\[\gdef{\S}{\mathcal{S}} % Note: overrides existing command \gdef{\O}{\mathcal{O}} % Note: overrides existing command \gdef{\F}{\mathcal{F}} \gdef{\N}{\mathcal{N}} \gdef{\R}{\mathbb{R}} \gdef{\Nat}{\mathbb{N}} \gdef{\sign}{sign} \gdef{\L}{\mathcal{L}} \gdef{\C}{\mathcal{C}} \gdef{\V}{\mathcal{V}} \gdef{\W}{\mathcal{W}} \gdef{\G}{\mathcal{G}} \gdef{\vle}{\preceq} % Source and fact orderings \gdef{\sle}{\sqsubseteq} \gdef{\slt}{\sqsubset} \gdef{\sge}{\ge} \gdef{\seq}{\simeq} \gdef{\fle}{\preceq} \gdef{\flt}{\prec} \gdef{\fgt}{\succ} \gdef{\fge}{\succeq} \gdef{\feq}{\approx} \gdef{\src}{\mathsf{src}} \gdef{\facts}{\mathsf{facts}} \gdef{\obj}{\mathsf{obj}} \gdef{\mut}{\mathsf{mut}} \gdef{\orderings}{\mathcal{L}} \gdef{\num}{\mathcal{T}_{Num}} \gdef{\rec}{\mathsf{rec}} \gdef{\norm}{\mathsf{norm}} \gdef{\ord}#1{\langle{#1}\rangle} % Shortcuts \gdef{\limn}{\lim_{n \to \infty}} \gdef{\voting}{\emph{Voting}} \gdef{\sums}{\emph{Sums}} \gdef{\usums}{\emph{UnboundedSums}} \gdef{\avlog}{\emph{Average$\cdot$Log}} \gdef{\scvoting}{\emph{SC-Voting}} \gdef{\scoh}{\mathrel{\lhd}} \gdef{\fcoh}{\mathrel{\blacktriangleleft}} \gdef{\tuple}#1{{\langle{#1}\rangle}}\]

Fuzzy TD demo¶

This is an idea to apply the Fuzzy Argumentation Frameworks (FAFs) and semantics of [JDCV08] to truth discovery.

Consider a fuzzy set $T$ of the trustworthy sources, i.e. $T(s)$ is a measure of how trustworthy $s$ is.
Create arguments corresponding to facts only, and build a FAF where the degrees of attacks are related to the trust degrees of the sources proposing them.
Using the notions of fuzzy extensions in [JDCV08], define which ones are “good”.
Step back and aim to find the fuzzy trust sets $T$ which have “good” resulting extensions $E$.
My hope is that this method would constrain the possibilities for trust and belief. E.g. if we trust $s$ strongly and $t$ not very much, then $f$ should be strongly believed etc.
i.e. we could find “good” combinations of $T$ and $E$; these combinations would be the output of truth discovery.

Constructing a FAF from a TD network¶

This demo uses the following idea to construct a FAF:

Consider a fixed fuzzy set $T: \S \to [0, 1]$.
Say that $f$ and $g$ attack each other when there is some source $s$ claiming $f$ and some source $t$ claiming $g$, but $s$ and $t$ disagree when it comes to some object.
The strength of $f \nrightarrow g$ is the maximal trust degree of such $s$.
We exclude an argument attacking itself.

Formally, write $C(s)$ for the set of sources conflicting with $s$, i.e.

\[C(s) = \{t \in \S : \exists f, g \in \F : \obj(f) = \obj(g), f \ne g, s \in \src(f), t \in \src(g)\}.\]

Then for $f \ne g$, write

\[K_{f, g} = \{s \in \S : s \in \src(f), C(s) \cap \src(g) \ne \emptyset \}\]

i.e. $K_{f, g}$ is the set of sources who believe in $f$ and conflict with a believer of $g$ (note that the disagreement may be due to another object; $f$ and $g$ need not be for the same object here).

Finally, define

\[\nrightarrow(f, g) = \begin{cases} 0 & \text{ if } f = g \text{ or } K_{f, g} = \emptyset \\ \max_{K_{f, g}}{T} \end{cases}\]

Demo¶

Enter the TD network in the textarea. Each line should be of the form

<source> - <fact> - <object>

Note that sources, facts and objects are drawing in the order of first appearance. There is no validation to check that sources do not claim multiple times for a single object.

The fuzzy set $T$ and the fuzzy “extension” are controlled with the sliders (updated once the textarea loses focus). See the paper [JDCV08] for the meaning of the semantics.

The demo uses the minimum t-norm and its induced S-implicator for the fuzzy logical operations.

Truth discovery network
s - f - o t - g - o s - h - p t - i - p u - i - p v - i - p

Trustworthy sources	Fact set

Fuzzy attack relation

Fuzzy semantics

PhD Notes

Fuzzy TD demo¶

Constructing a FAF from a TD network¶

Demo¶