=Paper=
{{Paper
|id=Vol-3835/paper12
|storemode=property
|title=On the Correspondence of Non-flat Assumption-based Argumentation and Logic Programming with Negation as Failure in the Head
|pdfUrl=https://ceur-ws.org/Vol-3835/paper12.pdf
|volume=Vol-3835
|authors=Anna Rapberger,Markus Ulbricht,Francesca Toni
|dblpUrl=https://dblp.org/rec/conf/nmr/Rapberger0T24
}}
==On the Correspondence of Non-flat Assumption-based Argumentation and Logic Programming with Negation as Failure in the Head==
https://ceur-ws.org/Vol-3835/paper12.pdf
On the Correspondence of Non-flat Assumption-based
Argumentation and Logic Programming with Negation as
Failure in the Head
Anna Rapberger1,* , Markus Ulbricht2 and Francesca Toni1
1
Imperial College London, Department of Computing
2
Leipzig University, ScaDS.AI
Abstract
The relation between (a fragment of) assumption-based argumentation (ABA) and logic programs (LPs) under stable model semantics
is well-studied. However, for obtaining this relation, the ABA framework needs to be restricted to being flat, i.e., a fragment where
the (defeasible) assumptions can never be entailed, only assumed to be true or false. Here, we remove this restriction and show a
correspondence between non-flat ABA and LPs with negation as failure in their head. We then extend this result to so-called set-
stable ABA semantics, originally defined for the fragment of non-flat ABA called bipolar ABA. We showcase how to define set-stable
semantics for LPs with negation as failure in their head and show the correspondence to set-stable ABA semantics.
Keywords
Computational Argumentation, Assumption-based Argumentation, Logic Programming, Stable Semantics
1. Introduction deployed in multi-agent settings to support dialogues [18]
and supports applications in, e.g., healthcare [19], law [20]
Computational argumentation and logic programming con- and robotics [21].
stitute fundamental research areas in the field of knowledge Research in ABA often focuses on the so-called flat ABA
representation and reasoning. The correspondence between fragment, which prohibits deriving assumptions from infer-
both research areas has been investigated extensively, re- ence rules. In this work, we show that generic (potentially
vealing that the computational argumentation and logic non-flat) ABA (referred to improperly but compactly as non-
programming paradigms are inextricably linked and pro- flat ABA [22]) captures the more general fragment of LPs
vide orthogonal views on non-monotonic reasoning. In with negation as failure in the head, differently from all of
recent years, researchers developed and studied various the aforementioned argumentation formalisms. This under-
translations between logic programs (LPs) and several ar- lines the increased and more flexible modelling capacities
gumentation formalisms, including translation from and of the generic ABA formalism.
to abstract argumentation [1, 2, 3], assumption-based argu- In this work, we investigate the relationship between
mentation [4, 5, 6, 7, 8], argumentation frameworks with col- non-flat ABA and LP with negation in the head, focusing
lective attacks [9], claim-augmented argumentation frame- on stable [4] and set-stable [23] semantics. While stable se-
works [10, 11], and abstract dialectical frameworks [12, 13]. mantics is well understood, the latter has not been studied
The multitude of different translations sheds light on the thoroughly so far. Set-stable semantics has been originally
close connection of negation as failure and argumentative introduced for a restricted non-flat ABA fragment (bipolar
conflicts. Apart from the theoretical insights, these transla- ABA [23]) only, with the goal to study the correspondence
tions are also practically enriching for both paradigms as between ABA and a generalisation of abstract argumenta-
they enable the application of methods developed for one tion that allows for support between arguments (bipolar
of the formalisms to the other. On the one hand, translat- argumentation [24]). In this paper we adopt it for any non-
ing logic programs to instances of formal argumentation flat ABA framework and study it in the context of LPs with
has been proven useful for explaining logic programs [14]. negation as failure in the head.
Translations from argumentation frameworks into logic In more detail, our contributions are as follows:
programs, on the other hand, allows to utilise the rich
toolbox for LPs, e.g., answer set programming solvers like β’ We show that each LP with negation as failure in
clingo [15], directly on instances of formal argumentation. the head corresponds to a non-flat ABA framework
Existing translations consider normal LPs [16], i.e., the under stable semantics.
class of LPs in which the head of each rule amounts pre- β’ We identify an ABA fragment (LP-ABA) in which
cisely to one positive atom. In this work, we take one the correspondence to LPs with negation as failure
step further and consider LPs with negation as failure in in the head is 1-1. We prove that each non-flat ABA
the head of rule [17]. We investigate the relation of this framework corresponds to an LPs with negation as
more general class of LPs to assumption-based argumenta- failure in the head by showing that each ABA frame-
tion (ABA) [4]. This is a versatile structured argumentation work can be mapped into an LP-ABA framework.
formalism which models argumentative reasoning on the ba- β’ We introduce set-stable model semantics for LPs
sis of assumptions and inference rules. ABA can be suitably with negation as failure in the head. We identify the
LP fragment corresponding to bipolar ABA under
22nd International Workshop on Nonmonotonic Reasoning, November 2- set-stable semantics. We furthermore consider the
4, 2024, Hanoi, Vietnam set-stable semantics for any LPs with negation as
*
Corresponding author.
failure in the head by appropriate adaptions of the
a.rapberger@imperial.ac.uk (A. Rapberger);
mulbricht@informatik.uni-leipzig.de (M. Ulbricht); ft@imperial.ac.uk reduct underpinning stable models [17].
(F. Toni)
Β© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
2. Background π πΌ2 : πβ π β β
βπ
We recall logic programs with negation as failure in the We see that πΌ1 is a minimal Herbrand model of π πΌ1 , whereas
head [17] and assumption-based argumentation [4]. πΌ2 is rendered invalid due to the rule β
β π . Thus, this rule
can be seen as a denial integrity constraint amounting to rul-
2.1. Logic programs with negation as ing out the atom π .
failure in head
2.2. Assumption-based Argumentation
A logic program with negation as failure (naf) in the
head [17] (LP in short in the remainder of the paper) consists We recall assumption-based argumentation (ABA) [4]. A
of a set of rules π of the form deductive system is a pair (β, β), where β is a formal lan-
guage, i.e., a set of sentences, and β is a set of inference
π0 βπ1 , . . . , ππ , not ππ+1 , . . . , not ππ rules over β. A rule π β β has the form
not π0 βπ1 , . . . , ππ , not ππ+1 , . . . , not ππ
π0 β π1 , . . . , ππ
for π β₯ 0, (propositional) atoms ππ , ππ , and naf operator not.
We write βπππ(π) = π0 and βπππ(π) = not π0 , respectively, for π β₯ 0, with ππ β β. We denote the head of π by
and ππππ¦(π) = {π1 , . . . , ππ , not ππ+1 , . . . , not ππ }. Fur- βπππ(π) = π0 and the (possibly empty) body of π with
thermore, we let ππππ¦ + (π) = {π1 , . . . , ππ } denote the ππππ¦(π) = {π1 , . . . , ππ }.
positive and ππππ¦ β (π) = {ππ+1 , . . . , ππ } denote the neg-
Definition 2.4. An ABA framework (ABAF) [22] is a tu-
ative atoms occuring in the body of π; moreover, we let
ple (β,β,π, ) for (β,β) a deductive system, π β β the
βπππβ (π) = {π0 } if βπππ(π) = not π0 and βπππβ (π) = β
assumptions, and :π β β a contrary function.
otherwise (analogously for βπππ+ (π)).
Definition 2.1. The Herbrand Base of an LP π is the set In this work, we focus on finite ABAFs, i.e., β, β, π are
HB π of all atoms occurring in π . By finite; also, β is a set of atoms or naf-negated atoms.
For a set of assumptions π β π, we let π = {π | π β π}
HB π = {not π | π β HB π } denote the set of all contraries of assumptions π β π.
Below, we recall the fragment of bipolar ABAFs [23].
we denote the set of all naf-negated atoms in HB π .
Definition 2.5. An ABAF (β,β,π, ) is bipolar iff for all
We call an LP π a normal program if βπππβ (π) = β
rules π β β, it holds that |ππππ¦(π)| = 1, ππππ¦(π) β π, and
for each π β π and a positive program if ππππ¦ β (π) =
βπππ(π) β π βͺ π.
βπππβ (π) = β
for each π β π . Given πΌ β HB π , the
reduct π πΌ of π is the positive program Next, we recall the crucial notion of tree-derivations. A
πΌ + + sentence π β β is tree-derivable from assumptions π β π
π = {βπππ (π) β ππππ¦ (π) | and rules π
β β, denoted by π β’π
π , if there is a finite
ππππ¦ β (π) β© πΌ = β
, βπππβ (π) β πΌ}. rooted labeled tree π s.t. the root is labeled with π ; the set
of labels for the leaves of π is equal to π or π βͺ {β€}, where
In contrast to the LP fragment that we consider in this β€ ΜΈβ β; for every inner node π£ of π there is exactly one rule
work, the reduct of a program can contain (denial integrity) π β π
such that π£ is labelled with βπππ(π), and for each
constraints, i.e., rules with empty head. π β ππππ¦(π) the node π£ has a distinct child labelled with
We are ready to define stable LP semantics. π; if ππππ¦(π) = β
, π£ has a single child labelled β€; for every
Definition 2.2. πΌ β HB π is a stable model [17] of an rule in π
there is a node in π labelled by βπππ(π). We often
LP π if πΌ is a β-minimal Herbrand model of π πΌ , i.e., πΌ is a write π β’π
π simply as π β’ π. Tree-derivations are the
β-minimal set of atoms satisfying arguments in ABA; we use both notions interchangeably.
Let π· = (β, β, π, ) be an ABAF. For a set of assump-
(a) π β πΌ iff there is a rule π β π πΌ s.t. βπππ(π) = π and tions π, by Th π· (π) = {π β β | βπ β² β π : π β² β’ π} we
ππππ¦(π) β πΌ; denote the set of all sentences derivable from (subsets of) π.
(b) there is no rule π β π πΌ with βπππ(π) = β
and Note that π β Th π· (π) since each π β π is derivable from
ππππ¦(π) β πΌ. {π} and rule-set β
({π} β’β
π). The closure of π is given
Negation as failure in the head can be also interpreted by cl (π) = Th π· (π) β© π. An ABAF is flat if each set π of
in terms of denial integrity constraints, as also observed assumptions is closed. We refer to an ABAF not restricted
by Janhunen [25]. Thus, naf literals and constraints are, to to be flat as non-flat.
some extent, two sides of the same coin. Let us consider the Definition 2.6. Let π· = (β, β, π, ) be an ABAF. An
following example. assumption-set π β π attacks an assumption-set π β π
Example 2.3. Consider the LP π given as follows. if π β Th π· (π) for some π β π . An assumption-set π is
conflict-free (π β cf (π·)) if it does not attack itself; it is
π : π β not π π β not π π β not π β π , not π. closed if ππ(π) = π.
Here, π models a choice between π and π. However, as π is We recall stable [22] and set-stable [23] ABA semantics
factual and not π entails not π (together with the fact π ), π (abbr. stb and sts, respectively). Note that, while set-stable
is rendered impossible. semantics has been defined for bipolar ABAFs only, we
For the sets of atoms πΌ1 = {π, π } and πΌ2 = {π, π } we generalise the semantics to arbitrary ABAFs.
obtain the following reducts:
Definition 2.7. Let π· = (β, β, π, ) be an ABAF. Further,
π πΌ1 : π β π β let π β cf (π·) be closed.
β’ π β stb(π·) if π attacks each {π₯} β π β π; We will prove that πΌ is a stable model (in π ) iff β(πΌ) is a
β’ π β sts(π·) if π attacks cl ({π₯}) for each π₯ β π β π. stable extension (in π·π ). First, we introduce a notion of
reachability in logic programs that is based on the construc-
Example 2.8. We consider an ABAF π· = (β, β, π, )
tion of arguments.
with assumptions π = {π, π, π}, their contraries π, π, and
π, respectively, and rules Definition 3.3. Let π be an LP. An atom π β HB π βͺ HB π
is reachable from a set of naf literals π β HB π iff there is
π β π. π β π. π β π, π.
a tree-based argument π β² β’ π with π β² β π in the corre-
The framework is non-flat because we can derive π from π. sponding ABAF π·π .
In π·, the set {π} is set-stable: Clearly, the assumption does
Note that the reachability target is defined for both pos-
not attack itself. It remains to show that the closure of π and
itive and negative atoms; the source on the other hand is
the closure of π is attacked. First note that π attacks π since
always a set of naf literals. The notion differs from reach-
{π} β’ π. Thus, π attacks also the closure of π. It follows that π
ability based on dependency graphs which is defined for
furthermore attacks the closure of π since cl ({π}) = {π, π}.
positive atoms only.
This shows that {π} is set-stable.
Below, we prove our first main result.
Moreover, the set {π, π} is stable and set-stable in π· be-
cause it is conflict-free and attacks the assumption π via the Theorem 3.4. Let π be an LP and π·π the ABAF correspond-
argument {π, π} β’ π. ing to π . Then πΌ is a stable model of π iff β(πΌ) β stb(π·π ).
Proof. By definition, a set πΌ is stable iff it is β-minimal
3. Stable Semantics model of π πΌ satisfying
Correspondence (π) π β πΌ iff there is π β π πΌ such that βπππ(π) = π
In this section, we show that non-flat ABA under stable and ππππ¦(π) β πΌ; and
semantics correspond to stable model semantics for logic (π) there is no π β π πΌ with βπππ(π) = β
and
programs with negation as failure in the head. First, we show ππππ¦ + (π) β πΌ.
that each LP can be translated into a non-flat ABAF; second, By definition of π πΌ we obtain πΌ is a stable model of π iff πΌ
we present a translation from a restricted class of ABAFs (LP- is a β-minimal model of π πΌ satisfying
ABA) into LPs; third, we extend the correspondence result
to general ABAFs by providing a translation from general (π) π β πΌ iff there is π β π such that βπππ(π) = π,
non-flat ABA into LP-ABA. We conclude this section by ππππ¦ + (π) β πΌ, and ππππ¦ β (π) β© πΌ = β
; and
discussing denial integrity constraints in non-flat ABA. (π) there is no π β π with βπππ+ (π) = β
, βπππβ (π) β
πΌ, ππππ¦ + (π) β πΌ, and ππππ¦ β (π) β© πΌ = β
.
3.1. From LPs to ABAFs
Below, we show that the first item and the β-minimality
Each LP π can be interpreted as ABAF with assumptions requirement captures conflict-freeness (no naf literal in πΌ
not π and contraries thereof, for each literal in the Herbrand is derived) and the requirement that all other assumptions
base HB π of π . We recall the translation from normal are attacked (all other naf literals outside πΌ are derived);
programs to flat ABA [4]. whereas the second item ensures closure of the program.1
First, Let πΌ be a stable model of π and let π = β(πΌ). We
Definition 3.1. The ABAF corresponding to an LP π is
show that π is stable in π·π , i.e., it is conflict-free, closed,
π·π = (β, β, π, ) with β = HB π βͺ HB π , β = π ,
and attacks all assumptions in π β π.
π = HB π , and not π₯ = π₯ for each not π₯ β π.
β’ π is conflict-free: π is conflict-free iff there is no
Example 3.2. Consider again the LP from Example 2.3.
π β HB π β πΌ such that π is reachable, i.e., can be
π : π β not π π β not π π β not π β π , not π derived from π. If such a derivation would exist,
then the assumption not π β π were attacked by π.
Here π·π = (β, β, π, ) is the ABAF with Towards a contradiction, suppose there is an atom
β = {π, π, π , not π, not π, not π } π β HB π β πΌ which is reachable from π. Let
β =π π = {π β HB π β πΌ | π β’ π}
π = {not π, not π, not π }
denote the set of atoms that are reachable from π
and contrary function not π₯ = π₯ for each π₯ β {π, π, π }. but lie βoutsideβ πΌ. We order π according the height
Recall that πΌ1 = {π, π } is a stable model of π . Naturally, this of the smallest tree-derivation.
set corresponds to the singleton assumption-set π = {not π}. Wlog, we can assume that our chosen atom π is min-
Indeed, since π is derivable from {not π} and π is factual, it imal in π, i.e., there is no other atom π β HB π β πΌ
holds that Th π·π (π) = {not π, π, π } which suffices to see which is reachable in less steps. Let π β² β’ π denote
that π β stb(π·π ). the smallest tree-derivation, and let π denote the
Let us generalize the observations we made in this exam- top-rule (the rule connecting the root π with the fist
ple. We translate a set of atoms πΌ (in HB π for an LP π ) into level of the tree) of the derivation. The rule satisfies
an assumption-set β(πΌ) (in the ABAF π·π ) by collecting all βπππ(π) = π, ππππ¦ β (π)β©πΌ = β
, and ππππ¦ + (π) β πΌ
assumptions βnot πβ corresponding to the atoms outside πΌ; 1
We note that in the case of normal logic programs without negation
that is, we set in the head, the second condition does not apply. It is well known and
has been discussed thoroughly in the literature that (a) holds iff Ξ(πΌ)
β(πΌ) = {not π | π β
/ πΌ}. is stable in π·π [5, 6].
(otherwise, there is an atom π β / πΌ with a smaller construct an argument for not π, contradiction to π
tree-derivation, contradiction to the minimality of π being closed.
in π). Consequently, we obtain that π β πΌ, contra- β’ It remains to show that πΌ is a β-minimal model of
diction to our initial assumption. π πΌ . Since each atom π β πΌ has an argument in
β’ π attacks all other assumptions: Suppose there is an π·π we obtain minimality: Towards a contradiction,
atom π β πΌ which is not reachable from π. We show suppose there is a model πΌ β² β πΌ of π πΌ . Let π β
that πΌ β² = πΌ β {π} is a model of π πΌ . That is, we πΌ β πΌ β² . Since there is an argument deriving π there is
show that πΌ β² satisfies each rule in π πΌ . By assump- some π β π πΌ with βπππ(π) = π and ππππ¦(π) β πΌ,
tion there is is no rule π β π such that βπππ(π) = π, showing that πΌ β² is not a model of π πΌ .
ππππ¦ + (π) β πΌ β² , and ππππ¦ β (π) β© πΌ β² = β
(otherwise,
π is reachable from π). Hence π β πΌ β² iff there is 3.2. From ABAFs to LPs
π β π πΌ such that βπππ(π) = π and ππππ¦ + (π) β πΌ β²
is satisfied. πΌ β² satisfies all constraints since, by as- For the other direction, we define a mapping so that each
sumption, there is no π β π πΌ with βπππ(π) = β
assumption corresponds to a naf-negated atom. However,
and ππππ¦ + (π) β πΌ. Thus πΌ β² is a model of π πΌ . Con- we need to take into account that ABA is a more general for-
sequently, πΌ cannot be a stable model, contradiction malism. Indeed, in LPs, there is a natural bijection between
to our initial assumption. ordinary atoms and naf-negated ones (i.e., π corresponds to
β’ π is closed: Towards a contradiction, suppose that not π). Instead, in ABAFs, assumptions can have the same
there is some π β πΌ such that the corresponding naf contrary, they can be the contraries of each other, and not
literal not π is reachable. Let π be the top-rule of the every sentence is the contrary of an assumption in general.
tree-derivation. It holds that ππππ¦ + (π) β πΌ (other- To show the correspondence (under stable semantics), we
wise, there is some π β HB π β πΌ which is reachable, proceed in two steps:
contradiction to the first item), ππππ¦ β (π) β© πΌ = β
1. We define the LP-ABA fragment in which i) no as-
and βπππ(π) = not π. Consequently, item (b) from sumption is a contrary, ii) each assumption has a
Definition 2.2 is violated. unique contrary, and iii) no further sentences exist,
i.e., each element in β is either an assumption or
This concludes the proof of the first direction. We have the contrary of an assumption. We show that the
shown that π = β(πΌ) is stable in π·π . translation from such LP-ABAFs to LPs is semantics-
Now, let π = β(πΌ) be a stable extension in π·π . We show preserving.
that πΌ is stable in π . 2. We show that each ABAF (whose underpinning lan-
β’ Let π β πΌ. Then we can construct an argument guage is restricted to atoms and their naf) can be
π β² β’ π, π β² β π in π·π , i.e., is reachable from π. transformed to an LP-ABAF whilst preserving se-
We show that there is a rule π with ππππ¦ + (π) β πΌ, mantics.
ππππ¦ β (π) β© πΌ = β
and βπππ(π) = π. We proceed
by induction over the height of the argument, that Relating LP and LP-ABA Let us start by defining the LP-
is, the height of the tree-derivation. ABA fragment. A similar fragment for the case of normal
LPs and flat ABAFs has been already considered [6, 22, 26].
β Base case: Suppose π β² β’ π has height 1.
Here, we extend it to the more general case.
Then there is π β π with βπππ(π) = π,
ππππ¦ + (π) = β
, and ππππ¦ β (π) β© π = β
. Definition 3.5. The LP-ABA fragment is the class of all
β π β¦β π + 1: Suppose now that the statement ABAFs π· = (β, β, π, ) where (1) π β© π = β
, (2) the
holds for all arguments of height smaller than contrary function is injective, and (3) β = π βͺ π.
or equal to π, and suppose π β² β’ π has height We show that each LP-ABAF corresponds to an LP, using
π + 1. Let π denote the top-rule of the tree- a translation similar to [6][Definition 11] (which is however
derivation. for flat ABA). We replace each assumption π with not π.
We derive the statement by applying the in- For an atom π β β, we let
duction hypothesis to all height-maximal sub- {οΈ
arguments (with claims in ππππ¦(π)) of our not π, if π β π
rep(π) =
fixed tree-derivation: Let πβ² β ππππ¦(π). The π, if π = π β π.
sub-tree with root πβ² is an argument of height
π. Hence, by induction hypothesis, β(πΌ) de- Note that in the LP-ABA fragment, this case distinction is
rives πβ² , i.e., there is πβ² β π with βπππ(πβ² ) = exhaustive. We extend the operator to ABA rules element-
πβ² , ππππ¦ + (πβ² ) β πΌ, and ππππ¦ β (πβ² ) β© πΌ = β
. wise: rep(π) = rep(βπππ(π)) β {rep(π) | π β ππππ¦(π)}.
In case πβ² is a positive literal, we obtain πβ² β πΌ Definition 3.6. For an LP-ABAF π· = (β, β, π, ), we de-
(by (a) from Definition 2.2); in case πβ² is a naf fine the associated LP ππ· = {rep(π) | π β β}.
literal, we obtain πβ² β β(πΌ) (by (b)). Since πβ²
Example 3.7. Let π· be an ABAF with π = {π, π, π } and
was arbitrary, we obtain ππππ¦ + (π) β πΌ and
ππππ¦ β (π) β© πΌ = β
. β : πβπ πβπ π β π β π , π.
β’ For the other direction, suppose there is a rule π β We replace e.g. the assumption π with not π and the contrary
π with ππππ¦ + (π) β πΌ, ππππ¦ β (π) β© πΌ = β
and π is left untouched. This yields the associated LP
βπππ(π) = π. We can construct arguments for all
ππππ¦ + (π) β πΌ and thus obtain π β πΌ. ππ· : π β not π π β not π π β not π β π , not π.
β’ Towards a contradiction, suppose there is a π β Striving to anticipate the relation between π· and ππ· , note
π with ππππ¦ + (π) β πΌ, ππππ¦ β (π) β© πΌ = β
and that π = {π} β stb(π·). Now we compute Th π· (π) β π =
βπππ(π) = not π for some π β πΌ. Then we can {π, π } noting that it is a stable model of ππ· .
It can be shown that, when restricting to LP-ABA, the Example 3.11. Consider the ABAF π· with literals β =
translations in Definitions 3.1 and 3.6 are each otherβs in- {π, π, π, π, π}, assumptions π = {π, π, π}, and their con-
verse. Below, we let traries π = π, π = π, and π = π, respectively, with rules
rep(π·) = (rep(β), rep(β), rep(π), ) β : π1 = π β π, π π2 = π β π, π π3 = π β π.
where rep(π) = π. First note that {π} β stb(π·). We construct the LP-ABAF π·β²
by adding rules ππ β π, ππ β π, and ππ β π; ππ , ππ , and
Lemma 3.8. For any LP π , it holds that π = ππ·π . ππ are the novel contraries. Moreover, π is neither a contrary
nor an assumption, so we add a novel assumption ππ with
Proof. Each naf atom not π corresponds to an assumption
contrary π. The stable extension {π} is only preserved under
in ππ· whose contrary is π. Applying the translation from
projection: we now have {π, ππ } β stb(π·β² ).
Definition 3.6, we map each assumption not π to the naf lit-
eral not not π = not π. Hence, we reconstruct the original We show that each ABAF π· can be mapped into an (under
LP π . projection) equivalent LP-ABAF π·β² . We furthermore note
that the translation can be computed efficiently.
We obtain a similar result for the other direction, under
the assumption that each literal is the contrary of an as- Proposition 3.12. For each ABAF π· = (β, β, π, ) there
sumption, i.e., if β = π βͺ π as it is the case for the LP-ABA is ABAF π·β² computable in polynomial time s.t. (i) π·β² is an
fragment. The translations from Definition 3.6 and 3.1 are LP-ABAF and (ii) π β stb(π·β² ) iff π β© π β stb(π·).
each otherβs inverse modulo the simple assumption renam-
ing operator rep as defined above. Note that we associate Proof. Let π· = (β, β, π, ) be an ABAF and let π·β² =
each assumption π β π with not π. (ββ² , ββ² , π, β² ) be ABAF constructed as described, i.e.,
Lemma 3.9. Let π· = (β, β, π, ) be an ABAF in the LP 1. For each assumption π β π we introduce a fresh
fragment. It holds that π·ππ· = rep(π·). atom ππ ; in the novel ABAF π·β² , ππ is the contrary
of π.
Proof. When applying the translation from ABA to LP ABA, 2. If π is the contrary of π in the original ABAF π·, then
we associate each assumption π β π with a naf literal we add a rule ππ β π.
not π. Applying the translation from Definition 3.1, each naf 3. For any atom π that is neither an assumption nor a
literal not π is an assumption in π·ππ· . We obtain π·ππ· = contrary in π·, we add a fresh assumption ππ and let
(rep(β), rep(β), rep(π), ) where rep(π) = π. π be the contrary of ππ in π·β² .
We are ready to prove the main result of this section. We First of all, the construction is polynomial. Towards the
make use of Theorem 3.4 and obtain the following result. semantics, let us denote the result of applying steps (1) and
(2) by π·* . We show that in π· and π·* the attack relation
Theorem 3.10. Let π· = (β, β, π, ) be an LP-ABAF and between semantics persists.
let ππ· be the associated LP . Then, π β stb(π·) iff Th π· (π)β Let π β π be a set of assumptions. In the following,
π is a stable model of ππ· . we make implicit use of the fact that entailment in π· and
π·* coincide except the additional rules deriving certain
Proof. It holds that π is stable in π· iff contraries in π·* .
(β) Suppose π attacks π in π· for some π β π. Then
rep(π) = {not π | π β π}
π β Th π· (π) where π = π. By construction, π β Th π·* (π)
is stable in rep(π·). This in turn is equivalent to rep(π) is as well and since π = π, the additional rule ππ β π is
stable in π·ππ· (by Proposition 3.9). Equivalently, applicable. Consequently, ππ β Th π·* (π), i.e., π attacks π
in π·* as well.
{π | not π β
/ rep(π)} = {π | π β
/ π} = Th π· (π) β π (β) Now suppose π attacks π in π·* for some π β π.
Then ππ β Th π·* (π) which is only possible whenever π β
is stable in ππ· (by Proposition 3.4). This in turn holds iff Th π·* (π) holds for π the original contrary of π. Thus π
Th π· (π) β π is stable in ππ· (by definition, ππ· = {rep(π) | attacks π in π·.
π β β} = ππ· ). We deduce
stb(π·) = stb(π·* ).
From ABA to LP-ABA To complete the correspondence Finally, for moving from π·* to π·β² we note that adding as-
result between ABA and LP, it remains to show that each sumptions ππ (which do not occur in any rule) corresponds
ABAF π· can be mapped to an LP-ABAF π·β² . To do so, we to adding arguments without outgoing attacks to the con-
proceed as follows: structed AF πΉπ·* . This has (under projection) no influence
1. For each assumption π β π we introduce a fresh on the stable extensions of π·* . Consequently
atom ππ ; in the novel ABAF π·β² , ππ is the contrary
of π. π β stb(π·β² ) β π β© π β stb(π·* ) β π β© π β stb(π·).
2. If π is the contrary of π in the original ABAF π·, then as desired.
we add a rule ππ β π to π·β² .
3. For any atom π that is neither an assumption nor a Given an ABAF π·, we combine the previous translation
contrary in π·, we add a fresh assumption ππ and let with Definition 3.6 to obtain the associated LP ππ· . Thus,
π be the contrary of ππ . each ABAF π· can be translated into an LP, as desired.
Example 3.13. Let us consider again the ABAF π· from 4. Set-Stable Model Semantics
Example 3.11. As outlined before, applying the translation
into an LP-ABA π·β² yields an ABAF π·β² with assumptions In this section, we investigate set-stable semantics in the
π = {π, π, π, ππ , ππ } their contraries π = ππ , π = ππ , context of logic programs.
π = ππ , ππ = π, and ππ = π, respectively, and with rules Set-stable semantics has been originally introduced for
bipolar ABAFs (where each rule is of the form π β π with
π β π, π. π β π, π. π β π.
π an assumption and π either an assumption or the contrary
ππ β π. ππ β π. ππ β π. thereof) for capturing existing notions of stable extensions
The resulting framework lies in the LP-ABA class. In the next for bipolar (abstract) argumentation; we will thus first iden-
step, we apply the translation from LP-ABA to LP and obtain tify the corresponding LP fragment of bipolar LPs and intro-
the associated LP ππ· with rules duce the novel semantics therefor. We then show that this
semantics corresponds to set-stable ABA semantics, even in
π β not ππ , not ππ . π β not ππ , not ππ . π β not ππ .
the general case. Interestingly, despite being the formally
ππ β π. ππ β π. ππ β not ππ . correct counter-part to set-stable ABA semantics, the novel
The set {π, ππ , ππ } is the stable model corresponding to our LP semantics exhibits non-intuitive behavior in the general
stable extension {π} from π· (under projection). case, as we will discuss.
3.3. Denial Integrity Constraints in ABA 4.1. Bipolar LPs and Set-Stable Semantics
Our correspondence results allow for a novel interpretation Recall that an ABAF π· = (β, β, π, ) is bipolar iff each
of the derivation of assumptions in ABA in the context of rule is of the form π β π where π is an assumption and π
stable semantics. Analogous to the correspondence of naf is either an assumption or the contrary of an assumption.
in the head and allowing for constraints (rules with empty We adapt this to LPs as follows.
head) in LP we can view the derivation of an assumption as
setting constraints: for a set of assumptions π β π and an Definition 4.1. The bipolar LP fragment is the class of LPs
assumption π β π, a derivation π β’ π intuitively captures π with |ππππ¦(π)| = 1 and ππππ¦(π) β HB π for all π β π .
the constraint β π, π, i.e., one of π βͺ {π} is false. We note that the head of a rule corresponds by definition
Thus, our results indicate that deriving assumptions is the either to an assumption (if it is a naf literal) or the contrary
same as imposing constraints. More formally, the following of an assumption (if it is a positive literal).
observation can be made. We set out to define our new semantics. In ABA, set-
Proposition 3.14. Let π· = (β, β, π, ) be an ABAF and stable semantics relaxes stable semantics: it suffices if the
let π·β² = (β, β βͺ {π}, π, ) for a rule π of the form π β π closure of an assumption π outside a given set is attacked;
with π βͺ {π} β π. Then, π β stb(π·β² ) iff (i) π β stb(π·) that is, it suffices if π βsupportsβ an attacked assumption π,
and (ii) π ΜΈβ π or π β π. e.g., if the ABAF contains the rule π β π. Let us discuss
this for bipolar LPs: given a set of atoms πΌ β HB π in
Proof. We first make the following observation. We have a program π , we can accept an atom π not only if it is
βπ β π : Th π· (π) β Th π·β² (π) reachable from β(πΌ), but also if there is some reachable π
and not π βsupportsβ not π. For instance, given the rule
by definition and
of the form not π β not π β π , we are allowed to add the
π β Th π·β² (π) β Th π· (π) β π β
/π contraposition π β π to the program π before evaluating
because the only additional way to make deriviations in π·β² our potential model πΌ.
is through a rule entailing π. This, however, implies To capture all βsupportsβ between naf-negated atoms, we
define their closure, amounting to the set of all positive and
π closed in π·β² β Th π· (π) = Th π·β² (π), (1)
naf-negated atoms obtainable by forward chaining.
i.e., for sets closed in π· , the derived atoms coincide.
β²
Now let us show the equivalence. Definition 4.2. For a bipolar LP π and a set π β HB π βͺ
(β) Suppose π β stb(π·β² ). Since π is closed, π ΜΈβ π HB π , we define
or π β / π, so condition (ii) is met. Moreover, by (1), π is
conflict-free and attacks each π β / π in π·, i.e., π· β stb(π·). supp(π) = π βͺ {π | βπ β π : ππππ¦(π) β π, βπππ(π) = π}.
Thus condition (i) is also met.
The closure of π is defined as cl (π) = π>0 suppπ (π).2
βοΈ
(β) Let π β stb(π·) and let π ΜΈβ π or π β π. Then π is
also closed in π·β² . We apply (1) and find π β stb(π·β² ). Note that cl (π) returns positive as well as negative atoms.
For a singleton {π}, we write cl (π) instead of cl ({π}).
Example 3.15. Consider the ABAF π· with assumptions
π = {π, π, π, π}, and their contraries π, π, π, and π, respec- Example 4.3. Consider the bipolar LP π given as follows.
tively, with rules
π1 = π β π, π. π2 = π β π. π : π β not π not π β not π π β not π .
The ABAF π· has two stable models: π1 = {π, π, π} and Then, cl ({not π}) = {π, not π, not π}, cl ({not π}) =
π2 = {π, π, π}. {not π}, and cl ({not π }) = {π, not π }.
Consider the ABAF π·β² where we add a new rule
We define a modified reduct by adding rules to make the
π3 = π β π. closure explicit: for each atom π β HB π , if not π can be
Intuitively, this rule encodes the constraint β π, π, i.e., π reached from not π, we add the rule π β π.
and π cannot be true both at the same time. Consequently,
the ABAF π·β² has a single stable model π1 . 2
suppπ (π) denotes the π-th application of supp(Β·) to π .
Definition 4.4. For a bipolar LP π and πΌ β HB π , the set- Lemma 4.9. For a bipolar LP π and a set π β HB π βͺHB π ,
stable reduct ππ πΌ of π is defined as ππ πΌ = π πΌ βͺ ππ where cl (π) is computable in polynomial time.
ππ = {π β π | π, π β HB π , π ΜΈ= π, not π β cl ({not π})}. It follows that the computation of a set-stable model of
a given program π is of the same complexity as finding a
Note that we require π ΜΈ= π to avoid constructing redun- stable model.
dant rules of the form βπ β πβ. In the case of general LPs, however, the novel seman-
tics exhibits counter-intuitive behavior, as the following
Example 4.5. Let us consider again the LP π from Exam- example demonstrates.
ple 4.3. Let πΌ1 = {π} and πΌ2 = {π, π}. We compute the
set-stable reducts according to Definition 4.4. First, we com- Example 4.10. Consider the following two LPs π1 and π2 :
pute the reducts π πΌ1 and π πΌ2 . Second, for each naf literal
not π₯, we add a rule π₯ β π¦, for each π¦ β HB π with π1 : π β not π β not π
not π¦ β cl ({not π₯}), to both reducts. Inspecting the com- π2 : π β not π β not π, not π .
puted closures of the naf literals of π , this amounts to adding
the rule (π β π) to each reduct. In π1 the set {π, π} is set-stable because we can take the con-
Overall, we obtain traposition of the rule and obtain π β π. This is, however,
not possible in π2 which in fact has no set-stable model.
ππ πΌ1 : π β β
β πβ πβπ
The example indicates that the semantics does not gen-
ππ πΌ2 : πβ πβπ eralise well to arbitrary LPs. We note that a possible and
arguably intuitive generalisation of set-stable model seman-
We are ready to give the definition of set-stable semantics.
tics would be to allow for contraposition for all rules that
Note that we state the definition for arbitrary (not only
derive a naf literal. This, however, requires disjunction in
bipolar) LPs.
the head of rules. Applying this idea to Example 4.10 yields
Definition 4.6. An interpretation πΌ β HB π is a set-stable the rule π β¨ π β π when constructing the reduct with re-
model of an LP π if πΌ is a β-minimal model of ππ πΌ satisfying spect to π2 . The resulting instance therefore lies in the class
of disjunctive LPs (a thorough investigation of this proposal
(a) π β πΌ iff there is π β ππ πΌ s.t. βπππ(π) = π and however is beyond the scope of the present paper).
ππππ¦(π) β πΌ;
(b) there is no rule π β ππ πΌ with βπππ(π) = β
and
4.3. Relating ABA and LP under set-stable
ππππ¦(π) β πΌ.
semantics
Example 4.7. Consider again the LP π from Example 4.3.
In the previous subsection, we identified certain shortcom-
It can be checked that π has no stable model. Indeed, the
ings of set-stable semantics when applied to general LPs.
reduct π πΌ1 contains the unsatisfiable rule (β
β); the set
This poses the question whether our formulation of set-
πΌ2 = {π, π} on the other hand is not minimal for π πΌ2 .
stable LP semantics is indeed the LP-counterpart of set-
If we consider the generalised set-stable reduct instead, we
stable ABA semantics. In this subsection, we show that,
find that the set πΌ2 is a β-minimal model for ππ πΌ2 . The atom
despite the unwanted behavior of set-stable model seman-
π is factual in ππ πΌ2 and the atom π is derived by π. Thus, πΌ2
tics for LPs, the choice of our definitions is correct: set-stable
is set-stable in π .
ABA and LP semantics correspond to each other. We show
that our novel LP semantics indeed captures the spirit of
4.2. Set-stable Semantics in general ABA set-stable semantics, even in the general case.
(non-bipolar) LPs We show that the semantics correspondence is preserved
under the translation presented in Definition 3.1. We prove
So far, we considered set-stable model semantics in the the following theorem.
bipolar LP fragment. As it is the case for the set-stable
ABA semantics, our definition of set-stable LP semantics Theorem 4.11. For an LP π and its associated ABAF π·π ,
generalises to arbitrary LPs, beyond the bipolar class. πΌ is set-stable in π iff β(πΌ) is set-stable in π·π .
Set-stable model semantics belong to the class of two-
valued semantics, that is, each atom is either set to true Proof. By definition, πΌ is set-stable iff it is a β-minimal
or false (no undefined atoms exist). Moreover, set-stable model of ππ πΌ satisfying
model semantics generalises stable model semantics: each
(a) π β πΌ iff there is π β ππ πΌ s.t. βπππ(π) = π and
stable model of an LP is set-stable, but not vice versa, as
ππππ¦(π) β πΌ;
Example 4.7 shows.
(b) there is no π β ππ πΌ with βπππ(π) = β
and
Proposition 4.8. Let π be an LP. Each stable model πΌ of π ππππ¦(π) β πΌ.
is set-stable (but not vice versa).
Equivalently, by definition of ππ πΌ ,
Proof. Let πΌ denote a stable model of π . By definition, the
(π) π β πΌ iff
generalised reduct ππ πΌ of π πΌ is a superset of all rules in π πΌ .
Thus (a) and (b) in Definition 4.6 are satisfied. Moreover, πΌ (1) there is π β π s.t. βπππ(π) = π, ππππ¦ + (π) β
is β-minimal by Definition 2.2. πΌ and ππππ¦ β (π) = β
; or
(2) there is π β πΌ such that not π β cl (not π)
We furthermore note that the support of a set of positive and there is π β π s.t. βπππ(π) = π,
and negative atoms can be computed in polynomial time. ππππ¦ + (π) β πΌ and ππππ¦ β (π) = β
; and
(b) there is no π β π with βπππ+ (π) = β
, βπππβ (π) β the non-intuitive behavior affects set-stable ABA semantics.
πΌ, πΌ β ππππ¦ + (π), and ππππ¦ β (π) = β
. However, we find that set-stable semantics generalise well
for ABAFs. The reason lies in the differences between deriv-
The second item (b) is analogous to the proof of Theorem 3.4; ing assumptions (in ABA) and naf literals (in LPs) beyond
item (a1) corresponds to item (a) of the proof of Theorem 3.4. classical stable model semantics.
Item (a2) formalises that it suffices to (in terms of ABA) Let us translate Example 4.10 in the language of ABA.
attack the closure of a set.
Let πΌ be a set-stable model of π . We show that π = β(πΌ) Example 4.13. The translation of the LPs π1 and π2 from
is set-stable in π·π , i.e., π is conflict-free, closed, and attacks Example 4.10 yields two ABAFs π·1 and π·2 . The ABAF π·1
the closure of all remaining assumptions. The first two has two assumptions π1 = {π, π} (representing not π and
points are analogous to the proof of Theorem 3.4. Below we not π, respectively) with contraries π and π, and rules
prove the last item.
β1 : π β . π β π.
β’ π attacks the closure of all other assumptions: Sup-
pose there is an atom π β πΌ which is not reachable The ABAF π·2 has three assumptions π2 = {π, π, π} (repre-
from π and there is no π β πΌ with notπ β cl (notπ). senting not π, not π, and not π , respectively) with contraries
Similar to the proof in Theorem 3.4, we can show π, π, π, and rules
that πΌ β² = πΌ β {π} is a model of ππ πΌ . By assumption
there is is no rule π β π such that βπππ(π) = π, β2 : π β . π β π, π.
ππππ¦ + (π) β πΌ β² , and ππππ¦ β (π) β© πΌ β² = β
(otherwise,
π is reachable from π); moreover, there is no rule By Theorem 4.11, we obtain the set-stable extensions of the
π β π in ππ (otherwise, not π is in the support ABAFs from our results from the original programs π1 and
from not π). We obtain that πΌ β² is a model of ππ πΌ , π2 . In π·1 , the empty set is set-stable because it attacks the
contradiction to our initial assumption. closure of each assumption. In π·2 , on the other hand, no set of
assumptions is set-stable: π and π are not attacked, although
Next, we prove the other direction. Let π = β(πΌ) be a they jointly derive π which is attacked by the empty set.
set-stable extension of π·π . We show that πΌ is set-stable in
π . Similar to the proof of Theorem 3.4 we can show that all In contrast to the LP formulation of the problem where
constraints are satisfied and that πΌ is indeed minimal. Also, taking the contraposition of each rule with a naf literal
the remaining correspondence proceeds similar as in the in the head would have been a more natural solution, the
case of stable semantics, as shown below. application of set-stable semantics in the reformulation of
Example 4.10 confirms our intuition. The set {π, π} derives
β’ Let π β πΌ. Then either we can construct an argu- the assumption π, however, the attack onto π is not propa-
ment π β² β’ π, π β² β π in π·π , or there is some π β πΌ gated to (the closure of) one of the members of {π, π}.
such that not π β cl (not π) for which we can con- The example indicates a fundamental difference between
struct an argument in π·π . If the former holds, then deriving assumptions and naf literals in ABA and LPs, re-
we proceed analogously to the corresponding part spectively. A rule in an LP with a naf literal in the head
in the proof of Theorem 3.4 and item (a1) is satisfied. is interpreted as denial integrity constraint (under stable
Now, suppose the latter is true. Analogously to the model semantics). As a consequence, the naf literal in the
the proof of Theorem 3.4, we can show that there is head of a rule is replaceable with any positive atom in the
a rule π β π with ππππ¦ + (π) β πΌ, ππππ¦ β (π)β©πΌ = β
body; e.g., the rules not π β π, π and not π β π, π are
and βπππ(π) = π, that is (a2) is satisfied. equivalent as they both formalise the constraint β π, π, π .
β’ For the other direction, suppose there is a rule Although a similar behavior of rules with assumptions in
π β π with ππππ¦ + (π) β πΌ, ππππ¦ β (π) β© πΌ = β
the head can be identified in the context of stable semantics
and βπππ(π) = π and there is π β πΌ with not π β in ABA, the derivation of an assumption goes beyond that; it
cl (not π) and βπππ(π) = π, ππππ¦ + (π) β πΌ and indicates a hierarchical dependency between assumptions.
ππππ¦ β (π) = β
for some π. We can construct argu-
ments for all ππππ¦ + (π) β πΌ and thus π β πΌ.
5. Discussion
Analogous to the case of stable semantics, we can show
that the LP-ABA fragment preserves the set-stable semantics In this work, we investigated the close relation between
and obtain the following result. non-flat ABA and LPs with negation as failure in the head,
focusing on stable and set-stable semantics. Research often
Theorem 4.12. Let π· be an LP-ABAF and let ππ· be the focuses on the flat ABA fragment in which each set of as-
associated LP. Then, π β sts(π·) iff Th π· (π) β π is a set- sumptions is closed. This restriction has however certain
stable model of ππ· . limitations; as the present work demonstrates, non-flat ABA
is capable of capturing a more general LP fragment, thus
Making use of the translation from general ABA to the LP- opening up more broader application opportunities. To the
ABA fragment outlined in the previous section, we obtain best of our knowledge, our work provides the first corre-
that the correspondence extends to general ABA. spondence result between an argumentation formalism and
a fragment of logic programs which is strictly larger than
4.4. Set-stable Semantics for General the class of normal LPs. We furthermore studied set-stable
(non-bipolar) ABAFs semantics, originally defined only for bipolar ABAFs, in
context of general non-flat ABAFs and LPs.
Recall that Example 4.10 indicates that the semantics does The provided translations have practical as well as the-
not generalise well in the context of LPs. In light of the oretical benefits. Conceptually, switching views between
close relation between ABA and LP, it might be the case that
deriving assumptions (as possible in non-flat ABA) and im- models in logic programming, Studia Logica 93 (2009)
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This research was funded in whole, or in part, by the Euro-
ment - Proceedings of COMMA 2022, Cardiff, Wales,
pean Research Council (ERC) under the European Unionβs
UK, 14-16 September 2022, volume 353 of Frontiers in
Horizon 2020 research and innovation programme (grant
Artificial Intelligence and Applications, IOS Press, 2022,
agreement No. 101020934, ADIX) and by J.P. Morgan and
pp. 212β223. URL: https://doi.org/10.3233/FAIA220154.
by the Royal Academy of Engineering under the Research
doi:10.3233/FAIA220154.
Chairs and Senior Research Fellowships scheme; by the Fed-
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ficial Intelligence Dresden/Leipzigβ, project identification
Santiago Compostela, Spain, August, 2020, volume
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