wanyu/IteraTeR_v2 huggingface dataset:
Original:  Still, we can build a representation of S of size O(\delta\log (n/\delta) ), which{\delta}) space that } supports direct access to any  S[i] in time O(\log (n/\delta) ){\delta}) } and finds the occ occurrences of any pattern P[1..m] in time O(m\log n + occ\log^\epsilon n) for any constant \epsilon>0.
Revised:  Still, we can build a representation of any string S in O(\delta\log (n/\delta) ), which{\delta}) space that } supports direct access to any S[i] in time O(\log (n/\delta) ){\delta}) } and finds the occ occurrences of any pattern P[1..m] in time O(m\log n + occ\log^\epsilon n) for any constant \epsilon>0.
Edit intent: clarity
Original:  Further, such representation is worst-case optimal because, in some families, S\le \le  can only be represented in \Omega(\delta\log n{\delta}} ) space.
Revised:  Further, such representation is worst-case optimal because, in some families, S\le \le can only be encoded in \Omega(\delta\log n{\delta}} ) space.
Edit intent: clarity
Original:  We complete our characterization of \delta by showing that \gamma, z and other measures of repetitiveness are always O(\delta\log(n/\delta)), but in some string families, the smallest context-free grammar is of size g= \Omega(\delta \log^2 n / \log\log n).
Revised:  We complete our characterization of \delta by showing that the smallest context-free grammar is of size g= \Omega(\delta \log^2 n / \log\log n).
Edit intent: clarity
Original:  We complete our characterization of \delta by showing that \gamma, z and other measures of repetitiveness are always O(\delta\log(n/\delta)), but in some string families, the smallest context-free grammar is of size g= \Omega(\delta \log^2 n / \log\log n).
Revised:  We complete our characterization of \delta by showing that \gamma, z and other measures of repetitiveness are always O(\delta\log(n/\delta)), but in some string families, the smallest context-free grammar can be of size \Omega(\delta \log^2 n / \log\log n).
Edit intent: clarity
Original:  No such a lower bound is known to hold for \gamma.
Revised:  No such separation is known for \gamma.
Edit intent: clarity
Original:  Since statistical entropy does not capture repetitiveness, ad-hoc measures like the size z of the Lempel--Ziv parse are frequently used to estimate repetitiveness.
Revised:  Since statistical entropy does not capture repetitiveness, ad-hoc measures like the size z of the Lempel--Ziv parse are frequently used to estimate it. The size b
Edit intent: clarity
Original:  The measure \gamma \le  lower bounds all the previous relevant ones, yet length-n strings can be represented and efficiently indexed within space O(\gamma\logn{\gamma}), which also upper bounds most measures.
Revised:  The measure \gamma \le b lower bounds all the previous relevant ones, yet length-n strings can be represented and efficiently indexed within space O(\gamma\logn{\gamma}), which also upper bounds most measures.
Edit intent: fluency
Original:  While \gamma is certainly a better measure of repetitiveness than z , it is  NP-complete to compute , and no o(\gamma\log n) -space representation of strings is known .
Revised:  While \gamma is certainly a better measure of repetitiveness than z , it is also NP-complete to compute , and no o(\gamma\log n) -space representation of strings is known .
Edit intent: clarity
Original:  While \gamma is certainly a better measure of repetitiveness than z , it is  NP-complete to compute , and no o(\gamma\log n) -space representation of strings is known .
Revised:  While \gamma is certainly a better measure of repetitiveness than z , it is NP-complete to compute , and no o(\gamma\log n) space .
Edit intent: clarity
Original:  For every length n and every value \delta%DIFDELCMD < \ge %%% 2, we construct a string such that \gamma = \Omega(\delta \log n{\delta (  \delta\log\frac{n}{\delta}) space that supports direct access to any character S i%DIFDELCMD < ] %%% in time O(\log\frac{n{\delta}) and finds the occ occurrences of any pattern P}%DIFDELCMD < [%%% 1..m in time O(m\log n + occ\log^\epsilon n).
Revised:  %DIFDELCMD < \ge %%% 2, we construct a string such that \gamma = \Omega(\delta \log n{\delta ( \delta\log\frac{n}{\delta}) space that supports direct access to any character S i%DIFDELCMD < ] %%% in time O(\log\frac{n{\delta}) and finds the occ occurrences of any pattern P}%DIFDELCMD < [%%% 1..m in time O(m\log n + occ\log^\epsilon n).
Edit intent: clarity
Original:  For every length n and every value \delta%DIFDELCMD < \ge %%% 2, we construct a string such that \gamma = \Omega(\delta \log n{\delta (  \delta\log\frac{n}{\delta}) space that supports direct access to any character S i%DIFDELCMD < ] %%% in time O(\log\frac{n{\delta}) and finds the occ occurrences of any pattern P}%DIFDELCMD < [%%% 1..m in time O(m\log n + occ\log^\epsilon n).
Revised:  For every length n and every value \delta%DIFDELCMD < \ge %%% Concretely, we show that ( \delta\log\frac{n}{\delta}) space that supports direct access to any character S i%DIFDELCMD < ] %%% in time O(\log\frac{n{\delta}) and finds the occ occurrences of any pattern P}%DIFDELCMD < [%%% 1..m in time O(m\log n + occ\log^\epsilon n).
Edit intent: coherence
Original:  For every length n and every value \delta%DIFDELCMD < \ge %%% 2, we construct a string such that \gamma = \Omega(\delta \log n{\delta (  \delta\log\frac{n}{\delta}) space that supports direct access to any character S i%DIFDELCMD < ] %%% in time O(\log\frac{n{\delta}) and finds the occ occurrences of any pattern P}%DIFDELCMD < [%%% 1..m in time O(m\log n + occ\log^\epsilon n).
Revised:  For every length n and every value \delta%DIFDELCMD < \ge %%% 2, we construct a string such that \gamma = \Omega(\delta \log n{\delta ( \delta\log\frac{n}{\delta}) space that supports direct access to any character S %DIFDELCMD < ] %%% in time O(\log\frac{n{\delta}) and finds the occ occurrences of any pattern P}%DIFDELCMD < [%%% 1..m in time O(m\log n + occ\log^\epsilon n).
Edit intent: fluency
Original:  For every length n and every value \delta%DIFDELCMD < \ge %%% 2, we construct a string such that \gamma = \Omega(\delta \log n{\delta (  \delta\log\frac{n}{\delta}) space that supports direct access to any character S i%DIFDELCMD < ] %%% in time O(\log\frac{n{\delta}) and finds the occ occurrences of any pattern P}%DIFDELCMD < [%%% 1..m in time O(m\log n + occ\log^\epsilon n).
Revised:  For every length n and every value \delta%DIFDELCMD < \ge %%% 2, we construct a string such that \gamma = \Omega(\delta \log n{\delta ( \delta\log\frac{n}{\delta}) space that supports direct access to any character S i%DIFDELCMD < ] %%% {\delta}) and finds the occ occurrences of any pattern P}%DIFDELCMD < [%%% 1..m in time O(m\log n + occ\log^\epsilon n).
Edit intent: fluency
Original:  Further, we prove that no o(\delta\log n)-space representation exists: for every n and every 2%DIFDELCMD < \le %%% \delta%DIFDELCMD < \le %%% n^{1-\epsilon, we exhibit a string family whose elements can only be encoded in \Omega}  (\delta\log\frac{n}{\delta}) space.
Revised:  Further, we prove that no o(\delta\log n)-space representation exists: for every n and %DIFDELCMD < \le %%% \delta%DIFDELCMD < \le %%% n^{1-\epsilon, we exhibit a string family whose elements can only be encoded in \Omega} (\delta\log\frac{n}{\delta}) space.
Edit intent: clarity
Original:  Further, we prove that no o(\delta\log n)-space representation exists: for every n and every 2%DIFDELCMD < \le %%% \delta%DIFDELCMD < \le %%% n^{1-\epsilon, we exhibit a string family whose elements can only be encoded in \Omega}  (\delta\log\frac{n}{\delta}) space.
Revised:  Further, we prove that no o(\delta\log n)-space representation exists: for every n and every 2%DIFDELCMD < \le %%% %DIFDELCMD < \le %%% n^{1-\epsilon, we exhibit a string family whose elements can only be encoded in \Omega} (\delta\log\frac{n}{\delta}) space.
Edit intent: fluency
Original: } . No such separation is known for \gamma .
Revised: } . .
Edit intent: coherence
Original:  This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms in the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems.
Revised:  This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems.
Edit intent: fluency
Original:  Some of the formulations presented are exact, and thus can be used to certify the absence of a minor embedding  .
Revised:  One of the formulations presented are exact, and thus can be used to certify the absence of a minor embedding .
Edit intent: fluency
Original:  Some of the formulations presented are exact, and thus can be used to certify the absence of a minor embedding  .
Revised:  Some of the formulations presented is exact, and thus can be used to certify the absence of a minor embedding .
Edit intent: fluency
Original: The Credit Networkis a model for transactions across a network of agents based on bilateral trust between agents . Credit Networks capture many aspects of traditional currencies as well as new virtual currencies and payment mechanisms.
Revised: The Credit Networkis a model for transactions across a network of bilateral trust relationships . Credit Networks capture many aspects of traditional currencies as well as new virtual currencies and payment mechanisms.
Edit intent: clarity
Original:  In a credit network , if an agent defaults, every other node that trusted it is vulnerable to loss.
Revised:  In the traditional credit network model , if an agent defaults, every other node that trusted it is vulnerable to loss.
Edit intent: clarity
Original:  Alternatively, in a cryptocurrency context, securing a payment channel requires putting capital into escrow to guarantee against default .
Revised:  In a cryptocurrency context, securing a payment channel requires putting capital into escrow to guarantee against default .
Edit intent: coherence
Original:  Surprisingly, aggregate borrowing constraints greatly simplify the analysis and achieve the optimal tradeoff between liquidity and the number of trust-edges .
Revised:  Surprisingly, aggregate borrowing constraints greatly simplify the analysis and achieve the optimal tradeoff between liquidity and amount of escrowed capital .
Edit intent: clarity
Original:  To verify the invertibility of the perturbed operator and explicitly evaluate its inverse norm, we evaluate the eigenvalues of the revealed problem based on Liu's approach with certain a priori error estimates.
Revised:  To verify the invertibility of the operator and evaluating the norm of its inverse based on Liu's approach with certain a priori error estimates.
Edit intent: clarity
Original: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce a notion of flat rational sets: if M is a monoid and N \leq M is its submonoid, then  flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i  s are rational subsets of N and g_i\in M.
Revised: In this work we extend previously known decidability results for 2\times 2 matrices over Q. For that , we introduce a notion of flat rational sets: if M is a monoid and N \leq M is its submonoid, then flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i s are rational subsets of N and g_i\in M.
Edit intent: coherence
Original: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce a notion of flat rational sets: if M is a monoid and N \leq M is its submonoid, then  flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i  s are rational subsets of N and g_i\in M.
Revised: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce the concept of flat rational sets: if M is a monoid and N \leq M is its submonoid, then flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i s are rational subsets of N and g_i\in M.
Edit intent: fluency
Original: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce a notion of flat rational sets: if M is a monoid and N \leq M is its submonoid, then  flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i  s are rational subsets of N and g_i\in M.
Revised: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce a notion of flat rational sets: if M is a monoid and N is a submonoid, then flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i s are rational subsets of N and g_i\in M.
Edit intent: clarity
Original: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce a notion of flat rational sets: if M is a monoid and N \leq M is its submonoid, then  flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i  s are rational subsets of N and g_i\in M.
Revised: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce a notion of flat rational sets: if M is a monoid and N \leq M is its submonoid, then `` flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i s are rational subsets of N and g_i\in M.
Edit intent: fluency
Original: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce a notion of flat rational sets: if M is a monoid and N \leq M is its submonoid, then  flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i  s are rational subsets of N and g_i\in M.
Revised: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce a notion of flat rational sets: if M is a monoid and N \leq M is its submonoid, then flat rational sets of M over N'' are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i s are rational subsets of N and g_i\in M.
Edit intent: clarity
Original: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce a notion of flat rational sets: if M is a monoid and N \leq M is its submonoid, then  flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i  s are rational subsets of N and g_i\in M.
Revised: In this work we extend previously known decidability results for 2\times 2 matrices over Q. Namely , we introduce a notion of flat rational sets: if M is a monoid and N \leq M is its submonoid, then flat rational sets of M relative to N are finite unions of the form L_0g_1L_1 \cdots g_t L_t where all L_i ' s are rational subsets of N and g_i\in M.
Edit intent: fluency
Original:  As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of \mathrm{GL (2,Q) over \mathrm{GL (2,\mathbb{Z}) is decidable .
Revised:  As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL (2,Q) over \mathrm{GL (2,\mathbb{Z}) is decidable .
Edit intent: fluency
Original:  As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of \mathrm{GL (2,Q) over \mathrm{GL (2,\mathbb{Z}) is decidable .
Revised:  As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of \mathrm{GL (2,Q) over GL (2,\mathbb{Z}) is decidable .
Edit intent: fluency
Original:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%%  , of infinite index.
Revised:  We also show a dichotomy for nontrivial group extension of GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%% , of infinite index.
Edit intent: fluency
Original:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%%  , of infinite index.
Revised:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%% , of infinite index.
Edit intent: fluency
Original:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%%  , of infinite index.
Revised:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%% , of infinite index.
Edit intent: fluency
Original:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%%  , of infinite index.
Revised:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%% , of infinite index.
Edit intent: fluency
Original:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%%  , of infinite index.
Revised:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%% , of infinite index.
Edit intent: fluency
Original:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%%  , of infinite index.
Revised:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%% , of infinite index.
Edit intent: fluency
Original:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%%  , of infinite index.
Revised:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group BS (1,q), with q %DIFDELCMD < \geq2%%% , of infinite index.
Edit intent: fluency
Original:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%%  , of infinite index.
Revised:  We also show a dichotomy for nontrivial group extension of \mathrm{GL (2,\mathbb{Z}) in \mathrm{GL (2,\mathbb{Q}): if G is a f.g. group such that \mathrm{GL (2,\mathbb{Z}) < G \leq \mathrm{GL (2,\mathbb{Q}), then either G\cong \mathrm{GL (2,\mathbb{Z})\times \mathbb{Z ^k, for some k\geq 1, or G contains an extension of the Baumslag-Solitar group \mathrm{BS (1,q), with q %DIFDELCMD < \geq2%%% \geq 2 , of infinite index.
Edit intent: fluency
Original:  It turns out that in the first case the membership problem for G is decidable but the equality problem for rational subsets of G is undecidable.
Revised:  In the first case the membership problem for G is decidable but the equality problem for rational subsets of G is undecidable.
Edit intent: coherence
Original:  In the second case,  the membership problem for G is an openproblem as it is open for \mathrm{BS . In the last section we prove new decidability results for flat rational sets that contain singular matrices.
Revised:  In the second case, decidability of the membership problem for G is an openproblem as it is open for \mathrm{BS . In the last section we prove new decidability results for flat rational sets that contain singular matrices.
Edit intent: clarity
Original:  With the widespread use of mobile devices, the vast diversity of mobile apps, and the massive adoption of encryption protocols (such as TLS), large-scale  traffic classification becomes inevitable and more difficult.
Revised:  With the widespread use of mobile devices, the vast diversity of mobile apps, and the massive adoption of encryption protocols (such as TLS), large-scale encrypted traffic classification becomes inevitable and more difficult.
Edit intent: clarity