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# A Python Script for Abstract Dialectical Frameworks

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## Abstract and Figures

We introduce a Python script for an easy and intuitive calculation of semantics of Abstract Dialectical Frameworks (ADFs) with arbitrary acceptance conditions. In addition, the script enables the evaluation of so-called single-node formulae with the help of Kleene's three-valued logic. The experimental results show that we achieve an enormous computational gain in this case.
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a a
a
D= (S, Φ) S
Φ = {ϕs|sS}
v:S7→ {t,f,u}
v0:S7→ {t,f}
iu
i
it f
u u it u if
v w v iw v(s)iw(s)
sS
v
w
uituif=u u uit=uuif=u
u
D= (S, Φ) ΓD:VD
37→ VD
3
ΓD(v) : S7→ {t,f,u}s7→ ui{w(ϕs)|w[v]D
2}.
σ
σΓD
D= (S, Φ) v∈ VD
3
vcmp(D)v= ΓD(v)
vprf (D)vicmp(D)
vgrd(D)vicmp (D)
h
b w
s
r
D= ({b, h, r, s, w},{φb=¬wb, φh=wb, φr=, φs=hb, φw=¬r∧ ¬b})
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K3
nodes = [["b","#w,b"],["h","w;b"],["r","?"], \
["s","h,b"],["w","#r,#b"]]
chooseinterpretations = ["a","c","p"]
[[’r’, [’ False ’], {}], [’b’, [’ not ’, ’w’, ’ and ’,
’b’, ’’], {’b’: [3], ’w’: [1]}], ...
Nr.1 [’b:False’, ’h:u’, ’r:False’, ’s:False’, ’w:u’]
Nr.2 [’b:False’, ’h:u’, ’r:False’, ’s:False’, ’w:True’]
...
D
vΓD(v) ΓD(v)
ΓD(v)
f,u,t0,0.5,1
v(AB) = min{v(A), v(B)}, v(AB) = max{v(A), v (B)}
v(¬A)=1v(B)
w v
w
u
vΓD(v)
viΓD(v)v= ΓD(v)
v
wcmp(D)\ {v}viw
ΓD
ii
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D
σ
ΓD(v)
ΓD(v)
v∈ VD
3
ΓD(v)v
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VD
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vVD
3
ΓD(v)
vΓD(v)σ
S
n a
S
0.5
> ⊥
S
1 10 100
σ σ
ΓD
1 0.0 0.0 1.37 0.0 0.0 1.17 0.0 0.0 1.13 0.0 0.0 1.33
2 0.0002 0.0002 1.31 0.0002 0.0002 1.27 0.0002 0.0002 1.28 0.0001 0.0 1.88
3 0.0012 0.0008 1.51 0.0012 0.0008 1.52 0.0012 0.0008 1.51 0.0002 0.0001 2.79
4 0.0065 0.0036 1.82 0.0065 0.0036 1.81 0.0066 0.0036 1.82 0.0004 0.0001 4.3
5 0.0347 0.015 2.31 0.0347 0.0151 2.3 0.0348 0.0151 2.3 0.0008 0.0001 8.21
6 0.1765 0.0605 2.92 0.1771 0.0605 2.93 0.1749 0.0601 2.91 0.0016 0.0001 13.6
7 0.8953 0.2378 3.77 0.8961 0.2377 3.77 0.8954 0.2376 3.77 0.0039 0.0001 27.57
8 4.3859 0.8921 4.92 4.3854 0.8932 4.91 4.3863 0.8934 4.91 0.0094 0.0002 54.65
9 21.3739 3.3403 6.4 21.3224 3.326 6.41 21.3296 3.3227 6.42 0.0234 0.0002 112.17
10 103.1276 12.3506 8.35 103.0094 12.3051 8.37 103.3044 12.3584 8.36 0.0535 0.0002 215.73
... In order to encode such expressions, we need to be able to distinguish between different time states related via a certain ordering. We, therefore, introduce so-called timed Abstract Dialectical Frameworks (tADFs) [3] 3 which are powerful enough to model many frequently occurring temporal restrictions. More precisely, a timed Abstract Dialectical Framework (tADF) will be a classical ADF equipped with a countable set T of time states. ...
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