wanyu/IteraTeR_v2 huggingface dataset:
Original: It is a major unsolved problem as to whether unknot recognition---that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle---has a polynomial time solution.
Revised: It is a major open problem as to whether unknot recognition---that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle---has a polynomial time solution.
Edit intent: clarity
Original: It is a major unsolved problem as to whether unknot recognition---that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle---has a polynomial time solution.
Revised: It is a major unsolved problem as to whether unknot recognition - that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle---has a polynomial time solution.
Edit intent: fluency
Original: It is a major unsolved problem as to whether unknot recognition---that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle---has a polynomial time solution.
Revised: It is a major unsolved problem as to whether unknot recognition---that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle - has a polynomial time solution.
Edit intent: fluency
Original:  Here we present the first unknot recognition algorithm which is  conclusive and, although exponential time in theory, exhibits a clear polynomial time behaviour in practice even for non-trivial knots.
Revised:  Here we present the first unknot recognition algorithm which is always conclusive and, although exponential time in theory, exhibits a clear polynomial time behaviour in practice even for non-trivial knots.
Edit intent: coherence
Original:  Specifically, through exhaustive testing over all 2977 non-trivial prime knots with <= 12 crossings, the algorithm runs in every case by solving only a linear number of linear programs.
Edit intent: coherence
Original: It is a major open problem as to whether unknot recognition - that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle - has a polynomial time solution .
Revised: It is a major unsolved problem as to whether unknot recognition - that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle - has a polynomial time solution .
Edit intent: clarity
Original: It is a major open problem as to whether unknot recognition - that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle - has a polynomial time solution .
Revised: It is a major open problem as to whether unknot recognition - that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle - has a polynomial time algorithm .
Edit intent: clarity
Original:  Here we present the first unknot recognition algorithm which is always conclusive and, although exponential time in theory, exhibits a clear polynomial time behaviour in practice even for non-trivial knots.
Revised:  Here we present the first unknot recognition algorithm which is always conclusive and, although exponential time in theory, exhibits a clear polynomial time behaviour under exhaustive experimentation even for non-trivial knots.
Edit intent: clarity
Original:  A protein's  3D  structure hence leaves an echo of correlations in the evolutionary record.
Revised:  A protein's three-dimensional ( 3D structure hence leaves an echo of correlations in the evolutionary record.
Edit intent: fluency
Original:  A protein's  3D  structure hence leaves an echo of correlations in the evolutionary record.
Revised:  A protein's 3D ) structure hence leaves an echo of correlations in the evolutionary record.
Edit intent: fluency
Original:  Within this task lies a statistical inference problem, rooted in the following: correlation between two sites in a protein sequence can arise from firsthand interaction , but can also be network-propagated via intermediate sites;
Revised:  Within this task lies a statistical inference problem, rooted in the following: correlation between two sites in a protein sequence can arise from firsthand interaction but can also be network-propagated via intermediate sites;
Edit intent: fluency
Original:  In the context of protein sequences, the approach is referred to as direct-coupling analysis.
Revised:  In the context of protein sequences, the approach has been referred to as direct-coupling analysis.
Edit intent: fluency
Original:  1. We improve the previously best O(n^(k^2+3k+3)) time algorithm for Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k) by showing that it is linear-time solvable for constant k. 2. We show that, for hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Revised:  -- Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k) by showing that it is linear-time solvable for constant k. 2. We show that, for hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Edit intent: clarity
Original:  1. We improve the previously best O(n^(k^2+3k+3)) time algorithm for Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k) by showing that it is linear-time solvable for constant k. 2. We show that, for hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Revised:  1. We improve the previously best O(n^(k^2+3k+3)) time algorithm for Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k) is linear-time solvable for constant k. 2. We show that, for hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Edit intent: coherence
Original:  1. We improve the previously best O(n^(k^2+3k+3)) time algorithm for Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k) by showing that it is linear-time solvable for constant k. 2. We show that, for hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Revised:  1. We improve the previously best O(n^(k^2+3k+3)) time algorithm for Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k) by showing that it is linear-time solvable for constant k. -- For hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Edit intent: coherence
Original: We introduce a method of applying Myhill-Nerode methods from formal language theory to hypergraphs and show how this method can be used to obtain the following parameterized complexity results . -- Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k ) is linear-time solvable for constant k. -- For hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Revised: We give an analog of the Myhill-Nerode methods from formal language theory to hypergraphs and show how this method can be used to obtain the following parameterized complexity results . -- Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k ) is linear-time solvable for constant k. -- For hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Edit intent: clarity
Original: We introduce a method of applying Myhill-Nerode methods from formal language theory to hypergraphs and show how this method can be used to obtain the following parameterized complexity results . -- Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k ) is linear-time solvable for constant k. -- For hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Revised: We introduce a method of applying Myhill-Nerode method from formal language theory to hypergraphs and show how this method can be used to obtain the following parameterized complexity results . -- Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k ) is linear-time solvable for constant k. -- For hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Edit intent: fluency
Original: We introduce a method of applying Myhill-Nerode methods from formal language theory to hypergraphs and show how this method can be used to obtain the following parameterized complexity results . -- Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k ) is linear-time solvable for constant k. -- For hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Revised: We introduce a method of applying Myhill-Nerode methods from formal language theory to hypergraphs and show how this method can be used to obtain the following parameterized complexity results . -- Hypergraph Cutwidth (deciding whether a hypergraph has cutwidth at most k ) is linear-time solvable for constant k. -- For hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Edit intent: clarity
Original: We introduce a method of applying Myhill-Nerode methods from formal language theory to hypergraphs and show how this method can be used to obtain the following parameterized complexity results . -- Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k ) is linear-time solvable for constant k. -- For hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Revised: We introduce a method of applying Myhill-Nerode methods from formal language theory to hypergraphs and show how this method can be used to obtain the following parameterized complexity results . -- Hypergraph Cutwidth (deciding whether a hypergraph on n vertices has cutwidth at most k that runs in linear time for constant k. -- For hypergraphs of constant incidence width (treewidth of the incidence graph), Hypertree Width and variants cannot be solved by simple finite tree automata.
Edit intent: clarity
Original: We give an analog of the Myhill-Nerode method from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems . 1. We provide an algorithm for testing whether a hypergraph has cutwidth at most k that runs in linear time for constant k.
Revised: We give an analog of the Myhill-Nerode methods from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems . 1. We provide an algorithm for testing whether a hypergraph has cutwidth at most k that runs in linear time for constant k.
Edit intent: clarity
Original: We give an analog of the Myhill-Nerode method from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems . 1. We provide an algorithm for testing whether a hypergraph has cutwidth at most k that runs in linear time for constant k.
Revised: We give an analog of the Myhill-Nerode method from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems : * We provide an algorithm for testing whether a hypergraph has cutwidth at most k that runs in linear time for constant k.
Edit intent: coherence
Original:  In terms of parameterized complexity theory, the problem is fixed-parameter linear parameterized by k. 2. We show that it is not expressible in monadic second-order logic whether a hypergraph has bounded (fractional, generalized) hypertree width.
Revised:  In terms of parameterized complexity theory, the problem is fixed-parameter linear parameterized by k. * We show that it is not expressible in monadic second-order logic whether a hypergraph has bounded (fractional, generalized) hypertree width.
Edit intent: coherence
Original: We present a Coding Theorem Method involving various notions from  computer science, particularly the concept of algorithmic probability and universal search, that together provide a solution to the problem of approximating the Kolmogorov complexity of a string--alternative to lossless compression algorithms.
Revised: Drawing on various notions from computer science, particularly the concept of algorithmic probability and universal search, that together provide a solution to the problem of approximating the Kolmogorov complexity of a string--alternative to lossless compression algorithms.
Edit intent: coherence
Original: We present a Coding Theorem Method involving various notions from  computer science, particularly the concept of algorithmic probability and universal search, that together provide a solution to the problem of approximating the Kolmogorov complexity of a string--alternative to lossless compression algorithms.
Revised: We present a Coding Theorem Method involving various notions from computer science, particularly the concept of algorithmic probability , to the problem of approximating the Kolmogorov complexity of a string--alternative to lossless compression algorithms.
Edit intent: coherence
Original:  We provide a thorough analysis and exact complexity values for all binary strings of length n<12 and for most strings of length 16>n> 12 \leq16 by running all the ~ 2.5 x10^13 (by using reduction techniques8x22^9) , Turing machineswith 5 states to calculate an output frequency distribution.
Revised:  We provide a thorough analysis and exact complexity values for all binary strings of length n<12 and for most strings of length 12 \leq16 by running all the ~ 2.5 x10^13 (by using reduction techniques8x22^9) , Turing machineswith 5 states to calculate an output frequency distribution.
Edit intent: clarity
Original:  We provide a thorough analysis and exact complexity values for all binary strings of length n<12 and for most strings of length 16>n> 12 \leq16 by running all the ~ 2.5 x10^13 (by using reduction techniques8x22^9) , Turing machineswith 5 states to calculate an output frequency distribution.
Revised:  We provide a thorough analysis and exact complexity values for all binary strings of length n<12 and for most strings of length 16>n> 12 \leq n\leq16 by running all the ~ 2.5 x10^13 (by using reduction techniques8x22^9) , Turing machineswith 5 states to calculate an output frequency distribution.
Edit intent: fluency
Original:  We also address the question of the stability of our approach, that is the sensitivity of K to the continuation of the application of the method for larger coverage and better accuracy, with numerical evidence suggesting robustness.
Revised:  We also address the question of the stability of our approach, that is the sensitivity of the continued application of the method for larger coverage and better accuracy, with numerical evidence suggesting robustness.
Edit intent: clarity
Original:  We also address the question of the stability of our approach, that is the sensitivity of K to the continuation of the application of the method for larger coverage and better accuracy, with numerical evidence suggesting robustness.
Revised:  We also address the question of the stability of our approach, that is the sensitivity of K to the continuation of the application of the method for wider coverage and better accuracy, with numerical evidence suggesting robustness.
Edit intent: clarity
Original:  We also address the question of the stability of our approach, that is the sensitivity of K to the continuation of the application of the method for larger coverage and better accuracy, with numerical evidence suggesting robustness.
Revised:  We also address the question of the stability of our approach, that is the sensitivity of K to the continuation of the application of the method for larger coverage and better accuracy, and provide statistical evidence suggesting robustness.
Edit intent: clarity
Original:  Just as for compression algorithms, this work promises to deliver a full range of applications, and to provide insight into the question of randomness for finite strings.
Revised:  As with compression algorithms, this work promises to deliver a full range of applications, and to provide insight into the question of randomness for finite strings.
Edit intent: coherence
Original:  Just as for compression algorithms, this work promises to deliver a full range of applications, and to provide insight into the question of randomness for finite strings.
Revised:  Just as for compression algorithms, this work promises to deliver a range of applications, and to provide insight into the question of randomness for finite strings.
Edit intent: clarity
Original:  Extracting such a set of vertices from a 3D mesh is important in applications such as digital watermarking, but as well as a part of optimization and triangulation.
Revised:  Extracting such a set of vertices from a 3D mesh is important in applications such as digital watermarking, but also as a component of optimization and triangulation.
Edit intent: clarity
Original: We consider a natural variation of the concept of stabbing a segment by a simple polygon: a segment is stabbed by a simple polygon P if at least one of its two endpoints is contained in P.
Revised: We consider a natural variation of the concept of stabbing a segment by a simple polygon: a segment is stabbed by a simple polygon P if at least one of its two endpoints is contained in P.
Edit intent: fluency
Original:  We tune the metric on our search space so that it suits best to minimizing the least square objective that we have and also respects the symmetry arising out of the factorized search space .
Revised:  We tune the metric on our search space to the particular least square cost function .
Edit intent: clarity
Original:  At another level, our algorithm can be considered as an improved version of  the state-of-the-art Gauss-Seidel algorithm namely, LMaFit .
Revised:  At another level, our algorithm can be considered as an improved version of LMaFit, the state-of-the-art Gauss-Seidel algorithm namely, LMaFit .
Edit intent: clarity
Original:  At another level, our algorithm can be considered as an improved version of  the state-of-the-art Gauss-Seidel algorithm namely, LMaFit .
Revised:  At another level, our algorithm can be considered as an improved version of the state-of-the-art Gauss-Seidel algorithm .
Edit intent: coherence
Original:  We also show that,  it is numerically cheap to perform an exact linesearch given a search direction, which makes our algorithms very competitive with the state-of-the-art on standard low-rank matrix completion instances.
Revised:  We also show that, it is numerically cheap to perform an exact linesearch given a search direction, which makes our algorithms competitive with the state-of-the-art on standard low-rank matrix completion instances.
Edit intent: clarity
Original: Quick Response Codes have made that encoded messages have entered our everyday life, what allows to embed a second layer of information to them : directly available to human receiver for informational or marketing purposes.
Revised: Quick Response Codes have made that encoded messages have entered our everyday life, what suggests to attach them a second layer of information to them : directly available to human receiver for informational or marketing purposes.
Edit intent: clarity
Original: Quick Response Codes have made that encoded messages have entered our everyday life, what allows to embed a second layer of information to them : directly available to human receiver for informational or marketing purposes.
Revised: Quick Response Codes have made that encoded messages have entered our everyday life, what allows to embed a second layer of information : directly available to human receiver for informational or marketing purposes.
Edit intent: clarity
Original:  There will be discussed a general problem of using codes with chosen statistical constrains, for example reproducing given grayscale picture in halftone technique.
Revised:  We will discuss a general problem of using codes with chosen statistical constrains, for example reproducing given grayscale picture in halftone technique.
Edit intent: clarity
Original:  There will be discussed a general problem of using codes with chosen statistical constrains, for example reproducing given grayscale picture in halftone technique.
Revised:  There will be discussed a general problem of using codes with chosen statistical constrains, for example reproducing given grayscale picture using halftone technique.
Edit intent: clarity
Original:  If both sender and receiver know these constrains, the optimal capacity for this purpose can be easily approached by entropy coder.
Revised:  If both sender and receiver know these constrains, the optimal capacity can be easily approached by entropy coder.
Edit intent: clarity
Original:  The problem is that this time only the sender knows the constrain - we will refer to these scenarios as constrained coding.
Revised:  The problem is that this time only the sender knows them - we will refer to these scenarios as constrained coding.
Edit intent: clarity
Original:  Kuznetsov and Tsybakov problem in which only the sender knows which bits are fixed can be seen as a special case, surprisingly approaching the same capacity as  both sides would know the constrains - we will analyze Correction Trees approach to do it in the general case , still combining with error correction capabilities .
Revised:  Kuznetsov and Tsybakov problem in which only the sender knows which bits are fixed can be seen as a special case, surprisingly approaching the same capacity as if both sides would know the constrains - we will analyze Correction Trees approach to do it in the general case , still combining with error correction capabilities .
Edit intent: clarity
Original:  Kuznetsov and Tsybakov problem in which only the sender knows which bits are fixed can be seen as a special case, surprisingly approaching the same capacity as  both sides would know the constrains - we will analyze Correction Trees approach to do it in the general case , still combining with error correction capabilities .
Revised:  Kuznetsov and Tsybakov problem in which only the sender knows which bits are fixed can be seen as a special case, surprisingly approaching the same capacity as both sides would know the constrains . We will analyze Correction Trees approach to do it in the general case , still combining with error correction capabilities .
Edit intent: fluency
Original:  Kuznetsov and Tsybakov problem in which only the sender knows which bits are fixed can be seen as a special case, surprisingly approaching the same capacity as  both sides would know the constrains - we will analyze Correction Trees approach to do it in the general case , still combining with error correction capabilities .
Revised:  Kuznetsov and Tsybakov problem in which only the sender knows which bits are fixed can be seen as a special case, surprisingly approaching the same capacity as both sides would know the constrains - we will analyze Correction Trees to approach analogous capacity in the general case , still combining with error correction capabilities .
Edit intent: clarity