The aim of this paper is to map out the relation between different approaches for handling preferences in assumption-based argumentation (in short, ) . The orthodox approach in , that we call direct, defines defeats (among sets of assumptions) as attacks from assumptions that are at least as preferred as the assumption under attack. The fact that admits asymmetric contrariness relations, though, makes preference-handling more difficult: this asymmetry is preserved on the level of attacks and then defeats, possibly leading to inconsistencies. In order to re-establish consistency, the framework was recently proposed in  to handle preferences in . adds reverse defeats as passive counterparts to direct defeats: if an assumption is attacked from less preferred assumptions a reverse attack is initiated. Therefore, it seems fruitful to investigate the exact relation between systems that are equipped with a reverse defeat and systems that only make use of direct defeats. In this paper, we contribute to this line of research by studying two questions. First, we investigate under which conditions equipped with direct but not reverse defeat satisfies the consistency postulate. Thereafter, we investigate the relationship between these two frameworks by providing translations.
Outline of the paper: In Section 2 we review the different versions for defined by: non-prioritized defeats —i.e. attacks (), preference-based defeats (), and preference-based defeats extended with reverse defeat (). In Section 3 we motivate the translations by showing first that is well-behaved and secondly that and give rise to incomparable outcomes. Then in Section 4, we provide first a translation from to . In Section 5 we show and are conservative extensions of . This result also extends the translation from Section 4 into . In Section 6, we complete the cycle by providing a direct translation from to . The contributions of this paper can be summarized in the following diagram:
2 Assumption-Based Argumentation
ABA, thoroughly described in , is a formal model on the use of plausible assumptions used “to extend a given theory” [2, p.70] unless and until there are good arguments for not using (some of) these assumptions.
Inferences are implemented in ABA by means of rules formulated over a formal language. Furthermore, defeasible assumptions are introduced, together with a contrariness operator to express argumentative attacks. We adapt the definition from  for an assumption-based framework as follows:
Definition 1 (Assumption-based framework).
An assumption-based framework is a tuple of the form , where:
is a formal language (consisting of countably many sentences).
is a set of inference rules of the form or , where .
is a non-empty set of candidate assumptions.
is a contrariness operator.
The members of are called values and we require that and .
is a preorder over the values.
is a function assigning values to the assumptions111In , a preference order is defined directly over the assumptions. It will, however, greatly increase readability to use values to express priorities in this paper. Clearly, these modes of expression are equivalent..
As usual, we define as the inverse of , and define iff and . An without priorities is simply defined as a tuple .222If needed, one can identify an without priorities with a trivial prioritized given by for all .
In some presentations of , deductions are obtained from a set of strict premises , a set of plausible assumptions and a set of rules . Here we follow , by rewritting each strict premise as an empty-bodied rule (contained in the set of rules ).
The previous definition generalizes the contrariness function in , from a single contrary , to a set of contraries . (Although in our examples, for the sake of simplicity, will denote an arbitrary member of .) The reason for this generalization is to avoid clutter for the translations presented. 333If one is interested in reducing a set of contraries to a single contrary , one can simply add the rule for every , cf. [11, p. 109].
Definition 2 (-deduction).
Given and a set , an -deduction from of , written , is a finite tree where
the root is ,
the leaves are either of the form , where , or elements from ,
the children of non-leaf nodes are the conclusions of rules in whose antecedents correspond to their own parents,
is the set of all that occur as nodes in the tree.
Note that for flat s, if then will be the set of all occurring as leaves in the tree. The following example shows that for non-flat s we also have to consider non-leaf nodes.
Let be given by: and the set of rules . Note that there is no deduction since appears as a node in any derivation of . We have both , whose tree only consists of the root , and with root and unique leaf .
We define various ways to lift to sets of assumptions.
Definition 3 (-minimal set).
Given an assumption-based framework and , we define and:
The intuition behind is to close under incomparable elements: includes all the elements that are incomparable to at least one element of .
Definition 4 (Lifting of ).
Given an assumption-based framework , , we define 444It is not necessary to consider the lifting: iff for some , . It can be proved that and coincide: iff . Furthermore, notice that .
|iff||for some ,|
|iff||for all ,|
|iff||for all ,|
For any such that over is total, the three liftings , and coincide. From here on, then, when is a total order, we will simply use to denote any of its liftings: . The following example shows that all of these lifting principles give rise to different outcomes when considering a non-total preorder.
Let be a set of values with and given by the following figure (where a line means that the upper value is more preferred than the lower value, e.g. ). We have the following:
Definition 5 (Attack, defeat, reverse defeat).
Given a framework , a lifting and ,
|attacks (with )||iff||there is such that for some|
|--defeats||iff||attacks with some such that|
We also say that attacks iff attacks some ; and similarly for --defeats . Finally, we say that
|--defeats555We follow  in letting reverse defeat only if . However, we do not see any conclusive reason why we should not let reverse defeat only if . We leave the investigation of this alternative form of --defeat for future work.||iff|
In the context of ABA without priorities, attack coincides with -defeat, so we will sometimes write -defeat instead of attack to avoid confusion. From here on, , and denote assumption-based argumentation using, respectively -, - and -defeats.
Definition 6 (-closure).
Given an , where and , we define:
|iff||there is a sequence with , and for|
|or is obtained by an application of a rule|
|(-closed sets within ).|
Finally, we say that is -closed iff .
The consequences of a given are determined by the argumentation semantics. On the basis of argumentative attacks, the semantics determine when a set of assumptions is acceptable. Informally, an acceptable set should at least not attack itself, and it should be able to defend itself against attacks from other sets of assumptions. Argumentation semantics, originally defined for abstract frameworks in , have been reformulated for ABA in e.g. .
Definition 7 (Argumentation semantics ).
Given a framework , a lifting and sets , we define for and each :
|is ---conflict-free||iff||for no , --defeats|
|is ---naive||iff||is -closed and -maximally ---conflict-free|
|---defends||iff||for any that ---defeats ,|
|there is such that --defeats|
|is ---admissible||iff||is -closed, ---conflict-free|
|and ---defends every|
|is ---complete||iff||is ---admissible|
|and contains every it ---defends|
|is ---preferred||iff||is -maximally ---admissible|
|is ---grounded||iff||is -minimally ---complete|
|is ---stable||iff||is -closed, ---conflict-free|
|and --defeats every|
We will denote naive, grounded, preferred resp. stable by , , , . For any semantics , we define as the sets of assumptions that are ---, as defined above. 666Since the order does not matter in any semantics or ---, we will simply write this as and, resp., --.
In many papers (e.g. [2, 5]), a set is admissible if it can defend itself from every -closed set of assumptions that defeats . In the context of priorities, however, this might not always be the most intuitive outcome, as demonstrated by Ex. 3. Therefore, we define both semantics where this requirement is enforced (setting in Def. 7) and semantics where defeaters are not required to be closed (setting in Def. 7).
Let be given by , , , , with and . For any , we have one ---complete set: . To see that is complete, observe that is the only closed set that --defeats . Since --defeats , defends itself from . When we move to ---complete sets, the situation changes: in that case only is ---complete. To see this, note that --defeats and does not --defeat .
One might ask if it is more intuitive to have and as complete extensions, or just (which contains the -maximal element ). Here we study both options. This example motivates studying defeaters that are not closed under the full set , as in Ex. 3. In Section 6, we will see an example of an whose defeaters should be closed under a proper subset of the set of rules . For another example of semantics parametrized with a set of rules, although for different purposes, see .
3 Some considerations on and
In this section we motivate the translations given in Sections 4 and 6. First we show that is well-behaved: it satisfies the postulate of Consistency under the assumption of contraposition. Secondly, we show that even with contraposition, and might produce incomparable outcomes.
and Conflict Preservation
In , several rationality postulates were proposed for structured argumentation systems. These postulates describe desirable properties to be satisfied by these systems. The only rationality postulate proposed in  that is non-trivial for and frameworks is the postulate of consistency:
no set of assumptions selected by a given semantics contains an assumption for which is derivable from
(see Theorem 1 below for a formal statement). One of the reasons for introducing reverse defeats in is to avoid violations of the postulate of consistency by preserving conflicts between assumptions even if the attacking assumptions are strictly less preferred then the attacked assumption. The following example shows that for , conflicts are not necessarily preserved:
Let , , , and . Note that does not --defeat . As a consequence, is ---conflict-free for both and , but at the same time it entails .
Accordingly, one might ask under which conditions consistency is preserved in the context of . As in ASPIC , one might start by looking at contraposition-like properties.
Definition 8 (Contraposition ).
is closed under contraposition if for every non-empty :
if for some
[1mm] then for every it holds that for some .
Indeed, contraposition guarantees consistency, as shown next. 777In ASPIC , contraposition together with various other conditions are sufficient for consistency. However, for it turns out that contraposition alone guarantees consistency.
Theorem 1 (Consistency).
Let be closed under contraposition. For any , and ,
if is ---conflict-free, then for no , we have and .
The proof of all results in this paper are left out due to space restrictions.
Note that Theorem 1 immediately implies that if is ---naive, -preferred, -grounded or -stable then for no , we have and .
On the relation between and
Since reverse-defeat is essentially a form of contrapositive reasoning (cf. Section 6), one might ask whether a given closed under contraposition gives the same outcomes under (i.e. without reverse-defeat) and . A partial answer is given by the following result:
Given some , a set and some , assume is closed under contraposition. Then,
any ---admissible set is an ---admissible set; and
every ---complete set is a subset of an ---complete set.
However, in general the two approaches produce different outcomes (as can be verified by inspection of [11, Ex. 13]):
not every ---complete set is ---complete, and moreover
not every ---admissible set is ---admissible (or extensible to such a set)
4 Translating into
The translation from into essentially embeds the priority ordering over into an expanded object language . The expanded language contains atoms for each atom and value , and we translate
(In fact, we expand the set with a maximum element , and abusing notation we denote again as .) With more detail, we translate into non-prioritized frameworks as follows: the assumptions encode the priority of the assumption ; the rules in carry over the antecedents’ priorities to the consequent by taking their minimal value; and the contrariness operator (again written as for simplicity) mirrors the idea of -defeat being an attack that succeeds by restricting the contrary pairs to those pairs satisfying .
We first discuss the translation for flat, totally ordered frameworks, thereafter explaining the complications when these restrictions are given up.
Definition 9 (Translation ).
Where is flat and is totally ordered by , its translation is defined as follows:
The translation of any set will also be denoted .
Let be given by , and ; and with and , .
Applying Def. 9 gives us defined by: , and .
The need for modifying Def. 9 in non-flat frameworks is shown next.
Let be given by: , , , , , with , and
In we have that does not defeat since . Using Def. 9, however, we obtain in : and thus , so defeats . The problem is that the translation from Def. 9 allows us to derive from (using ). This does not mirror the behaviour of , since there the only deduction of using would be . Since is used in this deduction, it does not defeat (since ). Consequently, the translation from Def. 9 is not adequate for non-flat frameworks.
Translation for under .
Let us proceed to define a translation for the lifting (see Def. 4) which is adequate for frameworks whose preorder is not necessarily total.
Definition 10 ( for --defeat).
Given , we define its translation as in Def. 9 except for:
|if , and otherwise|
An additional change to Def. 9 can be motivated by Ex. 6 as well. Indeed, note that the set would be closed in the translated framework with Def. 9 since but . However, is not closed in and it can be easily seen that this gives rise to non-adequacies in any of the semantics defined. This can be fixed by adding new rules in of the form: for any and .
Let be given, and let be as in Def. 10. For any semantics and any :
Translation for under and .
For these two liftings, further complications arise, the investigation of which is left for future work.
5 as a special case of and
Suppose a framework is given, where is defined by trivial priorities: for any . It can be easily verified that both the --- and --- sets of assumptions coincide with -- sets of assumptions for all liftings in Def. 4. This means that we can capture in both and , i.e. both and are conservative extensions of , generalizing Theorem 5 in .
Let be given, where for any . For any semantics , lifting , set of rules and any :
6 Translating and
The translation from into is based on the idea that reverse-defeat in is an instance of contrapositive reasoning: whenever but is strictly less preferred than , then we should instead reject . This means that an assumption can -defeat a set of assumptions without attacking any particular member of this set; e.g. it can be observed in Ex. 6 that -defeats without -defeating or . Note that this mechanism from is ruled out in : whenever -defeats , then for some . In order to capture it within , we proceed stepwise: first, we add a conjunction to to make explicit the way of defeating a set of assumptions; second, we translate frameworks with conjunction: from to . These two steps expand the set with, first, rules for the introduction and elimination of conjunction and, second, with contrapositive rules.
The and systems
In the following, let denote that is a finite subset of , and let .
Definition 11 (Conjunction).
We say that an has a conjunction if there is a connective such that:
is in , for every (-closure of )
where ,999In order not to clutter notation we omit brackets and assume the connective to be commutative and associative. An enumeration of the countably-many sentences in can be used to define a canonical form for conjunctions, e.g. in increasing order: for . and for any and any with , the set is closed under the following:
where and denote the sets of -introduction- and -elimination-rules. For any , let
From now on, we proceed as follows: if an has no conjunction we add one, otherwise we use the one present in . In either case, the -closure of the language is defined as , and the closure of the set of rules is denoted .
Definition 12 ( framework).