Jorge Luis

Fernandez Davila

Dominique Longin
email: dominique.longin@irit.fr

Emiliano Lorini
email: emiliano.lorini@irit.fr

Frédéric Maris
email: frederic.maris@irit.fr

A Simple Framework for Cognitive Planning

We present a novel approach to cognitive planning, i.e., an agent's planning aimed at changing the cognitive attitudes of another agent including her beliefs and intentions. We encode the cognitive planning problem in an epistemic logic with a semantics exploiting belief bases. We study a NP-fragment of the logic whose satisfiability problem is reduced to SAT. We provide complexity results for the cognitive planning problem. Moreover, we illustrate its potential for applications in human-machine interaction in which an artificial agent is expected to interact with a human agent through dialogue and to persuade the human to behave in a certain way.

Introduction

In social sciences, influence is defined as "change in an individual's thoughts, feelings, attitudes, or behaviors that results from interaction with another individual or a group" [START_REF] Rashotte | Social Influence[END_REF]. It is conceived as tightly connected with persuasion. The latter is the intentional form of influence in which an agent (the persuader) tries to make someone (the persuadee) do or believe something by giving her a good reason [START_REF] Cialdini | Influence: science and practice[END_REF][START_REF] Perloff | The Dynamics of Persuasion: Communication and Attitudes in the 21st Century[END_REF].

Models of persuasion in AI are mostly based on argumentation. (See [START_REF] Prakken | Formal Systems for Persuasion Dialogue[END_REF] for a general introduction to the research in this area.) Some of these models are built on Walton & Krabbe's notion of persuasion dialogue in which one party seeks to persuade another party to adopt a belief or point-of-view she does not currently hold [START_REF] Walton | Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning[END_REF]. There exist models based on abstract argumentation [START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF][START_REF] Bonzon | On the Outcomes of Multiparty Persuasion[END_REF][START_REF] Amgoud | Modelling dialogues using argumentation[END_REF] as well probabilistic models where the persuader's uncertainty about what the persuadee knows or believes is represented [START_REF] Hunter | Modelling the Persuadee in Asymmetric Argumentation Dialogues for Persuasion[END_REF]. There exist also models based on possibility theory in which a piece of information is represented as an argument which can be more or less accepted depending on the trustworthiness of the agent who proposes it [START_REF] Da Costa Pereira | Changing One's Mind: Erase or Rewind? Possibilistic Belief Revision with Fuzzy Argumentation Based on Trust[END_REF]. Persuasion has also been formalized with the support of logical tools, e.g., by combining abstract argumentation with dynamic epistemic logic (DEL) [START_REF] Proietti | Persuasive Argumentation and Epistemic Attitudes[END_REF] and epistemic logic with dynamic logic [START_REF] Budzyńska | A Logic for Reasoning about Persuasion[END_REF].

Epistemic planning is a generalization of classical planning that has been increasingly studied in AI in the last years. The goal to be achieved is not necessarily a state of the world but some belief states of one or more agents. This requires a theory of mind by the planning agent [START_REF] Goldman | Simulating Minds: The Philosophy, Psychology, and Neuroscience of Mindreading[END_REF]. A typical goal in epistemic planning is to make a certain agent believe something. Such a belief of the persuadee may be a higher-order belief, i.e., a belief about another agents' beliefs. The initial proposal was to use a standard logic of knowledge or belief together with a representation of actions in terms of event models of DEL [START_REF] Bolander | Epistemic planning for single-and multi-agent systems[END_REF][START_REF] Löwe | DEL planning and some tractable cases[END_REF]. While the DEL framework is very expressive, it turned out that the existence of a solution becomes quickly undecidable even for very simple kinds of event models [START_REF] Aucher | Undecidability in epistemic planning[END_REF][START_REF] Bolander | Complexity Results in Epistemic Planning[END_REF][START_REF] Lê Cong | Small Undecidable Problems in Epistemic Planning[END_REF]. Kominis and Geffner considered epistemic planning problems with very simple event models leading to a decidable fragment [START_REF] Kominis | Beliefs in multiagent planning: from one agent to many[END_REF]. They distinguish three kinds of actions: physical actions modifying the world, public updates (DEL-like public announcements), and sensing actions by means of which an agent learns whether a formula is true. Other researchers investigated another source of complexity, namely that of standard epistemic logic. There, reasoning is strictly more complex than in classical logic: the satisfiability problem is at least in PSPACE [START_REF] Halpern | A guide to completeness and complexity for modal logics of knowledge and belief[END_REF]. Based on earlier work by Levesque, Muise et al. studied epistemic planning in fragments of standard epistemic logic [START_REF] Muise | Planning over multiagent epistemic states: A classical planning approach[END_REF]. They considered state descriptions in terms of conjunctions of epistemic literals: formulas that do not contain any conjunction or disjunction. Cooper et al. considered another fragment: boolean combinations of 'knowing-whether' operators followed by propositional variables [START_REF] Cooper | A simple account of multi-agent epistemic planning[END_REF].

Our approach pushes the envelope of the above approaches to epistemic planning. Our main contribution is its generalization to cognitive planning: it is not only some belief state of a target agent that is to be achieved, but more generally a cognitive state. The latter could involve not only beliefs, but also intentions. Cognitive planning makes clear the distinction between persuasion on beliefs (i.e., inducing someone to believe that a certain fact is true) and persuasion on intentions (i.e., inducing someone to form a certain intention) and elucidates the connection between these two notions. Specifically, since beliefs are the input of decisionmaking and provide reasons for deciding and for acting, the persuader can indirectly change the persuadee's intentions by changing her beliefs, through the execution of a sequence of speech acts. In other words, in cognitive planning, the persuader tries to modify the persuadee's beliefs in order to affect persuadee's intentions. Moreover, cognitive planning takes resource boundedness and limited rationality of the persuadee seriously. For this reason, it is particularly wellsuited for human-machine interaction (HMI) applications in which an artificial agent is expected to interact with a human -who is by definition resource-bounded -through dialogue and to persuade her to behave in a certain way.

To meet the previous requirements of cognitive planning, we formalize it in a recently proposed variant of epistemic logic with operators for explicit and implicit belief [START_REF] Lorini | In Praise of Belief Bases: Doing Epistemic Logic Without Possible Worlds[END_REF][START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF]. The logic allows us to represent, at the same time: (i) the limited reasoning of the human agent (the persuadee), whose explicit beliefs are not necessarily closed under deduction, and (ii) the unbounded inferential capability of the artificial agent (the persuader), which is capable of computing the logical consequences of its explicit beliefs and of finding an optimal persuasion plan.

The paper is organized as follows. We first introduce the language of explicit and implicit belief for the specification of the cognitive planning problem. Given that the satisfiability problem for the full language is PSPACE-hard, we study an interesting NP-fragment of it that can be used in the context of a real HMI application. Then, we present an extension of the language by the notion of belief base expansion which is necessary for representing the actions of the planning agent. The second part of the paper is devoted to cognitive planning. We first formulate the cognitive planning problem and study its complexity. Then, we instantiate it in a concrete example in which an artificial assistant has to help a human user to choose a sport to practice in her leisure time. To achieve its goal, the agent needs to have a model of the user's beliefs and desires. Thanks to this model, the agent will be able to plan a sequence of speech acts aimed at persuading the user that a certain sport is the ideal one for her and, consequently, at inducing the user to form the intention to practice it.

A Language for Explicit and Implicit Belief

This section is devoted to present the language which will serve as a specification language for cognitive planning. We start by recalling the full language and the semantics presented in [START_REF] Lorini | In Praise of Belief Bases: Doing Epistemic Logic Without Possible Worlds[END_REF][START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF]. This language distinguishes explicit belief (a fact in an agent's belief base), from implicit belief (a fact that is deducible from the agent's explicit beliefs). Then, we study its NP-fragment. Finally, we focus on the extension of the latter by belief base expansion operators.

Full Language and Semantics

Assume a countably infinite set of atomic propositions Atm and a finite set of agents Agt = {1, . . . , n}. The language is defined in two steps. First, the language L 0 (Atm, Agt) is defined by the following grammar in BNF:

α ::= p | ¬α | α 1 ∧ α 2 | α 1 ∨ α 2 | i α,
where p ranges over Atm and i ranges over Agt. L 0 (Atm, Agt) is the language for representing agents' explicit beliefs. The formula i α is read "i explicitly believes that α". Then, the language L(Atm, Agt) extends the language L 0 (Atm, Agt) by modal operators of implicit belief and is defined by the following grammar:

ϕ ::= α | ¬ϕ | ϕ 1 ∧ ϕ 2 | ϕ 1 ∨ ϕ 2 | i ϕ | ♦ i ϕ,
where α ranges over L 0 (Atm, Agt) and i ranges over Agt. For notational convenience we write L 0 instead of L 0 (Atm, Agt) and L instead of L(Atm, Agt), when the context is unambiguous. The formula i ϕ is read "i implicitly believes that ϕ" and ♦ i ϕ is read "ϕ is compatible (or consistent) with i's explicit beliefs". The other Boolean constructions , ⊥, → and ↔ are defined in the standard way. We introduce here ∨ and ♦ i as primitive and do not define them from ∧ and i because at a later stage we will need them for translating formulas in negation normal form.

The interpretation of language L exploits the notion of belief base. While the notions of possible state (or world) and epistemic alternative are primitive in the standard Kripke semantics for epistemic logic, they are defined from the primitive concept of belief base in this semantics. In particular, a state is a composite object including a description of both the agents' belief bases and the environment. 1Definition 1 (State) A state is a tuple B = (B 1 , . . . , B n , V ) where: for every i ∈ Agt, B i ⊆ L 0 is agent i's belief base; V ⊆ Atm is the actual environment. The set of all states is noted S.

Note that an agent's belief base B i can be infinite. The sublanguage L 0 (Atm, Agt) is interpreted w.r.t. states, as follows:

Definition 2 (Satisfaction) Let B = (B 1 , . . . , B n , V ) ∈ S. Then: B |= p ⇐⇒ p ∈ V , B |= ¬α ⇐⇒ B |= α, B |= α 1 ∧ α 2 ⇐⇒ B |= α 1 and B |= α 2 , B |= α 1 ∨ α 2 ⇐⇒ B |= α 1 or B |= α 2 , B |= i α ⇐⇒ α ∈ B i .
Observe in particular the set-theoretic interpretation of the explicit belief operator: agent i explicitly believes that α if and only if α is included in her belief base.

A multi-agent belief model (MAB) is defined to be a state supplemented with a set of states, called context. The latter includes all states that are compatible with the common ground [START_REF] Stalnaker | Common ground[END_REF], i.e., the body of information that the agents commonly believe to be the case.

Definition 3 (Multi-Agent Belief Model) A multi-agent belief model (MAB) is a pair (B, Cxt), where B ∈ S and Cxt ⊆ S. The class of all MABs is noted M.

Note that we do not impose that B ∈ Cxt. When Cxt = S then (B, Cxt) is said to be complete, since S is conceivable as the complete (or universal) context which contains all possible states. We compute an agent's set of epistemic alternatives from the agent's belief base, as follows.

Definition 4 (Epistemic alternatives) Let i ∈ Agt. Then R i is the binary relation on the set S such that, for all

B = (B 1 , . . . , B n , V ), B = (B 1 , . . . , B n , V ) ∈ S: BR i B if and only if ∀α ∈ B i : B |= α.
BR i B means that B is an epistemic alternative for agent i at B. So i's set of epistemic alternatives at B includes exactly those states that satisfy all i's explicit beliefs.

Definition 5 extends Definition 2 to the full language L. Its formulas are interpreted with respect to MABs. We omit Boolean cases that are defined in the usual way.

Definition 5 (Satisfaction) Let (B, Cxt) ∈ M. Then:

(B, Cxt) |= α ⇐⇒B |= α, (B, Cxt) |= i ϕ ⇐⇒∀B ∈ Cxt, if BR i B then (B , Cxt) |= ϕ, (B, Cxt) |= ♦ i ϕ ⇐⇒∃B ∈ Cxt such that BR i B and (B , Cxt) |= ϕ. A formula ϕ ∈ L is valid in the class M, noted |= M ϕ, if and only if (B, Cxt) |= ϕ for every (B, Cxt) ∈ M; it is satisfiable in M if and only if ¬ϕ is not valid in M.
Theorem 1 Checking satisfiability of L(Atm, Agt) formulas in the class M is a PSPACE-hard problem.

This theorem is a consequence of the fact that our logic contains the basic modal logic K whose satisfiability problem is PSPACE-complete [START_REF] Halpern | A guide to completeness and complexity for modal logics of knowledge and belief[END_REF].

NP-Complete Fragment

In this section, we study the following fragment of the language L, called L Frag :

ϕ ::= α | ¬ϕ | ϕ 1 ∧ ϕ 2 | ϕ 1 ∨ ϕ 2 | m α | ♦ m α,
where α ranges over L 0 and m is a special agent in Agt called the 'machine'. In L Frag , all agents have explicit beliefs but only agent m has implicit beliefs, and moreover the latter are restricted to L 0 formulas of type α. So there are no nested implicit beliefs for agent m. Agent m is assumed to be the unique artificial agent in the system which is endowed with unbounded reasoning and planning capabilities. The cognitive planning problem will be modeled from agent m's perspective.

In the rest of this section, we are going to provide a polysize reduction of the satisfiability problem of L Frag to SAT. The reduction consists of three steps which are summarized in Figure 1. As a first step, we put L Frag formulas in negation

L Frag nnf L NNF Frag tr 1 L Mod tr 2 L Prop
nnf (p) = p, nnf ( i α) = i α, nnf ( m α) = m nnf (α), nnf (♦ m α) = ♦ m nnf (α), nnf (ϕ ∧ ψ) = nnf (ϕ) ∧ nnf (ψ), nnf (ϕ ∨ ψ) = nnf (ϕ) ∨ nnf (ψ), nnf (¬p) = ¬p, nnf (¬ i α) = ¬ i α, nnf (¬¬ϕ) = nnf (ϕ), nnf ¬(ϕ ∧ ψ) = nnf (¬ϕ ∨ ¬ψ), nnf ¬(ϕ ∨ ψ) = nnf (¬ϕ ∧ ¬ψ), nnf (¬ m α) = ♦ m nnf (¬α), nnf (¬♦ m α) = m nnf (¬α).
Let us define the NNF variant L NNF 0 of the language L 0 by the following grammar:

β ::= p | ¬p | i α | ¬ i α | β 1 ∧ β 2 | β 1 ∨ β 2 ,
where p ranges over Atm, i ranges over Agt and α ranges over L 0 . Furthermore, let us define the language L NNF Frag by the following grammar. For β ranging over L NNF 0

: ϕ ::= β | ϕ 1 ∧ ϕ 2 | ϕ 1 ∨ ϕ 2 | m β | ♦ m β.
Proposition 1 Let ϕ ∈ L Frag . Then, ϕ ↔ nnf (ϕ) is valid in the class M, and nnf (ϕ) ∈ L NNF Frag . Note that the size of nnf (ϕ) is polynomial in the size of ϕ.

As a second step, we define the following modal language L Mod into which the language L NNF Frag will be translated:

ω ::= q | ¬ω | ω 1 ∧ ω 2 | ω 1 ∨ ω 2 , ϕ ::= q | ¬ϕ | ϕ 1 ∧ ϕ 2 | ϕ 1 ∨ ϕ 2 | ω | ω ,
where q ranges over the following set of atomic formulas:

Atm + = Atm ∪ {p iα : i ∈ Agt and α ∈ L 0 (Atm, Agt)}.
So p iα is nothing but a special propositional variable.

We interpret the language L Mod w.r.t. a pair (M, w), called pointed Kripke model, where M = (W, ⇒, π), W is a non-empty set of worlds, ⇒ ⊆ W ×W and π : Atm + -→ 2 W . (Boolean cases are again omitted as they are defined in the usual way.) Definition 6 The semantic interpretation for formulas in L Mod w.r.t. a pointed Kripke model (M, w) is as follows:

(M, w) |= q ⇐⇒w ∈ π(q); (M, w) |= ω ⇐⇒∀v ∈ W, if w ⇒ v then (M, v) |= ω; (M, w) |= ω ⇐⇒∃v ∈ W s.t. w ⇒ v and (M, v) |= ω.
The class of pointed Kripke models is noted K. Satisfiability and validity of formulas in L Mod relative to the class K is defined in the usual way.

Let tr 1 : L NNF Frag -→ L Mod be a translation such that:

tr 1 (p) = p, tr 1 (¬p) = ¬p, tr 1 (ϕ 1 ∧ ϕ 2 ) = tr 1 (ϕ 1 ) ∧ tr 1 (ϕ 2 ), tr 1 (ϕ 1 ∨ ϕ 2 ) = tr 1 (ϕ 1 ) ∨ tr 1 (ϕ 2 ), tr 1 ( i α) = p m α ∧ tr 0 (α), if i = m, p iα , otherwise, tr 1 (¬ i α) = ¬p iα , tr 1 ( m β) = tr 0 (β), tr 1 (♦ m β) = tr 0 (β);
with tr 0 : L 0 -→ L Mod such that:

tr 0 (p) = p, tr 0 (¬α) = ¬tr 0 (α), tr 0 (α 1 ∧ α 2 ) = tr 0 (α 1 ) ∧ tr 0 (α 2 ), tr 0 (α 1 ∨ α 2 ) = tr 0 (α 1 ) ∨ tr 0 (α 2 ), tr 0 ( i α) = p iα .
As the following theorem indicates, the polynomial translation tr 1 guarantees the transfer of satisfiability from model class M to model class K.

Theorem 2 Let ϕ ∈ L NNF Frag . Then, ϕ is satisfiable in the class M if and only if tr 1 (ϕ) is satisfiable in the class K.

SKETCH OF PROOF. The proof relies on the fact that the belief base semantics for the language L Frag is equivalent to a "weaker" semantics exploiting pointed structures of the form (X, s) where X = (S, B, (⇒ i ) i∈Agt , τ ), S is a nonempty set of states, s ∈ S is the actual state, B : Agt ×S -→ 2 L0 is a belief base function, τ : Atm -→ 2 S is valuation function, ⇒ i ⊆ S × S is agent i's epistemic accessibility relation and with respect to which L-formulas are interpreted as follows (boolean cases are omitted for simplicity):

(i) (X, s) |= p iff s ∈ τ (p), (ii) (X, s) |= i α iff α ∈ B(i, s), (iii) (X, s) |= i ϕ iff ∀s ∈ S, if s ⇒ i s then (X, s ) |= ϕ.
In particular, for every ϕ ∈ L Frag , we have that

ϕ is satisfi- able in M iff ϕ is satisfiable in the subclass of pointed struc- tures (X, s) such that ⇒ m (s) ⊆ α∈B(m,s) ||α|| (X,s) with ||α|| (X,s) = {s ∈ X : (X, s ) |= α}.
As a last step, we provide a polysize reduction of L Modsatisfiability to SAT, where the underlying propositional logic language L Prop is built from the following set of atomic propositions:

Atm ++ ={q x : q ∈ Atm + and x ∈ N}∪ {r x,y : x, y ∈ N}.

The set Atm ++ includes two types of atomic propositions: one of the form q x denoting the fact that q is true at world x and the other of the form r x,y denoting the fact that world x is related to world y.

Let tr 2 : L Mod × N × N -→ L Prop be the following translation function:

tr 2 (q, x, y) = q x , tr 2 (¬ϕ, x, y) = ¬tr 2 (ϕ, x, y), tr 2 (ϕ 1 ∧ ϕ 2 , x, y) = tr 2 (ϕ 1 , x, y) ∧ tr 2 (ϕ 2 , x, y), tr 2 (ϕ 1 ∨ ϕ 2 , x, y) = tr 2 (ϕ 1 , x, y) ∨ tr 2 (ϕ 2 , x, y), tr 2 ( ω, x, y) = 0≤z≤y r x,z → tr 2 (ω, z, y) , tr 2 ( ω, x, y) = 0≤z≤y r x,z ∧ tr 2 (ω, z, y) .
Translation tr 2 is similar to the translation of modal logic S5 into propositional logic given in [START_REF] Caridroit | A SAT-Based Approach for Solving the Modal Logic S5-Satisfiability Problem[END_REF]) and, more generally, to the standard translation of modal logic into FOL in which accessibility relations are encoded by special predicates. The size of an L Mod formula, size(ϕ), is defined by:

size(p) = 1, size(ϕ 1 ∧ ϕ 2 ) = size(ϕ 1 ) + size(ϕ 2 ) + 1, size(ϕ 1 ∨ ϕ 2 ) = size(ϕ 1 ) + size(ϕ 2 ) + 1, size(¬ϕ) = size(ϕ) + 1, size( ω) = size( ω) = size(ω) + 1.
Note that the size of tr 2 ϕ, 0, size(ϕ) is polynomial in the size of ϕ.

Theorem 3 Let ϕ ∈ L Mod . Then, ϕ is satisfiable in the class K if and only if tr 2 ϕ, 0, size(ϕ) is satisfiable in propositional logic.

SKETCH OF PROOF. The theorem is proved in the same way as the standard translation of modal logic to FOL plus a straightforward adaptation of (Ladner 1977, Lemma 6.1) about polysize-model property for S5 to our case.

The size of tr 2 ϕ, 0, size(ϕ) being polynomial in the size of ϕ, thanks to Proposition 1, Theorem 2 and Theorem 3 we state the following complexity result.

Theorem 4 Checking satisfiability of formulas in L Frag in the class M is an NP-complete problem.

Dynamic Extension

In this section, we extend the language L Frag by belief expansion operations. Such an extension will allow us to represent the actions of the planning agent in the cognitive planning problem. Specifically, we introduce the following language L + Frag :

ϕ ::= α | ¬ϕ | ϕ 1 ∧ ϕ 2 | ϕ 1 ∨ ϕ 2 | m α | ♦ m α | [+ i α]ϕ,
where α ranges over L 0 and i ranges over Agt. The formula [+ i α]ϕ is read "ϕ holds after agent i has privately expanded her belief base with α". Events of type + i α are generically called informative actions.

Our extension has the following semantics relative to a MAB:

Definition 7 (Satisfaction relation, cont.) Let B = (B 1 , . . . , B n , V ) ∈ S and let (B, Cxt) ∈ M. Then:

(B, Cxt) |= [+ i α]ϕ ⇐⇒ (B +iα , Cxt) |= ϕ with V +iα = V , B +iα i = B i ∪ {α} and B +iα j = B j for all j = i.
Intuitively speaking, the private expansion of i's belief base by α simply consists in agent i adding the information that α to her belief base, while all other agents keep their belief bases unchanged.

The following equivalences are valid in the class ∈ M:

[+ i α]α ↔ , if α = i α, α , otherwise; [+ i α]¬ϕ ↔ ¬[+ i α]ϕ; [+ i α](ϕ 1 ∧ ϕ 2 ) ↔ [+ i α]ϕ 1 ∧ [+ i α]ϕ 2 ; [+ i α](ϕ 1 ∨ ϕ 2 ) ↔ [+ i α]ϕ 1 ∨ [+ i α]ϕ 2 ; [+ i α] m α ↔ m (α → α ), if i = m, m α , otherwise; [+ i α]♦ m α ↔ ♦ m (α ∧ α ), if i = m, ♦ m α , otherwise.
Thanks to these equivalences we can define the following reduction red transforming every L + Frag formula ϕ into an equivalent L Frag formula red (ϕ):

red (p) = p, red ( i α) = i α, red (¬ϕ) = ¬red (ϕ), red (ϕ 1 ∧ ϕ 2 ) = red (ϕ 1 ) ∧ red (ϕ 2 ), red (ϕ 1 ∨ ϕ 2 ) = red (ϕ 1 ) ∨ red (ϕ 2 ), red ( m ϕ) = m red (ϕ), red (♦ m ϕ) = ♦ m red (ϕ), red ([+ i α]α ) = , if α = i α, red (α ), otherwise; red ([+ i α]¬ϕ) = red (¬[+ i α]ϕ), red [+ i α](ϕ 1 ∧ ϕ 2 ) = red ([+ i α]ϕ 1 ∧ [+ i α]ϕ 2 ), red [+ i α](ϕ 1 ∨ ϕ 2 ) = red ([+ i α]ϕ 1 ∨ [+ i α]ϕ 2 ), red ([+ i α] m α ) = red m (α → α ) , if i = m, red ( m α ) otherwise; red ([+ i α]♦ m α ) = red ♦ m (α ∧ α ) if i = m, red (♦ m α ) otherwise; red ([+ i α 1 ][+ j α 2 ]ϕ) = red [+ i α 1 ]red ([+ j α 2 ]ϕ) .
Proposition 2 Let ϕ ∈ L + Frag . Then, ϕ ↔ red (ϕ) is valid in the class M, and red (ϕ) ∈ L Frag .

The following theorem is a consequence of Theorem 4, Proposition 2 and the fact that the size of red (ϕ) is polynomial in the size of ϕ.

Theorem 5 Checking satisfiability of formulas in L + Frag in the class M is an NP-complete problem.

Before concluding this section, we define the concept of logical consequence for the language L + Frag which will be used in the formulation of the cognitive planning problem at a later stage. Let Σ be a finite subset of L 0 and let ϕ ∈ L + Frag . We say that ϕ is a logical consequence of Σ in the class M, noted Σ |= M ϕ, if and only if, for every (B, Cxt) ∈ M such that Cxt ⊆ S(Σ) we have (B, Cxt) |= ϕ, with S(Σ) = {B ∈ S : ∀α ∈ Σ, B |= α}. We say that ϕ is Σ-satisfiable in the class M if and only if, ¬ϕ is not a logical consequence of Σ in M. Clearly, ϕ is valid if and only if ϕ is a logical consequence of ∅, and ϕ is satisfiable if and only if ϕ is ∅satisfiable.

As the following deduction theorem indicates, the logical consequence problem with a finite set of premises can be reduced to the satisfiability problem.

Theorem 6 Let ϕ ∈ L + Frag and let Σ ⊂ L 0 be finite. Then,

Σ |= M ϕ if and only if |= M α∈Σ m α → ϕ.

Cognitive Planning

This section is devoted to the formal specification of cognitive planning by way of the language L + Frag and to its illustration with an example of HMI.

Problem

The planning problem in the context of the logic L + Frag consists in finding a sequence of informative actions for agent m of type + m α which guarantees that it believes that its goal α G is satisfied. As we emphasized above, agent m is assumed to be an artificial agent which interacts with other resource-bounded human agents.

Let Act m = {+ m α : α ∈ L 0 } be agent m's set of informative actions and let elements of Act m be noted , , . . . Agent m's informative actions have executability preconditions that are specified by the following function: P : Act m -→ L Frag . So, we can define the following operator of successful occurrence of an informative action:

ϕ def = P( ) ∧ [ ]ϕ
with ∈ Act m . The formula ϕ has to be read "agent m's informative action can take place and ϕ holds after its occurrence".

Informative actions of type 'speech act' are of interest here. In particular, we consider speech acts of type 'to inform', where m is assumed to be the speaker and j ∈ Agt such that j = m is assumed to be the hearer. We identify the speech act "agent m informs agent j that α" with the perlocutionary effect (Searle 1969, Sect. 6.2) of the speaker learning that the hearer has learnt that α:2 inform(m,j,α)

def = + m j α.
The previous abbreviation and, more generally, the idea of describing actions of a plan with agent m's private belief expansion operations is justified by the fact that we model cognitive planning from the perspective of the planning agent m. Therefore, we only need to represent the effects of actions on agent m's beliefs.

Definition 8 (L +

Frag -planning problem) A L + Frag -planning problem is a tuple Σ, Op, α G where:

• Σ ⊂ L 0 is a finite set of agent m's available information, • Op ⊂ Act m is a finite set of agent m's operators, • α G ∈ L 0 is agent m's goal. A solution plan to a L + Frag -planning problem Σ, Op, α G is a sequence of operators 1 , . . . , k from Op for some k such that Σ |= M 1 . . . k m α G .
Theorem 7 Checking plan existence for a L + Frag -planning problem is in NP NP = Σ P 2 .

SKETCH OF PROOF. A L +

Frag -planning problem Σ, Op, α G has a solution plan if and only if it has a poly-size solution plan 1 , . . . , k with k ≤ |Op|. Indeed, it is easily seen that if an operator has been executed in a plan, another future occurrence of the same operator will not change the planning state due to the monotonicity of private belief expansion:

... B +iα i +iα1... +iα h +iα = ... B +iα i +iα1... +iα h .
Consider a poly-time non-deterministic Turing machine with an NP-oracle (Σ P 2 -Turing machine). It begins with an empty plan and branches over all poly-size plans of length k ≤ |Op| choosing non deterministically operators to add to the plan. It accepts if Σ |= M 1 . . . k m α G i.e., using Theorem 6, if ¬ α∈Σ m α → 1 . . . k m α G is unsatisfiable in the class M. Thanks to Theorem 5, unsatisfiability of this L + Frag formula can be checked by the NPoracle. When k = |Op| and the formula is satisfiable, the Turing machine rejects.

Example: Artificial Assistant

We consider a HMI scenario in which agent m is the artificial assistant of the human agent h. Agent h has to choose a sport to practice since her doctor recommended her to do a regular physical activity to be in good health. Agent m's aim is to help agent h to make the right choice, given her actual beliefs and desires. The finite set of sport activities from which h can choose is noted Opt. Elements of Opt are noted o, o , . . . Each option in Opt is identified with a finite set of variables Var . Each variable x in Var takes a value from its corresponding finite set of values Val x .

In this example, we suppose that Opt is composed of the following eight elements: swimming (sw), running (ru), horse riding (hr), tennis (te), soccer (so), yoga (yo), diving (di) and squash (sq). Moreover, there are exactly six variables in Var which are used to classify the available options: environment (env), location (loc), sociality (soc), cost (cost), dangerousness (dan) and intensity (intens). The set of values for the variables are:

Val env = {land , water }, Val loc = {indoor , outdoor , mixed }, Val soc = {single, team, mixed }, Val cost = {low , med , high}, Val dan = {low , med , high}, Val intens = {low , med , high}. The set of assignments for variable x is defined as follows:

Assign x = {x → v : v ∈ Val x }. The set of variable assignments is Assign = x∈Var Assign x .
Elements of Assign are noted a, a , . . .

We assume that the content of an atomic desire is a variable assignment or its negation. That is, agent h's atomic desire can be any element from the following set:

Des 0 = Assign ∪ {∼ a : a ∈ Assign}. Elements of Des 0 are noted d, d , . . .. For example, the fact that h has loc → indoor as a desire means that h would like to practice an indoor activity, while if h's desire is ∼ cost → high, then h would like to practice an activity whose cost is not high. Agent h's desires are either atomic desires or conditional desires. That is, h's desire can be any element from the following set:

Des = Des 0 ∪ [d 1 , . . . , d k ] d : d 1 , . . . , d k , d ∈ Des 0 .
Elements of Des are noted γ, γ , . . . For example, if agent h has [cost → high] dan → low as a desire, then she would like to practice a sport whose dangerousness level is low, if its cost is high. We define 2 Des * = 2 Des \ ∅.

Let us assume that the set Atm includes four types of atomic formulas, for every x → v ∈ Assign, o, o ∈ Opt and Γ ∈ 2 Des * : (i) val(o, x → v) standing for "option o has value v for variable x", (ii) ideal(h, o) standing for "o is an ideal option for agent h", (iii) justif(h, o) standing for "agent h has a justification for choosing option o", and (iv) des(h, Γ) standing for "Γ is agent h's set of desires".

The following function f comp specifies, for every option o ∈ Opt and possible desire γ ∈ Des, the condition guaranteeing that o satisfies (or, complies with) γ:

f comp (o, a) = val(o, a), f comp (o, ∼ a) = ¬val(o, a), f comp o, [d 1 , . . . , d k ] d = ¬f comp (o, d 1 ) ∨ . . . ∨ ¬f comp (o, d k ) ∨ f comp (o, d).
The following function f h comp specifies, for every option o ∈ Opt and possible desire γ ∈ Des, the condition guaranteeing that agent h believes that o satisfies γ:

f h comp (o, a) = h f comp (o, a), f h comp (o, ∼ a) = h f comp (o, ∼ a), f h comp o, [d 1 , . . . , d k ] d = h ¬f comp (o, d 1 ) ∨ . . . ∨
The previous formulation of f h comp o, [d 1 , . . . , d k ] d presupposes an understanding of conditional (goal) sentences by agent h. In particular, agent m does not need to provide information to agent h about the antecedent of the conditional, if the consequent is true.

We assume that the artificial agent m has the following pieces of information in its belief base:

α 1 def = o∈Opt x∈Var v,v ∈Valx:v =v val(o, x → v) → ¬val(o, x → v ) , α 2 def = o∈Opt x∈Var v,v ∈Valx:v =v h val(o, x → v) → h ¬val(o, x → v ) , α 3 def = Γ,Γ ∈2 Des * :Γ =Γ des(h, Γ) → ¬des(h, Γ ) , α 4 def = Γ∈2 Des * des(h, Γ), α 5 def = o∈Opt ideal(h, o) ↔ Γ∈2 Des * des(h, Γ)∧ γ∈Γ f comp (o, γ) , α 6 def = o∈Opt justif(h, o) ↔ Γ∈2 Des * des(h, Γ)∧ γ∈Γ f h comp (o, γ)
.

Formula α 1 captures the fact that a sport cannot have two different values for a given variable. Formula α 2 is its subjective version for agent h. Formulas α 3 and α 4 capture together the fact that agent h has exactly one non-empty set of desires. According to formula α 5 , an option o is ideal for agent h if and only if it satisfies all agent h's desires. Finally, according to formula α 6 , agent h has a reasonable justification for choosing option o if and only if she has all necessary information to conclude that option o satisfies all her desires. We also assume that agent m has in its belief base a complete representation of Table 1, which specifies the variable assignments for all options: In order to help agent h to select an activity, agent m also needs information about h's set of actual desires. The latter is captured by the following formula:

α o,x 7 def = val(o, x → v o,x
α 8 def = des(h, Γ h ), with Γ h ={env → land , intens → med , ∼ loc → indoor , [cost → high] soc → mixed }.
This means that, according to agent m, agent h would like to practice a land activity, with medium intensity, which is not exclusively indoor, and which can be practiced both in single and team mode, if its cost is high.

Let us now turn to the cognitive planning problem. We suppose agent m's set of operators Op is:

Op = inform m,h,val(o, a) : o ∈ Opt and a ∈ Assign ∪ inform m,h,ideal(h, o) : o ∈ Opt .
In other words, agent m can only inform agent h about an option's value for a certain variable or about the ideality of an option for her. We suppose the following executability precondition for every o ∈ Opt and a ∈ Assign:

P inform m,h,val(o, a) = m val(o, a) ∧ v∈Val dan val(o, dan → v) → h val(o, dan → v) if a ∈ Assign dan , P inform m,h,val(o, a) = mval(o, a) if a ∈ Assign dan , P inform m,h,ideal(h, o) = m ideal(h, o) ∧ justif(h, o) .
According to the first definition, agent m can inform agent h about an option's value for a certain variable, if and only if this information is believed by m and m believes that h has been already informed about the dangerousness level of the option. Indeed, we assume that, before being presented with an option's features, agent h must be informed about its the dangerousness level and agent m complies with this rule. The second definition simply stipulates that m can inform h about the dangerousness level of an option if and only if it believes what it says. Finally, according to the third definition, m can inform h about the ideality of an option only if it believes that h has a reasonable justification for choosing it. Indeed, we assume m will inform h about the ideality of an option only after having explained why the option is ideal for her. The three definitions presuppose that agent m cannot spread fake news (i.e., something that it does not implicitly believe).

We moreover suppose that, for agent h to have a potential intention to choose option o, denoted by potIntend(h, o), she must have a justified belief that o is an ideal option for her:

3 potIntend(h, o) def = h ideal(h, o) ∧ justif(h, o).
This abbreviation together with the abbreviation α 6 given above relate intention with belief and desire, in line with existing theories of intention [START_REF] Audi | Intending[END_REF][START_REF] Davidson | Essays on Actions and Events[END_REF].

It turns out that the sequence of speech acts 1 , 2 , 3 , 4 , 5 , 6 with

1 def = inform m,h,val(te, dan → med ) , 2 def = inform m,h,val(te, env → land ) , 3 def = inform m,h,val(te, intens → med ) , 4 def = inform m,h,val(te, loc → mixed ) , 5 def = inform m,h,val(te, soc → mixed ) , 6 def = inform m,h,ideal(h, te) .
provides a solution for the planning problem Σ, Op, α G , where Σ ={α 1 , α 2 , α 3 , α 4 , α 5 , α 6 , α 8 }∪ {α o,x 7 : o ∈ Opt and x ∈ Var }, Op has the previous specifications and agent h's persuasive goal α G is defined as follows:

α G def = o∈Opt potIntend(h, o).
This means that, by performing the sequence of operators 1 , 2 , 3 , 4 , 5 , 6 , agent m will induce agent h to form a potential intention to choose an activity. In other words, agent m will provide an effective recommendation to agent h.

We conclude this section with a general observation about the formulation of the planning problem for our example. Let Σ, Op, α G be the planning problem we want to solve. Let m's set of operators Op(o) for option o ∈ Opt relative to Σ, Op, α G be defined as follows: It is easy to verify that the planning problem Σ, Op, α G has a solution if and only if there exists o ∈ Opt such that the planning problem Σ, Op Σ,Op,α G o , α G has a solution. Therefore, in order to solve the planning problem Σ, Op, α G , we simply need to linearly order the options in Opt and solve the planning problems Σ, Op Σ,Op,α G o , α G in sequence one after the other according to the ordering.

Conclusion

We have presented a simple logic-based framework for cognitive planning which can be used to endow an artificial agent with the capability of influencing a human agent's beliefs and intentions. We have studied both complexity of satisfiability for the logic and complexity of the cognitive planning problem. Our approach relies on SAT, given the NPcompleteness of the satisfiability problem for the epistemic language we consider. We believe that this aspect opens up interesting perspectives of application in HCI based on the exploitation of existing SAT techniques. We are currently implementing a cognitive planning algorithm using a SATsolver as well as the HMI scenario we presented in the paper.

Directions of future work are manifold. The dynamic extension L + Frag considers belief expansion only. We plan to extend our approach by belief revision operators. This will allow the artificial planning agent to revise its belief base, in the light of new information received from the human agent that contradicts its explicit beliefs. We also plan to enrich our analysis of speech acts by taking 'yes-no' questions into consideration. To this aim, we plan to extend the language L +

Frag by program constructions of propositional dynamic logic (PDL) including non-deterministic choice (∪). Last but not least, we plan to combine our cognitive planning approach with inductive logic programming (ILP) (Muggleton and de Raedt 1994), in order to construct an agent's prior information Σ, as used in the formulation of the cognitive planning problem, through inductive methods. This will allow the persuader to predict the persuadee's beliefs, like in models of theory of mind based on neural networks [START_REF] Rabinowitz | Machine Theory of Mind[END_REF]).

Figure 1 :

 1 Figure 1: Summary of reduction process

  Op Σ,Op,α G o = inform m,h,val(o, a) : val(o, a) ∈ Σ ∪ inform m,h,ideal(h, o) .

Table 1 :

 1 ). Variable assignments. For every option o ∈ Opt and variable x ∈ Var , we denote by v o,x the corresponding entry in the table. For instance, we have v sw,env = water .

		env	loc	soc	cost dan intens
	sw water	mixed	single med low	high
	ru land outdoor single low med	high
	hr land outdoor single high high	low
	te	land	mixed	mixed high med	med
	so	land	mixed	team	med med	med
	yo land	mixed	single med low	low
	di water	mixed	single high high	low
	sq	land	indoor mixed high med	med

This is similar to the way states are modeled in the interpreted system semantics for multi-agent systems[START_REF] Fagin | Reasoning about Knowledge[END_REF][START_REF] Lomuscio | MCMAS: an open-source model checker for the verification of multiagent systems[END_REF]

Raimondi 2017).

We implicitly assume that, by default, m believes that j trusts its judgment, so that j will believe what it says.

h ¬f comp (o, d k ) ∨ h f comp (o, d).

Our account of potential intention is reminiscent of the JTB ('justified true belief') account to knowledge[START_REF] Goldman | What is Justified Belief?[END_REF]).

Acknowledgments

This work is supported by the ANR project CoPains ("Cognitive Planning in Persuasive Multimodal Communication"). Support from the ANR-3IA Artificial and Natural Intelligence Toulouse Institute is also acknowledged.


Fabrice Piazza
email: fpiazza@pucmm.edu.do

K Cruz

Marc Monthioux

Pascal Puech

Iann Gerber

Raman evidence for the successful synthesis of diamane

Diamane was prepared from the exposure of bi-layer graphene to H radicals produced by the hot-filament process. A sharp sp 3 -bonded carbon stretching mode was observed by UV Raman spectroscopy while no sp 2 -bonded carbon peak was simultaneously detected. This is the first time that Raman spectra of genuine diamane are reported, which, meanwhile, are the very first evidence for the successful synthesis of genuine diamane. First principle calculations support possible full hydrogenation and confirm the hydrogenated AB configuration to be the most stable one. We believe those results constitute a milestone in the path towards the synthesis of high-quality diamane and open the door to large-scale production. Genuine diamane consists of two crystalline sp 3 -bonded carbon layers for which half of the carbon atoms are hydrogenated while the other half bond the two layers to each other [1,2]. The material stability was first predicted by Chernozatonskii et al in 2009 [1]. The increasing interest

. This breakthrough was achieved from the chemisorption of H radicals produced by the hot-filament process at low temperature and pressure on the basal planes of few-layer graphene (FLG) and the subsequent interlayer bonding between sp 3 -bonded carbon atoms (sp 3 -C). sp 3 -C stretching peak was detected in visible and UV

2

Raman spectra. However, these reports cannot claim having evidenced genuine diamane, because of the poor suitably of the starting graphene material with too many layers, thereby inducing the formation of diamanoïd/graphene hybrids instead, and because of the lack of lateral resolution of Raman spectroscopy [START_REF] Piazza | Towards a better understanding of the structure of diamanoïds and diamanoïd/graphene hybrids[END_REF]. Following that work, the synthesis of F-diamane, where carbon atoms are fluorinated rather than hydrogenated as in genuine diamane, was claimed from the fluorination, at low pressure, of AB-stacked bi-layer graphene [START_REF] Bakharev | Chemically induced transformation of chemical vapour deposition grown bilayer graphene into fluorinated single-layer diamond[END_REF], the latter having been shown to be the most favorable configuration for the sp 2 -C to sp 3 -C conversion [START_REF] Chernozatonskii | Diamane: Simulation of the Structure and Properties[END_REF][START_REF] Piazza | Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films[END_REF]. However, notably, no sp 3 -C related peak was observed in Raman spectra [START_REF] Bakharev | Chemically induced transformation of chemical vapour deposition grown bilayer graphene into fluorinated single-layer diamond[END_REF]. Also, the method based on the intensity of the electron diffraction spots to discriminate between the graphite and the diamond structures is known to be unreliable because of the "interference error" effect which can occur in the case of ultrathin films that can be easily distorted [START_REF] Piazza | Towards a better understanding of the structure of diamanoïds and diamanoïd/graphene hybrids[END_REF][START_REF] Williams | Diffraction from Small Volumes[END_REF]. Hence, to the best of authors' knowledge, evidences of diamane synthesis are missing.

Raman spectroscopy is a critical technique to characterize carbon materials, including hydrogenated graphene, graphane [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF] and diamanoïds [START_REF] Piazza | Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films[END_REF][START_REF] Piazza | Towards a better understanding of the structure of diamanoïds and diamanoïd/graphene hybrids[END_REF]. However, in the case of those nanomaterials, since for visible energies, the Raman scattering of sp 2 -C is a resonant process between 50 and 230 times more effective than that of sp 3 -C [START_REF] Wagner | Resonance effects in Raman scattering from polycrystalline diamond films[END_REF], it is necessary to employ UV excitation to obtain a more evenly weighted probe of sp 3 -C and sp 2 -C sites. This critical point has not been considered in most reports on hydrogenated graphene and 2D sp 3 -C nanomaterials [START_REF] Piazza | Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films[END_REF]. Therefore, in this work, we studied the hydrogenation of bi-layer graphene by the chemisorption of H generated from the dissociation of H2 in a hot filament reactor at low temperature and low pressure, and the subsequent structure conversion into highly hydrogenated genuine diamane. UV Raman spectroscopy was used to track the structure conversion, which is unprecedented in the literature in the field but in our own work [START_REF] Piazza | Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films[END_REF][START_REF] Piazza | Towards a better understanding of the structure of diamanoïds and diamanoïd/graphene hybrids[END_REF]. First principle calculations and electronic diffraction are used to explain our findings. In addition, ultra-low voltage (5 kV) transmission electron microscopy (TEM) analysis was carried-out, mostly in electron diffraction mode.

As-received suspended bi-layer (2LG) graphene films grown by chemical vapor deposition and deposited on 3 mm diameter gold Quantifoil TEM grids were obtained from Graphenea and used as starting graphene materials. Each film is polycrystalline, with a maximum grain size of 20 m. Coverage of the grid is ~98%. 2LG films were obtained from the successive transfer of two individual single-layer graphene (1LG), hence the resulting 2LG stacking is random. However, considering the dimension of the 1LG layers (~3 mm in diameter) and their polycrystalline nature (20-µm-large grains for the most), chances for locally getting the favorable AB or AA stacking configurations [START_REF] Chernozatonskii | Diamane: Simulation of the Structure and Properties[END_REF][START_REF] Piazza | Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films[END_REF] at micrometer scale are high and can be actually observed as reported in Fig. 1b. A commercial hot filament reactor (BWI 1000 model from Blue Wave Semiconductors) was used for the hydrogenation process [START_REF] Piazza | Carbon nanotubes coated with diamond nanocrystals and silicon carbide by hot-filament chemical vapor deposition below 200 C substrate temperature[END_REF]. The specific experimental details related to the synthesis of diamanoïds were previously disclosed [START_REF] Piazza | Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films[END_REF]. 2LG grids were placed vertically on the substrate holder. The samples studied here were prepared at a pressure of 50 Torr and using a H2 flux of 1 sccm. The duration was of 6 minutes and 30 seconds. Filament temperature was within the range of 2350-2590 C while the maximum 2LG temperature was around 400 C.

The Raman scattering measurements were performed in the backscattering configuration using 244 nm line and analyzed using a 40 objective micro-Raman spectrometer system (inVia Reflex, Renishaw). The resolution was of 3.5 cm -1 . Typically, laser power was in the range of 1 mW with 1s exposure time. No visible damage and no change of the spectral shape during measurements have been observed. Highly-oriented pyrolytic graphite (HOPG) was used for peak position calibration. Raman mapping was employed using high-speed encoded mapping stage to generate high definition 2D chemical images over typically around 140,000 m 2 .

Step size was set at 3 m. TEM (mostly electron diffraction) was employed to further examine the material structure before and after hydrogenation. The converted material, as opposed to the behavior of graphene or diamond, appeared to be electron sensitive, possibly because of the high hydrogenation degree of the surface, even for electron energy as low as 80 keV [START_REF] Piazza | Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films[END_REF]. This is consistent with the photon sensitivity observed under certain conditions [START_REF] Piazza | Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films[END_REF] and the reported electron sensitivity of F-diamane [START_REF] Bakharev | Chemically induced transformation of chemical vapour deposition grown bilayer graphene into fluorinated single-layer diamond[END_REF]. This is the reason why an ultra-low-voltage (5 kV) benchtop transmission electron microscope from America Delong was used. The instrument includes a Schottky-type field emission gun and a 25602160 pixel Front Illuminated Scientific CMOS (6.5 m 2 pixel size). Electron diffraction patterns were obtained in selected area mode from 100 nm-large areas.

DFT calculations were performed with the VASP package [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF] with the same conditions than in our previous work [START_REF] Piazza | Towards a better understanding of the structure of diamanoïds and diamanoïd/graphene hybrids[END_REF]: Perdew-Burke-Ernzerhof (PBE) functional [START_REF] Perdew | Generalized gradient approximation made simple[END_REF], cut-off energy of 400 eV, force convergence below 0.005 eV/Å, vacuum height of 20 Å and supercell of (7×7) with a 7×7×1 grid for k-points sampling. VESTA software was used to generate optimized structure pictures [START_REF] Momma | VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data[END_REF].

Selected area electron diffraction of the starting 2LG films (Fig. 1b-d) exhibited various features depending on the location of the area analyzed, i.e., the evidence for (i) single graphenic domains (supposedly 2LG with AB or AA stacking) (Fig. 1b); (ii) two randomly stacked superimposed graphenic domains (supposedly 1LG each) (Fig. 1c) (iii) four randomly stacked graphenic domains (Fig. 1d), showing than the material is not 2LG only all over the grid. Micro-Raman mapping was performed before and after exposure to the hot-filament-promoted hydrogenation process, to track any structure conversion change induced by the above-mentioned process. Before hydrogenation process, the spectra of 2LG are characterized by a regular sharp G peak at around 1582 cm -1 which is due to bond stretching of all pairs of sp 2 -C in graphene sheets.

After the hydrogenation process, drastic changes can be observed in Raman spectra of various regions of the 2LG samples. Fig. 2a presents a typical spectrum of such a region, which exhibits specific features: (i) the G peak is no longer detected; (ii) the D peak which originates from defects in graphene sheets [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF] is not observed either; (iii) a sharp peak (FWHM of around 10-33 cm -1 ) at around 1344-1367 cm -1 has occurred. This type of spectrum can be observed over extended regions of several µm width and in various areas probed by the laser, as illustrated in Fig. 2b which displays a map of the intensity of this peak. The peak at around 1344-1367 cm -1 is not the D peak, since the G peak is not simultaneously observed; also, it is too sharp anyway and its spectral position does not correspond to the value observed in the UV spectra of nanocrystalline graphite [START_REF] Paulmier | Deposition of nanocrystalline-graphite films by cathodic plasma electrolysis[END_REF]. It is supposed to be due to bond stretching between sp 3 -C, even if the peak was predicted at near 1320 cm -1 from ab initio calculations for diamane, and found at lower wavenumber in diamanoïds (1319.5-1337 cm -1 ) [START_REF] Piazza | Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films[END_REF][START_REF] Piazza | Towards a better understanding of the structure of diamanoïds and diamanoïd/graphene hybrids[END_REF] and diamond (1332 cm -1 ). Therefore, Figs 2a and2b show that full sp 2 -C to sp 3 -C conversion, that is to say full conversion of 2LG into genuine diamane, can be obtained over 20 m large region. It is assumed that the conversion takes place in regions where graphene sheets are AB-or possibly AA-stacked, which are the most energetically favorable configurations [START_REF] Chernozatonskii | Diamane: Simulation of the Structure and Properties[END_REF][START_REF] Piazza | Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films[END_REF]. But because of the variability in the stacking sequence all over the polygranular film, many other configurations should appear which lack of symmetry makes them out of the range of calculation. Therefore, the presence of stress or strain is expected in hydrogenated yet randomly stacked layers for which C-C interlayer bonding cannot occur, or only partially, and at the grain boundaries between graphenic and diamane domains. The local modifications induced by the variety of interlayer bonding situation are probably at the origin of the variety of up-shifted positions of the sp 3 -C peak (Fig. 2c). Cracks which have appeared in the film (Fig. 1e, to be compared to Fig. 1a), and layers stacked with a very slight twist angle (Fig. 1f) are probably related to this occurrence of stress and strain.

In Fig. 2c, one can note a coarse correlation of the sp 3 -C peak width with its position, which ranges from 1340 to 1370 cm -1 . An upshift can be associated to strain due to the presence of several phases (diamane, graphene) in equilibrium, or to an inhomogeneous layer hydrogenation, or to a distorted bonding between two hydrogenated twisted layers. The Gruneisen parameter for optical phonon is around 1cm -1 /GPa [START_REF] Wu | Zone-center modes of cubic and hexagonal diamond under high pressure: A first-principles study[END_REF] leading to a conversion factor of 3 cm -1 /GPa for hydrostatic pressure. For biaxial strain, a splitting between singlet and doublet is expected and not observed, which is consistent with an average hydrostatic strain, which is an acceptable yet rough hypothesis. The in-plane lattice parameter of lonsdaleite is 0.249 nm, and 0.252 nm in diamond, while it is 0.246 nm in graphene, leading to a strain of -1 to -2%. In graphene or diamond, -1% is associated to a hydrostatic stress of about 10 GPa. Consequently, a wavenumber shift of 30 cm -1 is possible and corresponds to the range of the shift observed.

To test this hypothesis, we have performed first principle calculations on modified diamane structures. First, starting from the genuine diamane structure (one hydrogen atom every two carbon atoms), and then subsequently moving a hydrogen or a pair of hydrogen atoms to the neighboring C atoms, a spontaneous relocation of the hydrogen atoms to the right sites to form diamane is observed. So, hydrogen atoms self-organize when reacting with a graphene layer. Second, by adding extra hydrogens on C atoms already coordinated to 3 others carbon atoms so that two firstneighbor carbon atoms both wear a hydrogen, the excess H atoms always desorb spontaneously to reform H2 molecules. Over-hydrogenation is thus not favorable. Finally, while testing the case of under-hydrogenation by removing 7 H atoms from the top layer, a strong structural reorganization occurs, see Fig. 3a and Fig. 3b. The 19 C atoms involved in the extended defect tend to lose their sp 3 character to recover a sp 2 one, implying reductions of several C-C bond-lengths, confirmed by the histogram given in Fig. 3c. The phonon polarization imposed by the light polarization is in the plane, activating phonons without physical limitation. In such a case, the linewidth is not broadened and the wavenumbers of the optical modes calculated by DFT are close to the bulk value within the uncertainty range of the calculations. Thus, the lattice parameter remains the same while very local distortions can be present which could be at the origin of both linewidth broadening and frequency changes. Therefore, our calculations show that full hydrogenation for AB stacking is the most stable, final material, and that significant distortions can occur upon partial hydrogenation. Thus, sp 2 -C to sp 3 -C conversion induces huge stresses, specifically at the grain boundaries, either when the 2LG involved does not exhibit the AB or AA stacking needed for allowing the optimized hydrogenation of the layers (in case of turbostratic or commensurate stacking indeed, only partial hydrogenation may occur, locally promoted when carbon atoms happen to locally superimposed), or in case of under-hydrogenation. This is also consistent with the stress estimated in diamanoïd/graphene hybrids [START_REF] Piazza | Towards a better understanding of the structure of diamanoïds and diamanoïd/graphene hybrids[END_REF].

To summary, we have shown that the hot-filament promoted hydrogenation process can be successfully used to produce genuine diamane from bi-layer graphene. UV Raman spectroscopy reveals the sp 3 -C stretching mode peak in diamane showing that using a UV excitation is critical for the characterization of such sp 3 -C nanomaterials. It is believed that the key for producing homogenous diamane films now only relies on starting from a good quality 2LG material, that is to say, ideally, single-crystal AB flakes as large as possible. The present results open the door to mass-produced diamane by means of the hot-filament-assisted CVD process, as it is employed for the industrial production of other sp 3 -bonded carbon materials, such as diamond films, for about 40 years.

Figure 1 :

 1 Figure 1: Examples of TEM characterization of the 2LG films before (from (a) to (d)) and after hydrogenation ((e) and (f)). Diffraction patterns are from 100 nm-large areas. (a) Image of an asreceived 2LG film. Contrast variations are due to the presence of amorphous material partially covering the film. (b) Single graphenic domain, as an evidence for two superimposed 1LG in AB or AA stacking; the inner and outer 6 spots correspond to the 100 and 110 spot families in graphite, respectively. (c) Two slightly rotated graphenic domains, as an example of a random stacking of two superimposed 1LG. (d) Evidence for the presence of 4 superimposed graphenic

Figure 2 :

 2 Figure 2: Typical UV (at 244 nm) Raman data. (a) Example of a Raman spectrum obtained on a hydrogenated bilayer graphene, as the evidence for the occurrence of genuine diamane, as discussed in the text. (b) Example of the Raman map of the intensity of the sp 3 -C stretching mode of 2LG after exposure to the hot-filament-promoted hydrogenation process. (c) Full Width at Half Maximum (FWHM) as a function of wavenumber of the sp 3 -C stretching peak for graphene materials exposed to the hot-filament-promoted hydrogenation process.

Figure 3 :

 3 Figure 3: (a) Top view of the reconstructed unsaturated structure, with 7 missing hydrogen atoms on the top layer (position of the missing H atoms is given by the black circles, and (b) side view. Carbon atoms involved in the reconstruction are given in yellow, while the others are brown and H atoms are white. (c) Resulting C-C bond-length distribution.

Acknowledgements

This research was funded by the Ministry of Higher Education, Science and Technology of the Dominican Republic (2015, 2016 and 2018 FONDOCyT programs). F. Piazza would like to greatly acknowledge PUCMM for strong administrative support. Funding by the LabEx NEXT is acknowledged for enabling mutual scientific visits and research stays. I.G. acknowledges the Calcul en Midi-Pyrénées initiative CALMIP (Project p0812) for allocations of computer time, as well as GENCI-CINES and GENCI-IDRIS for Grant No. 2020-A008096649.


Special Issue on Transport in Narrow Channels

The transport of molecules, colloids and particulate matter in narrow channels is ubiquitous in natural, industrial and technological processes. While there is a long history of research relevant to the topic (Stokes/Poiseuille flow, Hele-Shaw cells, ion channels and pumps, etc.) the field has grown rapidly in the last few years attracting contributions from researchers with a variety of backgrounds. The 22 articles appearing in this special edition showcase a variety of approaches and applications of transport in narrow channels. We can identify a number of general topics including polymers, active particles, currents and current fluctuations. One common theme found in several articles is the problem of optimizing the transport efficiency.

Polymers

Synthetic or bio-polymers in channels have many important applications including, for example, protein and DNA sequencing. Chinappi and Cecconi [3] discuss nanopore-based devices for protein sequencing. They focus on the physical modelling of the transport process, emphasing its multiscale nature. Sakaue [9] analyses the behavior of a semiflexible polymer confined in nanochannel under compression in the axial direction. Sarabadani and Ala-Nissila [15] review recent developments in the theory of polymer translocation dynamics through a nanopore using their own iso-flux tension propagation theory. Tagliazucchi et al. [16] study translocation routes of nanoparticles through polymer-brush modified nanopores, including those that do not coincide with the axis of the pore. They show that there is an optimum particle size that maximizes translocation. Nomidis et al. [19] review existing models of the reaction limited capture process of biomolecules by a nanopore.

Active Particles

Active particles, which consume energy in order to move, have been the focus of intense interest in recent years. In this special issue, the are featured in several contributions: Bet et al. [5], motivated by design principles for self-steering particles, derive analytical solutions to the equation of motion of general particles that undergo strongly confined quasi-2D motion in Hele-Shaw cells. They study both symmetric and asymmetric particle shapes and they find excellent agreement with numerical simulations of the particle trajectories. The contribution of Daddi-Moussa-Ider et al. [11] explores the state diagram of a three-sphere microswimmer in a channel. They consider neutral, puller and pusher type swimmers. The latter always end up trapped at the channel walls, while the first two may also perform a gliding motion. Certain aspects may be captured analytically using the method of images. Cecconi et al. [13] examine the anomalous mobility of a driven active particle in a steady laminar flow. Yang et al. [14] study colloidal transport through an array of traps controlled by microswimmers and observe that the active micro-swimmers control the diffusion of passive particles.

Current and Current Fluctuations

The Fick-Jacobs equation provides an approximate, one-dimensional description of diffusion in channels (2D) and tubes (3D). This approach is used in two of the present contributions: Chávez et al.

[2] develop a theory for the position-dependent effective diffusivity for two-dimensional asymmetric varying-width channels, and for three-dimensional curved midline tubes, formed by straight walls. Kalinay and Slanina [8] present a formalism reducing the general 2D advectiondiffusion equation onto the longitudinal coordinate, also valid for nonconservative forces. The result is a generalized Fick-Jacobs equation. [START_REF] Chávez | Current fluctuations across a nano-pore, Mira Zorkot and Ramin Golestanian[END_REF] present a linearised theory of frequencey dependent fluctuations of current through nano channels. The predictions are compared with brownian dynamics simulations. Carusela and Rubi [7] study the current rectification of particles moving in a pulsating channel under the influence of an applied force. Berezhkovskii and Bezrukov [12] examine the effect of stochastic gating on the flux of solute molecules through a membrane channel. The spontaneous transitions of channel-forming proteins between their open and closed states is well known. Coalson and Jasnow [18] present discrete state models of single-file ion permeation through a narrow ion channel pore and use them to analyze the ratio of forward to backward tracer current. et al. [4] present a molecular dynamics study of the invasion of gas into mica nanopores. Liot et al. [6] present an experimental study of the influence of diffusion, confinement and particle nature on the transport of nano-objects in narrow channels. Two contributions are concerned with clogging phenomena: Reichhardt and Reichhardt [10] examine the flow and clogging of particles driven through asymmetric funnel arrays using brownian dynamics simulation. The clogged states remain robust at finite temperatures but develop intermittent bursts of flow in which a clog temporarily breaks apart but quickly reforms. Barré et al. [20] propose analytically soluble stochastic models of blockage in channel bundles and show that, in the case of reversible blockage, the exiting flux can be maximized. Bacchin [17] provides a general, unifying description of colloidal dispersion flow in confined systems. Ooshida and Otsuki [21] present a theoretical framework that allows analytical calculation of two-tag correlations of interacting brownian particles confined to a narrow channel. Bénichou et al. [22] present a review of tracer diffusion in crowded, narrow channels.

Zorkot and Golestanian

Colloidal and Other Topics

Fang

We thank the authors for their contributions and the editors of the Journal of Physics : Condensed Matter for the opportunity to serve as guest editors of this special issue.

Paolo Malgaretti, Gleb Oshanin, Julian Talbot