Conference PaperPDF Available

A Python Script for Abstract Dialectical Frameworks

Authors:

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.
Content may be subject to copyright.
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
vadm(D)viΓD(v)
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})
w
¬r∧ ¬b
b
¬wb
r
h
wb
s
hb
n ϕnn ϕn
, ”; ” ”#”
”!” ”?”
p c g
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]}], ...
Admissible Interpretations
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
i
D
σ
ΓD(v)
ΓD(v)
v∈ VD
3
ΓD(v)v
ΓD(v)σ
v0
D σ
tri
ΓD(v)
ΓD(v)
VD
3
vVD
3
ΓD(v)
vΓD(v)σ
S
n a
S
0.5
> ⊥
S
1 10 100
σ σ
ΓD
adm adm cmp cmp prf prf grd grd
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. ...
Article
Full-text available
Two different perspectives on argumentation have been pursued in computer science research, namely approaches of argument mining in natural language processing on the one hand, and formal argument evaluation on the other hand. So far these research areas are largely independent and unrelated. This article introduces the agenda of our recently started project “FAME – A framework for argument mining and evaluation”. The main project idea is to link the two perspectives on argumentation and their respective research agendas by employing controlled natural language as a convenient form of intermediate knowledge representation. Our goal is to develop a framework which integrates argument mining and formal argument evaluation to study patterns of empirical argumentation usage. If successful, this combination will allow for new types of queries to be answered by argumentation retrieval systems and large-scale content analysis. Moreover, feeding evaluation results as additional knowledge input to argument mining processes could be utilized to further improve their results.
Conference Paper
Full-text available
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.
Article
Full-text available
By using the notions of exact truth (‘true and not false’) and exact falsity (‘false and not true’), one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the (extended) Strong Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus (the SK calculus) that is sound and complete with respect to all of them. Two further sequent calculi (the \({{\bf SK}^\mathcal{P}}\) and \({\bf SK}^{\mathcal{N}}\) calculus) will be considered, which serve the same purpose and which are obtained by applying general methods (due to Baaz et al.) to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the \({\bf SK}^{\mathcal{P}}\) and the \({\bf SK}^{\mathcal{N}}\) calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the \({\bf SK}^{\mathcal{P}}\) and the \({\bf SK}^{\mathcal{N}}\) calculus, we also hint at its philosophical significance.
Article
Dialectical Frameworks (ADFs) generalize Dung's argumentation frameworks allowing various relationships among arguments to be expressed in a systematic way. We further generalize ADFs so as to accommodate arbitrary acceptance degrees for the arguments. This makes ADFs applicable in domains where both the initial status of arguments and their relationship are only insufficiently specified by Boolean functions. We define all standard ADF semantics for the weighted case, including grounded, preferred and stable semantics. We illustrate our approach using acceptance degrees from the unit interval and show how other valuation structures can be integrated. In each case it is sufficient to specify how the generalized acceptance conditions are represented by formulas, and to specify the information ordering underlying the characteristic ADF operator. We also present complexity results for problems related to weighted ADFs.
The DIAMOND System for Computing with Abstract Dialectical Frameworks
  • S Ellmauthaler
  • H Strass
Ellmauthaler S, Strass H. The DIAMOND System for Computing with Abstract Dialectical Frameworks. In: Proceedings of COMMA; 2014. p. 233240.