\section{Introduction}

%% Motivation for why HLS might be needed

The current approach to writing energy-efficient and high-throughput applications is to use application-specific hardware, instead of relying on a general-purpose CPU.\@  However, custom hardware designs come at the cost of having to design and produce them, which can be a tedious and error-prone process using hardware description languages (HDL) such as Verilog.  Especially with the size of hardware designs growing over the years, it can become difficult to verify that the hardware design behaves in the expected way, as simulation of HDLs can be quite inefficient.  Furthermore, the algorithms that are being accelerated in hardware often already have a software implementation, meaning they have to be reimplemented efficiently in a hardware description language which can be time-consuming.

%% Definition and benefits of HLS
One possible solution to the tedious design process of custom hardware is to use high-level synthesis (HLS), which is the process of generating custom hardware, represented in a HDL, based on a behavioural description, often in a subset of C.  This elevates the level of abstraction, because the description of the algorithm in C is inherently untimed, meaning actions don't have to be scheduled into clock cycles.  The higher level of abstraction makes it easier to reason about the algorithms and therefore also makes them easier to maintain.  This already reduces the time taken to design the hardware, especially if a software description of the algorithm already exists, because it doesn't have to be designed again at a lower level and directly in hardware.  However, another benefit of using HLS to design the hardware is that functional verification of the design is much simpler and more efficient than if it were done at the HDL stage, as the whole software ecosystem can be used to do that.  Instead of having to run simulations of the hardware, the C code can just be compiled and executed natively, as the hardware design after HLS should have the same behaviour.  Moreover, the abstraction that HLS provides helps to improve the productivity of hardware designers, as well as reduces the barrier of entry to hardware design for software programmers.  For hardware designers, checking that the created hardware does indeed behave properly may reduce productivity, whereas for software programmers it may be unfeasible to properly test the hardware as they are unaware of the standard tools used.  Both these audiences therefore stand to benefit from the guarantees provided by verified C-to-Verilog generation.
% \NR{The abstraction of HLS helps in two ways: improving productivity of hardware designers and reducing the entry barrier of hardware design for software programmers. Both these audiences stand to benefit from the guarantees provided by verified C-to-Verilog generation.} \JW{Yes, Nadesh makes a good point here. Worth incorporating.}\YH{Added.}

%% Unreliability of HLS

However, the fact that the behaviour is preserved after HLS cannot be guaranteed most existing tools,\YH{Mentor's catapult C can in some cases} meaning behavioural simulation of the hardware design still has to be performed. HLS tools are also known to be quite unreliable, for example, Intel's (formerly Altera's) OpenCL SDK compiler contained too many bugs to even be considered for random testing, as more than 25\% of the test cases failed~\cite{lidbury15_many_core_compil_fuzzin}. In addition to that, designers often feel like HLS tools are quite unreliable and fragile with respect to which language features that are supported.  However, most HLS tools cannot guarantee that compilation is behaviour-preserving. In fact, on the contrary, there is some evidence that current HLS tools are actually quite \emph{unreliable} in this regard. For instance, an attempt by \citet{lidbury15_many_core_compil_fuzzin} to fuzz Altera's (now Intel's) OpenCL compiler had to be abandoned because the compiler ``either crashed or emitted an internal compiler error'' on so many of their test inputs.  Meanwhile, Xilinx's Vivado HLS has been shown to apply pipelining optimisations incorrectly\footnote{\url{https://bit.ly/vivado-hls-pipeline-bug}} or to silently generate wrong code should the programmer stray outside the fragment of C that it supports\footnote{\url{https://bit.ly/vivado-hls-pointer-bug}}.  As HLS tools are extremely complex and can therefore incorrectly change the behaviour of the design, it is not possible to guarantee that all the properties of the code that were proven in software will also hold for the generated hardware.

% JW: Another candidate, probably less interesting:
% https://bit.ly/intel-hls-memory-bug

%% Current work in formal verification of HLS
%%\NR{This is a good paragraph, but we need to relate it more to this work and why this work is different.}
%%\NR{Focus on more high-level of "why this work is interesting"? Two key points we want to get across to the reader is that in existing works: validation is neccessary every time a new program is compiled and the verifying algorithm is not verified.}
%%\NR{Also define words like validation, verifying algorithm (can you use the word ``verifier'',mechanisation)}
%%\NR{Having said that, keep the text for related work section.}\YH{Added into related works.}
To mitigate the problems about the unreliability of synthesis tool, it is often required to check the generated hardware for functional correctness.  This can is commonly done by simulating the design with a large test-bench, however, the guarantees are only as good as the test-bench, meaning if all the inputs were not covered, there may still be bugs in the untested code.  To be sure that the hardware does indeed behave in the same way as the C code, it may therefore be necessary to prove that they are equivalent. \JW{I think that point could be strengthened by emphasising that simulation with a test-bench only provides guarantees that are as good as the test-bench! That is, if the test-bench does not cover all possible inputs then bugs may remain.}\YH{Added a sentence on that, it does motivate the equivalence proving more.} Translation validation~\cite{pnueli98_trans} is the main method which is used to prove that the HLS translation was successful, and has been successfully applied to many HLS optimisations~\cite{kim04_autom_fsmd,karfa06_formal_verif_method_sched_high_synth,chouksey20_verif_sched_condit_behav_high_level_synth,banerjee14_verif_code_motion_techn_using_value_propag,chouksey19_trans_valid_code_motion_trans_invol_loops}.  However, the main problem is that the validator itself has often not been mechanically proven correct, meaning that the implementation is quite separate from the proof.  In addition to that, with large designs it may not be feasible to perform translation validation, as the state space would grow exponentially. \JW{Does this link back to Mentor's Catapult-C, which you mentioned earlier? Does Catapult-C attempt to do translation validation as part of its HLS process? And if so, can you make the point that this effort is largely ineffective because once the design is a reasonable size, the translation validation usually fails anyway?}\YH{TODO: Currently I only have a whitepaper which goes over the translation validation in a very high level, but they do actually mention some flakiness and state that the user would have to manually change the code to fix that.  So I think I can actually make that point.  I just realised they have a pretty funny diagram of verification $\rightarrow$ differences $\rightarrow$ adjustments $\rightarrow$ ... until it is finally verified.} A mechanically verified HLS tool would remove the need to perform simulation after the synthesis process if one has proven desirable properties about the C code.  In addition to that, it would allow for the implementation of translation validated optimisation passes which are also proven correct mechanically, thereby greatly improving the reliability of these passes.

CompCert~\cite{leroy06_formal_certif_compil_back_end} is a C compiler that has been written and formally verified in the Coq theorem prover~\cite{bertot04_inter_theor_provin_progr_devel}.  First of all, most of the compiler passes in CompCert have been proven correct, meaning that once the compiler is built, the proofs can be erased as the algorithm has been shown to be correct independent of the input.  However, some optimisation passes such as software pipelining require translation validation~\cite{tristan08_formal_verif_trans_valid}, in which case the correctness of the compiler pass needs to be checked at runtime.  However, even in this case the verifier itself is proven to only verify code correct that does indeed behave in the same way.

%% Contributions of paper

In this paper we describe a fully verified HLS tool called CoqUp, which adds a Verilog back end to CompCert and proves that the behaviour of the C code is preserved with respect to an existing Verilog semantics.  The main contributions of the paper are the following:

\begin{itemize}
  \item First mechanised and formally verified HLS flow from C to Verilog.
  \item Proof by simulation mechanised in Coq between CompCert's intermediate language and Verilog.
  \item Description of the Verilog semantics integrated into CompCert and how this interacts with CompCert's intermediate language.
  \item Correct by construction Verilog code for four programs that are part of the CHStone benchmark suite, which is a well-known HLS benchmark.
%%  \item \NR{We implement our Verilog semantics in CompCert and we are able to generate correct-by-construction Verilog for all programs in the CHStone benchmark suite, which is a well-known HLS benchmark.}
\end{itemize}

The first section will describe the Verilog semantics that were used and extended to fit into CompCert's model.  The second section will then describe the HLS algorithm, together with its proof.

CoqUp is open source and is hosted on Github\footnote{https://github.com/ymherklotz/coqup}.

\NR{Other comments:}
\NR{Is both the translator and verifier written in Coq?}\YH{Currently there is no verifier, the algorithms themselves are proven directly in Coq.}
%%\NR{A tool-flow diagram here will be useful. I thought you had one previously?}
\NR{Do you think we have a very simple example of a program where wrong Verilog is generated in VHLS or LegUp, but not in CoqUp?}\YH{The LegUp example actually with the shift might be a good one.}

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