wanyu/IteraTeR_v2 huggingface dataset:
Original:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Edit intent: clarity
Original:  Our method shows that Positive instances of 1-in-3 SAT may be reduced to significantly smaller instances of I.P. in the following sense: any such instance of |V| variables and |C| clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with equality only, of size at most 2/3|V| variables and at most |C| clauses.
Revised:  Our method shows that Positive 1-in-3 SAT may be reduced to significantly smaller instances of I.P. in the following sense: any such instance of |V| variables and |C| clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with equality only, of size at most 2/3|V| variables and at most |C| clauses.
Edit intent: fluency
Original:  We then proceed to define formally the notion of a non-trivial kernel.
Revised:  We proceed to define formally the notion of a non-trivial kernel.
Edit intent: clarity
Original:  For this, we define the problems considered as Constraint Satisfaction Problems.
Revised:  We define the problems considered as Constraint Satisfaction Problems.
Edit intent: coherence
Original:  Considering recent advances in Computational Complexity relating to Sparsification and existence of non-trivial kernels , we conclude by showing that the method presented here, giving a non-trivial kernel for positive 1-in-3 SAT, implies the existence of a non-trivial kernel for 1-in-3 SAT  .
Revised:  Recent advances in Computational Complexity relating to Sparsification and existence of non-trivial kernels , we conclude by showing that the method presented here, giving a non-trivial kernel for positive 1-in-3 SAT, implies the existence of a non-trivial kernel for 1-in-3 SAT .
Edit intent: fluency
Original:  Considering recent advances in Computational Complexity relating to Sparsification and existence of non-trivial kernels , we conclude by showing that the method presented here, giving a non-trivial kernel for positive 1-in-3 SAT, implies the existence of a non-trivial kernel for 1-in-3 SAT  .
Revised:  Considering recent advances in Computational Complexity relating to sparsification and existence of non-trivial kernels , we conclude by showing that the method presented here, giving a non-trivial kernel for positive 1-in-3 SAT, implies the existence of a non-trivial kernel for 1-in-3 SAT .
Edit intent: fluency
Original:  Considering recent advances in Computational Complexity relating to Sparsification and existence of non-trivial kernels , we conclude by showing that the method presented here, giving a non-trivial kernel for positive 1-in-3 SAT, implies the existence of a non-trivial kernel for 1-in-3 SAT  .
Revised:  Considering recent advances in Computational Complexity relating to Sparsification and existence of non-trivial kernels are used to conclude by showing that the method presented here, giving a non-trivial kernel for positive 1-in-3 SAT, implies the existence of a non-trivial kernel for 1-in-3 SAT .
Edit intent: clarity
Original:  Thus we obtain an upper bound for the complexity of its counting version of O( 2K%DIFDELCMD < \R %DIFDELCMD < %%%  2^{(1-K)R}) for number of variables R and clauses-to-variables ratio K. Combining this method with previous results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O( %DIFDELCMD < \R %DIFDELCMD < %%% 2^{(1-K)R}) for number of variables R and clauses-to-variables ratio K. Combining this method with previous results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Edit intent: clarity
Original:  For 1-in-3 SAT instances with non-negated literals we  obtain considerably smaller equivalent instances of 0/1 Integer Programming restricted to equality.
Revised:  For 1-in-3 SAT with non-negated literals we obtain considerably smaller equivalent instances of 0/1 Integer Programming restricted to equality.
Edit intent: clarity
Original:  For 1-in-3 SAT instances with non-negated literals we  obtain considerably smaller equivalent instances of 0/1 Integer Programming restricted to equality.
Revised:  For 1-in-3 SAT instances with non-negated literals we are able to obtain considerably smaller equivalent instances of 0/1 Integer Programming restricted to equality.
Edit intent: clarity
Original:  Thus we obtain an upper bound for the complexity of its counting version of O(2KR 2^{(1-K)R . Our method shows that Positive  1-in-3 SAT may be reduced to significantly smaller instances of I.P.  in the following sense : any such instance of |V| variables and |C| clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with equality only , of size at most 2/3|V| variables and at most |C| clauses.
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O(2KR 2^{(1-K)R . Our method shows that Positive instances of 1-in-3 SAT may be reduced to significantly smaller instances of I.P. in the following sense : any such instance of |V| variables and |C| clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with equality only , of size at most 2/3|V| variables and at most |C| clauses.
Edit intent: fluency
Original:  Thus we obtain an upper bound for the complexity of its counting version of O(2KR 2^{(1-K)R . Our method shows that Positive  1-in-3 SAT may be reduced to significantly smaller instances of I.P.  in the following sense : any such instance of |V| variables and |C| clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with equality only , of size at most 2/3|V| variables and at most |C| clauses.
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O(2KR 2^{(1-K)R . Our method shows that Positive 1-in-3 SAT may be reduced to significantly smaller instances of I.P. in the following sense . Any such instance of |V| variables and |C| clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with equality only , of size at most 2/3|V| variables and at most |C| clauses.
Edit intent: fluency
Original:  Thus we obtain an upper bound for the complexity of its counting version of O(2KR 2^{(1-K)R . Our method shows that Positive  1-in-3 SAT may be reduced to significantly smaller instances of I.P.  in the following sense : any such instance of |V| variables and |C| clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with equality only , of size at most 2/3|V| variables and at most |C| clauses.
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O(2KR 2^{(1-K)R . Our method shows that Positive 1-in-3 SAT may be reduced to significantly smaller instances of I.P. in the following sense : any such instance of |V| variables and |C| clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with Equality , of size at most 2/3|V| variables and at most |C| clauses.
Edit intent: clarity
Original:  Recent advances in Computational Complexity relating to sparsification and existence of non-trivial kernels are used to conclude by showing that the method presented here, giving a non-trivial kernel for positive 1-in-3 SAT, implies the existence of a non-trivial kernel for 1-in-3 SAT.
Revised:  We conclude showing the methods presented here, giving a non-trivial kernel for positive 1-in-3 SAT, implies the existence of a non-trivial kernel for 1-in-3 SAT.
Edit intent: coherence
Original:  Recent advances in Computational Complexity relating to sparsification and existence of non-trivial kernels are used to conclude by showing that the method presented here, giving a non-trivial kernel for positive 1-in-3 SAT, implies the existence of a non-trivial kernel for 1-in-3 SAT.
Revised:  Recent advances in Computational Complexity relating to sparsification and existence of non-trivial kernels are used to conclude by showing that the method presented here, giving a non-trivial kernel for positive 1-in-3 SAT, imply the existence of a non-trivial kernel for 1-in-3 SAT.
Edit intent: fluency
Original:  This implies the structure known as Polynomial Hierarchy collapses  .
Revised:  Our proof shows the structure formerly known as the Polynomial Hierarchy collapses .
Edit intent: clarity
Original: We have entered the era of multidimensional large datasets represented by increasingly complex data structures.
Revised: We have entered the era of large multidimensional datasets represented by increasingly complex data structures.
Edit intent: clarity
Original:  Coaxioms allows us to interpret recursive definitions as fixed points which are not necessarily the least, nor the greatest one, and classical results, which smoothly extend to this generalized framework, ensure the existence of such fixed points .
Revised:  Coaxioms allow us to interpret recursive definitions as fixed points which are not necessarily the least, nor the greatest one, and classical results, which smoothly extend to this generalized framework, ensure the existence of such fixed points .
Edit intent: fluency
Original:  Coaxioms allows us to interpret recursive definitions as fixed points which are not necessarily the least, nor the greatest one, and classical results, which smoothly extend to this generalized framework, ensure the existence of such fixed points .
Revised:  Coaxioms allows us to interpret recursive definitions as fixed points which are not necessarily the least, nor the greatest one, whose existence is guaranteed by a smooth extension of classical results .
Edit intent: clarity
Original:  This notion nicely subsumes standard inference systems and their inductive and coinductive interpretation, thus allowing formal reasoning in cases where the inductive and coinductive interpretation do not provide the intended meaning, or are mixed together.
Revised:  This notion nicely subsumes standard inference systems and their inductive and coinductive interpretation, thus allowing formal reasoning in cases where the inductive and coinductive interpretation do not provide the intended meaning, but are rather mixed together.
Edit intent: coherence
Original:  The relaxed notion of ramp schemes requires the reconstruction of secret from any t+ 1+ g shares, for a  gap parameter g >0. Ramp secret sharing is possible with share lengths depending only on the gap ratio g/N.
Revised:  The more general notion of ramp schemes requires the reconstruction of secret from any t+ 1+ g shares, for a gap parameter g >0. Ramp secret sharing is possible with share lengths depending only on the gap ratio g/N.
Edit intent: clarity
Original:  The relaxed notion of ramp schemes requires the reconstruction of secret from any t+ 1+ g shares, for a  gap parameter g >0. Ramp secret sharing is possible with share lengths depending only on the gap ratio g/N.
Revised:  The relaxed notion of ramp schemes requires the reconstruction of secret from any t+ 1+ g shares, for a gap parameter g . Ramp secret sharing is possible with share lengths depending only on the gap ratio g/N.
Edit intent: clarity
Original:  We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminates the impossibility.
Revised:  We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminate the impossibility.
Edit intent: fluency
Original:  For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any \tau fraction of observed shares, and reconstruction from any \kappa fraction of shares, for any choices of 0 \leq \tau < \kappa \leq 1.
Revised:  For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any \tau fraction of observed shares, and reconstruction from any \rho fraction of shares, for any choices of 0 \leq \tau < \kappa \leq 1.
Edit intent: clarity
Original:  For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any \tau fraction of observed shares, and reconstruction from any \kappa fraction of shares, for any choices of 0 \leq \tau < \kappa \leq 1.
Revised:  For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any \tau fraction of observed shares, and reconstruction from any \kappa fraction of shares, for any choices of 0 \leq \tau < \rho \leq 1.
Edit intent: clarity
Original:  Our construction achieves secret length N( \kappa -\tau-o(1))  which we show to be optimal.
Revised:  Our construction achieves secret length N( \kappa -\tau-o(1)) , which we show to be optimal.
Edit intent: fluency
Original:  Finally , we construct explicit schemes against adaptive adversaries attaining a secret length \Omega(N( \kappa -\tau)) . Our work makes a new connection between secret sharing and coding theory, this time wiretap codes, that was not known before, and raises new interesting open questions .
Revised:  For adaptive adversaries , we construct explicit schemes against adaptive adversaries attaining a secret length \Omega(N( \kappa -\tau)) . Our work makes a new connection between secret sharing and coding theory, this time wiretap codes, that was not known before, and raises new interesting open questions .
Edit intent: clarity
Original:  Finally , we construct explicit schemes against adaptive adversaries attaining a secret length \Omega(N( \kappa -\tau)) . Our work makes a new connection between secret sharing and coding theory, this time wiretap codes, that was not known before, and raises new interesting open questions .
Revised:  Finally , we construct explicit schemes attaining a secret length \Omega(N( \kappa -\tau)) . Our work makes a new connection between secret sharing and coding theory, this time wiretap codes, that was not known before, and raises new interesting open questions .
Edit intent: clarity
Original:  Finally , we construct explicit schemes against adaptive adversaries attaining a secret length \Omega(N( \kappa -\tau)) . Our work makes a new connection between secret sharing and coding theory, this time wiretap codes, that was not known before, and raises new interesting open questions .
Revised:  Finally , we construct explicit schemes against adaptive adversaries attaining a secret length \Omega(N( \kappa -\tau)) .
Edit intent: coherence
Original:  Consistent with recent experimental observations, an enzyme modeled as a dimer of activated colloids shows self-propulsion and enhanced diffusion.
Edit intent: coherence
Original:  We present our deep RL-based DQ optimizer, which currently optimizes select-project-join blocks , and we evaluate DQ on the Join Order Benchmark.
Revised:  We present our RL-based DQ optimizer, which currently optimizes select-project-join blocks , and we evaluate DQ on the Join Order Benchmark.
Edit intent: clarity
Original:  And, we introduce new graph labellings for enriching Topsnut-matchings and show  these labellings can be realized for trees or spanning trees of networks.
Revised:  And, we introduce new graph labellings for enriching Topsnut-matchings and show that these labellings can be realized for trees or spanning trees of networks.
Edit intent: fluency
Original:  In theoretical works we explore Graph Labelling Analysis, and show first that every graph admits our extremal labellings and set-type labellings in graph theory.
Revised:  In theoretical works we explore Graph Labelling Analysis, and show that every graph admits our extremal labellings and set-type labellings in graph theory.
Edit intent: coherence
Original: Adaptive stochastic gradient descent methods, such as AdaGrad, Adam , AdaDelta, Nadam, AMSGrad,etc. , have been demonstrated efficacious in solving non-convex stochastic optimization, such as training deep neural networks.
Revised: Adaptive stochastic gradient descent methods, such as AdaGrad, Adam , etc. , have been demonstrated efficacious in solving non-convex stochastic optimization, such as training deep neural networks.
Edit intent: clarity
Original: Adaptive stochastic gradient descent methods, such as AdaGrad, Adam , AdaDelta, Nadam, AMSGrad,etc. , have been demonstrated efficacious in solving non-convex stochastic optimization, such as training deep neural networks.
Revised: Adaptive stochastic gradient descent methods, such as AdaGrad, Adam , AdaDelta, Nadam, AMSGrad,etc. , have been demonstrated efficacious in solving non-convex stochastic optimization, such as training deep neural networks.
Edit intent: fluency
Original:  However, their convergence rates have not been touched under the non-convex stochastic circumstance except recent breakthrough results on AdaGrad \mbox{ ward2018adagrad0pt and perturbed AdaGrad \mbox{ li2018convergence}0pt }  . In this paper, we propose two new adaptive stochastic gradient methods called AdaHB and AdaNAG which integrate  coordinate-wise AdaGrad with heavy ball momentum and Nesterov accelerated gradient momentum, respectively.
Revised:  However, their convergence rates have not been touched under the non-convex stochastic circumstance except recent breakthrough results on AdaGrad 0pt and perturbed AdaGrad \mbox{ li2018convergence}0pt } . In this paper, we propose two new adaptive stochastic gradient methods called AdaHB and AdaNAG which integrate coordinate-wise AdaGrad with heavy ball momentum and Nesterov accelerated gradient momentum, respectively.
Edit intent: clarity
Original:  The O(\log{T}{T}) non-asymptotic convergence rates of AdaHB and AdaNAG in non-convex stochastic setting are also jointly characterized by leveraging a newly developed unified formulation of these two momentum mechanisms  .
Revised:  The O(\log{T}{T}) non-asymptotic convergence rates of AdaHB and AdaNAG in non-convex stochastic setting are also jointly established by leveraging a newly developed unified formulation of these two momentum mechanisms .
Edit intent: clarity
Original: Adaptive stochastic gradient descent methods , such as AdaGrad, RMSProp, Adam, AMSGrad, etc. , have been demonstrated efficacious in solving non-convex stochastic optimization, such as training deep neural networks.
Revised: Adaptive stochastic gradient descent methods , such as AdaGrad, RMSProp, Adam, AMSGrad, etc. , have been demonstrated efficacious in solvingetc non-convex stochastic optimization, such as training deep neural networks.
Edit intent: clarity
Original:  In this paper , we propose two new adaptive stochastic gradient methods called AdaHB and AdaNAG which integrate a novel weighted coordinate-wise AdaGrad with heavy ball momentum and  Nesterov accelerated gradient momentum , respectively.
Revised:  In this paper , we propose two new adaptive stochastic gradient methods called AdaHB and AdaNAG which integrate a novel weighted coordinate-wise AdaGrad with heavy ball momentum and the Nesterov accelerated gradient momentum , respectively.
Edit intent: fluency
Original: It is shown how to construct an arbitrage-free short rate model under  uncertainty about the drift and the volatility.
Revised: We construct an arbitrage-free short rate model under uncertainty about the drift and the volatility.
Edit intent: coherence
Original: It is shown how to construct an arbitrage-free short rate model under  uncertainty about the drift and the volatility.
Revised: It is shown how to construct an arbitrage-free short rate model under uncertainty about the volatility.
Edit intent: clarity
Original:  The pricing of zero-coupon bonds in such a setting differs substantially from the traditional models, since the prices need to be chosen in a different way in order to exclude arbitrage.
Revised:  The pricing of zero-coupon bonds in such a setting differs substantially from traditional models, since the prices need to be chosen in a different way in order to exclude arbitrage.
Edit intent: fluency
Original:  The uncertainty  is represented by a set of priors , which naturally leads to a  G-Brownian motion.
Revised:  The uncertainty is represented by a set of beliefs , which naturally leads to a G-Brownian motion.
Edit intent: clarity
Original:  In addition, we provide constructive proof for the P vs NP problem under the assumption that the class NP comprises of problems solvable by non-deterministic algorithms .
Revised:  In addition, we argue that the P vs NP problem under the assumption that the class NP comprises of problems solvable by non-deterministic algorithms .
Edit intent: clarity
Original: We present LemmaTag, a featureless recurrent neural network architecture that jointly generates part-of-speech tags and lemmatizes sentences of languages with complex morphology, using bidirectional RNNs with character-level and word-level embeddings.
Revised: We present LemmaTag, a featureless neural network architecture that jointly generates part-of-speech tags and lemmatizes sentences of languages with complex morphology, using bidirectional RNNs with character-level and word-level embeddings.
Edit intent: clarity
Original: We present LemmaTag, a featureless recurrent neural network architecture that jointly generates part-of-speech tags and lemmatizes sentences of languages with complex morphology, using bidirectional RNNs with character-level and word-level embeddings.
Revised: We present LemmaTag, a featureless recurrent neural network architecture that jointly generates part-of-speech tags and lemmas for sentences by using bidirectional RNNs with character-level and word-level embeddings.
Edit intent: clarity
Original: In the problems of Genome-Wide Association Study (GWAS) , the objective is to associate subsequences of individuals' genomes to the observable characteristics called phenotypes .
Revised: The objective of a genome-wide association study (GWAS) , the objective is to associate subsequences of individuals' genomes to the observable characteristics called phenotypes .
Edit intent: coherence
Original: In the problems of Genome-Wide Association Study (GWAS) , the objective is to associate subsequences of individuals' genomes to the observable characteristics called phenotypes .
Revised: In the problems of Genome-Wide Association Study (GWAS) is to associate subsequences of individuals' genomes to the observable characteristics called phenotypes .
Edit intent: clarity
Original:  In particular, as the parameters N, G, and L grow, we observe a threshold effect at \frac{Gh(L/G) }  , where h( . ) is the binary entropy function.
Revised:  In particular, as the parameters N, G, and L grow, a threshold effect at \frac{Gh(L/G) } , where h( . ) is the binary entropy function.
Edit intent: clarity
Original:  This effect allows us to define the capacity of recovering the causal subsequence by denoting the rate of the GWAS problem as \frac{Gh(L/G) . We develop an achievable scheme and a matching converse for this problem, and thus characterize its capacity in two scenarios: the zero-error-rate and the \epsilon-error-rate.
Revised:  This effect allows us to define the capacity of associated subsequence retrieval . We develop an achievable scheme and a matching converse for this problem, and thus characterize its capacity in two scenarios: the zero-error-rate and the \epsilon-error-rate.
Edit intent: clarity