Conference PaperPDF Available

Timed Abstract Dialectical Frameworks: A Simple Translation-Based Approach

Abstract

dialectical frameworks (ADFs) are one of the most powerful generalization of classical Dung-style AFs. In this paper we show how to use ADFs if we want to deal with acceptance conditions changing over time. We therefore introduce so-called timed abstract di-alectical frameworks (tADFs) which are essentially ADFs equipped with time states. Beside a precise formal denition of tADFs and an illustrating example we prove that Kleene's three-valued logic K 3 facilitate the evaluation of acceptance functions if we do not allow multiple occurrences of atoms.
a a
a
K3
s t
φs4=a1a2a3s
4a
1 3
s a
φs4
D= (S, Φ) S
Φ = {ϕs|sS}
D= (S, Φ)
v D v :S7→ {t,f}v:S7→ {t,f,u}
VD
2VD
3D
u
t f
D= (S, Φ) it f u
<iu<it u <if
D
v1iv2sS:v1(s)∈ {t,f}=v1(s) = v2(s)
uituit=t f uif=f u
∈ VD
3(s) = usS v ∈ VD
3,iv
iVD
3
v∈ VD
3[v]D
2={w∈ VD
2|viw}
[v]D
2v
D= (S, Φ) ΓD:VD
37→ VD
3
ΓD(v) : S7→ {t,f,u}s7→ ui{w(ϕs)|w[v]D
2}.
t f
u
adm cmp prf grd
D= (S, Φ) v∈ VD
3
vadm(D)viΓD(v)
vcmp(D)v= ΓD(v)
vprf (D)vicmp(D)
vgrd(D)vicmp (D)
D
prf (D)cmp(D)adm (D)grd(D)cmp(D)adm(D)
D= ({a, b, c},{φa=¬b, φb=¬a, φc=a})
a
¬b
b
¬a
c
a
D:
{ } =grd(D) = ΓD( )
iiVD
3
I1I2I1(a) = I1(b) = I1(c) = tI2(a) =
I2(b) = I2(c) = fI1(φa)uiI2(φa) = uI1(φa) = I1(¬b) = f
I2(φa) = I2(¬b) = tI1(φb)uiI2(φb) = u
I1(φc)uiI2(φc) = u= ΓD( )
adm(D) = {v1, v2, v3, v4,}cmp(D) = {v1, v3,}prf (D) = {v1, v3}
v1={a:t, b :f, c :t}v2={a:t, b :f, c :u}v3={a:f, b :t, c :f}
v4={a:f, b :t, c :u}
T
T
T
n
s t st
φsts t
D= (S, T, Φ) S T
Φ = {ϕst|sS, t T}
sS t T
n m
n·m
D= (S, T, Φ)
a, c S[i, j]T
ϕct=a[i,j]
1:= W
ikj
ak
c t a
[i, j]a c
i j
ϕct=a[i,j]
n:= W
{k1,...,kn}⊆[i,j]
|{k1,...,kn}|=n
ak1. . . akn
c t a
n[i, j]a c n
[i, j]
ϕct=a[i,j]
n:= ¬(a[i,j]
n+1) = V
{k1,...,kn+1}⊆[i,j ]
|{k1,...,kn+1}|=n+1
¬ak1. . . ∨ ¬akn+1
c t a
n[i, j]n a
[i, j]c
ϕct=a[i,j]
1:= ¬(a[i,j]
2) = V
{k1,k2}⊆[i,j]
|{k1,k2}|=2
¬ak1∨ ¬ak2
a[i,j]
1c
t a [i, j]
ϕct=a[i,j]
=n:= ϕct=a[i,j]
na[i,j]
n
c t a
n[i, j]
v
s
p l
t
D= (S, T, Φ)
S={v, s, t, p, l}T={1,2,3,4,5}nT
nth
ϕl4
ϕl4
ϕt4
ϕp4
ϕsi=sii∈ {1,2,3,4,5}
v1
s1
t1
p1
l1
v2
s2
t2
p2
l2
v3
s3
t3
p3
l3
v4
s4
t4
p4
l4
v5
s5
t5
p5
l5
D
atϕat
v1, v2, v3, v5
v4s[1,3]
1(p4t4)
t1, t2, t3, t5>
t4>∧¬p4≡ ¬p4
p1, p2
p3l3
p4l[3,4]
1∧ ¬t4
p5l[3,5]
1
l1, l2
l3¬l[4,5]
1
l4¬l3∧ ¬l5
l5¬l[3,4]
1
sisi
D
D
D
v1
prf (D)l3l4l5p3p4p5s1s2s3t4v4
v1f f t f f t f f f t f
v2f f t f f t f f t t t
v3f f t f f t f t f t t
v4f f t f f t f t t t t
v5f f t f f t t f f t t
v6f f t f f t t f t t t
v7f f t f f t t t f t t
v8f f t f f t t t t t t
D
v v0
s
ϕsv
s
v0L3ϕs
L3
L3
ϕ v
vL3(ϕ) = ui{w(ϕ)|w[v]2}.
A={a, b, c, ...}
σ(ϕ)ϕ
ϕ=a∨ ¬a σ(ϕ) = {a}
F
A⊆F
ϕ∈ F ¬ϕ∈ F
ϕ, ψ ∈ F σ(ϕ)σ(ψ) = ϕψ, ϕ ψ∈ F
a b a b a b¬a
t t t t f
t f t f f
t u t u f
f t t f t
f f f f t
f u u f t
u t t u u
u f u f u
u u u u u
K3
ϕ∈ F
F K3
ϕ∈ F v
vK3(ϕ) = ui{w(ϕ)|w[v]2}.
... To the best of our knowledge, this work is the first systematic such study, but there are some works which contain some similar results or questions. In (Baumann and Heinrich 2020), it is shown that there is no truth-functional three-valued logic L s.t. for every v ∈ V(At) and every φ ∈ L(At), v L (φ) = i [v] 2 (φ). Lemma 1 is a generalization of this result. ...
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