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 Research in high-level synthesis (HLS) often concentrates on performance: trying to achieve the lowest area with the shortest run-time.  What is often overlooked is ensuring that the HLS tool is indeed correct, which means that it outputs hardware designs that are equivalent to the behavioural input.
 
-\JWreplace{It is often assumed when working with HLS tools,}{When working with HLS tools, it is often assumed} that they transform the behavioural input into a semantically equivalent design.  This assumption is made in \citet{lahti19_are_we_there_yet} for example.  However, this is not the case, and as with all complex pieces of software there are bugs in HLS tools as well.  For example, Vivado HLS was found to incorrectly apply pipelining optimisations\footnote{\url{https://bit.ly/vivado-hls-pipeline-bug}} or generate wrong designs with valid C code as input \JW{citation needed for this claim too?}.  These types of bugs are difficult to identify, and exist because firstly it is not quite clear what input these tools support, and secondly whether the output design actually behaves the same as the input.
+When working with HLS tools, it is often assumed that they transform the behavioural input into a semantically equivalent design~\cite{lahti19_are_we_there_yet} for example.  However, this is not the case, and as with all complex pieces of software there are bugs in HLS tools as well.  For example, Vivado HLS was found to incorrectly apply pipelining optimisations\footnote{\url{https://bit.ly/vivado-hls-pipeline-bug}} or generate incorrect designs silently when straying outside of the supported C\footnote{\url{https://bit.ly/vivado-hls-pointer-bug}}.  These types of bugs are difficult to identify, and exist because firstly it is not quite clear what input these tools support, and secondly whether the output design actually behaves the same as the input.
 
-\paragraph{Our position} We believe that a formally verified high-level synthesis tool could be the solution to these problems.  It not only guarantees that the output is correct, but also brings a formal specification of the input and output language semantics.  These are the only parts of the compiler that need to be trusted, and if these are well-specified, then the behaviour of the resulting design can be fully trusted.  In addition to that, if the semantics of the input semantics \JW{repeated `semantics'} are taken from a tool that is widely trusted already, then there should not be any strange behaviour; the resultant design will either behave exactly like the input specification, or the translation will fail early at compile time.  To this end, we have built a formally verified HLS tool called \vericert{}~\cite{herklotz21_formal_verif_high_level_synth}.
+\paragraph{Our position} We believe that a formally verified high-level synthesis tool could be the solution to these problems.  It not only guarantees that the output is correct, but also brings a formal specification of the input and output language semantics.  These are the only parts of the compiler that need to be trusted, and if these are well-specified, then the behaviour of the resulting design can be fully trusted.  In addition to that, if the semantics of the input language are taken from a tool that is widely trusted already, then there should not be any strange behaviour; the resultant design will either behave exactly like the input specification, or the translation will fail early at compile time.  To this end, we have built a formally verified HLS tool called \vericert{}~\cite{herklotz21_formal_verif_high_level_synth}.
 
 In what follows, we will argue our position by presenting several possible \emph{objections} to our position, and then responding to each in turn.
 
@@ -177,43 +177,45 @@ Formally verifying HLS of C is the wrong approach. C should not be used to desig
 
 \textbf{\textit{Response:}} Verifying HLS is also important.  Firstly, C is often the starting point for hardware designs, as initial models are written in those languages to get a quick prototype~\cite{gajski10_what_hls}, so it is only natural to continue using C when designing the hardware.  Not only is HLS from C becoming more popular, but much of that convenience also comes from the easy behavioural testing that HLS allows to ensure correct functionality of the design~\cite{lahti19_are_we_there_yet}.  This assumes that HLS tools are correct.
 
-\objection{Current HLS tools are already reliable enough}
+\objection{HLS tools are already commonly used in industry, so they are clearly already reliable enough}
 
-One might argue that as current HLS tool are already commonly used in industry, that they should be reliable enough.
-
-\textbf{\textit{Response:}} \citet{du21_fuzzin_high_level_synth_tools} showed that on average 2.5\% of randomly generated C programs, tailored to the specific HLS tool, end up with incorrect designs.  These bugs were reported and confirmed to be new bugs in the tools, demonstrating that existing internal tests did not catch them. And existing verification techniques for checking the output of HLS tools may not be enough to catch these bugs reliably.  Checking the final design against the original model using a test bench may miss many edge cases that produce bugs.
+\paragraph{Response:} They are widely used, but they are also widely acknowledged to be quite flaky. \citet{du21_fuzzin_high_level_synth_tools} showed that on average 2.5\% of randomly generated C programs, tailored to the specific HLS tool, end up with incorrect designs.  These bugs were reported and confirmed to be new bugs in the tools, demonstrating that existing internal tests did not catch them. And existing verification techniques for checking the output of HLS tools may not be enough to catch these bugs reliably.  Checking the final design against the original model using a test bench may miss many edge cases that produce bugs.
 
 \objection{Existing approaches for testing or formally verifying hardware designs are sufficient for ensuring reliability}
 
-Besides using test benches to test designs produced by HLS, which suffers from many missing edge cases, there has been research on performing equivalence checks between the output design and the behavioural input, focusing on creating translation validators~\cite{pnueli98_trans} to prove the equivalence between the design and input code, while supporting various optimisations such as scheduling~\cite{kim04_autom_fsmd,karfa06_formal_verif_method_sched_high_synth,chouksey20_verif_sched_condit_behav_high_level_synth} or code motion~\cite{banerjee14_verif_code_motion_techn_using_value_propag,chouksey19_trans_valid_code_motion_trans_invol_loops}.  However, these aren't perfect solutions either, as there is no guarantee that these proofs really compose with each other.  This means that equivalence checkers are often designed to check the translation from start to finish, which is computationally expensive, as well as possibly being highly incomplete due to a combination of optimisations producing an output that cannot be matched to the input, even though it is correct.
+Besides the use of test benches to test designs produced by HLS, which suffers from many missing edge cases, there has been research on performing equivalence checks between the output design and the behavioural input, focusing on creating translation validators~\cite{pnueli98_trans} to prove the equivalence between the design and input code, while supporting various optimisations such as scheduling~\cite{kim04_autom_fsmd,karfa06_formal_verif_method_sched_high_synth,chouksey20_verif_sched_condit_behav_high_level_synth} or code motion~\cite{banerjee14_verif_code_motion_techn_using_value_propag,chouksey19_trans_valid_code_motion_trans_invol_loops}.  However, these aren't perfect solutions either, as there is no guarantee that these proofs really compose with each other.  In response, equivalence checkers are often designed to check the translation from start to finish, but this is computationally expensive, as well as possibly being highly incomplete.
+
+%\JW{could end sentence here, as the rest is a little convoluted and perhaps unnecessary} due to a combination of optimisations producing \JWreplace{an}{a correct} output that cannot be matched to the input, \JWcouldcut{even though it is correct.}
 
-\textbf{\textit{Response:}} The radical solution to this problem is to formally verify the whole tool.  This has been shown to be successful in \compcert{}~\cite{leroy09_formal_verif_realis_compil}, for example, which is a formally verified C compiler written in Coq~\cite{coquand86}.  The reliability of a formally verified compiler was demonstrated by CSmith~\cite{yang11_findin_under_bugs_c_compil}, a random, valid C generator, which found more than 300 bugs in GCC and Clang, whereas no bugs were found in the verified parts of \compcert{}.
+\textbf{\textit{Response:}} The radical solution to this problem is to formally verify the whole tool.  This has been shown to be successful in \compcert{}~\cite{leroy09_formal_verif_realis_compil}, for example, which is a formally verified C compiler written in Coq~\cite{coquand86}.  The reliability of this formally verified compiler was demonstrated by CSmith~\cite{yang11_findin_under_bugs_c_compil}, a random, valid C generator, which found more than 300 bugs in GCC and Clang, but no bugs in the verified parts of \compcert{}.
 
 \objection{HLS applications don't require the levels of reliability that a formally verified compiler affords}
 
-One might argue that developing a formally verified tool in a theorem prover and proving correctness theorems about it might be too tedious and take too long, and that HLS tools specifically do not need that kind of reliability.  With our experience developing a verified HLS tool called \vericert{}~\cite{herklotz21_formal_verif_high_level_synth} based on \compcert{}, we found that it normally takes $5\times$ or $10\times$ longer to prove a translation correct compared to writing the algorithm.
+One might argue that developing a formally verified tool in a theorem prover and proving correctness theorems about it might be too tedious and take too long, and that HLS tools specifically do not need that kind of reliability.  Indeed, with our experience developing a verified HLS tool called \vericert{}~\cite{herklotz21_formal_verif_high_level_synth} based on \compcert{}, we found that it normally takes $5\times$ or $10\times$ longer to prove a translation correct compared to writing the algorithm.
+
+\textbf{\textit{Response:}} However, this could be seen as being beneficial, as proving the correctness of the HLS tool proves the absence of any bugs according to the language semantics, meaning much less time has to be spent on fixing bugs.  In addition to that, verification also forces the algorithm to deal with many different edge cases that may be hard to identify normally, and may even allow for more optimisations to be performed, which may at first glance seem unreliable.
 
-\textbf{\textit{Response:}} However, this could be seen as being beneficial, as proving the correctness of the HLS tool proves the absence of any bugs according to the language semantics, meaning much less time has to be spent on fixing bugs.  In addition to that, verification also forces the algorithm to deal with many different edge cases that may be hard to identify normally, and may even allow for more optimisations as one can be certain about assumptions one would usually have to make.
+%\JW{I can't quite understand what `being certain about assumptions one would usually have to make' means. Can it be said more plainly?}\YH{Yes, I was mainly trying to explain that proofs help you be certain that your optimisations are correct, and so you can apply them in more cases sometimes.}
 
 \objection{Any HLS tool that is simple enough for formal verification to be feasible won't produce sufficiently optimised designs to be useful}
 
 Another concern might be that a verified HLS tool might not be performant enough to be usable in practice.  If that is the case, then the verification effort could be seen as useless, as it could not be used.
 
-\textbf{\textit{Response:}} We think that even a verified HLS tool can be comparable in performance to a state-of-the-art unverified HLS tool.  Taking \vericert{} as an example, which does not currently include many optimisations, we found that performing comparisons between the fully verified bits of \vericert{} and \legup{}~\cite{canis11_legup}, we found that the speed and area was comparable ($1\times$ - $1.5\times$) that of \legup{} without LLVM optimisations and without operation chaining.  With those optimisations fully turned on, \vericert{} is around $4.5\times$ slower than \legup{}, with half of the speed-up being due to LLVM.\@
+\textbf{\textit{Response:}} We think that even a verified HLS tool can be comparable in performance to a state-of-the-art unverified HLS tool.  Taking \vericert{} as an example, which does not currently include many optimisations, we found that performing comparisons between the fully verified bits of \vericert{} and \legup{}~\cite{canis11_legup}, we found that the speed and area was comparable ($1\times$ - $1.5\times$) to that of \legup{} without LLVM optimisations and without operation chaining.  With those optimisations fully turned on, \vericert{} is around $4.5\times$ slower than \legup{}, with half of the speed-up being due to LLVM.\@
 
-There are many optimisations that need to be added to \vericert{} to turn it into a viable and competitive HLS tool.  First of all, the most important addition is a good scheduling implementation, which supports operation chaining and properly pipelining operators.  Our main focus is implementing scheduling based on system of difference constraint~\cite{cong06_sdc}, which is the same algorithm \legup{} uses.  With this optimisation turned on, \vericert{} is only $~2\times$ to $~3\times$ slower than fully optimised \legup{}, with a slightly larger area.  The scheduling step is implemented using verified translation validation, meaning the scheduling algorithm can be tweaked and optimised without ever having to touch the correctness proof.
+There are many optimisations that need to be added to \vericert{} to turn it into a viable and competitive HLS tool.  First of all, the most important addition is a good scheduling implementation that supports operation chaining and properly pipelining operators.  Our main focus is implementing scheduling based on systems of difference constraints~\cite{cong06_sdc}, which is the same algorithm \legup{} uses.  With this optimisation turned on, \vericert{} is only $~2\times$ to $~3\times$ slower than fully optimised \legup{}, with a slightly larger area.  The scheduling step is implemented using verified translation validation, meaning the scheduling algorithm can be tweaked and optimised without ever having to touch the correctness proof.
 
 \objection{Even a formally verified HLS tool can't give absolute guarantees about the hardware designs it produces}
 
-\textbf{\textit{Response:}} It is true that a verified tool is still allowed to fail at compilation time, meaning none of the correctness proofs need to hold if no output is produced.  However, this is mostly a matter of putting more engineering work into the tool to make it reliable.  Bugs are easier to identify as they will induce tool failures at compile time.
+\paragraph{Response:} It is true that a verified tool is still allowed to fail at compile time, meaning none of the correctness proofs need to hold if no output is produced.  However, this is mostly a matter of putting more engineering work into the tool to make it more complete.  Bugs are easier to identify as they will induce tool failures at compile time.
 
-In addition to that, specifically for an HLS tool taking C as input, undefined behaviour will allow the HLS tool to behave any way it wishes.  This becomes even more important when passing the C to a verified HLS tool, because if it is not free of undefined behaviour, then none of the proofs will hold.  Extra steps therefore need to be performed to ensure that the input is free of any undefined behaviour.
+In addition to that, specifically for an HLS tool taking C as input, undefined behaviour in the input code will allow the HLS tool to behave any way it wishes.  This becomes even more important when passing the C to a verified HLS tool, because if it is not free of undefined behaviour, then none of the proofs will hold.  Extra steps therefore need to be performed to ensure that the input is free of any undefined behaviour. 
 
-Finally, the input and output language semantics also need to be trusted, as the proofs only hold as long as the semantics hold are a true representation of the language.  In \vericert{} this comes down to trusting the C semantics developed by \compcert{}~\cite{blazy09_mechan_seman_cligh_subset_c_languag} and the Verilog semantics, which were adapted from \citet{loow19_proof_trans_veril_devel_hol}.  The latter semantics are designed to only model a small part of the semantics, ensuring that the behaviour for synthesis and simulation stays the same.
+Finally, the input and output language semantics also need to be trusted, as the proofs only hold as long as the semantics are a faithful representation of the languages.  In \vericert{} this comes down to trusting the C semantics developed by \compcert{}~\cite{blazy09_mechan_seman_cligh_subset_c_languag} and the Verilog semantics, which we adapted from \citet{loow19_proof_trans_veril_devel_hol}.
 
 \section{Conclusion}
 
-In conclusion, we have demonstrated that a realistic HLS tool should be proven correct, and even though the performance does not quite match state of the art HLS tools, as the optimisations are formally verified, and added, similar performance may be achieved.  In addition to that, a formally verified HLS tool, written in a theorem prover, would provide a good base to mechanically verify other HLS specific optimisations and help develop more reliable HLS tools in the future.
+In conclusion, we have argued that HLS tools should be formally verified, and through our Vericert prototype, we have demonstrated that doing so is feasible. Even though the performance does not \JWreplace{quite}{yet} match state-of-the-art HLS tools, as \JWreplace{the}{more and more} optimisations are implemented and formally verified, similar performance \JWreplace{may be achieved}{should be achievable}. In addition to that, a formally verified HLS tool, written in a theorem prover, would provide a good base to mechanically verify other HLS-specific optimisations and help develop more reliable HLS tools in the future.
 
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