## 4 Materials and Methods  
### Preliminaries  
Let us illustrate some basic concepts and notations, partially from [1]:  
**Definition 1** (Power set). : _The power set of a set \(A\) is the set of all possible subsets of \(A\), and is denoted with \(\mathcal{P}(A)\). \(\mathcal{P}_{NE}(A)\) denotes the set of all the non-empty subsets of \(A\): \(\mathcal{P}_{NE}(A)=\mathcal{P}(A)\setminus\{\mathcal{O}\}\)._  
**Definition 2** (Multiset). : _A multiset is an extension of the concept of set that keeps track of the cardinality of each element. \(\mathcal{B}(A)\) is the set of all multisets over some set \(A\). Multisets are denoted with square brackets, e.g., \(b=[x,x,y]\), or with the cardinality of the elements as superscript, e.g., \(b=[x^{2},y]\). We denote the empty multiset with \([\,]\). The operator \((\cdot)\) retrieves the cardinality of an element of the multiset, e.g., \(b(x)=2\), \(b(y)=1\), \(b(z)=0\). Over multisets we define \(x\in b\Leftrightarrow b(x)\geq 1\), and \(\text{set}(b)=\{x\in b\}\). The multiset union \(b=b_{1}\uplus b_{2}\) is the multiset \(b\) such that for all \(x\) we have \(b(x)=b_{1}(x)+b_{2}(x)\)._  
**Definition 3** (Sequence and permutation). : _Given a set \(X\), a finite sequence over \(X\) of length \(n\) is a function \(s\in X^{*}:\{1,\ldots,n\}\to X\), and is written as \(s=\langle s_{1},s_{2},\ldots,s_{n}\rangle\). For any sequence \(s\) we define \(|s|=n\), \(s[i]=s_{i}\), \(x\in s\Leftrightarrow x\in\{s_{1},s_{2},\ldots,s_{n}\}\) and \(s\oplus s_{0}=\langle s_{1},s_{2},\ldots,s_{n},s_{0}\rangle\). A permutation of the set \(X\) is a sequence \(x_{\mathcal{S}}\) that contains all elements of \(X\) without duplicates: \(x_{\mathcal{S}}\in X\), \(X\in x_{\mathcal{S}}\), and for all \(1\leq i\leq|x_{\mathcal{S}}|\) and for all \(1\leq j\leq|x_{\mathcal{S}}|\), \(x_{\mathcal{S}}[i]=x_{\mathcal{S}}[j]\to i=j\). We denote with \(\mathcal{S}_{X}\) all such permutations of set \(X\). We overload the notation for sequences: given a sequence \(s=\langle s_{1},s_{2},\ldots,s_{n}\rangle\), we will write \(\mathcal{S}_{s}\) in place of \(\mathcal{S}_{\{s_{1},s_{2},\ldots,s_{n}\}}\)._  
**Definition 4** (Transitive relation and correct evaluation order). : _Let \(X\) be a set of objects and \(R\) be a binary relation \(R\subseteq X\times X\). \(R\) is transitive if and only if for all \(x,x^{\prime},x^{\prime\prime}\in X\) we have that \((x,x^{\prime})\in R\land(x^{\prime},x^{\prime\prime})\in R\)._  
Figure 5: Three different process models for the uncertain event log \(\langle a,b,e,f,g,h\rangle\)[80], \(\langle a,\{b,c\},[e,f],g,i\rangle\)[15], \(\langle a,\{b,c,d\},[e,f],g,\overline{f}\rangle\)[5] obtained through inductive mining over an uncertain directly-follows graph. The different filtering parameters for the UDFG yield models with distinct features. \(R\to(x,x^{\prime\prime})\in R\). A correct evaluation order is a permutation \(s\in\mathcal{S}_{X}\) of the elements of the set \(X\) such that for all \(1\leq i<j\leq|s|\) we have that \((s[i],s[j])\in R\)._  
**Definition 5** (Strict partial order). : _Let \(S\) be a set of objects. Let \(s,s^{\prime}\in S\). A strict partial order \((\prec,S)\) is a binary relation that have the following properties:_  
* _Irreflexivity:_ \(s\prec s\) _is false._
* _Transitivity:_ \(s\prec s^{\prime}\) _and_ \(s^{\prime}\prec s^{\prime\prime}\) _imply_ \(s\prec s^{\prime\prime}\)_._
* _Antisymmetry:_ \(s\prec s^{\prime}\) _implies that_ \(s^{\prime}\prec s\) _is false. It is implied by irreflexivity and transitivity_ _[_10_]__._  
**Definition 6** (Directed graph). : _A directed graph \(G\in\mathcal{U}_{G}\) is a tuple \((V,E)\) where \(V\) is the set of vertices and \(E\subseteq V\times V\) is the set of directed edges. The set \(\mathcal{U}_{G}\) is the graph universe. A path in a directed graph \(G=(V,E)\) is a sequence of vertices \(p\) such that for all \(1<i<|p|-1\) we have that \((p_{i},p_{i+1})\in E\). We denote with \(P_{G}\) the set of all such possible paths over the graph \(G\).