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206 Artificial Intelligence for the Internet of Everything

The landlord contacted me, the tenant, and the rent was requested. However, I
refused the rent since I demanded repair to be done. I reminded the landlord
about necessary repairs, but the landlord issued the three-day notice confirming
that the rent was overdue. Regretfully, the property still stayed unrepaired.

A defeasible derivation of L from P consists of a finite sequence L 1 , L 2 , …,
L n ¼L of ground literals, such that each literal L i is in the sequence because:
(a)L i is a fact in Π,or
(b)there exists a rule R i in P (strict or defeasible) with head L i and body B 1 ,
B 2 , …, B k and every literal of the body is an element L j of the sequence
appearing before L j (j<i).
Let h be a literal, and P¼(Π, Δ) a DeLP program. We say that <A, h> is an
argument for h if A is a set of defeasible rules of Δ, such that:
1. there exists a defeasible derivation for h from (Π [ A);
2. the set (Π [ A) is noncontradictory; and
3. A is minimal: there is no proper subset A 0 of A such that A 0 satisfies con-
ditions (1) and (2).
Hence an argument <A, h> is a minimal noncontradictory set of defeasible
rules obtained from a defeasible derivation for a given literal h associated
with a program P.
We say that <A 1 , h 1 > attacks <A 2 , h 2 > if there exists a sub-argument
<A, h> of <A 2 , h 2 >,(A A 1 ) such that h and h 1 are inconsistent (i.e., Π [
{h, h 1 } derives complementary literals). We will say that <A 1 , h 1 > defeats
<A 2 , h 2 > if <A 1 , h 1 > attacks <A 2 , h 2 > at a sub-argument <A, h> and
<A 1 , h 1 > is strictly preferred (or not comparable to) <A, h>. In the first
case we will refer to <A 1 , h 1 > as a proper defeater, whereas in the second case
it will be a blocking defeater. Defeaters are arguments that in their turn can be
attacked by other arguments, as is the case with a human dialogue. An argu-
mentation line is a sequence of arguments where each element in a sequence
defeats its predecessor. In the case of DeLP there are a number of acceptability
requirements for argumentation lines in order to avoid fallacies (such as cir-
cular reasoning by repeating the same argument twice).
Target claims can be considered DeLP queries solved in terms of dialec-
tical trees, which subsumes all possible argumentation lines for a given query.
The definition of a dialectical tree provides us with an algorithmic view for
discovering implicit self-attack relations in users’ claims. Let <A 0 , h 0 > be an
argument (target claim) from a program P.A dialectical tree for <A 0 , h 0 > is
defined as follows:
1. The root of the tree is labeled with <A 0 , h 0 >.