1. Social Behavior of Agents and Stable Models Francesco Buccafurri and Gianluca Caminiti DIMET, Università degli Studi “Mediterranea” di Reggio Calabria Convegno Italiano di Logica Computazionale (CILC’05) Roma, 21-22 Giugno 2005

2. 2CILC'05, Roma, 21-22 Giugno 2005 MAS & Logic Programming Agents  Logic Programs Agents  Logic Programs Desires/Requests  Fixpoints The behavior of one agent can depend on that of the other agents. Social Ability: interaction enables reasoning on agents’ mental states.

3. 3CILC'05, Roma, 21-22 Giugno 2005 Social-Oriented Reasoning/0 a1a1a1a1 A |A| = N

4. 4CILC'05, Roma, 21-22 Giugno 2005 Social-Oriented Reasoning/1 head ← [l,h]{body} a1a1a1a1 A

5. 5CILC'05, Roma, 21-22 Giugno 2005 Social-Oriented Reasoning/3 head ← [l,h]{body} a1a1a1a1 S l  |S|  h A

6. 6CILC'05, Roma, 21-22 Giugno 2005 Social-Oriented Reasoning/4 head ← [l,h]{body} a1a1a1a1 S l  |S|  h A

7. 7CILC'05, Roma, 21-22 Giugno 2005 Social-Oriented Reasoning/4 head ← [l,h]{body} a1a1a1a1 S l  |S|  h A

8. 8CILC'05, Roma, 21-22 Giugno 2005 Social-Oriented Reasoning/5 head ← [a 2 ]{body} a1a1a1a1 A a2a2a2a2

9. 9CILC'05, Roma, 21-22 Giugno 2005 Social-Oriented Reasoning/5 a1a1a1a1 A a2a2a2a2 head ← [a 2 ]{body}

10. 10CILC'05, Roma, 21-22 Giugno 2005 Social-Oriented Reasoning/5 a1a1a1a1 A a2a2a2a2 head ← [a 2 ]{body}

11. 11CILC'05, Roma, 21-22 Giugno 2005 Social-Oriented Reasoning/6 head ← [l 1,h 1 ]{body 1, [l 2,h 2 ]{body 2 }} a1a1a1a1 A

12. 12CILC'05, Roma, 21-22 Giugno 2005 Social-Oriented Reasoning/6 a1a1a1a1 S1S1S1S1 l 1  |S 1 |  h 1 l 2  |S 2 |  h 2 S 2  S 1 A S2S2S2S2 head ← [l 1,h 1 ]{body 1, [l 2,h 2 ]{body 2 }}

13. 13CILC'05, Roma, 21-22 Giugno 2005 Example 1: a Wedding Party P 1 (Agent 1 ) party ← [ N/2 - 1, ]{party} P 2 (Agent 2 ) okay(party) ← okay(drive) ← party P 3 (Agent 3 ) party ← [Agent 2 ]{party, not drive} P 4 (Agent 4 ) empty program

14. 14CILC'05, Roma, 21-22 Giugno 2005 Example 1: Intended Models {},{}, {party P 1, party P 2, drive P 2 },{party P 1, party P 2, drive P 2 }, {party P 1, party P 2, party P 3 }.{party P 1, party P 2, party P 3 }.

15. 15CILC'05, Roma, 21-22 Giugno 2005 Example 2: a P2P Scenario download(X) ← [min, ]{ share(X), [1,]{not incomplete(X)} }, file(X) okay(share(X)) ← [0.33*N, ]{ share(X), [0.1*N,0.2*N]{high_bw} }, file(X)

16. 16CILC'05, Roma, 21-22 Giugno 2005 Syntax: Social Rules h ← body body = b 1, …, b m, s 1, …, s k (m ≥ 0, k ≥0) body = b 1, …, b m, s 1, …, s k (m ≥ 0, k ≥ 0) h - literal or okay(p) h - literal or okay(p) b i (1 ≤ i ≤ m) - (possibly NAF) literal b i (1 ≤ i ≤ m) - (possibly NAF) literal s j (1≤ j ≤ k) - (possibly NAF) SSC s j (1 ≤ j ≤ k) - (possibly NAF) SSC SOLP program = set of social rules SOLP collection = set of SOLP programs

17. 17CILC'05, Roma, 21-22 Giugno 2005 Syntax: SSC [selection_condition]{body} [selection_condition] [l, h] [agent_id] Warning:[1, 1] ≠ [id]. Note: SSCs can be nested.

18. 18CILC'05, Roma, 21-22 Giugno 2005 Semantics: Autonomy P - SOLP program Var(P) - atoms occurring in P AP - autonomous reduction of P AT P - extends classical T P to social rules AFP(P) = set of autonomous fixpoints of P

19. 19CILC'05, Roma, 21-22 Giugno 2005 Semantics: SOLP Collection C ={P 1,..., P n } - a SOLP collection Social Interpretation for C - I = I P 1  …  I P n I P j - a labeled interpretation for P j  C (1 ≤ j ≤ n) Candidate Social Interpretations for C – U(P 1, …, P n ) = {F 1 P 1  …  F n P n | F i  AFP(P i ), 1 ≤ i ≤ n}

20. 20CILC'05, Roma, 21-22 Giugno 2005 Semantics: literals C ={P 1,..., P n } - a SOLP collection I - a social interpretation for C a  Var(P i ), P i  C (resp. not a) is true w.r.t. I if a P i  I (resp. a P i  I) if a P i  I (resp. a P i  I)

21. 21CILC'05, Roma, 21-22 Giugno 2005 Semantics: SSCs [selection_condition]{body} is true w.r.t. I if a set of agents exists s.t. 1) selection_condition holds, and 2) body is true w.r.t. I C ={P 1,..., P n } - a SOLP collection I - a social interpretation for C

22. 22CILC'05, Roma, 21-22 Giugno 2005 Semantics: Social Rules C ={P 1,..., P n } - a SOLP collection I - a social interpretation for C h ← body - a social rule r of P  C r is true w.r.t. I if either: h is true w.r.t. I, or h is true w.r.t. I, or body is false w.r.t. I. body is false w.r.t. I.

23. 23CILC'05, Roma, 21-22 Giugno 2005 Semantics: Social Models I  U(P 1, …, P n ) is a social model of C if  r  P 1  …  P n, r is true w.r.t. I C ={P 1,..., P n } - a SOLP collection I - a social interpretation for C

24. 24CILC'05, Roma, 21-22 Giugno 2005 Translation: Overview Goal: to build a single logic program J(C) whose stable models are in 1:1 correspondence with the social models of C. C ={P 1,..., P n } - a SOLP collection

25. 25CILC'05, Roma, 21-22 Giugno 2005 Translation: Step-by-step Each SOLP program is rewritten as a classical one, P’, where the SSCs are represented by special literals. Each SOLP program is rewritten as a classical one, P’, where the SSCs are represented by special literals. Conventions on atom names are used in order to know which program a given atom comes from. Conventions on atom names are used in order to know which program a given atom comes from. All the SSCs in C are rewritten as sequences of DLP A aggregate functions in a logic program S. All the SSCs in C are rewritten as sequences of DLP A aggregate functions in a logic program S.

26. 26CILC'05, Roma, 21-22 Giugno 2005 Translation: Final Result J(C) = (  P  C P’)  S The logic program J(C) selects, among all the candidate social interpretations for C, those w.r.t. which all the SSCs in C are true. Feasible in polynomial time

27. 27CILC'05, Roma, 21-22 Giugno 2005 Translation: Theorem J(C) = (  P  C P’)  S A one-to-one correspondence exists between the social models in SOS(P 1,..., P n ) and the stable models of J(C). C ={P 1,..., P n } - a SOLP collection SOS(P 1,..., P n ) - the social models of C

28. 28CILC'05, Roma, 21-22 Giugno 2005 Social Models & Joint Fixpoints COLP* program = logic program + okay rules COLP program  desires/consents of a single agent Semantics of a set of COLP programs (Joint Fixpoint Semantics*): common agreement among the agents. * F. Buccafurri and G. Gottlob. Multiagent Compromises, Joint Fixpoints and Stable Models, volume 2407 of LNCS and LNAI. Springer, 2002

29. 29CILC'05, Roma, 21-22 Giugno 2005 Social Semantics extends JFP From: COLP programs (JFP Semantics) To:SOLP programs (Social Semantics) Feasible in polynomial time

30. 30CILC'05, Roma, 21-22 Giugno 2005 Social Models & JFP: Theorem A one-to-one correspondence exists between the Joint Fixpoints of P 1,..., P n and the social models of SOS(Q 1,..., Q n ). P 1,..., P n - COLP programs Q 1,..., Q n - SOLP programs (the translation of P 1,..., P n )

31. 31CILC'05, Roma, 21-22 Giugno 2005 SOS Existence: Theorem Theorem The problem SOS n is NP complete. Problem SOS n Instance: A SOLP collection P 1,..., P n Question: Is SOS(P 1,..., P n )   ?

32. 32CILC'05, Roma, 21-22 Giugno 2005 Conclusions SOcial Logic Programming (SOLP) enables social behavior among a community of agents. SOcial Logic Programming (SOLP) enables social behavior among a community of agents. SOLP extends COLP. SOLP extends COLP. Complexity results. Complexity results.

33. 33CILC'05, Roma, 21-22 Giugno 2005 Finally… Thank You!