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A logical framework for modelling eMAS. Pierangelo Dell’Acqua Dept. of Science and Technology - ITN Linköping University, Sweden. Luís Moniz Pereira Centro de Inteligência Artificial - CENTRIA Universidade Nova de Lisboa, Portugal. PADL’03. Motivation.
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A logical framework for modelling eMAS Pierangelo Dell’Acqua Dept. of Science and Technology - ITN Linköping University, Sweden Luís Moniz Pereira Centro de Inteligência Artificial - CENTRIA Universidade Nova de Lisboa, Portugal 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: • Reasonon 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.
makes observations • reciprocally updates other agents with goals and rules • thinks (rational) • selects and executes actions (reactive) 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:
Formulae: L0¬ L1 ,... , Ln every Li is an atom or a default atom Logic framework Atomic formulae: Aatom not Adefault atom generalized rule 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 Ps , its successor knowledge state Ps+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.
L2 L1 L1 L2 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: • How to combine these two dimensions of knowledge precedence? • DLP with Multiple Dimensions (MDLP)
Multi-Dimensional Logic Programming • In MDLP knowledge is given by a set of programs. • Each program represents a different piece of updating knowledge assigned to a state. • States are organized by a DAG (Directed Acyclic Graph) representing their precedence relation. • MDLP determines the composite semantics at each state according to the DAG paths. • MDLP 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 ?
Hierarchy of agents Temporal evolution of one agent Two basic dimensions of a MAS 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:
A sub-agent Hierarchy 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
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={Pv : vV} 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 v1 v2 to indicate a path from v1 to v2.
0.2 v1 v2 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 v1 and v2, the weights of v1 and v2 may resolve the conflict. • If the weights are the same both conclusions are false. (Or, two alternative conclusions can be made possible.) [ a ] v 0.1 {a} {not a}
Path dominance Def. Dominant path • Let a1 an be a path with vertices a1,a2,…,an. • a1 an is a dominant path if there is no other path b1,b2,…,bm such that: • b1= a1, bm= an, and • - i, j such that ai= bj and w((ai-1,ai)) < w((bj-1,bj)).
Example: path dominance a4 Let w((a5,a4)) < w((a3,a4)). Then, a1, a2 , a3, a4 is a dominant path. a3 a5 a2 a1
Example: formalizing agents • Epistemic agents can be formalized via MDLPs. • 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
A v1 v1 B v3 v3 C v4 v5 v5 v6 Example: formalizing agents A = (DA,DA) DA = ({v1},{},wA) Pv1 = {answerPhone secretary phoneRing} B = (DB,DB) DB = ({v3,v4},{(v4,v3)},wB) wB((v4,v3)) = 0.6 Pv3 = {} Pv4 = {phoneRing, secretary, not answerPhone} C = (DC,DC) DC = ({v5,v6},{(v6,v5)},wC) wC((v6,v5)) = 0.6 Pv5 = {} and Pv6 = Pv4 A B C
Logical framework F Def. Logical framework F • A logical framework F is a tuple (A, L, wL) where: • A={1,…,n} is a set of MDLPs • L is a set of links among the i • and wL : L R+.
s v2 v1 v2 v1 s Semantics of F • Declarative semantics of F is stable model based. Idea: The knowledge of a vertex v1overrides the knowledge of a vertex v2 wrt. a vertex s iff v1 prevails v2 wrt. s. Example: Pv1= {answerPhone} Pv2= {not answerPhone} if then Ms={answerPhone} • Procedural semantics based on a syntactic transformation.
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.
A v1 v1 B v3 v3 G C v2 v4 v5 v5 v6 Example: formalizing groups F = (A,L,wL) A = {A,B,C,G ) L = {(v1,v2), (v2,v3), (v2,v5)} wL((v1,v2)) = wL((v2,v5)) = 0.5 wL((v2,v3)) = 0.7 G = (DG,DG) DG = ({v2},{},wG) Pv2 = {} G F
A v1 v1 v1 v6 v5 B v3 v3 G 0.5 v4 v1 v3 0.5 0.6 C v4 v2 0.7 v5 v5 0.6 v6 Example: semantics Model of agent B: Mv3= {phoneRing, secretary, answerPhone} Model of agent C: Mv5= {phoneRing, secretary, not answerPhone}
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.
an an . . . ai . . . a2 bm ai-1 a1 . . . . . . b1 a1 Prevalence Def. Prevalence wrt. a vertex an Let a1 an be a dominant path with vertices a1,a2,…,an. Then, 1. every vertex aiprevails a1 wrt. an (1< i n). 2. if there exists a path b1 ai with vertices b1,…,bm,ai and w((ai-1,ai)) < w((bm,ai)), then every vertex bjprevails a1 wrt. an. 1. 2. a1 ai a1 bj an an
Links Def. Link Given two WDAGs, D1 and D2, a link is an edge between a vertice of D1 and a vertice D2.
Joining WDAGs Def. Link Given two WDAGs D1 and D2, a link is an edge between vertices of D1 and D2. Def. WDAGs joining Given n WDAGs Di = (Vi,Ei,wi), a set L of links, and a function wL : L R+, the joining({D1,…, Dn},L,wL) is the WDAG D=(V,E,w) obtained by the union of all the vertices and edges, and w(e) = wi(e) if eEi wL(e) if eL
Joined MDLP Def. Joined MDLP • Let F=(A,L,wL) be a logical framework. • Assume that A={1,…,n} and each i=(Di,Di). • The joined MDLP induced by F is the WDAG =(D,D) where: • - D= ({D1,…, Dn},L,wL) and • - D= iDi
v2 v1 s Stable models of MDLP Def. Stable models of MDLP Let =(D,D) be a MDLP, where D=(V,E,w) and D={Pv : vV}. Let s V. An interpretation M is a stable model of at s iff: M = least( X Default(X, M) ) where: Q = v sPv Reject(s,M)= { r Pv2 : r’ Pv1, 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}
Stable models of F Def. Stable models of F Let F=(A,L,wL) 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.
