# How is a bigram defined mathematically?

I know what a bigram is, intuitively (in a string it's two contiguous symbols). However, I found this mathematical definition in a book and there is one part that keeps tripping me up, namely the $$2-grams(w)$$ function.

Definition 3.2 (Bigrams). Given a string $$w$$ over alphabet $$\Sigma$$, its augmented counterpart $$\hat{w} := \ \cdot w \cdot \$$ is obtained by adding the left and right edge markers \$and \$ to $$w$$, where \$and \$ are distinguished symbols not contained in $$\Sigma$$. Furthermore, $$2-grams(w) := \{ab \; |\; \exists u, v \in \Sigma^* \; s.t. \; u \cdot ab \cdot v = \hat{w}\}$$ denotes the set of bigrams over $$\hat{w}$$, i.e. the smallest set that contains all substrings of $$\hat{w}$$ that consist of exactly 2 symbols.

I am reading this as "the function 2-grams is defined as the set of a string $$ab$$ such that there exist a symbol $$u$$ and a symbol $$v$$ in the alphabet Sigma star such that the concatenation of $$u$$, $$ab$$, and $$v$$ is equal to the augmented string $$\hat{w}$$"

My question is, if $$\hat{w}$$ is already the augmented string as we defined it above, why do we still concatenate it with $$u$$ and $$v$$? Am I supposed to assume that $$u$$ and $$v$$ are the edge markers $$\\;\$$? I am not seeing how this function gives us |the smallest set that contains all substrings of $$\hat{w}$$. What am I missing?

PS. Apologies for using the dollar symbol as edge marker symbols, I couldn't find the actual edge markers my textbook uses.

• Looks wrong to me - by the very definition of $\hat w,$ it contains a letter not in $\Sigma,$ so there is no way to have any $u,v\in \Sigma^*$ with $\hat w=uabv.$ Sep 23, 2021 at 4:29
• The intent seems to be that if $xyz$ is your string then $xy$ is a bigram, because $u=\$,v=z\$,$ but this is problematic because it is explicitly said that $\$ $is not in$\Sigma,$so these$u,v$are not in$\Sigma^*.$The definition needs a new alphabet,$\Sigma_1=\Sigma\cup\{\$\}$ and the function should use $\Sigma_1^*.$ Or the definition could just use $w.$ The definition should also say: $a,b\in \Sigma,$if you don’t want to allow $\$x$and$z\ as bigrams Sep 23, 2021 at 4:35
• Where are you seeing this definition? Sep 23, 2021 at 4:40
• those were exactly my thoughts on $u, v \in \Sigma$ which is why I think I was so confused. What you are saying sounds correct but I don't know, I am not a mathematician. This definition is in a mathematical linguistics book draft. I am now thinking it might be a mistake on the part of the author? Sep 23, 2021 at 4:55
• Be careful not to confuse $\Sigma$ for $\Sigma^*.$ $\Sigma$ is the set of single letters ($1-$grams?) and $\Sigma^*$ is all words that can be constructed in those letters, including the empty string. $u,v$ are not, in general, single letters. Sep 23, 2021 at 5:19

I think the definition says that we can obtain all $$|w|-1$$ bigrams in the string $$w$$ by partitioning the augmented string $$\hat{w}$$ into 3 non-empty parts: $$u$$, $$ab$$, $$v$$, assuming that $$|w|\geq 2$$.
Here, $$u, v \in \Sigma^*=\Sigma\cup{\{\text{.'}\}}$$, $$a,b \in \Sigma$$ and $$ab$$ is a length-2 substring of $$w$$.
Start from $$u=.\text{'}$$ and finish at $$v=`.\text{'}$$, to obtain all the bigrams in $$w$$ (slide $$ab$$ over $$w$$), as shown in the below figure.