Ensuring Consistency in the Joint Beliefs of Interacting Agents Shamimabi Paurobally Jim Cunningham Nicholas R. Jennings University of Southampton Southampton SO17 1BJ, UK. Imperial College London SW7 2BZ, UK. University of Southampton Southampton SO17 1BJ, UK. [email protected] [email protected] [email protected] ABSTRACT ! " Categories and Subject Descriptors #$%% & '( ) # General Terms * + , Keywords 1. INTRODUCTION Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. AAMAS’03, July 14–18, 2003, Melbourne, Australia. Copyright 2003 ACM 1-58113-683-8/03/0007 ... 5.00. - # ! ! . # ! ! &%/' 0 0 ! 1 # &2' 3 &2' ! 3 4 0 ! 3 3 1 0 ! interacting Interaction Protocols Speech−act like messages Synchronisation Protocols Knowledge−based protocols for message exchange and belief revision 1 0 ! 5 4 ! 67 6 %7 Æ 8 # 9 "! #: &;' &%%' 0 ! 6 7 # ! : ! < 4 3 &=' &2' ! agree offered reject agreed rejected counter−offer Communication Protocols Session, TCP/IP layers closed open = > ? &@' ! A ) + &%/' " @ ; 2. AN INTERACTING GROUP’S BELIEFS 3 # : ! 3 0 $ # A 2.1 Individual and Common Beliefs of a Group " # &?' 0 6 &='7 B ! ! &=' &2' 3 ! ! # ! ! ! ! &2'( 6 7 / 3 3 ! 1 2.2 Consistency of Joint Beliefs 4 ! 5 1 ! < 6 7 6 7 : 6 7 : 6 7 C 6 D%7 6 %7 ! # 6 D%7 6 7 ! ! ! # "# $ % % % * E < &$' 6 &F'7 $ ! 4 # 4 0 ! # # 1 ! 4 # 2.3 Shared Beliefs 3 0 % ! 4 ! % > 6 7 $ = 6 7 & > 6 7 > 6 7 # "# ' 6 7 # 3 $ 2.4 Sufficiency of Shared Beliefs $ # $ ( ! % )$* ! )'* # Æ 6 7 6 7 6 ! ! ! 7 6 7 4 6 7 6 7 6 7 ! 5 6 6 77 6 6 77 6 7 < 6%7 $ . 4 # # % + ! $ 6 7 3. DEFINITIONS AND ASSUMPTIONS # ! # : G 8 ! + : ! , # C ! ! 3 3.1 Assumptions 1 % $ 6 7 . 0 ! % ' 0 % 0 ! &%/' % - 6 %//H 7 % . 3.2 Acknowledgments and Message Structure 3 ( 6 7 0#A <+ IJ8+ &;' K 1 ! 3 ( / / ! + $ # 6 7 ! $ 8 ' $ 3 ' 6 D%7 $ ' $ # ! # &?' 3 )* ! : 6 7 4 6 7 4 6 7 3.3 Terminating an Interaction 3 ! ! 67 >%$ 6 7 6 7 3.4 Formalisation of Protocols 3 C8+ 6 C 8+7 &%/' 6 7 ! C8+ &>' ! ! 3 ! Æ! 0 ! K 1 4 B 2 4 0 ! L9 6 &F' C8+7 0 ! $ ( ! 6& "'7 " ! 6& '7 6 7 6&# ' 6# 7 &#' 6# 7 &# ' 6# 77 0 ! &@' ! C8+ 0 ! % = & # ' ! > 4. GUARANTEED RECEIPT OF MESSAGES # ! 6 = 7 6 ? D 7 K &=' 3 & # ' 6 7 # ! 6 &='7 ! # = A + 4 ! 34 6% $ %/7 %4 6% % %=7 = % # : ! # S1 67 ! S2 A2 S3 A1 A9 A3 S4 A7 A4 S7 A5 S8 A6 S9 A10 S5 termination A8 A11 S6 A12 4.1.1 Proof of Consistency of Joint Belief S 10 A13 open closed interacting % 6# 7 $ 6 7 6 7 = 6 7 > &" '6 7 ? '6# 7 @ &" 6# 7 ; " 6# 7 2 " '6 7 F 6# 7 %/ ' 6 7 % #'6 # 7 $ 6 L9 6 # 77 = #'6 # 7 > 6# 7L9 6 L9 #66 D % # 77 ? '6 D % # 7 @ & # '9 &6 D %7' ; &6 L9 #6# 77 2 '6# 7 F 6 L9 6 D % # 77 9 & ' %/ 6 7L9 6 L9 #6# 77 9 & '9 &/' %% '6 # 7 %$ 6 7L9 66 7L9 6 # 77 %= 6 7L ! # ! ! ! ! 4.1 Proof of Safety of Protocol ( 67 5 ! 3 6 7 ! 5 ( 0 $ + 6 ( + ( 6 D %7 63' %' =7 3 6 37 =7 ! 6 D$7 6 %. %8 7 6 D =7 6 3. 39 7 ! . 6 D =7 # ! 0 " * &%/' 4.1.2 Proof of Termination with Shared Belief 6 >7 % ):$* ( % % ):'* ( % % 6 7 6 7 % % ):-* % % % % % % ):.* % % % 6 7 6 7 % ):7* % 6 7 Group’s beliefs EG st Agent %% ' 6 7 %$ %= 6 7 %> '6 ' 6 77 %? 6# 7 %@ & ' %; ( # # 6 3$$ 7 6 %$7 7 6 3$- 7 5.1 Proof of Safety of Protocol >% 4 6< A7 ! # 6 A7 # # 0 0 ! = ! * # 0 X Agent Y BY s j BX s t−1 st B X st B X ( B Y st BY BX s ) t−1 k BY BX s t−1 B Y st B Y B X s t BY B ( B s X Y t BY BX s ) t−1 Ack 2 Ack 3 k+1 k+2 k+3 k+4 B Y st BY BX s t B Y B X B Y st BY BX BY BX s t B Y B X B Y B X B Y st i.e B Y E E st Ack 4 i.e B X EG EG st Point B Y st Ack 1 B Y st B X(st BY BX s ) t B X B Y B X B Y st EG st EG EG st k+5 ! " G G 5. % 67 % 6 %$ %$- 7 ? ( NON-GUARANTEED RECEIPT OF MESSAGES S1 S2 A2 S3 A1 A3 S5 A9 termination S4 S9 A10 A8 A11 S6 A7 A12 S 10 A4 A6 S7 A5 S8 A16 S13 A13 open A15 A14 S11 interacting A17 S12 closed # D K # = : ! 6 ?7 + K 63$ 3$; 7 9 # %$ %$- ? 5.2 Repeated Messaging and Timeouts A D G! A D 0 D ? %$ %$- % 3$$ 3$' 3$ %$. %$< %$8 D %$7 66 7L9 6# 77 D 3 D # 6 7 * ! # # D . D A D 6. RELATED WORK " &?' &=' &%' 4 &=' ) * # 4 4 1 4 # ! 1 ! ) * : MN 7. CONCLUSION AND FURTHER WORK ) 4 #: * 4 ! 6 7 ! ! 3 ! ! 8. REFERENCES &%' I . * A 3 ! ! %4 %$6?7 %F@F &$' 5 . * O1 ( < K A %F2F &=' * 0 O P 8 8 , = > 8# A %FF? &>' * 5 ( ? <+# %F2; &?' O + Q ( I @%4 =F6=7 %FF$ &@' ) 8 A 4 85 %FF2 &;' (RR A + % A+% % ( &2' O 8 3 , ) ( %+ 3 < K A %FF? &F' A = % 3 , A) # < $//$ &%/' A * < , # %+$7 >=>2 $//$ &%%' O * 5 Q = 0 " % , 8# A %FF> &%$' 3 , 8 3 8 # B + 3 C F?%%% $//$
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