Modelling Decision-making Behavior Based on keg-BDI Agents*

Xiaojun Zhang

College of Political Education，Sichuan Normal University

Institute of Logic and Information，Sichuan Normal University

Fujian Provincial Key Laboratory of Brain-Like Intelligent Systems

zhangxj566@163.com

Baoxiang Wu

College of Political Education，Sichuan Normal University

Institute of Logic and Information，Sichuan Normal University

495171973@qq.com



Modelling Decision-making Behavior Based on keg-BDI Agents*

Xiaojun Zhang

College of Political Education，Sichuan Normal University

Institute of Logic and Information，Sichuan Normal University

Fujian Provincial Key Laboratory of Brain-Like Intelligent Systems

zhangxj566@163.com

Baoxiang Wu

College of Political Education，Sichuan Normal University

Institute of Logic and Information，Sichuan Normal University

495171973@qq.com

Abstract.Inthispaperweextendtherangeoftruthvalueofinfinite-valuedŁukasiewiczlogic from［0，1］to［-1，1］，and propose an extended emotional graded BDI logic，i.e.，keg-BDI logic that is based on this extended Łukasiewicz logic and propositional dynamic logic to formalize knowledge states，mental states(such as belief，desire，intention)and emotional states(such as fear，anxiety and self-confidence)which influence on the keg-BDI agent’s decision-making behavior.This behavior is determined by the different measure of each context that is added by concrete conditions.After presenting the language and semantics of keg-BDI logic and illustratingrelationshipsbetween/amongcontextsforthekeg-BDIagent，anexampleofmilitary decision-making behavior is given.This study will provide a formal support for distributed artificial intelligence and military simulation.

1　Introduction

The formalization of BDI(Belief-Desire-Intention)agents is a topic of continuing interest in Artificial Intelligence about how mental states(such as belief(B)，desire(D)and intention(I))and actions influence each other.Research on this subject has held the limelight ever since the pioneering work of Bratman(［4］)which lays the foundation of BDI approach to describe artificial agent behavior.Then researches try to formalize Bratman’s theory using many logical ways.There are three well-known approaches：the first one is based on a linear-time style temporal logic(［8］)，the second one is based on the branching-time temporal logic(［17］)，the third one takes propositional dynamic logic as a basis instead of a temporal logic(［25］).Over the years important contributions have been made on both mental aspects like desire and intention(［17，18，19］)，and informational aspects like knowledge and belief(［11，23，24，25，26］).Recent developments include the work on graded BDI models for agent architectures(［5，6，7，9，28］)，formal modelling of emotions in BDI agents(［14，21］)，and modelling emotional agents based on graded BDI architectures(［27］)，and many others(［15，22］).

Whenbuildingemotionalagents，theBDImodelhasbeenprovedtobeoneofthe best options one can select.For example，Emotional-BDI logic presented by Pereira et al.illustrates that agents’behavior is guided not only by beliefs，desires and intentions，but also by the role of emotions in reasoning and decision-making(［13，12，14］).KARO logic for emotional agents based on dynamic logic deals with the behavior of rational agents(［20］).The strongly related work by Adam et al.is devoted to a formalization of OCC emotions in BDI terms(［1，2］).The Emotional-BDI agent models developed by Puică integrates emotions，resources，and personality features intoartificialintelligentsoastoobtainahuman-likebehaviorofthisagent(［16］).The generic model for decision making of virtual agents in relation to emotions and trust takes the BDI framework as point of departure，and extends this with mechanisms to represent the dynamics of emotions and trust.The model has been tested by means of simulation experiments，and then been successfully incorporated into the virtual agents with the RoboCup 2D soccer environment(［3］).

In our research(［27］)we defined an emotional graded BDI logic(eg-BDI logic forshort)forrationalagents，i.e.，alogicthatisusedtospecify，andtoreasonaboutthe behavior of rational agents.In our framework we concentrated on how mental states (such as belief，desire and intention)and emotional states(such as fear，anxiety and self-confidence)influence the agent’s behavior.In the basic architecture，we blend the infinite-valued Łukasiewicz logic and propositional dynamic logic to formulize the emotional graded BDI agents.

The aim of this paper is a formalization of decision-making behavior based on the framework mentioned above，i.e.，eg-BDI logic.This paper can for instance be modelled that an agent knows that some action is a correct plan to achieve her goal since she knows that performing the action will lead to the goal，and that she knows some action is a feasible plan since the agent knows of her ability to perform the action.In subsequent research，we extend the eg-BDI logic with modal operators for knowledge.

Not all knowledge is trivial.In order to simplify the model，we focus only on the critical knowledge which has a significant impact on the agent’s decision-making behavior.We directly use Kφ to say that the agent knows the critical knowledge φ. The knowledge which prompts the agent to take actions is called positive knowledge，and the knowledge which prevents the agent to take actions is called negative knowledge.Correspondingly，there are positive and negative critical knowledge.Similarly，there are positive and negative desires.At present，the vast majority of scholars suchas Pereira et al.(［23］)introduce positive and negative desire operators at the same time to handle desires.Therefore，even if the factors with which we are dealing are not too much，the model is too complicated.

In order to reduce the number of modal operators that we introduce，and to simplify the model and able to handle more factors，it is necessary for us to extend the range of value of the infinite-valued Łukasiewicz logic from［0，1］to［-1，1］.Thus，we need a unified provisions：the range of truth value corresponding to the positive factorswhichprompttheagenttotakeactionsis［0，1］，andtherangeoftruevaluecorresponding to the negative factors which prevent the agent to take actions is［-1，0］.

The rest of the paper is organized in the following way：in Section 2 we define the keg-BDI logic.This new logic is based on eg-BDI logic(［27］)and we begin by presenting the new modal operators for critical knowledge that were added.Besides the syntax and semantics of keg-BDI logic，we present the axiom systems for the new modal operators.In Section 3 we illustrate the relationship between/among contexts for knowledge states，mental states and emotional states.An application of the keg-BDI agent for military decision-making behavior is given in Section 4.In Section 5，and finally，we present some conclusions about this work and point some topics for ongoing and future lines of work in the keg-BDI logic.

2　keg-BDI Logic

The keg-BDI logic is a multi-modal and multi-valued logic which combines the above Extended infinite-valued Łukasiewicz Logic(ELL for short)and Propositional Dynamic Logic(PDL for short).(［13］)The formal semantics of keg-BDI logic is based on Kripke models with accessibility relations between possible worlds which correspond to different knowledge states，mental states and emotional states.

2.1The keg-BDI Language

Wedefinemodaloperatorsforrepresentingtheknowledgestatesofcriticalknowledge(K)，the mental states of belief(B)，desire(D)and intention(I)，and emotional states of fear，anxiety and self-confidence in ELL.Now we define the language Lkeg-BDIby adding action modalities of the form［α］where α is an action，and seven fuzzy modal operators K，B，D，I，Fear，Anx and SConf to the classical propositional language L：

Kφ，Bφ，Dφ and I mean that“φ is known”，“φ is believable”，“φ is desired”and“φ is intended”，respectively，and their truth degrees refer to the agent’s level of satisfaction when becomes true.The meanings of Fearφ，Anxφ and SConfφ are similar to Bφ，Dφ and Iφ.

Similar to the language Leg-BDI，the language Lkeg-BDIhas two sorts of expressions，that is，propositions φ，ψ，...and actions α，β，...，Φ0and Φ refer to the setof all atomic propositions and of all propositions，respectively，and Π0and Π refer to the set of all atomic actions and of all actions(including atomic actions and plans which are composite actions)，respectively.Formulae can be inductively built from the atomic ones using connectives and mixed operators，that is，¬(negation)，→(implication)，［］(necessity)and？(test).Otherconnectivescanbedefinedfrom¬and→. And then actions can be inductively built from the atomic ones using the following action operators：；(composition)，∩(nondeterministic choice)and∗(iteration)，and the mixed operator？.

Definition 1Formulae are definable inductively as follows：

φ：：=φ|¬φ|φ→ψ|［α］φ|Kφ|Bφ|Dφ|Iφ|Fearφ|Anxφ|SConfφ In definition 1，［α］φ means that φ is necessarily true after performing the action α.

Definition 2Actions are definable inductively as follows：

α：：=α0|α；β|α∪β|α∗|φ？

Here iterated actions αn(with n≥0)can be inductively defined by α0=id，and αn+1=α；αn.

Now we define a modal context language for ΣC(here Σ∈｛K，B，D，I，Fear，Anx，SConf｝)，and use the connectives of the above extended infinite-valued Łukasiewicz logic to build Σ-modal from elementary modal formulae and truth constants rc，for each rational r∈［-1，1］：

(1)if φ∈Lkeg-BDI，then φ，Σφ∈ΣC；

(2)if r∈Q∩［-1，1］，then rc∈ΣC(Q is the rational set)；

(3)if Σφ，Σψ∈ΣC，then Σφ∧Σψ∈ΣC and Σφ→LΣψ∈ΣC.

For example，if Σ=K and φ∈Lkeg-BDI，then φ，Kφ∈KC by clause(1).If Σ=K and Kφ，Kψ∈KC，then Kφ∧Kψ∈KC and Kφ→LKψ∈KC by clause(3).The other cases are similar.In clause(3)，→Land∧are similar to the implication and conjunction of Łukasiewicz logic，respectively.The truth value of Σφ→LΣψ is 1 if and only if the truth value of Σφ is greater or equal to that of Σψ. rc→LΣφ means that the probability of φ is at least rc，which is denoted as(Σφ，rc).

In order to make∆Σφ become a two-valued Boolean formula，we use Łukasiewicz logic extended with a new unary connective∆(known as Baaz’s connective). For any modal formula Σφ，if the truth value of Σφ is smaller than 1，then∆Σφ gets value 0；otherwise，if of Σφ is 1，then∆Σφ gets value 1.

2.2The keg-BDI Semantics

Differentpossibleworldscorrespondtodifferentknowledgestates，mentalstates and emotional states.As in usual in modal logics，the formal semantics of keg-BDIlogic is based on Kripke models.By adding the structure and function λ that handles knowledge to the Emotional Graded BDI model，that is，a 9-tuple Kripke structure F=〈W，υ，ρ，τ，θ，｛µw｝w∈W，ε，η，κ〉(［27］)，we can define a 10-tuple Kripke structure F=〈W，υ，ρ，τ，θ，｛µw｝w∈W，λ，ε，η，κ〉where：

(4)W is a set of possible worlds，and w，w′∈W ̸=∅；

(5)υ：Φ×W → ｛0，1｝assigns a Boolean evaluation to each φ∈Φ and w∈W，and υ(φ，w)∈｛0，1｝；

(6)ρ：2W→［0，1］is a finitely additive probability measure on subsets of W，and for φ∈Φ0，｛w|υ(φ，w)=1｝is measurable；

(7)τ：Π0→2W×Wgivesasetofpairsofworldsreferringtoworldtransitions for each atomic action；

(8)θ：W →［-1，1］is a distribution of pover possible worlds，and |θ(w)|<|θ(w′)|means that w′is more preferred than w；

(9)µw：W → ［-1，1］is a possibility distribution，for each w∈W.Where µw(w′)∈［-1，1］is the degree on which the agent may try to reach w′from w；

(10)λ：W→ ［-1，1］is a distribution of knowledge over possible worlds，|λ(w)|<|λ(w′)|means that w′is greater impacted on w；

(11)ε：W →［-1，1］isadistributionoffearoverpossibleworlds，and|ε(w)|< |ε(w′)|means that w′is more feared than w；

(12)η：W → ［-1，1］is a distribution of anxiety over possible worlds.And |η(w)|<|η(w′)|means that w′is more anxious than w；

(13)κ：W →［-1，1］is a distribution of self-confidence over possible worlds. And|κ(w)|<|κ(w′)|means that w′is more self-confident than w.

Lkeg-BDIcan be defined by extending L using action modalities and classical connectives.And Σ-formulae(here Σ∈｛K，B，D，I，Fear，Anx and SConf｝)are defined by extending Łukasiewicz logic as follows：

(14)υ(Σφ，w)=ξ(｛w′∈W|υ(φ，w′)=1｝，for each φ.Here ξ∈｛ρ，θ，µw，λε，η，κ｝；

(15)υ(rc，w)=r，for all r∈Q∩［-1，1］；

(16)υ(Σφ&Σψ，w)=max(υ(Σφ)+υ(Σψ)-1，0)；

(17)υ(Σφ→LΣψ，w)=min(1-υ(Σφ)+υ(Σψ)，1)；

(18)‖Σφ‖F=tdw∈Wυ(Σφ，w)，where‖Σφ‖Fis the truth degree of a formula Σφ in the Kripke structure F=〈W，υ，ρ，τ，θ，｛µw｝w∈W，λ，η，ε，κ〉；

(19)if υ(Σφ，w)=1，then υ(∆Σφ，w)=1；

(20)if υ(Σφ，w)̸=1，then υ(∆Σφ，w)=0；

(21)td∅=1；

(22)for all w∈W，υ(Σ⊥，w)=1.In clauses(18)and(21)td refers to the truth degree of a formula Σφ in the Kripke structure F= 〈W，υ，ρ，τ，θ，｛µw｝w∈W，λ，η，ε，κ〉.The evaluation of Σ-formulae just depends on the formula itself—represented in its corresponding measure over possible worlds where the rational agent is situated.

2.3Axioms，Rules，Soundness and Completeness for keg-BDI Logic

Similar to eg-BDI semantics，keg-BDI semantics is just augmented with modal operators for knowledge.The important difference is that the range of truth value in eg-BDI logic∈［0，1］，while the range of truth value in keg-BDI logic∈［-1，1］. Therefore，we can get the axioms and rules for keg-BDI logic only adding the ones aboutknowledge.Bothofthemarealmostexactlythesame.Andtheotherdifference is that in the eg-BDI logic Σ∈｛B，D，I，Fear，Anx，SConf｝(［27］)，while in the keg-BDI logic Σ∈｛K，B，D，I，Fear，Anx，SConf｝.In other words，Σ can get value K in the latter.The axioms for keg-BDI logic are composed of axioms of Classical Propositional Logic for the non-modal formulae and axioms of the Łukasiewicz logic for modal formulae，and axioms for Σ over propositional dynamic logic.

The keg-BDI logic is sound and complete.Its proof is almost exactly the same as that of eg-BDI logic，that is，by embedding of keg-BDI logic into Łukasiewicz logic and propositional dynamic logic which are sound and complete(cf.，［3］and ［10］，respectively).Of course，in terms of the soundness and completeness result of eg-BDI logic(［27］)，we can straightforwardly prove soundness and completeness for the keg-BDI logic by embedding of keg-BDI logic into eg-BDI logic.

3　Relationship between/among Contexts for the keg-BDI Agent

According to the knowledge states，mental states and emotional states of the keg-BDI agent，and the possible transformations by performing actions，the planner can build plans generated from actions to fulfill her desires.Relationships among K，D，B，Fear，Anx，SConf，and P contexts are as follows：

(23)ifK：(K(［α］φ，k))，D：¬∆¬(Dφ，d)，B：(B(［α］φ，b))，Fear：(Fear(φ，f))，Anx：(Anx(φ，a))，SConf：(SConf(φ，s))andP：action(α，Pre-，Costα)，then P：plan(φ，action(α，Pre-，Costα)，b)where：

α∈Π0is an atomic action.The knowledge degrees k in K(［α］φ，k)refers to the probability which the agent will take actions α after she knows φ.The other cases are similar.Pre-are the preconditions of the action α，and Costα∈［-1，1］is the associated cost according to the action α involved；k，b，f，a and s∈［-1，1］are respectively knowledge，belief，fear，anxiety，and self-confidence degree of actually achieving φ by performing α.Action(α，Pre-，Costα)expresses an atomic action，and plan(φ，action(α，Pre-，Costα)，r)a plan which is a composite action whichallows the agent to move from its current world to another.It is assumed that the current state of the world must satisfy the preconditions，and that the plan must make true the desire that the plan is built for.

We can use the intention degree to trade off the benefit and the cost of achieving a goal，thus for each composite action α which allows to reach the goal，the degree of Iφ can be deduced from the degree of Kφ，Dφ，Fearφ，Anxφ，SConfφ and the cost of a plan that satisfies the desire φ.That is，the degree of Iφ is calculated by a function F as follows：

(24)ifK：(K(［α］φ，k))，D：(Dφ，d)，B：(B［α］φ，b)，Fear：(Fear(［α］φ，f))，Anx：(Anx(［α］φ，a))，SConf：(SConf(［α］φ，s))and P ：plan(φ，action(α，Pre-，Costα)，b)，then I：(Iφ，F(k，d，b，f，a，s，Costα)).

Different functions F(k，d，b，f，a，s，Costα)may model different decision-making behavior.It is assumed that the agent full beliefs in achieving φ after performing α，the degree of the intention to bring about φ mainly depends on the satisfaction and the degree of Kφ，Dφ，Fearφ，Anxφ，SConfφ and Costαthat it bring the agent. It is needed to find what kind of the relationship between/among i，k，d，f，a，s，b and Costα.The degree of intention can be the ultimate embodiment of knowledge states，the other mental states，and emotional states.

We can assign different weights to knowledge states，mental states，and emotional states according to their influence on the degree of the intention.The weights allow simple revisions and frequent modifications according to the information about the keg-BDI agent.For example，after a preliminary study，for the military decisionmaking behavior examined in this paper，we assign respectively weights to k，d，b，f，a，s and Costα0.13，0.26，0.22，0.09，0.08，0.15 and 0.07.Thus，the function F(k，d，b，f，a，s，Costα)can be defined as follows：

(25)F(k，d，b，f，a，s，Costα)=0.13k+0.26d+0.22b+0.09f+0.08a+ 0.15s+0.07Costα.

If the agent intends φ at imax，then the rational agent will choose the best plan.Therefore：

(26)if I：(Iφ，imax)，and P ：bestplan(φ，action(α，Pre-，Costα)，r)，then C：C(does(α)).

4　An Example of Military Decision-making Behavior for the keg-BDI Agent

Now we instruct a keg-BDI agent to look for a military decision-making package.We assign to the agent the following critical knowledge：enemy reinforcements having arrived in time；having got the enemy garrison chart；and having got the enemy’s battle scheme.At the same time，we assign to the agent the following desires：fewer casualties；shorter battle time；and the battle sites from an assumed location no more than 500 nautical miles.And then，we instruct the agent with the three emotions：fear of being surrounded；anxiety of that command system is destroyed；and self-confidence of wining the final victory.In order to determine which battle scheme is better，the agent will have to take into consideration the critical knowledge，the benefit(with respect to fewer casualties，shorter battle time and wining the final victory)，fear，anxiety and the cost of the battle.In this scenario，different decisionmaking behavior is to choose a different battle scheme，and the chosen schemes are as follows：

(27)Π0=｛scheme-A，scheme-B，scheme-C，scheme-D，scheme-E｝.

In this case，KC，BC，DC，FearC，AnxC，SConfC and PC contexts are as follows：

Knowledge Contexts(KC)：The agent has the following critical knowledge：

(28)K(［scheme-A］enemy reinforcements having arrived in time，k1=-0.88)；

(29)K(［scheme-B］enemy reinforcements having arrived in time，k1=0.73)；

(30)K(［scheme-C］enemy reinforcements having arrived in time，k1=0.53)；

(31)K(［scheme-D］enemy reinforcements having arrived in time，k1=0.64)；

(32)K(［scheme-E］enemy reinforcements having arrived in time，k1=0.46)；

(33)K(［scheme-A］having got the enemy garrison chart，k2=0.75)；

(34)K(［scheme-B］having got the enemy garrison chart，k2=0.89)；

(35)K(［scheme-C］having got the enemy garrison chart，k2=0.72)；

(36)K(［scheme-D］having got the enemy garrison chart，k2=0.80)；

(37)K(［scheme-E］having got the enemy garrison chart，k2=0.92)；

(38)K(［scheme-A］having got the enemy’s battle scheme，k3=0.79)；

(39)K(［scheme-B］having got the enemy’s battle scheme，k3=0.93)；

(40)K(［scheme-C］having got the enemy’s battle scheme，k3=0.86)；

(41)K(［scheme-D］having got the enemy’s battle scheme，k3=0.79)；

(42)K(［scheme-E］having got the enemy’s battle scheme，k3=0.97).

Since‘enemy reinforcements having arrived in time’will prevent the agent to take action，the truth value which it corresponds to∈［-1，0］.Since‘having got the enemy garrison chart’will prompt the agent to take action，the truth value which it correspondsto∈［0，1］.Theothercasesaresimilar.Itisassumedthatthethreecritical knowledge are stochastically independent.In this scenario，we may assign to the following inference rule for knowledge contexts according to the critical knowledge in this practical situation influence on different weights：

(43)ifK(［α］enemyreinforcementshavingarrivedintime，k1)andK(［α］having got the enemy garrison chart，k2)and K(［α］having got the enemy’s battle scheme，k3)，then K(［α］enemy reinforcements having arrived in time∧having got the enemy garrison chart∧having got the enemy’s battle scheme，k=0.25k1+0.35k2+0.40k3).

Desire Contexts(DC)：The agent has desires as follows：

(44)(D(fewer casualties)，d=0.82)；

(45)(D(shorter battle time)，d=0.89)；

(46)(D(fewer casualties∧shorter battle time)，d=0.98)；

(47)(D(distance≤500nm)，d=0.85).

Belief Contexts(BC)：The keg-BDI agent has knowledge about the interrelations between possible actions that she can take and formulae made true by their execution.

ThedegreeofB(［α］fewercasualties)referstotheprobabilityoffewercasualties after performing α.The degree of B(［α］shorter battle time)is similar.The agent is assigned to the following beliefs：

(48)B(［scheme-A］fewer casualties，b1=0.68)；

(49)B(［scheme-B］fewer casualties，b1=0.72)；

(50)B(［scheme-C］fewer casualties，b1=0.89)；

(51)B(［scheme-D］fewer casualties，b1=0.53)；

(52)B(［scheme-E］fewer casualties，b1=0.40)；

(53)B(［scheme-A］shorter battle time，b2=0.65)；

(54)B(［scheme-B］shorter battle time，b2=0.78)；

(55)B(［scheme-C］shorter battle time，b2=0.86)；

(56)B(［scheme-D］shorter battle time，b2=0.79)；

(57)B(［scheme-E］shorter battle time，b2=0.90).

It is assumed that the desires are stochastically independent.We may add the following inference rule for belief contexts：

(58)if B(［α］fewer casualties，b1)and B(［α］shorter battle time，b2)，then B(［α］fewer casualties∧shorter battle time，b=0.56b1+0.44b2)

Fear Contexts(FearC)：In this case，the agent has the following measure of fear：

(59)Fear(［scheme-A］being surrounded，f=-0.66)；

(60)Fear(［scheme-B］being surrounded，f=-0.87)；

(61)Fear(［scheme-C］being surrounded，f=-0.70)；

(62)Fear(［scheme-D］being surrounded，f=-0.36)；

(63)Fear(［scheme-E］being surrounded，f=-0.48).

AnxietyContexts(AnxC)：Inthisscenario，weassigntotheagentthefollowing measure of anxiety：

(64)Anx(［scheme-A］command system is destroyed，a=-0.56)；

(65)Anx(［scheme-B］command system is destroyed，a=-0.84)；

(66)Anx(［scheme-C］command system is destroyed，a=-0.75)；

(67)Anx(［scheme-D］command system is destroyed，a=-0.62)；

(68)Anx(［scheme-E］command system is destroyed，a=-0.92).

Self-confidenceContexts(SConfC)：Inthisexample，theagent hasthefollowing measure of self-confidence：

(69)SConf(［scheme-A］wining the final victory，s=0.93)；

(70)SConf(［scheme-B］wining the final victory，s=0.82)；

(71)SConf(［scheme-C］wining the final victory，s=0.68)；

(72)SConf(［scheme-D］wining the final victory，s=0.79)；

(73)SConf(［scheme-E］wining the final victory，s=0.89).

Plan Contexts(PC)：In this scenario，a series of atomic actions are as follows：

(74)action(scheme-A，dist-=400 nm，cost=500 billions，Costα=-0.72)；

(75)action(scheme-B，dist-=300 nm，cost=400 billions，Costα=-0.65)；

(76)action(scheme-C，dist-=800 nm，cost=900 billions，Costα=-0.98)；

(77)action(scheme-D，dist-=700 nm，cost=800 billions，Costα=-0.92)；

(78)action(scheme-E，dist-=450 nm，cost=600 billions，Costα=-0.78).

Now the keg-BDI agent can determine which intention to adopt and which plan is associated with that intention.The agent’s desires are conveyed to plan contexts by desire contexts，and then the agent finds plans for each desire within plan contexts.The agent looks for a set of different battle schemes in terms of knowledge states，mental states and emotional states，and takes into comprehensive consideration various aspects of these contexts.Due to the restriction by the desire(47)，that is，the distance no more than 500nms，the agent gives up plans(76)and(77)，that is，the agentgivesuptochoosescheme-C andscheme-D.Therefore，plansaregeneratedfor each desire by(23).For instance，for the most preferred desire，i.e.fewer casualties shorter battle time，the generated plans are as follows：

(79)plan(fewercasualties∧shorterbattletime，action(scheme-A，｛dist-=400nm｝，｛cost=500 billion｝，Costα=-0.72)，b=0.56b1+0.44b2=0.6668)；

(80)plan(fewercasualties∧shorterbattletime，action(scheme-B，｛dist-=300nm｝，｛cost=400 billion｝，Costα=-0.65)，b=0.56b1+0.44b2=0.7464)；

(81)plan(fewercasualties∧shorterbattletime，action(scheme-E，｛dist-=450nm｝，｛cost=600 billion｝，Costα=-0.78)，b=0.56b1+0.44b2=0.62)；

The agent is now in conditions to determine the degree of intentions according to knowledge states，mental states，emotional states and the plans.Since the function f is monotonically increasing with respect to d by(25)，it is enough to take into consideration the most preferred desire，i.e.fewer casualties∧shorter battle time，which is preferred to a degree 0.98.In terms of(25)，using F(k，d，b，f，a，s，Costα)=0.13k+0.26d+0.22b+0.09f+0.08a+0.15s+0.07Costα，we successively have for α∈｛scheme-A，scheme-B，scheme-E｝as follows：

(82)I(fewer casualties∧shorter battle time，0.13k+0.26d+0.22b+0.09f+ 0.08a+0.15s+0.07Costα=0.429845)；

(83)I(fewer casualties∧shorter battle time，0.13k+0.26d+0.22b+0.09f+ 0.08a+0.15s+0.07Costα=0.416138)；

(84)I(fewer casualties∧shorter battle time，0.13k+0.26d+0.22b+0.09f+ 0.08a+0.15s+0.07Costα=0.43065).

The maximal degree of intention for fewer casualties∧shorter battle time by the plan scheme-E is 0.43065.At last，the agent can adopt the best plan and take the corresponding action according to the maximal degree of intention.Now，the action α=choosing scheme-E can be selected and passed to the communication context by ［2］.

5　Conclusions and Future Work

In this paper we extend the truth value of infinite-valued Łukasiewicz logic from ［0，1］to［-1，1］，and propose keg-BDIlogic that is anextended emotional graded BDI logic to formalize knowledge states，mental states and emotional states that influence ondecision-makingbehavior.Thisbehaviorisdeterminedbythedifferentmeasureof eachcontextwhichisaddedbyconcreteconditions.Afterpresentingthelanguageand semanticsofkeg-BDIlogicandillustratingrelationshipsbetween/amongcontextsfor the keg-BDI agent，an application of military decision-making behavior is given.It is hopedthatthisstudywillprovideaformalsupportfordistributedartificialintelligence and military simulation.

As a future work，it would be interesting to extend keg-BDI agent to include other mental states and other emotional states，and to extend a multi-agent scenario by introducing a social context，and explore other applications.

References

［1］ C.Adam，2007，Emotions:From Psychological Theories to Logical Formalization and Implementation in a BDI Agent，PhD thesis，Insititut National Polytechnique de Toulouse.

［2］ C.Adam，A.Herzig and D.Longin，2009，“A logical formalization of the OCC theory of emotions”，Sythese，168(2)：201-248.

［3］ T.Bosse and D.Höhle，2011，“Enhancing believability of virtual soccer players：Application of a BDI-model with emotions and trust，developing concepts in applied intelligence”，Studies in Computational Intelligence，363：119-128.

［4］ M.E.Bratman，1987，Intentions，Plans，and Practical Reason，Cambridge：Harvard University Press.

［5］ A.Casali，L.Godo and C.Sierra，2005，“Graded BDI models for agent architecture”，inJ.LeiteandP.Torroni(eds.)，ComputationalLogicinMulti-agentSystems，pp.126-143，Springer.

［6］ A.Casali，L.Godo and C.Sierra，2008，“A logical framework to represent and reason about graded preferences and intentions”，Knowledge Representation，pp.27-37，Menlo Park：AAAI Press.

［7］ A.Casali，L.Godo and C.Sierra，2011，“A graded BDI agent model to represent and reason about preferences”，Artificial Intelligence，175(7-8)：1468-1478.

［8］ P.R.Cohen and H.Levesque，1991，“Teamwork”，Noûs，24(4)：487-512.

［9］ N.Criado，E.Argente and V.Botti，2010，“Normative deliberation in graded BDI agents”，in J.D.C.Witteveen(ed.)，MATES 2010，LNAI，Vol.6251，pp.52-63.

［10］ D.Harel，D.Kozen and J.Tiuryn，2000，Dynamic Logic，Cambridge：The MIT Press.

［11］ J.-J.Ch.Meyer，W.van der Hoek and B.van Linder，1999，“A logical approach to the dynamics of commitments”，Artificial Intelligence，113：1-40.

［12］ D.Pereira，E.Oliveira and N.Moreira，2006，“Modelling emotional BDI agents”，Workshop on Formal Approaches to Multi-Agent Systems，Italy：Riva Del Garda.

［13］ D.Pereira，E.Oliveira，N.Moreira and L.Sarmento，2005，“Towards an architecture for emotional BDI agents”，12th Portu-guese Conference on Artificial Intelligence，pp.40-46.

［14］ D.Pereira，E.Oliveria and N.Moreira，2008，“Formal modeling of emotions in BDI agents”，in F.Sadri and K.Satoh(eds.)，Computational Logic in Multi-Agent Systems，pp.62-81，Springer.

［15］ I.Pezlar，2012，“Humean machine：When desire are in change”，in J.Romportl et al. (eds.)，Beyond AI:Artificial Dreams，pp.1-9.

［16］ M.A.Puic，2013，“Emotional Belief-Desire-intention agent model：Previous work and proposed architecture”，International Journal of Advanced Research in Artificial Intelligence，2(2)：1-8.

［17］ A.Rao and M.Georgeff，1991，“Modeling rational agents within a BDI-architecture”，Proceedings of the 2nd International Conference on Principles of Knowledge Representation and Reasoning(KR-92)，pp.473-484.

［18］ A.Rao and M.Georgeff，1995，“BDI agents：From theory to practice”，Proceedings of the 1st International Conference on Multi-Agents Systems，pp.312-319.

［19］ A.RaoandM.Georgeff，1998，“DecisionproceduresforBDIlogics”，JournalofLogic and Computation，8(3)：293-342.

［20］ R.A.Schmidt，D.TishkovskyandU.Hustadt，2004，“Interactionsbetweenknowledge，action，and commitment with dynamic logic”，Studia Logica，78(3)：381-415.

［21］ B.R.Steunebrink，M.Dastani and J.-J.C.Meyer，2011，“A formal model of emotion triggers：An approach for BDI agents”，Synthese，185：83-129.

［22］ G.S.Thumé and R.E.da Silva，2012，“An extended EBDI model applied to autonomous digital actors”，Proceedings of SBGames 2012.

［23］ W.van der Hoek，B.van Linder and J.-J.Ch.Meyer，1998，“An integrated modal approach to rational agents”，in M.Wooldridge and A.Rao(eds.)，Foundations of Rational Agency，Applied Logic Series，Vol.14，pp.133-168，Dordrecht：Kluwer.

［24］ B.vanLinder，1996，ModalLogicsforRationalagents，PhDthesis，UtrechtUniversity.

［25］ B.van Linder，W.van der Hoek and J.-J.Ch.Meyer，1995，“Actions that make you change your mind：Belief revision in an agent-oriented setting”，in A.Laux and H. Wansing(eds.)，KnowledgeandBeliefinPhilosophyandArtificialIntelligence，pp.103-146，Berlin：Akademie Verlag.

［26］ B.van Linder，W.van der Hoek and J.-J.Ch.Meyer，1997，“Seeing is believing(and so are hearing and jumping)”，Journal of Logic，Language and Information，6：33-61.

［27］ X.Zhang，2012，“Modeling emotional agents based on graded BDI architectures”，in J.Lei et al.(eds.)，Communications in Computer and Information Science，Vol.345，pp.606-616，Springer.

［28］ X.Zhang，M.Jiang，C.Zhou and Y.Hao，2012，“Graded BDI models for agent architectures based on Łukasiewicz logic and propositional dynamic logic”，in F.L.Wang et al.(eds.)，in Proceedings of International Conference on Web Information Systems and Mining，Lecture Notes in Computer Science，Vol.7529，pp.439-450，Springer.

2015-04-23

*This work was supported by the Humanities and Social Sciences Planning Foundation of Chinese Ministry of Education(Grant No.13YJA72040001)，and by the National Natural Science Foundation of China under Grant No.61273338/F030603.