Structured argumentation formalisms provide a rich framework to formalise and reason over situations where contradicting information is present. However, in most formalisms the integral step of constructing all possible arguments is performed in an unconstrained way and is thus not under direct control of the user. This can hinder a solid analysis of the behaviour of the system and makes explanations for the results dificult to obtain. In this work, we introduce a general approach that allows constraining the derivation of arguments for assumption-based argumentation.
Assumption-based argumentation (ABA) [
Example 1 (adapted from [16, Example 3]). Our protagonist Alice has been accepted to a study program with payment obligations. Every student whose application has been accepted counts as eligible student (constitutive norm). Moreover, every eligible student must pay their tuition fee (regulative norm) and every student who pays their tuition fee counts as a self-funding student (constitutive norm). We can derive that Alice must pay her tuition fee (since she is an eligible student), and hence she counts as a self-funding student.
However, Alice has furthermore received a study grant which means that she is not a self-funding student after all. Hence we derive a counter-intuitive conflict, deducing Alice to be both self-funding and have received a grant.
The underlying issue is that the application of constitutive rules after regulative ones may produce fallacious conclusions, called institutional wishful thinking. The undesired situation in Example 1 could be circumvented by preventing the application of the rule “tuition fee → self-funding student” after the rule “eligible student → tuition fee”. In the context of formal argumentation, similar issues have been addressed in recent works, based on an ASPIC-like formalism [17, 16, 18, 19]. Standard ABA is, however, not expressive enough to account for such a qualitative distinction among inference rules. Consequently, it is not possible to constraint the rule combinations on the basis of their kind.
ferences among them; (b) equip this extension of ABA with formal constraints (called derivation graphs) that regulate its deductive machinery, by encoding applicability conditions for inference rules; (c) investigate the role of derivation constraints within the argument construction process.
the underlying knowledge base, on the other hand, we present a prototype encoding of our formalism in Answer Set Programming (ASP). Finally, we point out to the relation between the possibility of expressing conflicts in normative reasoning and the expressive power of non-flat ABA.
Definition 1. An ABA framework (ABAF) is a tuple = (ℒ, ℛ, , ) where: (i) ℒ and ℛ form together a deductive system and are respectively a set of atomic sentences in a language and a set of inference rules; (ii) ⊆ ℒ is a non-empty set of atoms called assumptions; (iii) is a total mapping from into ℒ, where is said to be the contrary of , for each ∈ .
Following [20], we write rules as : ← 1, . . . , and we say that is the head of the rule and {1, . . . , } is its body, formally ℎ() = and () = {1, . . . , }. For a set of rules , we use ℎ() to indicate the set of atoms which are head of the rules contained in it. We consider here the finite flat version of ABAF, i.e. frameworks where ℒ and ℛ are ifnite and assumptions do not occur as conclusions of inference rules: there is no ∈ ℛ and ∈ for which = ℎ(). Arguments of an ABAF are based on proof-trees, constructed by forward-derivation from leaf-nodes to the root: Definition 2 (deduction). Let = (ℒ, ℛ, , ) be an ABAF. A deduction for ∈ ℒ supported by ⊆ and ⊆ ℛ , denoted ⊢ (or simply ⊢ ), is a finite rooted tree t with: i) a labelling function that assigns each vertex of an element from ℒ ∪ {⊤} such that the root is labelled by and leaves are labelled by either ⊤ or atoms in ; ii) a surjective mapping from the set of internal nodes of onto rules satisfying, for each vertex , that the label of is the head of the rule () and the children of are (one-to-one) labelled with the elements of the body of ().
Definition 3 (attack). Let = (ℒ, ℛ, , ) be an ABAF, let , ⊆ be two sets of assumptions. attacks ( → ) if there is a set ′ ⊆ such that ′ ⊢ for some ∈ .
Definition 4 (semantics). Given ABAF = (ℒ, ℛ, , ) and , ∈ . The set is conflictfree if it does not attack itself; and admissible if it is conflict-free and → implies → . stable if it is conflict-free and ∈ ∖ implies → {}; preferred if it is a ⊆ -maximal admissible set. We write ∈ () with ∈ {cf, adm, stb, prf} to say that is a conflict-free, admissible, stable or preferred set of assumptions (or extension) of .
Definition 5. Given an ABAF = (ℒ, ℛ, , that: • A = { ⊢ | ⊆ } ; • R = {( ⊢ , ⊢ ) | = for some ∈ } ⊆ A ×
), we call = (A, R) its corresponding AF such We can now define ABAFs with multiple rule-sets and derivation graphs. Jointly, these enable to trace rule kinds along with some constraint on their combination. We consider only frameworks where rule-sets are pairwise disjoint. Further, it is often desirable to evaluate scenarios where the same atom cannot be derived by rules of diferent kinds. Take for instance a legal debate built up using constitutive and regulative rules, as the one described in Example 1. The very diference between the two type of rules concern their output (i.e. their heads): constitutive rules produce institutional facts whereas regulative rules produce deontic statements such as obligations or permissions. It is therefore an intuitive requirement to separate these two heterogeneous groups of statements. For this, we focus on the class of separated -ABAFs, for which heads of rules in diferent rule-sets are pairwise disjoint.
Definition 6 (-rule-sets ABA). A -rule-sets ABAF (-ABAF) is a tuple = (ℒ, {ℛ | 1 ≤ ≤ }, , ) such that (ℒ, ⋃︀=1 ℛ, , ) is an ABAF. Moreover, we call separated whenever ℎ(ℛ) ∩ ℎ(ℛ ) = ∅ for all , with ̸= .
⋃︀≤ ℛ depending on the particular application domain. Inspired by input/output combinations presented in [17], we introduce the more expressive concept of derivation graph to formalise combination constraints: Definition 7 (derivation graph). Let = (ℒ, {ℛ | 1 ≤ ≤ }, , ) be an -ABAF. A derivation graph = (, ) for is a directed graph with | | ≥ + 1 such that contains: i) a distinct vertex s (called “starting node") with no incoming edges; ii) at least one vertex r for each ℛ (called “rule-node” for ℛ) such that there is a surjective mapping from rule-nodes onto the set of rule-sets {ℛ | 1 ≤ ≤ }.
possibility of rules chaining. This afects the derivation process from the underlying deductive system. In particular, the idea consists in allowing only those sequential combinations of rules for which there is a path within the derivation graph. As a result, we extend the usual notion of deduction presented in ABA to accommodate this additional requirement.
Definition 8. Let = (ℒ, {ℛ | 1 ≤ ≤ }, , ) be an -ABAF and let = (, ) be a derivation graph for . A -deduction for ∈ ℒ supported by ⊆ and = ⋃︀=1 with ⊆ ℛ , denoted ⊢ (or simply ⊢ ), is a deduction with a surjective mapping that maps every internal node of to a rule node in such that i) corresponds to a rule that is in the rule set of and ii) for each leaf-to-root-path 0 . . . in , the corresponding series of nodes 0, . . . , in form a path in with 0 = s.
the standard ones by employing the notion of -deduction instead of regular deduction.
Example 2. Consider again Example 1. We construct a 2-ABAF = (ℒ, ℛ1, ℛ2, , ) where ℛ1 and ℛ2 contain our constitutive and regulative rules, respectively. We assume the language ℒ to contain a modality where stands for “it is obligatory that ”. We let ℒ = { := _(), := _(), := _(), := (_ ()), := _ _()}; = {, }; ℛ1 = { ← , ← } and ℛ2 = { ← }. Moreover, = . One can build the following conflicting derivations: {} ⊢{}
{} ⊢{← ,← ,← }
We conclude that {} attacks {}. However, there should not be any conflict between a student being accepted and receiving a grant. By prohibiting the application of constitutive rules in the scope of regulative ones, is no longer derived from the assumption . Let us consider the following derivation graph : s r1
r2 {← ,← ,← } . Therefore, {} does not -attack {}
Since (r2, r1) ∈/ (), we get {} ⊬ and the assumption set {, } is an extension under any -semantics.
In doing so, we take three diferent paths: first, we show that under certain conditions it is possible to exploit the information given by derivation graphs to pre-process the knowledge base in order to obtain equivalent results in terms of ABAFs instantiation; Then, we present an encoding for our formalism in Answer Set Programming that captures derivation constraints and their efect on the procedure of argument construction; Finally, we devote a paragraph to show how our formalism hints towards a possible relationship among non-flat ABA and certain instances of normative reasoning. 4.1. Equivalence under Derivation Function
from a given knowledge base. This job is largely done by the notions of deduction and attack. In turn, derivation graphs work as a device for controlling and manipulate such instantiating process. An interesting research question would be that of asking under which conditions one can obtain an equivalent framework by pre-processing the knowledge base while leaving the derivation process untouched. Initial results show that if a derivation graph contains exactly one rule node for each rule set in the given -ABAF, it is possible to define a derivation function that works in such a way. This is an operation on the knowledge base which automatically identifies and removes rules that would not be allowed under a derivation graph , allowing unconstrained deductions. For each graph constraint , there is a derivation function that extracts from the rules ℛ of an -ABAF the subset of rules whose application is allowed under . Definition 9. Let D be an -ABAF, = (, ) be some derivation graph with = {s, r1, . . . , r}. The derivation function : 2ℛ → 2ℛ corresponding to is defined as (ℛ) = ℛ ∖ { ∪ } with • = { ∈ ℛ | ∈ ℛ, ∩ () ̸= ∅ or () = ∅ and (s, r) ∈/ }; • = { ∈ ℛ | ∈ ℛ, ∃ ≤ s.t. ℎ(ℛ ) ∩ () ̸= ∅ and (r , r) ∈/ }.
via the derivation function . As a result, the set of -deductions for is equivalent to the set of standard deductions that can be built using rules in (ℛ) only.
Lemma 1. Let be a separated -ABA framework, = (, ) some derivation-graph and : ℛ → ℛ its corresponding derivation function. For any set of atomic sentences ⊆ ℒ , ∈ ℒ and ⊆ ℛ , ⊢ is a -deduction for if and only if ⊢ () is a deduction for . Proof. (⇒) Assume ⊢ is a -deduction for . We show that ⊢ () is a deduction for . In order to do this, we first show that = (). We prove this by contradiction. Suppose there is an ∈ such that ∈/ (). Hence, either (a) ∈ or (b) ∈ . We proceed by case distinction.
(a) Suppose ∈ . By Definition 9, the following holds true: (i) ∈ ℛ and (ii) ∩() ̸= ∅ or () = ∅ and (iii) (s, r) ∈/ , where r denotes the node in corresponding to the class ℛ.
By hypothesis, it holds that ⊢ . By Definition 2, this is a finite rooted tree that assigns to each vertex an element in ℒ ∪ ⊤. is constructed in such a way that there is a surjective mapping from rules in onto the nodes in such that each of these nodes and their children respectively correspond to the head and the elements in the body of a rule of . Since ∈ , there is a node in the deduction tree that corresponds to and is labelled with ℎ() and its children are labelled with the elements in () or ⊤.
(ii). First, consider ∩ () ̸= ∅. In this case, there is at least one node ′, which is a child of , labelled with an assumption. Hence, it is a leaf (since we assume to be flat). Consider now the leaf-to-root path 0 . . . with 0 = ′ and 1 = . By Definition 8, this path can be mapped to a path 0 . . . in such that 0 = . Given that ∈ ℛ and corresponds to 1, we know that 1 corresponds to the class ℛ, i.e. 1 = r. Hence, we have shown that (, r) ∈ , in contradiction with the assumption (iii): (s, r) ∈/ . Next, assume () = ∅. In this case, the only child ′ of is labelled with ⊤. Again, the corresponding node ′ of ⊤ is a leaf in . Consider the leaf-to-root path 0 . . . with 0 = ′ and 1 = . By definition 8, this path can be mapped to a path 0 . . . in such that 0 = and 1 = r. Hence, (, r) ∈ , contradiction to the assumption (s, r) ∈/ . Consequently, ∈/ . (b) Suppose ∈ . By Definition 9, this means that (i) ∈ ℛ, (ii) ∃ ≤ such that ℎ(ℛ ) ∩ () ̸= ∅ and (iii)(r , r) ∈/ . We show that (i)-(iii) cannot be true at the same time.
Given that D is separated, there is no atom ℎ ∈ ℒ such that ℎ ∈ ℎ(ℛ) ∩ ℎ(ℛ ) with ̸= . Hence, for every atom in () there exists at most one ≤ such that ℎ(ℛ ) ∩ () ̸= ∅.
By hypothesis, it holds that ⊢ . By Definition 2, this is a finite rooted tree, denoted as , where each vertex is associated with an element from ℒ ∪ ⊤. The construction of tree ensures a surjective mapping from rules in onto its nodes. In each node and its children respectively correspond to the head and body elements of a rule in . For a given rule ∈ , there exists a node in the deduction tree that corresponds to . This node is labelled with ℎ(), and its children are labelled with elements from () or ⊤.
′ ∈ ℛ . Consider now the leaf-to-root path 0 . . . with = ′ and +1 = (with
0 = . Moreover, ∈ ℛ. Since and ′ correspond to and +1, the nodes and +1 in correspond to the classes ℛ and ℛ. Thus, = r and +1 = r. Hence, we have shown that (r , r) ∈ , in contradiction with the assumption (iii): (rj, r) ∈/ .
Consequently, ∈/ .
Since there is a rule ∈ such that ∈/ and ∈/ , we conclude that ∈ (), in contradiction with our initial assumption. We derive = (). Thus we replace with () in ⊢ , deriving that ⊢() is a -deduction in . Hence, ⊢ () is a deduction in .
(⇐) Suppose ⊢ () is a deduction in . We show ⊢ is a -deduction in . Since (ℛ) ⊆ ℛ , we know that each rule in () is contained in , i.e. we can use each rule in () under . Hence, we can assume that = (). We can thus replace () with in ⊢ () , thus showing that it is a deduction for (and a fortiori for ). It remains to show that ⊢ is a -deduction for .
∪ {⊤} and each node is associated to a rule ∈ . To prove further that ⊢ is a -deduction, we need to show that it is possible to map each leaf-to-root path 0 . . . in to a path 0 . . . in the graph (condition (ii) of Definition 8). Take one leaf-to-root-path 0 . . . in such that each node (corresponding to ) is associated with a rule ∈ ℛ.
is at least one edge (, +1) in for which the corresponding (, +1) ∈/ . To prove that this always leads to a contradiction, we distinguish the following two cases. (a) First suppose = 0, that is, (0, 1) ∈/ . Let denote the rule corresponding to 1.
Since ∈ () by hypothesis, is not deleted by . By Definition 9 we have that ∈/ , that is ∩ () = ∅ and () ̸= ∅. However, by Definition 2, it holds that each leaf is labelled with ∪ {⊤}. Hence, 0 corresponds either to an assumption or ⊤. In the ifrst case, ∩ () ̸= ∅. In the second case, () = ∅. Both cases contradict the assumption that ∈ (). Hence we obtain (0, 1) ∈ . (b) Now suppose > 0 and (, +1) ∈/ . Let ′ and denote the rules respectively corresponding to and +1. Since is not deleted by ( ∈ ()), by Definition 9 we have that ∈/ , that is for all ≤ it holds that ℎ(ℛ ) ∩ () = ∅.
satisfying, for each vertex , that the label of is the head of the rule corresponding to and the children of are (one-to-one) labelled with the elements of the body of the rule.
labelled with the head of rule ′ which is is an element in the body of . Let ℛ denote the rule set corresponding to ′. Then ℎ(ℛ ) ∩ () ̸= ∅. Therefore, ∈ , that is, is deleted in . But this is in contradiction with the assumption that ∈ (). Hence we obtain (, +1) ∈ . -ABAF
Restricted -ABAF deduction
Argument ⊢ ≡ ⊢ () consider the following example. found. We conclude that ⊢ is a -deduction.
{s, r1, r2} and = {(s, r1), (r1, r2)} as follows: Example 3. Let = (ℒ, ℛ1, ℛ2, , ) be a 2-ABAF with ℒ = {, , , , }, ℛ1 = { ← }, ℛ2 = { ← , ← } and = {, }. Let = (, ) be a derivation graph with = s r 1 r 2 Under , the set of -deduction that can (and cannot) be built are the following: {} ⊢ , {} ⊢ {} ⊢ {← } {} ⊢ {← ,← } , but {} ⊬ {← }
Let us now take the corresponding derivation function for . By definition 9, we have (ℛ) = ℛ ∖ { ∪ }, where = { ← Eventually, for we obtain the following: } and = ∅ such that = (ℒ, (ℛ), , ). {} ⊢ , {} ⊢ {} ⊢{← ,← } , but {} ⊬{← }
As it can be seen, every -deduction for is also a deduction for , and viceversa. = (, ) be the following derivation graph: Remark 1. In Lemma 1, we require that the -ABAF is separated, that is ℎ(ℛ) ∩ ℎ(ℛ ) = ∅ for all , with ̸= . The motivation behind this choice lies in the fact that the derivation function could in some occasions restrict the rule-set causing the set of deductions for to be a subset of the set of -deductions for . To show this, let us consider the following example: take such that ℒ = {, , , }, ℛ1 = { ← , ← }, ℛ2 = { ← } and = {, }. is not separated due to the fact that ∈ ℎ(ℛ1) ∩ ℎ(ℛ2). Moreover, let r1 s As it can be seen easily, the rule : ← would be eliminated by the function since ∈ . Indeed, we have ∈ ℎ(ℛ2) ∩ (), ∈ ℛ1 and (r2, r1) ∈/ . Thus, {} ⊢ (ℛ) is not a deduction for . However, we would at the same time allow the rule under since (s, r1) ∈ and (r1, r1) ∈ , so that {} ⊢ℛ is a -deduction for . In order to avoid such undesired behaviour, we restrict our study to separated ABAFs.
is equivalent to the one instantiated through standard deductions after its rule-set has been restricted by the derivation function. This assures that the same outcome is reached by limiting deductions via some derivation graph or by restricting the knowledge base accordingly. This is captured by the following theorem:
some derivation graph and : ℛ → ℛ some derivation function. Let = (A, R) be the corresponding AF with respect to under and ′ = (A′, R′) the corresponding AF with respect to . For these, we derive that ≡ ′, in the sense that: (1) A = A′; (2) R = R′.
Proof. We start by considering (1). By Definition 5, we have A = { ⊢ | ⊆ } . Given
be obtained through some derivation function such that { ⊢ () | ⊆ } = A′. Hence, A = A′.
The proof of (2) is similar. By Definition 5, we have R = {( ⊢ , ⊢⋆ ) | = for some ∈ }. Given Lemma 1, we know that for each pair of arguments in such set, there is an equivalent pair of arguments that can be obtained through some derivation function such that ⊢ ⇐⇒ ⊢ () and ⊢⋆ ⇐⇒ ⊢ (⋆) . Since these arguments are pairwise equivalent, the attack relation among them will be equivalent as well. Thus, we obtain {( ⊢ () , ⊢ (⋆) ) | = for some ∈ } = R′ as the set of attacks for ′. Finally, we can state R = R′, as desired.
Corollary 1 (Equivalence). Let D be a separated -ABAF, its restriction under and the corresponding derivation graph. Given any ABA semantics ∈ {cf , adm, stb, prf } and their constrained version , it holds that () = ( ). 4.2. Encoding constrained -ABA in ASP In addition, we present an encoding of the -ABA formalism and derivation graphs in ASP (available at: https://www.dbai.tuwien.ac.at/research/argumentation/abasp/), inspired by the one provided in [21] for regular ABA frameworks and semantics. Given an -ABAF and derivation graph as input, the encoding provides an answer set for each -extension of a given -ABAF under the graph constraint . Regarding the -ABAF in input, we extend the encoding presented in [21] by introducing a new predicate specifying for each rule the (unique) rule-set ℛ it belongs to. Let = (ℒ, {ℛ | 1 ≤ ≤ }, , ) be an -ABAF with ℛ be the -th set of rules. We use the following set of facts in ASP to represent the -ABAF : D ={assumption(). | ∈ }∪ {head(, ). | ∈ ℎ(), ∈ ℛ}∪ {body(, ). | ∈ (), ∈ ℛ}∪ {rule_set(, ). | ∈ ℛ}∪ {contrary(, ). | = , ∈ }.
and that is the contrary of . Moreover, head(, ) and body(, ) mean that is the head (resp. body) of the rule within ℛ. In addition, we introduce the predicate rule_set(, ) to specify that is in the rule-set ℛ.
specifying which node corresponds to the starting node and each rule set.
G ={node(). | ∈ }∪ {edge(1, 2). | (1, 2) ∈ }∪ {start_node(). | is the starting node}∪ {rule_node(, ). | corresponds to ℛ}.
an -ABAF under (see Listing 1). To present this in a more concise way, we say that “a rule is in a node ” whenever such rule is contained in the rule-set corresponding to the node in the derivation graph.
Listing 1: Module 1 in() ← assumption(), out(). 2 out() ← assumption(), in(). 3 fact_rule(, ) ← head(, ), non_fact_rule(, ). 4 non_fact_rule(, ) ← head(, ), body(, ). 5 supported_by_node(, ) ← in(), start_node( ). 6 supported_by_node(, ) ← head(, ), rule_set(, ), rule_node(, ) , non_supported_by_node_via_rule(, , ). 7 non_supported_by_node_via_rule(, , ) ← fact_rule(, ), rule_set(, ) , rule_node(, ), start_node( ), edge(, ). 8 non_supported_by_node_via_rule(, , ) ← head(, ), rule_set(, ), rule_node(, ) , non_supported_by_node_via_rule_because_of(, , , ), body(, ). 9 non_supported_by_node_via_rule_because_of(, , , ) ← head(, ), rule_set(, ) , rule_node(, ), body(, ), {supported_by_node(, ) : edge(, )} = 0. rule_set(, ), rule_set(, ), ! = .
Lines 1 and 2 encode a guess of some possible extension in the set of assumptions, specifying which of them are taken to be in and out respectively. We label facts via the predicate fact_rule, telling them apart from rules with non-empty body (Lines 4 and 5). Lines 5-9 encode the construction process of -deductions as forward derivations from subsets of to supported claims. These establish the connection between nodes of a derivation graph, rules and supported atoms, represented by the predicate supported_by_node. As for Line 5, the set of assumptions ⊆ that is guessed to be in is set to be supported by the starting node of the derivation graph. Subsequently, as in [21], for any atom that can be -deduced from (a subset of) , we obtain supported_by_node() in some answer set. In particular, Line 9 says that a rule in a node that might fire is blocked when its body is not supported by any node from an incoming edge (, ). Combinations of rules which are not allowed under the derivation graph in input are thus ruled out. Further, Line 8 enforces that all elements in the body of a rule in some rule-node have to be supported for its head to be supported as well. Thus, it prevents rules with partially supported body to fire, even when the derivation graph would allow for such combination. Line 7 blocks facts in a node to be derived when there is no incoming edge from the starting node. Finally, Line 6 makes use of the concept of negation as failure to establish that an atom is supported by a node if it occurs in the head of some rule in and no rule can be found in for which is non_supported. A constraint in Line 10 checks that the same rule is not contained in two diferent rule sets, encoding the requirement that rule sets are pairwise disjoint.
4.3. Non-flat ABA for Normative Reasoning In the area of normative reasoning, conflicts may occur not only in presence of inconsistent information regarding brute facts, but also regarding institutional facts and norms[16]. Scenarios that concern the detachment of conflicting institutional facts are called normative conflicts and arise when more agents agree on the same brute fact, but assign conflicting institutional values to it. For example, an homosexual couple counts as married after having signed the marriage contract in some legal systems, but not in others. Similarly, conflicts among obligations can arise, giving rise to so-called moral dilemmas. Instances of these arise in presence of an obligation for and for its opposite (formally, this translates to deriving ∧ ¬ from assumptions).
In assumption-based argumentation, conflicts between sentences are encoded by the contrary function over the assumption set. This represent a fundamental design property of ABA, because it allows in turn to define semantics directly on the assumption level. For this reason, in order to express normative conflicts and moral dilemmas, it is required that assumptions may not only consist of so-called brute facts, but also of institutional facts (produced by constitutive rules) or obligations (produced by regulative rules). Since these are always derived by rules in our knowledge base, flat ABA may not always be expressive enough to encompass such cases.
Therefore, we anticipate here that the full expressiveness of non-flat ABA may be required for capturing instances of normative reasoning. In order to see this, let us consider an instance of Forrester’s paradox [22]. In Standard Deontic Logic [23], Forrester’s paradox, also known in the literature as “gentle murderer paradox”, follows from the statements A: “One should not (under the law) commit murder" and B: “if someone commits murder, then they should do it gently". Moreover, B implies C: “if someone should commit murder gently, then they should commit murder". Under the assumption that D: “someone commits murder", this eventually creates a paradoxical situation whereby it is obligatory to commit and not to commit murder at the same time. Therefore, a moral dilemma is created where it is contradictory to assume that a law exists and someone violates it. In the following, we show that such an undesired outcome can be avoided by imposing some constraint by means of a derivation graph . Example 4. Take a non-flat 1-ABAF = (ℒ, ℛ1, , ) such that ℒ = {, , }, = {, } where := () and := (¬()) and ℛ1 = { ← , ← , ←} where := (_()). Then one could build the arguments: {} ⊢{←} {} ⊢{} {} ⊢{← ,← }
As it can be seen immediately, {} attacks {}. Hence the contrary-to-duty paradox: the assumptions that murdering is forbidden and that someone murders are mutually exclusive and their union is not conflict-free. For this example, not every derivation graph will prevent the paradox to arise. Consider the following: s
r1 1 s
r1 2 s
r1 r1 3 1 puts no restriction on deductions, 2 restricts deductions to using only one rule, and 3 restricts deductions to using only two rules per branch. In our example both 1 and 3 do not prohibit any of the possible deductions, while 2 does and in fact is the only graph which prevents the paradox. In fact, by forbidding the iteration for rules, it blocks the derivation of from the {← ,← } . Therefore, {} does not -attack {} and the assumption . Indeed, we get {} ⊬ assumption set {, } is an extension under any -semantics.
an exhaustive analysis of conflict resolution in the context of normative reasoning requires the full expressiveness of non-flat -ABA.
This work introduces an extension of assumption-based argumentation with multiple rule-sets together with some formal constraints on its deductive machinery. These constraints, called derivation graphs, regulate the argument construction process from the underlying knowledge base, thereby limiting the procedure for its instantiation into an ABA framework. While this allows to avoid undesired conclusions as shown in Examples 1 and 4, we are currently working on defining constraints that operate directly on the knowledge base. In addition, we presented an encoding of our formalism in ASP, building up on the work presented in [21]. Finally, we discussed the possibility to capture certain instances of normative reasoning in assumptionbased argumentation, using the full expressive power of non-flat ABA to represent and reason about normative conflicts and moral dilemmas.
The derivation constraints presented in this work successfully avoid some paradoxes and fallacies in the domain of normative reasoning, but they do so at the expense of the deductive power of the ABA formalism. As a general direction for future research we want to broaden our horizon and investigate diferent kinds of reasoning constraints that minimise this loss. In doing so, we aim at positioning our formalism with respect to related frameworks: the work by Pigozzi and Van der Torre on constitutive and regulative norms in argumentation [16]; modular ABA [24] as it was proposed in connection with normative reasoning; Deontic ASP [25] encoding input/output logics. Although we restricted our studies on flat ABAFs so far, we anticipate that the full expressiveness of non-flat ABAFs may be needed to capture general instances of normative reasoning (cf. Example 4). Equipping non-flat ABAFs with derivation graphs might pose additional challenges since non-flat ABAFs require certain closure conditions on the set of acceptable assumptions. In addition, we aim at studying how size and complexity of instantiated
project ICT19-065 and by the European Research Council (ERC) under the European Union’s
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