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Chapter 7[*] A Logical Model for Commitments and Arguments

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In this chapter, we develop a semantics of the pragmatic approach proposed in Chapters 5 and 6. We propose a logical model based on CTL* ( Extended Computation Tree Logic ) and on Dynamic logic that we call DCTL*CAN.. This logical model addresses 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 makes it possible 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 which agents are committed to perform.

In the domain of agent communication, semantics is one of the most important aspects particularly in the current context of open and interoperable multi-agent systems (MAS) (Chaib-draa and Dignum, 2002), (Dignum and Greaves, 2000). Although much significant research work was done in this field, for example (Singh, 2000), (Wooldridge, 2000), (Guerin and Pitt, 2001), (Amgoud et al., 2002), (Vericchio and Colombetti, 2003), 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.

While pragmatics deals with the way of using communication acts, semantics is interested in the meaning of these acts. Pragmatics is related to the dynamics of agent interactions and to the way of connecting the isolated acts to build complete conversations. Pragmatics was also addressed by many researchers, for example (Dastani et al., 2000), (Pitt and Mamdani, 2000), (Pasquier and Chaib-draa, 2003). However, little previous work tried to address these two facets of agent communication in the same framework, considering the difficulty of such a task. Even in this work, semantics and pragmatics are dealt with as a unique object of research whereas they are different in nature. In this context, we believe that the success of applications based on agent communication requires to address these two elements together but keeping them distinct.

The objective of this chapter is to develop the semantic part of our unified framework based on commitments and arguments for agent communication. Thus, the chapter deals with semantic issues in the approach proposed in Chapters 5 and 6 and the link with pragmatic ones. The semantics we define here addresses all the aspects that we use in our commitment and argument approach. This chapter presents two results: 1. it semantically establishes the link between commitments and arguments; 2. it uses a combination of temporal logic (CTL* with some additions) and a dynamic logic to define a complete and unambiguous semantics.

The rest of this chapter is organized as follows. In Section 7.2, we recall the taxonomy of social commitments we used in our pragmatic approach. In Sections 7.3 and 7.4, we present the syntax and the semantics of our logical model. In Section 7.5, we define some postulates. A discussion is presented in Section 7.6 and finally we conclude the chapter.

In the following section, we briefly recall the taxonomy we presented in Chapter 5. We use this taxonomy in the logical model presented in this chapter.

A. Absolute Commitments ( ABC )

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 [5] , expressed by assertives. They can be directed towards the past, the present, or the future.

A2. Action Commitments ( AC )

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 carrying out. The fulfillment and the lack of fulfillment of such commitments depend on the performance of the underlying action and the specified delay. 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 ( PCC ) and conditional commitments about actions ( ACC ). A conditional commitment about a proposition p’ expresses the fact that if a condition p is true, then the creditor will be committed towards the debtor that p’ is true.

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 carried out. 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 Ag1 requests another agent Ag2 to do something, we say that the first agent is trying to induce the other agent to make a commitment. A commitment attempt is thought of as a type of social commitment because it conveys content which is made public once the attempt is performed. However, in our approach, there is a true commitment only after the creditor agent reacts in response to the commitment attempt. We distinguish four types of commitment attempts: propositional commitment attempts ( PCT ), action commitment attempts ( ACT ), conditional commitment attempts about propossitions ( CCTP ), and conditional commitment attempts about actions ( CCTA ).

Figure 7.1 illustrates the taxonomy explained in this section.

In our framework, there is no explicit relation between propositional commitments and action commitments. When the current state of the world does not satisfy a propositional commitment, we speak about a violation of this commitment. There is no rule indicating that an agent develops an action commitment to make the content of its propositional commitment true when this commitment becomes violated. A propositional commitment is a commitment about a state of the world that the debtor agent can not realize. In contrast, an action commitment is a commitment about an action that the debtor commits to perform in the present or in the future.

In the two following sections we define the logical model (syntax and semantics) of our commitment and argument based-approach (CAN). We call this logical model DCTL*CAN because it is based on CTL* and Dynamic Logic.

In this section we specify the syntax of the different elements that we use in our framework. These elements are: propositional elements, actions, social commitments, actions applied to commitments and argumentation relations.

Our formal language £ (the metalanguage) is based on an extended version of CTL* (Emerson and Halpern, 1986), (Hafer and Thomas, 1987) and on dynamic logic (Harel, 1979). Temporal logic and dynamic logic are two powerful logics developed to specify and to prove properties of computational processes (Harel, 1984), (Pnueli, 1986). We use a branching time for the future and we suppose that the past is linear (Ben-Ari et al., 1983). Each node in the branching time model is represented by a state si and a time point tj (Figure 7.2). 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 (Belnap, 1991). The actions of agents 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 (or histories ) representing possible courses of the world with the same past and present but different historic continuations of that moment.

Let Φp be the set of atomic propositions and Φa be the set of atomic action symbols. The set of agents is denoted A and the set of time points is denoted TP . The various types of commitments, the agents’ actions on commitments and on their contents and the argumentation relations are introduced as modal operators. We distinguish between commitment formulae and commitment free formulae . In this chapter, a commitment formula, independently of the commitment type, is denoted: SC ( Ag1 , Ag2 , t, φ ) where t is the utterance time (time at which the commitment is created) and φ is a commitment free formula . A commitment free formula is a well-formed formula that does not have the form of a commitment formula. In a commitment formula Ag1 and Ag2 are two agents and φ is the commitment content. When t is unknown because the commitment is not yet created, we drop it from the commitment formula. In this case a commitment is denoted: SC ( Ag1 , Ag2 , , φ ). In this logical model we use the symbol ∧ in the object language and the symbol & in the metalanguage for "and". For "or" we use the symbol ∨ in the object language and the symbol | in the metalanguage. For "not" we use the same symbol ¬ in the two languages.

The language £ can be defined by the following syntactic rules.

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.

Let S be a set of states and R S S be a transition relation indicating branching time. A path Pa is an infinite sequence of states < s 0, s 1,...> where: and The function T gives us for each state si the corresponding moment t (this function will be specified later).

We use the notation si [ Pa to indicate that the state si belongs to the path Pa (i.e. si appears in the sequence < s 0, s 1,...> that describes the path Pa ). We denote the set of all paths by σ . The set of all paths traversing the state si are denoted: σsi . We suppose that all paths start from s 0 ( T ( s 0) = 0).

In our vision of branching future, we can have several states at the same moment. Thus, in Figure 7.2 we have two different states: s1 and s2 at the same time t1 . At moment t2 we have the states s3 , s4 , s5 , s6 , s7 . Along a given path (for example the real path) there is one and only one state at one moment. Indeed, in our framework, si does not indicate (necessarily) the state at moment i . Therefore, it is necessary to specify the state s and the moment t i.e. a pair ( s , t ) S × TP .

According to this formalization, we can use the notation: M, si, T(si) ψ to indicate that ψ is satisfied in the model M at state si at moment T ( si ). To simplify this notation, we will use in the rest of this chapter the following abreviation: M, si ψ. In this notation: M, si ψ there is a "hidden" time.

A formal model for £ is defined as follows:

M(S, R, A, TP, Np, Fap, T, Rsc, Rw)

where:

S  : a nonempty set of states.

R : R S S a transition relation that defines all the transitions of the model.

A  : a nonempty set of agents.

TP : a nonempty set of time points.

Np : S 2Φp : function relating each state s S to the set of the atomic propositions that are true in this state.

Fap : S × Φa 2S  : function that gives us the state transitions caused by the achievement of an action. For instance, in the Figure 7.3 we have : Fap ( si , α ) = { sj , sl }. The transitions defined by Fap are a sub-set of the transitions defined by R . This function allows us to represent what is known in philosophical logic by "moments of time that are related by virtue of the actions of the agents". As Chellas pointed out (1992), to each moment m there corresponds the set of alternative moments m’ which are compatible with all the actions that an agent Ag performs at moment m . These moments m’ as under the control of, or responsive to the actions of, agent Ag at the moment m .

T : S TP : function associating to any state si the corresponding time. For instance, in Figure 7.2 we have: T ( s5 ) = t2 .

Rsc : A × A × S : function producing the accessibility modal relations for social commitments. is a powerset of paths.

Rw : A × S : function producing the accessibility modal relations for agent’ desires about the commitments of the addressee.

The function Rsc gives us all the paths along which the commitment created by an agent Ag1 towards another agent Ag2 must be satisfied (fulfilled). These paths are conceived as merely "possible", and as paths when the content of a commitment should be true. Indeed, the outputs of the function Rsc are known only after the creation of the commitments. Thus, this depends on the state in which the commitment is created. For example, if we have: Pa Rsc ( Ag1 , Ag2 , si ), then this means that at moment T ( si ) agent Ag1 is committed towards agent Ag2 to satisfy a certain commitment along the path Pa . We can see that Rsc depends on the current moment T ( si ).

As operators, the social commitments we introduced in our model and whose semantics will be defined on the basis of this relation are modal operators like the operator (□) (Chellas, 1980). The reading of □ p is as follows: an agent Ag1 commits towards an agent Ag2 that p is true or an action will be performed making p true.

The function Rw ( Ag1 , si ) gives us the paths along which Ag1 wants that the addressee commits or justifies its commitment. This accessibility modal relation will be used to define the semantics of the commitment attempts and the challenge of a commitment attempt.

Our logical model of absolute and conditional commitments is a KD modal logic (D: serial). This logic allows us to capture interesting intuitions about the manipulation of commitments. The rule of necessitation in this model can be expressed as follows: if p is a theorem, then SC ( Ag1 , Ag2 , t , p ) is also a theorem. A commitment is a theorem iff it is satisfied in all states of the model. Semantically speaking, if the commitment-content is a theorem, then the commitment is always satisfied. However, expressed in such a way, this rule indicates that agents commit about all theorems. In the context of agent communication that we address in this thesis, this rule should be expressed as follows: if p is a theorem and an agent Ag1 creates at moment t a commitment towards another agent Ag2 about p , then SC ( Ag1 , Ag2 , t , p ) is a theorem. In addition, the N axiom can be expressed as follows: if an agent commits towards another agent about a proposition, then it commits that this proposition is true or false (i.e. PC ( Ag1 , Ag2 , ( p ∨ ¬ p ))).

The accessibility modal relation Rsc is serial , i.e.:

This property fits with the notion of infinite paths in CTL*. It means that if an agent commits towards another agent that a proposition is true or that an action will be performed, then this agent does not commit about the negation of this proposition or so that this action will not be performed (i.e. □ p ⇒ ¬□¬ p ). An agent cannot commit about some thing and its negation.

The accessibility modal relation Rw is serial :

Therefore, the logic of commitment attempts is a KD modal logic.

As in CTL*, we have in our model path formulae and state formulae. We propose to evaluate the different types of commitments as state formulae. These formulae can also be interpreted on paths in which case one considers satisfaction in the first state of a path. On the other hand, we propose to evaluate the actions on commitments and the argumentation relations on paths. These path formulae can be interpreted on states if they are true on all the paths traversing a given state. The notation M , si ψ indicates that the formula ψ is evaluated in the state si of the model M . The notation M , Pa , si ψ indicates that the formula ψ is evaluated at the state si along the path Pa where si [ Pa .

We can now define the semantics of the elements of £ in the model M .

In this section we define the semantics of different types of social commitments according to the taxonomy that we specified in Section 7.2.

Definitions

We notice here that we evaluate p along an accessible path Pa at a state sj that can be different from the current state si . 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.

This formula gives us the semantics of propositional commitments in terms of accessible paths. The commitment is satisfied in a model at a state si iff its content is satisfied in the model along all accessible paths. This formula gives us the meaning of a social commitment, but states nothing about the fact that the agent must commit that some thing is true. Consequently, the omniscience problem in the sense that the agent commits that all the theorems are true is not present in our logic. On the other hand, to capture the idea that the agent commits that some proposition is true, we use dynamic logic.

The formula S29 indicates that the commitment of agent Ag1 towards agent Ag2 about a proposition p is satisfied in the model iff along all accessible paths Pa p is true. The formula S30 indicates that agent Ag1 is committed towards agent Ag2 to do α and that along all accessible paths Pa performing α makes p true. According to formulae S29 and S30, 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 (Belnap and Perloff, 1992). 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 (violation and withdrawal) are explicitly included in our framework. Thus, it is possible to have wrong commitments because the accessibility relation Rsc gives us the paths along which the commitment created by an agent Ag1 towards another agent Ag2 must be satisfied.

This formula indicates that agent Ag1 commits that p is true only if the condition p is true (or is satisfied).

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

The Ag 1’s propositional commitment attempt towards Ag 2 is satisfied in the model iff Ag 1 commits that it wants that Ag 2 commits at a certain moment that one of the propositions p ( ci ) is true. This notion of commitment attempt captures open and yes/no questions.

The Ag 1’s action commitment attempt towards Ag 2 is satisfied in the model iff Ag 1 commits that it wants that Ag 2 commits to perform the action. In the same way we can define the semantics of conditional commitment attempts about propositions and about actions.

In this section we specify the semantics of different actions that agents can apply on their commitments. These actions are: creation, withdrawal, satisfaction, violation and reactivation. We also specify the relation between satisfaction and violation and we discuss the link between commitment states and these different actions.

Definitions

This formula indicates that the creation of a commitment is satisfied in the model M along a path Pa iff there is an action α whose performance makes true the commitment (i.e. the commitment holds after the performance of the action α ) and if the creation moment is equal to the time associated to the current state. This formula highlights the fact that the creation of a commitment is an action in itself. Indeed, the action α corresponds to the agent’s utterance which creates the commitment.

This formula indicates that an agent withdraws its commitment for φ iff the following conditions are satisfied:

The agent has already created this commitment in the past.

The commitment is not yet satidfied or violated in the past

The agent performs an action α so that this commitment does not hold at the current moment.

In addition, we add the following meaning postulate which is a constraint that agents must respect when communicating.

According to this constraint, if an agent withdraws its commitment, this means that before the current moment this commitment is not withdrawn since its creation or last reactivation.

On the other hand, commitments are persistent until their withdrawal. Formally, we have the following meaning postulate:

A propositional commitment is satisfied along a path Pa at a state si iff it was already created, and the path Pa is accessible via the relation Rsc . This means that, the path Pa corresponds to the satisfaction path of the commitment which is true at the state sj . Along this accessible path the commitment content is true.

In addition, we add the following meaning postulate indicating that globally in all paths, if a commitment is withdrawn and not reactivated in the future, globally it can not be satisfied or violated.

A conditional commitment is satisfied in the model M along the path Pa iff the underlying condition p is satisfied in the past and that the debtor satisfies in ( M , Pa , si ) the resulting commitment PC ( Ag1 , Ag2 , t , p ). In the same way we define the semantics of a conditional commitment about an action.

A propositional commitment attempt is satisfied by the creditor iff this agent satisfies the resulting propositional commitment. In the same way we define the semantics of the satisfaction of the other commitment attempt types.

In the same way, the violation of the different types of commitments can be formulated. We give here just the definition of the violation of a propositional commitment

A propositional commitment is violated along a path Pa at a state si iff it already exists, and the path Pa does not correspond to the satisfaction path of the commitment which is true at the state sj . Along this path the commitment content is false.

We have also the following proposition:

The proof is a consequence of the definitions.

A commitment is reactivated iff:

1- It was previously withdrawn.

2- The commitment is not yet satidfied or violated in the past

3- The agent performs an action making the commitment true at the current moment.

Like for withdrawl, we add the following meaning postulate which is a constrant that agents must respect when communicating.

According to this constraint, if an agent reactivates its commitment, this means that before the current moment this commitment is not reactivated since its last withdrawal.

Commitment states

The semantics of the actions that agents apply to commitment contents is related to the notion of commitment states (see Chapter 5). Thus, the semantics of these actions must be defined in terms of the semantics of these commitment states. Since a commitment state only holds as a result of the debtor’s action, the semantics of a commitment state is determined by the operation that leads to this state. For example, the operation "withdraw" leads to the state "withdrawn". The semantics of the actions applied on the commitment contents requires a combination of all possible commitment states. An agent cannot act on a commitment content whose state is withdrawn. Thus, to simplify the notation, we suppose that a commitment is either active, or not active (withdrawn).

After introducing the different actions that the debtor can apply to its commitment, we can define the semantics of an active commitment as follows:

This property indicates that a commitment is active iff the two following conditions are satisfied (see Figure 7.4):

1- This commitment was already created or reactivated.

2- Until the current moment, the commitment was not withdrawn.

Therefore, once the commitment is withdrawn, it becomes inactive.

The formula S53 explains a persistence property of social commitments. A social commitment is persistent while it is active. This means that, it is persistent in all the states following the state in which it was created until its withdrawal, violation or satisfaction. The active state is satisfied in the model in the state in which the commitment is created and in all the states until its withdrawal, satisfaction or violation. In addition, we have the following properties:

The proof of is straightforward. The proof of is a consequence of the semantics of Active .

In this section we define the semantics of the argumentation relations that we introduced in Section 7.3.6. These argumentation relations are: justification, attack, defend, defend+ and contradiction. We also formulate an interesting property that enables us to reflect the nonmonotonic nature of arguments.

Definition of basic notions

This formula indicates that the justification of the commitment content φ by an agent Ag1 is satisfied in the model M on a path Pa iff:

1- This commitment is active on this path.

2- This agent creates on this path a commitment whose content is p’ that supports the conclusion p .

In other words, a social commitment of an agent to another one to make a content p true is justified (by means of p' ) iff the social commitment exists (has been created) and moreover a social commitment is created to establish an argument ( p’ , p ), where p' is committed to be true because accordingly to the definition of the connector (∴), p’ is true for Ag1 . The fact that this operator is included in the commitment indicates that the agent is committed that p’ is true and then p is true, i.e. p is true because p’ is true. We define the semantics of the overloaded formula of Justify-content as follows:

We notice here that the purpose of this chapter is to give a semantics of the different actions that agents can perform when conversing. Thus, how do agents choose an argument among others and how do we ensure that the argumentation process terminates are questions that are addressed in Chapters 5 and 8.

The justification operation is the basis of other argumentation operations. As shown by the following definitions (formulae S54, S55, S56), this is due to the fact that all the other operations are defined using this operation.

This formula indicates that an agent contradicts its previous commitment whose content is p if it creates another commitment whose content is a logical conclusion of ¬ p , whereas its commitment for p is still active.

Definition of derived notions

This formula indicates that the attack of the commitment content p by an agent Ag2 is satisfied in the model M along a path Pa iff:

1- This commitment is active on this path.

2- This agent justifies along this path its commitment whose content is ¬ p .

This formula indicates that the defense of the commitment content p by an agent Ag1 is satisfied in the model M along a path Pa iff:

1- This commitment is active on this path.

2- This agent attacks the attacker of the content of its commitment.

This formula indicates that the strong defense of the commitment content φ by an agent Ag1 is satisfied in the model M in along a path Pa iff:

1- This commitment is active on this path.

2- This agent attacks all the attackers of the content of its commitment.

Ag 1’s desire about the justification of a commitment of Ag 2 is satisfied in the model iff along all accessible paths via Rw , Ag 2 justifies in the future this commitment.

Property of nonmonotonicity

According to the property of nonmonotonicity, adding arguments can lead to the defeat of existing arguments. An argument is defeated if it is attacked successfully by a counterargument. In other words, an argument becomes invalid when it is attacked and it cannot be defended. In our model, that results in the following meaning postulate: in all paths of the model M , if Ag2 attacks the content p of Ag1 ’s commitment and if Ag1 cannot defend this content or attack the content of Ag2 ’s commitment, then Ag1 ’s commitment becomes unsatisfied in the model M . Formally, we have the following meaning postulate:

In defeasible reasoning, an argument is valid until a counterargument attacks it. This property can be formally specified in our model by the following meaning postulate:

This property indicates that in all paths of the model if a commitment whose content is an argument is created, then in the next state this commitment is either globally valid or it is valid until a counterargument attacks it. This property can be formulated using the week until operator U+w of CTL* as follows:

In this section we defined the semantics of argumentation relations about propositional commitments. The argumention relation about the other types of commitments are related to the underlying propositional commitments. For example, the justification of a conditional commitment about a proposition is defined as the justification of the associated propositional commitment. Formally:

Until now we gave the syntax and semantics of all the elements of our formalism. We can now formally establish the link between commitments and argumentation. This link is shown by the two following formulae.

This formula is a rationality postulate that we impose in the model. It provides the conditions generated by the creation of a commitment on all paths. The agent must be in a position to check these conditions before creating commitments. Indeed, if an agent creates a commitment, then it should not contradict itself during the conversation. It must also be able to justify its commitment if it is challenged and to defend it if it is attacked. By establishing the link between commitments and arguments, this formula reflects the deontic aspect of commitments. These conditions are also valid for withdrawal, acceptance and refusal because their semantics is expressed in terms of the creation operation.

Because this formula holds on all paths of the model, it seems to be strong. However, this formula is defined as a constraint that software conversational agents must respect. When an agent participates in a conversation using some protocol, it must respect this constraint. If not, we conclude that this agent does not respect the semantics. Therefore, it is easy to verify whether agents respect or not the semantics by verifying if they respect the different constraints. The protocol they use must also respect these constraints. In Chapter 8, we propose a model checking technique addressing this issue. Computationally speaking, agents’ programs must include these constraints as rules, and the protocol can be implemented as a set of rules representing these constraints. In Chapter 9, we propose such an implementation using a set of dialogue games.

We notice that it is possible to relax this constraint by changing the model. The idea is to change the model when an agent creates a commitment (and in a general way when an agent performs an action). In this case, this constraint will hold on all paths of the new model and not of the original model. This means that, it is possible to capture, for example, the case in which an agent contradicts itself. However, our objective is not to model the different possibilities but to specify the constraints to be respected by agents. In other words, we are only interested in models respecting these constraints. In addition, changing the whole model increases the complexity of the model checking (see for example (Rao and Georgeff, 1993)).

On the other hand, an agent challenges a commitment content if it has no argument for or against this content. Therefore, an agent challenges a commitment content if it cannot accept or refuse such a commitment content. Formally:

In this section we give some additional propositions (P) of our logical model. Proofs of these propositions are based on the semantics we defined in the previous section.

This formula states that if an agent creates a commitment then:

1- The commitment was never active in the past (thus it does not exist).

2- The commitment will hold until the moment of its withdrawal.

In other words, a commitment becomes active after its creation, and it remains active until its withdrawal.

Proof

If an agent creates a commitment which is already active, then according to S53 this commitment has already been created or reactivated. If the commitment is reactivated, then according to S51 and S37 it has already been created. However, this is not possible according to the semantics of the creation action (S36).

In addition, according to S53, a commitment is active iff it has already been created or reactivated and not yet withdrawn. Consequently, one can check if a commitment is active at a given moment on a path Pa by checking if it was already created in the past and if since its creation, it has been not withdrawn. Thus, the creation of a commitment implies that it is active until withdrawal.

Formula P2 indicates that if a commitment is not active, then it can not be withdrawn.

Proof

This formula is a consequence of formula S53 and the semantics of U. Let us suppose that the commitment is inactive at a given moment. Consequently, either this commitment was not created or reactivated in the past, or, since its creation or reactivation, the commitment was already withdrawn. In these two cases, the commitment cannot be withdrawn.

This formula states that if a commitment is satisfied, then it remains always satisfied and it cannot be violated or withdrawn.

Proof

According to the semantics of satisfaction (S40), Satisfy formula is satisfied in the model along a path Pa at any state of this path. Consequently, if it is satisfied, it remains always satisfied. Because the path Pa is a satisfaction path in the sens of the accessibility relation Rsc , the commitment cannot be violated along this path. In addition, according to S53, if an agent satisfies a commitment, then this commitment becomes inactive. Therefore, the commitment cannot be withdrawn

In the same way we can prove the following proposition:

This formula indicates that if an agent creates or reactivates a commitment, then it must violate it, satisfy it, or still withdraw it. These operations can take place in the future of the moment following the creation of the commitment. The proof of this proposition follows from the semantics of theses operations.

This proposition states that a commitment remains withdrawn until an eventual reactivation. Thus, the only authorized operation after the withdrawal of a commitment is its reactivation. The proof of this proposition follows from the semantics of Withdraw and Reactivate and from the meaning postulates M39.

We have also the following meaning postulates:

This postulate states that it is not possible to have on a given path two active commitments of the same debtor whose contents are respectively φ and ¬ φ .

This formula indicates that if:

The agent Ag1 already is committed that φ ,

The commitment still holds,

This agent accepts the commitment of its interlocutor for ¬ φ ,

then this implies that the agent withdraws its commitment for φ . If Ag1 does not withdraw this commitment, we would have two active commitments on a given path whose contents are φ and ¬ φ . However, this is not possible according to M6.

The meaning of some important speech acts, especially the ones commonly used in multi-agent interactions, can be expressed using our framework. According to illocutionary logic (Searle and Vanderveken, 1985), the five illocutionary points of language use are: the assertive point , the commissive point , the directive point , the declaratory point and the expressive point . The assertive point consists in representing how things are in the world. The commissive point consists in committing the speaker to doing something. The directive point consists in trying to get the hearer to do something. The declaratory point consists in doing something by way of representing oneself as doing it. The expressive point consists in expressing attitudes.

Assertive acts can be represented by propositional commitments and by conditional commitments about propositions. For example, the performance of an Inform act can be defined as the creation of a propositional commitment. The inform act Inform ( Ag1 , Ag2 , t , p ) indicates that the speaker Ag1 wants to inform the addressee Ag2 that p is true. Formally, we can write:

Inform ( Ag1 , Ag2 , t, p ) = def Create ( Ag1 , PC ( Ag1 , Ag2 , t, p ))

The operations applied on the content of these commitments can be considered as assertive or directive acts. For example, the Assert act Assert ( Ag1 , Ag2 , t , p ) means that the speaker Ag1 is committed relatively to the addressee Ag2 that p is true. In our framework, this acts can be defined by the acceptance of a commitment content in the context where this commitment exists. Formally:

Assert ( Ag2 , Ag1 , t , p ) = def Accept-content ( Ag2 , PC ( Ag1 , Ag2 , , p ))

The assertive act about an argument can be defined by a justification relation:

Assert (( Ag2 , Ag1 , t , p p’ )) = def Justify-content ( Ag1 , PC ( Ag1 , Ag2 , , p ), p’ )

Commissive acts can be reflected by the action commitments and the conditional commitments about actions. The point of the commissive acts is to commit the debtor, relative to the creditor, to the performance of an action α with or without a certain condition. The performance of the action α makes a proposition p true. For example, a promise act Promise ( Ag1 , Ag2 , t , α , p ) means that agent Ag1 is committed towards agent Ag2 to do α without condition. This act can be defined either by the creation of an action commitment or by the acceptance of the content of a commitment attempt:

Directive acts can be represented by commitment attempts and by challenges of commitment contents. The operations applied to the content of commitment attempts can be considered as assertive, commissive or directive acts. Request is an example of a directive act that can be defined in our framework as follows:

Request ( Ag1 , Ag2 , t , α , p ) = def Create ( Ag1 , ACT ( Ag1 , Ag2 , t , ( α , p )))

The request act Request ( Ag1 , Ag2 , t , α ) indicates the fact that agent Ag1 asks agent Ag2 to do α . If Ag2 accepts the request, then it promises Ag1 to do α (see the previous definition of the promise act).

A declaratory act brings about a state of affairs that makes its content true (Colombetti, 2000). An example of declaration is "the auction is open" that is used to open an auction. In our framework, a delaratory act can be captured by the immediate satisfaction of a propositional commitent. Formally:

Expressive acts can also be captured using propositional social commitments.

In this section we showed that our formalism handles in a unified framework both pragmatic and semantic issues of agent conversation. In addition, the framework can capture many different types of illocutionary acts according to speech acts theory. Since the framework makes it possible to capture all these aspects, it can be used as a powerful means to specify, model and implement flexible and highly expressive protocols for agent communication.

According to several researchers in defeasible argumentation, using a model-theoretic semantics is not adapted to defining the meaning of the central notions of defeasible argumentation like attack, rebuttal, defense, etc. The purpose of this section is to show that such a model theory can be successfully used to capture the semantics of these notions.

According to Pollock (1991), Vreeswijk (1997), and Prakken and Vreeswijk (2000), the meaning of defeasible notions should not be found in a correspondence with reality by using a model theory, but in their role in dialectical inquiry. The reason is that these notions are not ‘propositional’, and consequently, their meaning is not naturally captured in terms of correspondence between a proposition and the world. We agree with the fact that the defeasible notions are not propositional, because in our framework they are actions applied to social commitments. Thus, these defeasible concepts (considered in this chapter as argumentation relations), can be captured in a model theory by using a dynamic logic within a global framework of temporal logic. Using these two logics enables us to represent the relation between arguments by taking into account the temporal and the dynamic characteristics of the argumentative interactions between agents. Our theoretical model semantics does not establish a correspondence between defeasible notions (as propositions) and the world, but defines the meaning of operations that agents can apply on their social commitments and the meaning of argumentative supports of these operations. This semantics allows us to capture the conditions on handling commitments and arguments (see S66 and S67). The branching temporal nature of our logical model makes it possible to capture the fact that an agent in a given state at a given moment has several strategies. Agents use their argumentation systems to choose a strategy among others.

On the other hand, the nonmonotonicity property of arguments can be captured in a model theory of branching temporal logic. The idea is that an argument is valid only in a given state, at a given moment for a given agent. An argument is not valid (not satisfied in the Kripke model) when it is attacked and cannot be defended. This idea can be formulated as a property in our logical model by using the path quantifiers A and E (see M65 (1), S65 (2)). In addition, an advantage of using a model theory of temporal and dynamic logics to define the semantics is that we can then use model-checking techniques (Clarke et al., 1986). These techniques enable us to verify some interesting properties of the formalism. In this context, we can use our approach to specify interaction protocols illustrating how agents interact by acting on commitments and on arguments. The automatic tools of model checking (called model checkers) make it possible to provide simulations and traces of execution of such protocols in order to verify properties that these protocols must satisfy (Clarke et al., 2000), (Wooldridge et al., 2002). These techniques are not offered for a logic based on dialectical systems.

In the context of agent interactions, using only an argumentative-based semantics is not sufficient to capture the nonmonotonic reasoning of agents. The reason is that in their conversations, agents do not use only an acceptance theory based on arguments and on attack and defense relations. Agents must also take into account social relations such as trustworthiness.

Finally, we think that a model theoretical semantics and a dialectical-based semantics are not contradictory but rather complementary in the context of agent communication. A model theoretical semantics using temporal and dynamic logics has the advantage of capturing actions and temporal issues of communicative acts. Dialectical-based semantics have the advantage of representing the interaction between arguments that agents use in their conversations.

Semantical considerations for agent interaction have recently begun to find a significant audience in the MAS community. We can distinguish four kinds of semantics for agent interactions:

1- Mentalistic semantics: This subjective semantics is based on so-called agent’s mental states (e.g. beliefs, desires and intentions). The best-known formalisms describing it are: Cohen and Levesque’s intention logic (1990), Rao and Georgeff’s BDI framework (1995), and the KARO framework proposed by van Linder et al. (1998). KQML (Finin et al., 1995) and FIPA-ACL (FIPA, 1997, 1999, 2001a) use this type of semantics to define a pre/post conditions semantic of communication acts. For example, the semantics of a KQML message is given by the following three ingredients: 1) a precondition on the mental states of the sender and the receiver before the communication of the message, 2) a postcondition that should hold after the communication and 3) a completion condition that indicates when the perlocutionary effect has been fulfilled. The advantage of this semantics is its compatibility with the formalisms used for reasoning about rational agents. Hence, the same formalism can be used to specify the agents’ mental states and the communication acts they perform. However, the verification of such a semantics is not possible if we cannot access to the agents’ programs. In this situation we cannot verify whether the agents’ behavior matches their private mental states. In this context, van Eijk and his colleagues (2003) proposed a verification method for agent communication using a framework called Agent Communication Programming Language (ACPL) (van Eijk et al., 2001). ACPL is designed to program systems of agents that communicate by exchanging information. The authors consider the operational semantics of this language which describes the agents’ behavior in terms of their computations. From this semantics, they identified a notion of observable behavior that captures those aspects of computations that are visible to an external observer, and they introduced an assertion language to express specifications of this behavior. To check if agents act in accordance with the behavior specification, the authors developed a verification calculus based on a compositional proof system.

Another limitation of KQML is the pre/post condition semantics. This semantics offers no dynamic or operational description of agent interactions. Because our approach is based on public and argumentative concepts, the compliance verification can be made without having access to the agents’ programs. The satisfaction and the violation of agents’ commitments make it possible to determine if the agent respects our semantics. In addition, the agents’ ability to argue and to justify their commitments facilitates this verification. Moreover, our semantics treats more explicitly the dynamic aspect of agent communication. This aspect is modeled not only by the agents’ actions on commitments and on their contents and by the argumentation relations, but also by the evolution of commitment states and commitment content states.

2- Social semantics: This objective semantics was proposed by Singh as an alternative to the mentalistic one (Singh, 2000). It is based on social commitments and it stresses the importance of conventions and the public aspects of agent interactions. Singh used CTL to propose a formal language and a formal model in which the notion of commitment is described by using an accessibility relation. Verdicchio and Colombetti proposed a logical model of social commitments by extending CTL* (Verdicchio and Colombetti, 2003). They introduced a number of predicates in order to represent events and actions. They specified some axioms to model agents that create commitments, create precommitments, and accept precommitments. They also studied the fulfillment and violation of commitments. Mallya et al. (2004) used the temporal commitment structure specified by Fornara and Colombetti (2002) to define some constraints in order to capture some operations on commitments. They dealt with temporal commitments by studying their satisfactions and breaches. Our logical model belongs to this class of semantics, but it differs from these proposals in the following respects:

a) In our approach the commitment semantics is defined as an accessibility relation that takes into account the satisfaction of the commitment. The commitment semantics is defined in terms of the paths along which the commitment must be satisfied. This way is more intuitive than the semantics defined by Singh.

b) We differentiate commitments as static structures evaluated in states from the operations applied to commitments as dynamic structures evaluated on paths. This enables us to describe more naturally the evolution of the communication as a system of states / transitions which reflects the interaction dynamics. Thus, our logical model allows us to describe the dynamics of agent interactions in terms of the actions that agents apply to commitments, commitment contents and to arguments. These actions are captured by the perform operator used in dynamic logic and that we introduce in our model.

c) In our model, the strength of commitments as a basic principle of agent communication does not result only from the fact that they are observable, but also from the fact that they are supported by arguments. The social commitment notion we formalize is not only a public notion but also a deontic one. The deontic aspect is captured by the fact that commitments are thought of as obligations. The agent is obliged to respect its commitments (i.e to satisfy them), to behave in accordance with these commitments and to justify them. The idea is to impose this constraint in the model we are interested in. The agent is also obliged not to contradict its commitment contents during the conversation. The creation operation and the argumentation relations capture this deontic aspect. Formulae S66 and S67 which supplement our semantics show how our approach makes it possible to capture this aspect. Indeed, the link we establish between commitments and arguments enables us to formally express the following idea: by committing towards other agents that a certain formula is true, the agent is compelled not to contradict itself during the conversation. It must also be able to explain, argue, justify and defend itself if another participant contradicts it.

d) In our semantics, we capture not only propositional commitments, but the various othertypes of commitments. This enables us to have a greater expressivity and to capture many different types of speech acts. In addition, all the elements constituting our commitment and argument approach are expressed using the same logical framework. The different types of commitments, the different operations on them, and the different argumentation relations are semantically specified in a clear and unambiguous way.

3- Argumentation-based semantics: This type of semantics is defined in (Amgoud et al., 2002), (Parsons et al., 2002), (Parsons et al., 2003) to capture the meaning of certain communication acts. It is based upon an argumentation system in which the agents’ reasoning capabilities are often linked to their ability to argue. These reasoning capabilities are mainly based on the agent’s ability to establish a link between different facts, to determine if a fact is acceptable, to decide which arguments support which facts, etc. The authors proposed a two-layered semantics. The first layer captures the reasoning level of agents. Agents must check some preconditions in order to use a communication act. These preconditions are described in terms of arguments. For example, before using an assertion act that p , an agent checks whether it has an argument in favor of p . The second layer relies upon the formal dialectics introduced by Mackenzie (1970). Dialectical models are rule-governed structures of organized conversations in which two parties (in the simplest case) speak in turn in an orderly way. These models associate to each agent a commitment store (CS), which holds the information given by the interlocutors during the dialogue. This layer describes the rules which define how the CS is updated. For example, after an assertion act that p is true, the CS of the speaker is updated by adding p to it. This semantics has the advantages of being simple and of taking into account the argumentation aspect of agent communication. In addition to the fact that this semantics does not take into account the temporal and dynamic aspects of communicative acts in its formalization, it is different from our approach on several points. The fact that it uses a logic without theoretical model makes a formal verification impossible. On the other hand, the semantics is described in terms of pre/post conditions and it does not capture the meaning of the different communication acts. The commitment notion used in this semantics is different from the one we use in our semantics. In Amgoud et al’s approach, this concept captures only the propositions stated by the agents. Contrary to our approach, the satisfaction, violation, cancellation and reactivation notions do not appear. Moreover, in terms of argumentation, only the argue operation is captured. The attack and defense operations are not addressed in this semantics. Finally, the dynamic aspect of agent communication is reduced to the sole update operations of the CS. These operations reflect only the history without clearly reflecting the current state of the communication. On the other hand, in our approach this state is well captured by the states of different commitments and arguments handled in the conversation.

4- Protocol based semantics: Developed by Pitt and Mamdani (2000), this type of semantics is based on the notion of protocol. The communication between two (or more) agents is viewed as a conversation. The meaning of communication acts is specified by describing an input-output relationship. The meaning of a speech act (as input) is defined to be the intention to perform another speech act (as output). This meaning then matches the set of the possible following answers. This semantics has the advantage of taking into account the context and the conversation state. However, technically, protocols are used as a practical tool and not as a means to define semantics. For example, by using only protocols, we cannot define the meaning of some notions like satisfaction, violation, contradiction, justification, etc. Protocols must be specified in accordance with a given semantics in such a way that a compliance verification is possible. In our approach we can define protocols by using our semantics and verify whether some properties (that we have to specify yet) are satisfied. For instance, such a property can be stated as follows: "It is not possible to withdraw a commitment that is previously satisfied". Because our semantics is expressed in a temporal logic, we can use protocols specifying that an action cannot take place before another. For example, a commitment cannot be cancelled before its creation. A protocol can also specify that when a commitment is created by an agent Ag1 , several paths are possible for its interlocutor Ag2 (acceptance, refusal, challenge). However, the choice of the path cannot be made without returning to the semantics. For example, acceptance indicates that the agent is also committed towards the accepted content. Thus, agent Ag2 must be able to justify this content and to satisfy the commitment.

Finally, we notice that although our accessibility relation Rsc is a dynamic function, we do not need to change the Kripke model M to capture this dynamics. This way of modeling is different from that used for example in KARO framework (Meyer et al., 1999). In KARO, the whole Kripke model must be changed as illustrated by the following formula:

Where <doi ( α ) represents the fact that agent i has the opportunity to do the action α and that doing α leads to φ , and r is a function defined from another function r0 as follows :

where A is a set of agents, At is a set of atomic actions and S a set of states. r0 ( i , α )( s ) yields the (possibly empty) state transition in s caused by the event doi ( α ). A successful performance of an atomic action always results in a state transition to another state in the model. r is defined as follows:

r ( i , α )( M , s ) yields the model change and the state transition caused by the event doi ( α ). In our model that fits in naturally with the use of CTL* the whole dynamics is represented in one unique model. Thus, we do not need to define a function like r0 . Indeed, we can capture all the actions that agents apply to commitments and to their contents without changing the model, but simply by changing the states of the model. This solution increases the number of states in the model. However, it enables us to reduce the complexity of the underlying decision procedure, and it gives rise to more efficient model-checking.

In this chapter, we developed a logic and formal semantics for our pragmatic approach based on commitments and arguments to model agents’ interactions. We proposed a logical model based on a combination of CTL* and dynamic logic. The model captures the different commitment types, the different actions that agents apply to these commitments and the various argumentation relations. In addition, the model captures the link between commitments and arguments that enables us to express the deontic aspect of commitments. Our semantic framework can also be used to express the meaning of some important speech acts, especially the ones commonly used in multi-agent interactions. Finally, we argued that our model-theoretic semantics can be successfully used to capture the semantics of defeasible arguments.



[5] Propositional commitments can also be expressed by speech acts of declaratory and expressive types.

© Jamal Bentahar, 2005