source: Deliverables/D4.2-4.3/reports/D4-3.tex @ 1405

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\title{
INFORMATION AND COMMUNICATION TECHNOLOGIES\\
(ICT)\\
PROGRAMME\\
\vspace*{1cm}Project FP7-ICT-2009-C-243881 \cerco{}}

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\includegraphics[width=0.6\textwidth]{../../style/cerco_logo.png}
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\begin{LARGE}
\textbf{
Report n. D4.3\\
Formal semantics of intermediate languages
}
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\begin{large}
Version 1.0
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\vspace*{0.5cm}
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\begin{large}
Main Authors:\\
Dominic P. Mulligan and Claudio Sacerdoti Coen
\end{large}
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\vspace*{\fill}

\noindent
Project Acronym: \cerco{}\\
Project full title: Certified Complexity\\
Proposal/Contract no.: FP7-ICT-2009-C-243881 \cerco{}\\

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\markright{\cerco{}, FP7-ICT-2009-C-243881}

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\vspace*{7cm}
\paragraph{Abstract}
We describe the encoding in the Calculus of Constructions of the semantics of the CerCo compiler's backend intermediate languages.
The CerCo backend consists of five distinct languages: RTL, RTLntl, ERTL, LTL and LIN.
We describe a process of heavy abstraction of the intermediate languages and their semantics.
We hope that this process will ease the burden of Deliverable D4.4, the proof of correctness for the compiler.

\newpage

\tableofcontents

\newpage

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\section{Task}
\label{sect.task}

The Grant Agreement states that Task T4.3, entitled `Formal semantics of intermediate languages' has associated Deliverable D4.3, consisting of the following:
\begin{quotation}
Executable Formal Semantics of back-end intermediate languages: This prototype is the formal counterpart of deliverable D2.1 for the back end side of the compiler and validates it.
\end{quotation}
This report details our implementation of this deliverable.

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\subsection{Connections with other deliverables}
\label{subsect.connections.with.other.deliverables}

Deliverable D4.3 enjoys a close relationship with three other deliverables, namely deliverables D2.2, D4.3 and D4.4.

Deliverable D2.2, the O'Caml implementation of a cost preserving compiler for a large subset of the C programming language, is the basis upon which we have implemented the current deliverable.
In particular, the architecture of the compiler, its intermediate languages and their semantics, and the overall implementation of the Matita encodings has been taken from the O'Caml compiler.
Any variations from the O'Caml design are due to bugs identified in the prototype compiler during the Matita implementation, our identification of code that can be abstracted and made generic, or our use of Matita's much stronger type system to enforce invariants through the use of dependent types.

Deliverable D4.2 can be seen as a `sister' deliverable to the deliverable reported on herein.
In particular, where this deliverable reports on the encoding in the Calculus of Constructions of the backend semantics, D4.2 is the encoding in the Calculus of Constructions of the mutual translations of those languages.
As a result, a substantial amount of Matita code is shared between the two deliverables.

Deliverable D4.4, the backend correctness proofs, is the immediate successor of this deliverable.

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\section{The backend intermediate languages' semantics in Matita}
\label{sect.backend.intermediate.languages.semantics.matita}

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\subsection{Abstracting related languages}
\label{subsect.abstracting.related.languages}

As mentioned in the report for Deliverable D4.2, a systematic process of abstraction, over the O'Caml code, has taken place in the Matita encoding.
In particular, we have merged many of the syntaxes of the intermediate languages (i.e. RTL, ERTL, LTL and LIN) into a single `joint' syntax, which is parameterised by various types.
Equivalent intermediate languages to those present in the O'Caml code can be recovered by specialising this joint structure.

As mentioned in the report for Deliverable D4.2, there are a number of advantages that this process of abstraction brings, from code reuse to allowing us to get a clearer view of the intermediate languages and their structure.
However, the semantics of the intermediate languages allow us to concretely demonstrate this improvement in clarity, by noting that the semantics of the LTL and the semantics of the LIN languages are identical.
In particular, the semantics of both LTL and LIN are implemented in exactly the same way.
The only difference between the two languages is how the next instruction to be interpreted is fetched.
In LTL, this involves looking up in a graph, whereas in LTL, this involves fetching from a list of instructions.

As a result, we see that the semantics of LIN and LTL are both instances of a single, more general language that is parametric in how the next instruction is fetched.
Furthermore, any prospective proof that the semantics of LTL and LIN are identical is not almost trivial, saving a deal of work in Deliverable D4.4.

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\subsection{Type parameters, and their purpose}
\label{subsect.type.parameters.their.purpose}

We mentioned in the Deliverable D4.2 report that all joint languages are parameterised by a number of types, which are later specialised to each distinct intermediate language.
As this parameterisation process is also dependent on designs decisions in the language semantics, we have so far held off summarising the role of each parameter.

We begin the abstraction process with the \texttt{params\_\_} record.
This holds the types of the representations of the different register varieties in the intermediate languages:
\begin{lstlisting}
record params__: Type[1] ≝
{
  acc_a_reg: Type[0];
  acc_b_reg: Type[0];
  dpl_reg: Type[0];
  dph_reg: Type[0];
  pair_reg: Type[0];
  generic_reg: Type[0];
  call_args: Type[0];
  call_dest: Type[0];
  extend_statements: Type[0]
}.
\end{lstlisting}
We summarise what these types mean, and how they are used in both the semantics and the translation process:
\begin{center}
\begin{tabular*}{\textwidth}{p{4cm}p{11cm}}
Type & Explanation \\
\hline
\texttt{acc\_a\_reg} & The type of the accumulator A register.  In some languages this is implemented as the hardware accumulator, whereas in others this is a pseudoregister.\\
\texttt{acc\_b\_reg} & Similar to the accumulator A field, but for the processor's auxilliary accumulator, B. \\
\texttt{dpl\_reg} & The type of the representation of the low eight bit register of the MCS-51's single 16 bit register, DPL.  Can be either a pseudoregister or the hardware DPL register. \\
\texttt{dph\_reg} & Similar to the DPL register but for the eight high bits of the 16-bit register. \\
\texttt{pair\_reg} & Various different `move' instructions have been merged into a single move instruction in the joint language.  A value can either be moved to or from the accumulator in some languages, or moved to and from an arbitrary pseudoregister in others.  This type encodes how we should move data around the registers and accumulators. \\
\texttt{generic\_reg} & The representation of generic registers (i.e. those that are not devoted to a specific task). \\
\texttt{call\_args} & The number of arguments to a function.  For some languages this is irrelevant. \\
\texttt{call\_dest} & \\
\texttt{extend\_statements} & Instructions that are specific to a particular intermediate language, and which cannot be abstracted into the joint language.
\end{tabular*}
\end{center}

As mentioned in the report for Deliverable D4.2, the record \texttt{params\_\_} is enough to be able to specify the instructions of the joint languages:
\begin{lstlisting}
inductive joint_instruction (p: params__) (globals: list ident): Type[0] :=
  | COMMENT: String → joint_instruction p globals
  | COST_LABEL: costlabel → joint_instruction p globals
  ...
\end{lstlisting}

We extend \texttt{params\_\_} with a type corresponding to labels, \texttt{succ}, obtaining a new record type of parameters called \texttt{params\_}:
\begin{lstlisting}
record params_: Type[1] ≝
{
  pars__ :> params__;
  succ: Type[0]
}.
\end{lstlisting}
The type \texttt{succ} corresponds to labels, in the case of control flow graph based languages, or is instantiated to the unit type for the linearised language, LIN.
Using \texttt{param\_} we can define statements of the joint language:
\begin{lstlisting}
inductive joint_statement (p:params_) (globals: list ident): Type[0] :=
  | sequential: joint_instruction p globals → succ p → joint_statement p globals
  | GOTO: label → joint_statement p globals
  | RETURN: joint_statement p globals.
\end{lstlisting}
Note that in the joint language, instructions are `linear', in that they have an immediate successor.
Statements, on the other hand, consist of either a linear instruction, or a \texttt{GOTO} or \texttt{RETURN} statement, both of which can jump to an arbitrary place in the program.

For the semantics, we need further parametererised types.
In particular, we parameterise the result and parameter type of an internal function call in \texttt{params0}:
\begin{lstlisting}
record params0: Type[1] ≝
 { pars__' :> params__
 ; resultT: Type[0]
 ; paramsT: Type[0]
 }.
\end{lstlisting}
We further extend \texttt{params0} with a type for local variables in internal function calls:
\begin{lstlisting}
record params1 : Type[1] ≝
 { pars0 :> params0
 ; localsT: Type[0]
 }.
\end{lstlisting}
Again, we expand our parameters with types corresponding to the code representation (either a control flow graph or a list of statements).
Further, we hypothesise a generic method for looking up the next instruction in the graph, called \texttt{lookup}.
Note that \texttt{lookup} may fail, and returns an \texttt{option} type:
\begin{lstlisting}
record params (globals: list ident): Type[1] ≝
 { succ_ : Type[0]
 ; pars1 :> params1
 ; codeT: Type[0]
 ; lookup: codeT → label → option (joint_statement (mk_params_ pars1 succ_) globals)
 }.
\end{lstlisting}
We now have what we need to define internal functions for the joint language.
The first two `universe' fields are only used in the compilation process, for generating fresh names, and do not affect the semantics.
The rest of the fields affect both compilation and semantics.
In particular, we have parameterised result types, function parameter types and the type of local variables.
Note also that we have lifted the hypothesised \texttt{lookup} function from \texttt{params} into a dependent sigma type, which combines a label (the entry and exit points of the control flow graph or list) combined with a proof that the label is in the graph structure:
\begin{lstlisting}
record joint_internal_function (globals: list ident) (p:params globals) : Type[0] :=
{
  joint_if_luniverse: universe LabelTag;
  joint_if_runiverse: universe RegisterTag;
  joint_if_result   : resultT p;
  joint_if_params   : paramsT p;
  joint_if_locals   : localsT p;
  joint_if_stacksize: nat;
  joint_if_code     : codeT … p;
  joint_if_entry    : $\Sigma$l: label. lookup … joint_if_code l ≠ None ?;
  joint_if_exit     : $\Sigma$l: label. lookup … joint_if_code l ≠ None ?
}.
\end{lstlisting}
Naturally, a question arises as to why we have chosen to split up the parameterisation into so many intermediate records, each slightly extending earlier ones.
The reason is because some intermediate languages share a host of parameters, and only differ on some others.
For instance, in instantiating the ERTL language, certain parameters are shared with RTL, whilst others are ERTL specific:
\begin{lstlisting}
...
definition ertl_params__: params__ :=
 mk_params__ register register register register (move_registers × move_registers)
  register nat unit ertl_statement_extension.
...
definition ertl_params1: params1 := rtl_ertl_params1 ertl_params0.
definition ertl_params: ∀globals. params globals ≝ rtl_ertl_params ertl_params0.
...
definition ertl_statement := joint_statement ertl_params_.

definition ertl_internal_function :=
  $\lambda$globals.joint_internal_function … (ertl_params globals).
\end{lstlisting}
Here, \texttt{rtl\_ertl\_params1} are the common parameters of the ERTL and RTL languages:
\begin{lstlisting}
definition rtl_ertl_params1 := $\lambda$pars0. mk_params1 pars0 (list register).
\end{lstlisting}

\begin{lstlisting}
record more_sem_params (p:params_): Type[1] :=
{
  framesT: Type[0];
  regsT: Type[0];
  succ_pc: succ p → address → res address;
  greg_store_: generic_reg p → beval → regsT → res regsT;
  greg_retrieve_: regsT → generic_reg p → res beval;
  acca_store_: acc_a_reg p → beval → regsT → res regsT;
  acca_retrieve_: regsT → acc_a_reg p → res beval;
  accb_store_: acc_b_reg p → beval → regsT → res regsT;
  accb_retrieve_: regsT → acc_b_reg p → res beval;
  dpl_store_: dpl_reg p → beval → regsT → res regsT;
  dpl_retrieve_: regsT → dpl_reg p → res beval;
  dph_store_: dph_reg p → beval → regsT → res regsT;
  dph_retrieve_: regsT → dph_reg p → res beval;
  pair_reg_move_: regsT → pair_reg p → res regsT;
  pointer_of_label: label → $\Sigma$p:pointer. ptype p = Code
}.
\end{lstlisting}

\begin{lstlisting}
record sem_params: Type[1] :=
{
  spp :> params_;
  more_sem_pars :> more_sem_params spp
}.
\end{lstlisting}

\begin{lstlisting}
record more_sem_params2 (globals: list ident) (p: params globals) : Type[1] :=
{
  more_sparams1 :> more_sem_params p;
  fetch_statement: genv … p → state (mk_sem_params … more_sparams1) → res (joint_statement (mk_sem_params … more_sparams1) globals);
  fetch_ra: state (mk_sem_params … more_sparams1) → res ((state (mk_sem_params … more_sparams1)) × address);
  result_regs: genv globals p → state (mk_sem_params … more_sparams1) → res (list (generic_reg p));
  init_locals : localsT p → regsT … more_sparams1 → regsT … more_sparams1;
  save_frame: address → nat → paramsT … p → call_args p → call_dest p → state (mk_sem_params … more_sparams1) → res (state (mk_sem_params … more_sparams1));
  pop_frame: genv globals p → state (mk_sem_params … more_sparams1) → res ((state (mk_sem_params … more_sparams1)));
  fetch_external_args: external_function → state (mk_sem_params … more_sparams1) → res (list val);
  set_result: list val → state (mk_sem_params … more_sparams1) → res (state (mk_sem_params … more_sparams1));
  exec_extended: genv globals p → extend_statements (mk_sem_params … more_sparams1) → succ p → state (mk_sem_params … more_sparams1) → IO io_out io_in (trace × (state (mk_sem_params … more_sparams1)))
 }.
\end{lstlisting}

\begin{lstlisting}
record sem_params2 (globals: list ident): Type[1] :=
{
  p2 :> params globals;
  more_sparams2 :> more_sem_params2 globals p2
}.
\end{lstlisting}

The \texttt{state} record holds the current state of the interpreter:
\begin{lstlisting}
record state (p: sem_params): Type[0] :=
{
  st_frms: framesT ? p;
  pc: address;
  sp: pointer;
  carry: beval;
  regs: regsT ? p;
  m: bemem
}.
\end{lstlisting}
Here \texttt{st\_frms} represent stack frames, \texttt{pc} the program counter, \texttt{sp} the stack pointer, \texttt{carry} the carry flag, \texttt{regs} the generic registers and \texttt{m} external RAM.
We use the function \texttt{eval\_statement} to evaluate a single joint statement:
\begin{lstlisting}
definition eval_statement:
  ∀globals: list ident.∀p:sem_params2 globals.
    genv globals p → state p → IO io_out io_in (trace × (state p)) :=
...
\end{lstlisting}
We examine the type here.

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\subsection{Use of monads}
\label{subsect.use.of.monads}

Monads are a categorical notion that have recently gained an amount of traction in functional programming circles.
In particular, it was noted by Moggi that monads could be used to sequence \emph{effectful} computations in a pure manner.
Here, `effectful computations' cover a lot of ground, from writing to files, generating fresh names, or updating an ambient notion of state.

In the semantics of both front and backend intermediate languages, we make use of monads.
In particular, we make use of two forms of monad:
\begin{enumerate}
\item
An `error monad', which signals that a computation either has completed successfully, or returns with an error message.
The sequencing operation of the error monad ensures that the result of chained computations in return the error message of the first failed computation.
This monad is used extensively in the semantics to signal a state which cannot be recovered from.
For instance, in the semantics of RTLabs, we make use of the error monad to signal bad final states:
\begin{lstlisting}
XXX
\end{lstlisting}
\item
An `IO' monad, signalling the emission or reading of data to some external location or memory address.
Here, the monads sequencing operation ensures that emissions and reads are maintained in the correct order (i.e. it maintains a `trace', or ordered sequence of IO events).
Most functions in the intermediate language semantics fall into the IO monad.
\end{enumerate}
This monadic infrastructure is shared between the frontend and backend languages.

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\section{Future work}
\label{sect.future.work}

A few small axioms remain to be closed.
These relate to fetching the next instruction to be interpreted from the control flow graph, or linearised representation, of the language.
Closing these axioms should not be a problem.
No further work remains, aside from `tidying up' the code.

\newpage

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\section{Code listing}
\label{sect.code.listing}

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\subsection{Listing of files}
\label{subsect.listing.files}

Semantics specific files (files relating to language translations ommitted):
\begin{center}
\begin{tabular*}{0.9\textwidth}{p{5cm}p{8cm}}
Title & Description \\
\hline
\texttt{RTLabs/syntax.ma} & The syntax of RTLabs \\
\texttt{RTLabs/semantics.ma} & The semantics of RTLabs \\
\texttt{joint/Joint.ma} & Abstracted syntax for backend languages \\
\texttt{joint/SemanticUtils.ma} & Generic utilities used in the semantics of all `joint' intermediate languages \\
\texttt{RTL/RTL.ma} & The syntax of RTL \\
\texttt{RTL/semantics.ma} & The semantics of RTL \\
\texttt{ERTL/ERTL.ma} & The syntax of ERTL \\
\texttt{ERTL/semantics.ma} & The semantics of ERTL \\
\texttt{LTL/LTL.ma} & The syntax of LTL \\
\texttt{LTL/semantics.ma} & The semantics of LTL \\
\texttt{LIN/LIN.ma} & The syntax of LIN \\
\texttt{LIN/semantics.ma} & The semantics of LIN
\end{tabular*}
\end{center}

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\subsection{Listing of important functions and axioms}
\label{subsect.listing.important.functions.axioms}

We list some important functions and axioms in the backend semantics:

\end{document}