated impact noise level over the frequency range [100 Hz : 3150 Hz] (see Equation 3). In other words, the introduction of the frequency range [50 Hz: 100 Hz] finally leads to an increase of the order of the decibel of the "tapping machine" noise level. For the case where the A-weighting is considered, the mean deviation is close to zero (0.1 dB(A)). The influence of the frequency range [50 Hz : 100 Hz] is finally greatly reduced by the introduction of the A-weighting. This is logical since the A-weighting greatly reduces the level of impact noise measured at low frequencies.

Concerning the acoustic indicator associated with the "impact ball" excitation source (see Table 2), for the case where the A-weighting is not considered, the mean deviation is equal to 4.8 dB. The introduction of the frequency range [50 Hz : 100 Hz] thus generates a mean deviation much greater than that obtained for the other acoustic indicator. This acoustic indicator is therefore much more sensitive to low frequencies. The fact that it is positive induces that the estimated impact noise level over the frequency range [50 Hz : 3150 Hz] is higher than the estimated impact noise level over the frequency range [100 Hz : 3150 Hz]. For the case where the A-weighting is considered, this mean deviation becomes close to zero (0.3 dB(A)). As before, the introduction of the A-weighting strongly attenuates the possible influence of the low frequencies.

In a similar way to the study of the influence of the correction terms (see section 3.1), a statistical analysis based on the estimated deviations ∆ low is carried out here. Hypothesis H0 is identical (zero mean deviation). If the hypothesis H0 is here verified, then this means that the value of the impact noise level L [50 Hz : 3150 Hz] is equal to the value of the impact noise level L [100 Hz : 3150 Hz] . In other words, the introduction of the frequency range [50 Hz : 100 Hz] has no impact on the value of the acoustic indicator tested. Another possible interpretation is that this acoustic indicator is not sensitive to low frequencies. If the hypothesis H0 is not verified (hypothesis H1), the conclusion is reversed.

The obtained resulsts are summarized in the Table 4. Finally, the hypothesis of a zero mean deviation is rejected by Wilcoxon's test for the two acoustic indicators, A-weighted or A-unweighted. The introduction of the frequency range [50 Hz : 100 Hz] therefore has an impact on the value of the acoustic indicators. For the case of the A-weighted acoustic indicator associated with the "tapping machine" excitation source, this result is however to qualify because the value of the p-value is much higher than for the other cases. Thus, in order to better underline the influence of low frequencies in light wood constructions, it would be undoubtedly more reliable to consider the acoustic indicator associated with the "impact ball" excitation source. This result is however to be taken with caution since the study of the influence of the term of correction showed that its expression is not necessarily adequate on this range of frequencies (see section 3.1).

Comparison of acoustic indicators

The following study concerns the influence of the choice of acoustic indicator. For each acoustic indicator, the difference between ∆ ind is calculated between the standardized impact noise level associated with the "impact ball" excitation source and the standardized impact noise level associated with the "tapping machine" source excitation:

∆ ind = L ball standardized -L machine standardized . (4)
These deviations were estimated for the cases where the indicators are calculated over the frequency ranges [50 Hz: 3150 Hz] and [100 Hz: 3150 Hz], A-weighted and Aunweighted.

Concerning the results based on the frequency range [50 Hz : 3150 Hz] (see Figure 4), the mean deviation is equal to -1.5 dB(A) for the case where A-weighting is taken into account. It is equal to 2.2 dB for the case where the A-weighting is excluded. If the frequency range is [100 Hz : 3150 Hz], the mean deviation is equal to -1.7 dB(A) for the case where A-weighting is taken into account and -1.5 dB otherwise. This suggests that the choice of the acoustic indicator here has an influence in all cases. If the low frequencies [50 Hz : 100 Hz] are not taken into account, then the value of the A-weighted acoustic indicator associated with the "impact ball" excitation source is weaker than that of the acoustic indicator associated with the "tapping machine" excitation source. If the low frequencies are taken into account and the A-weighting is excluded, this result is reversed. Table 5: Sensitivity to the choice of the acoustic indicator.

The statistical analysis based on the deviations ∆ ind is summarized in table 5. Hypothesis H0 is always the same. If it is accepted here, then this means that the impact noise levels L ball standardized and L machine standardized are equal to each other. In other words, the choice of the acoustic indicator to be used for acoustical studies in wood constructions is not important. If the hypothesis H0 is not verified (hypothesis H1), the conclusion is reversed. Finally, the test results associated with the frequency range [100 Hz : 3150 Hz] show that the choice of the acoustic indicator has no influence, A-weighted or A-unweighted. The same applies to the frequency range [50 Hz : 3150 Hz] if A-weighting is considered. In other words, if the A-weighting is applied, the introduction of the low frequencies [50 Hz : 100 Hz] doesn't make it possible to differentiate these two standardized acoustic indicators. For the frequency range [50 Hz : 3150 Hz], if the A-weighting is not considered, then the choice of the standardized acoustic indicator has an influence on the results. In other words, if the A-weighting is not applied, the introduction of the low frequencies [50 Hz : 100 Hz] can make it possible to differentiate these two standardized acoustic indicators.

SYNTHESIS AND CONCLUSION

At first, the influence of the correction term and that of the low frequencies [50 Hz : 100 Hz] were considered. The following results have been highlighted:

-A-weighting doesn't seem to influence the value of acoustic indicators; -The acoustic indicators are sensitive to the correction term, except that associated with the "impact ball" excitation source and estimated by introducing the low frequencies [50 Hz : 100 Hz]. In this case, this seems to mean that the standardized impact noise level is equal to the non-standardized impact noise level, ie the impact noise level is independent of the receiving room. But this is impossible. This suggests that the expression of the associated correction term should probably be revisited over the frequency range [50 Hz : 100 Hz];

-The introduction of the frequency range [50 Hz : 100 Hz] has an influence on the value of acoustic indicators. This influence seems slightly more pronounced for the acoustic indicator associated with the "impact ball" excitation source. This could suggest that, in order to better highlight the influence of low frequencies in light wood constructions, the use of this acoustic indicator would be more appropriate. This result is to be taken with caution with regard to the previous point relating to the correction term. -For this specific case, the value of the acoustic indicator associated with the "impact ball" excitation source is greater than that associated with the "tapping machine"excitation source. As previously, these results are to be taken with caution with regard to the previous point relating to the correction term.

Figure 1 :

 1 Figure 1: Old house in France: wood construction L (A)eq,i, j(A-weighted) sound pressure level for the i thirdoctave band and for the j measure

Figure 2 :

 2 Figure 2: Tapping machine: deviation between the standardized impact noise level and the non-standardized impact noise level for the case where the acoustic indicator is estimated over the frequency range [50 Hz : 3150 Hz], A-weighted and A-unweighted. The mean deviation is equal to -1.4 dB(A) and -1.3 dB, respectively A-weighted and Aunweighted.

  50 Hz : 3150 Hz] -L [100 Hz : 3150 Hz] .

Figure 3 :

 3 Figure 3: Tapping machine: deviations between the impact noise level based on the frequency range [50 Hz : 3150 Hz] and that related to the frequency range [50 Hz : 3150 Hz]. The acoustic indicator is estimated, A-weighted and A-unweighted. The mean deviation is equal to 0.1 dB(A) if A-weighting is taken into account and to 1.1 dB otherwise.

Figure 4 :

 4 Figure4: Deviation between the standardized impact noise level related to the "impact ball" excitation source and the standardized impact noise level related to the "tapping machine" excitation source. These levels are estimated over the frequency range [50 Hz : 3150 Hz], A-weighted and A-unweighted. The mean deviation is equal to -1.5 dB(A) if A-weighting is taken into account and to 2.2 dB in the opposite case.

Finally, this study

  focused on the relevance of the choice of acoustic indicators in the context of measurements in wood constructions. The following results have been highlighted: -The choice of the A-weighted acoustic indicator has no influence that the low frequencies [50 Hz : 100 Hz] are taken into account or not; -The choice of the A-unweighted acoustic indicator has no influence if the low frequencies [50 Hz : 100 Hz] are not considered; -The choice of the A-unweighted acoustic indicator has influence if the low frequencies [50 Hz : 100 Hz] are considered;

Table 3 :

 3 Sensitivity to the correction term.

Table 4 :

 4 Sensitivity at low frequencies [50Hz : 100Hz].

ACKNOWLEDGEMENTS

The author of this article wishes to thank Mr FALWISANNER and Mr BRENDEL who carried out the measurements.