Andreas Herzig

Ivan Varzinczak

Domain descriptions should be modular

This work is about the metatheory of actions, and here we address the problem of what a good domain description for reasoning about actions should look like. We state some postulates concerning this sore spot, which establishes the notion of a modular domain description. We point out the problems that arise when modularity is violated and propose algorithms to overcome them.

INTRODUCTION

In logic-based approaches to reasoning about actions a domain is described by a set of logical formulas Σ. At first glance satisfiability is the only criterion logic provides to check the quality of such a description. In this paper we go beyond that, and argue that we should require more than the mere existence of a model for Σ. Our starting point is that in reasoning about actions one usually distinguishes several kinds of logical formulas. Among these are effect axioms, precondition axioms, and domain constraints.

We prefer here to speak of effect, executability, and static laws, respectively. Moreover we separate inexecutability laws from effect laws. Given these ingredients, suppose the language is powerful enough to state that action α is inexecutable in contexts where A holds, and executable in contexts where B holds. It follows that there can be no context where A ∧ B holds. Now ¬(A ∧ B) is a static law that is independent of α. It is therefore natural to expect that it follows from these laws alone! By means of examples we show that if this is not the case then unexpected conclusions might follow from Σ. A similar case can be made against implicit inexecutability laws.

This motivates postulates requiring that the different ingredients of domain descriptions should be arranged in a modular way, such that interactions between them are limited and controlled. It turns out that in all existing accounts which allow for these four kinds of laws [START_REF] Lin | Embracing causality in specifying the indirect effects of actions[END_REF][START_REF] Mccain | A causal theory of ramifications and qualifications[END_REF][START_REF] Thielscher | Computing ramifications by postprocessing[END_REF][START_REF] Castilho | Formalizing action and change in modal logic I: the frame problem[END_REF][START_REF] Zhang | EPDL: A logic for causal reasoning[END_REF], consistent domain descriptions can be written that violate some of these postulates. We here give algorithms that allow one to check whether a domain description satisfies the postulates. With such algorithms, the task of correcting badly written descriptions can be made easier.

This paper is organized as follows: after the preliminary definitions (Sections 2 and 3) we state (Section 4) and study (5-7) three postulates. Finally we discuss strengthenings (Section 8) and assess related work (Section 9).

DOMAIN DESCRIPTIONS

In this section we establish the ontology of domain descriptions. 1 The authors are with the Institut de Recherche en Informatique de Toulouse (IRIT), Toulouse, France. e-mail: {herzig,ivan}@irit.fr

Static laws

Frameworks which allow for indirect effects make use of logical formulas that link invariant propositions about the world. Such formulas characterize the set of possible states. They do not refer to actions, and we suppose they are expressed as formulas of classical propositional logic. PFOR = {A, B, . . .} is the set of all classical formulas. A static law2 is a formula A ∈ PFOR that is consistent. An example is Walking → Alive, saying that if a turkey is walking, then it must be alive [START_REF] Thielscher | Computing ramifications by postprocessing[END_REF].

Effect laws Let ACT = {α, β, . . .} be the set of all actions of a given domain. To speak about action effects we use the syntax of propositional dynamic logic (PDL) [START_REF] Harel | Dynamic logic[END_REF]. The formula [α]A expresses that A is true after every possible execution of α.

An effect law3 for α is of the form A → [α]C , where A, C ∈ PFOR, with A and C both classically consistent. ('Classically consistent' is a shorthand for 'consistent in classical propositional logic'.) The consequent C is the effect which obtains when α is executed in a state where the antecedent A holds. An example is Loaded → [shoot]¬Alive, saying that whenever the gun is loaded, after shooting the turkey is dead. Another one is [tease]Walking: in every situation, the result of teasing is that the turkey starts walking.

Note that the consistency requirements for A and C make sense: if A is inconsistent then the effect law is superfluous; if C is inconsistent then we have an inexecutability law, that we consider to be a separate entity.

Inexecutability laws

We suppose that effect laws with inconsistent consequents are a particular kind of law. This allows us to avoid mixing things that are conceptually different: for an action α, an effect law mainly associates it with a consequent C , while an inexecutability law only associates it with an antecedent A.

An inexecutability law for α is of the form A → [α]⊥, where A ∈ PFOR is classically consistent. For example ¬HasGun → [shoot]⊥ expresses that shoot cannot be executed if the agent has no gun.

Executability laws

With only static and effect laws one cannot guarantee that shoot is executable if the agent has a gun. Whereas all the extant approaches in the literature that allow for indirect effects of actions contain static and effect laws, the status of executability laws is less consensual. Some authors [START_REF] Schubert | Monotonic solution of the frame problem in the situation calculus: an efficient method for worlds with fully specified actions[END_REF][START_REF] Doherty | Explaining explanation closure[END_REF][START_REF] Mccain | A causal theory of ramifications and qualifications[END_REF][START_REF] Thielscher | Computing ramifications by postprocessing[END_REF] more or less tacitly consider that executability laws should not be made explicit but rather inferred by the reasoning mechanism. Others [START_REF] Lin | Embracing causality in specifying the indirect effects of actions[END_REF][START_REF] Zhang | EPDL: A logic for causal reasoning[END_REF] have executability laws as first class objects one can reason about.

It seems strange to us just stating information about necessary conditions for execution of an action (inexecutabilities) and saying nothing about the sufficient ones. This is the reason why we think we need executability laws. Indeed, in several domains one wants to explicitly state under which conditions a given action is guaranteed to be executable, e.g. that a robot should never get stuck and should always be able to execute a move action. In any case, allowing for executability laws gives us more flexibility and expressive power.

In dynamic logic the dual α A, defined as ¬[α]¬A, can be used to express executability. α thus reads "the execution of action α is possible". An executability law 4 for α is of the form A → α , where A ∈ PFOR is classically consistent. For instance HasGun → shoot says that shooting can be executed whenever the agent has a gun, and tease establishes that the turkey can always be teased. Domain descriptions S ⊆ PFOR denotes the set of all static laws of a given domain. For a given action α ∈ ACT, Eα is the set of its effect laws, Xα is the set of its executability laws, and Iα is the set of its inexecutability laws. We define E = α∈ACT Eα, X = α∈ACT Xα, and I = α∈ACT Iα. A domain description is a tuple of the form S, E, X , I .

DYNAMIC LOGIC AND THE FRAME PROBLEM

Given a domain description S, E, X , I , we need a consequence relation solving the frame problem. To this end we now give the semantics of PDL, and extend it with dependence relations. P1, P2, . . . denote propositional constants, L1, L2, . . . PDL alone does not solve the frame problem. For instance, if S, E, X , I describes our shooting domain then S, E, X , I |= PDL HasGun → [load]HasGun. The deductive power of PDL has to be augmented in order to ensure that the relevant frame axioms follow. The presence of static constraints makes that this is a delicate task, and starting with [START_REF] Lin | Embracing causality in specifying the indirect effects of actions[END_REF][START_REF] Mccain | A causal theory of ramifications and qualifications[END_REF], several authors have argued that some notion of causality is needed. We here opt for the dependence based approach presented in [START_REF] Castilho | Formalizing action and change in modal logic I: the frame problem[END_REF], where dependence information has been added to PDL. In [START_REF] Demolombe | Regression in modal logic[END_REF] it has been shown how Reiter's solution to the frame problem can be recast in PDL. α ; L denotes that the execution of action α may change the truth value of the literal L. In our example we have

; = shoot, ¬Loaded , shoot, ¬Alive , shoot, ¬Walking , tease, Walking
Because load, ¬HasGun / ∈ ;, we have load ; ¬HasGun, i.e., ¬HasGun is never caused by load. We also have tease ; Alive and tease ; ¬Alive. 4 Some approaches (most prominently Reiter's) use biconditionals A ↔ α , called precondition axioms. This is equivalent to ¬A ↔ [α]⊥, which illustrates that they thus merge information about inexecutability with information about executability.

We here suppose that ; is finite. A given dependence relation ; defines a class of possible worlds models M;: every M ∈ M; must satisfy that whenever wRαw then • α ; P and w ∈ I(P ) implies w ∈ I(P ); • α ; ¬P and w ∈ I(P ) implies w ∈ I(P ).

The associated consequence relation is noted |=;. In our example we obtain S, E, X , I |=; HasGun → [load]HasGun.

POSTULATES

Our central hypothesis is that the different types of laws should be neatly separated, and should only interfere in one sense: static laws allow one to infer action laws that do not follow from the action laws alone. The other way round, action laws should not allow to infer new static laws, effect laws should not allow to infer inexecutability laws, etc. Here are the postulates for that: P0. Logical consistency: S, E, X , I |=; ⊥ A domain description should be logically consistent.

P1. No implicit executability laws

: if S, E, X , I |=; A → α , then S, X |= PDL A → α
If an executability law can be inferred from the domain description, then it should already "be" in X , in the sense that it should also be inferable in PDL from the set of executability and static laws alone. If a static law can be inferred from the domain description, then it should be inferable in PDL (and even classically) from the set of static laws alone.

Postulate P0 is obvious. P1 can be ensured by maximizing X . This suggests a stronger version of P1:

P4. Maximal executability laws: if S, E, X , I |=; A → [α]⊥, then S, X |= PDL A → α
It expresses that if in context A no inexecutability for α can be inferred, then the respective executability follows in PDL from the executability and static laws. P4 generally holds in nonmonotonic frameworks, and can be enforced in monotonic approaches such as ours by maximizing X .

Things are less obvious for Postulates P2 and P3. They are violated by domain descriptions designed in all approaches in the literature that allow to express the four kinds of laws. We therefore discuss each of them in the subsequent sections by means of examples, and give algorithms to decide whether they are satisfied.

Consider the following domain description (and ; as above): In the literature, such laws are also known as implicit qualifications [START_REF] Ginsberg | Reasoning about actions II: The qualification problem[END_REF], and it has been argued that it is a positive feature of reasoning about actions frameworks to leave them implicit and provide mechanisms for inferring them [START_REF] Lin | Embracing causality in specifying the indirect effects of actions[END_REF][START_REF] Thielscher | Computing ramifications by postprocessing[END_REF]. The other way round, one might argue as well that implicit qualifications indicate that the domain has not been described in an adequate manner: inexecutability laws have a form simpler than that of effect laws, and it might be reasonably expected that it is easier to exhaustively describe them. 5Thus, all the inexecutabilities should be explicitly stated, and this is what Postulate P2 says.

S1 = {Walking → Alive}, E1 = [tease]Walking, Loaded → [shoot]¬Alive , X1 = I1 = ∅ From [
How can we check whether P2 is violated? First we need a definition. Given classical formulas A and B, the function NewCons A(B) computes the set of strongest clauses that follow from A ∧ B, but do not follow from A alone (cf. e.g. [START_REF] Inoue | Linear resolution for consequence finding[END_REF]). It is known that NewConsA(B) can be computed by subtracting the prime implicates of A from those of A ∧ B. For example, the set of prime im-

plicates of P is just {P }, that of P ∧ (¬P ∨ Q) ∧ (¬P ∨ R ∨ T ) is {P, Q, R ∨ T }, hence NewCons P ((¬P ∨ Q) ∧ (¬P ∨ R ∨ T )) =
{Q, R∨T }. And for our example, NewConsWalking→Alive(Walking) = {Alive, Walking}.

Algorithm 1 (Finding implicit inexecutability laws)

input: S, E, I, ; output: a set of implicit inexecutability laws I I I I := ∅ for all α ∈ ACT do for all J ⊆ Eα do

A J := {A i : A i → [α]C i ∈ J} C J := {C i : A i → [α]C i ∈ J} if S ∪ {A J } is classically consistent then for all L i ∈ NewCons S (C J ) do if ∀i, α ; L i and S, I |= PDL (A J ∧ ¬L i ) → [α]⊥ then I I := I I ∪ {(A J ∧ ¬L i ) → [α]⊥}
Example 1 Consider S1, E1, I1 and ; as given above. Then Algorithm 1 returns

I I = {¬Alive → [tease]⊥}.
Theorem 1 S, E, X , I satisfies Postulate P2 if and only if

I I = ∅.
This is the key algorithm of the paper. We are aware that it comes with considerable computational costs: first, the number of formulas AJ and CJ is exponential in the size of Eα, and second, the computation of NewConsS (CJ ) might result in exponential growth. While we might expect Eα to be reasonably small in practice, the size of NewConsS(CJ ) is more difficult to control. Note that the algorithm terminates because we have assumed ; finite.

The algorithm not only decides whether the postulate is satisfied, its output I I also can provide a way to "repair" the domain description. Basically there are three options, that we illustrate with our example: 1) add ¬Alive → [tease]⊥ to I1; 2) add the (unintuitive) dependence tease, Alive to ;; or 3) weaken the law [tease]Walking to Alive → [tease]Walking. It is easy to see that whatever we opt for, the new domain description will satisfy P2.

NO IMPLICIT STATIC LAWS

Executability laws increase expressive power, but might conflict with inexecutability laws. For instance, let S2 = S1, E2 = E1, X2 = { tease }, and I2 = {¬Alive → [tease]⊥}. (Note that Postulate P2 is satisfied.) We have the unintuitive X2, I2 |= PDL Alive: the turkey is immortal! This is an implicit static law because Alive does not follow from S2 alone: P3 is violated.

How can we find out whether there are implicit static laws? We assume that Postulate P2 is satisfied, i.e., all inexecutabilities are captured by I. The existence of implicit static laws may thus indicate too strong executability laws: in our example, we wrongly assumed that tease is always executable. It may also indicate that the inexecutability laws are too strong, or that the static laws are too weak:

Example 3 Suppose a computer representation of the line of integers, in which we can be at a strictly positive number, Positive, or at a negative one or zero, ¬Positive. Let MaxInt and MinInt, respectively, be the largest and the smallest representable integer number. goleft is the action of moving to the biggest integer smaller than the one at which we are. Consider the following domain description for this scenario (Ati means we are at number i):

S3 = {Ati → Positive : i > 0} ∪ {Ati → ¬Positive : i ≤ 0} E3 = {AtMinInt → [goleft]Underflow}∪ {Ati → [goleft]Ati-1 : i > MinInt} X3 = { goleft }, I3 = ∅ with the dependence relation (MinInt ≤ i ≤ MaxInt): ; = goleft, Ati , goleft, Positive , goleft, ¬Positive , goleft, Underflow
In order to satisfy Postulate P2, we run Algorithm 1 and obtain I3 = {(At1 ∧ At2) → [goleft]⊥}. Now applying Algorithm 2 to this action theory gives us the implicit static law ¬(At1 ∧ At2), i.e., we cannot be at 1 and 2 at the same time.

Theorem 2 Suppose S, E, X , I satisfies P2. Then Postulate P3 is satisfied if and only if S I = ∅.

What shall we do with an implicit static law? Again, several options show up: whereas in the latter example the implicit static law should be added to S, in the former the implicit static law is due to an executability law that is too strong and should be weakened.

So, in order to satisfy Postulate P3, a domain description should contain a complete set of static laws or, alternatively, should not make so strong assumptions about executability. This means that eliminating implicit static laws may require revision of S or completion of X . In the next section we approach the latter option.

MAXIMAL EXECUTABILITY LAWS

Implicit static laws only show up when there are executability laws. Which executability laws can be consistently added to a given domain description? What Theorem 3 says is that it suffices to take the 'complement' of I to obtain all the executability laws of the domain. Note that this counts as a solution to the qualification problem given that all preconditions for guaranteeing executability of actions are thus known.

DISCUSSION

In this section we discuss other properties related to consistency and modularity of domain descriptions. Some will follow from ours, while some others look natural at first glance, but turn out to be too strong.

Theorem 4 If S, E, X , I satisfies P3, then S, E, X , I |=; ⊥ iff S |= PDL ⊥.
This means that if there are no implicit static laws then consistency of a domain description (P0) can be checked by just checking consistency of S.

Theorem 5 If S, E, X , I satisfies P3, then S, E, X , I |=; A → [α]C iff S, Eα, Iα |=; A → [α]C .
This means that under P3 we have modularity inside E, too: when deducing the effects of α we need not consider the action laws for other actions. Versions for executability and inexecutability can be stated as well.

Remark 8.1 Although in the present paper concurrency is not taken into account, we conjecture that Theorem 5 holds when we have concurrent action execution. Now we turn to postulates that are too strong. First, it seems to be in line with the other postulates to require domain descriptions not to allow for the deduction of new effect laws: if an effect law follows from a domain description, and no inexecutability law for the same action in the same context can be derived, then it should follow from the set of static and effect laws alone. This means we should have:

P5. No implicit effect laws: if S, E, X , I |=; A → [α]C and S, E, X , I |=; A → [α]⊥, then S, E |=; A → [α]C
But consider the following intuitively correct domain description: Another though obvious possibility of amending our modularity criteria could be by stating the following postulate: P6. No unattainable effects:

S5 = ∅, E5 = Loaded → [shoot]¬Alive, (¬Loaded ∧ Alive) → [shoot]Alive X5 = {HasGun → shoot }, I5 = {¬HasGun → [shoot]⊥}
if A → [α]C ∈ E, then S, E, X , I |=; A → [α]⊥
This expresses that if we have explicitly stated an effect law for α in some context, then there should be no inexecutability law for the same action in the same context. We do not investigate this further here, but just observe that the slightly stronger version below leads to unintuitive consequences: (¬HasGun ∧ Loaded) → [shoot]⊥. This is certainly too strong. Our example also illustrates that it is sometimes natural to have some 'redundancies' or 'overlaps' between I and E.

RELATED WORK

Pirri and Reiter have investigated the metatheory of the situation calculus [START_REF] Pirri | Some contributions to the metatheory of the situation calculus[END_REF]. In a spirit similar to ours, they simplify the entailment problem for this calculus, and show for several problems such as consistency or regression that only some of the modules of a domain description are necessary. Note that in their domain descriptions S = ∅. This allows them to show that such theories are always consistent.

Zhang et al. [START_REF] Zhang | Consistency of action descriptions[END_REF] have also proposed an assessment of what a good domain description should look like. They develop the ideas in the framework of EPDL [START_REF] Zhang | EPDL: A logic for causal reasoning[END_REF], an extended version of PDL which allows for propositions as modalities to represent causal connection between literals. We do not present the details of that, but concentrate on the main metatheoretical results.

Zhang et al. propose a normal form for describing action theories, and investigate three levels of consistency. Roughly speaking, a domain description Σ is uniformly consistent if it is globally consistent

(i.e., Σ |= EPDL ⊥); a formula Φ is Σ-consistent if Σ |= EPDL ¬Φ, for Σ a uniformly consistent theory; Σ is universally consistent if Σ |= EPDL A implies |= EPDL A.
Given these definitions, they propose algorithms to test the different versions of consistency for a domain description Σ that is in normal form. This test essentially amounts to checking whether Σ is safe, i.e., whether Σ |= EPDL α , for every α. Success of this check should mean the domain description under analysis satisfies the consistency requirements.

Nevertheless, this is only a necessary condition: it is not hard to imagine domain descriptions that are uniformly consistent but in which we can still have implicit inexecutabilities that are not caught by the algorithm. Consider for instance a scenario with a lamp that can be turned on and off by a toggle action, and its EPDL representation given by {On →

[toggle]¬On, Off → [toggle]On, [On]¬Off, [¬On]Off}.
The causal statement [On]¬Off means On causes ¬Off. Such a domain description satisfies each of the consistency requirements (in particular it is uniformly consistent, as Σ |= EPDL ⊥). However, Σ is not safe because the implicit static law ¬(On∧Off) cannot be proved.

Lang et al. [START_REF] Lang | Causal theories of action -a computational core[END_REF] address consistency in the causal laws approach [START_REF] Mccain | A causal theory of ramifications and qualifications[END_REF], focusing on the computational aspects. They suppose an abstract notion of completion of a domain description solving the frame problem. Given a domain description Σα containing logical information about α's direct effects as well as the indirect effects that may follow, the completion of Σα roughly speaking is the original theory Σα amended of logical axioms stating the persistence of all non-affected (directly nor indirectly) literals.

Their EXECUTABILITY problem is to check whether α is executable in all possible initial states (Zhang et al.'s safety property). This amounts to testing whether every possible state w has a successor w reachable by α such that w and w both satisfy the completion of Σα. For instance, still considering the lamp scenario, the representation of the domain description for toggle is {On toggle -→ Off, Off toggle -→ On, Off -→ ¬On, On -→ ¬Off}, where the first two formulas are conditional effect laws for toggle, and the latter two causal laws in McCain and Turner's sense. We will not dive in the technical details, and just note that the executability check will return "no" for this example as toggle cannot be executed in a state satisfying On ∧ Off.

CONCLUSION

We have tried to point out some of the problems that can arise when domain descriptions are not modular. In particular we have argued that the non-dynamic part of domain descriptions should not be influenced by the dynamic one. 6We have put forward several postulates, and have in particular tried to demonstrate that when there are implicit inexecutability and static laws then one has slipped up in designing the domain description under consideration. As shown, a possible solution comes into its own with Algorithms 1, 2 and 3, which can give us some guidelines in correcting a domain description if needed.

Given the difficulty of exhaustively enumerating all the preconditions under which a given action is executable and also those under which such an action cannot be executed, there is always going to be some executability precondition A or some inexecutability precondition B that together lead to a contradiction, forcing, thus, an implicit static law ¬(A ∧ B). This is the reason we propose to state some information about both executabilities and inexecutabilities, complete the latter and then, after deriving all implicit static laws, complete the former. As a final result we will have complete S, X and I.

Throughout this work we used a weak version of PDL, but our notions and results can be applied to other frameworks as well. It is worth noting however that for first-order based frameworks the consistency check of Algorithm 1 is undecidable. (We can get rid of this by assuming that S, E, X , I is finite and there is no function symbol in the language. In this way, the result of NewCons is finite and the algorithm terminates.)

Our postulates do not take into account causality statements linking propositions. This could be a topic for further investigation.

  literals, and Φ, Ψ, . . . formulas. (We recall that A, B, . . . denote classical formulas.) If L = ¬P then we identify ¬L with P . A PDL-model is a triple M = W, R, I where W is a set of possible worlds, R maps action constants α to accessibility relations Rα ⊆ W × W , and I maps propositional constants to subsets of W . Given a PDL-model M = W, R, I , |=M Φ if for all w ∈ W , w |=M Φ; w |=M [α]Φ if w |=M Φ for every w such that wRαw . A formula Φ is a consequence of the set of global axioms {Φ1, . . . , Φn} in the class of all PDL-models (noted Φ1, . . . , Φn |= PDL Φ) if and only if for every PDL-model M , if |=M Φi for every Φi, then |=M Φ.

P2.

  No implicit inexecutability laws: if S, E, X , I |=; A → [α]⊥, then S, I |= PDL A → [α]⊥ If an inexecutability law can be inferred from the domain description, then it should be inferable in PDL from the static and inexecutability laws alone. P3. No implicit static laws: if S, E, X , I |=; A, then S |= PDL A.

Algorithm 2 (Example 2

 22 Finding implicit static laws) input: S, X , I output: a set of implicit static laws S I S I := ∅ for all α ∈ ACT do for all A → [α]⊥ ∈ I and A → α ∈ X do if S |= PDL ¬(A ∧ A ) then S I := S I ∪ {¬(A ∧ A )} For S2, E2, X2, I2 , Algorithm 2 returns S I = {Alive}.

Algorithm 3 (Example 4 Theorem 3

 343 Finding implicit executability laws) input: S, X , I output: a set of implicit executability laws X I X I := ∅ for all α ∈ ACT do Aα:= {A i : A i → [α]⊥ ∈ Iα} if S |= PDL Aα and S, X |= PDL ¬Aα → α then X I := X I ∪ {¬Aα → α } Suppose S4 = {Walking → Alive}, X4 = ∅ and I4 = {¬Alive → [tease]⊥}. Then Algorithm 3 yields X I = {Alive → tease }. Suppose S, E, X , I satisfies P2 and P3. Postulate P4 is satisfied if and only if X I = ∅.

Theorem 6

 6 There exist domain descriptions S, E, X , I not satisfying P3 such that S, E, X , I |=; A → [α]C and S, Eα, Iα |=; A → [α]C . For example, we have S2, E2, X2, I2 |=; ¬Alive → [shoot]Alive, but S2, E2 shoot , I2 shoot |=; ¬Alive → [shoot]Alive.

  together with the dependence relation ; of Example 1. It satisfies Postulates P1, P2, P3, and P4, but does not satisfy P5. Indeed, we have that S5, E5, X5, I5 |=; ¬HasGun ∨ Loaded → [shoot]¬Alive and S5, E5, X5, I5 |=; ¬HasGun ∨ Loaded → [shoot]⊥, but S5, E5 |=; ¬HasGun ∨ Loaded → [shoot]¬Alive. So, Postulate P5 would not help us to deliver the goods.

P6.

  No unattainable effects (strong version): if S, E |=; A → [α]C , then S, E, X , I |=; A → [α]⊥ Indeed, for the above domain description we have that E5 |=; (¬HasGun ∧ Loaded) → [shoot]¬Alive, but S5, E5, X5, I5 |=;

  tease]Walking it follows with S1 that [tease]Alive, i.e., in every situation, after teasing the turkey is alive: S1, E1 |= PDL [tease]Alive. Now as tease ; Alive, the status of Alive is not modified by the tease action, and we have S1, E1 |=; ¬Alive → [tease]¬Alive. From the above, it follows S1, E1, X1, I1 |=; ¬Alive → [tease]⊥, i.e., the turkey cannot be teased if it is dead. But S1, I1 |= PDL ¬Alive → [tease]⊥, hence Postulate P2 is violated. The formula ¬Alive → [tease]⊥ is an example of what we call an implicit inexecutability law.

Static laws are often called domain constraints, but the different laws for actions that we shall introduce in the sequel could in principle also be called like that.

Effect laws are often called action laws, but we prefer not to use that term here because it would also apply to executability laws that are to be introduced in the sequel.

Note that nevertheless this is not related to the qualification problem, which basically says that it is difficult to state all the executability laws of a domain.

It might be objected that it is only by doing experiments that one learns the static laws that govern the universe. But note that this involves learning, whereas here -as always done in the reasoning about actions field -the static laws are known once forever, and do not evolve.

ACKNOWLEDGMENTS

Ivan Varzinczak has been supported by a fellowship from the government of the Federative Republic of Brazil, grant CAPES.


Thi-Hoa Nguyen

Marylène Vayer

Christophe Sinturel

PS-b-PMMA/PLA blends for advanced nano-lithographic templates

Keywords: thin films, homopolymer/copolymer blends, solvent annealing, sub-10 nm porosity, nanolithography

Blends of poly(styrene)-block-poly(methyl methacrylate) (PS-b-PMMA) and poly(lactide) (PLA) were deposited in the form of thin films on the surface of modified silicon wafers and exposed to tetrahydrofuran (THF) vapor annealing. It was shown that in specific experimental conditions, a coreshell morphology consisting in cylinders with a PMMA shell and a PLA core, within a continuous matrix of PS, was formed. In this case, PLA naturally segregated in the core of the PMMA cylinders, minimizing the PS/PLA interaction, which constitutes the most incompatible pair (the interaction strength between the various components was confirmed in thin films of the corresponding polymer blends). Compared to other block copolymer/homopolymer blends described in the literature, this system exhibits unexpected high increase of the characteristic lengths of the system (center-to-center distance and diameter). This was attributed to a partial solubilization of the PLA in the PMMA corona (the two polymers are highly compatible), inducing an enhanced level of PS and PLA stretching caused by the strong repulsion between these two polymers. The selective extraction of the PLA yielded to porous domains with small dimensions (6 ± 2.5 nm), reaching the performances that are currently attained in highly incompatible block polymers with low molecular weight. Further PMMA removal revealed a second porosity level, with higher pores diameter and center-to-center distance compared to the neat PS-b-PMMA system. This work highlights how PS-b-PMMA, that currently represents one of the industrial standards nanoporous template precursors, can be modified in an easy and costless approach using PLA homopolymer addition.

Introduction

The phase behavior of block polymers can be notably modified by the addition of homopolymers. When the macrophase is avoided, i.e. when the homopolymer is incorporated into the existing domains, changes in the dimensions can arise (one of the domains is swollen) but also new morphologies can be formed. This has been carefully addressed by numerous studies, from a theoretical point of view [1][2][3][4] and experimentally demonstrated in the bulk [5][6][7][8][9] and in thin films [9][10][11][12][13][14][15][16][17][18][19][20]. Starting from a given block polymer composition that normally dictates the type of morphology at equilibrium, it is thus possible to tune the properties of the self assembly by homopolymer addition, expanding the possibility to generate various template geometries with tailored dimensions. From a practical point of view, this is particularly interesting in the field of the elaboration of nanoporous templates where a simple costless homopolymer addition would render possible a fine tuning of the morphology (instead of a cumbersome and costly library of block polymers with various dimensions and compositions). Among the various block polymer systems considered for applications within such approach, PS-b-PMMA currently represents the industrial standard [20][21][22][23][24]. Despite a relatively moderate incompatibility (preventing the possibility to obtain feature size in the sub-10 nm range unlike high  materials [25]), the familiarity in the microelectronic industry with the two homopolymers, the relative ease of the polymer synthesis, the possibility to find solvent compatible with the industrial requirements, the improvement of the long range order and reduction of defects density obtained with the DSA approach have made PS-b-PMMA a very attractive system. For this polymer, it has been well demonstrated that homopolymer addition such as PMMA [10,14,15] or PEO [13] in a cylinder-forming PS-b-PMMA was able to modify the dimension of the domains formed. Interestingly, depending on the system composition and molar weight of the added homopolymer, new core/shell morphology could be formed and from such organization, Jeong et al.demonstrated the possibility to generate sub-10 nm porosity with selective extraction of the PMMA [11] or PEO [13]. In this work, we have examined the possibility to modify the morphology of a typical PS-b-PMMA system with PLA. Despite an abundant literature devoted to block polymer/homopolymer blends, such system has not been yet considered to our knowledge. Compared to other types of modifiers, PLA represents a material with a growing interest due to its renewable source and the ease of its selective degradation with dilute base, that would leave totally unaffected the PS and PMMA domains in contrast to PS-b-PMMA/PMMA or/PEO blends where the extraction of the PMMA would potentially result in surface reconstruction due to the PMMA block swelling. In addition, PLA displays higher level of incompatibility towards PS than PMMA, allowing for sharper behavior in comparison to PS-b-PMMA/PMMA(or even PS-b-PMMA/PEO) system. Our purpose is to examine how small amounts of such affordable homopolymer can have a profound impact on established block polymer morphologies. The stability of the obtained morphologies upon solvent annealing has been examined.

Experimental

Materials and chemicals

Poly(styrene) (PS), poly(lactide) (PLA), poly(methyl methacrylate) (PMMA) homopolymers and PS-b-PMMA, P(S-r-MMA) block polymers were purchased from Polymer Source Inc. TetraEthylOrthoSilicate (TEOS) and all used solvents were purchased from Sigma Aldrich and used as received. Si(100) substrates of 10*10 mm2were cleaned by sonication in dichloromethane, methanol and distilled water for 10 min each.

Thin films preparation

PS-b-PMMA/PLA blend: a 10 mg•mL -1 solution of PLA(16 kg•mol -1 ) in acetone or toluene was prepared and mixed in appropriate amounts to a 20 mg•mL -1 solution of PS-b-PMMA(101 kg•mol-1, f PMMA (vol)= 0.3) in toluene to prepare blends with homopolymer concentrations (vol/vol%) of 1, 5, 10 and 15% in the dry state (based on the density of each component). The resulting solution mixtures were agitated overnight before being deposited by spin coating (2500 rpm) onto modified silicon wafers with a P(S-r-MMA) (14 kg•mol -1 ) brush on top (to prepare the modified substrates, a thin layer (approx. 10 nm) of P(S-r-MMA) was firstly deposited onto clean silicon wafers, heated under vacuum at 170°C for 48 h and rinsed in toluene). Homopolymer/block polymer thin films with a thickness between 60 and 70 nm were obtained using this procedure (thicknesses were measured by imaging a scratched area in AFM tapping mode). After deposition, thin filmswere exposed at 25•C to THF vapors in a closed vessel (150 mL) containing 5 mL of THF for 5 and 10 min. PS/PMMA/PLA blend:PS (67 kg•mol -1 ), PMMA (30 kg•mol -1 ), and PLA (16 kg•mol -1 ) 3/1/1 (weight fraction) ternary blends were prepared by mixing in appropriate amounts a toluene solution of PS (10 mg•mL -1 ) with PMMA/PLA (10 mg•mL -1 ) blend in acetone.

Atomic force microscopy (AFM)

AFM in the tapping mode was carried out in air at room temperature with a Nanoscope III from Digital Instruments Corp. Silicon cantilevers Tap300 from Budget Sensors with integrated symmetrical pyramidal tips (15 µm high) with no Al coating backside, a nominal spring constant of 42 N.m -1 and a resonance frequency between 300 and 400 kHz were used.

All the displayed AFM images are height images taken in tapping mode. Characteristic lengths (diameter and center-to-center distance) were extracted from 2D line cut. Each dimension provided is the result of multiple measurements with a typical standard deviation of 2.5 nm taken as uncertainty (error bar on Fig. 3).

Thickness measurement upon swelling

The evaluation of the solvent/polymer interaction strength was achieved by exposing each polymer thin films (separately) to solvent vapors. For that purpose, a closed PTFE box with a cover equipped with a sealed optical glass window was used. The swelling ratio d/d0 is defined by the ratio between the thickness (d) of the film at time (t) and the thickness (d0) at time (t = 0). The film thickness was measured continuously using an interference-based film thickness measurement system (F20, Filmetrics) until the thickness reach a plateau (typically 5-10 min). The film was illuminated with a white light source at a normal incidence. The intensity of the reflected light was recorded as a function of the incident wavelength and was fitted using a model taking into account the interferences of the light waves reflected from the upper and lower surfaces of the film at different light wavelengths.

Selective removal of the component

PLA was selectively degraded by placing the sample in a0.5 M sodium hydroxide solution containing 40/60 (by volume) methanol/water for 30 min. After being removed from the solution, the samples were washed with a 40/60 (by volume) methanol/water solution. PMMA was selectively removed by exposing the thin films to UV radiation (254 nm) during 60 h (lamp power: 0.10 mJ/s) and further immersion of the irradiated films into concentrated acetic acid for 20 min and finally rinsed in distillated water.

Inorganic replication of the porous films

The silica precursor solution (TEOS:H2O:EtOH:HCl) with a molar proportion of 1:5.5:21:0.005 was prepared by mixing 26.5 mL EtOH,1 mL deionised water, 1.25 mL HCl 0.1 M and 5 mL TEOS and stirring at least during 16 h at room temperature. The porous PS films were immerged in the solution allowing for the infiltration of the porosity by the liquid silica precursors [26]. After withdrawal, samples were then heated at 450•C during 5 min to provoke the precursor condensation and the elimination of the polymer template to yield the silica replicas. Depending on the deposition conditions (withdrawal speed), the formation of a dense silica roof layer above the porous replica could be obtained [27]. This was exploited to prepared mechanically robust samples for the cross sectional views.

Results and discussions

Film morphology as function of homopolymer addition

In this section, we examine the influence of the homopolymer (PLA) content on the structuration of the block polymer (PS-b-PMMA). For that purpose, PS-b-PMMA (101 kg•mol -1 ,f PMMA = 0.3)/PLA (16 kg•mol -1 ) blends were firstly prepared by mixing a solution of PS-b-PMMA in toluene with a solution of PLA in acetone. The homogenous mixture was then spin-coated on silicon wafer. Fig. 1 shows the surface morphology of the resulting films, as a function of the homopolymer content. Surface topography is shown for the as casted samples and after 5 and 10 min of solvent vapor exposure. The absence of macroscopic phase segregation suggests that the incorporation of the PLA is homogenous in the studied range of the composition (up to 15%). We observed that the incorporation of the homopolymer in the block polymer was dependent on the deposition conditions. Fig. S1 (Supplementary Materials) shows several examples of as spun morphologies with homogenous and heterogeneous dispersions. Using chlorobenzene as the solvent for both the block polymer and the homopolymer led to a macroscopic phase separation (Fig. S1c andS1f). In contrast, solvent mixture (toluene/THF and toluene/acetone) promoted a homogenous dispersion of the homopolymer in the block polymer self-assembled pattern (Fig. S1a-b and Fig. S1d-e). In fact, we will demonstrate later that the PLA is incorporated in the minor PMMA domains (as one can already deduce from the clear increase in size of the segregated phases observed in Fig. 1). The absence of macrophase separation in the blend can be explained on the basis of the miscibility properties of the components. Acetone or THF, which are good solvents for PMMA and PLA (but not for PS), will promote the segregation of these two components, forming micelles like structures (with a PS corona) at nanoscopic scale as the solvent evaporates. As observed in Fig. 1, as spun neat PS-b-PMMA thins films do not exhibit clear ordered nanostructuration, due the fast evaporation of the solvent and the rather low chemical incompatibilityof the PS and PMMA blocks. Adding PLA in the system does not improve the order in the as spun state, but clearly the microphase separation is enhanced as judged by the increase in contrast of the images (particularly after 5%). After exposure to solvent vapors, the nanostructuration is improved.

Hexagonal array of dots or fingerprint morphologies are formed depending on the exposure time and amount of PLA added. The formation of the hexagonal array is driven by the natural properties of the PS-b-PMMA to self-assemble into such morphology. However, despite the use of a neutralized substrate that normally promotes the formation of a perpendicular orientation of the PMMA domains, [28] the presence of PLA favored the formation of parallel orientation. When the proportion of PLA is above a given threshold (between 1 and 5% for 10 min exposure; 5 and 10% for 5 min), the presence of the latter drives the orientation of the domains towards a parallel orientation. This suggests that the neutralized substrate is specific for PS and PMMA composition but not for PLA. We, and others, have already demonstrated that such transition is strongly driven in the swollen state, by the affinity of the domains towards the interfaces. [29,30] When polymers display different swelling extent they will exhibit different response towards a surface field even if the surface energy of the polymers is similar. The swelling extent of PS, PLA and PMMA measured under THF vapors (same conditions than Fig. 1) showed that PLA swells slightly more than the other counterparts (1.8 for PLA vs 1.6 for PS and PMMA) indicating that the delicate surface energy balance between PS/PMMA domains and the interface favoring the perpendicular orientation is prone to perturbation in presence of PLA. The characterization of the actual morphology, as well as the localization of the polymer phases was carried out using specific polymer extraction (hydrolysis under mild alkaline solution for PLA and UV exposure followed by acetic acid extraction for PMMA). The resulting porous polymer film was examined by AFM and subjected to replication in order to assess the internal structuration of the film and fully confirm the morphology adopted [26]. This is shown in Fig. 2 where the two typical In the case of the dots morphology, the first PLA extraction forms small depressions located in the center of the circular domains, indicating that the PLA is located in the middle of the PMMA domains (Fig. 2b). Further extraction of the PMMA domains enlarges the pores diameters (Fig. 2c). The final resulting porosity (after PLA and PMMA removal) was replicated by infiltration with sol-gel precursors followed by a brief thermal treatment in order to provoke the condensation of the precursors and the pyrolysis of the polymer phase. The top (Fig. 2d) and side view (Fig. 2e) of the obtained replica reveal an array of perpendicular pillars (covered by an upper layer in the case of the cross sectional view to ensure the mechanical stability of the replica upon fracture -see experimental part), indicating that the parent porosity and therefore, the initial morphology, can be depicted as an array of vertical cylinders. For the fingerprint morphology, the selective removal of PLA (Fig. 2g), followed by the PMMA extraction (Fig. 2h) indicates that the PLA is similarly located in the center of the PMMA. The replication confirmed the presence of parallel cylinders as seen on the top (Fig. 2i) and side (Fig. 2j) views. For this latter, the observed structure can be described as an array of collapsed solid cylinders in contact, resulting from the elimination of the continuous phase.

Analysis of the characteristic dimensions of PS-b-PMMA/PLA blends.

Naturally, the characteristic dimensions of the features formed by the self-assembly process increased with the amount of PLA added in the blends, as a direct consequence of the incorporation of the PLA into the PMMA domains. From the AFM images, we measured the variation of the center-to-center distance (d) and cylinder diameter (D) with the PLA content (Fig. 3). As expected and already demonstrated by others [10][11][12][13][14] increased amount of homopolymer leads to increased values of the characteristics distances of the system. As demonstrated by the selective extraction of the homopolymer, localization of the PLA phase occurs in the center of the PMMA blocks which corresponds to the so-called "dry-brush" regime described by the seminal work of Hashimoto [5]. In this specific case, the conformation of the two blocks (PS, PMMA) and the junction density (or average area per junction) at interface should remain unaffected by the homopolymer addition (this is actually rigorously true only in flat interface, but can be approximated in the case of cylindrical features where the dimension increase is small).In these conditions, simple volumetric considerations assuming a core-shell cylinders of PLA model, normally represents the highest increase value that can be achievable. Indeed, in the "wet-brush" model, addition of homopolymer tends to produce lower distance increase (eventually a decrease).This latter phenomenon is well documented for A-B block copolymer mixed with low molecular weight of A and/or B block and is commonly attributed to the necessity for the non-swollen block to contract in order to accommodate the change in junction density at the interface produced by the swelling of the other block (incompressibility requirement) [5][6][7]31]. This is why distance increase in the "dry brush" regime is generally higher than in the "wet brush" configuration. However, as shown in Fig. 3, both center-tocenter distances and diameters evaluated using this simple volumetric consideration are still underestimated, indicating that a simple picture of PLA being localized in the center of the PMMA domains without changing the conformation of the PS and PMMA blocks is not realistic. Higher level of chain stretching needs to be introduced to account for such behavior.

In our case, even if the localization of the PLA has been clearly demonstrated, a partial PLA partition inside the PMMA phase, producing a swollen corona of PMMA, cannot be ruled out.

Because PS and PLA are much more incompatible that PS and PMMA, a strong increase of the effective Flory-Huggins parameters between PS and (PMMA+PLA) phases would occur in this case, causing the chains to stretch more in both domains. Such trend has been already reported, through the observation that distance in the "dry brush" regime was following empirical relationship of the form:

(3) where takes values >0.5 leading to increased values of d. For asymmetric compositions, a value of = 0.83 has been reported by Stuen et al. [14] for a PS-b-PMMA/PMMA in the "dry brush" regime. In our case the best fit occurs for = 2.5 suggesting a more pronounced stretching effect, in good agreement with the strong incompatibility of the PS and PLA chains. (It is important to note that the stretching of the blocks (and consequently the porosity dimension), could be influenced by the type of solvent used in the annealing step.) In order to confirm the PS/PMMA/PLA interactions, we have examined the behavior of the corresponding homopolymer ternary blend.

Segregation strength in ternary homopolymer blends

Ternary blends of PS (67 kg•mol -1 ), PMMA (30 kg•mol -1 ), et PLA(16 kg•mol -1 ) were prepared by mixing a toluene solution of PS with PMMA/PLA blend in acetone (PS/PMMA/PLA 3/1/1 wt/wt). As observed in Fig. 4a, the as casted film exhibits a macrophase separated morphology, with important height difference between the segregated phases that can be related to different rates of drying as already proposed in previous work [32]. The film topography can be described as a continuous layer, with segregated domains of lower elevation (depression).

Again, a selective extraction of PS(with cyclohexane) and PLA (hydrolysis) allowed us to clarify the polymer repartition in the film. As seen in Fig. 4b, the continuous phase is removed by cyclohexane, thus assigned to PS. On the other hand, the depressions are removed by hydrolysis and then ascribe to PLA (Fig. 4c). PMMA localization is revealed by the successive extraction of PS and PLA (Fig. 4d). This results in the formation of vertical walls creating hollow structures, indicating that the PMMA is located at the interface PS/PLA. This behavior can be explained on the basis of the interaction strength for these components. This can be deduced from the known value of the Flory-Huggins parameters of the PS/PLA and PS/PMMA pairs, respectively 0.13 [33] and 0.04 [34] at 25•C, showing interaction strength in the order PS/PLA < PS/PMMA. Although reliable Flory-Huggins value of the PLA/PMMA is not available, several work [35][36][37] reported the good compatibility of this system that can reasonably place the interaction strength of the PLA/PMMA pair above the values of the two others (PS/PLA < PS/PMMA < PLA/PMMA). Consequently, the PMMA will locate at the interface PS/PLA, screening the interaction between those two polymers which constitute the most incompatible pair. This behavior corroborates the results obtained in the PS-b-PMMA/PLA system where the PLA naturally segregates in the core of the PMMA cylinders, minimizing the PS/PLA interaction. A schematic view of the blend morphology is provided in Fig. 5 (the volume fraction estimated form such schematic view, using the dimensions and high of the domains extracted from the AFM measurement was found to be in good agreement with the volume fraction of the ternary blend). In our work, we have estimated this porosity to be 6 ± 2.5 nm in the blend containing 5% of PLA, which corresponds to the level that are currently obtained using highly incompatible polymers, with increased segregation strength [25]. However, it is worth noting that the domain density increase which is normally attained with feature size reduction is here counterbalanced by the increase of the inter-domain spacing that results from the homopolymer addition. This approach, which is thus not ideal for the generation of templates with ultrahigh features density, is much in line with current technology such as contact hole shrink where the main objective is the domain size reduction [38]. From the application standpoint, we anticipate that this method could be interestingly combined with the directed self-assembly (DSA) approach [39,40] in order to favor increased long range ordering and lower defectivity. In addition, increase of the characteristic lengths could find interesting applications in templates for nanopatterning where particle-particle distance should be sufficiently high to suppress coupling between particles, like in the case of magnetic particles [41].Further extraction of the PMMA phases led to a second level of porosity, with pores diameters larger compared to the situation of the neat PS-b-PMMA. We observed that the molar mass of PLA was determinant to obtained this core-shell morphology. The segregation effect was herein observed for a PLA molar mass of16 kg•mol-1. Lower molar mass (6,5 kg•mol-1) resulted into hexagonally packed cylinders with no evidence of segregation of a PLA phase in the center on the PMMA cylinders. However, increased cylinder size and center-to-center distance suggests the incorporation of the PLA. Higher molar mass (48 kg•mol-1) resulted into a macroscopic separation of the components. These results are in line with previous observation in PS-b-PMMA/PMMA [6] blends where the core/shell morphology was only obtained for PMMA molar weight close to the one of the PMMA block, whereas values lower and higher respectively led to complete solubilization and macroscopic phase separation.

Conclusion

In this work we have examined the morphology of thin films of PS-b-PMMA/PLA blends.

For a PS-b-PMMA with a standard molar weight (101 kg•mol-1) we examined the influence of the type of solvent used for the deposition, the concentration and molar weight of PLA as well as the behavior of the obtained films upon solvent vapor annealing. In some conditions, hexagonally packed core(PLA)-shell(PMMA) cylinders, oriented perpendicularly to the substrate, within a continuous matrix of PS were formed. This allowed the formation of porous domains with extremely small dimensions (6 ± 2.5 nm) after selective extraction of the PLA, reaching the performances that is currently attained in highly incompatible block polymers with low molecular weight. Such core/shell morphology was obtained when PLA segregated in the core of the PMMA cylinders, minimizing the PS/PLA interaction, which constitutes the most incompatible pair. This result originates not only from the miscibility properties of the three polymers which are driven by their relative incompatibility but also by the solvent used for the deposition and the molar weight. Compare to other block copolymer/homopolymer blends described in the literature, this system exhibits unexpected high increase of the characteristic lengths of the system (center-to-center distance and diameter). This was attributed to a partial solubilization of the PLA in the PMMA corona (the two polymer are highly compatible), inducing an enhanced level of PS and PLA stretching caused by the strong repulsion between these two polymers. Another difference with similar system already described in the literature (PS-b-PMMA/PMMA or/PEO) relies in the mode of extraction of the center part of the cylinders. Because it implies a selective degradation of the PLA, it leaves totally unaffected the PS and PMMA domains in contrast to a solvent extraction method that would potentially result in surface reconstruction of the surface.

Figure 1 :

 1 Figure 1 : AFM height images of PS-b-PMMA/PLA thin films deposited on P(S-r-MMA) modified substrates as a function of the amount of PLA and THF vapor annealing time. All images are 0.5 × 0.5 µm 2 (scale bar is 100 nm), z scale bar is 0-10 nm.

figure 2 :

 2 figure 2 : AFM height images of PS-b-PMMA/PLA: after 5 min of exposure in THF vapors (a), after 10 min of exposure in THF vapors (f), after PLA extraction (b,g), after PLA and PMMA extraction (c, h). SEM image of the silica replica of the porous films obtained after PLA and PMMA extraction for the dot (top view (d) and lateral view (e)) and the fingerprint morphology (top view (i) and lateral view (j)). All AFM images are 0.5 × 0.5 µm 2 (scale bar 100 nm), with z scale 0-10 nm (except c and j: 25 nm). Scale bar for SEM images is 300 nm in all images. In Figure e and j, the dashed lines are guides for the eyes for the silicon surface and the silica upper layer (from bottom to top) and arrows point the cylinder silica replicas.

Figure 3 .

 3 Figure 3. Center-to-center distance measured by AFM after solvent annealing (square, error bars are ±2.5 nm), estimated from volumetric consideration (Eq(1), solid line)(a) and cylinder diameter features measured by AFM after solvent annealing (square, error bars are ±2.5 nm), estimated from volumetric consideration (Eq(2), solid line),fitted by Eq(3) with = 2.5.

(

  core) and PMMA (shell) allow to calculated the center-to-center distance (d) and diameter (D) of the resulting cylindrical domains as follows: where d 0 and D 0 are respectively the center-to-center and the diameter of the PMMA cylinders in the neat PS-b-PMMA film, f is the volume fraction of the PMMA bloc in the PSb-PMMA copolymer and is the volume fraction of the PLA added. It is important to note that this estimation, based on such simple volumetric considerations derived from the "dry brush"

Fig. 4 .

 4 Fig. 4. AFM height images of PS/PMMA/PLA blends after spin coating (a), after selective

Figure 5 :

 5 Figure 5 : Schematic representation of the morphology (cross sectional view) of the PS/PMMA/PLA ternary blend (a) deduced from the specific extraction of PS (b, top view), PLA(c, top view), PS and PLA (d, top view). PS is in blue, PLA in yellow, PMMA in green. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Acknowledgments

We thank Joël Puibasset (ICMN) for fruitful discussions and Annie Richard (University of Orléans, CME) for SEM observations.


Ahmed Altaher
email: ahmed.ah.altaher@gmail.com

Jehad Melad
email: jd.melad@gmail.com

Hassan Madi
email: madihassan@gmail.com

Sharafedeen Alkawan
email: sharaf.alkawan@gmail.com

On benchmarking of networked cyber-physical systems

Keywords: Ethernet, Networked automation, Real-Time, Testing. I

Benchmarking and testing are important steps toward ensuring reliable service of cyber-physical systems. Site acceptance testing of communication networks in smart grids is difficult in terms of safety due to the electrical power nature and real-time constraints. Ethernet has become de facto for smart automation systems in electric grid applications. Engineers, therefore, should consider benchmark testing regarding quality of service networking parameters to avoid degradation of the system reliability. Hence, testing methodology should consider both the dynamics of the Ethernet network and the automation system, i.e. testers should take into account the behavior of automation when communication faults take place. This paper describes how an integrated benchmarking methodology provides empirical evaluation of Ethernet services in industrial automation networks. The results prove the technical feasibility of this method and motivates using commercial-off-the-shelf components (COTS).

INTRODUCTION

Modern automation and control systems utilize Information and Communication Technologies (ICT). Networked automation and control employ Ethernet to exchange data between sensors, control, monitoring and actuators. This networking enables delivering information from/to field equipment as well as from/to human machine interfaces. Many functions benefit from data exchange, such as operation and maintenance, i.e. control coordination and periodic diagnostics that are vital to determine failures [START_REF] Altaher | Reliability Investigation Of Digital Substation Networks Design Using FMEA Technique[END_REF]. Hence, several status and events can be transferred through Ethernet frames in vertical and horizontal architectures to support Real-Time control functions. Particularly, in electrical power automation, sending commands from control room devices to field equipment in early days design require physical hardwired connections. In contrast, these days, vertical communications exist as Manufacturing Message Services (MMS), Fig. 1 shows these services as interfaces 1 and 6 [START_REF] Mohagheghi | Communication Protocols and Networks for Power Systems -Current Status and Future Trends[END_REF]. Data networks are also placed vertically (Fig. 1, interfaces 2 and 4), and horizontally (interface 3) in the bay and field process between protective relays. In this hierarchy, operators and technicians can operate commands either locally at control rooms or remotely at control centers that are connected via wide area networks [START_REF]Communication networks and systems for power utility automation -Part 1: Introduction and overview[END_REF]. Noticeably, communication networks exist at many levels to carry real-time data and other functional information. In figure 1, GOOSE frames exchange status and events data (interface 3) between protective relays in horizontal multicast communication. Recently, the protection and automation communications are extended beyond the local substation to control centers [START_REF]Communication networks and systems for power utility automation -Part 90-2: Using IEC 61850 for communication between substations and control centres[END_REF]. Software enabled functions bring smartness to these communications, i.e. diagnostics of connections, but add efforts complexity. This paper aims to propose a benchmarking method for cyberphysical systems then experimentally evaluate this method considering new failure types, various traffic scenarios and dynamics of Ethernet based communications. Initially, the request for comments (RFCs) issued by the (IETF) and the internet society have been used to offer benchmarking methodologies that can be used to evaluate Ethernet service performance. Generally, the RFC 1242 (Benchmarking Terminology for Network Interconnection Devices) provides benchmarking terminology and definitions for interconnection devices [START_REF]IETF RFC 1242 Benchmarking Terminology for Network Interconnection Devices[END_REF], although RFC 2544 (Benchmarking Methodology for Network Interconnect Devices) was published and specifically designed as a benchmarking methodology for internetworking devices in laboratory settings [START_REF]IETF RFC 2544 Benchmarking Methodology for Network Interconnect Devices[END_REF]. Additionally, RFC 2285 (Benchmarking Terminology for LAN Switching Devices) and RFC 2889 (Benchmarking Methodology for LAN Switching Devices) are commonly used for benchmarking of network switching devices [7 & 8]. RFC 2544 recommends generating traffic that overloads network devices' resources in order to assess their capacity [START_REF] Morton | Applicability Statement for RFC 2544: Use on Production Networks Considered Harmful[END_REF]. Industry and academia have introduced many modifications to the RFC 2544 methodologies and guidelines to describe specific issues in production environments. Bonica and Bryant suggested and adapted a method to production service activation [START_REF] Bonica | RFC 2544 Testing in Production Networks[END_REF]. Other researchers have presented a novel methodology based on the control and the network disciplines, aiming to benchmark Network Controlled Systems (NCS). Their evaluation method used a platform based on Lego Mindstorms where they identified two sets of performance indicators to describe the quality of control for CPS systems or the quality of transport of the underlying network [START_REF] Gallenmüller | Benchmarking Networked Control Systems[END_REF]. Some methodologies are limited in terms of telecommunication services testing due to some artifacts such as:

1. Validation of service configuration, e.g. the committed information rate (CIR).

Validation of performance metrics in a Service Level

Agreement (SLA), e.g. frame loss and latency. 3. Service activation testing, when traffic that shares network resources, with the test, could be adversely affected [START_REF] Bonica | RFC 2544 Testing in Production Networks[END_REF]. To overcome the RFC 2544 limitations, the ITU released systematic methods for benchmarking of Ethernet services. The ITU-T Y.1564 specifies a standardized methodology to measure the performance parameters. The standards covers assessment of information rate, Service Level Agreement (SLA) and service activation test [START_REF]Ethernet service activation test methodology[END_REF]. In fact, ITU-T Y.1564 is more comprehensive and applicable than RFC 2544, e.g. inter frame delay variation is not part of the RFC 2544 benchmarking. Both ITU-T Y.1563 and ITU-T Y.1564 involve extra definitions for vital metrics covering Ethernet service such as throughput, bandwidth, frame loss, delay and frame delay variation [START_REF]Ethernet service activation test methodology[END_REF]. The recommendation of ITU-T Y.1564 fills the methodological gap for measurement of operational Ethernet network services. It covers new benchmarking metrics applicable to Ethernet service activation that include:

1. Multiple time durations for tests, as often performed in operational networks with time-varying impairments. 2. Measuring committed information rate (CIR) and excess information rate (EIR) with several frame rates and load types.

Identifies bandwidth component profiles with a color

mode representing CIR and EIR. 4. Measuring latency and delay variation across several traffic profiles. The bandwidth profile is a concept, related to the expected frame service rate, which defines four traffic parameters (Fig. 2

) [13]: a) Committed information rate (CIR), b) Committed burst size (CBS), c) Excess information rate (EIR) and d) Excess burst size (EBS)

The CIR can be defined as the maximum sustained information rate (IR) the network is committed to transfer while meeting the performance level guaranteed in the SLA.

Performance metrics in terms of frame delay, frame delay variation and frame loss are applicable only to those frames that are transmitted at or below the CIR [START_REF]Ethernet service activation test methodology[END_REF]. EIR shall represent the maximum sustained IR by which a user can exceed its CIR with some expectations that the excess traffic might be carried through the network. Fig. 2 shows the relation between CIR, EIR and color-coding of the traffic. In addition, the standard recommends two tests, namely a service configuration test that aims to validate service configurations, and a service performance test to validate the quality of service over time. During these tests, the frame size can be constant, or a distribution of multiple frame sizes. Besides, user-defined frame sizes can be used during the configuration and performance tests. The test duration should be 15 minutes, 2 hours or 24 hours.

The ITU-T Y.1564 standard has detailed frames types, durations and amounts. These features thus motivate us to adapt them for our benchmarking method.

III. NETWORKED AUTOMATION SYSTEMS

Power automation functions are necessary for maintaining safe and reliable service of the electric grids. Automation systems provide programmed control functions that replace manual operations to increase the safety of operators and the reliability of these grids. Recently, protection functionalities are coordinated through data communication. For instance, a protection relay recognizes faults and informs other connected devices to take actions, then tripping a relevant circuit breaker that often has connections to other relays, i.e. intelligent electronic devices. Clearance of field faults requires real-time performance. Hence, protective IEDs communicate and coordinate protection and control in timely adjusted constraints to achieve critical missions in a real time manner.

A. Subtation Automation systems

Smart substations are complex systems that involve communication networks, automation systems, and software based functions. In modern substations, IEC 61850-based

Time (s) Bandwidth (Mbps) Conform Conform Non-conform 100% Link CIR+EIR CIR Figure 2
Important parameters for benchmarking [START_REF]Ethernet service activation test methodology[END_REF] systems use intelligent devices to exchange digital data. Measurements are carried digitally from the field process to protective IEDs. Protection data is embedded into Generic Object Oriented Substation Events (GOOSE), to coordinate protection, automation and control. Field measurements are digitally carried through Sampled Value frames that transport electrical digital quantities, e.g. voltage and ampere [3 & 4].

B. Measurment, status and events transmission

The GOOSE service uses a multicasting mechanism to inform protection and control devices, i.e. IEDs, about other devices and equipment status as well as the overall substation events. While field process devices called Merging Units (MU) multicast digital measurements with SV frames to bay IEDs. Both services use and send Ethernet frames containing status or measurements, in a publishing/subscribing paradigm. Hence, automation functionalities depend on Ethernet based IEC 61850-information exchange. For instance, measurement functions transfer power quantities and measurements to protection and control IEDs with SV frames. Protection functions at IEDs receive SV frames, extract measurements and status datasets, calculate metrics and issue GOOSE events according to them.

C. Operation technology requirements

Automation system communications at the control room require low speed where users perform slow tasks, while message transfer at the protection and field process requires fast speed for protection and control applications. Real-Time performance is necessary for protection status communications to guarantee reliable automation coordination and to avoid safety issues. These requirements place constraints for delay time, i.e. transmission time, between publisher and subscriber IEDs when GOOSE messages are used.

Table 1 shows several performance classes, where a fast real-time class of performance is assigned to GOOSE trip messages. The transmission of process measurement, i.e. SV, and timestamping of substation events needs less class of performance, but four microseconds as precise synchronization of time messages and fast real-time as transmission class of performance, i.e. transfer time in terms of three milliseconds (Fig. 3). Ethernet communications should respect these requirements, especially time constraints and performance levels. The GOOSE transfer is normally faster than hardwired based signaling due to the transfer of digital datasets within the GOOSE frames instead of classical hardwired analog signals. Similarly, SV based measurements reduce wiring complexity, increase speed and flexibility of installations compared to traditional current and voltage (CT/VT) instrumentation [START_REF]2020 CSV, Consolidated version, Communication networks and systems for power utility automation -Part 9-2: Specific communication service mapping (SCSM) -Sampled values over ISO/IEC 8802-3[END_REF]. Table 1 summarize timing requirements according to the IEC 61850 standard that sets several performance levels, e.g. P2/P3 for transmission substation. In details, the standard assigns low class for file transfer, fast class for trip GOOSE and precise synchronization for SV streams. Hence, End-to-End transfer time of fast class must not exceed 3 ms where 20% of this time is allocated to network transmission and 80% is divided between publisher and subscriber nodes (Fig. 3), which means 600 µs for the communication channel and 1.2 ms for each end, i.e. sender and receiver IEDs [START_REF]Communication networks and systems for power utility automation -Part 10: Conformance testing[END_REF]. When digital communications are used for teleprotection, then designers must respect security and dependability requirements according to known standards [START_REF]1999 Teleprotection equipment of power systems -Performance and testing -Part 1: Command systems[END_REF]. Although these standards do not identify methodologies of assessment. Table 2 defines performance requirements and shows testing parameters for teleprotection communications, e.g. a nominal transmission delay of a protection message, i.e. GOOSE trip message, in a noise-free channel. While, the actual transmission time (Tac) is the transmission delay of a protection message in a noisy communication channel. Furthermore, the standard defines the probability of unwanted commands (Puc), which is related to the safety, i.e. the ability to prevent interference from issuing a command state at the receiver, which is given by:

= ( 1
)
Where Nuc is the number of unwanted commands, NB is the bit error rate bursts. However, dependability is the ability to issue a valid command during interference and noisy conditions, which is signified by a low probability of missing commands. (Pmc) [START_REF]2020 CSV, Consolidated version, Communication networks and systems for power utility automation -Part 9-2: Specific communication service mapping (SCSM) -Sampled values over ISO/IEC 8802-3[END_REF].

= = 1 - (2)
Where NT and NR represent the number of transmitted and received commands. The test procedure shall use noisytype fault injections. In digital networks, bit error rates can be correlated to equivalent traffic impairments [START_REF] Scheer | Speed and Reliability of Ethernet Networks for Teleprotection and Control[END_REF].

Figure 3 Transfer, transmission and processing requirements

Table 2 represents requirements for both missed and unwanted commands within varied channel quality and noise duration if digital communications are used. The table depicts the actual transmission time, i.e. transmission where impairments such as traffic load and noise do exist. This time must be in a range of 10 ms with pulsed and continuous noise. Some terms used in the power standards, namely security and dependability, differ from those used in ICT where security and dependability are synonymous with safety and reliability in electrical power.

IV. METHODOLOGY

The method is an experimental practice to benchmark the designed automation platform. Table 3 shows the main objectives that aim to evaluate the system together with the dynamic presence of power transients and communication perturbations. Thus, adapting ITU-T Y.1564 to benchmarking the Ethernet service shall provide results that can show various performance classes. The table briefly explains our steps to determine the undesirable impacts on safety as well as dependability when the system enters the operation service, e.g. unavailability due to interruption of delivered communication services that may lead to unreliable protection of the electrical power equipment. In the automation platform, protective relays (IEDs) communicate through a connected Ethernet switch (blue boxes in Fig. 4). Fig. 4 shows IEDs with Ethernet based networking to exchange data. In the experimental platform, we implement these communications to achieve the protection automation strategy and to determine average time of coordination between devices.

The designated platform will contain three IEDs, 2 from the same supplier and the third from another supplier (Fig. 4). The network enables the exchange of GOOSE messages within 100 Mbps bandwidth limitations. The IEDs are equipped with three interfaces: one for management and configuration, the second and the third interfaces for protection and control purposes. The GOOSE messages, therefore, are normally sent in this setup through the second interface, while the IEDs use the third interface for redundancy purposes. The network architecture illustrates the synchronization of the devices' time through an NTP server, i.e. using an intranet access. Moreover, the experimental platform consists of more workstations and industrial devices in accordance with the real industry applications. The management and the configuration of all IEDs are achieved remotely via the network management interfaces that are accessed via the engineering workstations. The architecture similarly contains four computers; three of them use virtual machines to emulate three MUs per PC, while the fourth generates background traffic. In this platform, implemented protection and control applications incorporate three devices that publish functional GOOSE messages with a fixed frame size (table 4).

V. RESULTS AND DISCUSSIONS

Considering the standards requirements as the SLA, i.e. delay, jitter, altering and loss constraints, the traffic profiles are observed during all experiments. Then, the committed and excess information rates are identified when the GOOSE message frames do not use any type of service quality such as policies of frames priority and VLAN tagging techniques. Fig 5 shows the maximum sustained information rate (CIR), which is 50 Mbps, for the Ethernet network to transfer GOOSE frames while meeting the 20% constraint, 0.6 ms of 3 ms transfer time, as performance level guaranteed in these tests. The Ethernet network can exceed the CIR, up to 60 Mbps, but some observations prove that the EIR might not guarantee the required performance level, i.e. transfer time of GOOSE less or equal to 0.6 ms as required by the standards. The figure also shows the red colored area where the performance level cannot be guaranteed. In the experimental setup, VLAN based priority is used to overcome this issue, i.e. to overpass the EIR traffic profile. Table 5 shows KPIs, GOOSE transfer time is less or equal to 0.6 ms with CIR profile. The table also shows the CIR+EIR profiles when the performance is inappropriate. SV jitter is observed when background traffic surpasses 60 Mbps. Although received GOOSE frames are not altered in both CIR and EIR profiles. Overpassing the EIR traffic profile may lead to significant GOOSE delay and loss as well as SV jitter, i.e. leading to non-precise measurements.

VI. CONCLUSION

Communication network faults could degrade reliability of the automation and control systems. This paper presents a method with an experimental technique to benchmark communication services, especially the Ethernet LAN. The method explains how poorly network conditions might slow down data exchange and could cause faults. The method adapted ICT related standard, ITU-Y 1564, with CPS operation technology requirements. The paper shall help system engineers test networked automation systems. Engineers can evaluate different network architectures and settings, e.g. substation automation functions when Ethernet network devices such as switches, IEDs and network interfaces are used. In the future, we shall use redundant strategy to improve communications reliability, i.e. The Parallel Redundancy Protocol (PRP) and the Highavailability Seamless Redundancy (HSR). Hence, simulating faults of primary link disconnections is important to investigate and benchmark alternative links and paths.

Figure 1

 1 Figure 1 Communications in electric automation and control II. BENCHMARKING OF LOCAL COMMUNICATION NETWORKS Local communication networks have entered several levels in the industrial automation and control such as the electrical power stations, i.e. at control rooms, field process and intermediate protection levels. Many electric utilities use Ethernet based networks to deploy protection and control applications at these substations that have evolved significantly with the use of Ethernet in substation automation. In this section, available Ethernet benchmarking techniques from the ICT sector, such as ITU (International Telecommunication Union) and IETF (Internet Engineering Task Force) technical norms are explained. Industrial communities have established specific techniques that can be used for testing and evaluating Ethernet based services.Initially, the request for comments (RFCs) issued by the (IETF) and the internet society have been used to offer benchmarking methodologies that can be used to evaluate Ethernet service performance. Generally, the RFC 1242

5 Figure 5 Figure 4

 554 Figure 5 The network part of the designed automation platform

Table 1

 1 Real-Time communication requirements for automation

	Application	Time requirement
	Time-critical, e.g. trip.	T <= 10 ms
		T<= 3 ms
	Other fast message but not critical	T <= 100 ms
		T <= 20 ms
	Medium speed messages	T <= 100 ms
	Low speed messages, e.g. settings	T <= 500 ms
	Raw data, e.g. synchronized SV	T <= 10 ms
		T<= 3 ms
	File transfer	T >= 1000 ms
	Time synchronization for IEDs and	High precision accuracy (±):
	SV metric, e.g. MUs	1ms, 0.1ms, 25µs, 4µs, 1µs

Table 2

 2 Requirements when digital communications are employed

	Protection	Maximum actual	Security	Dependability
	function	transmission time	Puc	Pmc
		Tac (ms)
	Blocking	10	N/A	< 10 -3
	Blocking	10	< 10 -4	N/A
	Intertripping	10	N/A	< 10 -4
	Intertripping	10	< 10 -8	N/A

Transfer Time Transmission Time 20%= 0.6 ms 40%= 1.2 ms 40%= 1.2 ms

Table 3

 3 main objectives, method and procedures

	Objective	Procedure	Why
	(Measure)	(method)	(aim)
	Delay	Difference between frames timestamp (two IEDs)	Calculate the GOOSE transfer time (delay)
	Jitter	Variances of frames delay	Calculate SV jitter
	Loss rate	Number of published and received frames	Determine (Pmc) missed commands percentage
	Altering rate	Number of altered and correct frames	Determine (Puc) unwanted commands percentage

Table 4

 4 settings of frame size and IP addresses for protection IEDs

	Equipment	IP Address	Frame Size
	Transformer relay	10.10.20.5	1272 bits
	Feeder A relay	10.10.20.6	1288 bits 1248 bits
	Feeder B relay	10.10.20.9	1269 bits

Table 5 Resulted

 5 KPIs for the Ethernet based automation platform

	Link rate = 100Mbps
	CIR	CIR+EIR
	50 Mbps	60 Mbps
	GOOSE and SV loss = 0%	GOOSE and SV loss > 1%
	Transfer delay <= 0.6 ms	Transfer delay > 0.6 ms
	SV jitter = 0%	SV jitter > 4µs is > 1%
	Altered GOOSE = 0%	Altered GOOSE = 0%