Luís Moniz Pereira Centro de Inteligência Artificial - CENTRIA Universidade Nova de Lisboa, Portugal Pierangelo Dell’Acqua Dept. of Science and Technology - ITN Linköping University, Sweden PADL’03

Motivation  To provide control over the epistemic agents in a Multi- Agent System (eMAS) the need arises to: - explicitly represent its organizational structure, - and its agent interactions.  We introduce a logical framework F, suitable for representing organizational structures of eMAS. - we provide its declarative and procedural semantics. - F having a formal semantics, it permits us to prove properties of eMAS structures.

Our agents We have been proposing a LP approach to agents which can:  Reason on their own or in collaboration  React to other agents and to the environment  Update their own knowledge, reactions, and goals  Interact by updating the theory of any other agent  Decide whether to accept an update subject to the requesting agent  Capture the representation of social evolution

Framework This framework builds on the works:  Updating Agents P. Dell’Acqua & L. M. Pereira - MAS’99  Multi-dimensional Dynamic Logic Programming L. A. Leite & J. J. Alferes & L. M. Pereira - CLIMA’01 and subsequent ones.

Updating agent’s cycle  Updating agent: a rational, reactive agent that dynamically changes its own knowledge and goals. In its cycle, in some order, it:  makes observations  reciprocally updates other agents with goals and rules  thinks (rational)  selects and executes actions (reactive)

Logic framework Atomic formulae: A atom not A default atom generalized rule Formulae: every L i is an atom or a default atom L 0  L 1  L n Integrity constraints Action rules

Agents’ knowledge state sequences  Knowledge states represent dynamically evolving states of an agent’s knowledge. They undergo change due to updates (DLP).  Given the current knowledge state P s, its successor knowledge state P s+1 is produced as a result of the occurrence of a set of parallel updates.  Update actions do not modify the current or any of the previous knowledge states. They affect only the successor state: the precondition of an action is evaluated in the current state, and its postcondition updates the successor state.

MDLP Motivating Example  Parliament issues law L1 at time t1  A local authority issues law L2 at time t2 > t1  Parliamentary laws override local laws, but not vice-versa:  More recent laws have precedence over older ones: L2L1 L2  How to combine these two dimensions of knowledge precedence? ë DLP with Multiple Dimensions (MDLP)

Multi-Dimensional Logic Programming FIn MDLP knowledge is given by a set of programs. FEach program represents a different piece of updating knowledge assigned to a state. FStates are organized by a DAG ( Directed Acyclic Graph ) representing their precedence relation. FMDLP determines the composite semantics at each state according to the DAG paths. FMDLP allows for combining knowledge updates that evolve along multiple dimensions.

MDLP for Agents  Flexibility, modularity, and compositionality of MDLP makes it suitable for representing the evolution of several agents’ combined knowledge How to encode, in a DAG, the relationships among every agent’s evolving knowledge, along its multiple dimensions ?

Two basic dimensions of a MAS Hierarchy of agents Temporal evolution of one agent How to combine these dimensions into one DAG ?

Equal Role Representation  Assigns equal role to the two dimensions:

Time Prevailing Representation  Assigns priority to the time dimension:

Hierarchy Prevailing Representation  Assigns priority to the hierarchy dimension:

Inter- and Intra- Agent Relationships  The above representations refer to a community of agents  But they can be employed as well for relating the several sub-agents of an agent A sub-agent Hierarchy

Intra- and Inter- Agent Example  Prevailing hierarchy for inter-agents  Prevailing time for sub-agents

MDLPs revisited Def. MDLP – Multi-Dimensional Logic Program A MDLP  is a pair (  D,D), where: D=(V,E,w) is a WDAG - Weighted directed acyclic graph and,  D ={P v : v  V} is a set of generalized logic programs indexed by the vertices of D.

Weighted directed acyclic graphs Def. Weighted directed acyclic graph (WDAG) A weighted directed acyclic graph is a tuple D=(V,E,w) : - V is a set of vertices, - E is a set of edges, - w : E  R + maps edges into positive real numbers, - no cycle can be formed with the edges of E. We write v 1  v 2 to indicate a path from v 1 to v 2.

This paper: MDLPs revisited  We generalize the definition of MDLP by assigning weights to the edges of a DAG.  In case of conflictual knowledge, incoming into a vertex v by two vertices v 1 and v 2, the weights of v 1 and v 2 may resolve the conflict.  If the weights are the same both conclusions are false. (Or, two alternative conclusions can be made possible.) v 0.1 v1v1 v2v2 0.2 {a}{not a} [ a ]

Path dominance Def. Dominant path Let a 1  a n be a path with vertices a 1,a 2,…,a n. a 1  a n is a dominant path if there is no other path b 1,b 2,…,b m such that: - b 1 = a 1, b m = a n, and -  i, j such that a i = b j and w((a i-1,a i )) < w((b j-1,b j )).

Example: path dominance Let w((a 5,a 4 )) < w((a 3,a 4 )). Then, a 1, a 2, a 3, a 4 is a dominant path. a1a1 a2a2 a3a3 a4a4 a5a5

Example: formalizing agents Example: Formalize three agents A, B, and C, where: B and C are secretaries of A B and C believe it is not their duty to answer phone calls A believes it the duty of a secretary to answer phone calls  Epistemic agents can be formalized via MDLPs.

Example: formalizing agents  A = (  DA,D A ) D A = ({v 1 },{},w A ) P v1 = {answerPhone  secretary  phoneRing}  B = (  DB,D B ) D B = ({v 3,v 4 },{(v 4,v 3 )},w B ) w B ((v 4,v 3 )) = 0.6 P v3 = {} P v4 = {phoneRing, secretary, not answerPhone}  C = (  DC,D C ) D C = ({v 5,v 6 },{(v 6,v 5 )},w C ) w C ((v 6,v 5 )) = 0.6 P v5 = {} and P v6 = P v4 A B C v3v3 v4v4 B v3v3 v5v5 v6v6 C v5v5 v1v1 A v1v1

Logical framework F Def. Logical framework F A logical framework F is a tuple (A, L, w L ) where: A={  1,…,  n } is a set of MDLPs L is a set of links among the  i and w L : L  R +.

Semantics of F  Declarative semantics of F is stable model based. v 2  v 1 s  Procedural semantics based on a syntactic transformation. Idea: The knowledge of a vertex v 1 overrides the knowledge of a vertex v 2 wrt. a vertex s iff v 1 prevails v 2 wrt. s. Example: P v1 = {answerPhone} P v2 = {not answerPhone} if then M s ={answerPhone} v1v1 v2v2 s

Modelling eMAS  Multi-agent systems can be understood as computational societies whose members co-exist in a shared environment.  A number of organizational structures have been proposed: - coalitions, groups, institutions, agent societies, etc.  In our approach, agents and organizational structures are formalized via MDLPs, and glued together via F.

Modelling eMAS: groups  A group is a system of agents constrained in their mutual interactions.  A group can be formalized in F in a flexible way: - the agents’ behaviour can be restricted to different degrees. - formalizing norms and regulations may enhance trustfulness of the group.

Example: formalizing groups Secretaries example: Formalize group G, of agents A, B, and C, where: B must operate (strictly) in accordance with A, while C has a certain degree of freedom.

Example: formalizing groups  G = (  DG,D G ) D G = ({v 2 },{},w G ) P v2 = {} v2v2 G G F F = (A,L,w L ) A = {  A,  B,  C,  G ) L = {(v 1,v 2 ), (v 2,v 3 ), (v 2,v 5 )} w L ((v 1,v 2 )) = w L ((v 2,v 5 )) = 0.5 w L ((v 2,v 3 )) = 0.7 v3v3 v4v4 B v3v3 v5v5 v6v6 C v5v5 v1v1 A v1v1

Example: semantics v2v2 G v3v3 v4v4 B v3v3 v5v5 v6v6 C v5v5 v1v1 A v1v Model of agent B: M v3 = {phoneRing, secretary, answerPhone} Model of agent C: M v5 = {phoneRing, secretary, not answerPhone} v 1  v 6 v5v5 v 4  v 1 v3v3

Conclusions and future work  Novel logical framework to model structures of epistemic agents: - declarative semantics is stable model based, - procedural semantics based on a syntactical transformation.  To represent F within the theory of each agent: - to empower the agents with the ability to reason about and modify the agents’ structure, - to handle open societies where agents can enter/leave the system.

The End

Prevalence a1a1 anan a2a a1a1 a i-1 anan aiai bmbm b1b Def. Prevalence wrt. a vertex a n Let a 1  a n be a dominant path with vertices a 1,a 2,…,a n. Then, 1. every vertex a i prevails a 1 wrt. a n (1< i  n). 2. if there exists a path b 1  a i with vertices b 1,…,b m,a i and w((a i-1,a i )) < w((b m,a i )), then every vertex b j prevails a 1 wrt. a n a 1  a i anan a 1  b j anan

Links Def. Link Given two WDAGs, D 1 and D 2, a link is an edge between a vertice of D 1 and a vertice D 2.

Joining WDAGs Def. Link Given two WDAGs D 1 and D 2, a link is an edge between vertices of D 1 and D 2. Def. WDAGs joining Given n WDAGs D i = (V i,E i,w i ), a set L of links, and a function w L : L  R +, the joining  ({D 1,…, D n },L,w L ) is the WDAG D=(V,E,w) obtained by the union of all the vertices and edges, and w(e) = w i (e) if e  E i w L (e) if e  L

Joined MDLP Def. Joined MDLP Let F=(A,L,w L ) be a logical framework. Assume that A={  1,…,  n } and each  i =(  Di,D i ). The joined MDLP induced by F is the WDAG  =(  D,D) where: - D=  ({D 1,…, D n },L,w L ) and -  D =  i  Di

Stable models of MDLP Def. Stable models of MDLP Let  =(  D,D) be a MDLP, where D=(V,E,w) and  D ={P v : v  V}. Let s  V. An interpretation M is a stable model of  at s iff: M = least( X  Default(X, M) ) where: Q =  v  s P v Reject(s,M) = { r  P v2 :  r’  P v1, head(r)=not head(r’), M |= body(r’), } X = Q - Reject(s,M) Default(X,M) = {not A :  (A  Body) in X and M |  Body } v 2  v 1 s

Stable models of F Def. Stable models of F Let F=(A,L,w L ) be a logical framework and  the joined MDLP induced by F. M is a stable model of F at state s iff M is a stable model of  at state s.