Update on Overleaf.

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 There are various limitations in \vericert{} compared to other HLS tools due to the fact that our main focus was on formally verifying the translation from 3AC to Verilog. In this section, we outline the current limitations of our tool and suggest how they can be overcome in future work.
 
+\NR{You have very different types of limitations. I wonder if grouped them into software and hardware limitations respectively might simplify this section. Just a suggestion.}
+
 \paragraph{Lack of instruction-level parallelism}
 
 The main limitation of \vericert{} is that it does not perform instruction scheduling, meaning that instructions cannot be gathered into the same state and executed in parallel.  However, the design of the intermediate languages in \vericert{} take this optimisation into account and are designed to support scheduling in the future. For instance, our HTL language allows arbitrary Verilog to appear in each state of the FSMD, including parallel assignments to registers. Our plan for adding scheduling support involves adding a new intermediate language after 3AC, tentatively called 3ACPar. This would be similar to 3AC but rather than mapping program counters to instructions, it would map program counters to \emph{lists} of instructions that can all be executed in parallel. The translation from 3AC to 3ACPar would be performed by a scheduling tool. Following \citet{tristan08_formal_verif_trans_valid} and \citet{six+20}, we expect to use translation validation to verify that each generated schedule is correct (rather than verifying the scheduling tool itself). The translation from 3ACPar to HTL would not change conceptually, except for the fact that multiple instructions can now be translated into the same state.
+\NR{It is best to explain why we didn't focus on scheduling with a positive/future work spin. For example, ``It was more intuitive to handle one instruction per cycle at the initial stage of our project as we want to focus our efforts on correctness, which has been the main weakness of HLS, rather than performance. However, we were very much aware during the design stage that in order for our compiler to be able to perform better, supporting of scheduling was inevitable. Hence, we intentionally left room for support of scheduling. In essence, instead of supporting one instruction per cycle, we must be able to support a list of instructions per cycle. To do so, we envision extensions to our work in several ways.'' We might not even need to specify the details of how. You can keep the tricks in your sleeves for the next publication. :)}.
 
 %To simplify the proof of the scheduling algorithm, and to minimise the changes necessary for the current translation from 3AC to HTL, a new language must be introduced, called 3ACPar, which would be equivalent to 3AC but instead of consisting of a map from program counter to instruction, it would consist of a map from program counter to list of instructions, which can all be executed in parallel.  A new proof for the scheduling algorithm would have to be written for the translation from 3AC to 3ACPar, for which a verified translation validation approach might be appropriate, however, the translation form 3ACPar to HTL would not change conceptually, except for the fact that multiple instructions can now be translated into the same state.  This small difference means that most of the proof can be reused without any changes, as the translation of instructions was the main body of the proof and did not change.
 
 \paragraph{Lack of pipelined division}
-In a similar vein, the introduction of pipelined operators, especially for division, would alleviate the slow clock speed experienced in the \polybench{} benchmarks with divisions included in Section~\ref{sec:evaluation}.  Pipelined operators can execute different stages of the operation in parallel, and therefore run long operations in parallel while sharing the same hardware.  In HTL, operations like this can be represented in a similar fashion to the load and store instructions by using wires to communicate with an abstract computation block modelled in HTL and later replaced by a hardware implementation. %JW I've chopped the following sentence because it felt like it was going into too much detail. 
+In a similar vein, the introduction of pipelined operators, especially for division, would alleviate the slow clock speed experienced in the \polybench{} benchmarks with divisions included in Section~\ref{sec:evaluation}.  Pipelined operators can execute different stages of the operation in parallel, and therefore run long operations in parallel while sharing the same hardware.  In HTL, operations like this can be represented in a similar fashion to the load and store instructions by using wires to communicate with an abstract computation block modelled in HTL and later replaced by a hardware implementation. \NR{Are you describing using IP blocks? If so, you can generalise it to any IP block rather than just division.}
+%JW I've chopped the following sentence because it felt like it was going into too much detail. 
 %However, 3ACPar would have to be modified to also describe such instructions so that these can be placed optimally using the external scheduling algorithm.
 
 \paragraph{Limitations with I/O}
@@ -17,7 +21,7 @@ In a similar vein, the introduction of pipelined operators, especially for divis
 \vericert{} is currently limited in terms of I/O as well, because the main function cannot accept any arguments for the Clight program to be well-formed. However, just like \compcert{}, \vericert{} can compile main functions with arbitrary arguments and will handle the inputs appropriately.  However, the main correctness theorem in \compcert{} assumes that the main function does not have any arguments, so it may be possible that unexpected behaviour is introduced.  In addition to that, external function calls that produce traces have also not been implemented yet, however, these could enable the C program to read and write values to a bus that is shared by various other components in the hardware design.
 
 \paragraph{Lack of support for global variables}
-In \compcert{}, each global variable is stored in its own memory.  A generalisation of the stack translation into a RAM block could therefore be performed to translate global variables in the same manner.  This would require a slight generalisation of pointers so that they store provenance information to ensure that each pointer accesses the right RAM. It would also be necessary to generalise the RAM interface so that it decodes the provenance information and indexes the right array.
+In \compcert{}, each global variable is stored in its own memory.  A generalisation of the stack translation into a RAM block could therefore be performed to translate global variables in the same manner.  This would require a slight generalisation of pointers so that they store provenance information to ensure that each pointer accesses the right RAM. It would also be necessary to generalise the RAM interface so that it decodes the provenance information and indexes the right array. \NR{Curiously, is memory analysis in your to-do list?}
 
 \paragraph{Other language restrictions}
 C and Verilog handle signedness quite differently. By default, all operators and registers in Verilog (and HTL) are unsigned, so to force an operation to handle the bits as signed, both operators have to be forced to be signed. Moreover, Verilog implicitly resizes expressions to the largest needed size by default, which can affect the result of the computation.  This feature is not supported by the Verilog semantics we adopted, so to match the semantics to the behaviour of the simulator and synthesis tool, braces are placed around all expressions to inhibit implicit resizing.  Instead, explicit resizing is used in the semantics, and operations can only be performed on two registers that have the same size.