Emiliano Lorini
email: lorini@irit.fr

Dominique Longin
email: longin@irit.fr

Benoit Gaudou
email: benoit.gaudou@alumni.enseeiht.fr

Anchoring the institutional dimension of speech acts in agents' attitudes: a logical approach

Keywords: Modal logic, institutions, acceptance, speech act, commitment, multi-agent systems

The focus of this contribution is to define speech acts as institutional actions where modeled institutions are social or informal institutions. With social or informal institution, we mean an institution which is grounded on the acceptances of its members. In the first part of the paper we present a logic of acceptance, goal and action. Then, we specify how agents can create and maintain normative and institutional facts on the basis of their acceptances qua members of a certain institution. In particular we focus on obligations and social commitments and provide an original reductionist characterization of these concepts, anchoring them in agents' attitudes. Finally, we propose a formal characterization of the speech act promise. In this last part, we first define the constitutive rule which creates the relation between an utterance (as a physical action) and the speech act promise (as an institutional action). Then, we specify the deontic dimension of promise. Finally, we establish the relationship between promise and social commitment.

I. INTRODUCTION

Saussure [START_REF] De Saussure | Course in General Linguistics[END_REF] distinguishes language (la langue) from speech (la parole). The "meaning" of the former is defined by institutions (social aspect), by a set of conventions outside the linguistic activity (static aspect). The latter corresponds to the use of the language (dynamical aspect) by individuals, and it is the intention of these individuals that assigns a "value" to the act of utterance (individual aspect). Metaphorically, language is a sort of score while speech is the execution of this score by musicians. The Austin's concept of "performative utterance" [START_REF] Austin | How To Do Things With Words[END_REF] is the first step to bring together language and speech because: first, performative utterances are regulated by social conventions that determine not only the "meaning" of the utterance, but also the "value" of its enunciation 1 ; second, it is practically impossible to attribute a meaning to a performative utterance independently of the value of its enunciation.

Through the concept of illocutionary act, speech act theory [START_REF] Austin | How To Do Things With Words[END_REF], [START_REF] Searle | Speech acts: An essay in the philosophy of language[END_REF] generalizes the concept of performative utterance. In such a theory, the value of an enunciation cannot be no longer considered as a consequence of a preliminary meaning; but contrary to the linguistic nihilism of the wittgensteinian trend and their slogan "Meaning is use" (See [START_REF] Searle | Speech acts: An essay in the philosophy of language[END_REF]Section 6.4]), speech act theory does not deny that the use of language is founded on a prior knowledge: it claims that the meaning of an utterance integrates some shared rules that fixe the effects of its enunciation on the speech situation.

"It is essential to see [that] (...) language is the fundamental social institution." [4, Section 5]. Language has a constitutive role in every social institution, it is a necessary condition to the existence of such institutions. The difference between a crude fact and an institutional fact is that the latter has some status associated to some function, and that there exists a collective representation of this status and of this function [START_REF]What is an institution?[END_REF] : the "language (...) provides the means of this representation".

Language is thus the carrier associating a concept to its meanings into a particular institution. This mapping is made through constitutive rules by (in)formal institutions. Contrary to normative rules that govern existing behaviors, constitutive rules do not have a purely normative role but create or define new forms of behavior whose the existence depends of these rules (see [START_REF] Searle | Speech acts: An essay in the philosophy of language[END_REF]Section 2.5]). For instance, politeness rules govern relationships between individuals although these relationships exist independently of politeness. Thus, politeness is a set of normative rules.

Subsequently, to speak a language, it is to adopt a form of intentional behavior (by performing speech acts) governed by constitutive rules related to a particular institution. These rules create or define new forms of behavior such as promises, orders, requests, assertions, etc. (see [START_REF] Searle | Speech acts: An essay in the philosophy of language[END_REF]Section 2.5]). Such speech acts are institutional facts defined by a set of (constitutive) rules of the form "X count as Y in the situation S". (See [3, Section 2.7] for more details.) For example, in the context of ordinary communication [START_REF] Jones | Conventional signalling acts and conversation[END_REF], a promise is defined and created on the basis of a constitutive rule (shared by the speaker and the hearer) of the form: a speaker's enunciation of the utterance: "I am going to perform action a!" counts as a promise to the hearer to perform action a, under the condition that the hearer wants the speaker to perform action a. Subsequently, to speak a language, it is to adopt a form of intentional behavior (by performing a speech act) governed by such a kind of constitutive rules.

The focus of this contribution is to define speech acts as institutional actions where modeled institutions are social or informal institutions. With social or informal institution, we mean an institution which is grounded on the acceptances of its members. Differently from formal (legal) institutions, informal institutions are rule-governed social practices in which no member with 'special' powers to create and eliminate institutional facts is introduced (see Section IV). 2 After a short introduction about the concept of acceptance (Section II), we present a modal logic of acceptance, goal and action (Section III). On the basis of the notion of acceptance, we specify how a group of agents can create normative and institutional facts which hold only in an attitude-dependent way (Section IV). In Section V we define the promissive speech act.

II. THE CONCEPT OF ACCEPTANCE

Whereas beliefs have been studied for decades, acceptances have only been examined since [START_REF] Stalnaker | Inquiry[END_REF] and [START_REF] Cohen | An essay on belief and acceptance[END_REF] while studying the nature of argument premises or reformulating Moore's paradox [START_REF] Cohen | An essay on belief and acceptance[END_REF]. If a belief that p is an attitude constitutively aimed at the truth of p, an acceptance is the output of "a decision to treat p as true in one's utterances and actions" [START_REF] Hakli | Group beliefs and the distinction between belief and acceptance[END_REF] without being necessarily connected to the actual truth of the proposition. Another difference between belief and acceptance is that beliefs are context-independent, whilst acceptance depends on context [START_REF] Bratman | Practical reasoning and acceptance in context[END_REF]. In fact, one can decide (say for prudential reasons) to reason and act by "accepting" the truth of a proposition in a specific context, and possibly rejecting the very same proposition in a different one. Although, usually, this aspect of acceptance is studied in private contexts, here we continue the work initiated in [START_REF] Gaudou | Anchoring Institutions in Agents' Attitudes: Towards a Logical Framework for Autonomous MAS[END_REF] by exploring the role of acceptance in institutional contexts. Institutional contexts are rule-governed social practices on the background of which the agents reason. For example, take the case of a game like Clue. The institutional context is the rule-governed social practice which the agents conform to in order to be competent players. On the background of such contexts, we are interested in the individual and collective attitudes (individual and collective acceptances) that can be formally captured. In the context of Clue, for instance, agents accept that something has happened (see Example 2) qua players of Clue. The state of acceptance qua member of an institution is the kind of acceptance one is committed to when one is "functioning as member of the institution" [START_REF] Tuomela | The Philosophy of Sociality[END_REF].

III. THE LOGICAL FRAMEWORK

A. Syntax

The syntactic primitives of our logic of acceptance, actions and goals are the following:

• a finite set of n > 0 agents AGT = {1, 2, ..., n};

2 For the distinction between formal and informal institutions, see also [START_REF] North | Institutions, Institutional Change, and Economic Performance[END_REF].

• a nonempty finite set of atomic actions AT = {a, b, ...}; • a finite set of atomic formulas ATM = {p, q, ...};

• a finite set of labels denoting institutional contexts

INST = {inst 1 , inst 2 , ..., inst m }.
Moreover, we note 2 AGT ⋆ = 2 AGT \ {∅} the set of all non empty subsets of agents, ∆ = C:x|C ∈ 2 AGT ⋆ , x ∈ INST the set of all couples of non empty subsets of agents and institutional contexts.

The language LAN G of the logic L is defined as the smallest superset of ATM such that: if ϕ, ψ ∈ LAN G, i, j ∈ AGT , α ∈ ACT and C:x ∈ ∆ then ¬ϕ, ϕ ∨ ψ, [i:α:j] ϕ, Goal i ϕ and Accept C:x ϕ ∈ LAN G, where the set ACT is the smallest superset of AT such that: if i, j ∈ AGT , a ∈ AT then inf (i:a:j) ∈ ACT (informative actions). The classical boolean connectives ∧, →, ↔, ⊤ (tautology) and ⊥ (contradiction) are defined from ∨ and ¬ in the usual manner.

The set ACT includes all informative actions of type "informing that agent i is going to perform (atomic) action a for agent j", where, with informative action, we mean the performance of an utterance, the act of making an enunciation. Since AT and AGT are finite sets, it follows that ACT as well is finite.

Operators of the form Accept C:x have been introduced in [START_REF] Gaudou | Anchoring Institutions in Agents' Attitudes: Towards a Logical Framework for Autonomous MAS[END_REF] where a logic of acceptance has been proposed. These operators allow to express those facts that the agents in a group C accept while identifying themselves with a certain institution x.

Formula Accept C:x ϕ has to be read "the agents in C accept that ϕ while functioning as members of the institution x". EXAMPLE 1. Accept C:Greenpeace protectEarth is read "the agents in C accept that the mission of Greenpeace is to protect the Earth while functioning as activists of Greenpeace".

For C:x ∈ ∆, Accept C:x ⊥ has to be read "agents in C are not functioning as members of the institution x" because we assume that functioning as a member of an institution is, at least in this minimal sense, a rational activity; conversely, ¬Accept C:x ⊥ has to be read "agents in C are functioning as members of the institution x"; ¬Accept C:x ⊥ ∧ Accept C:x ϕ stands for "agents in C are functioning as members of the institution x and they accept that ϕ while functioning as members of x" or simply "agents in C accept that ϕ qua members of the institution x" (i.e. group acceptance).

The dynamic operators of the form [i:α:j] are just a generalization of standard operators of dynamic logic [START_REF] Harel | Dynamic Logic[END_REF] where both the author (initiator) and the addresse of a certain action α are specified. Formula [i:α:j] ϕ has to be read "after agent i does action α for agent j, it is the case that ϕ". Operators of the form Goal i are standard operators for agents' goals [START_REF] Cohen | Intention is choice with commitment[END_REF]. Goal i ϕ has to be read "agent i has the goal that ϕ holds" (or "i wants ϕ to be true").

The following abbreviations are given for any i, j ∈ AGT , a ∈ AT and α ∈ ACT :

i:α:j ϕ def = ¬ [i:α:j] ¬ϕ Int i,j (α) def = Goal i i:α:j ⊤ Inf i,j (a)
def = i:inf (i:a:j):j ⊤ Formula i:α:j ϕ is meant to stand for "agent i performs action a and ϕ is true afterward". Int i,j (α): agent i intends to perform action α for j. Inf i,j (a): agent i informs agent j that he is going to perform the (atomic) action a for him.

B. Semantics

We use a possible worlds semantics. A model of the logic L is a tuple M = W, A , R, G V where:

• W is a set of possible worlds;

• A : ∆ -→ (W -→ 2 W ) associates each C:x ∈ ∆ and
possible world w with the set A C:x (w) of possible worlds accepted by the group C in w, where the agents in C are functioning as members of the institution x;

• R : AGT × ACT × AGT -→ (W -→ 2 W ) associates
each two agents i, j ∈ AGT , action α ∈ ACT and possible world w with the set R i:α:j (w) of possible worlds that are reachable from w through the occurrence of action α performed by i for agent j;

• G : AGT -→ (W -→ 2 W ) associates each agent i ∈
AGT and possible world w with the set G i (w) of worlds that are compatible with agent i's goals in w; • V : W -→ 2 ATM is a truth assignment which associates each world w with the set V (w) of atomic propositions true in w.

To the standard truth conditions for atomic formulas, negation and disjunction we add :

• M, w |= Accept C:x ϕ iff all w ′ ∈ W , if w ′ ∈ A C:x (w) then M, w ′ |= ϕ; • M, w |= [i:α:j] ϕ iff for all w ′ ∈ W , if w ′ ∈ R i:α:j (w) then M, w ′ |= ϕ; • M, w |= Goal i ϕ iff for all w ′ ∈ W , if w ′ ∈ G i (w) then M, w ′ |= ϕ.

C. Axiomatization

Every operator of type Accept C:x and [i:α:j] is supposed to be a normal modal operator satisfying standard axioms and rules of inference of system K. Every operator Goal i is supposed to be a normal modal operator satisfying standard axioms and rules of inference of system KD. 3 Axiom D for goals corresponds to the following constraint of seriality over L models. For every i ∈ AGT and w ∈ W :

G i (w) = ∅ S1 3 Axiom D for Goal i is: ¬(Goal i ϕ ∧ Goal i ¬ϕ).
The rest of the section contains other axioms for acceptance, action and intention and corresponding semantic constraints over L models. We suppose that given a set of agents C, all B ⊆ C have access to all the facts that are (not) accepted by agents in C while functioning as members of the institution x. In particular, we suppose that: if agents in C (do not) accept that ϕ while functioning as members of the institution x then for every subset B of C and institution y while functioning as members of the institution y, agents in B accept that agents in C (do not) accept that ϕ while functioning as members of the institution x. Such properties are captured by the following two axiom schemas. For every C:x, B:y ∈ ∆, if B ⊆ C then:

Accept C:x ϕ → Accept B:y Accept C:x ϕ 4 Accept ¬Accept C:x ϕ → Accept B:y ¬Accept C:x ϕ 5 Accept
Axioms 4 Accept and 5 Accept together correspond to the following semantic constraint over L models. For every w ∈ W and C:x, B:y

∈ ∆, if B ⊆ C then: if w ′ ∈ A B:y (w) then A C:x (w ′ ) = A C:x (w) S2
We also suppose that if agents in C accept that ϕ qua members of the institution x then, for every subset B of C, it holds that agents in B accept ϕ qua members of the institution x. Formally, for every C:x, B:x ∈ ∆, if B ⊆ C then:

(¬Accept C:x ⊥ ∧ Accept C:x ϕ) → (¬Accept B:x ⊥ ∧ Accept B:x ϕ) Inc Accept EXAMPLE 2.
Imagine three agents i, j, k that, qua Clue players, accept that someone called Mrs. Red, has been killed: ¬Accept {i,j,k}:Clue ⊥ ∧ Accept {i,j,k}:Clue killedMrsRed . This implies that also the two agents i, j qua Clue players accept that someone called Mrs. Red has been killed: ¬Accept {i,j}:Clue ⊥ ∧ Accept {i,j}:Clue killedMrsRed .

Axiom Inc Accept corresponds to the following semantic constraint over L models. For every w ∈ W , C:x, B:

x ∈ ∆, if B ⊆ C: if A C:x (w) = ∅ then A B:x (w) = ∅ and A B:x (w) ⊆ A C:x (w) S3
We suppose the following additional constraint over L models. For every w ∈ W , i, j, i ′ , j ′ ∈ AGT and α, β ∈ ACT :

if w ′ ∈ R i:α:j (w) and w ′′ ∈ R i ′ :β:j ′ (w) then w ′ = w ′′ S4
The property S4 says that all actions occurring in a world w lead to the same world. Thus, all actions occur in parallel and they do not have non-deterministic effects. This explains why we have phrased i:α:j ϕ "i does α for j and ϕ holds afterward" rather than "it is possible that i does α for j and ϕ holds afterward". Constraint S4 corresponds to the following axiom of our logic. For every i, j, i ′ , j ′ ∈ AGT and α, β ∈ ACT :

i:α:j ϕ → [i ′ :β:j ′ ] ϕ Det

We also suppose that the world is never static in our framework, that is, for every world w there exists at least two agents i, j and action α such that i performs α for j at w. Formally, given a L model M , for every w ∈ W we have that:

∃i, j ∈ AGT , ∃α ∈ ACT , ∃w ′ ∈ W s.t. w ′ ∈ R i:α:j (w) S5
Property S5 of L models corresponds to the following axiom of our logic. i,j∈AGT ,α∈ACT i:α:j ⊤ Active

Axiom Active ensures that for every world w there is a next world of w which is reachable from w by the occurrence of some action performed by some agent for another agent. This is the reason why the operator X for next of LTL (linear temporal logic) can be defined as follows: 4Xϕ

def = i,j∈AGT ,α∈ACT i:α:j ϕ
where Xϕ is meant to stand for "ϕ will be true in the next state".

The following axiom relates intentions with actions. For every i, j ∈ AGT and α ∈ ACT :

i:α:j ⊤ → Int i,j (α) IntAct
According to Axiom IntAct, an agent i performs action α for agent j only if he has the intention to do α for j. In this sense we suppose that an agent's doing is by definition intentional. IntAct corresponds to the following semantic constraint over L models. For every i, j ∈ AGT , α ∈ ACT and w ∈ W :

if ∃v ′ such that v ′ ∈ R i:α:j (w) then ∀w ′ ∈ G i (w), ∃w ′′ such that w ′′ ∈ R i:α:j (w ′ ) S6
We call L the logic axiomatized by the principles presented above and we write ⊢ L ϕ iff formula ϕ is a theorem of L. Moreover, we write |= L ϕ iff formula ϕ is valid in all L models, i.e. M, w |= ϕ for every L model M and world w in M. Finally, we say that a formula ϕ is satisfiable if there exists an L model M and a world w in M such that M, w |= ϕ.

IV. INSTITUTIONAL CONCEPTS

Normative and institutional facts are a class of facts that are typical of institutional contexts. According to [START_REF] Searle | The Construction of Social Reality[END_REF], such facts have the peculiar feature of being dependent on the agents' attitudes in a way that we are in the position to specify in detail in the logic L. More precisely it has been noted that these facts are characterized at least by two features [START_REF] Lagerspetz | The opposite mirrors[END_REF], [START_REF] Searle | The Construction of Social Reality[END_REF], [START_REF] Tuomela | The Philosophy of Social Practices: A Collective Acceptance View[END_REF]: performativity (an attitude of certain type shared by a group of agents towards a normative or an institutional fact may contribute to the truth of a sentence describing the fact); reflexivity (if a sentence describing a normative or an institutional fact is true, the relevant attitude is present). EXAMPLE 3. If the agents qua group members accept a certain piece of paper as money (an institutional fact), then, in the appropriate context, this piece of paper is money for that group (performativity). At the same time, if it is true that a certain piece of paper is money for a group, then the agents qua group members accept the piece of paper as money (reflexivity).

In order to represent in the logic L these kind of facts, we need first to define the concept of truth with respect to an institutional context in way that respects these two principles.

A. Truth in an institutional context

We formalize the notion of truth w.r.

t. a certain institutional context with the operator [[x]]. A formula [[x]

] ϕ is read "within the institutional context x, it is the case that ϕ". Here we suppose that "within the institutional context x it is the case that ϕ" if and only if "for every set of agents C, the agents in C accept that ϕ while functioning as members of the institution x". We introduce the following formal definition.

Definition 1:

[[x]] ϕ def = C∈2 AGT⋆ Accept C:x ϕ
It is straightforward to prove that every [[x]] is a normal modal operator. Given the previous analysis, a fact is true w.r.t. an institutional context x if and only if such fact is accepted by all the agents while they function as members of x (hence the performativity and the reflexivity principles are maintained).

At this point, it might be objected that there are facts which are true in an institutional context but only "special" members of the institution are aware of them and can change them. For instance, there are laws in every country which are known and can be changed only by the specialists of the domain (lawyers, judges, members of the parliament, etc.). In order to resist to this objection recall that, at this stage, our model applies to informal institutions of a society, in particular to language. Relative to this restriction, the proposed assumption is justified because, for informal institutions, there is no special agent who has the power to create and eliminate institutional facts characterizing the institution itself (i.e. nobody has the power to change the rules of the speech act promise in the context of language).

Finally, the following abbreviation is given. Definition 2:

[[Univ ]] ϕ def = x∈INST [[x]] ϕ [[Univ ]
] ϕ stands for "ϕ is universally accepted as true".

B. Constitutive rules

From the concept of truth with respect to an institutional context a notion of constitutive rule of the form "ϕ counts as ψ in the institutional context x" can be defined. We conceive a constitutive rule as a material implication of the form ϕ → ψ in the scope of an operator [[x]]. We suppose that a constitutive rule is intrinsically local, that is, a rule that is not universally valid while it is accepted by the members of a certain institution. More generally, for every x ∈ INST the following abbreviation (that stands for "ϕ counts as ψ in the institutional context x") is given. 5 Definition 3:

ϕ x ⊲ ψ def = [[x]] (ϕ → ψ) ∧ ¬ [[Univ ]] (ϕ → ψ) EXAMPLE 4.
In the context of gestural language there exists a constitutive rule according to which, the nodding gesture "counts as" an endorsement of what the speaker is suggesting, noted nodding gesture ⊲ yes. But in different contexts the same gesture may express exactly the opposite fact (viz. ¬ [[Univ ]] (nodding → yes) holds).

C. Obligations

Informal institutions such as language involve a deontic dimension that up to now we have ignored. In order to capture this core feature we extend the logic L by introducing a violation atom V as in Anderson's reduction of deontic logic to alethic logic [START_REF] Anderson | A reduction of deontic logic to alethic modal logic[END_REF]. By means to this new formal construct we can formally characterize the obligations which are valid in a certain institution by anchoring them in the acceptances of the members of that institution.

We say that "in the institutional context x, agent i has the obligation to perform action α for agent j under the condition ϕ" (noted Obl x (i, α, j, ϕ)) if and only if "in the institutional context x, the fact that i does not perform action α for j under the condition ϕ counts as a violation at the next step of time". 6 Formally:

Definition 4:

Obl x (i, α, j, ϕ) def = (ϕ ∧ [i:α:j]⊥) x ⊲ XV EXAMPLE 5.
Formula Obl EBay (i, sendGoods, j, paid (j , i)) expresses that, in the context of EBay, it is obligatory for an agent i to send certain goods to agent j under the condition that j has paid the goods to i.

We here distinguish an obligation which is valid in a certain institution x (viz. institutional obligation) from the instantiations of this obligation in specific groups of agents which are 5 Our notion of constitutive rule of the form ϕ x ⊲ ψ is similar to the notion of proper classificatory rule given in [START_REF] Grossi | Classificatory aspects of counts-as: An analysis in modal logic[END_REF]. 6 Formulas of type Oblx(i, α, j, ϕ) can be conceived as particular instances of so-called regulative rules in Searle's sense [START_REF] Searle | The Construction of Social Reality[END_REF]. On the distinction between regulative rule and constitutive rule see also [START_REF] Boella | Regulative and constitutive norms in normative multiagent systems[END_REF]. members of this institution. We say that i's obligation to do action α for j is instantiated in the group C of members of institution x (noted IObl C:x (i, α, j)) if and only if the agents in C, qua members of the institution x, accept that if i does not perform action α for j then i will incur a violation in the next state. Formally: Definition 5:

IObl C:x (i, α, j) def = ¬Accept C:x ⊥ ∧ Accept C:x ([i:α:j]⊥ → XV )
The following Theorem highlights the relationships between institutional obligations and instantiations of obligations in groups.

Theorem 1:

⊢ L (Obl x (i, α, j, ϕ) ∧ ¬Accept C:x ⊥∧Accept C:x ϕ) → IObl C:x (i, α, j)
According to Theorem 1, if the agents in C, qua members of institution x, accept that ϕ and, in x, i has the obligation to perform action α for agent j under the condition ϕ then, i's obligation to do action α for j is instantiated in the group C of members of x. EXAMPLE 6. From the fact that in EBay, it is obligatory for an agent i to send certain goods to agent j under the condition that j has paid the goods to i (noted Obl EBay (i, sendGoods, j, paid (j , i))), and the fact that the group C of EBay surfers accept that j has paid certain goods to i (noted ¬Accept C:EBay ⊥ ∧ Accept C:EBay paid (j , i)), we infer that i's obligation to send the goods to j is instantiated in the group C of EBay surfers (noted IObl C:EBay (i, sendGoods, j)).

D. Social commitment

Social commitment has a fundamental role in the interaction between agents in an institution. It fixes the relations between agents, bounding an agent toward another. Social commitment is thus a relational notion: it relies at least two agents, the agent who is committed (the debtor [START_REF] Fornara | Agent communication and artificial institutions[END_REF]) and the agent to whom the debtor is committed (the creditor). According to Castelfranchi [START_REF] Castelfranchi | Commitment: from individual intentions to groups and organizations[END_REF], there are two crucial aspects of social commitment: a motivational aspect and a deontic aspect. If the debtor i is committed to the creditor j to perform action α for him then, the creditor and the debitor must mutually know that the creditor is interested in the fact that the debtor performs α (motivational aspect) and, the creditor must have specific rights on the debtor being entitled to ask to the debtor to perform action α (deontic aspect).

In the present analysis we specify these two conditions for social commitment on the basis of the notions of acceptance and instantiated obligation. Differently from Castelfranchi, we make explicit the institutional dimension of social commitment by specifying the institution in which the commitment is established. On the one hand, we characterize the deontic aspect of i's commitment to j to perform action α by the fact that i and j function as members of a certain institution x and, as members of x, i and j accept that i is obliged to perform action α for j. That is, i's obligation to do action α for j is instantiated in the group {i, j} of members of x. On the other hand, we characterize the motivational aspect of i's commitment to j by the fact that i and j, qua members of the institution x, accept that j wants i to do α for him. Formally: Definition 6:

SC x (i,α, j) def = IObl {i,j}:x (i, α, j)∧
¬Accept {i,j}:x ⊥ ∧ Accept {i,j}:x Goal j i:α:j ⊤ SC x (i, α, j) stands for "i is committed to j to do α w.r.t. institution x." a) Related works on social commitment: We can note that Castelfranchi also requires in his definition of social commitment the condition of i and j's mutual knowledge about i's intention to perform α for j. We do not include this condition here since in our view it is too strong because it logically implies that i's intention actually holds (under the commonly accepted hypothesis of introspection on intentions). But as Castelfranchi remarks himself, the actual intention of i to perform the action α is neither necessary nor sufficient for his social commitment to do α: the entailment link between social commitment and individual intention is represented by the hypothesis that the agent is honest. Moreover we consider that, after a promise, the social commitment can persist even if the debtor drops the intention that he manifested.

In opposition to Singh [START_REF] Singh | A social semantics for agent communication languages[END_REF] and works on ACLs [START_REF] Fornara | Agent communication and artificial institutions[END_REF], [START_REF] Mallya | An algebra for commitment protocols[END_REF], [START_REF] Flores | Conversational semantics with social commitments[END_REF], in this work social commitment is not taken primitive and is anchored in group attitudes. In particular, our notion of social commitment is grounded on the acceptance of the debtor and the creditor qua members of a certain institution.

Colombetti et al. [START_REF] Fornara | Operational specification of a commitment-based agent communication language[END_REF], [START_REF] Fornara | Agent communication and artificial institutions[END_REF] have developed an ACL semantics based on a primitive notion of commitment and its dynamics (through its states change). For example, when an agent promises to another one to perform an action, he creates a pending commitment (coinciding to the social commitment in the commonly used sense), that becomes fulfilled (resp. violated) if the agent performs (resp. do not perform) the action. The commitment can also be canceled by the creditor. When an agent is requested to performed an action, an unset commitment is created, which can be accepted (resp. refused) by the agent (the commitment becomes pending (resp. canceled)).

An unset commitment of c toward d represents the particular social relation that results from the performance by agent d of a request, an order or another directive speech act. This kind of commitment is quite disputable in particular in the case of an order. When I am driving my car and a policeman motions me to stop, I have automatically the obligation to stop my car. I have no refusal opportunity, neither do I have to accept to stop. This example can hardly be represented in the Colombetti's account. We cannot say that I have an unset commitment to stop my car and that I can accept or refuse this commitment. We neither can say that an pending commitment is automatically created, because indeed I am not committed but only obliged to stop. Our framework is able to handle such a fine example because it contains both notions of commitment and obligation.

In Colombetti et al.'s account, nothing is said about the consequences of the cancelation of a commitment. In our characterization, after the creditor utters that he does not want anymore the action to be performed by the debtor, the debtor's commitment is dropped. But the the debtor's obligation might remain instantiated. Indeed, formula ¬SC x (i, α, j) ∧ IObl {i,j}:x (i, α, j) is satisfiable in our logic.

V. APPLICATION TO THE PROMISE Speech acts can be viewed as institutional facts represented by constitutive rules. In this section, we focus on a particular kind of speech act: the promissive. We first define the constitutive rule which creates the relation between an utterance (as a physical action) and the speech act promise (as an institutional action). Then, we specify the deontic dimension of promise. Finally, we establish the relationship between promise and social commitment.

According to [START_REF] Searle | Speech acts: An essay in the philosophy of language[END_REF], in the context of ordinary communication (noted OC ) the speech act promise is defined on the basis of a constitutive rule of the form: a speaker's enunciation of the utterance: "I am going to perform action a" counts as a promise to the hearer to perform action a, under the condition that the hearer wants the speaker to perform action a. We here consider the constitutive rule defining the speech act promise with respect to the institutional context of ordinary communication (OC ) as a global axiom [START_REF] Fitting | Proof Methods for Modal and Intuitionistic Logics[END_REF]. For every i, j ∈ AGT and a ∈ AT we suppose that:

(Inf i,j (a) ∧ Goal j i:a:j ⊤) OC ⊲ X Promise(i,a,j ) (1)
where Promise(i,a,j ) is an atom denoting the institutional fact that i has promised to j to do action a for him. The previous constitutive rule says that: in the context of ordinary communication, for every agents i, j and atomic action a, i's act of informing j that is going to perform a for him, under the condition in which j wants i to do a, counts as a promise of i to j to perform a for him. The reason why we have X Promise(i,a,j ) instead of Promise(i,a,j ) in the consequent of the previous counts-as assertion is that the atom Promise(i,a,j ) represents the institutional effect of i's act of informing j that is going to perform a for him. Thus, Promise(i,a,j ) must necessarily hold after the occurrence of i's act of informing.

Note that the previous constitutive rule defining the speech act promise in the context of ordinary communication can not be generalized to all institutional contexts. For example, in the context of the card game of poker the enunciation of the utterance: "I am going to perform a certain action" does not necessarily count as a promise of the speaker's to the hearer to perform the action in question. Indeed, in the context of poker players are allowed to bluff. More generally, the following formula should be acceptable for the context of poker: ¬((Inf i,j (a) ∧ Goal j i:a:j ⊤) Poker ⊲ X Promise(i,a,j )) By definition, institutional facts are connected to a deontic dimension. In particular, an institutional fact is intrinsically connected to certain normative facts expressed in terms of obligations and permissions. For example, "being of age" is an institutional fact in many countries to which a certain number of permissions and obligations are associated (e.g. in many countries if you are of age you have the permission to vote and the obligation to fulfill the military duties). Therefore, since Promise(i,a,j ) is an institutional fact, it must be connected to certain normative facts. The following global axiom is given in order to establish such a connection with the normative level of promise. For every i, j ∈ AGT and a ∈ AT we suppose that:

Obl OC (i, a, j, Promise(i,a,j ))

Obl OC (i, a, j, Promise(i,a,j )) means that: in the context of ordinary communication, under the condition that i has promised to j to perform action a for him, i is obliged to perform action a for j. Theorem 2:

⊢ L (¬Accept {i,j}:OC ⊥ ∧ Accept {i,j}:OC Goal j i:a:j ⊤∧ Accept {i,j}:OC Promise(i,a,j )) → SC OC (i, a, j)

This theorem highlights the relationship between promise and social commitment in the context of ordinary communication: the fact that in the context of ordinary communication i and j accept that i has promised to agent j to do action a for him and that j wants i to do a for him entails the fact that i is committed to j to do a.

VI. CONCLUSION

We have introduced a logic of acceptance, action and goal and provided an institution-based semantics of the commissive act of promise. As far as we know, the only approach which is similar to ours is Colombetti's approach [START_REF] Colombetti | An analysis of agent speech acts as institutional actions[END_REF] where speech acts are also modeled in terms of constitutive rules of the form "X count as Y " and of the institutional effects brought about their performances. However, in Colombetti's approach, there is no connection between institutional level and agentive level of speech acts and the relationship between constitutive rules defining speech acts and agents' attitudes is not investigated. In our approach this relationship is established by anchoring constitutive rules and obligations in agents' acceptances.

In the future will extend our analysis to directive speech acts. To this end, we will need to model the hierarchy between roles in an institution which entitles an agents playing a certain role (e.g. the employer of a company) to make orders to other agents playing other roles (e.g. the employees).

By "enunciation", we mean "the performance of an utterance".

Note that X satisfies the standard property Xϕ ↔ ¬X¬ϕ.


Rafika Boutalbi
email: rafika.boutalbi@gmail.com

Lazhar Labiod
email: lazhar.labiod@u-paris.fr

Mohamed Nadif
email: mohamed.nadif@u-paris.fr

TensorClus: A Python Library for Tensor (Co)-clustering

Keywords: Tensors, (Co)-clustering, Multiple Graphs, Tensor Decomposition

Tensor data analysis is the evolutionary step of data analysis to more than two dimensions. Dealing with tensor data is often based on tensor decomposition methods. The present paper focuses on unsupervised learning and provides a python package referred to as TensorClus including novel co-clustering algorithms of three-way data. All proposed algorithms are based on the latent block models and suitable to different types of data, sparse or not. They are successfully evaluated on challenges in text mining, recommender systems, and hyperspectral image clustering. TensorClus is an open-source Python package that allows easy interaction with other python packages such as NumPy and TensorFlow; it also offers an interface with some tensor decomposition packages namely Tensorly and TensorD on the one hand, and on the other, the co-clustering package Coclust. Finally, it provides CPU and GPU compatibility. The TensorClus library is available at https://pypi.org/project/TensorClus/ 1 .

Introduction

The amount of data collected in fields, as different as social networks, online shopping, or medicine has grown exponentially over the last decade. Nowadays, the extraction of knowledge from such data can be based on data organized in the form of tensors instead of matrices. A tensor is a multidimensional array, which is also known as the N -way and Nth-order tensor; a third-order tensor has three dimensions. The use of tensors in data analysis applications was pioneered by researchers in psychometrics and chemometrics [START_REF] Smilde | Multi-way analysis: applications in the chemical sciences[END_REF]. Two recent effective opensource Tensorly [START_REF] Kossaifi | Tensorly: Tensor learning in python[END_REF] and TensorD [START_REF] Hao | TensorD: A tensor decomposition library in tensorflow[END_REF] are available. They offer a state-of-theart tensor decomposition approach, including algorithms such as PARAFAC and Tucker decomposition.

Here, we are interested in three-way data that are present in many applications. In medical domain, for instance, we could have a tensor patients × images × f eatures, and the objective could be analyzing patient images based on extracted features. To deal with such data we focus on co-clustering that can be viewed as an extension of clustering [START_REF] Govaert | Co-clustering: models, algorithms and applications[END_REF] devoted to reorganizing a data matrix into homogeneous blocks. This objective has attracted many authors these two decades through different approaches based on information-theoretic [START_REF] Dhillon | Information-theoretic co-clustering[END_REF], spectral co-clustering [START_REF] Labiod | Co-clustering under nonnegative matrix trifactorization[END_REF][START_REF] Labiod | A unified framework for data visualization and coclustering[END_REF], matrix factorization [START_REF] Affeldt | Ensemble block co-clustering: A unified framework for text data[END_REF][START_REF] Nadif | Unsupervised and self-supervised deep learning approaches for biomedical text mining[END_REF], or probabilistic models [START_REF] Ailem | Model-based co-clustering for the effective handling of sparse data[END_REF][START_REF] Ailem | Sparse poisson latent block model for document clustering[END_REF][START_REF] Govaert | Mutual information, phi-squared and model-based co-clustering for contingency tables[END_REF][START_REF] Salah | Directional co-clustering[END_REF][START_REF] Bock | Co-clustering for object by variable data matrices[END_REF]. The recent Coclust package [START_REF] Role | Coclust: A python package for co-clustering[END_REF] provides the implementation of co-clustering algorithms designed to efficiently handle count data matrices [START_REF] Role | Coclust: A python package for co-clustering[END_REF]. However, despite the great interest in co-clustering techniques on the one hand and the tensor decomposition methods on the other, few works tackle co-clustering from tensor data. To date, we can cite [START_REF] Wu | General tensor spectral co-clustering for higher-order data[END_REF][START_REF] Feizi | Tensor biclustering[END_REF] based on tensor-based decomposition while aiming to extract co-clusters. In contrast with these methods that require parameters tuning, in our proposal, the co-clustering objective is derived from flexible tensor latent block models which present many advantages described in [START_REF] Boutalbi | Tensor latent block model for coclustering[END_REF] and illustrated in section 2.2. Previously, we proposed [START_REF] Boutalbi | Sparse tensor co-clustering as a tool for document categorization[END_REF][START_REF] Boutalbi | Tensor latent block model for coclustering[END_REF] Tensor Latent Block Model (TLBM) for the co-clustering of tensor data as illustrated in Figure 1. TLBM exploits the flexibility of the latent block model [START_REF] Govaert | Co-clustering: models, algorithms and applications[END_REF] and is able to consider any type of data i.e. continuous, binary, count tables. We also showed that the derived algorithms can be also used for the clustering of multiple graphs or multi-view clustering. The package TensorClus that we propose is the first free python package for tensor (co)-clustering and it is open-source.

TensorClus package

TensorClus is a Python library composed of five modules dedicated to each step of three-way data analysis, from data loading to the visualization of results. Figure 2 shows the structures of the library and the packages that interface with TensorClus, namely Tensorly, TensorD, and Coclust available in Python.

Next, we describe in details the five modules depicted in Figure 2.

Reader module

To load tensor data, we built a Reader module that interacts with NumPy and Pandas packages. The module allows the following three ways of data loading:

• Load data from a text file: The user should save the tensor in a text file where the three first columns represent the tensor indices of entries and the last column the value of each tensor entry. For this, we can use the read txt tensor function.

• Load data from datasets: The user can import tensor datasets. We illustrate this step with datasets having different characteristics (see Table 1). The true partitions are also available for all datasets; they will be used just to evaluate the algorithms in this package in terms of clustering. For loading a dataset, the user can use the function load dataset by specifying the dataset name.

• Create a NumPy array: The user can also create tensor data as an NDarray using the NumPy package and use it for tensor clustering. The detailed description of the integrated datasets is available in a public github repository2 .

Decomposition and clustering modules

There are four popular implemented tensor decomposition methods, namely Parafac, Non-negative Parafac, Tucker decomposition, and Non-negative Tucker decomposition [START_REF] Tucker | Some mathematical notes on three-mode factor analysis[END_REF][START_REF] Xu | Infinite tucker decomposition: Nonparametric bayesian models for multiway data analysis[END_REF]. Note that these methods are not devoted to clustering, however, they return factor matrices that can be used for clustering. The Notice that, both learning representations and clustering tasks are performed successively -not simultaneously-. In contrast with these techniques, in our proposal with TensorClus the clustering procedure is carried out directly on threeway data and therefore does not require any learning representations.

Co-clustering module

Before describing the functions available in this module, we briefly present some essential points. From TLBM, different derived co-clustering algorithms are implemented. TLBM considers a three-way tensor data

X = [x ij ] ∈ R n×d×v
where n, d, and v are the dimensions; x ij is (v × 1) vector (Figure 3). To estimate the parameters of TLBM, we rely on variational EM, which optimizes the lower band of log-likelihood [START_REF] Boutalbi | Co-clustering from tensor data[END_REF][START_REF] Boutalbi | Sparse tensor co-clustering as a tool for document categorization[END_REF][START_REF] Boutalbi | Tensor latent block model for coclustering[END_REF]. The implemented algorithms take as input the tensor X and the number of row clusters g and columns clusters m. It alternates two steps E and M ( Algorithm 1), until the objective function value change is small or there is no change. The Expectation (E) step consists in computing the posterior probabilities and Θ which depends on the chosen probability distribution. Finally, at convergence, the algorithms return the row and column partitions and the estimated parameters Ω.

Z (t) = (z ik ) ∈ [0, 1] n×g with g k=1 z ik = 1 and W (t) ∈ [0, 1] d×m with m =1 w j = 1,
Algorithm 1: TLBM Input: X , g, m.

Initialization: Randomly generate (Z (0) , W (0) ) and compute Ω (0) repeat E-Step: Compute the posterior probabilities Z (t) and W (t) M-Step: Update parameters Ω (t) until Convergence; return Z, W, Ω With TensorClus, binary, continuous, and count data can be analyzed from Bernoulli, Gaussian, and Poisson models respectively. The co-clustering module provides the three following functions: tensorCoclusteringBernoulli, tensorCoclustringGaussian, and tensorCoclusteringPoisson that have the following arguments:

• n clusters denotes the number of clusters.

• init row and init col are the initial partitions Z and W, respectively. This means that the partitions are not randomly generated.

• max iter denotes the number of iterations.

• fuzzy is a boolean value to choose if the final partition is hard or soft partition.

• gpu is a boolean value to select the type of execution, with or without GPU. This is achieved by specifying the index of slices and the selected algorithm.

Note that

Furthermore, a version dedicated for sparse three-way data referred to as TSPLBM is also proposed. The TSLBM algorithm tackles the clustering of multiple graphs. It is devoted to co-clustering of a three-way sparse data. Given

X = [x ij ] ∈ R n×n×v
where n is the number of nodes, and v the number of graphs (slices). We can view the derived algorithm as an implicit consensus clustering for multiple graphs.

With the co-clustering module, sparseTensorCoclustering allows to apply sparse tensor co-clustering.

Visualization module

TensorClus also offers a module for data visualization to illustrate and analyze the results of co-clustering. Figure 5 shows the three visualizations proposed by the Visualization module.

• plot logLikelihood evolution plots the log-likelihood in function of iterations.

• plot parameter evolution provides the evolution of Θ at each iteration.

At the convergence, this allows to compare and interpret the obtained coclusters.

• plot slice reorganisation reorganizes each slice of a three-way data according the obtained co-clusters.

Comparison of tensor co-clustering and tensor decomposition

This package allows to evaluate different algorithms dedicated to three-way data in terms of clustering. To reach this objective, we rely on datasets where a partition of one dimension is available, this is the case of the used three-way datasets. We propose to use external measurements such as accuracy (ACC), Normalized Mutual Information (NMI) [START_REF] Strehl | Cluster ensembles-a knowledge reuse framework for combining multiple partitions[END_REF], Adjusted Rand Index (ARI) [START_REF] Liu | Properties of the hubertarabie adjusted rand index[END_REF].

These last two are less sensitive to heavily imbalanced clusters. These measures are equal to 1 if the resulting clustering is identical to the true one.

Applied on DBLP1, we compared the sparse tensor co-clustering algorithm TSPLBM with Parafac and Tucker decomposition combined with clustering algorithms. We use different ranks (10, 50, and 100) for tensor decomposition. We performed 30 runs with random initializations. Then we computed ACC, NMI, ARI, and computing time by averaging all runs. All experiments were performed using a PC with the following characteristics: Intel® Core 9e gen,a RAM(64 Gb), and GPU NVIDIA® GeForce® GTX 1650 Max-Q. Figure 4 shows the performances of TSPLBM and the two algorithms Parafac and Tucker decomposition (with different ranks) followed with the four clustering algorithms.

The experiments were performed using CPU version. It should be emphasized that TSPLBM gives better results, in terms of NMI than tensor-based decomposition algorithms combined with clustering. Tucker decomposition with a rank equal to In terms of time execution, TSPLBM is equivalent to Parafac combined with each clustering algorithm using a rank number equal to 50 and better than using the rank number equal to 100. Figure 5 shows the pictures obtained by the visualization module. We observe the log-likelihood increase at each iteration, and the algorithm converges at the 15th iteration (the plot on the left). In the middle figure, we observe the density evolution of co-clusters (densities of 3 diagonal co-clusters and one common density on outside of these co-clusters) given by plot parameter evolution. Finally, the figure on the right represents the slice reorganization based on the obtained co-clustering. We note that the three coclusters with higher parameter values in the previous plot, are the three diagonal co-clusters; for details see [START_REF] Boutalbi | Tensor latent block model for coclustering[END_REF].

TensorClus offers CPU and GPU compatibility. The CPU version uses the classical matrix operations from NumPy package. And for GPU, we rely to CuPy package wich is a NumPy-compatible array library accelerated by CUDA [START_REF] Okuta | Cupy: A numpycompatible library for nvidia gpu calculations[END_REF].

We compared the CPU and GPU versions of TSPLBM to evaluate computing time with both versions. In Figure 6 are reported the obtained results of CPU and GPU versions in performing 10 runs for each version. We observe that TensorClus shows a slight performance using GPU implementation. These performances can be improved using a more powerful GPU. The experimentation's source code is available in a public github repository of TensorClus. For future work, we intend to extend the library by introducing tri-clustering methods and targeting further improvements in performance using GPU computations.

Conclusion

Figure 1 :

 1 Figure 1: Objective of Tensor Co-clustering.

Figure 2 :

 2 Figure 2: TensorClus library structure.

  decomposition with clustering function is dedicated for this task. It has an argument algorithm for choosing which clustering algorithm among a list of suitable algorithms for the clustering of continuous data: Kmeans++, Spherical Kmeans, Spectral clustering (SC), and the EM algorithm derived from Gaussian Mixture Model (GMM) available in the Scikit-Learn package.

Figure 3 :

 3 Figure 3: Three-way Data structure

  and Maximization (M) step consists in updating model parameters Ω(t) . The parameter Ω is formed by proportions of row clusters π = (π 1 , . . . , π g ), proportions of column clusters ρ = (ρ 1 , . . . , ρ m ),

  TensorClus interfaces with Coclust. Therefore the user can also consider carrying out a co-clustering by slice. Coclust has been designed to complete and easily interface with popular machine learning libraires such as scikitlearn. Using the sliceMatrixCoclustering function of the co-clustering module, the user can perform different co-clustering algorithms with Coclust.

Figure 4 :

 4 Figure 4: Comparison results using DBLP1 dataset.

Figure 5 :

 5 Figure 5: Tensor co-clustering results analysis for DBLP1 dataset.

Figure 6 :

 6 Figure 6: Comparison results of CPU and GPU version of TSPLBM on the three datasets.

TensorClus

  is a Python library for three-way co-clustering. It is convenient and straightforward by proposing a panel of the tensor (co)-clustering methods, under a permissive license. It is simple and provides several tools for data loading and visualization. The library offers some illustrative examples to compare TensorClus with tensor-decomposition approaches combined to popular clustering methods. Thereby, the proposed implementation allows to easily interface with other python packages such as Numpy, Tensorly, TensorD and Coclust.

Table 1 :

 1 Characteristics of datasets.

	Datasets	Type	#Slices #Node #Cluster
	DBLP1	Text	4	1995	3
	DBLP2	Text	4	2223	3
	PubMed-Diabets-4K	Text	4	4354	3
	Nus-Wide-8	Text+Images	6	2738	8
	Amazon-Products-10 Text+Images	7	9897	10

https://github.com/boutalbi/TensorClus/blob/master/data_ description.md