# Using Conceptual Recurrence and Consistency Metrics for Topic Segmentation in Debate
## 6 CSseg Algorithm
### Debate Consistency Metrics
#### 6.3.3 Chain of Arguments and Counterarguments
 \tag{5}\]  
\(chain(i,j)\) is a function that searches for conceptual similarities when the \(i\)-th utterance contains an argument and the \(j\)-th utterance contains a counterargument. A weight \(chainWeight\) is given to conceptual similarities with conditions corresponding to argument and counterargument; this weight is user-configurable. \(\wedge\) stands for \"logical and\"; it returns \"true\" only if both the left and right conditions are true, and \"false\" for all other conditions. The formulas for the conditional functions \(side(i,j)\), \(negative(j)\), \(long(i)\), \(similarity(i,j)\), \(close(i,j)\), \(right(i,j)\) that search for assertions and refutations are as follows:  
\[side(i,j)= \begin{cases}True,\;\;\text{if speaker of $u$ and speaker of $u$j$ are not on the same side}\\ False,\;\;\text{otherwise}\end{cases}  \tag{6}\] \[negative(j)= \begin{cases}True,\;\;\text{if $\sum_{k=1}^{l} sentenceSentiment(k)\leqq-\alpha$}\\ False,\;\;\text{otherwise}\end{cases} \] (7) \[\text{where, }sentenceSentiment(k)= \begin{cases} sentimentScore,\;\text{if $sentimentScore\leqq-\beta$}\\ 0,\;\text{otherwise}\end{cases} \] (8) \[long(i)= \begin{cases}True,\;\;\text{if (number of characters of i-th utterance)}\geqq\gamma\\ False,\;\;\text{otherwise}\end{cases} \] (9) \[similarity(i,j)= \begin{cases}True,\;\;\text{if $S(i,j)\geqq\delta$}\\ False,\;\text{otherwise}\end{cases} \] (10) \[close(i,j)= \begin{cases}True,\;\;\text{if $distance(i,j)\leqq e$}\\ False,\;\text{otherwise}\end{cases} \] (11) \[right(i,j)= \begin{cases}True,\;\;\text{if $\sum_{k=i+1}^{j-1}isChain(k,j)=0$}\\ False,\;\text{otherwise}\end{cases} \] (12) \[\text{where, }isChain(i,j)= \begin{cases}1,\;\text{if $side(i,j)\wedge negative(j)\wedge\; long(i)$}\\ \;\;\;\;\wedge similarity(i,j)\wedge close(i,j)\\ \;\;\;\;\wedge right(i,j)\\ 0,\;\;\text{otherwise}\end{cases}  \tag{13}\]  
\(side(i,j)\) is a function that returns \"true\" if the speaker of the \(i\)-th utterance \(u\)i and the \(j\)-th utterance \(u\)j are on the other side, and \"false\" otherwise. This is specified manually rather than automatically calculated by the machine. \(negative(j)\) is a function to judge whether the \(j\)-th utterance has negative sentiment. In \(negative(j)\), \(l\) is the number of sentences in an utterance. \(sentenceSentiment(k)\) is a function that returns a negative sentiment value or probability of \(k\)-th sentence less than \(\beta\). Its value or probability varies depending on the type of sentiment analysis model. If the sum of all negative sentiment scores of the sentences in the utterance is less than \(\alpha\), it is determined that the utterance has a negative sentiment. For the prototype, \(\alpha=0.5\) and \(\beta=0.25\) are specified based on Google's natural language sentiment analysis model. \(long(i)\) is a function that determines whether the length of the \(i\)-th utterance is long enough to be refuted. The unit of utterance's length in \(long(i)\) is the number of characters. In the prototype, \(\gamma=100\). \(similarity(i,j)\) is a function that determines whether the conceptual similarity between the argument's utterance and the counterargument's utterance is higher than a certain level of \(\delta\). In the prototype, \(\delta=0.3\). \(close(i,j)\) is a function that determines whether an utterance containing a counterargument is at a certain distance from an utterance containing an argument. In \(close(i,j)\), \(distance(i,j)\) is a function that calculates the distance between the \(i\)-th and \(j\)-th utterances. The distance between adjacent utterances returns \(0\) and increases by \(1\) each time another utterance is entered between the \(i\) and \(j\)-th utterances, and then returns. \(\epsilon\) is a value that determines whether the distance is small, and it is set to \(10\) for the prototype. \(right(i,j)\) is a function that searches for the conceptual similarity of argument and counterargument among the utterances \(u_{i+1}\)\(\sim\)\(u_{j-1}\).