,
, 7.4.2 Propositional Encoding of Constraint Enforcement Operators, p.135
,
, AF
, ? 1 ? Ext ? (AF ) and ? 2 ? Ext ? (AF )
, Case 1 If ? 1 ? Ext ? (AF )
, Ext ? (AF ? form ? (? 1 , ? 2 )) = {? 2 } holds. Therefore, ? 1 AF ? 2 since neither ? 1 ? Ext ? (AF ) nor ? 1 ? Ext ? (AF ? form ? (? 1 , ? 2 )) hold, Case 2 If ? 1 ? Ext ? (AF ) and ? 2 ? Ext ? (AF )
, Ext ? (AF ? form ? (? 1 , ? 2 , ? 3 )) is a non-empty subset of {? 1, Case 3 If ? 1 ? Ext ? (AF ) and ? 2 ? Ext ? (AF ), then since ? satisfies (AE1) and (AE3)
, Ext ? (AF ?form ? (? 1 , ? 2 , ? 3 ))?{? 2 , ? 3 } = Ext ? (AF ?form ? (? 2 , ? 3 )) = {? 3 } which is in contradiction with the fact that ? 2 ? AF ? 3 and ? 2 ? Ext ? (AF ). So this case is impossible. (2) Suppose that Ext ? (AF ? form ? (? 1 , ? 2 , ? 3 )) ? {? 1 , ? 2 } = ?. Since ? 1 ? AF ? 2 and ? 1 ? Ext ? (AF ), ? 1 ? Ext ? (AF ?form ? (? 1 , ? 2 )) holds. From (AE5) and (AE6), Ext ? (AF ?form ? (? 1 , ? 2 , ? 3 ))? {? 1 , ? 2 } = Ext ? (AF ? form ? (? 1 , ? 2 )) holds, = {? 3 } (because from (AE1) and (AE3) it is a non-empty subset of {? 1 , ? 2 , ? 3 }), and from (AE5) and (AE6)
, Ext ? (AF ? form ? (? 1 , ? 2 , ? 3 )) ? {? 1 , ? 3 } = Ext ? (AF ? form ?, ? {? 1 , ? 2 } = ?, from (AE5) and (AE6)
, Let us now show that associating this a pre-order with an argumentation framework is a faithful assignment. 1. Given an argumentation system AF = A, R , let ? 1 , ? 2 be two elements in P (A) such that ? 1 ? Ext ? (AF ) and ? 2 ? Ext ? (AF )
, Like previously, ? 1 ? AF ? 2 holds. Moreover, now ? 2 ? Ext ? (AF ) holds. From this and (AE2), Ext ? (AF ? form ? (? 1 , ? 2 )) = {? 1 } holds, Given an argumentation system AF = A, R , let ? 1
Let ? be a propositional formula over the set of variables {x 1 ,
, If in addition we have AF ? ?| ? gr y, then it must be the case that Ext gr (AF ) = ?. Then, from (AE3), it must be the case that ? is inconsistent, which implies that ? is inconsistent. Conversely, if ? is consistent, then ? is inconsistent as well, From (AE1), it must be the case that AF ? ?| ? gr ?, hence we must have AF ? ?| ? gr ¬y
10 1.2 Inclusion between Dung ,
, An Argumentation Framework with Non-Matching Semantics
, , vol.21
, , p.22
, 27 2.2 Revision of K by ? through a System of Spheres centered on, p.33
, Minimal Models of ? with respect to ? ?
, The Argumentation Framework F 1 and Different Possible Expansions of it, p.50
,
, Enforcing {a 1 } in F 8 is impossible with weak expansion, while F 9 is a succesful strong enforcement
, The Result of an Intervention on the Causal Bayesian Network
, The Argumentation Framework after an Intervention
, Two Possible Explanations for the Observation
, The Input Framework F 1 and the Revising Framework F
, The Result of the Removed Set-based Revision
65 3.15 The k(st)-model F ? of F and its Stable Kernel, the Stable Kernel of ,
, The Argumentation Frameworks F 1 and F
, The k(gr)-expansion F 1 + k(gr) F 2
, 99 5.2 Schematic Explanation of the Revision Process, The k(st)-revision F 1 ? k(str) F
, 7 The Revised Argumentation Framework F ? acc d ?
, Two Possible Results of the Argument-Fixed Enforcement
, Average Time for Strong (×-curve) and Strict Argument-Fixed (+-curve)
, Average Time, n = 200, m Varying from 1 to 70
, Strict Argument-Fixed (+-curve) and Non-Strict Argument-Fixed (?-curve) Enforcement, Average Change for Strong (×-curve)
, The Semantics Dependence Graph of ?
The Non Oriented Graph G 1 ,
,
Schematic Representation of the Polynomial Hierarchy ,
, One of the revised frameworks from AF 18 ? ? 7
, Complexity of Inference Problems for the Usual Semantics
, Hamming Distance between the Models of ? and the Possible Models after the Change . 37 2.2 Semantics of Program Connectives in DL-PA
,
,
,
Rounded at 10 ?2 s, for Non-Strict Strong and Strict ArgumentFixed Enforcement, for n ? {300, 400, 500}, Average Computing Times ,
, 137 7.2 Summary of Existing Approaches of Change in Argumentation, Summary of Existing Approaches of Change in Argumentation, p.137, 2014.
, A.1 Truth Table of the Negation
, Truth Table of the Binary Connectives
147 A.4 The Transition Function ? for the Turing Machine M, A.3 Truth Table of formula ? for the interpretations ? and ? ,
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