=Paper=
{{Paper
|id=Vol-1287/lanmr2014_paper_6
|storemode=property
|title=Computing Semi-Stable Semantics of AF by 0-1 Integer Programming
|pdfUrl=https://ceur-ws.org/Vol-1287/lanmr2014_paper_6.pdf
|volume=Vol-1287
|dblpUrl=https://dblp.org/rec/conf/lanmr/OsorioDS14
}}
==Computing Semi-Stable Semantics of AF by 0-1 Integer Programming==
https://ceur-ws.org/Vol-1287/lanmr2014_paper_6.pdf
Computing Semi-Stable Semantics of AF by 0-1
Integer Programming
Mauricio Osorio, Juan Dı́az, and Alejandro Santoyo
Universidad de las Americas en Puebla,
Sta. Catarina Martir, Cholula, Puebla, 72820 Mexico
osoriomauri@gmail.com, juana.diaz@udlap.mx, alejandro1.santoyo@gmail.com
http://www.udlap.mx
Abstract. Dung’s abstract argumentation frameworks has been object
of intense study not just for its relationship with logical reasoning but
also for its uses within artificial intelligence. One research branch in ab-
stract argumentation has focused on finding new methods for comput-
ing its different semantics. We present a novel method, to the best of
our knowledge, for computing semi-stable semantics using 0-1 integer
programming, and also experimentally compare it with an answer set
programming approach. Our results indicate that this new method per-
formed well, and it has a great opportunity space for improving.
Keywords: Argumentation Frameworks, Binary Programming, Answer
Set Programming, Semi Stable Semantics
1 Introduction
The main purpose of argumentation theory is to study the fundamental mech-
anism humans use in argumentation and to explore ways to implement this
mechanism on computers. Currently formal argumentation research has been
strongly influenced by abstract argumentation theory of Dung [6]. This approach
is mainly orientated to manage the interaction of arguments by introducing a
single structure called Argumentation Framework (AF). An AF basically is a
pair of sets: a set of arguments and a set of disagreements between arguments
called attacks. Indeed an AF can be regarded as a directed graph in which the
arguments are represented by nodes and the attack relations are represented by
arcs.
In [6], four argumentation semantics were introduced: grounded, preferred,
stable, and complete semantics. The central notion of Dung’s semantics is the
acceptability of the arguments. Even though each of these argumentation se-
mantics represents different patterns of selection of arguments, all of them are
based on the basic concept of admissible set. Informally speaking, an admissi-
ble set presents a coherent and defendable point of view in a conflict between
arguments.
One research branch in abstract argumentation has been to find new meth-
ods for computing its different semantics, i.e. the search for acceptable (w.r.t.
2 Mauricio Osorio, Juan Dı́az and Alejandro Santoyo
certain criteria) sets of arguments. Charwat et al. [10] surveys the approaches
that has been used so far for computing AF semantics, and divided them into re-
duction and direct approaches. The direct approach consists in developing new
algorithms for computing AF semantics, but our interest is in the reduction
approach.
The reduction approach consists in using the software that was originally
developed for other formalisms [10]. Thus, a given AF has to be formalized in
the targeted formalism, like: constraint-satisfaction [5], propositional logic [1],
or answer-set programming [3], [13].
To the best of our knowledge, there is just a previous work [11] where au-
thors indirectly used 0-1 integer programming for computing preferred semantics,
since their approach was based on a mapping from an argumentation framework
AF into a logic program with negation as failure ΠAF to Clark’s completion
Comp(ΠAF ) [12] to 0-1 integer program lc(ΠAF ) [2], which then was solved by
a mathematical programming solver.
In this work, we present a novel method, to the best of our knowledge, for
directly computing semi-stable (SS) semantics using 0-1 integer programming
with no mapping. Our results indicate that this new method performed well.
The paper is organized as follows. Section 2 gives some background on argu-
mentation. Section 3 presents a procedure based on solving a series of 0-1 integer
programming problems for computing SS semantics. Finally, Section 4 presents
a brief description of results, some conclusions and future work.
2 Background
For space reasons, we assume that readers are familiar with the basic notions
of 0-1 integer programming and ASP. We will use some concepts of Dung’s
argumentation approach. An AF captures the relationships between arguments.
Definition 1. [6] An AF is a pair AF := hAR, attacksi, where AR is a finite set
of arguments, and attacks is a binary relation on AR, i.e. attacks ⊆ AR × AR.
We say that a attacks b (or b is attacked by a) if attacks(a, b) holds. Similarly,
we say that a set S of arguments attacks b (or b is attacked by S) if b is attacked
by an argument in S.
Dung defined his argumentation semantics based on the basic concept of
admissible set, which can be understood in terms of defense of arguments and
in terms of conflict-free sets, as follows:
Definition 2. [4] Let AF := hAR, attacksi be an argumentation framework,
A ∈ AR and S ⊆ AR, then:
1. A+ as {B ∈ AR | A attacks B} and
2. S + as {B ∈ AR | A attacks B f or some A ∈ S}.
3. A− as {B ∈ AR | B attacks A} and
4. S − as {B ∈ AR | B attacks A f or some A ∈ S}.
5. S is conflict-free iff S ∩ S + = ∅.
6. S defends an argument A iff A− ⊆ S + .
Computing Semi Stable Semantics of AF by BIP 3
7. F : 2AR → 2AR as F (S) = {A ∈ AR | A is def ended by S}.
It is possible to define the semantics in terms of admissible sets as follows:
Definition 3. [4] Let AF := hAR, attacksi be an argumentation framework and
S ⊆ AR be a conflict-free set of arguments, then:
1. S is admissible iff S ⊆ F (S).
2. S is a complete extension iff S = F (S).
3. S is a preferred extension iff S is a maximal (w.r.t. set inclusion) complete
extension.
The SS semantics is similar to the preferred semantics [4], but instead of
maximizing S it is required to maximize S ∪S + , as states the following definition:
Definition 4. [4] Let AF := hAR, attacksi be an argumentation framework and
S ⊆ AR be a conflict-free set of arguments, then: S is called a SS extension iff
S is a complete extension where S ∪ S + is maximal
The semi-stable semantics accepts an equivalente statements, as follows:
Proposition 1. [4] Let AF := hAR, attacksi be an argumentation framework
and let S ⊆ AR. The following statements are equivalent: 1).- S is a complete
extension such that S ∪ S + is maximal (Definition No. 4), and 2).- S is an
admissible set such that S ∪ S + is maximal.
3 Computing SS Semantics by 0-1 Integer Programming
In this section we show: the 0-1 integer programming formulation for the SS
semantics problem, its encoding in FICO Xpress Mosel1 language and also ex-
plain the objective function and constraints, as well as the iterative process for
computing all the SS semantics’ extensions.
3.1 Semi-Stable Semantics Problem Formulation
First of all, consider that it is required to have a mechanism to work with attacks
more suitable than working with the adjacency matrix of a given AF, then we
define:
Definition 5. Let AF := hAR, attacksi be an argumentation framework, then:
Ri− = {j ∈ AR : (j, i) ∈ attacks} ∀i ∈ AR, is the set of nodes attacking node i,
and R− = {Ri− : i ∈ AR}.
Considering Definitions 2, 3, and 5 we restate the admissible set definition in
order to be able to derive the linear constraint that assures admissibility.
Definition 6. Let AF := hAR, attacksi be an argumentation framework, and
set S ⊆ AR, then S is admissible iff ∀i ∈ S, ∀j ∈ Ri− , ∃k ∈ S, ((k, j) ∈ attacks)).
1
http://www.fico.com/en/wp-content/secure_upload/
Xpress-Mosel-User-Guide.pdf
4 Mauricio Osorio, Juan Dı́az and Alejandro Santoyo
Now consider that decision variables in a mathematical program are a set of
quantities that need to be determined by the solver in order to solve the problem,
i.e. when the best values of the decision variables have been identified in order
to maximize or minimize (optimal solution) the objective function. Therefore,
the first step is to define the required binary decision variables.
Considering that Definition 4 and Proposition 1 state a SS extension in terms
of an admissible set S such that S ∪ S + is maximal, then it is required a decision
variable for S and other for S + , as follows:
(
0 if i 6∈ S
Si = ∀ i ∈ AR (1)
1 if i ∈ S
Definition 7. The set M = {i ∈ AR : Si = 1 in the optimal solution}, and C =
{i ∈ AR : Si = 0 in the optimal solution} is M’s complement.
Accordingly, we define the decision variable for S + , as follows:
(
0 if i 6∈ S +
Si+ = ∀ i ∈ AR (2)
1 if i ∈ S +
This way, the optimal solution to the 0-1 integer problem formulation for the
SS semantics can be stated in terms of maximizing S ∪ S + .
Definition 8. The set M + = {i ∈ AR : Si+ = 1 in the optimal solution}, and
C + = {i ∈ AR : Si+ = 0 in the optimal solution} is M + ’ complement.
It is required to have a decision variable for the union of this sets, as follows:
(
0 if i 6∈ S ∪ S +
Ui = ∀i ∈ AR (3)
1 if i ∈ S ∪ S +
Definition 9. The set M u = {i ∈ AR : Ui = 1 in the optimal solution}, and
Cu = {i ∈ AR : Ui = 0 in the optimal solution} is Mu’s complement.
If the 0-1 program formulation with a given AF has an optimal solution, then
we can think of the SS semantics in terms of a serie of solutions that the model
finds in each iteration, as follows:
{M u1 , M u2 , ..., M uq } : |M u1 | ≥ |M u2 | ≥ ... ≥ |M uq |
which states that the M u1 has the largest possible cardinality, and that |M u1 | ≥
|M u2 | ≥ ... ≥ |M uq | such that |M uq | has the smallest possible cardinality.
Thus, It is also required a decision variable for a couple of either/or con-
straints [9] which we will add per each iteration (see section 3.3) in order to help
to assure S + S + maximality w.r.t. set inclusion, i.e. they help us just to avoid
that solution M ut+1 be different than solution found in iteration t, t − 1, . . . , 1
or any subset of them, as follows:
Computing Semi Stable Semantics of AF by BIP 5
(
1 if |M ut | > |M ut+1
Yr = r = 1, . . . , t − 1 (4)
0 otherwise
Taking into account that a mathematical solver searches in the solution space
for one optimal solution for a given problem formulation, then if we want all the
extensions of a given AF, it is required to solve a series of binary programming
models, one for each extension. Since an AF semantics is made up of several
sets, we use M ut to denote the solution of the binary subproblem in iteration t,
and q to denote the amount of preferred extensions that a given AF has, such
that q ≥ 1 and q ∈ N. The 0-1 integer problem formulation to compute the tth
semi-stable extension of a given AF is the following, including (1), (2), (3), (4):
X
max f (S) = (Si + Si+ ) (5)
i∈AR
Subject to:
Si + Sj ≤ 1 ∀i ∈ AR, ∀j ∈ Ri− (6)
X
Sk ≥ Si ∀i ∈ AR, ∀j ∈ Ri− (7)
k∈Rj−
Si+ ≥ Sj ∀i ∈ AR, ∀j ∈ Ri− (8)
X
Si+ ≤ Sj ∀i ∈ AR (9)
j∈Ri
Ui = Si + Si+ ∀i ∈ AR (10)
X
Si ≥ 1 ∀r = 1, . . . , t − 1 (11)
i∈C r
X
−( Ui ) + |M ur | ≤ |AR| ∗ Yr ∀r = 1, . . . , t − 1 (12)
i∈M ur
X
−( Ui ) + 1 ≤ |AR| ∗ (1 − Yr ) ∀r = 1, . . . , t − 1 (13)
i∈Cur
Si ∈ {0, 1} i = 1, ..., |AR| (14)
Si+ ∈ {0, 1} i = 1, ..., |AR| (15)
Ui ∈ {0, 1} i = 1, ..., |AR| (16)
Yi ∈ {0, 1} i = 1, ..., t − 1 (17)
In (1), (2), (3) and (4) we have the decision variables definition and con-
straints (14), (15), (16), (17) define the problem’s domain. The following para-
graphs explain each constraint of the SS semantics problem formulation:
Maximality with regard to set inclusion. The model’s objective function
(OF)(5) guarantees us that we will find a maximum cardinality set, which will
be the solution M ut . This set is made up of S ∪ S + . Constraints (11), (12),
and (13) along with the objective function will avoid that M t+1 and M ut+1 be
any subset of M t and M ut respectively, thus they guarantee us maximality with
6 Mauricio Osorio, Juan Dı́az and Alejandro Santoyo
regard to set inclusion. Subsection 3.3 explains how constraints (12) and (13)
were defined and why they are added after the computation of each additional
extension in order to compute the whole semantics.
Conflict-Freeness Note that the definition of a conflict-free set in Definition 2
item 5 is not stated in terms of attacks’s directions but just in terms of attacks
between arguments, without considering the directions of them. In this way, such
a definition is considering an arc just as an edge, and therefore the whole AF
can be regarded as an undirected graph, at least with regard to the conflict-free
set problem.
Note that the expression Si + Sj ≤ 1, in constraint (6), will be just fulfilled
when Si = 1 or Sj = 1 but not both and when Si = 0 and Sj = 0, therefore at
most one arguments will be selected which guarantees us that solution will be a
conflict-free set.
Admissibility. The intuition of Definitions 1, 2 and 3 is that an admissible set
S should defend each of its arguments, and Definition 6 just restates it in terms
of Definition 5. Note that P
in Definition 6 the existential quantifier
P suggests that
constraint (7) should be − Sk ≥ 1, but we used − Sk ≥ Si since
k∈R j k∈R j
the constraint must be fulfilled ∀i ∈ AR 2 . This way, the translation from this
definition to constraint (7) is a straightforward task, which guarantees us that
the set M t is admissible.
Creation of S + . So far, we have a conflict-free and admissible set, and it is
still required to build Si+ as in Definition 2 item 2 ∀i ∈ M + . It should be done
by decision variables (2) such that the objective function can be executed. This
way, Si+ should take on value 1 if argument i is attacked by some argument
j ∈ Ri− such that Sj = 1. This is the same that d = di ∨ d2 . . . ∨ dn as a logical
expression which can be linearized as follows [9]:
d ≥ di i = 1, . . . , n (18)
X
d≤ di i = 1, . . . , n (19)
i
d≤1 (20)
Note that (18) and (19) become (8) and (9) respectively while (20) is redundant
due to (15). Thus, constraints (8), (9) and (15) will determine the values of
decision variables (2) from values on decision variables (1).
Thus, M 1 is a conflict-free and admissible set, considering that (5) guarantees
a maximum cardinality set, and according to Definition 4 and Preposition 1, M 1
is a SS extension.
Construction of U = S + S + . In order to avoid that M ut+1 be a subset of
M ut , it is required to have as decision variables the union of (1) and (2), as
defined in (3). To this end, and in order to ease this process, consider that it
is not possible that Si = 1 and Si+ = 1 at the same time, since it would mean
2
There are two
P special cases: Si = 0 and Si = 1. In the first one, it does not matter
the total of k∈R− Sk because Si 6∈ Solution, thus in the second case we will have
P j
k∈R− k
S ≥ 1 which means that there will be at least one argument defending
j
argument i since Si ∈ Solution.
Computing Semi Stable Semantics of AF by BIP 7
that M is not a conflict-free set. Thus, the value that Ui will take on should be
Si + Si+ . Considering Definition 9, constraint (10) guarantees that M u will have
the whole solution.
3.2 Semi Stable Extensions Program
Notice that the problem formulation made up of (1)-(10), and (14)-(17) already
can be used for computing the first SS extension of a given AF. To this end,
this program should be coded using a mathematical programming language like
mosel 3 , which is a straightforward task, since the mathematical language was
developed for expressing mathematical formulas, and even though we can not
show the complete code for space reasons, the following code stands for the whole
mathematical model:
z:= sum(i in arguments) (S(i) + Sp(i))
forall(i in arguments, j in R(i)) S(i) + S(j) <= 1
forall(i in arguments, j in R(i)) sum(k in R(j)) S(k) >= S(i)
forall(i in arguments, j in R(i)) Sp(i) >= S(j)
forall(i in arguments) Sp(i) <= sum(j in R(i)) S(j)
forall(i in arguments) U(i) = S(i) + Sp(i)
forall(i in arguments) S(i) is_binary
forall(i in arguments) Sp(i) is_binary
forall(i in arguments) U(i) is_binary
forall(i in t-1) Y(i) is_binary
maximize(z)
We will denote this program as BIP in order to make reference to it.
3.3 Semi Stable Semantics
Note that once the model is implemented in a mathematical programming lan-
guage, the program just compute one SS extension. In order to compute an
additional extension it is required to solve the model again, but adding addi-
tional constraints to avoid getting previous solutions, which will force to get
another different extension. In this setting, it is required to iterate to find all
the extensions of a given AF until there is no a feasible solution. Thus, we have
to take care of getting no subsets of M t (Case No. 1), and no proper subsets of
M ut (Case No. 2).
Case No. 1. Then, in order to find the constraint that we have to add to avoid
that M t+1 ⊆ M t , consider Definitions 7, and let P be the solution in iteration
t + 1, and M the solution in iteration t, thus:
P ⊆ M ↔ ∀x(x ∈ P → x ∈ M ) ↔ ∀x(x 6∈ P ∨ x ∈ M ) (21)
P 6⊆ M ↔ ∃x(x ∈ P ∧ x 6∈ M ) ↔ ∃x(x ∈ P ∧ x ∈ C) (22)
3
http://www.fico.com/en/products/fico-xpress-optimization-suite/
8 Mauricio Osorio, Juan Dı́az and Alejandro Santoyo
The intuition of this result is that it is required that solution in iteration t+1
has at least one element from solution’s complement in iteration t. Constraint
(11) is defined from this intuition. Note that the mosel code must take care of
the special case where |M | = |AR|.
Case No. 2. Additionally, we have to add other constraint to avoid that
M ut+1 ⊂ M ut . In order to find such a constraint(s), consider Definition 9 and
let P be the solution in iteration t + 1, and M u the solution in iteration t, thus:
P ⊂ M u ↔ ∀x(x ∈ P → x ∈ M u) ∧ ∃x(x 6∈ P ∧ x ∈ M u) (23)
↔ ∀x(x ∈
/ P ∨ x ∈ M u) ∧ ∃x(x 6∈ P ∧ x ∈ M u) (24)
P 6⊂ M u ↔ ∃x(x ∈ P ∧ x 6∈ M u) ∨ ∀x(x ∈ P ∨ x 6∈ M u) (25)
↔ ∃x(x ∈ P ∧ x ∈ Cu) ∨ ∀x(x ∈ P ∨ x 6∈ M u) (26)
Note that the intuition of the expression in (22) should be the same as that of
the first part of the disjunction in (26). Consider also that we wanted to find an
expression that led us to a linearized constraint to avoid getting proper subsets
of previous solutions. Thus, we can have a proper subset when |P | < |M u| holds,
and therefore apply the expression ∃x(x ∈ P ∧ x ∈ Cu), otherwise apply the
expression ∀x(x ∈ P ∨ x 6∈ M u), whose intuition is that the new solution P can
have any element, which means that it requires no constraint. Thus, we have just
to work with the first part of the disjunction. Therefore, we have to linearize the
expression [9]:
X X
if |P | < |M u| then Ui ≥ 1 ↔ not (|P | < |M u|) ∨ Ui ≥ 1
i∈Cu i∈Cu
X
↔ |P | ≥ |M u| ∨ Ui ≥ 1
i∈Cu
X X
↔ Ui ≥ |M ur | ∨ Ui ≥ 1
i∈M ur i∈Cur
which means that the model should apply either i∈M ur Ui ≥ |M ur | or i∈Cu Ui ≥
P P
1, but to satisfy the simultaneousness assumption of binary integer programming,
they must be transformed considering the following general format [9]:
f (x1 , x2 , . . . , xn ) ≤ By g(x1 , x2 , . . . , xn ) ≤ B(1 − y)
where B is a big number, in our case B = |AR|, and y is the binary variables
defined in (17). This transformation becomes constraints (12) and (13).
Now, let SSE a set of all the SS extensions of a given AF, and M C a set
of additional (11), (12), and (13) constraints, then the algorithm for computing
the q extensions of a given AF is the following:
Computing Semi Stable Semantics of AF by BIP 9
1 Set SSE = ∅, M C = ∅;
2 Solve BIP ∪ M C;
3 while optimal solution found do
4 Let M, M + , and M u the optimal solution, and C, C + , and Cu its
complements respectively.;
5 Add MP to SSE;
6 Add i∈CP Si ≥ 1 to M C;
7 Add −(Pi∈M ur Ui ) + |M ur | ≤ |AR| ∗ Y (r)∀r = 1, . . . , t − 1 to M C;
8 Add −( i∈Cur Ui ) + 1 ≤ |AR| ∗ (1 − Y (r))∀r = 1, . . . , t − 1 to M C;
9 Solve BIP ∪ M C;
10 end
Algorithm 1: For Computing all Semi-Stable Extensions of a Given AF
Now it is possible to state the following theorem:
Theorem 1. Let AF be an argumentation framework, R− as defined in 5,
M, M + , and M u is a solution of BIP , and SSE is computed as described in
Algorithm No. 1, then SSE is the set of all SS extensions of AF .
Proof. Sketch: From the above discussion consider the following items:
1. By OF (5) we know that M ∪ M + is a maximum cardinality set.
2. By constraints(6), (7) we know that M is a conflict-free and admissible set.
3. By constraints (8) and (9) we know that M + is made up from M .
4. By constraint (10) we know that M u = M ∪ M +.
5. By Definition 4 and Proposition 1 we know that if a set M is a conflict-free
and admissible set, and M ∪ M + is maximal then M is a SS extension.
6. Now, notice that in each iteration, due to the constraint added to M C in
steps 6, 7 and 8 in iteration t, the solution M u (if exists), obtained in step
4 must not be a superset or subset of any previous solutions already in SSE,
and M u must be of maximum cardinality among the solutions that satisfy
BIP ∪ M C, therefore M u is maximal w.r.t. set inclusion.
In order to measure the performance of the 0-1 integer program, it was com-
pared with an ASP approach: the ASPARTIX 4 [8] which we will call just ASP1,
and we used the solver Clingo5 . Additionally, the approach based on 0-1 inte-
ger programming will be called BIP, and we used the ad-hoc Xpress6 solver.
The instances that were used during all the experiments were taken from the
ASPARTIX web page 7 .
As a summary, BIP approach had a performance quit similar to the ASP ap-
proach for arbitrary instances8 , but had problems until 60-arguments instances
for 4-grid and 8-grid instances. This result shows that the BIP approach per-
formed well and it has a great opportunity space for improving.
4
The program code is available at ASPARTIX’s web page. It is worth mentioning
that ASPARTIX is the de facto benchmark for argumentations systems
5
http://potassco.sourceforge.net
6
http://www.fico.com/en/Products/DMTools/Pages/FICO-Xpress-Optimization-
Suite.aspx
7
http://www.dbai.tuwien.ac.at/research/project/argumentation/systempage
8
There is a complete explanation of each kind of instance in [7]
10 Mauricio Osorio, Juan Dı́az and Alejandro Santoyo
4 Conclusions
We have presented a novel method for computing SS semantics using binary
integer programming, the performance was good although the ASP approach
outperformed it.
However, it is well known that binary integer programs can be improved, in
order to compute more efficiently its objective function, by using mathematical
programming techniques such as relaxation or adding strong valid inequalities.
It means that it is possible to improve the performance of this novel approach
for computing SS semantics.
This new approach constitutes an alternative for computing AF semantics
using mathematical programming, and even though we used an state of the art
mathematical programming solver, there exists several libraries for java, C++
and other general purpose languages.
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