A New Logical Semantics for Agent Communication

A New Logical Semantics for Agent Communication

Jamal Bentahar ${ }^{1}$, Bernard Moulin ${ }^{2}$, John-Jules Ch. Meyer ${ }^{3}$,
and Yves Lespérance ${ }^{4}$
${ }^{1}$ Concordia University, Concordia Institute for Information Systems Engineering
(CIISE), Canada
bentahar@encs.concordia.ca
${ }^{2}$ Laval University, Depart. of Computer Science and Software Engineering, Canada
bernard.moulin@ift.ulaval.ca
${ }^{3}$ University Utrecht, Depart. of Computer Science, The Netherlands
jj@cs.uu.nl
${ }^{4}$ York University, Depart. of Computer Science, Canada
lesperan@cs.yorku.ca

Abstract

In this paper we develop a semantics of our approach based on commitments and arguments for conversational agents. We propose a logical model based on CTL* (Extended Computation Tree Logic) and on dynamic logic. Called Commitment and Argument Network (CAN), our formal framework based on this hybrid approach uses three basic elements: social commitments, actions that agents apply to these commitments and arguments that agents use to support their actions. The advantage of this logical model is to gather all these elements and the existing relations between them within the same framework. The semantics we develop here enables us to reflect the dynamics of agent communication. It also allows us to establish the important link between commitments as a deontic concept and arguments. On the one hand CTL* enables us to express all the temporal aspects related to the handling of commitments and arguments. On the other hand, dynamic logic enables us to capture the actions that agents are committed to achieve.

1 Introduction

Recent years have seen an increasing interest in specifying and verifying multiagent systems (MAS) using computational logics [24]. Indeed, modeling agent interactions in logic-based MAS has attracted the attention of several researchers [716] (see [25] for a good synthesis). In this context, semantics of agent communication is one of the most important aspects, particularly in the current state of open and interoperable MAS. Although a certain number of significant proposals were done in this field, for example [11217232627, the definition of a clear and global semantics (i.e. dealing with the various aspects of agent communication) is an objective yet to be reached.

The objective of this paper is to propose a general framework capturing semantic issues of an approach based on social commitments (SCs) and arguments

for agent communication. Indeed, this work is a continuation of our preceding research in which we developed this approach and addressed in detail the pragmatic aspects [6. Pragmatics deals with the way of using the communicative acts correctly. It is related to the dynamics of agent interactions and to the way of connecting the isolated acts to build complete conversations. Thus, the paper highlights semantic issues of our approach and the link with pragmatic ones. The semantics we define here deals with all the aspects we use in our SC and argument approach. The purpose is to propose a complete, clear and unambiguous semantics for agent communication.

In addition to proposing a unified framework for pragmatic and semantic issues, this work presents two results: 1. it semantically establishes the link between SCs and arguments; 2. it uses both a temporal logic (CTL* with some additions) and a dynamic logic to define an unambiguous semantics. We notice here that the semantics presented in this paper is different from the one that we have developed in 89. The logical model presented here is more expressive because the content of SCs are path formulae and not state formulae and their semantics is expressed in terms of satisfaction paths and not in terms of deadlines. This makes the semantics more clear and easy to verify. In addition, this semantics expresses explicitly the relation between SCs and actions by using the philosophical literature on actions and by introducing the new Happens operator.

Paper overview. In Section 2, we introduce the main ideas of our pragmatic approach based on commitments and arguments. In Section 3, we present the syntax and the semantics of our logical model for agent communication. In Section 4, we conclude the paper by comparing our approach to related work.

2 Commitment and Argument-Based Approach

2.1 Social Commitments

A social commitment $S C$ is an engagement made by an agent (called the debtor), that some fact is true or that some action will be performed. This commitment is directed to a set of agents (called creditors). A commitment is an obligation in the sense that the debtor must respect and behave in accordance with this commitment. Commitments are social in the sense that they are expressed publicly and governed by some rules. This means that they are observable by all the participants. The main idea is that a speaker is committed to a statement when he made this statement or when he agreed upon this statement made by another participant and acts accordingly. What is important here is not that an agent agrees or disagrees upon a statement, but rather the fact that the agent expresses agreement or disagreement. Consequently, SCs are different from the agent’s private mental states like beliefs, desires and intentions. This notion allows us to represent agent conversations as observed by the participants and by an external observant, and not on the basis of the internal agents’ states.

We denote a SC as follows: $S C\left(A g_{1}, A^{*}, t, \varphi\right)$ where $A g_{1}$ is the debtor, $A^{*}$ is the set of the creditors $\left(A^{*}=A \backslash\left\{A g_{1}\right\}\right.$, where $A$ is the set of participants), $t$ is

the time associated with the commitment, and $\varphi$ its content. Logically speaking, a commitment is a public propositional attitude. The content of a commitment can be a proposition or an action. A detailed taxonomy of SCs that we use in our approach will be discussed later. To simplify the notation, we suppose throughout this paper that $A=\left\{A g_{1}, A g_{2}\right\}$.

In order to model the dynamics of conversations, we interpret a speech act as an action performed on a commitment or on a commitment content. A speech act is an abstract act that an agent, the speaker, performs when producing an utterance $U t$ and addressing it to another agent, the addressee. According to speech act theory, the primary units of meaning in the use of language are not isolated propositions but rather speech acts of the type called illocutionary acts. Assertions, questions, orders and declarations are examples of these illocutionary acts. In our framework, a speech act can be defined as follows.

Definition 1 (Speech Acts). $S A\left(i_{k}, A g_{1}, A g_{2}, t_{u t}, U t\right)={ }_{d e f}$

$$\begin{aligned} & \operatorname{Act}\left(A g_{1}, t_{u t}, S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \\ & \mid \operatorname{Act}-\operatorname{cont}\left(A g_{1}, t_{u t}, S C\left(A g_{i}, A g_{j}, t, \varphi\right)\right) \\ & \mid \operatorname{Act}\left(A g_{1}, t_{u t}, S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \& \\ & \operatorname{Act}-\operatorname{cont}\left(A g_{1}, t_{u t}, S C\left(A g_{i}, A g_{j}, t, \varphi\right)\right) \end{aligned}$$

where $S A$ is the abbreviation of “Speech Act”, $i_{k}$ is the identifier of the speech act, $A g_{1}$ is the speaker, $A g_{2}$ is the addressee, $t_{u t}$ is the utterance time, $U t$ is the utterance, Act indicates the action performed by the speaker on the commitment: Act $\in\{$ Create, Withdraw, Violate, Satisfy $\}$, Act-cont indicates the action performed by the speaker on the commitment content: Act-cont $\in\{$ Acceptcont, Refuse-cont, Chal-cont, Justify-cont, Defend-cont, Attack-cont}, $i, j \in\{1,2\}, i \neq j$, the meta-symbol " $\&$ " indicates the logical conjunction.

The definiendum $S A\left(i_{k}, A g_{1}, A g_{2}, t_{u t}, U t\right)$ is defined by the definiens $\operatorname{Act}\left(A g_{1}, t_{u t}\right.$, $\left.S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right)$ as an action performed by the speaker on its SC. The definiendum is defined by the definiens $\operatorname{Act}-\operatorname{cont}\left(A g_{1}, t_{u t}, S C\left(A g_{i}, A g_{j}, t, \varphi\right)\right)$ as an action performed by the speaker on the content of its SC $(i=1, j=2)$ or on the content of the addressee’s SC $(i=2, j=1)$. Finally, the definiendum is defined as an action performed by the speaker on its SC and as an action performed by the speaker on the content of its SC or on the content of the addressee’s SC. These actions are similar to the moves proposed in [22]. The following example illustrates this idea.

Example 1. Let us consider the following utterances:
$U t_{1}$ : Quebec is the capital of Canada
$U t_{2}$ : No, the capital of Canada is Ottawa
The utterance $U t_{1}$ leads to the creation of a new commitment:

$$\begin{aligned} & S A\left(i_{0}, A g_{1}, A g_{2}, t_{u t_{1}}, U t_{1}\right)={ }_{d e f} \\ & \quad \text { Create }\left(A g_{1}, t_{u t_{1}}, S C\left(A g_{1}, A g_{2}, t_{1}, \text { Capital }(\text { Canada, Quebec })\right)\right) \end{aligned}$$

The utterance $U t_{2}$ leads at the same time to an action performed on the content of the commitment created following the utterance $U t_{1}$ and to the creation of another commitment. Formally:

$$\begin{aligned} & S A\left(i_{1}, A g_{2}, A g_{1}, t_{u t_{2}}, U t_{2}\right)={ }_{d e f} \\ & \text { Refuse-cont }\left(A g_{2}, t_{u t_{2}}, S C\left(A g_{1}, A g_{2}, t_{1}, \text { Capital }(\text { Canada }, \text { Quebec })\right)\right) \& \\ & \text { Create }\left(A g_{2}, t_{u t_{2}}, S C\left(A g_{2}, A g_{1}, t_{2}, \text { Capital }(\text { Canada }, \text { Ottawa })\right)\right) \end{aligned}$$

2.2 Taxonomy

The types of commitments we use in our agent communication framework are:
A. Absolute Commitments ( $A B C$ ). Absolute commitments are commitments whose fulfillment does not depend on any particular condition. Two types can be distinguished: propositional commitments and action commitments.
A1. Propositional Commitments (PC). Propositional commitments are related to the state of the world. They are generally, but not necessarily, expressed by assertives. They can be directed towards the past, the present, or the future.
A2. Action Commitments ( $A C$ ). Action commitments (also called commitments to a course of action) are directed towards the present or the future and are related to actions that the debtor is committed to carry out. The fulfillment and the lack of fulfillment of such commitments depend on the performance of the underlying action. This type of commitment is typically conveyed by promises.
B. Conditional Commitments (CC). Absolute commitments do not consider conditions that may make relative the need for their fulfillment. However, in several cases, agents need to make commitments not in absolute terms but under given conditions. Another commitment type is therefore required. These commitments are said to be conditional. We distinguish between conditional commitments about propositions (CCP) and conditional commitments about actions (CCA). A conditional commitment about a proposition $p^{\prime}$ under the condition $p$ expresses the fact that if the condition $p$ is true, then the creditor commits that the content $p^{\prime}$ is true (there is an implication link between $p$ and $p^{\prime}$ ).
C. Commitment Attempts (CT). The commitments described so far directly concern the debtor who commits either that a certain fact is true or that a certain action will be performed. For example, these commitments do not allow us to explain the fact that an agent asks another one to be committed to carrying out an action (by a speech act of a directive type). To solve this problem, we propose the concept of commitment attempt. We consider a commitment attempt as a request made by a debtor to push a creditor to be committed. Thus, when an agent $A g_{1}$ requests another agent $A g_{2}$ to do something, we say that $A g_{1}$ is trying to induce $A g_{2}$ to make a commitment. We distinguish four types of commitment attempts: propositional commitment attempts (PCT), action commitment attempts (ACT), conditional commitment attempts about propositions (CCTP), and conditional commitment attempts about actions (CCTA).

2.3 Argumentation and Social Commitments

Contrary to monotonic logics, in a nonmonotonic logic, adding premises can lead to the withdrawal of conclusions. Argumentation systems are studied in the context of this kind of reasoning. Several models of defeasible argumentation have been proposed [13|14|19|21]. In these models, adding arguments can lead to the defeat of arguments. An argument is defeated if it is attacked successfully by a counterargument. A defeasible argumentation system essentially includes a logical language $\mathcal{L}$, a definition of the argument concept, a definition of the attack relation between arguments, and finally a definition of acceptability. Here $\Gamma$ indicates a possibly inconsistent knowledge base with possibly no deductive closure (that is the deductive closure is not necessarily included in $\Gamma$ ), and $\vdash$ stands for classical inference. The propositions of the language $\mathcal{L}$ are denoted by $a, b, \ldots$

Definition 2 (Argument). An argument is a pair $(H, h)$ where $h$ is a formula of $\mathcal{L}$ and $H$ a subset of $\Gamma$ such that: $i$ ) $H$ is consistent, ii) $H \vdash h$ and iii) $H$ is minimal, so that no subset of $H$ satisfying both $i$ and ii exists. $H$ is called the support of the argument and $h$ its conclusion.

Example 2. Let $\Gamma=\{a, a \rightarrow b, c \rightarrow \neg b, c\}$. Then, $(\{a, a \rightarrow b\}, b)$ and $(\{a \rightarrow$ $b\}, \neg a \vee b)$ are two arguments.

Definition 3 (Attack). Let $\left(H_{1}, h_{1}\right),\left(H_{2}, h_{2}\right)$ be two arguments. $\left(H_{1}, h_{1}\right)$ attacks $\left(H_{2}, h_{2}\right)$ iff $H_{2} \vdash \neg h_{1}$. In other words, an argument is attacked if and only if there exists an argument for the negation of its conclusion.

Example 3. Let $\Gamma=\{a, a \rightarrow b, c \rightarrow \neg b, c, \neg b \rightarrow \neg d, \neg c\}$. Then, the argument $(\{a, a \rightarrow b\}, b)$ attacks the argument $(\{c, c \rightarrow \neg b, \neg b \rightarrow \neg d\}, \neg d)$ and also the argument $(\{\neg c\}, \neg c)$ attacks the argument $(\{c, c \rightarrow \neg b, \neg b \rightarrow \neg d\}, \neg d)$.

The link between commitments and arguments enables us to capture both the public and reasoning aspects of agent communication. This link is explained as follows. Before committing to some fact $h$ being true (i.e. before creating a commitment whose content is $h$ ), the speaker agent must use its argumentation system to build an argument $(H, h)$. On the other side, the addressee agent must use its own argumentation system to select the answer it will give (i.e. to decide about the appropriate manipulation of the content of an existing commitment). For example, an agent $A g_{1}$ accepts the commitment content $h$ proposed by another agent $A g_{2}$ if it is able to build an argument which supports this content from its knowledge base. If $A g_{1}$ has an argument $\left(H^{\prime}, \neg h\right)$, then it refuses the commitment content proposed by $A g_{2}$ by attacking the conclusion $h$. Now, if $A g_{1}$ has an argument neither for $h$, nor for $\neg h$, then it must ask for an explanation. The social relationship that exists between agents, their reputations and trusts also influence the acceptance of the arguments by agents. However, this aspect will not be dealt with in this paper. The argumentation relations that we use in our model are thought of as actions applied to commitment contents. The set of these relations is: $\{$ Justify, Defend, Attack $\}$.

We used this approach in [6] to propose a formal framework called Commitment and Argument Network (CAN). The idea of this formalism is to reflect the dynamics of agent communication by a network in which agents manipulate commitments and arguments. In the following section, we will propose a formal semantics of our framework in the form of a logical model. The elements that we use in our framework and in our logical model are: SCs, actions and arguments. These elements are separated in three levels. The first level includes SCs that agents use in their conversations. The second level includes actions that agents apply to the commitments. The third level is composed of arguments that agents use to support their actions applied to the commitments.

3 The Logical Model of Agent Communication

3.1 Syntax

In this section we specify the syntax of the different elements that we use in our agent communication framework. These elements are: propositional elements, actions, social commitments, actions applied to commitments and argumentation relations. Our formal language $L$ (the object language) is based on an extended version of CTL* 15] and on dynamic logic [18. We use a branching time for the future and we suppose that the past is linear. Each node in the branching time model is represented by a state $s_{i}$ and a time point $t_{j}$. We also suppose that time is discrete. In our model, temporal logic enables us to express all the temporal aspects related to the handling of commitments and arguments. On one hand, we use the branching time in order to formalize the different choices that agents have when they participate in conversations. On the other hand, dynamic logic allows us to capture the actions that agents are committed to perform and the actions that agents perform on different commitments and commitment contents when they participate in these conversations. Indeed, from a philosophical point of view, action and branching time are logically related 4. The agents’ actions are not fully determined. Moreover, these actions can have many different possible future effects. For this reason, it is preferable to work out a logic of action that is compatible with indeterminism. According to indeterminism, several moments of time might follow the same moment in the future of the world. Any moment of time can belong to several paths representing possible courses of the world with the same past and present but different historic continuations of that moment.

Let $\Phi_{p}$ be the set of atomic propositions and $\Phi_{a t}$ be the set of atomic actions. The set $\Phi_{a}$ denotes a set of complex actions (a complex action is composed of a number of atomic actions). The set of agents is denoted by $A$ and the set of time points is denoted by $T P$. The agents’ actions on commitments and on their contents and the argumentation relations are introduced as modal operators. In this paper, a commitment formula, independently of the SC type, is denoted by $S C\left(A g_{1}, A g_{2}, t, \varphi\right)$ where $\varphi$ is a well-formed formula of $L$. When $t$ is unknown (unspecified), we drop it from the commitment formula. In this case a commitment is denoted by $S C\left(A g_{1}, A g_{2}, *, \varphi\right)$. In this logical model we use the symbol $\wedge$ in the object language and the symbol $\&$ in the metalanguage for

“and”. For “or” we use the symbol $\vee$ in the object language and the symbol | in the metalanguage. The language $L$ can be defined by the following syntactic rules.

Propositional Elements

R1 (Atomic formula). $\forall \psi \in \Phi_{p}, \psi \in L$
R2 (Conjunction). $p, q \in L \Rightarrow p \wedge q \in L$
R3 (Negation). $p \in L \Rightarrow \neg p \in L$
R4 (Arguments). $p, q \in L \Rightarrow p \therefore q \in L$
R4 means that $p$ is an argument for $q$. We can read this formula: $p$, so $q$. We notice that the difference between the operator $\therefore$ and the formal notation of an argument as a pair (see Definition 2) is that $\therefore$ is a language construct. In the notation $p \therefore q, p$ is a formula of the language, however in the notation $(H, h)$, $H$ is a sub-set of a knowledge base.

Example 4. Let $p$ and $q$ be two formulae of $L$. Then, $(p \wedge p \rightarrow q) \therefore q$ is a formula of $L$ saying that $p \wedge p \rightarrow q$ is an argument for $q$.

We notice that the property of defeasibility of arguments does not appear at this level. The reason is that R4 introduces only argumentation as a logical relation between propositions. As Prakken and Vreeswijk argued, argumentation systems are able to incorporate the monotonic notions of logical consequence as a special case in their definition of what an argument is 21]. In our model, we capture the property of defeasibility by the argumentation relations (attack, defense, justification).

R5 (Universal path-quantifier). $p \in L \Rightarrow A p \in L$
R6 (Existential path-quantifier). $p \in L \Rightarrow E p \in L$
R7 (Until). $p, q \in L \Rightarrow p U^{+} q \in L$
R8 (Next moment). $p \in L \Rightarrow X^{+} p \in L$
R9 (Since). $p, q \in L \Rightarrow p U^{-} q \in L$
R10 (Previous moment). $p \in L \Rightarrow X^{-} p \in L$
Informally, $A p$ means that $p$ holds along all paths, $E p$ means that there is a path along which $p$ holds, and $p U^{+} q(p$ until $q$ ) means that on a given path from the given moment, there is some future moment in which $q$ will eventually hold and $p$ holds at all moments until that future moment. $X^{+} p$ holds at the current moment, if $p$ holds at the next moment. The intuitive interpretation of $p U^{-} q(p$ since $q)$ is that on a given path from the given moment, there is some past moment in which $q$ eventually held and $p$ holds at all moments since that past moment. $X^{-} p$ holds at the current moment, if $p$ held at the previous moment.

Actions

R11 (Action performance). $\alpha \in \Phi_{a t} \Rightarrow$ Happens $(\alpha) \in L$
R12 (Sequential composition). $\alpha \in \Phi_{a t}, \alpha^{\prime} \in \Phi_{a} \Rightarrow$ Happens $\left(\alpha ; \alpha^{\prime}\right) \in L$
R13 (Non-deterministic choice). $\alpha, \alpha^{\prime} \in \Phi_{a} \Rightarrow$ Happens $\left(\alpha \mid \alpha^{\prime}\right) \in L$
R14 (Test). $p \in L \Rightarrow$ Happens $(p ?) \in L$

Happens( $\alpha$ ) is an operator from dynamic logic. It allows us to represent the actions that agents perform and the effects of these actions. It expresses the fact that the action symbol $\alpha$ happens. We also introduce constructors for action expression which are similar to the constructors used in dynamic logic. Happens $\left(\alpha ; \alpha^{\prime}\right)$ allows us to combine two action symbols. It means that $\alpha$ is followed by $\alpha^{\prime}$. Happens $\left(\alpha \mid \alpha^{\prime}\right)$ means that $\alpha$ or $\alpha^{\prime}$ happens. Happens $(p ?)$ allows us to build action expressions that depend on the truth or falsity of $p$. This constructor is interesting to express the fact that by way of performing actions, agents bring about facts in the world [11]. This fact can be expressed as follows: Happens $(\alpha ; p ?)$. This formula says that after $\alpha$ is performed, $p$ becomes true.

Creation of Social Commitments

In the rest of this paper we assume that $\left\{A g_{1}, A g_{2}\right\} \subseteq A$ and $t \in T P$
$\mathrm{R} 15(\mathrm{CrPC}) . p \in L \Rightarrow \operatorname{Create}\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t, p\right)\right) \in L$
R16 (CrAC). $\alpha \in \Phi_{a} \& p \in L \Rightarrow \operatorname{Create}\left(A g_{1}, A C\left(A g_{1}, A g_{2}, t,(\alpha, p)\right)\right) \in L$
R17 (CrCCP). $p, p^{\prime} \in L \Rightarrow \operatorname{Create}\left(A g_{1}, C C P\left(A g_{1}, A g_{2}, t,\left(p, p^{\prime}\right)\right)\right) \in L$
R18 (CrCCA). $\alpha \in \Phi_{a} \& p, p^{\prime} \in L \Rightarrow$

$$\operatorname{Create}\left(A g_{1}, C C A\left(A g_{1}, A g_{2}, t,\left(p,\left(\alpha, p^{\prime}\right)\right)\right)\right) \in L$$

$C r P C$ formula says that $A g_{1}$ commits towards $A g_{2}$ at the moment $t$ that $p$ is true. $C r A C$ formula says that $A g_{1}$ commits towards $A g_{2}$ to do $\alpha$, and by doing $\alpha, p$ becomes true. $C r C C P$ formula says that if $p$ is true, $A g_{1}$ commits towards $A g_{2}$ that $p^{\prime}$ is true. $C r C C A$ formula says that if $p$ is true, $A g_{1}$ commits towards $A g_{2}$ to do $\alpha$, and by doing $\alpha, p^{\prime}$ becomes true.

Commitment attempts. In order to formally introduce the notion of commitment attempt (syntax and semantics) we introduce the following definition.

Definition 4 (some predicate). Let $x$ be a variable term and $c_{1}, \ldots, c_{n}$ be constant terms. A constant term can be a number, a name, etc. some $\left(x,\left\{c_{1},, c_{n}\right\}, p(x)\right)=_{\text {def }} p\left(c_{1}\right) \vee \ldots \vee p\left(c_{n}\right)$

Example 5. Let $P R C$ be a predicate indicating the price of a given car, for example Mazda3, and let $x$ be a variable representing this price. Then, some $\left(x,\{20 K \$, 25 K \$, 30 K \$\}, \operatorname{PRC}(\operatorname{Mazda3}, x)\right)=_{\text {def }}$

$$\operatorname{PRC}(\operatorname{Mazda3}, 20 K \$ ) \vee \operatorname{PRC}(\operatorname{Mazda3}, 25 K \$ ) \vee \operatorname{PRC}(\operatorname{Mazda3}, 30 K \$)$$

In this example, the predicate some indicates that the price of a Mazda3 is in the set $\{20 K \$, 25 K \$, 30 K \$\}$.

We can define the syntax of propositional commitment attempts, action commitment attempts, conditional commitment attempts about propositions and conditional commitment attempts about actions as follows.

R19 (CrPCT). $p \in L \Rightarrow$
$\operatorname{Create}\left(A g_{1}, P C T\left(A g_{1}, A g_{2}, t\right.\right.$, some $\left.\left.\left(x,\left\{c_{1}, \ldots, c_{n}\right\}, p(x)\right)\right)\right) \in L$
R20 (CrACT). $\alpha \in \Phi_{a} \& p \in L \Rightarrow \operatorname{Create}\left(A g_{1}, A C T\left(A g_{1}, A g_{2}, t,(\alpha, p)\right)\right) \in L$

$$\begin{aligned} & \mathrm{R} 21(\mathrm{CrCCTP}) . p, p^{\prime} \in L \Rightarrow \\ & \quad \operatorname{Create}\left(A g_{1}, C C T P\left(A g_{1}, A g_{2}, t,\left(p, \operatorname{some}\left(x,\left\{c_{1}, \ldots, c_{n}\right\}, p^{\prime}(x)\right)\right)\right)\right) \in L \\ & \mathrm{R} 22(\mathrm{CrCCTA}) . \alpha \in \Phi_{\alpha} \& p, p^{\prime} \in L \\ & \quad \Rightarrow \operatorname{Create}\left(A g_{1}, C C T A\left(A g_{1}, A g_{2}, t,\left(p,\left(\alpha, p^{\prime}\right)\right)\right)\right) \in L \end{aligned}$$

$\operatorname{CrPCT}$ formula says that $A g_{1}$ asks $A g_{2}$ at the moment $t$ to commit about some $\left(x,\left\{c_{1},, c_{n}\right\}, p(x)\right) . \operatorname{CrACT}$ formula says that $A g_{1}$ asks $A g_{2}$ to commit to perform $\alpha$, which makes $p$ true. $\operatorname{CrCCTP}$ and $\operatorname{CrCCTA}$ formulae are conditional. Their meanings are intuitive.

Agent’s Desire about a Commitment from the Addressee

R23 (Want PC). $p \in L \Rightarrow$ Want $_{P}\left(A g_{1}, P C\left(A g_{2}, A g_{1}, t, p\right)\right) \in L$
This formula means that $A g_{1}$ wants that $A g_{2}$ commits that $p$ is true. The Want $_{P}$ modality will be used to define the semantics of open questions. In the same way we can define the Want $_{P}$ modality for the other commitment types.

Action Occurrences applied to Commitments

We use the abbreviation $S C\left(A g_{1}, A g_{2}, t, \varphi\right)$ to indicate a SC. The syntactical form of the commitment content $\varphi$ depends of the commitment type. For example, for a $\mathrm{PC}, \varphi$ has the syntactical form of $p$, and for an $\mathrm{AC}, \varphi$ has the syntactical form of $(\alpha, p)$, etc.

R24 (Withdrawal). $\varphi \in L \Rightarrow$ Withdraw $\left(A g_{1}, S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \in L$
R25 (Satisfaction). $\varphi \in L \Rightarrow \operatorname{Satisfy}\left(A g_{1}, S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \in L$
R26 (Violation). $\varphi \in L \Rightarrow \operatorname{Violate}\left(A g_{1}, S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \in L$
R27 (Non-persistence). $\varphi \in L \Rightarrow N$ Persist $\left(S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \in L$
R28 (Active). $\varphi \in L \Rightarrow \operatorname{Active}\left(S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \in L$
The notion of non-persistence is used to define the semantics of the withdrawal action. The commitment becomes not persistent after withdrawal. The notion of active commitment is defined in terms of the withdrawal. This notion is used to define the semantics of acceptation, refusal, and challenge. For example, a commitment cannot be accepted if it is not active.

Action Occurrences applied to Commitment Contents

R29 (Acceptation). $\varphi \in L \Rightarrow$ Accept-cont $\left(A g_{2}, S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \in L$
R30 (Refusal). $\varphi \in L \Rightarrow$ Refuse-cont $\left(A g_{2}, S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \in L$
R31 (Challenge). $\varphi \in L \Rightarrow$ Chal-cont $\left(A g_{2}, S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \in L$

Argumentation Relations

R32 (Attack). $p, p^{\prime} \in L \Rightarrow$ Attack-cont $\left(A g_{2}, P C\left(A g_{1}, A g_{2}, t, p\right), p^{\prime}\right) \in L$
R33 (Defense). $p, p^{\prime} \in L \Rightarrow$ Defend-cont $\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t, p\right), p^{\prime}\right) \in L$
R34 (Justification). $p, p^{\prime} \in L \Rightarrow$ Justify - cont $\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t, p\right), p^{\prime}\right) \in L$

Attack formula says that $A g_{2}$ attacks the $A g_{1}$ 's commitment by using the content $p^{\prime}$. Defense and justification formulae are defined by the same way.

Agent’s Desire about the Justification of a Commitment Content from the Addressee

R35 (Want Justify). $p \in L \Rightarrow$

$$\text { Want }_{J}\left(A g_{1}, \text { Justify }-\operatorname{cont}\left(A g_{2}, P C\left(A g_{2}, A g_{1}, t, p\right)\right)\right) \in L$$

This formula means that $A g_{1}$ wants that $A g_{2}$ justifies its commit content. This formula will be used to define the semantics of the challenge action.

3.2 Semantics

In this section, we define the formal model in which we evaluate the well-formed formulae of our framework. Thereafter, we give the semantics of the different elements that we specified syntactically in the previous section.

The Formal Model

Let $S$ be a set of states and $R \subseteq S \times S$ be a transition relation indicating branching time. A path $P a$ is an infinite sequence of states $\left\langle s_{0}, s_{1}, \ldots\right\rangle$ where: $\forall i \in \mathbb{N},\left(s_{i}, s_{i+1}\right) \in R$ and $T\left(s_{i+1}\right)=T\left(s_{i}\right)+1$. The function $T$ gives us for each state $s_{i}$ the corresponding moment $t$ (this function will be specified later).

We use the notation $s_{i} \mid P a$ to indicate that the state $s_{i}$ belongs to the path $P a$ (i.e. $s_{i}$ appears in the sequence $\left\langle s_{0}, s_{1}, \ldots\right\rangle$ that describes the path $P a$ ). We denote the set of all paths by $\sigma$. The set of all paths traversing the state $s_{i}$ are denoted by $\sigma^{s_{i}}$. We suppose that all paths start from $s_{0}\left(T\left(s_{0}\right)=0\right)$. In our vision of branching future, we can have several states at the same moment. Along a given path (for example the real path) there is one and only one state at one moment. Indeed, in our framework, $s_{i}$ does not indicate (necessarily) the state at moment $i$.

The formal model for $L$ is defined as follows: $M=\langle S, R, A, T P, N p, F a p, T$, $G a p, R s c, R w\rangle$ where: $S$ is a nonempty set of states; $R \subseteq S \times S$ is the transition relation; $A$ is a nonempty set of agents; $T P$ is a nonempty set of time points; $N p: S \rightarrow 2^{\Phi_{p}}$ is a function relating each state $s \in S$ to the set of the atomic propositions that are true in this state; $F a p: S \times \Phi_{a t} \rightarrow 2^{S}$ is a function that gives us the state transitions caused by the achievement of an action; $T: S \rightarrow T P$ is a function associating to any state $s_{i}$ the corresponding time; Gap : $A \times S \rightarrow 2^{\Phi_{a}}$ is a function that gives us for each agent the set of performed actions in a given state; $R s c: A \times A \times S \rightarrow \wp(\sigma)$ is a function producing the accessibility modal relations for SCs $(\wp(\sigma)$ is the powerset of paths $) ; R w: A \times A \times S \rightarrow \wp(\sigma)$ is a function producing the accessibility modal relations for agent’ desires about the commitments of the addressee.

The function Fap allows us to indicate the accessible states from a given state by transitions labeled with actions. This represents the Chellas’ view 11]: to each moment $m$ there corresponds the set of alternative moments which are compatible with all the actions that an agent performs at moment $m$. We notice

that $s_{j} \in \operatorname{Fap}\left(s_{i}, \alpha\right)$ does not necessarily imply that $\left(s_{i}, s_{j}\right) \in R$. The function $R s c$ associates to a state $s_{i}$ the set of paths along which an agent commits towards another agent. These paths are conceived as merely “possible”, and as paths where the contents of commitments made in a given state should be true. For example, if we have: $P a \in R s c\left(A g_{1}, A g_{2}, s_{i}\right)$, then this means that the commitments that are made in the state $s_{i}$ by $A g_{1}$ towards $A g_{2}$ should be satisfied along the path $P a$. The function $R w$ gives us the paths along which an agent wants that the addressee commits or justifies its commitment. These paths represent the agents’ desires about the addressees’ commitments. This accessibility modal relation will be used to define the semantics of the commitment attempts and the challenge of a commitment attempt. The logic of absolute and conditional commitments (using $R s c$ ) is a KD4 modal logic and the logic of commitment attempts (using $R w$ ) is a KD modal logic (see [5] for more details).

As in CTL*, we have in our model path formulae and state formulae. The notation $M, s_{i} \models \Psi$ indicates that the formula $\Psi$ is evaluated in the state $s_{i}$ of the model $M$. The notation $M, P a, s_{i} \models \Psi$ indicates that the formula $\Psi$ is evaluated at the state $s_{i}$ along the path $P a$ where $s_{i} \mid P a$ (we recall here that $s_{i} \mid P a$ means that the state $s_{i}$ belongs to the path $P a$ ). We can now define the semantics of the elements of $L$ in the model $M$. For space limit reasons, we only define the semantics of formulae that are not in CTL*. In addition, in the examples of this section, we suppose that the model $M$ is given. Thus, we only illustrate the path along which a formula is satisfied.

Arguments

$M, s_{i} \models p \therefore q$ iff $M, s_{i} \models p \&\left(\forall M^{\prime} \in \mathcal{M} \& \forall s_{j} \in S_{M^{\prime}} M^{\prime}, s_{j} \models p \Rightarrow M^{\prime}, s_{j} \models q\right)$ where $\mathcal{M}$ is the set of models, and $S_{M^{\prime}}$ is the set of states of the model $M^{\prime}$.

We add the first clause to capture the following aspect: when an agent presents an argument $p$ for $q$ for this agent $p$ is true and if $p$ is true then $q$ is true.

Actions

$M, P a, s_{i} \models$ Happens $(\alpha)$ iff $\exists s_{j}: s_{j} \mid P a \& s_{j} \in \operatorname{Fap}\left(s_{i}, \alpha\right)$
$M, P a, s_{i} \models$ Happens $\left(\alpha ; \alpha^{\prime}\right)$ iff
$\exists s_{j}: s_{j} \mid P a \& s_{j} \in \operatorname{Fap}\left(s_{i}, \alpha\right) \& M, P a, s_{j} \models$ Happens $\left(\alpha^{\prime}\right)$
$M, P a, s_{i} \models$ Happens $\left(\alpha \mid \alpha^{\prime}\right)$ iff $M, P a, s_{i} \models$ Happens $(\alpha) \vee$ Happens $\left(\alpha^{\prime}\right)$
$M, P a, s_{i} \models$ Happens $(p ?)$ iff $M, P a, s_{i} \models p$
Happens $(\alpha)$ is satisfied in the model $M$ iff there is an accessible state $s_{j}$ from the current state $s_{i}$ using the function $F a p$. Thus, this formula is satisfied iff there is a labeled transition with $\alpha$ between $s_{i}$ and a state $s_{j}$. To express the fact that a formula becomes true only after the performance of an action, we introduce the following abbreviation: Happens ${ }^{\prime}(\alpha ; p ?)=_{\text {def }} \neg p \wedge$ Happens $(\alpha ; p ?)$

Example 6. The formula Happens $(\alpha)$ is satisfied along the path $P a$ from the state $s_{i}$ as illustrated in Fig. 1

Fig. 1. Happens $(\alpha)$ along the path $P a$

Creation of Social Commitments

$M, s_{i} \models \operatorname{Create}\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t, p\right)\right)$ iff

$$\begin{aligned} & \exists \alpha \in \operatorname{Gap}\left(A g_{1}, s_{i}\right): \\ & \forall P a \operatorname{Pa} \in \operatorname{Rsc}\left(A g_{1}, A g_{2}, s_{i}\right) \Rightarrow \\ & \quad \exists s_{j}\left[P a: T\left(s_{j}\right)=T\left(s_{i}\right) \& M, P a, s_{j} \models \operatorname{Happens}(\alpha ; p ?) \& t=T\left(s_{i}\right)\right. \end{aligned}$$

We notice here that we evaluate the formula Happens $(\alpha ; p ?)$ along an accessible path $P a$ at a state $s_{j}$ that can be different from the current state $s_{i}$. This allows us to model agents’ uncertainty about this current state. This means that we do not assume that agents know the current state. However, we assume that these agents know which time is associated to each state. The semantics of propositional commitments is defined in terms of accessible paths. The commitment is satisfied in a model at a state $s_{i}$ iff there is an action $\alpha$ performed by $A g_{1}$ and this performance makes true the commitment content along all accessible paths and if the commitment moment is equal to the time associated to the current state. This semantics highlights the fact that committing is an action in itself. Indeed, the action $\alpha$ corresponds to the agent’s utterance which creates the commitment.

Example 7. The formula $\operatorname{Create}\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t, p\right)\right)$ is satisfied at the state $s_{i}$ as illustrated in Fig. 2

Fig. 2. Create $\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t, p\right)\right)$ at the state $s_{i}$

$$\begin{aligned} & M, s_{i} \models \operatorname{Create}\left(A g_{1}, A C\left(A g_{1}, A g_{2}, t,(\alpha, p)\right)\right) \text { iff } \\ & \quad M, s_{i} \models \operatorname{Create}\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t, F^{+} \text {Happens }^{\prime}(\alpha ; p ?)\right)\right) \& \\ & \quad \forall P a P a \in \operatorname{Rsc}\left(A g_{1}, A g_{2}, s_{i}\right) \Rightarrow \exists s_{j}\left[P a: T\left(s_{j}\right) \geq T\left(s_{i}\right) \& \alpha \in \operatorname{Gap}\left(A g_{1}, s_{j}\right)\right. \end{aligned}$$

This formula indicates that $A g_{1}$ is committed towards $A g_{2}$ to do $\alpha$ and that along all accessible paths $P a$ performing $\alpha$ makes $p$ true. The semantics we give to the commitments requires their fulfillment. Thus, if it is created, a commitment must be held. This satisfaction-based semantics reflects the idea of "prior

possible choices of agents" that Belnap and Perloff used in their logic of agency [3]. In this logic, agents make choices in time. In our model, these choices are represented by the commitments created by these agents. The notion of acting or choosing at a moment $m$ is thought of in Belnap and Perloff’s logic as constraining the course of events to lie within some particular subset of the possible histories available at that moment. This subset of the possible histories is represented by the set of paths along which the commitment must be satisfied. However, it is always possible to violate or withdraw such a commitment. For this reason, these two operations are explicitly included in our framework.

$$\begin{aligned} & M, s_{i} \models \operatorname{Create}\left(A g_{1}, C C P\left(A g_{1}, A g_{2}, t,\left(p, p^{\prime}\right)\right)\right) \text { iff } \\ & \forall P a \in \sigma^{s_{i}} \& \forall s_{j}\left[P a: T\left(s_{j}\right) \geq T\left(s_{i}\right)\right. \\ & \left(M, s_{j} \models p \Rightarrow M, s_{j} \models \operatorname{Create}\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t, p^{\prime}\right)\right)\right) \\ & M, s_{i} \models \operatorname{Create}\left(A g_{1}, C C A\left(A g_{1}, A g_{2}, t,\left(p,\left(\alpha, p^{\prime}\right)\right)\right)\right) \text { iff } \\ & \forall P a \in \sigma^{s_{i}} \& \forall s_{j}\left[P a: T\left(s_{j}\right) \geq T\left(s_{i}\right)\right. \\ & \left(M, s_{j} \models p \Rightarrow M, s_{j} \models \operatorname{Create}\left(A g_{1}, A C\left(A g_{1}, A g_{2}, t,\left(\alpha, p^{\prime}\right)\right)\right)\right) \end{aligned}$$

A conditional commitment about a proposition $p^{\prime}$ under condition $p$ is satisfied in the model iff the debtor commits that $p^{\prime}$ is true if the condition $p$ is true.

$$\begin{aligned} M, s_{i} \models & \operatorname{Want}_{P}\left(A g_{1}, P C\left(A g_{2}, A g_{1}, t, p\right)\right) \text { iff } \forall P a \operatorname{Pa} \in \operatorname{Rw}\left(A g_{1}, A g_{2}, s_{i}\right) \Rightarrow \\ & \exists s_{j}\left[P a: T\left(s_{j}\right)=T\left(s_{i}\right) \& M, P a, s_{j} \models \operatorname{Create}\left(A g_{2}, P C\left(A g_{2}, A g_{1}, t, p\right)\right)\right. \end{aligned}$$

$A g_{1}$ 's desire about a propositional commitment of $A g_{2}$ whose content is $p$ is satisfied in the model iff along all accessible paths via $R w, A g_{2}$ commits towards $A g_{1}$ that $p$. In the same way we can define the semantics of an agent’s desire about the other commitment types.

By using this formula, we can define the semantics of propositional commitment attempts as follows:

$$\begin{aligned} & M, s_{i} \models \operatorname{Create}\left(A g_{1}, P C T\left(A g_{1}, A g_{2}, t, \operatorname{some}\left(x,\left\{c_{1}, \ldots, c_{n}\right\}, p(x)\right)\right)\right) \text { iff } \\ & M, s_{i} \models \operatorname{Create}\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t\right.\right. \\ & \left.\left.\operatorname{Want}_{P}\left(A g_{1}, P C\left(A g_{2}, A g_{1}, *, p\left(c_{1}\right) \vee \ldots \vee p\left(c_{n}\right)\right)\right)\right)\right) \end{aligned}$$

The $A g_{1}$ 's propositional commitment attempt towards $A g_{2}$ is satisfied in the model iff $A g_{1}$ commits that it wants that $A g_{2}$ commits at a certain moment, which is not necessarily specified, that one of the propositions $p\left(c_{i}\right)$ is true. This notion of commitment attempt captures open and yes/no questions. We recall that the time argument is dropped from the propositional commitment formula because the moment at which $A g_{1}$ wants that $A g_{2}$ commits is not specified.

Example 8. Let us consider the example of an agent $A g_{1}$ asking an agent $A g_{2}$ about the price of a Mazda3. This request is captured in our model as a commitment attempt created by $A g_{1}$ towards $A g_{2}$ whose content can be expressed using the some predicate. If we assume that this price is in the set $\{20 K \$ , 25 K \$ , 30 K \$\}$, then we have:

\(\operatorname{Create}\left(A g_{1}, \operatorname{PCT}\left(A g_{1}, A g_{2}, t\right.\right.\)
    \(\operatorname{some}(x,\{20 K \$, 25 K \$, 30 K \$\}, \operatorname{PRC}(\operatorname{Maxda3}, x)))) \equiv\)
\(\operatorname{Create}\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t, \operatorname{Want}_{P}\left(A g_{1}, P C\left(A g_{2}, A g_{1}, *\right.\right.\right.\right.\)
    \(\left.\left.\operatorname{PRC}(\operatorname{Maxda3}, 20 K \$\}\vee \operatorname{PRC}(\operatorname{Maxda3}, 25 K \$\}\vee \operatorname{PRC}(\operatorname{Maxda3}, 30 K \$\right)))\right)\)

According to the semantics of propositional commitment attempts, this example is interpreted as an $A g_{1}$ 's propositional commitment that it wants that $A g_{2}$ commits about the price of the Mazda3.

The semantics of action commitment attempts is also defined using the Want $_{P}$ formula as follows:

$$\begin{aligned} & M, s_{i} \models \operatorname{Create}\left(A g_{1}, A C T\left(A g_{1}, A g_{2}, t,(\alpha, p)\right)\right) \text { iff } \\ & \quad M, s_{i} \models \operatorname{Create}\left(A g_{1}, P C\left(A g_{1}, A g_{2}, t, \operatorname{Want}_{P}\left(A g_{1}, A C\left(A g_{2}, A g_{1}, *,(\alpha, p)\right)\right)\right)\right) \end{aligned}$$

The $A g_{1}$ 's action commitment attempt towards $A g_{2}$ is satisfied in the model iff $A g_{1}$ commits that it wants that $A g_{2}$ commits at a certain moment to perform the action. In the same way we can define the semantics of conditional commitment attempts about propositions and about actions.

Example 9. Let us consider the example of an agent $A g_{1}$ asking an agent $A g_{2}$ to open the door. Formally:

ParseError: KaTeX parse error: Expected '\right', got '\end' at position 340: …\right)\right) \̲e̲n̲d̲{aligned}

The $A g_{1}$ 's request is expressed by an action commitment attempt created by $A g_{1}$ towards $A g_{2}$. By asking $A g_{2}$ to open the door, $A g_{1}$ wants that $A g_{2}$ commits to open the door.

Action Occurrences applied to Commitments and related Notions

$M, P a, s_{i} \models \operatorname{Withdraw}\left(A g_{1}, S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right)$ iff
$\exists \alpha \in \operatorname{Gap}\left(A g_{1}, s_{i}\right): M, P a, s_{i} \models \operatorname{Happens}\left(\alpha ; N\right.$ Persist $\left(S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right)$ ? $)$
$\&[\exists P a^{\prime} \in \sigma \& \exists s_{j} \mid P a^{\prime}: T\left(s_{j}\right) \leq T\left(s_{i}\right) \& P a^{\prime} \in \operatorname{Rsc}\left(A g_{1}, A g_{2}, s_{j}\right)$
$\& M, P a^{\prime}, s_{j} \models \operatorname{Create}\left(A g_{1}, S C\left(A g_{1}, A g_{2}, t, \varphi\right)\right) \& P a_{s_{j}, s_{i}}=P a_{s_{j}, s_{i}}^{\prime}$
$\& M, P a_{s_{j}, s_{i}}^{\prime}, s_{j} \models \neg \varphi]$
$P a_{s_{j}, s_{i}}=P a_{s_{j}, s_{i}}^{\prime}$ means that the paths $P a$ and $P a^{\prime}$ are similar on the fragment $s_{j}, s_{j+1}, \ldots, s_{i} . M, P a_{s_{j}, s_{i}}^{\prime}, s_{j} \models \neg \varphi$ means that the content $\varphi$ is not satisfied on the path $P a^{\prime}$ on the fragment $s_{j}, \ldots, s_{i}$. The semantics of NPersist formula is defined as follows.