wanyu/IteraTeR_v2 huggingface dataset:
Original: We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs H_1 and H_2 . By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to (H_1,H_2)-free graphs for all but a finite number of pairs (H_1,H_2)  , but without explicitly giving the number of open cases.
Revised: We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs H_1 and H_2 . By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to (H_1,H_2)-free graphs for all but a finite number of pairs (H_1,H_2) , but without explicitly giving the number of open cases.
Edit intent: clarity
Original: We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs H_1 and H_2 . By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to (H_1,H_2)-free graphs for all but a finite number of pairs (H_1,H_2)  , but without explicitly giving the number of open cases.
Revised: We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs H_1 and H_2 for all but a finite number of pairs (H_1,H_2) , but without explicitly giving the number of open cases.
Edit intent: coherence
Original: We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs H_1 and H_2 . By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to (H_1,H_2)-free graphs for all but a finite number of pairs (H_1,H_2)  , but without explicitly giving the number of open cases.
Revised: We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs H_1 and H_2 . By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to (H_1,H_2)-free graphs for all but six pairs (H_1,H_2) , but without explicitly giving the number of open cases.
Edit intent: clarity
Original: We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs H_1 and H_2 . By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to (H_1,H_2)-free graphs for all but a finite number of pairs (H_1,H_2)  , but without explicitly giving the number of open cases.
Revised: We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs H_1 and H_2 . By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to (H_1,H_2)-free graphs for all but a finite number of pairs (H_1,H_2) , but without specifying the open cases.
Edit intent: clarity
Original:  By exploiting the strong relationship between Graph Isomorphism and clique-width, we also prove that the class of (gem,P_1+2P_2)-free graphs has unbounded clique-width.
Revised:  By exploiting a relationship between Graph Isomorphism and clique-width, we also prove that the class of (gem,P_1+2P_2)-free graphs has unbounded clique-width.
Edit intent: clarity
Original:  As a reward for the user, we propose an incentive system based on Electrocoins crypto-currency to encourage the participants to host IoT sensors  .
Revised:  We propose the use of a virtual accounting system based on Electrocoins crypto-currency to encourage the participants to host IoT sensors .
Edit intent: clarity
Original:  As a reward for the user, we propose an incentive system based on Electrocoins crypto-currency to encourage the participants to host IoT sensors  .
Revised:  As a reward for the user, we propose an incentive system based on our tokens named Electrocoins to encourage participants to host IoT sensors .
Edit intent: clarity
Original:  These systems are however limited to applications of interest for governamental organizations or telecom providers, and only provide aggregated information about spectrum statistics.
Revised:  These systems are however limited to applications of interest for governamental organizationsor telecom providers, and only provide aggregated information about spectrum statistics.
Edit intent: fluency
Original:  The result is that there is a lack of incentives for layman users to participate, which limits its widespread deployment .
Revised:  Theresult is that there is a lack of incentives for layman users to participate, which limits its widespread deployment .
Edit intent: fluency
Original:  The result is that there is a lack of incentives for layman users to participate, which limits its widespread deployment .
Revised:  The result is that there is a lack of interest for layman users to participate, which limits its widespread deployment .
Edit intent: clarity
Original:  The result is that there is a lack of incentives for layman users to participate, which limits its widespread deployment .
Revised:  The result is that there is a lack of incentives for layman users to participate, which limits its widespreaddeployment .
Edit intent: fluency
Original:  We present Electrosense+ which addresses these challenges and creates a general-purpose and open platform  using low-cost, embedded, and software-defined spectrum IoT sensors.
Revised:  We present Electrosense+ which addresses this challenge and creates a general-purpose and open platform using low-cost, embedded, and software-defined spectrum IoT sensors.
Edit intent: fluency
Original:  Electrosense+ allows users to remotely decode specific parts of the radio spectrum.
Revised:  Electrosense+ allows users to remotely decode specific parts ofthe radio spectrum.
Edit intent: fluency
Original:  It builds on the centralized architecture of its predecessor, Electrosense, for controlling and monitoring the spectrum IoT sensors, but implements a real-time and peer-to-peer communication system for scalable spectrum data decoding.
Revised:  It builds on the centralized architecture of its predecessor, Electrosense, forcontrolling and monitoring the spectrum IoT sensors, but implements a real-time and peer-to-peer communication system for scalable spectrum data decoding.
Edit intent: fluency
Original:  It builds on the centralized architecture of its predecessor, Electrosense, for controlling and monitoring the spectrum IoT sensors, but implements a real-time and peer-to-peer communication system for scalable spectrum data decoding.
Revised:  It builds on the centralized architecture of its predecessor, Electrosense, for controlling and monitoring the spectrum IoT sensors, but implements a real-time and peer-to-peercommunication system for scalable spectrum data decoding.
Edit intent: fluency
Original:  We propose  the use of a virtual accounting system based on our tokens named Electrocoins to encourage participants to host IoT sensors and regula te the access to the sensors in a fair manner .
Revised:  We propose the participants to host IoT sensors and regula te the access to the sensors in a fair manner .
Edit intent: clarity
Original:  We propose  the use of a virtual accounting system based on our tokens named Electrocoins to encourage participants to host IoT sensors and regula te the access to the sensors in a fair manner .
Revised:  We propose the use of a virtual accounting system based on our tokens named Electrocoins to encourage participants to host IoT sensors .
Edit intent: coherence
Original:  We present the new Electrosense + system architecture and evaluate its performance at decoding various wireless signals , including FM radio, AM radio, ADS-B, AIS, LTE, and ACARS.
Revised:  We present the newElectrosense + system architecture and evaluate its performance at decoding various wireless signals , including FM radio, AM radio, ADS-B, AIS, LTE, and ACARS.
Edit intent: fluency
Original:  We present the new Electrosense + system architecture and evaluate its performance at decoding various wireless signals , including FM radio, AM radio, ADS-B, AIS, LTE, and ACARS.
Revised:  We present the new Electrosense + system architecture and evaluate its performance at decoding various wireless sig-nals , including FM radio, AM radio, ADS-B, AIS, LTE, and ACARS.
Edit intent: fluency
Original: We introduce \Psiec, a local spectral exterior calculus that provides a discretization of Cartan's exterior calculus of differential forms using wavelet functions .
Revised: We introduce \Psiec, a discretization of Cartan's exterior calculus of differential forms using wavelet functions .
Edit intent: clarity
Original: We introduce \Psiec, a local spectral exterior calculus that provides a discretization of Cartan's exterior calculus of differential forms using wavelet functions .
Revised: We introduce \Psiec, a local spectral exterior calculus that provides a discretization of Cartan's exterior calculus of differential forms using wavelets .
Edit intent: clarity
Original:  Our construction consists of differential form wavelets with flexible directional localization , between fully isotropic and curvelet- and ridgelet-like, that provide tight frames for the spaces of k-forms in R^2 and R^3. By construction, these wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of a k + 1-form is a k-form .
Revised:  Our construction consists of wavelet differential r-forms with flexible directional localization , between fully isotropic and curvelet- and ridgelet-like, that provide tight frames for the spaces of k-forms in R^2 and R^3. By construction, these wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of a k + 1-form is a k-form .
Edit intent: clarity
Original:  Our construction consists of differential form wavelets with flexible directional localization , between fully isotropic and curvelet- and ridgelet-like, that provide tight frames for the spaces of k-forms in R^2 and R^3. By construction, these wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of a k + 1-form is a k-form .
Revised:  Our construction consists of differential form wavelets with flexible directional localization that provide tight frames for the spaces of k-forms in R^2 and R^3. By construction, these wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of a k + 1-form is a k-form .
Edit intent: coherence
Original:  Our construction consists of differential form wavelets with flexible directional localization , between fully isotropic and curvelet- and ridgelet-like, that provide tight frames for the spaces of k-forms in R^2 and R^3. By construction, these wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of a k + 1-form is a k-form .
Revised:  Our construction consists of differential form wavelets with flexible directional localization , between fully isotropic and curvelet- and ridgelet-like, that provide tight frames for the spaces of k-forms in R^2 and R^3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of a k + 1-form is a k-form .
Edit intent: fluency
Original:  Our construction consists of differential form wavelets with flexible directional localization , between fully isotropic and curvelet- and ridgelet-like, that provide tight frames for the spaces of k-forms in R^2 and R^3. By construction, these wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of a k + 1-form is a k-form .
Revised:  Our construction consists of differential form wavelets with flexible directional localization , between fully isotropic and curvelet- and ridgelet-like, that provide tight frames for the spaces of k-forms in R^2 and R^3. By construction, these wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of an (r + 1-form is a k-form .
Edit intent: fluency
Original:  Our construction consists of differential form wavelets with flexible directional localization , between fully isotropic and curvelet- and ridgelet-like, that provide tight frames for the spaces of k-forms in R^2 and R^3. By construction, these wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of a k + 1-form is a k-form .
Revised:  Our construction consists of differential form wavelets with flexible directional localization , between fully isotropic and curvelet- and ridgelet-like, that provide tight frames for the spaces of k-forms in R^2 and R^3. By construction, these wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of a k + 1)-form is an r-form .
Edit intent: clarity
Original:  Our construction is based on the intrinsic geometric properties \Psiec  of the exterior calculus in the Fourier domain.
Revised:  \Psiec of the exterior calculus in the Fourier domain.
Edit intent: clarity
Original:  To reveal these , we extend existing results on the Fourier transform of differential forms to a frequency domain description of the exterior calculus, including, for example, a Parseval theorem for forms and a description of the symbols of all important operators.
Revised:  To be able to use it , we extend existing results on the Fourier transform of differential forms to a frequency domain description of the exterior calculus, including, for example, a Parseval theorem for forms and a description of the symbols of all important operators.
Edit intent: coherence
Original:  Our construction consists of wavelet differential r-forms with flexible directional localization that provide tight frames for the spaces \Omega^r(R^n) of r-forms in R^2 and R^3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the  integral of an (r+1 )-form is an r-form .
Revised:  Our construction consists of differential r-form wavelets with flexible directional localization that provide tight frames for the spaces \Omega^r(R^n) of r-forms in R^2 and R^3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of an (r+1 )-form is an r-form .
Edit intent: clarity
Original:  Our construction consists of wavelet differential r-forms with flexible directional localization that provide tight frames for the spaces \Omega^r(R^n) of r-forms in R^2 and R^3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the  integral of an (r+1 )-form is an r-form .
Revised:  Our construction consists of wavelet differential r-forms with flexible directional localization that provide tight frames for the spaces \Omega^r(R^n) of forms in R^2 and R^3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of an (r+1 )-form is an r-form .
Edit intent: fluency
Original:  Our construction consists of wavelet differential r-forms with flexible directional localization that provide tight frames for the spaces \Omega^r(R^n) of r-forms in R^2 and R^3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the  integral of an (r+1 )-form is an r-form .
Revised:  Our construction consists of wavelet differential r-forms with flexible directional localization that provide tight frames for the spaces \Omega^r(R^n) of r-forms in R^2 and R^3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the k-dimensional integral of an (r+1 )-form is an r-form .
Edit intent: clarity
Original:  Our construction consists of wavelet differential r-forms with flexible directional localization that provide tight frames for the spaces \Omega^r(R^n) of r-forms in R^2 and R^3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the  integral of an (r+1 )-form is an r-form .
Revised:  Our construction consists of wavelet differential r-forms with flexible directional localization that provide tight frames for the spaces \Omega^r(R^n) of r-forms in R^2 and R^3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the integral of an (r+1 )-form .
Edit intent: clarity
Original:  They also enforce Stokes' theorem for differential forms, which is approximated most efficiently using highly anisotropic , curvelet- or ridgelet-like forms. %DIFDELCMD < \Psiec %%% is build on the geometric simplicity of the exterior calculus in the Fourier domain.
Revised:  They also verify Stokes' theorem for differential forms, which is approximated most efficiently using highly anisotropic , curvelet- or ridgelet-like forms. %DIFDELCMD < \Psiec %%% is build on the geometric simplicity of the exterior calculus in the Fourier domain.
Edit intent: clarity
Original:  To be able to use it, we extend existing results on the Fourier transform of differential forms to a frequency domain description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators.
Revised:  We establish this structure by extending existing results on the Fourier transform of differential forms to a frequency domain description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators.
Edit intent: clarity
Original:  To be able to use it, we extend existing results on the Fourier transform of differential forms to a frequency domain description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators.
Revised:  To be able to use it, we extend existing results on the Fourier transform of differential forms to a frequency description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators.
Edit intent: clarity
Original:  One is coalgebra for an functor G, where a bisimulation is defined as a span of G-coalgebra homomorphism.
Revised:  One is coalgebra for a functor G, where a bisimulation is defined as a span of G-coalgebra homomorphism.
Edit intent: fluency
Original:  In turn, those can be split in a span of Boolean spaces and in a span of semigroups.
Revised:  In turn, those can be split in a span of Boolean spaces and a span of semigroups.
Edit intent: fluency
Original:  In this note, we  study these separately.
Revised:  In this note, we first study these separately.
Edit intent: clarity
Original:  This is closely related to a part of the earlier work on existential quantification in first-order logic on words by Gehrke, Petrisan and Reggio.
Revised:  This is closely related to a part of the earlier work on existential quantification in first-order logic on words by Gehrke, Petri\c san and Reggio.
Edit intent: fluency
Original:  We then use these bounds to obtain polynomial time smoothed analysis guarantees for the following three important problems in unsupervised learning: 1. Robust subspace recovery, when the fraction \alpha of inliers in the d-dimensional subspace T \subset R^n is at least \alpha > (d/n)^\ell for any constant integer \ell>0.
Revised:  We then use these bounds to design algorithms with polynomial time smoothed analysis guarantees for the following three important problems in unsupervised learning: 1. Robust subspace recovery, when the fraction \alpha of inliers in the d-dimensional subspace T \subset R^n is at least \alpha > (d/n)^\ell for any constant integer \ell>0.
Edit intent: clarity
Original:  We prove that any function arising from a transducer with s states , that is input symbols which are encoded by \ell bits , has a hazard-free circuit of size 2^{ %DIFDELCMD < \BO%%%  (s+\ell)}\cdot n and depth %DIFDELCMD < \BO%%%  (\ell+ s\cdot \log n);
Revised:  We prove that any function arising from a transducer with s states receiving input symbols encoded by \ell bits , has a hazard-free circuit of size 2^{ %DIFDELCMD < \BO%%% (s+\ell)}\cdot n and depth %DIFDELCMD < \BO%%% (\ell+ s\cdot \log n);
Edit intent: clarity
Original:  We prove that any function arising from a transducer with s states , that is input symbols which are encoded by \ell bits , has a hazard-free circuit of size 2^{ %DIFDELCMD < \BO%%%  (s+\ell)}\cdot n and depth %DIFDELCMD < \BO%%%  (\ell+ s\cdot \log n);
Revised:  We prove that any function arising from a transducer with s states , that is input symbols which are encoded by \ell bits has a hazard-free circuit of size 2^{ %DIFDELCMD < \BO%%% (s+\ell)}\cdot n and depth %DIFDELCMD < \BO%%% (\ell+ s\cdot \log n);
Edit intent: fluency
Original:  We prove that any function arising from a transducer with s states , that is input symbols which are encoded by \ell bits , has a hazard-free circuit of size 2^{ %DIFDELCMD < \BO%%%  (s+\ell)}\cdot n and depth %DIFDELCMD < \BO%%%  (\ell+ s\cdot \log n);
Revised:  We prove that any function arising from a transducer with s states , that is input symbols which are encoded by \ell bits , has a hazard-free circuit of size 2^{ %DIFDELCMD < \BO%%% O (s+\ell)}\cdot n and depth %DIFDELCMD < \BO%%% (\ell+ s\cdot \log n);
Edit intent: fluency
Original:  We prove that any function arising from a transducer with s states , that is input symbols which are encoded by \ell bits , has a hazard-free circuit of size 2^{ %DIFDELCMD < \BO%%%  (s+\ell)}\cdot n and depth %DIFDELCMD < \BO%%%  (\ell+ s\cdot \log n);
Revised:  We prove that any function arising from a transducer with s states , that is input symbols which are encoded by \ell bits , has a hazard-free circuit of size 2^{ %DIFDELCMD < \BO%%% (s+\ell)}\cdot n and depth %DIFDELCMD < \BO%%% O (\ell+ s\cdot \log n);
Edit intent: clarity
Original:  Combined with our main result, we obtain a hazard-free adder circuit of size 2^{ %DIFDELCMD < \BO%%%  (k)}n and depth %DIFDELCMD < \BO%%%  (k\log n) with respect to this code, i.e., a k-recoverable adder circuit that adds two codewords of n bits each.
Revised:  Combined with our main result, we obtain a hazard-free adder circuit of size 2^{ %DIFDELCMD < \BO%%% (k)}n and depth %DIFDELCMD < \BO%%% O (k\log n) with respect to this code, i.e., a k-recoverable adder circuit that adds two codewords of n bits each.
Edit intent: clarity
Original: Recently, an unconditional exponential separation between the hazard-free complexity and (standard) circuit complexity of explicit functions has been shown~\mbox{ ikenmeyer18complexity .
Revised: Recently, an unconditional exponential separation between the hazard-free complexity and (standard) circuit complexity of explicit functions .
Edit intent: clarity
Original:  We utilize our main result to derive efficient circuits for\emph{k-recoverable addition .
Revised:  \emph{k-recoverable addition .
Edit intent: clarity
Original:  Informally speaking, a code is\emph{k-recoverable if it does not increase uncertainty regarding the encoded value, so long as it is guaranteed that it is from \{x,x+1,\ldots,x+k\} for some x\in \mathbb{N_0.
Revised:  Informally speaking, a code is\emph{ if it does not increase uncertainty regarding the encoded value, so long as it is guaranteed that it is from \{x,x+1,\ldots,x+k\} for some x\in \mathbb{N_0.
Edit intent: clarity
Original:  Thus, if the uncertainties of each of the addends span intervals of length O(n/\log n), there is an }\emph{asymptotically optimal adder that attains the best possible output uncertainty .
Revised:  Thus, if the uncertainties of each of the addends span intervals of length O(n/\log n), there is an }\emph{ adder that attains the best possible output uncertainty .
Edit intent: clarity
Original:  Unlike most existing approaches which adopt the GAN framework, our method is based on the recently introduced Implicit Maximum Likelihood Estimation  framework.
Revised:  Unlike most existing approaches which adopt the GAN framework, our method is based on the recently introduced Implicit Maximum Likelihood Estimation (IMLE) framework.
Edit intent: clarity