Docs: Fix typos

Author: InfiniteSynthesis

content/introduction/main.tex

@@ -43,7 +43,7 @@ \section{Introduction}
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 This objective, similar to block order fairness, needs no shared notion of time to be meaningful, but it also translates to an eminently practical guarantee in the SMR setting that block order fairness lacks: the system will strive to minimize the number of (inevitable) unfair state updates that happen before the processing of any transaction.
 
-To illustrate the importance of bounded unfairness with an example, consider a simple automated market maker (AMM) where it maintains a constant product $XY = C$ for swapping two tokens $X$ and $Y$ ($Y$ is the native token) and one transaction can only sell a fixed number of tokens.
+To illustrate the importance of bounded unfairness with an example, consider a simple automated market maker (AMM) that maintains a constant product $XY = C$ for swapping two tokens $X$ and $Y$ ($Y$ is the native token) and one transaction can only sell a fixed number of tokens.
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 When the market loses confidence in $X$, every stakeholder would like to sell $X$ to minimize her loss.
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content/introduction/our-results.tex

@@ -22,7 +22,7 @@ \subsection{Our Results}
 
 In order to minimize $B$, we allow it to be a function of the parties' input profiles and the given pair of transactions.
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-The input profiles define a transaction dependency graph $G$ which includes an edge $(\tx, \tx')$  if and only if $\tx \before^\varphi \tx'$. Given this, we observe that the problem of $(\varphi, B)$ fairness relates to the concept of \emph{graph bandwidth} over $G$, cf. \cite{FSTTCS:JKLSS19}.
+The input profiles define a transaction dependency graph $G$ which includes an edge $(\tx, \tx')$  if and only if $\tx \before^\varphi \tx'$. Given this, we observe that the problem of $(\varphi, B)$-fairness relates to the concept of \emph{graph bandwidth} over $G$, cf. \cite{FSTTCS:JKLSS19}.
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 The bandwidth problem asks for a vertex ordering $\sigma: V \rightarrow \mathbb{N}$ that minimizes the \emph{maximum difference} $\sigma(u) - \sigma(v)$ across all edges $(u,v) \in E$.
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content/order-fairness/fairness-versus-liveness.tex

@@ -88,7 +88,7 @@ \subsection{Fairness versus Liveness}
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 We show that the results in \cref{thm:liveness-fairness-async} can be mitigated in this scenario.
 
-The core observation is, if a Condorcet cycle spans for a long period of time, we can perform partition on the set of transactions in this cycle, and these partitions correspond to a good partition on the dependency graph such that we can figure out an upper bound on the \textsc{DirectedBandwidth} problem.
+The core observation is, if a Condorcet cycle spans for a long period of time, we can split the set of transactions in this cycle, and the resulting partitions correspond to a good partition on the dependency graph such that we can figure out an upper bound on the \textsc{DirectedBandwidth} problem.
 
 The partition rule on the dependency graph goes as follows.
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content/order-fairness/order-fairness-directed-bandwidth.tex

@@ -171,7 +171,7 @@ \subsection{Bounded Unfairness from Directed Bandwidth}
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 Next, \textsf{Themis} improves the transaction linearization in a Condorcet cycle to a Hamiltonian-cycle-based order.
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-We point out that this treatment will always produce an order such that $\tx, \tx'$ are at the head and rear respectively but it holds $\tx' \before^\varphi \tx$.
+We point out that this treatment will always produce an order such that $\tx' \before^\varphi \tx$ however $\tx, \tx'$ are at the head and rear respectively.
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 Regarding \textsf{pompe} and \textsf{wendy}, note that in order to be resistant to possible adversarial manipulation, transactions are ordered by their median timestamp.
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content/related-works/timed-order-fairness.tex

@@ -10,7 +10,7 @@ \subsection{Timed Order Fairness.}
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 Now, consider two (local) timestamp vectors of transaction $\tx, \tx'$ respectively.
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-We can decide the their order as $\tx, \tx'$ if these two vectors are ``separated'' by a timestamp $\tau$.
+We can decide their order as $\tx, \tx'$ if these two vectors are ``separated'' by a timestamp $\tau$.
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 More precisely, ``separate'' means that all timestamps assigned to \tx are earlier than $\tau$ and all timestamps assigned to $\tx'$ are later than $\tau$.