Update on Overleaf.

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@@ -69,7 +69,7 @@
 \newtheorem*{remark}{Remark}
 
 \newcounter{objections}
-\newcommand{\objection}[2]{\refstepcounter{objections}\medskip\noindent{\itshape {\bfseries Objection \theobjections: #1.} #2}}
+\newcommand{\objection}[2]{\refstepcounter{objections}\bigskip\noindent{\itshape {\bfseries Objection \theobjections: #1.} #2}}
 \newcommand\response{\par\medskip\textbf{Our response:~}}
 
 \hypersetup{draft}
@@ -94,9 +94,9 @@
 \email{j.wickerson@imperial.ac.uk}
 
 \begin{abstract}
-  With hardware designs becoming ever more complex and demand for custom accelerators ever growing, high-level synthesis (HLS) is increasingly being relied upon.  However, HLS is known to be quite flaky, with each tool supporting subtly different fragments of the input language and sometimes even generating incorrect designs.
+  With hardware designs becoming ever more complex\JW{,} and demand for custom accelerators ever growing, high-level synthesis (HLS) is increasingly being relied upon.  However, HLS is known to be quite flaky, with each tool supporting subtly different fragments of the input language and sometimes even generating incorrect designs. \JW{Probably no paragraph break needed here.}
 
-  We argue that a formally verified HLS tool could solve this issue by dramatically reducing the amount of trusted code, and providing a formal description of the input language that is supported. To this end, we have developed \vericert{}, a formally verified HLS tool, based on \compcert{}, a formally verified C compiler.
+  We argue that a formally verified HLS tool could solve this issue by dramatically reducing the amount of trusted code, and providing a formal description of the input language that is supported. To this end, we have developed \JW{are developing?} \vericert{}, a formally verified HLS tool, based on \compcert{}, a formally verified C compiler.
 \end{abstract}
 
 \maketitle
@@ -112,49 +112,54 @@
 
 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 correct, which means that it outputs hardware designs that are equivalent to the behavioural 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.
+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. \JW{Not sure how to parse the `for example' part there.}  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 \JWcouldcut{of} the supported \JW{fragment of} C.\footnote{\url{https://bit.ly/vivado-hls-pointer-bug}}  These types of bugs are difficult to identify, and exist because \JWcouldcut{firstly} it is not quite clear \JW{firstly} what input these tools support, and secondly whether the output design actually behaves the same as the input.
 
-\paragraph{\bf 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}.
+\paragraph{\bf Our position:} We believe that a formally verified \JWreplace{high-level synthesis}{HLS} 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 \JWreplace{have built}{are building} 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.
 
 \section{Arguments against formalised HLS}
 
-\objection{People should not be designing hardware in C to begin with}{Formally verifying HLS of C is the wrong approach.  C should not be used to design hardware, let alone hardware where reliability is crucial.  First of all, HLS tools are not unreliable in terms of correctness, but more so in terms the quality of the hardware which can be quite unpredictable~\cite{nigam20_predic_accel_desig_time_sensit_affin_types}.  For example, writing C for an HLS tool, small changes in the input could result in large differences in area and performance for the resulting hardware, leading to a kind of flakiness in the output.  Instead, there have been many efforts to formally verify the translation of high-level hardware description languages like Bluespec with K\^{o}i\-ka~\cite{bourgeat20_essen_blues}, formalising the synthesis of Verilog into technology-mapped net-lists with Lutsig~\cite{loow21_lutsig}, or formalising circuit design in Coq itself to ease design verification~\cite{choi17_kami,singh_silver_oak}.}
+\objection{People should not be designing hardware in C to begin with}{Formally verifying HLS of C is the wrong approach.  C should not be used to design hardware, let alone hardware where reliability is crucial.  First of all, HLS tools are not unreliable in terms of correctness, but more so in terms \JW{of} the quality of the hardware\JW{,} which can be quite unpredictable~\cite{nigam20_predic_accel_desig_time_sensit_affin_types}.  For example, \JW{when} writing C for an HLS tool, small changes in the input could result in large differences in area and performance \JWreplace{for}{of} the resulting hardware, \JWcouldcut{leading to a kind of flakiness in the output}.  \JWreplace{Instead}{On the other hand}, there have been many efforts to formally verify the translation of high-level hardware description languages like Bluespec with K\^{o}i\-ka~\cite{bourgeat20_essen_blues}, \JWreplace{formalising}{to formalise} the synthesis of Verilog into technology-mapped net-lists with Lutsig~\cite{loow21_lutsig}, or \JWreplace{formalising}{to formalise} circuit design in Coq itself to ease design verification~\cite{choi17_kami,singh_silver_oak}.}
 
-\response{} Verifying HLS is also important.  First, 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 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.  Finally, even though unpredictability of the output of HLS tools might seem like the largest issue, working on a functionally correct HLS tool provides a good baseline to work on improving the predictability of the output as well.  Reasoning about correctness could maybe be extended with proofs about the variability of the generated output hardware, thereby also improving the quality of the output.  \compcert{}~\cite{leroy09_formal_verif_realis_compil}, an existing formally verified C compiler, has recently been extended with a proof of preservation of constant-time~\cite{barthe20_formal_c}, and a similar approach could be taken to improve to prove properties about the preservation of the hardware area or performance.
+\response{} Verifying HLS is also important.  First, 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 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.  Finally, even though unpredictability of the output of HLS tools might seem like the largest issue, working on a functionally correct HLS tool provides a good baseline to work on improving the predictability of the output as well.  Reasoning about correctness could maybe be extended with proofs about the variability of the generated output hardware, thereby also improving the quality of the output.  \JW{In this vein, }\compcert{}~\cite{leroy09_formal_verif_realis_compil}, an existing formally verified C compiler, has recently been extended with a proof of preservation of constant-time~\cite{barthe20_formal_c}, and a similar approach could be taken to \JWcouldcut{improve} to prove properties about the preservation of the hardware area or performance. % JW: Yes, this is great stuff.
 
 \objection{HLS tools are already commonly used in industry, so they are clearly already reliable enough}{}
 
-\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 edge cases that produce bugs.
+\response{} They are widely used, but they are also widely acknowledged to be quite flaky. In prior work~\cite{du21_fuzzin_high_level_synth_tools}, we have \JWreplace{showed}{shown} 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 edge cases that produce bugs.
 
 \objection{Existing approaches for testing or formally verifying hardware designs are sufficient for ensuring reliability}{
-Besides the use of test benches to test designs produced by HLS, 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.}
+Besides the use of test benches to test designs produced by HLS, 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 \JWcouldcut{the} equivalence between the design and \JW{the} 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{Hm, the flow doesn't quite make sense around here. Objection 3 starts by saying that `existing approaches for testing or formally verifying hardware designs are sufficient' but Response 2 has already said that `existing verification techniques for checking the output of HLS tools may not be enough'. And the text at the end of Objection 3 is already saying that translation validators aren't very good -- it feels like that belongs in the `response' rather than the `objection' itself.
+
+One idea: you could end Response 2 after `catch them'. Then Objection 3 suggests a variety of techniques for checking the output of HLS tools (translation validation of each stage, equivalence checking of the whole thing, etc). Then Response 3 demolishes each in turn, ending up with formal verification as the only way forward.}
 
 %\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.}
 
 \response{} The radical solution to this problem is to formally verify the whole tool. This proved successful for \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, finding 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 take too long, and that HLS tools specifically do not need that kind of reliability.  Indeed, in 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.}
+\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 take too long, and that HLS tools specifically do not need that kind of reliability.  Indeed, in 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 \JW{I find `a translation' confusing because it sounds like `a translation of a particular source program' like in `translation validation'. Perhaps you could say `compiler pass' or something?} correct compared to writing the algorithm.}
 
-\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.
+\response{} However, \JWcouldcut{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. \JW{Nice. I wonder if there's anything we could cite for that. Any examples where formal verification has led to improved performance?}
 
 %\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 is 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.}
+{\JWcouldcut{Another concern is 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.}
 
-\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 the speed up being due to LLVM.\@
+\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 \JWcouldcut{the fully verified bits of} % I deleted that because it looked like you were referring to the fully verified bits of both LegUp and Vericert
+\vericert{} and \legup{}~\cite{canis11_legup}, we found that the speed and area \JWreplace{was}{were} 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 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, a good scheduling implementation that supports operation chaining and pipelined operators is critical.  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.\\
+There are many optimisations that need to be added to \vericert{} to turn it into a viable and competitive HLS tool.  First of all, a good scheduling implementation that supports operation chaining and pipelined operators is critical.  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, \JWreplace{meaning}{which means that} 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 it produces}{}
 
-\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.
+\response{} It is true that a verified tool is still allowed to fail at compile time, \JWreplace{meaning none of the correctness proofs need to hold if no output is produced}{which means that no output is produced despite the input being valid}.  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 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, by using a tool like VST~\cite{appel11_verif_softw_toolc} for example.
 
-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}.
+Finally, the input and output language semantics \JWcouldcut{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 that we adapted from \citet{loow19_proof_trans_veril_devel_hol}.
 
 \section{Conclusion}