Introduction
Consider the following scenario: the members of a board of directors are gathered in a meeting to decide the future general strategy of their company.

: “We should focus on improving our business organization structure, because it determines our economic conduct.” (focusOnOs and OsDeterminesEc, to shorten) is advanced by one member.

: “We should focus on improving our market performance, because it determines our business organization structure.” (focusOnMp and MpDeterminesOs) is then advanced by another member, as an attack on .

: “We should focus on improving our economic conduct, because it determines our market performance.” (focusOnEc and EcDeterminesMp), is then given in response to .

The first member attacks , however, with , to an inconclusive argumentation.

: “Our firm needs 1 billion dollars revenue this fiscal year.”, meanwhile, is an argument expressed by another member.

: “Let our company just sink into bankruptcy!” (focusOnLiq), another member impatiently declares in response, against which, however, all the first three speakers promptly express dissent with their arguments.
We can represent this argumentation as of Figure A.
In Dung’s abstract argumentation theory [Dung1995], an admissible
set of arguments is such that (1) no argument in the set is
attacking an argument of the same set and (2)
each argument attacking an argument in the set is attacked back
by some argument in the set.
Clearly, there is no
nonempty admissible set in .
With labelling [Caminada2006] (an argument is in if all its attackers are out, is out if there exists an attacker
that is in, and is undecided, otherwise), every
argument is labelled undecided,
and so we gain almost no information on the
acceptability of the arguments.
However, notice that , and
are arguments for the benefit
of their company’s growth. So, by aggregating
them into a new argument : “We should
focus on our company’s further growth” (focusOnImp),
and by thus deriving the framework as in Figure B,
we could obtain some more useful information on the
acceptability of arguments, namely, that is in,
is out, and is in. Hence
we have sharpened acceptability statuses of and,
in particular, of .
Abstract interpretation for cycles
What we saw is effectively abstract interpretation [Cousot and Cousot1977],
a powerful methodology known
in static program analysis to map concrete space semantics to
abstract space semantics
and to do inferences in the latter space to say
something about the former space. The abstract semantics
is typically coarser than the concrete semantics;
in our example, the detail of what exactly their company
should focus on was abstracted away. In return, we were
able to conclude that is in and,
moreover, that
is out. Compared to
existing approaches to deal with cycles e.g.
[Baroni, Giacomin, and Guida2005], which gives in state
to by enforcing acceptance of either of , and
to reject , this approach that we propose does not require one to
accept any of the arguments within the cycle, even provisionally, in order to
be able to reject , and thus accept .
By abstracting away some or all of the cyclic arguments, we avoid
having to accept any of them while rejecting others.
In general, abstract interpretation is applicable
to cycles of any length. There
can be more than one way of interpreting an argumentation
framework abstractly, however, and the key for obtaining
a good outcome
is to find properties sufficiently fine for abstraction.
For the attacks among , , and in ,
we observe that, specifically: ’s
focusOnOs, ’s
focusOnEc, and ’s
focusOnMp
attacks ’s focusOnLiq; and
’s focusOnOs
’s focusOnEc and
’s focusOnEc form a cycle of attacks.
For these,
the semantic information fine enough to abstract
, , into is shown below (only
focusOnX expressions are explicitly stated here).
In this figure, (focusOnImp: focus on further
growth), sits as
more abstract an argument of
, and .
While there may be other alternatives for
abstracting them, focusOnImp
belongs to a class of good abstractions,
as it satisfies the property that
the three arguments but not focusOnLiq
fall into it, which gives justification
as to why the attack relation in the initial argumentation framework
is expected to preserve
in the abstractly interpreted argumentation
framework (from to ). This is
what we might describe the condition of attackpreservation.
Further,
the three concrete arguments exhibit
a kind of competition for the objective:
their company’s business focus.
While
“organisation structure”, “market performance”, and
“economic conduct” all vie for it and in that sense they
indeed oppose, neither of them
actually contradicts the objective, which is why abstraction
of the three arguments is possible here.
We will formulate abstract interpretation for argumentation
frameworks, the first study of the kind, as far as we are aware.
We will go through technical preliminaries (in Section 2), and
develop our formal frameworks and make comparisons
to Dung preferred and cf2 semantics (in Section 3),
before drawing conclusions.
The discovery we ultimately make is that whether
Dung preferred or cf2 semantics is adequate or problematic
depends not only on the argumentation framework’s structure, but also
on the semantic relation between its arguments.
We will show that our
methodology
is one viable way
of enhancing accuracy in judgement as to
which set of arguments should be accepted.
Technical preliminaries
Abstract argumentation
Let be a class of abstract entities. An argumentation framework according to Dung’s argumentation theory is a 2tuple for and . Let with a subscript refer to a member of , and let with or without a subscript refer to a subset of . An argument is said to attack another argument if and only if, or iff, . A subset is said to accept, synonymously to defend, iff, for each attacking , it is possible to find some such that attacks . A subset is said to be: conflictfree iff no element of attacks an element of ; an admissible set iff it is conflictfree and defends all the elements of ; and a preferred set (extension) iff it is a settheoretically maximal admissible set. There are other classifications to admissibility, and an interested reader will find details in [Dung1995]. An argument is skeptically accepted iff it is in all preferred sets and credulously accepted iff it is in at least one preferred set.
Order and Galois connection for abstract interpretation
Let and (each) be an ordered set, ordered in and respectively. Let be an abstraction function that maps each element of onto an element of , and let be a concretisation function that maps each element of onto an element of . for is said to be an abstraction of in , and for is said to be a concretisation of in . If implies and vice versa for every and every , then the pair of and is said to be a Galois connection. Galois connection is contractive: for every , and extensive: for every . Also, both and are monotone with and . An ordered set , ordered by a partial order , is a complete lattice just when it is closed under join and meet for every . Every finite lattice is a complete lattice.
Argumentation frameworks for abstraction
While, for our purpose, Dung’s theory is not expressive enough, all we have to do is to detail the components of the tuple so that we gain access to some internal information of each argument.
Lattices
Let be a
finite lattice. Let be the class
of expressions as abstract entities.
We denote each element of by with or without
a subscript and a superscript.
Those focusOnMP and others
in our earlier example are expressions.
Let
be a function that maps an expression onto
an element in the lattice. This function
is basically a semantic interpretation
of , which could be
some chosen ontology representation with annotations
of generalspecific relation among entities. For example,
in Introduction, focusOnImp
was more general than focusOnMp,
focusOnEc and focusOnOs,
which should enforce
focusOnImp mapped
onto an upper part in than
the three, i.e. . In the rest,
rather than itself, we will
talk of the subcompletelattice of all
for as well as its top and its bottom.
form abstract space arguments with
the relation as defined in .
Concrete space arguments, in comparison, are just a
set of expressions that can possibly be arguments.
Let be such that:
if (the bottom element);
else
.
We let
be another complete lattice
where and satisfies:

if .

and iff: and
with
.
The lattices shown in Figure D illustrate the second condition. Notice that in . and are equivalent in which is indeed a quotient lattice. This equivalence reflects the following interpretation of ours of expressions. Any expression has concrete instances if are children of in abstract lattice. If, here, are all the children of , our interpretation is that mentioning is just a shorthand of mentioning all , i.e. both mean the same thing with respect to the structure of . It is because of this that we place all equivalent sets of expressions at the same node in .
Argumentation frameworks
We call an expression with an ID  which we
just take from 
an argumentlet, so the class of all argumentlets
is .
Each argumentlet shall be denoted by with
a subscript.
We update Dung’s
into
for ,
and . To readers interested in knowing compatibility
with
Dung’s argumentation frameworks,
Dung’s argument corresponds to
a set of all argumentlets in
that share the same ID. For example,
we may have , in which case
maps into if projected
into Dung’s
argumentation framework.
For compatibility of attack relation, too,
Dung’s , i.e. attacks , (assuming
that both and are in )
corresponds to
for
some and some (assuming
both and
are in ).
For convenience hereafter, by argument with or without a subscript,
we refer to a set of argumentlets in that
share
the ID . We do not consider any more structured arguments
than a finite subset of in this work; further
structuring, while of interest,
is not the main focus, which
is left to a future work.
All notations
around extensions: conflictfreeness, acceptance and defence,
admissible and preferred sets, are carried over here for
arguments (note, not for argumentlets).
Abstraction and concretisation
Now, let be the abstraction function, and let be the concretisation function. We require: ; and . Intuition is as we described earlier in 3.1. Note is an empty set when does not contain any for . We say that is the best abstraction of iff , but more generally we say that is an abstraction of iff . We say that is the most general concretisation of iff . More generally, we say that is a concretisation of iff .
Proposition 1 (Galois connection)
For every and every , we have iff .
Proof
If: Suppose , i.e.
by definition of . is a standard abbreviation.
Then we have , contradiction.
Suppose and are not comparable
in , then clearly ,
contradiction.
Only if: Suppose ,
then there exists at least one in
which is not in any set equivalent
to under . Then by definition of ,
we have , contradiction.
Example 1
In Figure D, . We see that, for instance, is mapped to by as . focusOnImp is hence (the best) abstraction of . Meanwhile, . Since is a Galois connection, again.
Let with or without a subscript denote a set of expressions. Each argumentlet comes with a singleton set of expression, so an argument comes with a set of expressions. When we write , we mean to refer to all expressions associated with . For abstraction, we say that an argument is:
 abstractioncovering

for a set of arguments iff, if is an abstraction of , then it is an abstraction of where is a nonempty subset of .
 abstractiondisjoint

for a set of arguments iff, for each , , if is an abstraction of , then , , is not ’s abstraction.
 abstractionsound

for a set of arguments iff, for each , , there is no that is not abstracted by any .
 abstractioncomplete

for a set of arguments iff, for each , is an abstraction of .
Figure E illustrates these 4 conditions. comes with , with , and with . In [A], is abstractioncovering for because abstracts from a nonempty subset of both and . If should abstract only from a nonempty subset of but not of , abstractioncoveringness would not be satisfied. In [B], satisfies abstractiondisjointness because any expression is abstracted at most by one expression in . If, here, should abstract both from and , this condition would fail to hold. In [C], satisfies abstractionsoundness because all expressions in are abstracted. If there should be even one expression in that is not abstracted, this condition would fail to hold. In [D], satisfies abstractioncompleteness because each expression in abstracts expressions in . If there should be even one expression in that does not abstract any expressions in , this condition would fail to hold.
Proposition 2 (Independence)
Let be
one of the propositions is abstractioncovering,
is abstractionsound,
is abstractiondisjoint,
is abstractioncomplete.
Then materially implies
iff .
As for motivation of the four conditions, our goal for abstraction of a given set of arguments dictates that, in whatever manner we may abstract, eventually we abstract from all expressions of all the arguments in . Hence we have abstractionsoundness. However, consider an extreme case where each abstracts from a single argument . Then, certainly, such abstraction weakens each member of , but there is no guarantee that the weakening is a weakening of , because abstraction of is necessarily abstraction of each of , but abstraction of is not necessarily that of . Not abstracting from each and every is problematic for this reason. Abstractioncoveringness is therefore desired. Abstractiondisjointness discourages redundancy in abstraction. Abstractioncompleteness ensures relevance of abstraction to a given set of arguments to be abstracted. In the rest, whenever we state is an abstraction of , will be assumed to be abstractioncovering, abstractiondisjoint, abstractionsound and abstractioncomplete for them. We say that the abstraction is the best abstraction iff it is the best abstraction of all expressions associated with .
Proposition 3
If is the best abstraction of a set of arguments, then every abstraction of is an abstraction of .
Proposition 4 (Existence)
There exists at least one abstraction for every set of arguments.
Proof
is a complete lattice.
However,
some abstraction, including the top element of
if it is in ,
can be so general
that all arguments are abstracted by it.
In the first example in Introduction,
“Argumentation is taking place.”
could be such an argument, in which
one may not
be normally interested for reasoning about argumentation:
the whole point of argumentation theory is for us to be
able to judge which set(s) of arguments may be acceptable
when the others are unacceptable,
so we should not trivialise a given argumentation by
a big summary argument.
Conditions for conservative abstraction
Hence, a few conditions ought to be defined
in order to ensure
conservative abstraction.
Assume an argumentation framework
. We assume
that those elements of that are so abstract
that they could abstract all argumentlets
in into a single argumentlet
are forming a nonempty upper set of :
() is an upper set iff, if and
, then .
Intuition is that once we find some in
that is so general, then any
such that is also.
For example, if in
is so general, then
“Argumentation is taking place.” is also so general.
Let us say that
there is a path from an argument to
an argument iff either attacks ,
or else there is a path from
to some argument which attacks .
Let us say that
a set of arguments is strongly
connected component in
iff (1) there is a path from any
to any and (2) there exists
no such that
satisfies (1).
For an argumentation framework , we define that
abstraction of a set of arguments
is: valid iff there exists
a strongly connected component
such that: (1) ; and
(2) there exists no
such that is an abstraction of
(abstraction is over as many members of a strongly
connected component as possible);
nontrivial iff
(abstraction
cannot be too general); and
compatible iff: there exist
no argumentlets that satisfy
both (1) and
(2) and are comparable in
(abstraction cannot be over arguments that contain an attack
from more abstract an argument on more concrete an argument or vice
versa).
What compatibility expresses is:
a pair of arguments and with an attack
between them is not suited for abstraction if (or )
is more, if not equally, abstract than (or ).
For justification, suppose firstly that is .
Then it is a selfattack. Let us suppose that
abstraction of and is feasible, then we can
get rid of all selfattacks by means of abstraction.
But that would render all such selfattacks not playing
any role in argumentation, which cannot be the case [Baumann et al.2017].
In a more general setting where is not , if it is
that attacks , given that is more concrete an argument
of , the attack is again a type of selfattack. Still,
it is not safe to compile away the attack by taking abstraction
of and , because the abstraction is more, if not equally,
abstract than which was attacking. The second condition of
validity is motivated in a way by compatibility.^{1}^{1}1We, however,
have a more recent result on abstraction of selfattacks.
An interested reader can contact either of the authors for detail.
Let us consider
the example in Introduction again (a part of it is relisted in Figure
F on the left).
There are three arguments in the cycle, and , and are not comparable in (Cf. Figure D). The least upper bound of any two among the three, by the way, is the same element in . Hence, by taking abstraction of only two among them, say and , we obtain an abstract space argumentation framework as in the right figure of Figure F. As the attacks between and are both of abstractconcrete and of concreteabstract, the compatibility condition prevents any further abstraction on this argumentation framework. This, however, is amiss, because such abstractconcrete (concreteabstract) relation among the participants of the cycle were not present (they were not comparable in ) in the original argumentation framework. The validity condition precludes this anomaly.
Proposition 5 (Independence)
Let be one of the propositions: is valid, is nontrivial, is compatible. materially implies iff .
These are conditions that apply for abstraction of a given set of arguments alone. In an argumentation framework, however, we also consider attacks between a set of arguments and the arguments that are not in the set. We say that abstraction of a set of arguments is attackpreserving iff, for each and each , if at least either or , then and are not comparable in (abstraction of and external attackers/attackees shall not be in abstractconcrete (concreteabstract) relation). For intuition behind this condition, let us consider Figure G.
For simplicity, let us assume that
, , is a singleton . The abstract
lattice is shown in Figure G.
See to it that the attack of on is not
of absoluteconcrete (concreteabsolute). Now, with (the best) abstraction
of , and , we obtain with .
While, in general, there is no continuity between
some argument attacking
some argument and some abstraction of
attacking , an exception is taken when
there exists some pivotal point in that strongly
distinguishes and
(which, by the definition
of abstraction,
means and are equally distinguished), to one
group, and to another distinct group. In such a case,
as the attack of on can be viewed as
the attack of the group that belongs to on the group that
belongs to, and as and belong to the same group,
abstraction of into does not modify the attack.
This strong distinction holds just when and
are not comparable in . This justifies
retention of the attack by on (the pivot is
) in the abstract space
argumentation framework.
We say that abstraction
of a set of arguments is
conservative iff
it is valid, nontrivial, compatible, and
attackpreserving.
Theorem 1
For a given , if an abstraction of is conservative, then each abstraction of such that is an abstraction of is conservative.
Proof Suffice it to check the four conditions one by one.
Computation of abstract space argumentation frameworks from a concrete space argumentation framework
All abstract space argumentation frameworks with conservative and best abstraction can be computed for a given argumentation framework, , , and with Algorithm 1 which, informally, just keeps replacing, where possible at all, a part of, or an entire, cycle with an abstract argument for all possibilities. Concerning Line 9, for a set of arguments in a given argumentation framework, we say that is a maximal subset of that satisfies conservative abstraction iff (1) the best abstraction of is conservative and (2) there exists no that satisfy both (2A): and (2B): the best abstraction of is conservative.
Proposition 6 (Complexity)
Algorithm 1 runs at worst in exponential time.
Proof. Strongly connected components are known to be computable in linear time (Line 5). Line 9 is computable at worst in exponential time. With argumentlets (with possibly less than
arguments), we can overestimate that the for loop executes at most
times, the 1st while loop at most times, and the 2nd while loop at most times.Preferred sets in concrete and abstract spaces
We now subject preferred sets in concrete space to those in abstract space for more clues on arguments acceptability in concrete space. Let us denote Algorithm 1 by , and a function with [inputs = a set of argumentation frameworks] and [output = a set of all preferred sets for each given argumentation framework] by (the procedure can be found in the literature). Further, let be a projection function with [inputs = a set of sets of sets of arguments (i.e. all preferred sets for each argumentation framework)] and [outputs = a set of sets of sets of arguments], with the following description. Let be a function with [inputs = a set of sets of arguments arguments] and [outputs = a set of sets of arguments] such that . Then . For example, if , then is .
Figure H illustrates on one hand for an argumentation framework in concrete space, which gives us all preferred sets of , and on the other hand , which also gives us a set of all preferred sets in concrete space but through abstraction. The abstract transformations proceed by transforming the given concrete space argumentation framework into a set of abstract space argumentation frameworks (), deriving preferred sets for them (), and projecting them to concrete space preferred sets () so that comparisons to the preferred sets obtained directly within concrete space can be done. In particular, we can learn: (1) an argument deemed credulously/skeptically acceptable within concrete space is positively approved by abstract space preferred sets, thus we gain more confidence in the set members being acceptable; (2) arguments not deemed acceptable within concrete space, i.e. those that are not in any preferred set, are negatively approved also by abstract space preferred sets, thus we gain more confidence in those arguments not acceptable. But also: (3) arguments deemed credulously/skeptically acceptable within concrete space may be questioned when their acceptability is not inferred from any abstract space preferred set; and, on the other hand, (4) arguments deemed not acceptable within concrete space may be credulously/skeptically implied by abstract space preferred set(s). To summarise formally, given an argumentation framework , we say that an argument that is deemed credulously/skeptically acceptable in concrete space is:
 +approved

iff, for some/every element of , it belongs to some/every element of .
 questioned

iff, for every element of , it belongs to no element of .
And we say that an argument that is deemed not acceptable in concrete space is:
 approved

iff, for every element of , it belongs to no element of .
 credulously/skeptically implied

iff, for some/every element of , it belongs to some/all member(s) of .
Comparisons to Dung preferred semantics and cf2 semantics, and observations
We conclude this section with comparisons to
Dung preferred semantics and cf2 semantics [Baroni, Giacomin, and Guida2005].
Let us first consider in Figure A and
the lattices as shown in Figure D.
Let us denote by ,
then we have: for (i.e. Dung preferred set);
for cf2();
while for (as we have already
shown the only one abstract space argumentation framework,
in Figure B, we omit the derivation process).
By comparisons between and ,
we observe that all are approved, while
is implied. Hence in this case, with respect to
the semantic structure of , we might say that
Dung preferred set behaves more conservative than necessary.
On the other hand, by comparisons between cf2 and ,
we observe that cf2 accepts either of the arguments in the odd
cycle,
which is more liberal than necessary with respect to
 since no arguments in could break
the preference
preorder
of the three arguments.
Therefore, for , Dung semantics seems to give falsenegative
to acceptability, while cf2 seems to give falsepositives
to either of , , acceptability. If
those acceptability semantics aim to answer “Which arguments
should be (credulously) accepted?”, falsenegatives only signal
omission, but falsepositives signal unintuitive results and are less
desirable.
Let us, however, consider another argumentation framework
in Figure I borrowed from
[Baroni, Giacomin, and Guida2005].
Consider:

The downpour has been relentless since the morning.

It was burning hot today.

All our employees ran a pleasant full marathon today.

Nobody stayed indoor.

Many enjoyed TV shows at home.
We assume the abstract lattice as shown in Figure I
for . We assume , and any
nodes below , , ,
not explicitly
shown there are still assumed to be there. W, A, M, H, Id,
Fm, Dp, Br each abbreviates Weather, Activity,
Mild, Hard, Indoor, Fullmarathon, Downpour and Burning.
The lattice expresses in particular that a downpour and
the burning sun relate under the hard weather, and
the hard weather and indoor activities such as watching TV shows
relate under hard weather activity (that is, an activity to do
under a hard weather condition), but that
hard weather and mild weather activities do not go together. Also,
indoor and noindoor oppose. Here we have: for ; for cf2. Meanwhile,
for , is first of all
the set of a maximal subset of .
It is attackpreserving: ,
which is not comparable with or ,
valid because does not abstract Fm, nontrivial,
and compatible. Hence the argumentation framework shown
under in Figure I is
the abstract space argumentation framework with respect to .
Consequently,
.
Therefore, in this example, ,
too, credulously accepts an argument in the oddcycle as cf2() does.
Notice, however,
that we still obtain the Dung conservative preferred set which
obtains from .
It is safe to observe that
the traditional Dung, or cf2, which is more appropriate depends
not just on an argument graph but also the semantic relation among
the arguments in the graph; and that combination
of abstract argumentation and abstract interpretation is one viable
methodology to address this problem around cycles in argumentation
frameworks.
Related work
As far as we are aware, this is the first study that incorporates abstract
interpretation into abstract argumentation theory. Oddsized cycles
have been a popular topic of research in the literature for some time,
as they tend to prevent the acceptability of all subsequent
arguments with respect to directionality.
Noting the difference between preferred and the grounded semantics,
Baroni et al. [Baroni, Giacomin, and Guida2005] proposed to
accept maximal
conflictfree subsets of a cycle
for gaining more acceptable arguments off an oddlength
cycle, which led to cf1/cf2 semantics. They
are regarded as improvements on more traditional naive semantics
[Bondarenko et al.1997]. They also weaken Dung defence around strongly
connected components of an argumentation framework
into SCCrecursiveness.
The stage2 semantics that took inspiration
from cf2 is another approach with a similar SCCrecursive aspect, but
which is based on the stage semantics [Verheij1996] rather than
the naive semantics, the incentive being to maximise
range (the range of a set of arguments is itself plus all arguments
it attacks).
The fundamental motivation behind those semantics was to treat
an oddlength cycle in a similar manner to an evenlength cycle.
As we showed, however, specialisation of Dung semantics
without regard to semantic relation among arguments in a given argumentation
framework seems not fully generalisable.
To an extent, that any such systematic resolution of
acceptability of cyclic arguments based only on a Dung
argumentation graph is tricky relates
to the fact that attacking arguments
in a cycle can be contrarily [Horn2001] but not necessarily
contradictorily
opposing.
As the study in [Baroni, Giacomin, and Liao2015] shows and as is known in linguistics,
dealing with contrary relations is difficult in Fregean logic.
However, with abstract interpretation, we can take advantage
of semantic information of arguments in partitioning
those attacking arguments in a cycle into mutually incompatible
subsets, by which uniform treatment of cycles come into reach.
Conclusion
We introduced abstract interpretation into argumentation frameworks. Our formulation shows it is also a powerful methodology in AI reasoning. We believe that more and more attention will be directed towards semanticargumentgraph hybrid studies within argumentation community, and we hope that our work will provide one fruitful research direction.
Acknowledgement
We thank Leon van der Torre and Ken Satoh for discussion on related topics which greatly influeced this work of ours.
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[Dung1995]
Dung, P. M.
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