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artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech r ...
and related fields, an argumentation framework is a way to deal with contentious information and draw conclusions from it using formalized arguments. In an abstract argumentation framework, entry-level information is a set of abstract arguments that, for instance, represent data or a proposition. Conflicts between arguments are represented by a binary relation on the set of arguments. In concrete terms, you represent an argumentation framework with a
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such that the nodes are the arguments, and the arrows represent the attack relation. There exist some extensions of the Dung's framework, like the logic-based argumentation frameworks or the value-based argumentation frameworks.


Abstract argumentation frameworks


Formal framework

Abstract argumentation frameworks, also called argumentation frameworks ''à la Dung'', are defined formally as a pair: * A set of abstract elements called ''arguments'', denoted A * A binary relation on A, called ''attack relation'', denoted R For instance, the argumentation system S = \langle A, R \rangle with A = \ and R = \ contains four arguments (a, b, c and d) and three attacks (a attacks b, b attacks c and d attacks c). Dung defines some notions : * an argument a \in A is acceptable with respect to E \subseteq A if and only if E defends a, that is \forall b \in A such that (b,a) \in R, \exists c \in E such that (c,b) \in R, * a set of arguments E is conflict-free if there is no attack between its arguments, formally : \forall a, b \in E, (a,b) \not\in R, * a set of arguments E is admissible if and only if it is conflict-free and all its arguments are acceptable with respect to E.


Different semantics of acceptance


Extensions

To decide if an argument can be accepted or not, or if several arguments can be accepted together, Dung defines several semantics of acceptance that allows, given an argumentation system, sets of arguments (called ''extensions'') to be computed. For instance, given S = \langle A, R\rangle, * E is a complete extension of S only if it is an admissible set and every acceptable argument with respect to E belongs to E, * E is a preferred extension of S only if it is a maximal element (with respect to the set-theoretical inclusion) among the admissible sets with respect to S, * E is a stable extension of S only if it is a conflict-free set that attacks every argument that does not belong in E (formally, \forall a \in A \backslash E, \exists b \in E such that (b,a) \in R, * E is the (unique) grounded extension of S only if it is the smallest element (with respect to set inclusion) among the complete extensions of S. There exists some inclusions between the sets of extensions built with these semantics : * Every stable extension is preferred, * Every preferred extension is complete, * The grounded extension is complete, * If the system is well-founded (there exists no infinite sequence a_0,a_1,\dots,a_n,\dots such that \forall i>0, (a_,a_) \in R), all these semantics coincide—only one extension is grounded, stable, preferred, and complete. Some other semantics have been defined. One introduce the notation Ext_\sigma(S) to note the set of \sigma-extensions of the system S. In the case of the system S in the figure above, Ext_\sigma(S) = \ for every Dung's semantic—the system is well-founded. That explains why the semantics coincide, and the accepted arguments are: a and d.


Labellings

Labellings are a more expressive way than extensions to express the acceptance of the arguments. Concretely, a labelling is a mapping that associates every argument with a label ''in'' (the argument is accepted), ''out'' (the argument is rejected), or ''undec'' (the argument is undefined—not accepted or refused). One can also note a labelling as a set of pairs (\mathit, \mathit). Such a mapping does not make sense without additional constraint. The notion of reinstatement labelling guarantees the sense of the mapping. L is a reinstatement labelling on the system S = \langle A, R \rangle if and only if : * \forall a \in A, L(a) = \mathit if and only if \forall b \in A such that (b,a) \in R, L(b) = \mathit * \forall a \in A, L(a) = \mathit if and only if \exists b \in A such that (b,a) \in R and L(b) = \mathit * \forall a \in A, L(a) = \mathit if and only if L(a) \neq \mathit and L(a) \neq \mathit One can convert every extension into a reinstatement labelling: the arguments of the extension are ''in'', those attacked by an argument of the extension are ''out'', and the others are ''undec''. Conversely, one can build an extension from a reinstatement labelling just by keeping the arguments ''in''. Indeed, Caminada proved that the reinstatement labellings and the complete extensions can be mapped in a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
way. Moreover, the other Datung's semantics can be associated to some particular sets of reinstatement labellings. Reinstatement labellings distinguish arguments not accepted because they are attacked by accepted arguments from undefined arguments—that is, those that are not defended cannot defend themselves. An argument is ''undec'' if it is attacked by at least another ''undec''. If it is attacked only by arguments ''out'', it must be ''in'', and if it is attacked some argument ''in'', then it is ''out''. The unique reinstatement labelling that corresponds to the system S above is L = \.


Inference from an argumentation system

In the general case when several extensions are computed for a given semantic \sigma, the agent that reasons from the system can use several mechanisms to infer information: * ''Credulous inference'': the agent accepts an argument if it belongs to at least one of the \sigma-extensions—in which case, the agent risks accepting some arguments that are not acceptable together (a attacks b, and a and b each belongs to an extension) * ''Skeptical inference'': the agent accepts an argument only if it belongs to every \sigma-extension. In this case, the agent risks deducing too little information (if the intersection of the extensions is empty or has a very small cardinal). For these two methods to infer information, one can identify the set of accepted arguments, respectively Cr_\sigma(S) the set of the arguments credulously accepted under the semantic \sigma, and Sc_\sigma(S) the set of arguments accepted skeptically under the semantic \sigma (the \sigma can be missed if there is no possible ambiguity about the semantic). Of course, when there is only one extension (for instance, when the system is well-founded), this problem is very simple: the agent accepts arguments of the unique extension and rejects others. The same reasoning can be done with labellings that correspond to the chosen semantic : an argument can be accepted if it is ''in'' for each labelling and refused if it is ''out'' for each labelling, the others being in an undecided state (the status of the arguments can remind the epistemic states of a belief in the AGM framework for dynamic of beliefs).


Equivalence between argumentation frameworks

There exists several criteria of equivalence between argumentation frameworks. Most of those criteria concern the sets of extensions or the set of accepted arguments. Formally, given a semantic \sigma : * \mathit : two argumentation frameworks are equivalent if they have the same set of \sigma-extensions, that is S_1 \equiv_1 S_2 \Leftrightarrow Ext_\sigma(S_1) = Ext_\sigma(S_2) ; * \mathit : two argumentation frameworks are equivalent if they accept skeptically the same arguments, that is S_1 \equiv_2 S_2 \Leftrightarrow Sc_\sigma(S_1) = Sc_\sigma(S_2) ; * \mathit : two argumentation frameworks are equivalent if they accept credulously the same arguments, that is S_1 \equiv_3 S_2 \Leftrightarrow Cr_\sigma(S_1) = Cr_\sigma(S_2). The strong equivalence says that two systems S_1 and S_2 are equivalent if and only if for all other system S_3, the union of S_1 with S_3 is equivalent (for a given criterion) with the union of S_2 and S_3.


Other kinds

The abstract framework of Dung has been instantiated to several particular cases.


Logic-based argumentation frameworks

In the case of logic-based argumentation frameworks, an argument is not an abstract entity, but a pair, where the first part is a minimal consistent set of formulae enough to prove the formula for the second part of the argument. Formally, an argument is a pair (\Phi,\alpha) such that * \Phi \nvdash \bot * \Phi \vdash \alpha * \Phi is a minimal set of \Delta satisfying \alpha where \Delta is a set of formulae used by the agent to reason. One calls \alpha a consequence of \Phi, and \Phi a support of \alpha. In this case, the attack relation is not given in an explicit way, as a subset of the Cartesian product A \times A, but as a property that indicates if an argument attacks another. For instance, * Relation ''defeater'' : (\Psi,\beta) attacks (\Phi,\alpha) if and only if \beta \vdash \neg (\phi_1 \wedge \dots \wedge \phi_n) for \ \subseteq \Phi * Relation ''undercut'' : (\Psi,\beta) attacks (\Phi,\alpha) if and only if \beta =\neg (\phi_1 \wedge \dots \wedge \phi_n) for \ \subseteq \Phi * Relation ''rebuttal'' : (\Psi,\beta) attacks (\Phi,\alpha) if and only if \beta \Leftrightarrow \neg \alpha is a tautology Given a particular attack relation, one can build a graph and reason in a similar way to the abstract argumentation frameworks (use of semantics to build extension, skeptical or credulous inference), the difference is that the information inferred from a logic based argumentation framework is a set of formulae (the consequences of the accepted arguments).


Value-based argumentation frameworks

The value-based argumentation frameworks come from the idea that during an exchange of arguments, some can be ''stronger'' than others with respect to a certain value they advance, and so the success of an attack between arguments depends on the difference of these values. Formally, a value-based argumentation framework is a tuple VAF = \langle A, R, V, \textit, \textit \rangle with A and R similar to the standard framework (a set of arguments and a binary relation on this set), V is a non empty set of values, \textit is a mapping that associates each element from A to an element from V, and \textit is a preference relation (transitive, irreflexive and asymmetric) on V \times V. In this framework, an argument a defeats another argument b if and only if * a attacks b in the "standard" meaning: (a,b) \in R ; * and (\textit(b),val(a)) \not\in \textit, that is the value advanced by b is not preferred to the one advanced by a. One remarks that an attack succeeds if both arguments are associated to the same value, or if there is no preference between their respective values.


Assumption-based argumentation frameworks

In assumption-based argumentation (ABA) frameworks, arguments are defined as a set of rules and attacks are defined in terms of assumptions and contraries. Formally, an assumption-based argumentation framework is a tuple \langle \mathcal, \mathcal, \mathcal, \overline \rangle, where * \langle \mathcal, \mathcal \rangle is a deductive system, where \mathcal is the language and \mathcal is the set of inference rules in the form of s_0\leftarrow s_1, \dotsc, s_m, for m > 0 and s_0, s_1, \dotsc, s_m \in \mathcal; * \mathcal, where \mathcal\subseteq\mathcal is a non-empty set, named the ''assumptions''; * \overline is a total mapping from \mathcal to \mathcal, where \overline is defined as the contrary of a. As a consequence of defining an ABA, an argument can be represented in a tree-form. Formally, given a deductive system \langle \mathcal, \mathcal \rangle and set of assumptions \mathcal\subseteq\mathcal, an argument for claim c\in supported by S\subseteq\mathcal, is a tree with nodes labelled by sentences in \mathcal or by symbol \tau, such that: * The root is labelled by c * For each node N, ** If N is a
leaf node In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be con ...
, then N is labelled by either an assumption or by \tau ** If N is not a leaf node, then there is an inference rule l_N\leftarrow s_1, ... , s_m, (m \geq 0), where l_Nis the label of N and *** If m = 0, then the rule shall be l_N\leftarrow \tau (i.e. child of N is \tau) *** Otherwise, N has m children, labelled by s_1, ... , s_m * S is the set of all assumptions labeling the leave nodes An argument with claim c supported by a set of assumption S can also be denoted as S\vdash c


See also

*
Argument map An argument map or argument diagram is a visual representation of the structure of an argument. An argument map typically includes the key components of the argument, traditionally called the '' conclusion'' and the ''premises'', also called ''con ...
*
Argumentation theory Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be supported or undermined by premises through logical reasoning. With historical origins in logic, dialectic, and rhetoric, argumentation theory, incl ...
*
Defeater A defeater of a belief is evidence that this belief is false. Defeaters are of particular interest to epistemology since they affect whether a belief is justified. An important distinction is between undercutting and rebutting defeaters. Undercu ...
* Diagrammatic reasoning *
Dialogical logic Dialogical logic (also known as the logic of dialogues) was conceived as a pragmatic approach to the semantics of logic that resorts to concepts of game theory such as "winning a play" and that of "winning strategy". Since dialogical logic was the ...
*
Logic and dialectic Formal scientists have attempted to combine logic and dialectic through formalisation. These attempts include pre-formal and partially formal treatises on argument and dialectic, systems based on defeasible reasoning, and systems based on game ...
*
Logic of argumentation The logic of argumentation (LA) is a formalised description of the ways in which humans reason and argue about propositions. It is used, for example, in computer artificial intelligence systems in the fields of medical diagnosis and prognosis, ...
*
Knowledge representation and reasoning Knowledge representation and reasoning (KRR, KR&R, KR²) is the field of artificial intelligence (AI) dedicated to representing information about the world in a form that a computer system can use to solve complex tasks such as diagnosing a medic ...
* Paraconsistent logic *
Probabilistic argumentation Probabilistic argumentation refers to different formal frameworks pertaining to probabilistic logic. All share the idea that qualitative aspects can be captured by an underlying logic, while quantitative aspects of uncertainty can be accounted for b ...


Notes


References

* * * * * * * * * * * {{refend Artificial intelligence