wanyu/IteraTeR_v2 huggingface dataset:
Original:  Finally, we show how to use weighted automata and linear recurrence equations to calculate the density function of a regular language, which assigns to every n the number of words of length n in the language.
Revised:  Finally, we show how to use linear recurrence equations to calculate the density function of a regular language, which assigns to every n the number of words of length n in the language.
Edit intent: coherence
Original:  We accomplish this by showing how to count the number of successful paths of a given weighted automaton.
Revised:  We also show how to count the number of successful paths of a given weighted automaton.
Edit intent: clarity
Original:  We accomplish this by showing how to count the number of successful paths of a given weighted automaton.
Revised:  We accomplish this by showing how to count the number of successful paths of a weighted automaton.
Edit intent: clarity
Original: We consider the following problem: Given a weight x and a graph with n vertices, count the independent sets such that each set of size k contributes x^k.
Revised: We consider weighted counting of independent sets using a rational weight x: Given a graph with n vertices, count the independent sets such that each set of size k contributes x^k.
Edit intent: clarity
Original: We consider the following problem: Given a weight x and a graph with n vertices, count the independent sets such that each set of size k contributes x^k.
Revised: We consider the following problem: Given a weight x and a graph with n vertices, count its independent sets such that each set of size k contributes x^k.
Edit intent: clarity
Original:  We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables  needs time 2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in time 2^{cn}), counting the independent sets of  an n-vertex graph needs time 2^{\Omega(n)} and weighted counting of independent sets needs time 2^{\Omega(n/ \log ^3 n)} for all rational weights x\neq 0.
Revised:  We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time 2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in time 2^{cn}), counting the independent sets of an n-vertex graph needs time 2^{\Omega(n)} and weighted counting of independent sets needs time 2^{\Omega(n/ \log ^3 n)} for all rational weights x\neq 0.
Edit intent: fluency
Original:  We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables  needs time 2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in time 2^{cn}), counting the independent sets of  an n-vertex graph needs time 2^{\Omega(n)} and weighted counting of independent sets needs time 2^{\Omega(n/ \log ^3 n)} for all rational weights x\neq 0.
Revised:  We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables needs time 2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in time 2^{cn}), counting the independent sets of an n-vertex graph needs time 2^{\Omega(n)} and weighted counting of independent sets needs time 2^{\Omega(n/ log ^3 n)} for all rational weights x\neq 0.
Edit intent: fluency
Original:  This graph transformation enables us to adapt a recent technique by Dell, Husfeldt, and Wahl\'en which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.
Revised:  This graph transformation allows us to adapt a recent technique by Dell, Husfeldt, and Wahl\'en which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.
Edit intent: fluency
Original:  This graph transformation enables us to adapt a recent technique by Dell, Husfeldt, and Wahl\'en which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.
Revised:  This graph transformation enables us to adapt a recent technique by Dell, Husfeldt, and Wahlen which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.
Edit intent: fluency
Original:  We present a polynomial-time algorithm for minimizing functions in the new class in the case of binary trees .
Revised:  We present a polynomial-time algorithm for minimizing functions in the new class .
Edit intent: coherence
Original:  These are functions f satisfying f( %DIFDELCMD < \bx%%%  )+f( %DIFDELCMD < \by%%%  )\ge f( %DIFDELCMD < \bx %%% \sqcap%DIFDELCMD < \by%%%  )+f( %DIFDELCMD < \bx %%%  \sqcup %DIFDELCMD < \by%%%  ) where the domain of each variable x_i corresponds to nodes of a rooted binary tree, and operations \sqcap,\sqcup are defined with respect to this tree.
Revised:  These are functions f satisfying f( %DIFDELCMD < \bx%%% )+f( %DIFDELCMD < \by%%% )\ge f( %DIFDELCMD < \bx %%% %DIFDELCMD < \by%%% )+f( %DIFDELCMD < \bx %%% \sqcup %DIFDELCMD < \by%%% ) where the domain of each variable x_i corresponds to nodes of a rooted binary tree, and operations \sqcap,\sqcup are defined with respect to this tree.
Edit intent: fluency
Original:  We show that two spectral functions: the algebraic connectivity and the total effective resistance, when extended to a suitable class of directed weighted graphs, yield important information about convergence and robustness to noise of the corresponding CPs.
Revised:  We show how the spectral properties, such as algebraic connectivity and the total effective resistance, when extended to a suitable class of directed weighted graphs, yield important information about convergence and robustness to noise of the corresponding CPs.
Edit intent: clarity
Original:  We show that two spectral functions: the algebraic connectivity and the total effective resistance, when extended to a suitable class of directed weighted graphs, yield important information about convergence and robustness to noise of the corresponding CPs.
Revised:  We show that two spectral functions: the algebraic connectivity and total effective resistance, when extended to a suitable class of directed weighted graphs, yield important information about convergence and robustness to noise of the corresponding CPs.
Edit intent: fluency
Original:  We show that every randomized algorithm with uniform random sampling needs \Omega({n\over i=1^n a_i}) time to give a (1+\epsilon)-approximation.
Revised:  We show that every randomized algorithm with uniform random sampling needs \Omega({n\over i=1^n a_i}) time to give an (1+\epsilon)-approximation.
Edit intent: fluency
Original:  For a constant b\in (0,1), every problem in \sum(n^{b}) has an O(n^{1-b}(\log n)(\log\log n)+({1\over \epsilon})^{O({1\over\epsilon})}) time (1+\epsilon) approximation for an arbitrary constant \epsilon.
Revised:  For a constant b\in (0,1), every problem in \sum(n^{b}) has an O(n^{1-b}(\log n)(\log\log n)+({1\over \epsilon})^{O({1\over\epsilon})}) time (1+\epsilon) -approximation for an arbitrary constant \epsilon.
Edit intent: fluency
Original:  On the other hand, there is no o(n^{1-b}) time (1+\epsilon) approximation scheme for the bin packing problems in \sum(n^{b}) for some constant \epsilon>0.
Revised:  On the other hand, there is no o(n^{1-b}) time (1+\epsilon) -approximation scheme for the bin packing problems in \sum(n^{b}) for some constant \epsilon>0.
Edit intent: fluency
Original:  We also show a randomized streaming approximation scheme for the bin packing problem such that it needs only constant updating time and constant space, and outputs an (1+\epsilon)-approximation in ( 1%DIFDELCMD < \over %DIFDELCMD < %%% \epsilon )^{O({1\over\epsilon)} time.
Revised:  We also show a randomized streaming approximation scheme for the bin packing problem such that it needs only constant updating time and constant space, and outputs an (1+\epsilon)-approximation in ( %DIFDELCMD < \over %DIFDELCMD < %%% \epsilon )^{O({1\over\epsilon)} time.
Edit intent: fluency
Original:  We also show a randomized streaming approximation scheme for the bin packing problem such that it needs only constant updating time and constant space, and outputs an (1+\epsilon)-approximation in ( 1%DIFDELCMD < \over %DIFDELCMD < %%% \epsilon )^{O({1\over\epsilon)} time.
Revised:  We also show a randomized streaming approximation scheme for the bin packing problem such that it needs only constant updating time and constant space, and outputs an (1+\epsilon)-approximation in ( 1%DIFDELCMD < \over %DIFDELCMD < %%% )^{O({1\over\epsilon)} time.
Edit intent: fluency
Original: It is proven that the connected pathwidth of any graph G is at most 2pw\pw (G)+1, where pw \pw (G) is the pathwidth of G. The method is constructive, i.e. it yields an efficient algorithm that for a given path decomposition of width k computes a connected path decomposition of width at most 2k+1.
Revised: It is proven that the connected pathwidth of any graph G is at most 2pw\pw (G)+1, where \pw (G) is the pathwidth of G. The method is constructive, i.e. it yields an efficient algorithm that for a given path decomposition of width k computes a connected path decomposition of width at most 2k+1.
Edit intent: clarity
Original:  One of the advantages of the above bound for connected pathwidth is an inequality csn \csn (G)\leq 2sn\sn (G)+ 3.\csn\sn Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a  connected one using 2k+3 searchers and starting at  arbitrary homebase.
Revised:  One of the advantages of the above bound for connected pathwidth is an inequality \csn (G)\leq 2sn\sn (G)+ 3.\csn\sn Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a connected one using 2k+3 searchers and starting at arbitrary homebase.
Edit intent: fluency
Original:  One of the advantages of the above bound for connected pathwidth is an inequality csn \csn (G)\leq 2sn\sn (G)+ 3.\csn\sn Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a  connected one using 2k+3 searchers and starting at  arbitrary homebase.
Revised:  One of the advantages of the above bound for connected pathwidth is an inequality csn \csn (G)\leq 2\sn (G)+ 3.\csn\sn Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a connected one using 2k+3 searchers and starting at arbitrary homebase.
Edit intent: fluency
Original:  One of the advantages of the above bound for connected pathwidth is an inequality csn \csn (G)\leq 2sn\sn (G)+ 3.\csn\sn Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a  connected one using 2k+3 searchers and starting at  arbitrary homebase.
Revised:  One of the advantages of the above bound for connected pathwidth is an inequality csn \csn (G)\leq 2sn\sn (G)+ 3.\csn(G) and\sn Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a connected one using 2k+3 searchers and starting at arbitrary homebase.
Edit intent: coherence
Original:  One of the advantages of the above bound for connected pathwidth is an inequality csn \csn (G)\leq 2sn\sn (G)+ 3.\csn\sn Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a  connected one using 2k+3 searchers and starting at  arbitrary homebase.
Revised:  One of the advantages of the above bound for connected pathwidth is an inequality csn \csn (G)\leq 2sn\sn (G)+ 3.\csn\sn Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a connected one using 2k+3 searchers and starting at an arbitrary homebase.
Edit intent: fluency
Original:  The MFA formalism provides a framework for modeling individual protein behaviors and the systems-level dynamics via construction of programmable and executable machines.
Revised:  The MFA formalism provides a framework for modeling individual protein behaviors and systems-level dynamics via construction of programmable and executable machines.
Edit intent: fluency
Original:  Yet no efficient algorithm is know to find a satisfying assignment for densities as low as m/n ~ r(k).ln(k)/k with a non-vanishing probability.
Revised:  Yet no efficient algorithm is know to find a satisfying assignment for densities as low as m/n r(k).ln(k)/k with a non-vanishing probability.
Edit intent: clarity
Original:  In fact, the density m/n ~ r(k).ln(k)/k seems to form a barrier for a broad class of local search algorithms.
Revised:  In fact, the density m/n r(k).ln(k)/k seems to form a barrier for a broad class of local search algorithms.
Edit intent: clarity
Original:  At the beginning of a game, a set of numbered cards going from 1 to n is split into two parts (hands), and at each round of the game the players lay down the top card from their hand .
Revised:  At the beginning of the game, the pack is split into two parts (hands), and at each round of the game the players lay down the top card from their hand .
Edit intent: clarity
Original:  At the beginning of a game, a set of numbered cards going from 1 to n is split into two parts (hands), and at each round of the game the players lay down the top card from their hand .
Revised:  At the beginning of a game, a set of numbered cards going from 1 to n is split into two parts , then on each move the players reveal their top cards .
Edit intent: clarity
Original:  The player laying the highest card then takes both cards and transfers them to the bottom of their hand.
Revised:  The player having the highest card then takes both cards and transfers them to the bottom of their hand.
Edit intent: clarity
Original:  The player laying the highest card then takes both cards and transfers them to the bottom of their hand.
Revised:  The player laying the highest card collects both and returns them to the bottom of their hand.
Edit intent: clarity
Original:  We settle this question by investigating the equivalent maximum matching size of a  random bipartite graph , with a constant left-side vertex degree d=2.
Revised:  We settle this question by investigating the equivalent maximum matching size of a given random bipartite graph , with a constant left-side vertex degree d=2.
Edit intent: clarity
Original:  In particular, we provide an exact expression for any finite bipartite graph size and also deduce asymptotic results as the number of vertices goes to infinity.
Revised:  In particular, we provide an exact expression for any finite system, and also deduce asymptotic results as the number of vertices goes to infinity.
Edit intent: clarity
Original: The well known algorithm of VOLKER STRASSEN for matrix multiplication can only be used for (m2^k \times m2^k) matrices.
Revised: The well known algorithm of Volker Strassen for matrix multiplication can only be used for (m2^k \times m2^k) matrices.
Edit intent: fluency
Original:  For arbitrary (n \times n) matrices one has to add zero rows and columns to the given matrices to use STRASSEN 's algorithm.
Revised:  For arbitrary (n \times n) matrices one has to add zero rows and columns to the given matrices to use Strassen 's algorithm.
Edit intent: fluency
Original:  STRASSEN gave a strategy of how to set m and k for arbitrary n to ensure n\leq m2^k.
Revised:  Strassen gave a strategy of how to set m and k for arbitrary n to ensure n\leq m2^k.
Edit intent: fluency
Original:  The aim of this work is to give a detailed analysis of the number of additional zero rows and columns and the additional work caused by STRASSEN 's bad parameters.
Revised:  The aim of this work is to give a detailed analysis of the number of additional zero rows and columns and the additional work caused by Strassen 's bad parameters.
Edit intent: fluency
Original:  STRASSEN used the parameters m and k to show that his matrix multiplication algorithm needs less than 4.7n^{\log_2 7} flops.
Revised:  Strassen used the parameters m and k to show that his matrix multiplication algorithm needs less than 4.7n^{\log_2 7} flops.
Edit intent: fluency
Original:  The three-permutations-conjecture of Beck is the following.
Revised:  The three-permutations-problem of Beck is the following.
Edit intent: clarity
Original:  Given any 3 permutations on n symbols, one can color the symbols red and blue, such that in any interval of any of those permutations, the number of red and blue symbols differs only by a constant.
Revised:  Given any 3 permutations on n symbols, color the symbols red and blue, such that in any interval of any of those permutations, the number of red and blue symbols differs only by a constant.
Edit intent: clarity
Original:  We establish a surprising connection between bin packing and Beck's conjecture: If the latter holds true, then the additive integrality gap of the 3-partition linear programming relaxation is bounded by a constant.
Revised:  We establish a surprising connection between bin packing and Beck's problem: The additive integrality gap of the 3-partition linear programming relaxation is bounded by a constant.
Edit intent: clarity
Original: Site-specific recombination is an important cellular process that yields a variety of knotted and catenated DNA products on supercoiled circular DNA  . Twist knots are some of the most common conformations of these products  .
Revised: Site-specific recombination on supercoiled circular DNA . Twist knots are some of the most common conformations of these products .
Edit intent: clarity
Original:  Here we present a topological model of site-specific recombination characterising all possible products of site-specific recombination on twist knot substrates, extending previous work of Buck and Flapan.
Revised:  Here we present a topological model of site-specific recombination characterising all possible products of this reaction on twist knot substrates, extending previous work of Buck and Flapan.
Edit intent: clarity
Original:  We illustrate how to use our model to examine previously uncharacterized experimental data.
Revised:  We illustrate how to use our model to examine previously uncharacterised experimental data.
Edit intent: fluency
Original:  We  show how our model can help determine the sequence of products in multiple rounds of processive recombination and distinguish between products of processive and distributive recombination.
Revised:  We also show how our model can help determine the sequence of products in multiple rounds of processive recombination and distinguish between products of processive and distributive recombination.
Edit intent: coherence
Original: Addressing the ongoing controversy over aggressive high-frequency trading practices in financial markets, we report the results of an extensive empirical study estimating the maximum possible profitability of  such practices, and arrive at figures that are surprisingly modest  .
Revised: Addressing the ongoing examination of high-frequency trading practices in financial markets, we report the results of an extensive empirical study estimating the maximum possible profitability of such practices, and arrive at figures that are surprisingly modest .
Edit intent: clarity
Original: How do living cells achieve sufficient abundances of functional protein complexes while minimizing promiscuous non-functional interactions between their proteins ?
Revised: How do living cells achieve sufficient abundances of functional protein complexes while minimizing promiscuous non-functional interactions ?
Edit intent: clarity
Original:  Such functors are uniquely determined by functors from an%DIFDELCMD < {\it %%% enriching category to an%DIFDELCMD < {\it %%% enriched category.
Revised:  Such functors are uniquely determined by functors from an%DIFDELCMD < {\it %%% category to an%DIFDELCMD < {\it %%% enriched category.
Edit intent: fluency
Original:  Such functors are uniquely determined by functors from an%DIFDELCMD < {\it %%% enriching category to an%DIFDELCMD < {\it %%% enriched category.
Revised:  Such functors are uniquely determined by functors from an%DIFDELCMD < {\it %%% enriching %DIFDELCMD < {\it %%% enriched category.
Edit intent: clarity
Original:  Such functors are uniquely determined by functors from an%DIFDELCMD < {\it %%% enriching category to an%DIFDELCMD < {\it %%% enriched category.
Revised:  Such functors are uniquely determined by functors from an%DIFDELCMD < {\it %%% enriching category to an%DIFDELCMD < {\it %%% category.
Edit intent: fluency
Original:  In such a scenario, the proof of correspondence yields a translation from well typed multi-level terms to well typed one-level terms which are parameterized over an instance of a%DIFDELCMD < {\it %DIFDELCMD < %%% generalized arrow (GArrow).
Revised:  In such a scenario, the proof of correspondence yields a translation from well typed multi-level terms to well typed one-level terms %DIFDELCMD < {\it %DIFDELCMD < %%% generalized arrow (GArrow).
Edit intent: clarity
Original:  Object language%DIFDELCMD < {\it %%% providers need only implement the functions of the GArrow type class in point-free style.
Revised:  Object language%DIFDELCMD < {\it %%% need only implement the functions of the GArrow type class in point-free style.
Edit intent: fluency
Original:  Object language%DIFDELCMD < {\it %%% users may write metaprograms over these object languages in a pointful style, sharing the same binding, scoping, abstraction, and application mechanisms across both the object language and metalanguage .
Revised:  Object language%DIFDELCMD < {\it %%% may write metaprograms over these object languages in a pointful style, sharing the same binding, scoping, abstraction, and application mechanisms across both the object language and metalanguage .
Edit intent: clarity
Original: We describe a reduction from  problem of unordered search (with a unique solution) to the problem of inverting a permutation.
Revised: We describe a reduction from the problem of unordered search (with a unique solution) to the problem of inverting a permutation.
Edit intent: fluency
Original:  Since there is a straighforward reduction in the reverse direction, the problems are essentially equivalent.
Revised:  Since there is a straightforward reduction in the reverse direction, the problems are essentially equivalent.
Edit intent: fluency
Original: We describe a reduction from the problem of unordered search (with a unique solution) to the problem of inverting a permutation .
Revised: We describe a reduction from the problem of unordered search (with a unique solution) .
Edit intent: clarity
Original:  The reduction  helps us bypass the Bennett-Bernstein-Brassard-Vazirani hybrid argument (1997) and the Ambainis quantum adversary method  (2002) that were earlier used to derive lower bounds on the quantum query complexity of the problem of inverting permutations.
Revised:  The reduction we present helps us bypass the Bennett-Bernstein-Brassard-Vazirani hybrid argument (1997) and the Ambainis quantum adversary method (2002) that were earlier used to derive lower bounds on the quantum query complexity of the problem of inverting permutations.
Edit intent: clarity
Original:  The reduction  helps us bypass the Bennett-Bernstein-Brassard-Vazirani hybrid argument (1997) and the Ambainis quantum adversary method  (2002) that were earlier used to derive lower bounds on the quantum query complexity of the problem of inverting permutations.
Revised:  The reduction helps us bypass the Bennett-Bernstein-Brassard-Vazirani hybrid argument (1997) and the quantum adversary method (2002) that were earlier used to derive lower bounds on the quantum query complexity of the problem of inverting permutations.
Edit intent: clarity
Original:  It investigates the pricing  of variance swaps under the 3/2 volatility model.
Revised:  It investigates the pricing of variance swaps under a 3/2 volatility model.
Edit intent: fluency
Original:  Carr with Itkin and Sun have discussed the pricing of variance swaps under this type of model.
Revised:  Explicit pricing and hedging formulas of variance swaps under this type of model.
Edit intent: clarity