T14 Argumentation for agent societies

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14th European Agent Systems Summer School

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T14 Argumentation for agent societies

  1. 1. Argumentation for Agent Societies Part I4%33(5()(%$/&6(%7/5($& !"#"$%&())%*%& 8$(9"#3(*%:&;(&<#"37(%& +,-+.&!/01(%&.$20/)(3& 1
  2. 2. Introduction to the tutorialArgumentation for Agent Societies (Some) answers to the following two questions: 1.  What’s argumentation? ! mainly today 2.  What is argumentation good for (in the MAS context)? ! mainly tomorrow Let’s start with the first question… 2
  3. 3. An informal example (1) We should run Large Hadron Collider The conclusion LHC allows us to Understandingunderstand the Laws the Laws of the The reason of the Universe Universe is good We are justified in believing that we should run LHC ! 3
  4. 4. An informal example (2) We should run Large Hadron Collider The conclusion LHC allows us to Understandingunderstand the Laws the Laws of the The reason of the Universe Universe is good We are justified in believing that we should run LHC ! BUT In Argumentation (and in real life as well): - reasons are not necessary “conclusive” (they don’t logically entail conclusions) - arguments and conclusions can be “retracted” in front of new information, i.e. counterarguments 4
  5. 5. An informal example (3) We should run Large Hadron Collider We should not run LHC LHC allows us to Understanding LHC will generate Destroyingunderstand the Laws the Laws of the black holes Earth of the Universe Universe is good destroying Earth is bad Now we are justified in believing that we should not run LHC " 5
  6. 6. An informal example (4) We should run Large Hadron Collider We should not run LHC LHC allows us to Understanding LHC will generate Destroyingunderstand the Laws the Laws of the black holes Earth of the Universe Universe is good destroying Earth is bad Black holes will not destroy Earth Black holes will evaporate because of Hawking radiation Now we are again justified in believing that we should run LHC ! 6
  7. 7. An informal example (5) We should run Large Hadron Collider We should not run LHC LHC allows us to Understanding LHC will generate Destroyingunderstand the Laws the Laws of the black holes Earth of the Universe Universe is good destroying Earth is bad Hawking radiation Black holes will does not exist not destroy Earth Black holes will Dr Azzeccagarbugli evaporate because says so of Hawking radiation Now we are again justified in believing that we should not run LHC " 7
  8. 8. An informal example (6) We should run Large Hadron Collider We should not run LHC LHC allows us to Understanding LHC will generate Destroyingunderstand the Laws the Laws of the black holes Earth of the Universe Universe is good destroying Earth is bad Hawking radiation Black holes will does not exist not destroy Earth Black holes will Dr Azzeccagarbugli evaporate because says so of Hawking radiation Dr Azzeccagarbugli Now we are again justified is not expert in physics in believing that we should He is a lawyer run LHC ! 8
  9. 9. What s argumentation? (1)[Prakken 2011] Argumentation is the process of supporting claims with grounds and defending them against attack.[van Eemeren et al, 1996] Argumentation is a verbal and social activity of reason aimed at increasing (or decreasing) the acceptability of a controversial standpoint for the listener or reader, by putting forward a constellation of propositions intended to justify (or refute) the standpoint before a rational judge.•  A framework for practical and uncertain reasoning able to cope with partial and inconsistent knowledge - philosophical roots: Aristotle, Toulmin (1958) - in AI: R.P. Loui (1987), J. Pollock (1987), G. Simari & Loui (1992) 9
  10. 10. What s argumentation? (2)The elements of an argumentation system •  The definition of argument (possibly including an underlying logical language + a notion of logical consequence) •  The notion of attack and defeat (successful attack) between arguments •  An argumentation semantics selecting acceptable (justified) arguments 10
  11. 11. Definition of argument: several possibilities (1)•  ASSUMPTION-BASED ARGUMENTATION Given a knowledge base (K, Ass) Consistent theory Set of assumptions ARGUMENT for p: (A, p) such that - A " Ass - A # K is consistent and entails p - There is no A’$A such that A’ # K entails p ATTACKS to an argument: on its assumptions [see Besnard&Hunter, Dung-Kowalski-Toni] 11
  12. 12. Definition of argument: several possibilities (2)•  ARGUMENT SCHEMES - correspond to recurring patterns of reasoning - have associated “critical questions”Example: Expert Testimony [WALTON 1996] E is expert on D E says P P is in D Therefore, P is the case Critical questions: Is E biased? Is P consistent with what other experts say? Is P consistent with known evidence? 12
  13. 13. Definition of argument: several possibilities (3)•  ARGUMENT SCHEMES IN A MEDICAL APPLICATION Viability Scheme Organ O of donor D is available No contraindications are known for donating O to recipient R Therefore, organ O is viable CRITICAL QUESTIONS: Does donor D have a contraindication for donating organ O? Nonviability Scheme Donor D of organ O has condition C C is a contraindication are for donating O Therefore, organ O is nonviable [Tolchinsky et al, 2006] 13
  14. 14. Definition of argument: several possibilities (4)•  STABLE MARRIAGE PROBLEM - Arguments of the kind <Alice, John> - <Barbara, John> attacks <Alice, John> if John prefers Barbara to Alice•  PLANNING - Plans as arguments (that a goal will be achieved) - Defeat between plans as attacks……… In general Arguments take different forms (domain-independent vs. domain dependent) Today examples will refer to rule-based approaches… 14
  15. 15. Rule-based approaches•  ARGUMENT a tree made up of rules of inference constructed from a set of premises to reach a conclusion•  Two kinds of rules: A (0.7) % ¬C (0.7)   A % B: deductive - indefeasible B (0.9)   A ! B: non-deductive - defeasible D (0.9) ! C (0.8)•  A strength value may be associated to premises and rules, giving rise to argument strength See [J.Pollock, 1992], [G. Vreeswijk, 1997], … 15
  16. 16. Rule-based approaches (2)Notion of conflict –  Rebutting: an argument attacks another one by denying its [possibly intermediate] conclusion –  Undercutting: an argument attacks the applicability of a defeasible rule of inference Notion of defeat A An argument defeats ( iff: % ¬C B - undercuts (, or - rebuts ( and D!C is not weaker than ( E!(D&C) [Pollock 92] 16
  17. 17. Rule-based approaches (3)EXAMPLE REBUTTING DEFEAT UNDERCUTTING It’s It’s not DEFEAT Bob raining raining is unreliable Smith says Bob says Bob is drunk it’s raining it’s not raining 17
  18. 18. The ASPIC framework•  One result of the European ASPIC Project (2004-2006)•  Generalizes Pollock’s rule-based approach in several respects: - any logical language (and an associated ‘contrariness function’ generalizing classical negation) can be adopted - can be instantiated by a partial preorder on defeasible rules - premises are distinguished into necessary, ordinary and assumption premises (ordinary and assumption premises partially preordered) - a partial preorder is assumed between arguments•  Besnard & Hunter’s approach, Pollock’s system… can be obtained as instances of ASPIC framework•  See [H. Prakken, “An abstract framework for argumentation with structured arguments”, Argument and Computation, 2010] for details. 18
  19. 19. Argumentation in the context of MAS (1)Advantageous features •  Several kinds of arguments can be represented - epistemic reasoning - practical reasoning •  Able to handle uncertain and partial knowledge - nonmonotonic notion of warrant: 1) wrt further information 2) wrt further reasoning steps (anytime reasoning framework) •  A natural representation + justification of choices (in terms of argument , rebuttal , counterargument …) •  Argumentation has a dialogical side (in terms of argument , attack , defence …) 19
  20. 20. Argumentation in the context of MAS (2)The uses of argumentation (examples) AUTONOMOUS MULTI-AGENT REASONING INTERACTION EPISTEMIC - Belief Revision REASONING (arguing over beliefs) - Trust management (arguing over other - Dialectics in agents reputation) Multiagent Interaction PRATICAL - Decision making REASONING (arguing about the expected value of possible actions) 20
  21. 21. What s argumentation? (3)The elements of an argumentation system •  The definition of argument (possibly including an underlying logical language + a notion of logical consequence) •  The notion of attack and defeat (successful attack) between arguments •  An argumentation semantics selecting acceptable (justified) arguments 21
  22. 22. What s abstract argumentation?Usually “abstract” stands for a difficult thing… Here it means “simple”!The elements of an argumentation system •  The definition of argument (possibly including an underlying logical language + a notion of logical consequence) •  The notion of attack and defeat (successful attack) between arguments •  An argumentation semantics selecting acceptable (justified) arguments Abstract argumentation focuses on this aspect 22
  23. 23. Dung s argumentation framework [Dung ’95] AF = <A, %> attack (or defeat) relation [unspecified definition] Arguments [origin and structure not specified]•  Graphical representation as a directed graph [defeat graph], e.g. Representation of LHC example Representation of weather example 23
  24. 24. Dung s argumentation framework (2) So, what remains to be done? ARGUMENT EVALUATION: GIVEN AN ARGUMENTATION FRAMEWORK, DETERMINE THE JUSTIFICATION STATE (ALSO CALLED DEFEAT STATUS) OF ARGUMENTS, IN PARTICULAR: WHAT ARGUMENTS EMERGE UNDEFEATED FROM THE CONFLICT, I.E. ARE ACCEPTABLE? 24
  25. 25. Argumentation semantics•  Specification of a method for argument evaluation, or of criteria to determine, given a set of arguments, their defeat status SemanticsArgumentation Framework Defeat status Undefeated Defeat status Defeated Provisionally Defeated 25
  26. 26. Labelling vs. extension-based semanticsLABELLING-BASED SEMANTICS - Based on the notion of labelling [assignment to each argument of a label from a predefined set] - Specifies how to derive from an argumentation framework a set of labellings - Justification of arguments derived from the set of labellingsEXTENSION-BASED SEMANTICS - Less general (at least in theory), but more common kind of semantics -  Based on the notion of extension [set of arguments collectively acceptable ] 26
  27. 27. Extension-based semantics Semantics SArgumentation framework AF Set of extensions S(AF) 27
  28. 28. From extensions to defeat statusSet of extensions S(AF) Defeat/Justification StatusA common definition •  Skeptically justified argument: belongs to all of the extensions •  Credulously justified argument: belongs to at least one •  Indefensible argument: does not belong to any extension 28
  29. 29. Unique-status vs. multiple-status semantics Unique-Status Semantics Unique extension: empty set ) () and ( directly unjustified (provisionally defeated)) () Multiple-Status Semantics ) () ) () ! and ( unjustified (provisionally defeated) 29
  30. 30. Relationship between labelling and extension-based approaches•  Almost all approaches adopt the set {IN, OUT, UNDEC} - IN = belonging to the extension - OUT = attacked by the extension - UNDEC= not belonging to nor attacked by the extension Unique-Status Semantics ) () UNDEC UNDEC ) () Multiple-Status Semantics ) () ) () IN OUT OUT IN 30
  31. 31. The core of Dung’s theory: complete “semantics”Acceptability acceptable w.r.t. (“defended by”) S •  all attackers of are attacked by SAdmissible set S S •  conflict-free •  every element acceptable w.r.t. S (defends all of its elements) Complete semantics Complete IF extension also includes all acceptable elements w.r.t. itself All traditional semantics select complete extensions 31
  32. 32. Complete “semantics”: examplesChain Admissible sets: ø, {}, {, *} ) () *) Only one complete extension: CO(AF) = {{, *}}Nixon Diamond ) () All admissible sets are complete ) () CO(AF) = ) () { ø, {}, {(} } ) () 32
  33. 33. Complete “semantics”: examples (2)Nixon Diamond + node Admissible sets: ) () *) ø, {}, {(}, {, *} CO(AF) = { ) () *) ø CO(AF) ) () *) {, *}, ) () *) {(} } 33
  34. 34. The Grounded Semantics: a unique status approach Grounded extension GE(AF): Least complete extension included in all extensions of any traditional semantics Grounded semantics is the “most skeptical” one Undefeated Defeat status Defeated Provisionally Defeated 34
  35. 35. Grounded semantics: examplesChain ) () *) GE(AF) = {, *}Nixon Diamond ) () GE(AF) = øNixon Diamond + node ) () *) GE(AF) = ø 35
  36. 36. Floating arguments: a problem for grounded semantics•  Actually, grounded semantics is polynomially computable•  But sometimes a more discriminative behavior is desirableTHE CASE OF FLOATING ARGUMENTS () () *) +) VS *) +) ) ) Grounded Semantics What we (may) want •  A problem for all possible unique status approaches Let us consider multiple status approaches! 36
  37. 37. Stable SemanticsStable extension = conflict-free set attacking all outside argumentsTHE CASE OF FLOATING ARGUMENTS () () *) +) *) +) ) ) ST(AF) = { {, +}, {(, +} } ! + is justifiedODD-LENGTH CYCLES: A PROBLEM FOR STABLE SEMANTICS ) No stable extension exists! () *) (and also imposing ø is not satisfactory) 37
  38. 38. Stable Semantics: an unsatisfactory patchStable extensions = - conflict-free sets attacking all outside arguments, if there is one - {ø}, otherwise 1) 3) () *) 2) ST(AF) = {ø } ! ( NOT justified!!! 38
  39. 39. Preferred semanticsStable extensions are maximal complete extensions •  conflict-free: by definition •  admissible: every argument attacking an extension is outside ! attacked by the extension itself •  maximal: no argument can be included!Preferred semantics [P.M. Dung, 95]Preferred extension Maximal complete extension = max Set: •  is conflict-free •  defends all of its elements 39
  40. 40. Preferred semantics and floating arguments () *) +)() () *) +) ) *) +)) () ) *) +) )PR(AF) = ST(AF) = { {, +}, {(, +} } ! + is justified () Grounded semantics: *) +) ) 40
  41. 41. Preferred semantics and odd-length cycles) PR(AF) = {ø} () A big difference, isn’t it? ST(AF) = ø*) GE(AF) = {ø}No argument justified w.r.t. grounded and preferred semantics •  As stable semantics, preferred semantics handles the case of floating arguments (differently wrt grounded semantics) •  W.r.t. stable semantics it behaves “better” in the case of odd-length cycles (as the grounded semantics) So, what remains to be done? 41
  42. 42. Semi-stable semantics (1)•  Stable semantics - clashes in some cases (odd-length cycles), however: - a widely applied approach (default logic, stable models of logic programming, answer set programming, etc.) - a very credulous approach: stable extensions are preferred but not viceversa ! justified arguments w.r.t. stable semantics are a (sometimes strict) superset of arguments justified w.r.t. preferred semantics, e.g. ,) ,) *) ) () *) ) () +) +) PR(AF)={{, !}, {"}} ST(AF) = {{, !}} 42
  43. 43. Semi-stable semantics (2)•  Aims at guaranteeing existence of extensions [Verheij’96, (differently from stable semantics) Caminada’06] + coinciding with stable semantics when stable extensions exist (differently from preferred semantics)•  Definition: E- SST(AF) iff E is a complete extension such that (E U {| E% }) is maximal •  Main properties: -  A semistable extension always exists (in the finite case!) since a maximization requirement replaces “aggressive attack” -  If a stable extension E exists, then (E U {| E% }) includes all arguments, therefore semistable extensions # stable extensions -  In any case, semistable extensions are preferred extensions, but the opposite is not always true 43
  44. 44. Semi-stable semantics: examplesExample for existence ) The unique admissible set is empty () ! trivially maximizes (E U {| E% } ) *)Example for backward compatibility (and difference w.r.t. preferred semantics) ,) PR(AF)={{, !}, {"}} *) ) () SST(AF)={{, !}} )= ST(AF) +) 44
  45. 45. CF2 semantics: motivationPreferred/stable/semistable semantics and cycles ) ) () () ) () *) ) ) () () *) A different treatment for even and odd-length cycles. Is it just a matter of symmetry and elegance? 45
  46. 46. Preferred/Semistable Semantics and cycles PR(AF) =) () +1) +2) {{, +1}, {, +2}, VS {(, +2} } 46
  47. 47. Preferred/Semistable Semantics and cycles PR(AF) =) () +1) +2) {{, +1}, {, +2}, VS {(, +2} }() *) +1) +2) PR(AF) = {{+2}}) VS 47
  48. 48. Preferred/Semistable Semantics and cycles PR(AF) = ) () +1) +2) {{, +1}, {, +2}, VS {(, +2} } () *) +1) +2) PR(AF) = {{+2}} ) VS () PR(AF) = {{, *, +2}, ) *) +1) +2) {(, +, +1}, {(, +, +2} } +)NOTE: grounded semantics yields the empty set in all cases 48
  49. 49. Pollock example revisited (1) Jones Smith It’s It’s not unreliable unreliable raining raining Rob unreliable Rob says Smith says Jones says Smith says Bob saysJones unrel. Rob unrel. Smith unrel. it’s raining it’s not raining 49
  50. 50. Pollock example revisited (2) Fred says Jones unrel. Jones unreliable Fred Smith It’s It’s notunreliable unreliable raining raining Rob unreliable Rob says Smith says Jones says Smith says Bob saysFred unrel. Rob unrel. Smith unrel. it’s raining it’s not raining 50
  51. 51. Preferred Semantics and Floating Arguments again… () [ two preferred *) +) extensions] ) VS () [empty set is the unique *) +) .) preferred extension] )NB: same behavior for semistable semantics, stable semantics clashes, grounded semantics yields the empty set in both cases 51
  52. 52. Strongly connected components (SCCs) Equivalence classes under the relation of path-equivalence (mutual reachability)() *) .1) .2)) () *) .1) .2) ) () *) .1) .2) ) 52
  53. 53. Strongly connected components (SCCs) SCCs form an acyclic graph S3 S6S1 S4S2 S5 S7 S1 and S2 are initial SCCs S1 is sccparent of S3, S4 and S5 all other SCCs precede S7 53
  54. 54. CF2 semantics: the definitionE- CF2(AF) iff: - E - MCF(AF) if |SCCSAF| = 1 - / S - SCCSAF (E0S) - CF2(AF UP_AF(S,E)) otherwise S UP_AF(S,E) 54
  55. 55. CF2 semantics and odd-length cycles (1) () *) ) () () () *) *) *) ) ) ) Maximal conflict-free sets 55
  56. 56. CF2 semantics and odd-length cycles (2) () *) .1) .2) ){*,.2}, {,.1}, {,.2}, {(,.1}, {(,.2} Yields several extensions ! all arguments not justified in both cases () ) *) .1) .2) +) {,*,.2}, {(,+,.1}, {(,+,.2} 56
  57. 57. Floating arguments with a three-length cycle() *) +) .)) ()() *) +) .) *) +) .) ))() *) +) .)) Extensions: {*,.}, {,.}, {(,.} Defeat status 57
  58. 58. since we do not want problems in one in relation toknowledge base to affect other, advantageous part of the consistency requirements, as explained in the following. ly unrelated parts of the knowledge base. generates an argumentation framework based on a set of propositional formul Suppose one Stable semantics is therefore not an option. semantics have to be admissibility based? That is,rules desirable that each the propositional formulas express informatio P and a set of defeasible is it D. The idea is that extension an admissible (or even complete) one? Again, it ofthe defeasible rules an ultimate of thumb that can be subject that isA problem is difficult to provide express(1) beyond doubt and CF2 semantics rulesneral: one has to refer to specific contexts. In particular, following knowledge base: exceptions. Now consider the in the context of instantiated nerated from an underlying logical knowledge base, admissibility can be regarded as P ¼ fjw; mw; sw; :ðjt ^ mt ^ stÞg s in relation to consistency requirements, as explained in the following. ne generates•  Considering some examples with structuredmw ) mt; sw ) it turns out that an argumentation framework based on a set ¼ fjw ) jt; arguments, stg D of propositional formulas f defeasible rules D. The idea is that the propositional formulas express information and Suzy want to go cycling on conflict-freenessexpress not entail consistency, e.g. Mary, This example can be interpreted as follows: John,nd doubt and the defeasible rules doesthat John wants to thaton thebe subject tois a reason to believe that John will b tandem. The fact rules of thumb get can tandem (jw) An introduction to argumentation semanticsNow consider the following knowledge base: The same holds for Mary and Suzy. However, since the tandem only has tw 407 on the tandem (jt). seats, they :ðjt ^ mt ^ it with the three of them: :(jt 4 mt 4 st). From this knowledge base, w P ¼ fjw; mw; sw;cannot be onstÞg A5 can then construct the 10 following arguments, based on an argument construction scheme D ¼ fjw ) jt; mw ) mt; sw ) stg Amgoud (2007) and Prakken (2010): presented in Caminada and A1 5 :(jt 4 mt 4 st) ple can be interpreted as follows: John, Mary, and Suzy want to go cycling on a A2 A8 A3 A 5 tandem (jw) is a reason to believe that John will befact that John wants to get on2thejw A3 5 mwm (jt). The same holds for Mary and Suzy. However, since the tandem only has two nnot be on it with the three A4 5 sw :(jt 4 mt 4 st). From this knowledge base, we of them: A1 A5 5 A2 ) jt on an argument construction scheme as struct the 10 following arguments, based A10 A9Caminada and Amgoud (2007)5 A3 ) mt A6 and Prakken (2010): A7 5 A4 ) st4 mt 4 st) A8 5 A6, A7, A1 - :jt A9 5 A5, A7, A1 - :mt A4 A7 A6 A10 5 A5, A6, A1 - :st) jt Assuming the principle of restrictednot enough to obtain consistent conclusions Figure 20 Conflict-freeness is rebutting23 it would then follow that A8 attacks A5, A9, an A10, that A9 attacks A6, A8, and A10, and that A10 attacks A7, A8, and A9. This yields th 58 mt argumentation also semi-stable and preferred extensions). It should be mentioned that the sets of conclusions framework of Figure 20.
  59. 59. A problem of CF2 semantics (2)•  By slightly complicating example, one can find a CF2 extension which is not consistent•  On the other hand, admissibility entails consistency (proved by Caminada & Amgoud 2007, AIJ) ADMISSIBILITY CAN BE A DESIRED REQUIREMENT 59
  60. 60. TO CONCLUDE… GROUNDED PRUDENT STABLE STAGE PREFERRED ROBUST SEMISTABLETOLERANT CF2 SUSTAINABLE IDEALEACH SEMANTICS HAS ITS OWN ROLE… … WHICH ONE IS A GOOD RESEARCH QUESTION…! 60
  61. 61. What about general principles?Here we consider only some semantics - see [Baroni & Giacomin ’06] Grounded Preferred CF2 Semistable CF-principle Yes Yes Yes Yes Admissibility Yes Yes No Yes Reinstatement Yes Yes No Yes Weak reinstatement Yes Yes Yes Yes CF-reinstatement Yes Yes Yes Yes I-maximality Yes Yes Yes Yes Directionality Yes Yes Yes No Weak Skepticism Yes No Yes No Adequacy [all forms] Weak Resolution No Yes No Yes Adequacy [all forms] 61
  62. 62. Applications and principles PRINCIPLES TO BE TO BE STUDIED DEEPENEDAPPLICATION DOMAINS SEMANTICS 62
  63. 63. Semantics and attitudeSKEPTICAL REASONING CREDULOUS REASONING E EE1 W E2 : E1 C E2 : / E2 -E2, 1 E1 -E1 : E1" E2 / E1 -E1, 1 E2 -E2 : E1" E2 63 Fig. 3. S +, S → and S relations for any argumentatio
  64. 64. MANY THANKSFOR YOUR KIND ATTENTION 64
  65. 65. Selected references (1)Landmark argumentation papers and books S. Toulmin, “The Uses of Argument” Cambridge University Press, 1958. R. P. Loui, “Defeat Among Arguments: a System of Defeasible Inference”, Computational Intelligence, vol. 3(3), 1987. J. Pollock, “Defeasible Reasoning”, Cognitive Science, vol. 11(4), 1987. G. Simari & R. P. Loui, “A mathematical treatment of defeasible reasoning and its implementation , Artificial Intelligence, vol. 53(2-3), 1992.Argumentation surveys H. Prakken & G.A.W. Vreeswijk, “Logics for Defeasible Argumentation”, in Handbook of Philosophical Logic, 2nd Edition, Kluwer Academic Publishers, 2001. C.I. Chesnevar, A.G. Maguitman, R.P. Loui, “Logical models of argument”, ACM Computing Surveys, vol. 32(4), 2000. 65
  66. 66. Selected references (2)Argumentation semantics survey P. Baroni, M. Caminada, M. Giacomin “An introduction to argumentation semantics , The Knowledge Engineering Review, vol. 26(4),2011.Books D. Walton, “Fundamentals of critical argumentation , Cambridge University Press, 2006. P. Besnard & A. Hunter, “Elements of Argumentation , MIT Press, 2008. “Argumentation in Artificial Intelligence , edited by I. Rahwan and G. R. Simari, Springer, 2009.Dung s influential paper on abstract argumentation P.M. Dung, “On the Acceptability of Arguments and Its Fundamental Role in Nonmonotonic Reasoning, Logic Programming, and n-Person Games , Artificial Intelligence, vol. 77(2), 1995. 66
  67. 67. Selected references (3)Semantics P. Baroni, M. Giacomin, G. Guida, “SCC-recursiveness: a general schema for argumentation semantics” Artificial Intelligence, vol. 168(1-2), 2005. B. Verheij, “Two approaches to dialectical argumentation:admissible sets and argumentation stages”, Proc. of the 8th Dutch Conference on Artificial Intelligence, 1996 M. Caminada, “Semi-Stable Semantics”, Proc. of 1st International Conference on Computational Models of Arguments (COMMA 2006), 2006 P.M. Dung, P. Mancarella, F. Toni, “A dialectic procedure for sceptical, assumption-based argumentation”, Proc. of 1st International Conference on Computational Models of Arguments (COMMA 2006), 2006 S. Coste-Marquis, C. Devred, P. Marquis, "Prudent Semantics for Argumentation Frameworks", Proc. of 17th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2005), 2005 67
  68. 68. Selected references (4)Semantics H. Jakobovits & D. Vermeir, "Robust Semantics for Argumentation Frameworks", Journal of Logic and Computation 9(2), 1999 P. Baroni, M. Giacomin, G. Guida, “SCC-recursiveness: a general schema for argumentation semantics” Artificial Intelligence, vol. 168(1-2), 2005. P. Baroni, M. Giacomin, “Resolution-based argumentation semantics”, Proc. of 2nd International Conference on Computational Models of Arguments (COMMA 2008), 2008 G.A. Bodanza, F.A. Tohmé, “Two approaches to the problems of self-attacking arguments and general odd-length cycles of attack” Journal of Applied Logic, to appear. P. Baroni, P. Dunne, M. Giacomin, “Computational Properties of Resolution-based Grounded Semantics”, IJCAI 2009, to appear. 68
  69. 69. Selected references (5)General criteria for semantics evaluation and comparison M. Caminada & L. Amgoud, “On the evaluation of argumentation formalisms”, Artificial Intelligence, vol. 171(5-6), 2007. P. Baroni, M. Giacomin, G. Guida, “On principle-based evaluation of extension-based argumentation semantics”, Artificial Intelligence, vol. 171(10-15), 2007. P. Baroni, M. Giacomin, Skepticism relations for comparing argumentation semantics , International Journal of Approximate Reasoning, vol. 50(6), 2009. 69
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  76. 76. !"#$%&()*+,-+",=&/010+,%)<0#,•  M+C5&=#&,)=,2"&-&"&/&1>, –  :01("08$*+,-$/*+,)11&110#,H5)$108050(I,+-,(?&, 1()(&1,+-,(?&,C+"5=@, –  N*50(I,-$/*+,&/+=0#,H"&-&"&/&1,(+,&1*%)(&, (?&,#++=&11,+-,),/+1&G$&/&A,•  2"0/0H5&1,$1&=,(+,/+%H)"&,H)0"1,+-, )5(&")*7&1A,
  77. 77. !"#$%&()*+,-+",=&/010+,%)<0#,•  !)5I*/)5,&OH"&110+1,1$%%)"0P&,(?&, =&/010+A,•  Q)"=,(+,$=&"1()=,C?I,),H"+H+1&=, )5(&")*7&,01,#++=@,+",8&R&"A,•  B&,&&=,),)HH"+)/?,(+,$=&"1()=,(?&, $=&"H00#,+-,(?&,&7)5$)*+A,
  78. 78. !"#$%&()*+,-+",=&/010+,%)<0#,•  B?I,$10#,)"#$%&()*+S, –  TOH5))(+"I,H+C&"@, –  :&/010+1,8)1&=,+,)"#$%&(1,)=,,,,,,,,,,,,,, /+$(&"U)"#$%&(1@, –  !85&,(+,&OH5)0,!"#,),/?+0/&,?)1,8&&,%)=&A,
  79. 79. 9C+,)5(&")*7&,)HH"+)/?&1,•  !%#+$=@,2")=&A,83($E&%#E?5"$*3&A/#&5%C($E& %$;&"F0)%($($E&;"7(3(/$3G,!"*-A,3(&55A,VWXDXUYE>, YVXUYXZ,D[]EA,•  M)<)1@,;+")0*1A,.#E?5"$*%2/$&H%3";&;"7(3(/$& 5%C($E&A/#&%?*/$/5/?3&%E"$*3G&3,2"+/1,+-, !!;!.,[X>,^^XU^]A,
  80. 80. !%#+$=,_,2")=&,)HH"+)/?,•  :&`&,),D/+%H5&(&E,H"&U+"=&"0#,+,),1&(,+-, H+11085&,+H*+1,:A,•  !"#$%&()*+U8)1&=,=&/010+,%)<0#>, –  F+1("$/*#,)"#$%&(1,0,-)7+"a)#)01(,1()(&%&(1, DH&"()00#,(+,8&50&-1,+",=&/010+1E@, –  T7)5$)*#,(?&,1("&#(?,+-,&)/?,)"#$%&(@, –  :&(&"%00#,(?&,=0J&"&(,/+b0/(1,)%+#,)"#$%&(1@, –  T7)5$)*#,(?&,)//&H()8050(I,+-,)"#$%&(1@,, –  F+%H)"0#,=&/010+1,+,(?&,8)101,+-,"&5&7)(, c)//&H(&=d,)"#$%&(1A,
  81. 81. $%&()*+•  ,*-./.012,?)70#,),1$"#&"I,D1#E,+",+(,De1#EA,•  310!)*45*2,(?&,H)*&(,?)1,/+5+0/,H+5IH1A,•  9?&,M4,/+()01,(?&,-+55+C0#,0-+"%)*+>, –  ?)70#,),1$"#&"I,?)1,10=&U&J&/(1f, –  +(,?)70#,1$"#&"I,)7+0=1,?)70#,10=&U&J&/(1f,, –  C?&,?)70#,),/)/&"@,?)70#,),1$"#&"I,)7+0=1,5+11,+-,50-&f, –  0-,),H)*&(,?)1,/)/&",)=,?)1,+,1$"#&"I@,(?&,H)*&(,C+$5=,5+1&,?01, 50-&f, –  (?&,H)*&(,?)1,/+5+0/,H+5IH1f, –  ?)70#,/+5+0/,H+5IH1,%)I,5&)=,(+,/)/&"f, g+)51,50<&>,c+,10=&,&J&/(1d,)=,c(+,+(,5+1&,?01,50-&dA,, ;+"&,0%H+"()(,-+",?0%,(+,+(,5+1&,?01,50-&,(?),(+,+(,?)7&,10=&,&J&/(1A,
  82. 82. 9?&,-")%&C+"<,•  !81(")/(,)"#$%&(U8)1&=,-")%&C+"<A,•  9C+,1(&H1>, –  .(&H,V, •  !"#$%&(1,-+",8&50&-1@, •  !"#$%&(1,-+",+H*+1@, •  4$05=0#,)=,&7)5$)*+,$=&",/5)110/)5,1&%)*/1A, –  .(&H,[, •  F+%H)"01+,+-,H)0"1,+-,+H*+1,$10#,=&/010+, H"0/0H5&1A,
  83. 83. 9?&,-")%&C+"<,•  9?"&&,/5)11&1,+-,=&/010+,H"0/0H5&1>, –  N0H+5)">,, •  L5I,)"#$%&(1,H"+1@, •  L5I,)"#$%&(1,/+1A, –  40H+5)", •  4+(?,(IH&1A, –  h+UH+5)", •  !##"&#)*+,+-,H"+1,)=,/+1,)"#$%&(1,0(+,%&()U )"#$%&(A,
  84. 84. 9?&,-")%&C+"<,•  :&/010+1,1$HH+"(&=,8I,)"#$%&(1>, –  &%H?)10P0#,0(1,H+10*7&,/+1&G$&/&1@,, –  #+)51,1)*1`&=@, –  "&i&/*+1,)7+0=&=A,•  :&/010+1,)R)/<&=,8I,)"#$%&(1>,, –  &%H?)10P0#,0(1,&#)*7&,/+1&G$&/&1@,, –  %011&=,#+)51@, –  /&"()0,"&i&/*+1A,
  85. 85. 9C+,)5(&")*7&,)HH"+)/?&1,•  !%#+$=@,2")=&A,83($E&%#E?5"$*3&A/#&5%C($E& %$;&"F0)%($($E&;"7(3(/$3G&!"*-A,3(&55A,VWXDXUYE>, YVXUYXZ,D[]EA,•  M)<)1@,;+")0*1A,.#E?5"$*%2/$&H%3";&;"7(3(/$& 5%C($E&A/#&%?*/$/5/?3&%E"$*3G&3,2"+/1,+-, !!;!.,[X>,^^XU^]A,
  86. 86. M)<)1,_,;+")0*1,)HH"+)/?,•  ;+=$5)",)#&(,)"/?0(&/($"&>, –  ;+=$5&1,=&=0/)(&=,(+,)#&(j1,/)H)8050*&1@, –  ;+=$5&1,)"&,0=&H&=&(A,•  L7&")55,8&?)70+$",+-,(?&,)#&(>, –  3(&")/*+1,)%+#,=0J&"&(,%+=$5&1@, –  3(&")/*+1,)%+#,(?&0",1&H)")(&,=&/010+1@, –  :&508&")*+,H"+/&11,C0(?0,&)/?,%+=$5&A,
  87. 87. 9?&,%+=&5,•  !"#$%&()*7&,=&508&")*+,%+=$5&A,•  k)"0+$1,=&/010+,%)<0#,("&)(&=,$0-+"%5IA,•  Q0#?,5&7&5,+-,)=)H()8050(I,C?&,(?&, &70"+%&(,/?)#&1A,•  T/+%H)110#,(?&,0b$&/&,+-, –  ,=0J&"&(,"&5)*7&,"+5&1,+-,0(&")/*#,)#&(1@,)=, –  ,(?&,/+(&O(,+-,(?&,H)"*/$5)",0(&")/*+A,
  88. 88. 9?&,%+=&5,•  6+5&1,)=,/+(&O(,)1,=I)%0/,H"&-&"&/&1,+, (?&,=&/010+,H+50/0&1A,•  6+5&1,)=,/+(&O(,"&H"&1&(&=,0,(C+,%+=$5)", H)"(1A,•  !0%0#,(+,H"+70=&,(?&,)#&(,C0(?,"+8$1(&11>, –  0/+%H5&(&,0-+"%)*+,-"+%,&70"+%&(,!, )8=$/*+,C0(?0,)"#$%&()*+,-")%&C+"<f, –  )#&(,)85&,(+,=&508&")(&,+,)5(&")*7&,/?+0/&f, –  )#&(,()<&,=&/010+1,/+=0*+)5,+,)11$%H*+1, )8+$(,&70"+%&(f,
  89. 89. 9?&,%+=&5,•  2&"1+)50(I,%+=$5&, –  8)1&=,+,(?&,1)%&,)"#$%&()*+,-")%&C+"<, $1&=,-+",(?&,+(?&",%+=$5&1@, –  "&H"&1&*#,(?&,H&"1+)50(I,+-,),)#&(,)1,), =&/010+,H+50/I,)//+"=0#,(+,&&=1,)=, %+*7)*+1,+-,)#&(1@, –  )==0*+)5,i$=#%&(,+,=&/010+,H"+85&%1,+-,)I, +&,+-,(?&,+(?&",%+=$5&1,=&H&=0#,+,7)"0+$1, &&=1A,
  90. 90. 9?&,%+=&5,•  9?"&&,5&7&51,0,),)#&(j1,(?&+"I>, –  K0"1(,5&7&5>,"$5&1,9,(?)(,"&-&",=0"&/(5I,(+,(?&,1$8i&/(, =+%)0,+-,(?&,)#&(,D678*-9:)*;*)+,*-./.01+<=)*/+ 0>+9"*+&5*19?+ –  .&/+=,)=,(?0"=,5&7&5>,"$5&1,"&5)(&,(+,(?&,H+50/I, $=&",C?0/?,(?&,)#&(,$1&1,?01,+8i&/(U5&7&5, =&/010+,"$5&1,)//+"=0#,(+,"+5&1,@AB+C+<0)*+ AB.0B.D*/?E+)=,/+(&O(,@A-+:+F019*%9+AB.0B.D*/?G,
  91. 91. $%&()*+"V,>,"&G$&1(l?&5HD!@9@!VE,m,&&=l/++H&")*+D!@9E@, "&5&7)(D9@!VE,"[,>,e"&G$&1(l?&5HD!@9@,!VE,m,"&G$&1(l?&5HD!@9@,![E@,, ,!V,n,![,,6V,>,?lHD"VD!@9@!VE@"VD!@9@![EE,m,%))#&%&(l()1< D9E@%))#&"D!VE,,6[,>,?lHD"VD!@9@,![EE@,"VD!@9@,!VEE,m,(&/?0/)5l()1< D9E@&OH&"(D![E,,FV,>,?lHD6V@6[E,m,%)"<&(l1?)"&l0/"&)1&lH&"0+=,F[,>,?lHD6[@6VE,m,e%)"<&(l1?)"&l0/"&)1&lH&"0+=,
  92. 92. 4&50&-1,6&7010+,
  93. 93. !"#$%&()*+,-+",8&50&-,"&7010+,•  H*).*>+B*;./.01+01,(?&,H"+/&11,+-,/?)#0#,8&50&-1, (+,)=)H(,(?&,&H01(&%0/,1()(&,+-,),)#&(,(+,),&C, H0&/&,+-,0-+"%)*+A,•  IB5=*19&D01,01,/+/&"&=,C0(?,(?&,&7)5$)*+, +-,/5)0%1,8)1&=,+,H"&%01&1,0,+"=&",(+,"&)/?, /+/5$10+1A,•  H09"+0>+9"*>,, –  H"+70=&,8)10/,)=,1$81()*)5,(&/?0G$&1,-+",(?&,)"(,+-, "&)1+0#@,)1,H&"-+"%&=,8I,?$%),8&0#1,0,&7&"I=)I, 50-&,10($)*+1@, –  #+,-)",8&I+=,5+#0/)5,=&=$/*+A,
  94. 94. !"#$%&()*+,-+",8&50&-,"&7010+,•  !#&(,&70"+%&(@,•  4&50&-,"&7010+,=&1/"08&1,, –  (?&,C)I,0,C?0/?,),)#&(,01,1$HH+1&=,(+,/?)#&,?&", 8&50&-1,C?&,&C,0-+"%)*+,)""07&1@,+",/?)#&1,0, (?&,C+"5=,)"&,+81&"7&=f,,•  !"#$%&()*+,=&)51,, –  C0(?,1(")(&#0&1,)#&(1,&%H5+I,-+",(?&0",+C,"&)1+0#@,, –  (+,/?)#&,(?&,8&50&-1,+-,+(?&",)#&(1@,8I,H"+70=0#, "&)1+1,-+",1$/?,/?)#&A,
  95. 95. !"#$%&()*+,-+",8&50&-,"&7010+, !"#$%&()*+,)=,8&50&-,"&7010+,)1, /+%H5&%&()"I,=01/0H50&1A, T)/?,&&=1,(?&,+(?&"j1,1$HH+"(,(+,%+=&5, 1$//&11-$5,=&/010+,%)<0#,0,"&)5,C+"5=, )HH50/)*+1A,
  96. 96. !"#$%&()*+,_,4&50&-,6&7010+,•  !,)"#$%&(,!,-+",o,01,),1&(,+-,0(&""&5)(&=, H0&/&1,+-,<+C5&=#&,1$HH+"*#,o,-"+%, &70=&/&A,•  F5)110/)5,8&50&-,"&7010+>,`O&=,`0(&,5)#$)#&,p, C0(?,),/+%H5&(&,1&(,+-,8++5&),/+&/*7&1A,•  ;)I,=0J&"&(,-")%&C+"<1,-+",8&50&-,"&7010+, C0(?,(?&0","&1H&/*7&,&H01(&%0/,%+=&51A,,
  97. 97. 4&50&-,6&7010+,•  $(./9*.-+04*)>,-+"%)501%,0,C?0/?,8&50&-1,)"&, "&H"&1&(&=@,)=,0,C?0/?,=0J&"&(,<0=1,+-, +H&")(+"1,/),8&,=&`&=A,,•  4)10/,"&H"&1&()*+,+-,&H01(&%0/,1()(&1>, –  7*).*>+/*9/>,1&(1,+-,1&(&/&1,/5+1&=,$=&",5+#0/)5, /+1&G$&/&@,+",, –  7*).*>+7&/*/>,1&(1,+-,1&(&/&1,+(,&/&11)"05I,/5+1&=A, –  LH&")(+"1,H"&1&(&=,0,(C+,C)I1>,, •  8I,#070#,),&OH50/0(,/+1("$/*+,D)5#+"0(?%E,-+",(?&, +H&")(+"@,+", •  8I,#070#,),1&(,+-,")*+)50(I,H+1($5)(&1,(+,8&,1)*1`&=, D/+1(")0(1EA,,
  98. 98. !"#$%&()*+,_,4&50&-,6&7010+,•  <*-*.;.15+1*!+.1>0B&D01>, –  &C,0-+"%)*+,0,=0J&"&(,1?)H&1,)=,-+"%1@,&A#A@,3,01,),H"+H+10*+)5,-)/(f, –  8)10/,!g;,(?&+"I@,)11$%0#,(?&,&H01(&%0/,1()(&,+-,(?&,)#&(,(+,8&,#07&,8I,),8&50&-,1&(f, –  3,%0#?(,8&,%+"&,/+%H5&O@,&G$0HH&=,C0(?,=&#"&&,+-,H5)$108050(I@,+",?)7&,-+"%,+-,),"$5&@,+",), /+%H5&(&,)"#$%&(@,+",+-,),1&(,+-,1$/?,&**&1f,•  $;&)=&D15+1*!+.1>0B&D01>,, –  -+",-$"(?&",H"+/&110#,3@,/"$/0)5,-+",(?&,)#&(,(+,<+C,0(1,+"0#0@,)1,(?01,<+C5&=#&,0b$&/&1, C0550#&11,(+,)=+H(,3f, –  3,8)1&=,+,),+81&"7)*+,%)=&,8I,)#&(,?&"1&5-@,1?&,C055,$1$)55I,8&,/+70/&=,+-,0(,8&0#,("$&f,, –  0-,3,01,/+7&I&=,8I,)+(?&",)#&(@,(?&,)#&(,C055,"&G$0"&,1+%&,i$1*`/)*+,-+",3f, –  )1,%)=)(+"I,1(&H,-+",")*+)5,(?0<0#@,1?&,&7)5$)(&,8+(?,3,)=,H+11085&,i$1*`/)*+,+,(?&, 8)101,+-,?&",+C,8&50&-1@,)=,=&/0=&,0-,3,01,(+,8&,0/+"H+")(&=,0(+,?&",8&50&-1,+",+(f,•  +F"&15.15+7*).*>/>,, –  0-,(?&,)#&(,=&/0=&=,(+,)=+H(,3@,1?&,&%H5+I1,1(")(&#0&1,(+,0/+"H+")(&,3,/+101(&(5I,0(+,?&", 8&50&-1A,K+",(?01@,1?&,?)1,(+,$1&,8&50&-,"&7010+,(&/?0G$&1,(+,/?)#&,?&",&H01(&%0/,1()(&f,•  J1>*B*1-*>,, –  K"+%,&C,&H01(&%0/,1()(&@,(?&,)#&(,=&"07&1,H5)$1085&,8&50&-1,(?)(,#$0=&,?&",8&?)70+$"f,
  99. 99. !"#$%&()*+,_,4&50&-,6&7010+,•  T%8&==0#,0(+,/+%H5&O,"&)1+0#,H"+/&11A,•  9?&,/+%H5&%&()"I,/?)")/(&"1,+-,)"#$%&()*+, )=,8&50&-,"&7010+,8&/+%&,&70=&(>, –  )"#$%&()*+,%)<&1,/+("08$*+1,(+,(?&,&7)5$)*+, 1(&H@, –  8&50&-,"&7010+,01,&%H5+I&=,0,(?&,8&50&-,/?)#&,H)"(A,,–  T7)5$)*+,0/5$=0#,?IH+(?&*/)5,/?)#&, H"+/&11&1>, •  C?)(,C+$5=,?)HH&,0-,(?&,&C,0-+"%)*+,C&"&,(+,8&, 8&50&7&=@,, •  8&50&-,/?)#&,0%H50/0(5I,"&50&1,+,5+#0/)5,50<1,8&(C&&,H0&/&1, +-,0-+"%)*+,C?0/?,/),8&,"&H"&1&(&=,8I,)"#$%&(1A,,
  100. 100. !"#$%&()*+,_,4&50&-,6&7010+,•  4+(?,-"+%,)"#$%&()*+,H"+/&11&1,)=,-"+%, 8&50&-,"&7010+,H"+/&11&1@,H5)$1085&,8&50&-1,/), 8&,+8()0&=A,•  4+(?,)"&)1,-+/$1,+5I,+,H)"(1,+-,(?&,=I)%0/, "&)1+0#,H"+/&11,C?05&,)(,(?&,1)%&,*%&, H"+70=0#,#&&")5,)=,7&"1)*5&,-")%&C+"<1A,,
  101. 101. F+%H)"01+,•  ,.K*B*1-*/+7*9!**1+&B5=*19&D01+&14+7*).*>+B*;./.01>, –  "&H"&1&()*+)5,011$&1@,1I()/*/,)=,1&%)*/,-+$=)*+1,+-,8+(?,)"&)1f,, –  0,1()=)"=,8&50&-,"&7010+@,5+#0/)5,-+"%$5)1,$1&=,-+",M6@,"&1$5(1,+-,/?)#&, H"+/&11&1,)"&,5+#0/)5,-+"%$5)1f, –  &H01(&%0/,1()(&1,)"&,/?)#&=,C0(?,"&1H&/(,(+,&("&/?%&(@,H5)$108050(I@,qf, –  (+,7&"0-I,"&1$5(1,+-,/?)#&,H"+/&11&1@,/5)110/)5,5+#0/)5,1&%)*/1,01,$1&=f, –  )"#$%&()*+,-+/$1&1,+,0(&")/*+1,+-,)"#$%&(1,)1,H0&/&1,+-,0-+"%)*+, (?)(,%)I,)R)/<,+&,)+(?&"@,)=,"&5)*+,8&(C&&,)"#$%&(1,%)I,#07&, H"0+"0(I,(+,+&,)"#$%&(,+",)+(?&"f, –  )"#$%&(1,(?&%1&57&1,)"&,7&"I,?&(&"+#&&+$1f, –  1&%)*/1,%)<&1,H"&/01&,C?)(,#++=,)"#$%&(1,)"&f,•  F001+5B0=14/>, –  8+(?,=01/0H50&1,)0%,)(,"&1+570#,/+b0/(1,C?0/?,)"&,$1$)55I,8)1&=,+,5+#0/)5, #"+$=1@,0A&A@,+,/+(")=0/*+1f, –  %)<&,$1&,+-,H"&-&"&/&,"&5)*+1,(+,)/?0&7&,(?01,)0%f, –  8&50&-,"&7010+,H"+70=&1,),?0#?5I,=&/5)")*7&,-")%&C+"<,-+",(?)(@,8)1&=,+, H+1($5)(&1f, –  )"#$%&()*+,01,%+"&,/+/&"&=,C0(?,H")/*/)5@,i$1*`/)*+U8)1&=, (&/?0G$&1f,
  102. 102. !"#$%&()*+,0,4&50&-,6&7010+,•  Q+C,)"#$%&()*+,(&/?0G$&1,/),8&,$1&=,0, 8&50&-,"&7010+,(?&+"IS, –  r$1*`/)*+U8)1&=,("$(?,%)0(&)/&,1I1(&%1,s:+I5&t>, 0(&")/*+1,8&(C&&,i$1*`/)*+1,C?&,),&C, i$1*`/)*+,?)1,8&&,)==&=@,(+,`=,+$(,C?0/?,/+/5$10+1, /),8&,i$1*`&=A, –  !11$%H*+U8)1&=,("$(?,%)0(&)/&,1I1(&%1,sK)5)HH),&(, )5At>,%))#0#,)11$%H*+1,01(&)=,+-,0%H5&%&*#, /?)#&,H"+/&11&1@,/+%80&,!9;.,0=&),C0(?,8)1&,"&7010+, )=,1I1(&%,$10#,)"#$%&()*7&,1("$/($"&1,0,(?&,-+"%,+-, &OH5))*+1,-+","&7010+1,+-,8&50&-,8)1&A,
  103. 103. 4&50&-,6&7010+,0,!"#$%&()*+,•  Q+C,8&50&-,"&7010+,(&/?0G$&1,/),8&,$1&=,0, )"#$%&()*+,(?&+"IS,•  B+"<1,8I,s6+(1(&0t@,s;+#$055)1<I,&(,)5At,)=,s4+&55),&(,)5At, )%+#,%+1(,/+%H"&?&107&,)HH"+)/?&1,(+,)=="&11,), "&7010+,(?&+"I,-+",)"#$%&(,1I1(&%1A,•  .&7&")5,C)I1,+-,)HH5I0#,8&50&-,"&7010+,0,)"#$%&()*+>, –  F?)#0#,8I,)==0#,+",=&5&*#,),)"#$%&(A, –  F?)#0#,8I,)==0#,+",=&5&*#,),1&(,+-,)"#$%&(1A, –  F?)#0#,(?&,)R)/<,D)=a+",=&-&)(E,"&5)*+,)%+#,)"#$%&(1A, –  F?)#0#,(?&,1()($1,+-,8&50&-1,D)1,/+/5$10+1,+-,)"#$%&(1EA, –  F?)#0#,(?&,(IH&,+-,),)"#$%&(,D-"+%,1("0/(,(+,=&-&)1085&@,+", 70/&,7&"1)EA,
  104. 104. 9?"&&,)5(&")*7&,)HH"+)/?&1,•  K)5)HH)@,M&"U318&"&"@,)=,.0%)"0A,<")("A& -"9(3(/$I&JF0)%$%2/$3&%$;&K"A"%3(H)"& -"%3/$($EG,!"*`/0)5,3(&550#&/&,r+$")5@, VYV>Vu[^@,[[A,•  F)I"+5@,=&,.)0(,FI"@,)=,p)#)1G$0&,./?0&OA, -"9(3(/$&/A&%$&.#E?5"$*%2/$&!L3*"5A,3,2"+/1, +-,M6,[^@,H)#&1,V[YuVXY@,[^A,•  4&-&"?)(@,:$8+01@,)=,2")=&A,=/>&*/&($A"#& A#/5&($7/$3(3*"$*&H")("A3&>(*1/?*&#"9(3($EA,3, 2"+/1,+-,3rF!3,V]]v@,H)#&1,VYY]uVYvv@,V]]vA,
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  108. 108. 9?"&&,)5(&")*7&,)HH"+)/?&1,•  K)5)HH)@,M&"U318&"&"@,)=,.0%)"0A,<")("A& -"9(3(/$I&JF0)%$%2/$3&%$;&K"A"%3(H)"& -"%3/$($EA,!"*`/0)5,3(&550#&/&,r+$")5@, VYV>Vu[^@,[[A,•  F)I"+5@,=&,.)0(,FI"@,)=,p)#)1G$0&,./?0&OA, -"9(3(/$&/A&%$&.#E?5"$*%2/$&!L3*"5A,3,2"+/1, +-,M6,[^@,H)#&1,V[YuVXY@,[^A,•  4&-&"?)(@,:$8+01@,)=,2")=&A,=/>&*/&($A"#& A#/5&($7/$3(3*"$*&H")("A3&>(*1/?*&#"9(3($EA,3, 2"+/1,+-,3rF!3,V]]v@,H)#&1,VYY]uVYvv@,V]]vA,
  109. 109. F)I"+5,&(,)5A,)HH"+)/?,•  2"+H+1&,),:$#U1(I5&,)81(")/(,)"#$%&()*+, 1I1(&%,)55+C0#,(?&,)==0*+,+-,),&C, )"#$%&(,C?0/?,%)I,0(&")/(,C0(?,H"&70+$1, )"#$%&(1A,,•  !"#$%&()*+,-")%&C+"<,⟨!@6⟩,0=&*`&=, C0(?,),)11+/0)(&=,)R)/<,#")H?,gA,•  6&7010+,H"+/&11,H"+=$/&1,),&C,-")%&C+"<, "&H"&1&(&=,8I,),#")H?,gj,)=,),&C,1&(,+-, &O(&10+1A,
  110. 110. 9?&,%+=&5,•  Q+C,(?&,1&(,+-,&O(&10+1,01,%+=0`&=,$=&",(?&,"&7010+, H"+/&11S,•  9IH+5+#I,+-,=0J&"&(,"&7010+1>, –  4*-./.;*+B*;./.01>,+5I,+&,)//&H()85&,1&(,+-,)"#$%&(1,0,(?&, "&701&=,-")%&C+"<@, –  *%(&1/.;*+B*;./.01>,)==1,(?&,&C,)"#$%&(,(+,(?&,&O01*#, &O(&10+1@, –  /*)*-D;*+B*;./.01>,=&/"&)1&,+-,(?&,$%8&",+-,/?+0/&1@, –  L=*/D01.15+B*;./.01>,")01&,)%80#$0(I@,8I,0/"&)10#,(?&,$%8&", +-,&O(&10+1@, –  4*/9B=-D;*+B*;./.01>,"&%+70#,&7&"I,&O(&10+@, –  &)9*B.15+B*;./.01>,1+%&,&O(&10+1,D)55,+-,(?&%E,)"&,)5(&"&=A,
  111. 111. 9?"&&,)5(&")*7&,)HH"+)/?&1,•  K)5)HH)@,M&"U318&"&"@,)=,.0%)"0A,<")("A& -"9(3(/$I&JF0)%$%2/$3&%$;&K"A"%3(H)"& -"%3/$($EG,!"*`/0)5,3(&550#&/&,r+$")5@, VYV>Vu[^@,[[A,•  F)I"+5@,=&,.)0(,FI"@,)=,p)#)1G$0&,./?0&OA, -"9(3(/$&/A&%$&.#E?5"$*%2/$&!L3*"5G,3,2"+/1, +-,M6,[^@,H)#&1,V[YuVXY@,[^A,•  4&-&"?)(@,:$8+01@,)=,2")=&A,=/>&*/&($A"#& A#/5&($7/$3(3*"$*&H")("A3&>(*1/?*&#"9(3($EG&3, 2"+/1,+-,3rF!3,V]]v@,H)#&1,VYY]uVYvv@,V]]vA,
  112. 112. 4&-&"?)(,&(,)5A,)HH"+)/?,•  ;+=&5,+"0&(&=,(+C)"=1,(?&,("&)(%&(,+-, 0/+101(&/I,/)$1&=,8I,(?&,$1&,+-,%$5*H5&, 1+$"/&1,+-,0-+"%)*+A,,•  M+C5&=#&,8)1&1,)"&,1(")*`&=>, –  &)/?,-+"%$5),0,(?&,M4,01,)11+/0)(&=,C0(?,0(1,5&7&5, +-,/&"()0(I,/+""&1H+=0#,(+,(?&,5)I&",(+,C?0/?,0(, 8&5+#1A,
  113. 113. 9?&,%+=&5,•  9C+,/5)11&1,+-,)HH"+)/?&1,(+,=&)5,C0(?,0/+101(&/I,0, M4>,/+?&"&/&,(?&+"0&1,)=,-+$=)*+,(?&+"0&1A,, –  K0"1(,0101(1,+,"&7010#,(?&,M4,)=,"&1(+"0#,/+101(&/I@, –  p)R&",)//&H(1,0/+101(&/I,)=,/+H&1,C0(?,0(A,•  F+?&"&/&,(?&+"0&1,H"+H+1&,, –  (+,#07&,$H,1+%&,-+"%$5)1,+-,(?&,M4,0,+"=&",(+,#&(,+&,+", 1&7&")5,/+101(&(,1$8U8)1&1@, –  (+,)HH5I,/5)110/)5,&()05%&(,+,(?&1&,/+101(&(,1$8U8)1&1, (+,=&=$/&,H5)$1085&,/+/5$10+1,+-,(?&,M4A,,•  K+$=)*+,(?&+"0&1,"&()0,, –  )55,)7)05)85&,0-+"%)*+,, –  &)/?,H5)$1085&,/+/5$10+,0-&""&=,-"+%,(?&,M4,01,i$1*`&=, 8I,1("+#,)"#$%&()*7&,"&)1+1,-+",8&50&70#,0,0(A,
  114. 114. 9?&,%+=&5,•  F5)0%>,, –  0(,=+&1,+(,)5C)I1,%)<&,1&1&,(+,"&701&,), 0/+101(&(,M4@,0,H)"*/$5)"@,0-,0-+"%)*+,/+%&1, -"+%,%$5*H5&,1+$"/&1f, •  +(,&7&,&/&11)"I,(+,"&1(+"&,/+101(&/I,0,+"=&", (+,%)<&,1&1085&,0-&"&/&1,-"+%,),0/+101(&(, M4@,10/&,0-&"&/&,8)1&=,+,)"#$%&()*+,/), =&"07&,/+/5$10+1,)=,"&)1+1,(+,8&50&7&,(?&%@, 0=&H&=&(5I,+-,/+101(&/I,+-,(?&,M4f,
  115. 115. 6&)1+0#,)8+$(,9"$1(,
  116. 116. 9"$1(,0,;!.,•  ;&/?)01%,-+",%))#0#,$/&"()0(I,)8+$(, )$(++%+$1,&**&1,)=,(?&,0-+"%)*+,(?&I, 1(+"&A,•  !#&(1,?)7&,(+,"&)1+,)8+$(,, •  )%+$(,(?)(,(?&I,("$1(,(?+1&,+(?&",&**&1@, •  C?&(?&",(?&I,)"&,("$1*#,(?+1&,&**&1,(+,/)""I, +$(,1+%&,()1<@,+",, •  C?&(?&",(?&I,)"&,("$1*#,(?+1&,&**&1,(+,+(, %01$1&,/"$/0)5,0-+"%)*+A,
  117. 117. 9"$1(,•  F)1(&5-")/?0,)=,K)5/+&>,c%&5"$*%)&3*%*"I&%& 7/50)"F&%M*?;"&/A&%$&%E"$*&F&*/>%#;3& %$/*1"#&%E"$*&L&%H/?*&*1"&H"1%9(/?#N%72/$&%& #")"9%$*&A/#&*1"&E/%)&EdA,•  g)%8&R)>,c*#?3*&(3&*1"&3?HO"729"&0#/H%H()(*L& HL&>1(71&%$&($;(9(;?%)&.&"F0"7*3&*1%*&%$/*1"#& ($;(9(;?%)&<&0"#A/#53&%&E(9"$&%72/$&/$&>1(71& (*3&>")A%#"&;"0"$;3dA,,
  118. 118. 9"$1(,•  p0)$>,c(A&%E"$*&(&H")("9"3&*1%*&%E"$*&O&1%3&*/);& 1(5&*1"&*#?*1&/$&0I&%$;&1"&*#?3*3&*1"& O?;E"5"$*&/A&O&/$&0I&*1"$&1"&>())&%)3/&H")("9"& 0dA,,•  F+%%+,&5&%&(1,)"&, –  /+101(&(,=&#"&&,+-,$/&"()0(I@,)=,, –  /+b0/*#,0-+"%)*+,)11+/0)(&=,C0(?,("$1(A,,
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  127. 127. 9?&,-")%&C+"<,•  K")%&C+"<,(+,0("+=$/&,(?&,1+$"/&1,0,)"#$%&()*+@, )=,(+,&OH50/0(5I,&OH"&11,=&#"&&1,+-,("$1(A,,•  !55,)#&(1,(?)(,?)7&,/+%%$0/)(&=,0-+"%)*+,(+,!#, )"&,%&%8&"1,+-,!#j1,1+/0)5,&(C+"<,•  3(,01,H+11085&,, –  (+,/+1("$/(,),#")H?,C?0/?,"&5)(&1,!#,(+,)55,(?&1&,)#&(1f,, –  (+,)R)/?,),$%&"0/)5,%&)1$"&,(+,&)/?,50<,0,(?01,1+/0)5, &(C+"<,(+,G$)*-I,(?&,&O(&(,(+,C?0/?,),)#&(,("$1(1, (?+1&,(+,C?0/?,0(,01,50<&=,0,(?&,1+/0)5,&(C+"<f,•  .("$/($"&,/)55&=,("$1(,&(C+"<A,
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  131. 131. 9?"&&,)5(&")*7&,)HH"+)/?&1,•  2)"1+1@,9)#@,.<5)"@,F)0@,;/4$"&IA, .#E?5"$*%2/$PH%3";&#"%3/$($E&($&%E"$*3&>(*1& 9%#L($E&;"E#""3&/A&*#?3*G&3,2"+/1,+-,!!;!.,[VVA,•  =),F+1(),2&"&0")@,9&R)%)P0@,k055)()A,F?)#0#, +&j1,;0=>,T")1&,+",6&C0=S,2+11080501*/,4&50&-, 6&7010+,C0(?,K$PPI,!"#$%&()*+,4)1&=,+, 9"$1(A,3,2"+/1,+-,3rF!3,[VV@,H)#&1,VZYUVWV@, [VVA,•  ;)R@,;+"#&@,9+0A,Q/5H($($E&3*%23273&%$;& %#E?5"$*3&*/&7/50?*"&*#?3*G,3,2"+/1,+-,!!;!., [V@,H)#&1,[]U[VZ@,[VA,
  132. 132. =),F+1(),2&"&0"),&(,)5A,)HH"+)/?, ,•  p+11,+-,0-+"%)*+,0,/)1&,+-,"&01()(&%&(,+-, H"&70+$1,0-+"%)*+A,•  2"0/0H5&,+-,cH"0+"0(I,(+,0/+%0#,0-+"%)*+d,•  :")C8)/<1,0,%$5*)#&(,1I1(&%1>, –  /?"++5+#0/)5,1&G$&/&,+-,)""07)5,+-,0-+"%)*+@, –  +(?0#,(+,=+,C0(?,(?&0",("$1()8050(IA,
  133. 133. 9?&,%+=&5 ,•  !"#$%&(1,⟨x@,y⟩,1$HH+"(,)#&(1j,8&50&-1@,•  4&50&-1,)"&,/+/5$10+1,+-,)"#$%&(1@,•  9"$1(C+"(?0&11,%&)1$"&=,8I,$10#, H"+8)8050*&1@4N9,+5I,0-,=)(),)7)05)85&@,•  2+11080501*/,5+#0/,C&55,1$0(&=,(+,=&)5,C0(?, 0/+%H5&(&,0-+"%)*+A,
  134. 134. 9?&,%+=&5 ,•  K$PPI,&7)5$)*+,+-,(?&,)"#$%&(1@,,•  !"#$%&(1,)11+/0)(&=,C0(?,=&#"&&,+-, H5)$108050(I@,,•  9"$1(C+"(?0&11,+-,(?&,1+$"/&,+-,0-+"%)*+,•  !"#$%&(1,&7)5$)(&=,0,#")=$)5,C)I, =&H&=0#,+,=&#"&&,+-,("$1(A,
  135. 135. 9?&,%+=&5 , (#")*$ +,-.,/ ! ! "!"#$% & " 0 !
  136. 136. TO)%H5&, B I saw John killing Mary, thus John killed Mary. A Wit1 If John did not kill Mary, then John is innocent.Judge D Mary was killed before 6 p.m., thus when Mary was killed the show was still to begin C John was at the theater with Corner me when Mary was killed, thus John did not kill Mary. Wit2
  137. 137. TO)%H5&, B A(B) = 0.2 I saw John killing Mary,A(A) = 1.0 thus John killed Mary. A Wit1 If John did not kill Mary, then John is innocent. Judge A(D) = 0.3 D Mary was killed before 6 p.m., thus when Mary was killed the show was still to begin C John was at the theater with Corner me when Mary was killed, thus John did not kill Mary. Wit2 A(C) = 1.0
  138. 138. TO)%H5&, B α(B) = 0.2 I saw John killing Mary,α(A) = 0.8 thus John killed Mary. A Wit1 If John did not kill Mary, then John is innocent. Judge α(D) = 0.3 D Mary was killed before 6 p.m., thus when Mary was killed the show was still to begin C John was at the theater with Corner me when Mary was killed, thus John did not kill Mary. Wit2 α(C) = 0.7
  139. 139. 9?"&&,)5(&")*7&,)HH"+)/?&1,•  2)"1+1@,9)#@,.<5)"@,F)0@,;/4$"&IA, .#E?5"$*%2/$PH%3";&#"%3/$($E&($&%E"$*3&>(*1& 9%#L($E&;"E#""3&/A&*#?3*G&3,2"+/1,+-,!!;!.,[VVA,•  =),F+1(),2&"&0")@,9&R)%)P0@,k055)()A,F?)#0#, +&j1,;0=>,T")1&,+",6&C0=S,2+11080501*/,4&50&-, 6&7010+,C0(?,K$PPI,!"#$%&()*+,4)1&=,+, 9"$1(A,3,2"+/1,+-,3rF!3,[VV@,H)#&1,VZYUVWV@, [VVA,•  ;)R@,;+"#&@,9+0A,Q/5H($($E&3*%23273&%$;& %#E?5"$*3&*/&7/50?*"&*#?3*G,3,2"+/1,+-,!!;!., [V@,H)#&1,[]U[VZ@,[VA,
  140. 140. ;)R,&(,)5A,)HH"+)/?,•  .()"*#,-"+%,z$,)=,.0#?>,)HH"+)/?,(+,("$1(, $10#,:&%H1(&"U.?)-&",8&50&-,-$/*+,=&"07&=, -"+%,1()*1*/)5,=)(),/+/&"0#,()"#&(j1, 8&?)70+$"A,•  TO(&10+,+-,z$,)=,.0#?j1,)HH"+)/?,8I, )55+C0#,&7)5$)(+",(+,()<&,0(+,)//+$(@,0, )==0*+,(+,1()*1*/)5,=)()@,i$1*`&=,/5)0%1, /+/&"0#,&OH&/(&=,8&?)70+$",+-,(?&,()"#&(A,
  141. 141. 9?&,%+=&5,•  F5)0%1,-+"%,8)101,+-,&7)5$)(+"j1,+H00+1A,•  K+"%)55I,"&H"&1&(&=,8I,)"#$%&(1,0,)81(")/(, )"#$%&()*+A,•  9C+,/5)11&1,+-,)"#$%&(1>,, –  >0B*-&/9+&B5=*19/@,0,-)7+$",+",)#)01(,("$1*#, (?&,()"#&(@,)=,, –  .D5&D01+&B5=*19/@,)R)/<0#,-+"&/)1(, )"#$%&(1,+",+(?&",%0*#)*+,)"#$%&(1A,
  142. 142. 9?&,%+=&5,•  ;&(?+=,-+",/+1("$/*#,:&%H1(&"U.?)-&", 8&50&-,-$/*+1,-"+%,1()*1*/)5,=)(),)=,(?&1&, )"#$%&(1A,•  h&C,)"#$%&()*+U8)1&=,8&50&-,-$/*+,0, (&"%1,+-,&C,)"#$%&()*+U8)1&=,&70=&/&, %)11,-$/*+,%l),/+%800#,1()*1*/)5, &70=&/&,)=,)"#$%&(1,)1,&70=&/&A,
  143. 143. 9?&,%+=&5,•  K+",&)/?,=0%&10+,=@,/+10=&",),)81(")/(,!K,K=, /+101*#,+-,(?&,-+55+C0#,)"#$%&(1>, –  K+"&/)1(,)"#$%&(,e(,1$HH+"*#,e9,D0A&A,{)De(E,|,e9E, +,(?&,#"+$=,(?)(,(?&"&,01,+,#$)")(&&,0,(?&,-+"%, +-,C"0R&,/+(")/(,/5)$1&,/+/&"0#,=f, –  K+"&/)1(,)"#$%&(,(,1$HH+"*#,9,D0A&A,{),D(E|9E,+,(?&, #"+$=,(?)(,(?&"&,&O01(1,),#$)")(&&,0,(?&,-+"%,+-,), /+(")/(,/5)$1&,/+/&"0#,=f, –  ;0*#)*+,)"#$%&(,7,)R)/<0#,(,+,(?&,#"+$=,(?)(, (?&,()"#&(,?)1,0,(?&,H)1(,c%+1(,+}&d,70+5)(&=, &O01*#,/+(")/(,/5)$1&1,/+/&"0#,=A,

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