Changeset 3659


Ignore:
Timestamp:
Mar 16, 2017, 12:54:45 PM (6 weeks ago)
Author:
mulligan
Message:

more cannibalising, adding paolo's report on indexed labelling technique

Location:
Papers/jar-cerco-2017
Files:
2 edited

Legend:

Unmodified
Added
Removed
  • Papers/jar-cerco-2017/cerco.tex

    r3657 r3659  
    2626\usepackage{amsfonts}
    2727\usepackage{amsmath}
    28 \usepackage{amssymb}
     28\usepackage{amssymb}
     29\usepackage{array}
    2930\usepackage[british]{babel}
    3031\usepackage{color}
     32\usepackage{enumerate}
    3133\usepackage{fancybox}
    3234\usepackage{fancyvrb}
     
    3941\usepackage{mdwlist}
    4042\usepackage{microtype}
     43\usepackage{multirow}
    4144\usepackage{stmaryrd}
    4245\usepackage{url}
     
    7174\DeclareUnicodeCharacter{9002}{\ensuremath{\rangle}}
    7275
     76\newcolumntype{b}{@{}>{{}}}
     77\newcolumntype{B}{@{}>{{}}c<{{}}@{}}
     78\newcolumntype{h}[1]{@{\hspace{#1}}}
     79\newcolumntype{L}{>{$}l<{$}}
     80\newcolumntype{C}{>{$}c<{$}}
     81\newcolumntype{R}{>{$}r<{$}}
     82\newcolumntype{S}{>{$(}r<{)$}}
     83\newcolumntype{n}{@{}}
     84\newcommand{\spanr}[2]{\multicolumn{1}{Rn}{\multirow{#1}{*}{(#2)}}}
     85\def\nocol{\multicolumn{1}{ncn}{}}
     86
    7387\newcommand{\cerco}{CerCo}
    7488\newcommand{\ocaml}{OCaml}
     
    7690\newcommand{\matita}{Matita}
    7791\newcommand{\sdcc}{\texttt{sdcc}}
     92
     93\newcommand{\tern}[3]{#1\mathrel ? #2 : #3}
     94\newcommand{\sop}[1]{\s{#1}\ }
     95\newcommand{\sbin}[1]{\ \s{#1}\ }
     96\newcommand{\Ell}{\mathcal L}
     97\newcommand{\alphab}{\boldsymbol\alpha}
     98\newcommand{\betab}{\boldsymbol\beta}
     99\newcommand{\gramm}{\mathrel{::=}}
     100\newcommand{\ass}{\mathrel{:=}}
     101
     102\renewcommand{\to}[1][]{\stackrel{#1}{\rightarrow}}
     103
     104\newcommand{\eg}{\emph{e.g.\ }}
     105\newcommand{\ie}{\emph{i.e.\ }}
     106
     107\newcommand{\inde}{\hspace{20pt}}
     108
     109\usetikzlibrary{decorations.pathreplacing}
     110\newcommand{\tikztarget}[2]{%
     111  \tikz[remember picture, baseline={(#1.base)}]{
     112  \node (#1) [inner sep = 0pt]{#2};}}
     113\newcommand{\tikztargetm}[2]{%
     114  \tikz[remember picture, baseline={(#1.base)}]{
     115  \node (#1) [inner sep = 0pt]{$#2$};}}
     116 
     117  \newenvironment{comment}{{\bf MORE WORK:}}
     118 
     119\newenvironment{restate-proposition}[2][{}]{\noindent\textbf{Proposition~{#2}}
     120\;\textbf{#1}\ 
     121}{\vskip 1em}
     122 
     123\newenvironment{restate-theorem}[2][{}]{\noindent\textbf{Theorem~{#2}}\;\textbf{
     124#1}\ 
     125}{\vskip 1em}
     126 
     127\newenvironment{restate-corollary}[2][{}]{\noindent\textbf{Corollary~{#2}}
     128\;\textbf{#1}\ 
     129}{\vskip 1em}
     130 
     131\newcommand{\myparagraph}[1]{\medskip\noindent\textbf{#1}}
     132 
     133\newcommand{\Proofitemb}[1]{\medskip \noindent {\bf #1\;}}
     134\newcommand{\Proofitemfb}[1]{\noindent {\bf #1\;}}
     135\newcommand{\Proofitem}[1]{\medskip \noindent $#1\;$}
     136\newcommand{\Proofitemf}[1]{\noindent $#1\;$}
     137\newcommand{\Defitem}[1]{\smallskip \noindent $#1\;$}
     138\newcommand{\Defitemt}[1]{\smallskip \noindent {\em #1\;}}
     139\newcommand{\Defitemf}[1]{\noindent $#1\;$}
     140 
     141 
     142%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     143%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     144 
     145 
     146 
     147%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     148%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     149 
     150\newcommand{\eqdef}{=_{\text{def}}}
     151\newcommand{\concat}{\cdot}%%{\mathbin{+}}
     152\newcommand{\Int}{\mathit{int}}
     153\newcommand{\nat}{\mathit{nat}}
     154\newcommand{\String}{\mathit{string}}
     155\newcommand{\Ident}{\mathit{ident}}
     156\newcommand{\Block}{\mathit{block}}
     157\newcommand{\Signature}{\mathit{signature}}
     158 
     159\newcommand{\pc}{\mathit{pc}}
     160\newcommand{\estack}{\mathit{estack}}
     161\newcommand{\Error}{\epsilon}
     162 
     163%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     164%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     165 
     166% --------------------------------------------------------------------- %
     167% Proof rule.                                                           %
     168% --------------------------------------------------------------------- %
     169 
     170\newcommand{\staterule}[3]{%
     171  $\begin{array}{@{}l}%
     172   \mbox{#1}\\%
     173   \begin{array}{c}
     174   #2\\ 
     175   \hline
     176   \raisebox{0ex}[2.5ex]{\strut}#3%
     177   \end{array}
     178  \end{array}$}
     179 
     180\newcommand{\GAP}{2ex}
     181 
     182\newcommand{\recall}[2]{%
     183 $\begin{array}{c}
     184 #1\\ 
     185 \hline
     186 \raisebox{0ex}[2.5ex]{\strut}#2%
     187 \end{array}$}
     188 
     189\newcommand{\hbra}{\noindent\hbox to \textwidth{\leaders\hrule height1.8mm
     190depth-1.5mm\hfill}}
     191\newcommand{\hket}{\noindent\hbox to \textwidth{\leaders\hrule
     192height0.3mm\hfill}}
     193\newcommand{\ratio}{.3}
     194 
     195\newenvironment{display}[1]{\begin{tabbing}
     196  \hspace{1.5em} \= \hspace{\ratio\linewidth-1.5em} \= \hspace{1.5em} \= \kill
     197  \noindent\hbra\\[-.5em]
     198  {\ }\textsc{#1}\\[-.8ex]
     199  \hbox to \textwidth{\leaders\hrule height1.6mm depth-1.5mm\hfill}\\[-.8ex]
     200  }{\\[-.8ex]\hket
     201  \end{tabbing}}
     202 
     203 
     204\newcommand{\sbline}{\hfill\smash[t]{\rule[1.5em]{\textwidth}{0.2ex}
     205\hfill\hspace*{0ex}}}
     206\newcommand{\sline}{\hfill\smash[t]{\rule[1.5em]{\textwidth}{0.1ex}
     207\hfill\hspace*{0ex}}}
     208\newcommand{\sentry}[2]{\>$#1$\>\ \smash[t]{\vrule width 0.2mm height
     209    1.2\baselineskip depth 1.5\baselineskip}\>#2}
     210 
     211\newcommand{\entry}[2]{\>$#1$\>\>#2}
     212\newcommand{\clause}[2]{$#1$\>\>#2}
     213\newcommand{\category}[2]{\clause{#1::=}{#2}}
     214\newcommand{\subclause}[1]{\>\>\>#1}
     215\newcommand{\redrule}[3]{$#1$\>\>$#2$\>\>\>#3}
     216% proofs 
     217 
     218\newcommand{\Proof}{\noindent {\sc Proof}. }
     219\newcommand{\Proofhint}{\noindent {\sc Proof hint}. }
     220\newcommand{\EndProof}{\qed}
     221 
     222% figure environment
     223 
     224\newcommand{\Figbar}{{\center \rule{\hsize}{0.3mm}}}
     225 %horizontal thiner line for figures
     226\newenvironment{figureplr}[1][t]{\begin{figure}[#1] \Figbar}{\Figbar \end{figure}}
     227%environment for figures
     228%       ************Macros for mathematical symbols*************
     229% Style
     230 
     231\newcommand{\cl}[1]{{\cal #1}}          % \cl{R} to make R calligraphic
     232\newcommand{\la}{\langle}               % the brackets for pairing (see also \pair)
     233\newcommand{\ra}{\rangle}
     234 
     235\newcommand{\lf}{\lfloor}
     236\newcommand{\rf}{\rfloor}
     237\newcommand{\ul}[1]{\underline{#1}}     % to underline
     238\newcommand{\ol}[1]{\overline{#1}}      % to overline
     239\newcommand{\ok}{~ok}                   % well formed context
     240 
     241% Syntax
     242 
     243\newcommand{\Gives}{\vdash}             % in a type judgment
     244\newcommand{\IGives}{\vdash_{I}}        % intuitionistic provability
     245\newcommand{\AIGives}{\vdash_{{\it AI}}} %affine-intuitionistic provability
     246\newcommand{\CGives}{\vdash_{C}}        % affine-intuitionistic confluent provability
     247
     248
     249\newcommand{\Models}{\mid \! =}              % models
     250
     251\newcommand{\emptycxt}{\On}              % empty context
     252\newcommand{\subs}[2]{[#1 / #2]}
     253\newcommand{\sub}[2]{[#2 / #1]}         % substitution \sub{x}{U} gives [U/x]
     254 
     255\newcommand{\Sub}[3]{[#3 / #2]#1}       % Substitution with three arguments \Sub{V}{x}{U}
     256
     257\newcommand{\lsub}[2]{#2 / #1}          % substitution \lsub{x}{U} gives U/x, to  be used in a list.
     258 
     259\newcommand{\impl}{\supset}
     260\newcommand{\arrow}{\rightarrow}        % right thin arrow
     261\newcommand{\trarrow}{\stackrel{*}{\rightarrow}}        % trans closure
     262%\newcommand{\limp}{\makebox[5mm]{\,$- \! {\circ}\,$}}   % linear
     263                                % implication
     264\newcommand{\limp}{\multimap} %linear implication
     265\newcommand{\bang}{\, !}
     266% LNCS
     267%\newcommand{\bang}{\oc}
     268\newcommand{\limpe}[1]{\stackrel{#1}{\multimap}}
     269%\newcommand{\hyp}[3]{#1:(#2, #3)}
     270\newcommand{\letm}[3]{{\sf let} \ ! #1 = #2 \ {\sf in} \ #3}    % modal let
     271\newcommand{\lets}[3]{{\sf let} \ #1 = #2 \ {\sf in} \ #3}    % simple let
     272\newcommand{\letp}[3]{{\sf let} \ \S #1 = #2 \ {\sf in} \ #3}    % paragraph let
     273\newcommand{\tertype}{{\bf 1}}
     274\newcommand{\behtype}{{\bf B}}
     275\newcommand{\bt}[1]{{\it BT}(#1)}       % Boehm tree
     276\newcommand{\cxt}[1]{#1[~]}             % Context with one hole
     277\newcommand{\pr}{\parallel}             % parallel ||
     278\newcommand{\Nat}{\mathbf{N}}                 % natural numbers
     279\newcommand{\Natmax}{\mathbf{N}_{{\it max}}}  % natural numbers with minus infinity
     280\newcommand{\Rat}{\mathbf{Q}^{+}}                 % non-negative rationals
     281\newcommand{\Ratmax}{\mathbf{Q}^{+}_{{\it max}}}  % non-negative rationals with minus infinity
     282%\newcommand{\Alt}{ \mid\!\!\mid  }
     283\newcommand{\isum}{\oplus}
     284\newcommand{\csum}{\uplus}              %context sum
     285\newcommand{\dpar}{\mid\!\mid}
     286\newcommand{\todo}[1]{}
     287                                        % for the production of a grammar containing \mid
     288\newcommand{\infer}[2]{\begin{array}{c} #1 \\ \hline #2 \end{array}}
     289                                        % to make a centered inference rule
     290 
     291% (Meta-)Logic
     292 
     293\newcommand{\bool}{{\sf bool}}          % boolean values
     294\newcommand{\Or}{\vee}                  % disjunction
     295\newcommand{\OR}{\bigvee}               % big disjunction
     296\newcommand{\AND}{\wedge}               % conjunction
     297\newcommand{\ANDD}{\bigwedge}           % big conjunction
     298\newcommand{\Arrow}{\Rightarrow}        % right double arrow
     299\newcommand{\IFF}{\mbox{~~iff~~}}       % iff in roman and with spaces
     300\newcommand{\iffArrow}{\Leftrightarrow} % logical equivalence
     301 
     302% Semantics
     303 
     304\newcommand{\dl}{[\![}                  % semantic [[
     305\newcommand{\dr}{]\!]}                  % semantic ]]
     306
     307
     308% The equivalences for this paper
     309
     310% the usual ones
     311\newcommand{\ubis}{\approx^u}          % usual labelled weak bis
     312\newcommand{\uabis}{\approx^{u}_{ccs}} % usual labelled weak bis on CCS
     313
     314% the contextual conv sensitive
     315\newcommand{\cbis}{\approx}        % convergence sensitive bis
     316\newcommand{\cabis}{\approx_{ccs}}  % convergence sensitive bis on CCS
     317
     318% the labelled conv sensitive
     319\newcommand{\lcbis}{\approx^{\ell}} %
     320\newcommand{\lcabis}{\approx^{\ell}_{ccs}} % labelled convergence sensitive bis on CCS
     321\newcommand{\lcbiswrong}{\approx^{\ell \Downarrow}} %
     322
     323\newcommand{\maytest}{=_{\Downarrow}}
     324\newcommand{\musttest}{=_{\Downarrow_{S}}}
     325 
     326% Sets
     327 
     328\newcommand{\prt}[1]{{\cal P}(#1)}      % Parts of a set
     329\newcommand{\finprt}[1]{{\cal P}_{fin}(#1)}% Finite parts
     330\newcommand{\finprtp}[1]{{\cal P}_{fin}^{+}(#1)}% Non-empty Finite parts
     331\newcommand{\union}{\cup}               % union
     332\newcommand{\inter}{\cap}               % intersection
     333\newcommand{\Union}{\bigcup}            % big union
     334\newcommand{\Inter}{\bigcap}            % big intersection
     335\newcommand{\cpl}[1]{#1^{c}}            % complement
     336\newcommand{\card}{\sharp}              % cardinality
     337\newcommand{\minus}{\backslash}         % set difference
     338\newcommand{\sequence}[2]{\{#1\}_{#2}}  % ex. \sequence{d_n}{n\in \omega}
     339\newcommand{\comp}{\circ}               % functional composition
     340%\newcommand{\oset}[1]{\{#1\}}            % set enumeration
     341\newcommand{\mset}[1]{\{\! | #1 |\!\}}  % pseudo-set notation {| |}
     342 
     343% Domains
     344 
     345\newcommand{\two}{{\bf O}}              % Sierpinski space
     346\newcommand{\join}{\vee}                % join
     347\newcommand{\JOIN}{\bigvee}             % big join 
     348\newcommand{\meet}{\wedge}              % meet
     349\newcommand{\MEET}{\bigwedge}           % big meet
     350\newcommand{\dcl}{\downarrow}           % down closure
     351\newcommand{\ucl}{\uparrow}             % up closure
     352\newcommand{\conv}{\downarrow}          % synt. conv. pred. (postfix)
     353\newcommand{\diver}{\uparrow}           % diverging term
     354\newcommand{\Conv}{\Downarrow}          % sem. conv. pred. (postfix)
     355\newcommand{\SConv}{\Downarrow_{S}}          % sem. conv. pred. (postfix)
     356\newcommand{\CConv}{\Downarrow_{C}}
     357\newcommand{\Diver}{\Uparrow}           % diverging map
     358\newcommand{\cpt}[1]{{\cal K}(#1)}      % compacts, write \cpt{D}
     359\newcommand{\ret}{\triangleleft}        % retract
     360\newcommand{\nor}{\succeq}
     361\newcommand{\prj}{\underline{\ret}}     % projection
     362\newcommand{\parrow}{\rightharpoonup}   % partial function space
     363\newcommand{\ub}[1]{{\it UB}(#1)}       % upper bounds
     364\newcommand{\mub}[1]{{\it MUB}(#1)}     % minimal upper bounds
     365\newcommand{\lift}[1]{(#1)_{\bot}}      % lifting
     366\newcommand{\forget}[1]{\underline{#1}} % forgetful translation
     367
     368%\newcommand{\rel}[1]{\;{\cal #1}\;}     % infix relation (calligraphic)
     369\newcommand{\rl}[1]{\;{\cal #1}\;}             % infix relation
     370\newcommand{\rel}[1]{{\cal #1}}         %calligraphic relation with no
     371                                        %extra space
     372\newcommand{\per}[1]{\;#1 \;}
     373\newcommand{\wddagger}{\natural}  % weak suspension
     374%\newcommand{\wddagger}{=\!\!\!\!\parallel}  % weak suspension
     375% Categories
     376 
     377\newcommand{\pair}[2]{\langle #1 , #2 \rangle} % pairing \pair{x}{y}, do not use < >.
     378 
     379%               *******  Notation for the $\pi$-calculus *********
     380% Syntax:
     381 
     382\newcommand{\fn}[1]{{\it fn}(#1)}                       % free names
     383\newcommand{\bn}[1]{{\it bn}(#1)}                       % bound names
     384\newcommand{\names}[1]{{\it n}(#1)}                     % names
     385\newcommand{\true}{{\sf t}}                             % true
     386\newcommand{\false}{{\sf f}}                            % false
     387\newcommand{\pio}{\pi_1}                                % 1 receptor calculus
     388\newcommand{\pioo}{\pi_{1}^{r}}
     389\newcommand{\piom}{\pi_{1}^{-}}                         % 1 receptor calculus wo match
     390\newcommand{\pioi}{\pi_{1I}}                    % 1 receptor I-calculus
     391\newcommand{\pifo}{\pi_{\w{1f}}}                                % functional calculus
     392\newcommand{\pilo}{\pi_{\w{1l}}}                                % located calculus
     393\newcommand{\sort}[1]{{\it st}(#1)}                     % sort
     394\newcommand{\ia}[1]{{\it ia}(#1)}                     % sort
     395\newcommand{\ite}[3]{{\sf if~} #1 {\sf ~then~} #2 {\sf ~else~} #3}      %if then else
     396\newcommand{\casep}[2]{{\sf case}^{\times}(#1, \pair{x}{y}\Arrow#2)}      %case on pairs
     397\newcommand{\casel}[3]{{\sf case}^{L}(#1, #2, \s{cons}(x,y)\Arrow#3)}      %case on lists
     398\newcommand{\caseb}[3]{{\sf case}^{b}(#1, #2, \s{cons}(x,y)\Arrow#3)}      %case on lists
     399\newcommand{\nil}{{\sf nil}}
     400\newcommand{\cons}{{\sf cons}}
     401\newcommand{\idle}[1]{{\it Idle}(#1)}                   %idle process
     402\newcommand{\conf}[1]{\{ #1 \}}                         %configuration
     403\newcommand{\link}[2]{#1 \mapsto #2}                    %likn a ->b
     404\newcommand{\mand}{\mbox{ and }}
     405\newcommand{\dvec}[1]{\tilde{{\bf #1}}}                 %double vector
     406\newcommand{\erloc}[1]{{\it er}_{l}(#1)}                % location erasure
     407\newcommand{\w}[1]{{\it #1}}    %To write in math style
     408\newcommand{\vcb}[1]{{\bf #1}}
     409\newcommand{\lc}{\langle\!|}
     410\newcommand{\rc}{|\!\rangle}
     411\newcommand{\obj}[1]{{\it obj}(#1)} 
     412\newcommand{\move}[1]{{\sf move}(#1)} 
     413\newcommand{\qqs}[2]{\forall\, #1\;\: #2}
     414\newcommand{\qtype}[4]{\forall #1 :  #2 . (#4,#3)}
     415\newcommand{\xst}[2]{\exists\, #1\;\: #2}
     416\newcommand{\xstu}[2]{\exists\, ! #1\;\: #2}
     417\newcommand{\dpt}{\,:\,}
     418\newcommand{\cond}[3]{\mathsf{if}\ #1\ \mathsf{then}\ #2\ \mathsf{else}\ #3}
     419\newcommand{\s}[1]{{\sf #1}}    % sans-serif 
     420\newcommand{\vc}[1]{{\bf #1}}
     421\newcommand{\lnorm}{\lbrack\!\lbrack}
     422\newcommand{\rnorm}{\rbrack\!\rbrack}
     423\newcommand{\sem}[1]{\underline{#1}}
     424\newcommand{\tra}[1]{\langle #1 \rangle}
     425\newcommand{\trb}[1]{[ #1 ]}
     426\newcommand{\squn}{\mathop{\scriptstyle\sqcup}}
     427\newcommand{\lcro}{\langle\!|}
     428\newcommand{\rcro}{|\!\rangle}
     429\newcommand{\semi}[1]{\lcro #1\rcro}
     430\newcommand{\sell}{\,\ell\,}
     431\newcommand{\SDZ}[1]{\marginpar{\textbf{SDZ:} {#1}}}
     432 
     433\newcommand{\when}[3]{{\sf when}~#1~{\sf then}~#2~{\sf else}~#3} 
     434\newcommand{\wthen}[2]{{\sf when}~#1~{\sf then}~#2~} 
     435\newcommand{\welse}[1]{{\sf else}~#1} 
     436
     437%Pour la fleche double, il faut rajouter :
     438%      \usepackage{mathtools}
     439
     440\newcommand{\act}[1]{\xrightarrow{#1}} %labelled actionlow %high
     441
     442\newcommand{\lact}[1]{\stackrel{#1}{\makebox[5mm]{\,$- \! {\circ}\,$}}}
     443
     444\newcommand{\set}[1]{\{#1\}}
     445\newcommand{\pst}[2]{{\sf pset}(#1,#2)}
     446\newcommand{\st}[2]{{\sf set}(#1,#2)}
     447\newcommand{\wrt}[2]{{\sf w}(#1,#2)}
     448
     449\newcommand{\chtype}[2]{{\it Ch_{#1}(#2)}}
     450\newcommand{\rgtype}[2]{{\it {\sf Reg}_{#1} #2}}
     451
     452\newcommand{\get}[1]{{\sf get}(#1)}
     453
     454%\newcommand{\wact}[1]{\xRightarrow{#1}} %weak labelled action low high
     455
     456%\newcommand{\mact}[1]{\xrightarrow{#1}_{m}} %labelled action low %high
     457
     458%\newcommand{\wmact}[1]{\xRightarrow{#1}_{m}} %weak labelled action low high
     459
     460%\newcommand{\act}[1]{\stackrel{#1}{\rightarrow}} %labelled action low
     461                                %%%high
     462
     463\newcommand{\acteq}[1]{\stackrel{#1}{\leadsto}} %labelled action low
     464                                %%%high
     465
     466
     467%\newcommand{\actI}[1]{\stackrel{#1}{\rightarrow_{1}}} %labelled action low
     468\newcommand{\actI}[1]{\xrightarrow{#1}_{1}}
     469
     470%\newcommand{\actII}[1]{\stackrel{#1}{\rightarrow_{2}}} %labelled action low
     471\newcommand{\actII}[1]{\xrightarrow{#1}_{2}}
     472
     473
     474 \newcommand{\wact}[1]{\stackrel{#1}{\Rightarrow}} %weak labelled action low high
     475\newcommand{\wactI}[1]{\stackrel{#1}{\Rightarrow_{1}}} %weak labelled action low high
     476\newcommand{\wactII}[1]{\stackrel{#1}{\Rightarrow_{2}}} %weak labelled action low high
     477
     478
     479\newcommand{\mact}[1]{\stackrel{#1}{\rightarrow_{m}}} %labelled action low
     480%high
     481\newcommand{\wmact}[1]{\stackrel{#1}{\Rightarrow_{m}}} %weak labelled action low high
     482 
     483%\newcommand{\lact}[1]{\stackrel{#1}{\leftarrow}}
     484\newcommand{\lwact}[1]{\stackrel{#1}{\Leftarrow}}
     485 
     486 
     487 
     488\newcommand{\eval}{\Downarrow}
     489\newcommand{\Eval}[1]{\Downarrow^{#1}}
     490 
     491 
     492\newcommand{\Z}{{\bf Z}}
     493\newcommand{\Real}{\mathbb{R}^{+}} 
     494\newcommand{\Return}{\ensuremath{\mathtt{return}}\xspace}                 
     495\newcommand{\Stop}{\ensuremath{\mathtt{stop}}\xspace}
     496\newcommand{\Wait}{\ensuremath{\mathtt{wait}}\xspace}
     497\newcommand{\Read}{\ensuremath{\mathtt{read}}\xspace}
     498\newcommand{\Write}{\ensuremath{\mathtt{write}}\xspace}
     499\newcommand{\Yield}{\ensuremath{\mathtt{yield}}\xspace}
     500\newcommand{\Next}{\ensuremath{\mathtt{next}}\xspace}
     501\newcommand{\Load}{\ensuremath{\mathtt{load}}\xspace}
     502\newcommand{\Call}{\ensuremath{\mathtt{call}}\xspace}
     503\newcommand{\Tcall}{\ensuremath{\mathtt{tcall}}\xspace}
     504\newcommand{\Pop}{\ensuremath{\mathtt{pop}}\xspace}
     505\newcommand{\Build}{\ensuremath{\mathtt{build}}\xspace}
     506\newcommand{\Branch}{\ensuremath{\mathtt{branch}}\xspace}
     507\newcommand{\Goto}{\ensuremath{\mathtt{goto}}\xspace}
     508 
     509\newcommand{\hatt}[1]{#1^{+}}
     510\newcommand{\Of}{\mathbin{\w{of}}}
     511 
     512\newcommand{\susp}{\downarrow}
     513\newcommand{\lsusp}{\Downarrow_L}
     514\newcommand{\wsusp}{\Downarrow}
     515\newcommand{\commits}{\searrow}
     516 
     517 
     518\newcommand{\spi}{S\pi}
     519 
     520
     521 \newcommand{\pres}[2]{#1\triangleright #2} %TCCS else next (alternative)
     522% \newcommand{\pres}[2]{ \lfloor #1 \rfloor (#2)}  %TCCS else next
     523\newcommand{\present}[3]{{\sf present} \ #1 \ {\sf do } \ #2 \ {\sf  else} \ #3}
     524
     525
     526\newcommand{\tick}{{\sf tick}}          %tick action
     527
     528 
     529 
     530\newcommand{\sbis}{\equiv_L}
     531\newcommand{\emit}[2]{\ol{#1}#2} 
     532%\newcommand{\present}[4]{#1(#2).#3,#4}
     533\newcommand{\match}[4]{[#1=#2]#3,#4}       %pi-equality
     534
     535\newcommand{\matchv}[4]{[#1 \unrhd #2]#3,#4}
     536
     537\newcommand{\new}[2]{\nu #1 \ #2}
     538\newcommand{\outact}[3]{\new{{\bf #1}}{\emit{#2}{#3}}}
     539\newcommand{\real}{\makebox[5mm]{\,$\|\!-$}}% realizability relation
     540
     541\newcommand{\regterm}[2]{{\sf reg}_{#1} #2}
     542\newcommand{\thread}[1]{{\sf thread} \ #1}
     543\newcommand{\store}[2]{(#1 \leftarrow #2)}
     544\newcommand{\pstore}[2]{(#1 \Leftarrow #2)}
     545
     546\newcommand{\regtype}[2]{{\sf Reg}_{#1} #2}
     547\newcommand{\uregtype}[3]{{\sf Reg}_{#1}(#2, #3)}
     548\newcommand{\urtype}[2]{{\sf Reg}(#1, #2)}
     549
     550\newcommand{\upair}[2]{[#1,#2]}
     551\newcommand{\letb}[3]{\mathsf{let}\;\oc #1 = #2\;\mathsf{in}\;#3}
     552
     553\newcommand{\vlt}[1]{{\cal V}(#1)}
     554\newcommand{\prs}[1]{{\cal P}(#1)}
     555
     556\newcommand{\imp}{{\sf Imp}}            %imp language
     557\newcommand{\vm}{{\sf Vm}}              %virtual machine language
     558\newcommand{\mips}{{\sf Mips}}          %Mips language
     559\newcommand{\C}{{\sf C}}                % C language
     560\newcommand{\Clight}{{\sf Clight}}        %C light language
     561\newcommand{\Cminor}{{\sf Cminor}}
     562\newcommand{\RTLAbs}{{\sf RTLAbs}}
     563\newcommand{\RTL}{{\sf RTL}}
     564\newcommand{\ERTL}{{\sf ERTL}}
     565\newcommand{\LTL}{{\sf LTL}}
     566\newcommand{\LIN}{{\sf LIN}}
     567\newcommand{\access}[1]{\stackrel{#1}{\leadsto}}
     568
     569\newcommand{\codeex}[1]{\texttt{#1}}   % code example
    78570
    79571\title{CerCo: Certified Complexity\thanks{The project CerCo acknowledges the
  • Papers/jar-cerco-2017/proof.tex

    r3657 r3659  
    55%   Technical issues in front end (Brian?)
    66%   Main theorem statement
     7
     8\section{Introduction}
     9In~\cite{D2.1}, Armadio \emph{et al} propose an approach for building a compiler for a large fragment of the \textsc{c} programming language.
     10The novelty of their proposal lies in the fact that their proposed design is capable of lifting execution cost information from the compiled code and presenting it to the user.
     11This idea is foundational for the CerCo project, which strives to produce a mechanically certified version of such a compiler.
     12
     13To summarise, Armadio's proposal consisted of `decorations' on the source code, along with the insertion of labels at key points.
     14These labels are preserved as compilation progresses, from one intermediate language to another.
     15Once the final object code is produced, such labels should correspond to the parts of the compiled code that have a constant cost.
     16
     17Two properties must hold of any cost estimate.
     18The first property, paramount to the correctness of the method, is \emph{soundness}, that is, that the actual execution cost is bounded by the estimate.
     19In the labelling approach, this is guaranteed if every loop in the control flow of the compiled code passes through at least one cost label.
     20The second property, optional but desirable, is \emph{preciseness}: the estimate \emph{is} the actual cost.
     21In the labelling approach, this will be true if, for every label, every possible execution of the compiled code starting from such a label yields the same cost before hitting another one.
     22In simple architectures such as the 8051 micro-controller this can be guaranteed by placing labels at the start of any branch in the control flow, and by ensuring that no labels are duplicated.
     23
     24The reader should note that the above mentioned requirements must hold when executing the code obtained at the end of the compilation chain.
     25So even if one is careful about injecting the labels at suitable places in the source code, the requirements might still fail because of two main obstacles:
     26\begin{itemize}
     27\item
     28The compilation process introduces important changes in the control flow, inserting loops or branches.
     29For example, the insertion of functions in the source code replacing instructions that are unavailable in the target architecture.
     30This require loops to be inserted (for example, for multi-word division and generic shift in the 8051 architecture), or effort spent in providing unbranching translations of higher level instructions~\cite{D2.2}.
     31\item
     32Even when the compiled code \emph{does}---as far as the the syntactic control flow graph is concerned---respect the conditions for soundness and preciseness, the cost of blocks of instructions might not be independent of context, so that different passes through a label might have different costs.
     33This becomes a concern if one wishes to apply the approach to more complex architectures, for example one with caching or pipelining.
     34\end{itemize}
     35The first point unveils a weakness of the current labelling approach when it comes to some common code transformations performed along a compilation chain.
     36In particular, most \emph{loop optimisations} are disruptive, in the sense outlined in the first bulletpoint above.
     37An example optimisation of this kind is \emph{loop peeling}.
     38This optimisation is employed by compilers in order to trigger other optimisations, for example, dead code elimination or invariant code motion.
     39Here, a first iteration of the loop is hoisted out of the body of the loop, possibly being assigned a different cost than later iterations.
     40
     41The second bulletpoint above highlights another weakness. Different tools allow to predict up to a certain extent the behaviour of cache.
     42For example, the well known tool \s{aiT}~\cite{absint}---based on abstract interpretation---allows the user to estimate the worst-case execution time (\textsc{wcet}) of a piece of source code, taking into account advanced features of the target architecture. While
     43such a tool is not fit for a compositional approach which is central to CerCo's project\footnote{\s{aiT} assumes the cache is empty at the start of computation, and treats each procedure call separately, unrolling a great part of the control flow.},
     44\s{aiT}'s ability to produce tight estimates of execution costs would sthill enhance the effectiveness of the CerCo compiler, \eg{} by integrating such techniques in its development.
     45A typical case where cache analysis yields a difference in the execution cost of a block is in loops: the first iteration will usually stumble upon more cache misses than subsequent iterations.
     46
     47If one looks closely, the source of the weakness of the labelling approach as presented in~\cite{D2.1} is common to both points: the inability to state different costs for different occurrences of labels, where the difference might be originated by labels being duplicated along the compilation, or the costs being sensitive to the current state of execution.
     48The preliminary work we present here addresses this weakness by introducing cost labels that are dependent on which iteration of its containing loops it occurs in.
     49This is achieved by means of \emph{indexed labels}; all cost labels are decorated with formal indices coming from the loops containing such labels.
     50These indices allow us to rebuild, even after multiple loop transformations, which iterations of the original loops in the source code a particular label occurrence belongs to.
     51During the annotation stage of the source code, this information is presented to the user by means of \emph{dependent costs}.
     52
     53We concentrate on integrating the labelling approach with two loop transformations.
     54For general information on general compiler optimisations (and loop optimisations in particular) we refer the reader to the vast literature on the subject (\eg\cite{muchnick,morgan}).
     55
     56\paragraph{Loop peeling}
     57As already mentioned, loop peeling consists in preceding the loop with a copy of its body, appropriately guarded.
     58This is used, in general, to trigger further optimisations, such as those that rely on execution information which can be computed at compile time, but which is erased by further iterations of the loop, or those that use the hoisted code to be more effective at eliminating redundant code.
     59Integrating this transformation in to the labelling approach would also allow the integration of the common case of cache analysis explained above; the analysis of cache hits and misses usually benefits from a form of \emph{virtual} loop peeling~\cite{cacheprediction}.
     60
     61\paragraph{Loop unrolling}
     62This optimisation consists of the repetition of several copies of the body of the loop inside the loop itself (inserting appropriate guards, or avoiding them altogether if enough information about the loop's guard is available at compile time).
     63This can limit the number of (conditional or unconditional) jumps executed by the code and trigger further optimisations dealing with pipelining, if appropriate for the architecture.
     64\\\\
     65Whilst we cover only two loop optimisations in this report, we argue that the work presented herein poses a good foundation for extending the labelling approach, in order to cover more and more common optimisations, as well as gaining insight into how to integrate advanced cost estimation techniques, such as cache analysis, into the CerCo compiler.
     66Moreover loop peeling itself has the fortuitous property of enhancing and enabling other optimisations.
     67Experimentation with CerCo's untrusted prototype compiler, which implements constant propagation and partial redundancy elimination~\cite{PRE,muchnick}, show how loop peeling enhances those other optimisations.
     68
     69\paragraph{Outline}
     70We will present our approach on a minimal `toy' imperative language, \imp{} with \s{goto}s, which we present in Section~\ref{sec:defimp} along with formal definitions of the loop transformations.
     71This language already presents most of the difficulties encountered when dealing with \textsc{c}, so we stick to it for the sake of this presentation.
     72In Section~\ref{sec:labelling} we summarize the labelling approach as presented in~\cite{D2.1}.
     73Section~\ref{sec:indexedlabels} presents \emph{indexed labels}, our proposal for dependent labels which are able to describe precise costs even in the presence of the various loop transformations we consider.
     74Finally Section~\ref{sec:conc} goes into more detail regarding the implementation of indexed labels in CerCo's untrusted compiler and speculates on further work on the subject.
     75
     76\section{\imp{} with goto}\label{sec:defimp}
     77We briefly outline the toy language, \imp{} with \s{goto}s.
     78The language was designed in order to pose problems for the existing labelling approach, and as a testing ground for our new notion of indexed labels.
     79
     80The syntax and operational semantics of our toy language are presented in~\ref{fig:minimp}.
     81Note, we may augment the language further, with \s{break} and \s{continue}, at no further expense.
     82\begin{figureplr}
     83$$\begin{gathered}
     84\begin{array}{nlBl>(R<)n}
     85\multicolumn{4}{C}{\bfseries Syntax}\\
     86\multicolumn{4}{ncn}{
     87  \ell,\ldots \hfill \text{(labels)} \hfill x,y,\ldots \hfill
     88\text{(identifiers)}
     89\hfill e,f,\ldots \hfill \text{(expression)}
     90}\\
     91P,S,T,\ldots &\gramm& \s{skip} \mid s;t
     92\mid \sop{if}e\sbin{then}s\sbin{else}t
     93\mid \sop{while} e \sbin{do} s \mid
     94  x \ass e
     95\\&\mid&
     96\ell : s \mid \sop{goto}\ell& \spanr{-2}{statements}\\
     97\\
     98\multicolumn{4}{C}{\bfseries Semantics}\\
     99K,\ldots  &\gramm& \s{halt} \mid S \cdot K & continuations
     100\end{array}
     101\\[15pt]
     102\s{find}(\ell,S,K) \ass
     103\left\{\begin{array}{lL}
     104\bot & if $S=\s{skip},\sop{goto} \ell'$ or $x\ass e$,\\
     105(T, K) & if $S=\ell:T$,\\
     106\s{find}(\ell,T,K) & otherwise, if $S = \ell':T$,\\
     107\s{find}(\ell,T_1,T_2\cdot K) & if defined and $S=T_1;T_2$,\\
     108\s{find}(\ell,T_1,K) & if defined and
     109$S=\sop{if}b\sbin{then}T_1\sbin{else}T_2$,\\
     110\s{find}(\ell,T_2,K) & otherwise, if $S=T_1;T_2$ or
     111$\sop{if}b\sbin{then}T_1\sbin{else}T_2$,\\
     112\s{find}(\ell,T,S\cdot K) & if $S = \sop{while}b\sbin{do}T$.
     113\end{array}\right.
     114\\[15pt]
     115\begin{array}{lBl}
     116(x:=e,K,s)  &\to_P& (\s{skip},K,s[v/x]) \qquad\mbox{if }(e,s)\eval v \\ \\
     117
     118(S;T,K,s)  &\to_P& (S,T\cdot K,s) \\ \\
     119
     120(\s{if} \ b \ \s{then} \ S \ \s{else} \ T,K,s)
     121&\to_P&\left\{
     122\begin{array}{ll}
     123(S,K,s) &\mbox{if }(b,s)\eval v \neq 0 \\
     124(T,K,s) &\mbox{if }(b,s)\eval 0
     125\end{array}
     126\right. \\ \\
     127
     128
     129(\s{while} \ b \ \s{do} \ S ,K,s)
     130&\to_P&\left\{
     131\begin{array}{ll}
     132(S,\s{while} \ b \ \s{do} \ S \cdot K,s) &\mbox{if }(b,s)\eval v \neq 0 \\
     133(\s{skip},K,s) &\mbox{if }(b,s)\eval 0
     134\end{array}
     135\right. \\ \\
     136
     137
     138(\s{skip},S\cdot K,s)  &\to_P&(S,K,s) \\ \\
     139
     140(\ell : S, K, s)  &\to_P& (S,K,s) \\ \\
     141
     142(\sop{goto}\ell,K,s)  &\to_P& (\s{find}(\ell,P,\s{halt}),s) \\ \\
     143\end{array}
     144\end{gathered}$$
     145\caption{The syntax and operational semantics of \imp.}
     146\label{fig:minimp}
     147\end{figureplr}
     148The precise grammar for expressions is not particularly relevant so we do not give one in full.
     149For the sake of conciseness we also treat boolean and arithmetic expressions together (with the usual \textsc{c} convention of an expression being true iff non-zero).
     150We may omit the \s{else} clause of a conditional if it leads to a \s{skip} statement.
     151
     152We will presuppose that all programs are \emph{well-labelled}, \ie every label labels at most one occurrence of a statement in a program, and every \s{goto} points to a label actually present in the program.
     153The \s{find} helper function has the task of not only finding the labelled statement in the program, but also building the correct continuation.
     154The continuation built by \s{find} replaces the current continuation in the case of a jump.
     155
     156\paragraph{Further down the compilation chain}
     157We abstract over the rest of the compilation chain.
     158We posit the existence of a suitable notion of `sequential instructions', wherein each instruction has a single natural successor to which we can add our own, for every language $L$ further down the compilation chain.
     159
     160\subsection{Loop transformations}
     161We call a loop $L$ \emph{single-entry} in $P$ if there is no \s{goto} to $P$ outside of $L$ which jumps into $L$.\footnote{This is a reasonable aproximation: it defines a loop as multi-entry if it has an external but unreachable \s{goto} jumping into it.}
     162Many loop optimisations do not preserve the semantics of multi-entry loops in general, or are otherwise rendered ineffective.
     163Usually compilers implement a single-entry loop detection which avoids the multi-entry ones from being targeted by optimisations~\cite{muchnick,morgan}.
     164The loop transformations we present are local, \ie they target a single loop and transform it.
     165Which loops are targeted may be decided by some \emph{ad hoc} heuristic.
     166However, the precise details of which loops are targetted and how is not important here.
     167
     168\paragraph{Loop peeling}
     169$$
     170\sop{while}b\sbin{do}S \mapsto \sop{if}b\sbin{then} S; \sop{while} b \sbin{do} S[\ell'_i/\ell_i]
     171$$
     172where $\ell'_i$ is a fresh label for any $\ell_i$ labelling a statement in $S$.
     173This relabelling is safe for \s{goto}s occurring outside the loop because of the single-entry condition.
     174Note that for \s{break} and \s{continue} statements, those should be replaced with \s{goto}s in the peeled body $S$.
     175
     176\paragraph{Loop unrolling}
     177$$
     178\sop{while}b\sbin{do}S\mapsto
     179\sop{while} b \sbin{do} (S ;
     180  \sop{if} b \sbin{then} (S[\ell^1_i/\ell_i] ;
     181  \cdots
     182  \sop{if} b \sbin{then} S[\ell^n_i/\ell_i]) \cdots)
     183$$
     184where $\ell^j_i$ are again fresh labels for any $\ell_i$ labelling a statement in $S$.
     185This is a wilfully na\"{i}ve version of loop unrolling, which usually targets less general loops.
     186The problem this transformation poses to CerCo's labelling approach are independent of the sophistication of the actual transformation.
     187
     188\begin{example}
     189In \autoref{fig:example1} we show a program (a wilfully inefficient computation of of the
     190sum of the first $n$ factorials) and a possible transformation of it, combining loop
     191peeling and loop unrolling.
     192\begin{figureplr}
     193$$
     194\fbox{$\begin{array}{l}
     195s\ass 0;\\
     196i\ass 0;\\
     197\sop{while}i<n\sbin{do}\\
     198\inde p\ass 1;\\
     199\inde j\ass 1;\\
     200\inde \sop{while}j \le i\sbin{do}\\
     201\inde \inde p\ass j*p\\
     202\inde \inde j\ass j+1;\\
     203\inde s\ass s+p;\\
     204\inde i\ass i+1;\\
     205\end{array}
     206$}
     207\mapsto
     208\fbox{$\begin{array}{l}
     209s\ass 0;\\
     210i\ass 0;\\
     211\tikztargetm{a}{\s{if}}\ i<n\sbin{then}\\
     212\inde p\ass 1;\\
     213\inde j\ass 1;\\
     214\inde \sop{while}j \le i\sbin{do}\\
     215\inde \inde p\ass j*p\\
     216\inde \inde j\ass j+1;\\
     217\inde s\ass s+p;\\
     218\inde i\ass i+1;\\
     219\inde \tikztargetm{d}{\s{while}}\ i<n\sbin{do}\\
     220\inde \inde p\ass 1;\\
     221\inde \inde j\ass 1;\\
     222\inde \inde \tikztargetm{b}{\s{if}}\ j \le i\sbin{then}\\
     223\inde \inde \inde p\ass j*p\\
     224\inde \inde \inde j\ass j+1;\\
     225\inde \inde \inde \sop{if}j \le i\sbin{then}\\
     226\inde \inde \inde \inde p\ass j*p\\
     227\inde \inde \inde \inde j\ass j+1;\\
     228\inde \inde \inde \inde \s{while}\ j \le i\sbin{do}\\
     229\inde \inde \inde \inde \inde p\ass j*p\\
     230\inde \inde \inde \inde \inde j\ass j+1;\\
     231\inde \inde \inde \inde \inde \sop{if}j \le i\sbin{then}\\
     232\inde \inde \inde \inde \inde \inde p\ass j*p\\
     233\inde \inde \inde \inde \inde \inde \tikztargetm{c}j\ass j+1;\\
     234\inde \inde s\ass s+p;\\
     235\inde \inde i\ass i+1;\\
     236\inde \inde \sop{if}i<n\sbin{then}\\
     237\inde \inde \inde p\ass 1;\\
     238\inde \inde \inde j\ass 1;\\
     239\inde \inde \inde \tikztargetm{e}{\s{while}}\ j < i\sbin{do}\\
     240\inde \inde \inde \inde p\ass j*p\\
     241\inde \inde \inde \inde j\ass j+1;\\
     242\inde \inde \inde \inde \s{if}\ j < i\sbin{then}\\
     243\inde \inde \inde \inde \inde p\ass j*p\\
     244\inde \inde \inde \inde \inde \tikztargetm{f}j\ass j+1;\\
     245\inde \inde \inde s\ass s+p;\\
     246\inde \inde \inde i\ass i+1\tikztargetm{g};{}
     247\end{array}
     248$}\tikztargetm{right}{}
     249\begin{tikzpicture}[overlay, remember picture, thick,
     250brace/.style = {decorate, decoration={brace, amplitude = 15pt}},
     251label/.style = {sloped, anchor = base, yshift = 17pt, font = \large}]
     252\draw [brace, transform canvas={xshift=5pt}] (b.north-|right) -- node[label]{peeled} (c.south-|right);
     253\draw [brace, transform canvas={xshift=30pt}] (b.north-|right) -- node[label]{unrolled} (c.south-|right);
     254\draw [brace, transform canvas={xshift=5pt}] (e.north-|right) -- node[label]{unrolled} (f.south-|right);
     255\draw [brace, transform canvas={xshift=55pt}] (d.north-|right) -- node[label]{unrolled} (g.south-|right);
     256\draw [brace, transform canvas={xshift=80pt}] (a.north-|right) -- node[label]{peeled} (g.south-|right);
     257\end{tikzpicture}
     258\hspace{85pt}{}
     259$$
     260\caption{An example of loop transformations done on an \imp{} program. Parentheses are omitted in favour of
     261blocks by indentation.}
     262\label{fig:example1}
     263\end{figureplr}
     264\end{example}
     265
     266\section{Labelling: a quick sketch of the previous approach}
     267\label{sec:labelling}
     268Plainly labelled \imp{} is obtained by adding to the code \emph{cost labels} (with metavariables $\alpha,\beta,\ldots$), and cost-labelled statements:
     269$$
     270S,T\gramm \cdots \mid \alpha: S
     271$$
     272Cost labels allow us to track some program points along the compilation chain.
     273For further details we refer to~\cite{D2.1}.
     274
     275With labels the small step semantics turns into a labelled transition system along with a natural notion of trace (\ie lists of labels) arises.
     276The evaluation of statements is enriched with traces, so that rules follow a pattern similar to the following:
     277$$
     278\begin{array}{lblL}
     279(\alpha: S, K,s) &\to[\alpha]_P (S,K,s)\\
     280(\s{skip}, S \cdot K,s) &\to[\varepsilon]_P (S, K, s)\\
     281& \text{etc.}
     282\end{array}$$
     283Here, we identify cost labels $\alpha$ with singleton traces and we use $\varepsilon$ for the empty trace.
     284Cost labels are emitted by cost-labelled statements only\footnote{In the general case the evaluation of expressions can emit cost labels too (see~\ref{sec:conc}).}.
     285We then write $\to[\lambda]\!\!^*$ for the transitive closure of the small step semantics which produces by concatenation the trace $\lambda$.
     286
     287\paragraph{Labelling}
     288Given an \imp{} program $P$ its \emph{labelling} $\alpha:\Ell(P)$ in $\ell-\imp$ is defined by putting cost labels after every branching statement, at the start of both branches, and a cost label at the beginning of the program. Also, every labelled statement gets a cost label,
     289which is a conservative approach to ensuring that all loops have labels inside them, as a loop might be done with \s{goto}s.
     290The relevant cases are
     291$$\begin{aligned}
     292  \Ell(\sop{if}e\sbin{then}S\sbin{else}T) &=
     293    \sop{if}e\sbin{then}\alpha:\Ell(S)\sbin{else}\beta:\Ell(T)\\
     294  \Ell(\sop{while}e\sbin{do}S) &=
     295    (\sop{while}e\sbin{do}\alpha:\Ell(S));\beta:\s{skip}\\
     296  \Ell(\ell : S) &=
     297    (\ell : \alpha : \Ell(S))
     298  \end{aligned}$$
     299where $\alpha,\beta$ are fresh cost labels.
     300In all other cases the definition just passes to substatements.
     301
     302\paragraph{Labels in the rest of the compilation chain}
     303All languages further down the chain get a new sequential statement $\sop{emit}\alpha$ whose effect is to be consumed in a labelled transition while keeping the same state.
     304All other instructions guard their operational semantics and do not emit cost labels.
     305
     306Preservation of semantics throughout the compilation process is restated, in rough terms, as:
     307$$
     308\text{starting state of $P$}\to[\lambda]\!\!^*\;\text{halting state} \iff
     309\text{starting state of $\mathcal C(P)$} \to[\lambda]\!\!^*\;\text{halting state}
     310$$
     311Here $P$ is a program of a language along the compilation chain, starting and halting states depend on the language, and $\mathcal C$ is the compilation function\footnote{The case of divergent computations needs to be addressed too.
     312Also, the requirement can be weakened by demanding some sort weaker form of equivalence of the traces than equality.
     313Both of these issues are beyond the scope of this presentation.}.
     314
     315\paragraph{Instrumentations}
     316Let $\mathcal C$ be the whole compilation from $\ell\imp$ to the labelled version of some low-level language $L$.
     317Supposing such compilation has not introduced any new loop or branching, we have that:
     318\begin{itemize}
     319\item
     320Every loop contains at least a cost label (\emph{soundness condition})
     321\item
     322Every branching has different labels for the two branches (\emph{preciseness condition}).
     323\end{itemize}
     324With these two conditions, we have that each and every cost label in $\mathcal C(P)$ for any $P$ corresponds to a block of sequential instructions, to which we can assign a constant \emph{cost}\footnote{This in fact requires the machine architecture to be `simple enough', or for some form of execution analysis to take place.}
     325We therefore may assume the existence of a \emph{cost mapping} $\kappa_P$ from cost labels to natural numbers, assigning to each cost label $\alpha$ the cost of the block containing the single occurrance of $\alpha$.
     326
     327Given any cost mapping $\kappa$, we can enrich a labelled program so that a particular fresh variable (the \emph{cost variable} $c$) keeps track of the summation of costs during the execution.
     328We call this procedure \emph{instrumentation} of the program, and it is defined recursively by:
     329$$
     330\mathcal I(\alpha:S) = c \ass c + \kappa(\alpha) ; \mathcal I(S)
     331$$
     332In all other cases the definition passes to substatements.
     333
     334\paragraph{The problem with loop optimisations}
     335Let us take loop peeling, and apply it to the labelling of a program without any prior adjustment:
     336$$
     337(\sop{while}e\sbin{do}\alpha:S);\beta:\s{skip}
     338\mapsto
     339(\sop{if}b\sbin{then} \alpha : S; \sop{while} b \sbin{do} \alpha :
     340S[\ell'_i/\ell_i]);
     341\beta:\s{skip}
     342$$
     343What happens is that the cost label $\alpha$ is duplicated with two distinct occurrences.
     344If these two occurrences correspond to different costs in the compiled code, the best the cost mapping can do is to take the maximum of the two, preserving soundness (\ie the cost estimate still bounds the actual one) but losing preciseness (\ie the actual cost could be strictly less than its estimate).
     345
     346\section{Indexed labels}
     347\label{sec:indexedlabels}
     348This section presents the core of the new approach.
     349In brief points it amounts to the following:
     350\begin{enumerate}[\bfseries~\ref*{sec:indexedlabels}.1.]
     351\item
     352\label{en:outline1}
     353Enrich cost labels with formal indices corresponding, at the beginning of the process, to which iteration of the loop they belong to.
     354\item
     355\label{en:outline2}
     356Each time a loop transformation is applied and a cost labels is split in different occurrences, each of these will be reindexed so that every time they are emitted their position in the original loop will be reconstructed.
     357\item
     358\label{en:outline3}
     359Along the compilation chain, alongside the \s{emit} instruction we add other instructions updating the indices, so that iterations of the original loops can be rebuilt at the operational semantics level.
     360\item
     361\label{en:outline4}
     362The machinery computing the cost mapping will still work, but assigning costs to indexed cost labels, rather than to cost labels as we wish.
     363However, \emph{dependent costs} can be calculated, where dependency is on which iteration of the containing loops we are in.
     364\end{enumerate}
     365
     366\subsection{Indexing the cost labels}
     367\label{ssec:indlabs}
     368
     369\paragraph{Formal indices and $\iota\ell\imp$}
     370Let $i_0,i_1,\ldots$ be a sequence of distinguished fresh identifiers that will be used as loop indices.
     371A \emph{simple expression} is an affine arithmetical expression in one of these indices, that is $a*i_k+b$ with $a,b,k \in \mathbb N$.
     372Simple expressions $e_1=a_1*i_k+b_1$, $e_2=a2*i_k+b_2$ in the same index can be composed, yielding $e_1\circ e_2\ass (a_1a_2)*i_k + (a_1b2+b_1)$, and this operation has an identity element in $id_k \ass 1*i_k+0$.
     373Constants can be expressed as simple expressions, so that we identify a natural $c$ with $0*i_k+c$.
     374
     375An \emph{indexing} (with metavariables $I$, $J$, \ldots) is a list of transformations of successive formal indices dictated by simple expressions, that is a mapping\footnote{Here we restrict each mapping to be a simple expression \emph{on the same index}.
     376This might not be the case if more loop optimisations are accounted for (for example, interchanging two nested loops).}
     377$$
     378i_0\mapsto a_0*i_0+b_0,\dots, i_{k-1} \mapsto a_{k-1}*i_{k-1}+b_{k-1}
     379$$
     380
     381An \emph{indexed cost label} (metavariables $\alphab$, $\betab$, \ldots) is the combination of a cost label $\alpha$ and an indexing $I$, written $\alpha\la I\ra$.
     382The cost label underlying an indexed one is called its \emph{atom}.
     383All plain labels can be considered as indexed ones by taking an empty indexing.
     384
     385\imp{} with indexed labels ($\iota\ell\imp$) is defined by adding to $\imp$ statements with indexed labels, and by having loops with formal indices attached to them:
     386$$
     387S,T,\ldots \gramm \cdots i_k:\sop{while}e\sbin{do}S\mid \alphab : S
     388$$
     389Note than unindexed loops still exist in the language: they will correspond to multi-entry loops which are ignored by indexing and optimisations.
     390We will discuss the semantics later.
     391
     392\paragraph{Indexed labelling}
     393Given an $\imp$ program $P$, in order to index loops and assign indexed labels, we must first distinguish single-entry loops.
     394We sketch how this can be computed in the sequel.
     395
     396A first pass of the program $P$ can easily compute two maps: $\s{loopof}_P$ from each label $\ell$ to the occurrence (\ie the path) of a $\s{while}$ loop containing $\ell$, or the empty path if none exists; and $\s{gotosof}_P$ from a label $\ell$ to the occurrences of \s{goto}s pointing to it.
     397Then the set $\s{multientry}_P$ of multi-entry loops of $P$ can be computed by
     398$$
     399\s{multientry}_P\ass\{\, p \mid \exists \ell,q.p =\s{loopof}_P(\ell),q\in\s{gotosof}_P(\ell), q \not\le p\,\}
     400$$
     401Here $\le$ is the prefix relation\footnote{Possible simplifications to this procedure include keeping track of just the while loops containing labels and \s{goto}s (rather than paths in the syntactic tree of the program), and making two passes while avoiding building the map to sets $\s{gotosof}$}.
     402
     403Let $Id_k$ be the indexing of length $k$ made from identity simple expressions, \ie the sequence $i_0\mapsto id_0, \ldots , i_{k-1}\mapsto id_{k-1}$.
     404We define the tiered indexed labelling $\Ell^\iota_P (S,k)$ in program $P$ for occurrence $S$ of a statement in $P$ and a natural $k$ by recursion, setting:
     405$$
     406\Ell^\iota_P(S,k)\ass
     407\left\{
     408\begin{array}{lh{-100pt}l}
     409 (i_k:\sop{while}b\sbin{do}\alpha\la Id_{k+1}\ra : \Ell^\iota_P(T,k+1));\beta\la Id_k \ra : \s{skip}
     410\\& \text{if $S=\sop{while}b\sbin{do}T$ and $S\notin \s{multientry}_P$,}\\[3pt]
     411(\sop{while}b\sbin{do}\alpha\la Id_k \ra : \Ell^\iota_P(T,k));\beta\la Id_k \ra : \s{skip}
     412\\& \text{otherwise, if $S=\sop{while}b\sbin{do}T$,}\\[3pt]
     413\sop{if}b\sbin{then} \alpha\la Id_k \ra : \Ell^\iota_P(T_1,k) \sbin{else} \beta\la Id_k \ra : \Ell^\iota_P(T_2,k)
     414\\&\text{if $S=\sop{if}b\sbin{then}T_1\sbin{else}T_2$,}\\[3pt]
     415\ell:\alpha\la Id_k\ra : \Ell_P^\iota(T,k) & \text{if $S = \ell : T$,}\\[3pt]
     416\ldots
     417\end{array}
     418\right.
     419$$
     420Here, as usual, $\alpha$ and $\beta$ are fresh cost labels, and other cases just keep making the recursive calls on the substatements.
     421The \emph{indexed labelling} of a program $P$ is then defined as $\alpha\la \ra : \Ell^\iota_P(P,0)$, \ie a further fresh unindexed cost label is added at the start, and we start from level $0$.
     422
     423In plainer words: each single-entry loop is indexed by $i_k$ where $k$ is the number of other single-entry loops containing this one, and all cost labels under the scope of a single-entry loop indexed by $i_k$ are indexed by all indices $i_0,\ldots,i_k$, without any transformation.
     424
     425\subsection{Indexed labels and loop transformations}\label{ssec:looptrans}
     426We define the \emph{reindexing} $I \circ (i_k\mapsto a*i_k+b)$ as an operator on indexings by setting:
     427\begin{multline*}
     428(i_0\mapsto e_0,\ldots, i_k \mapsto e_k,\ldots,i_n\mapsto e_n)
     429\circ(i_k\mapsto a*i_k+b)
     430\ass\\
     431i_0\mapsto e_0,\ldots, i_k \mapsto e_k \circ(a*i_k+b),\ldots,i_n\mapsto e_n,
     432\end{multline*}
     433We further extend to indexed labels (by $\alpha\la I\ra\circ(i_k\mapsto e)\ass \alpha\la I\circ (i_k\mapsto e)\ra$) and also to statements in $\iota\ell\imp$ (by applying the above transformation to all indexed labels).
     434
     435We can then redefine loop peeling and loop unrolling, taking into account indexed labels.
     436It will only be possible to apply the transformation to indexed loops, that is loops that are single-entry.
     437The attentive reader will notice that no assumptions are made on the labelling of the statements that are involved.
     438In particular the transformation can be repeated and composed at will.
     439Also, note that after erasing all labelling information (\ie indexed cost labels and loop indices) we recover exactly the same transformations presented in~\ref{sec:defimp}.
     440
     441\paragraph{Indexed loop peeling}
     442$$
     443i_k:\sop{while}b\sbin{do}S\mapsto
     444\sop{if}b\sbin{then} S\circ (i_k\mapsto 0); i_k : \sop{while} b \sbin{do} S[\ell'_i/\ell_i]\circ(i_k\mapsto i_k + 1)
     445$$
     446As can be expected, the peeled iteration of the loop gets reindexed, always being the first iteration of the loop, while the iterations of the remaining loop are shifted by $1$. Notice that this transformation can lower the actual depth of some loops, however their index is left untouched.
     447
     448\paragraph{Indexed loop unrolling}
     449$$
     450\begin{array}{l}
     451\begin{array}{ncn}
     452i_k:\sop{while}b\sbin{do}S\\
     453\tikz\node[rotate=-90,inner sep=0pt]{$\mapsto$};
     454\end{array}\\
     455i_k:\sop{while} b \sbin{do}\\
     456\quad (S\circ(i_k\mapsto n*i_k) ;\\
     457\quad \sop{if} b \sbin{then}\\
     458\quad\quad (S[\ell^1_i/\ell_i]\circ(i_k\mapsto n*i_k+1) ;\\
     459\quad\quad\quad \vdots \\
     460\quad\quad \quad \sop{if} b \sbin{then}\\
     461\quad \quad \quad \quad S[\ell^n_i/\ell_i]\circ(i_k\mapsto n*i_k+n-1)
     462)\cdots )
     463\end{array}
     464$$
     465Again, the reindexing is as expected: each copy of the unrolled body has its indices remapped so that when they are executed, the original iteration of the loop to which they correspond can be recovered.
     466
     467\subsection{Semantics and compilation of indexed labels}
     468In order to make sense of loop indices, one must keep track of their values in the state.
     469A \emph{constant indexing} (metavariables $C,\ldots$) is an indexing which employs only constant simple expressions.
     470The evaluation of an indexing $I$ in a constant indexing $C$, noted $I|_C$, is defined by:
     471$$
     472I\circ(i_0\mapsto c_0,\ldots, i_{k-1}\mapsto c_{k-1}) \ass \alphab\circ(i_0\mapsto c_0)\circ\cdots\circ(i_{k-1}\mapsto c_{k-1})
     473$$
     474Here, we are using the definition of ${-}\circ{-}$ given in~\ref{ssec:indlabs}.
     475We consider the above defined only if the the resulting indexing is a constant one too\footnote{For example $(i_0\mapsto 2*i_0,i_1\mapsto i_1+1)|_{i_0\mapsto 2}$ is undefined, but $(i_0\mapsto 2*i_0,i_1\mapsto 0)|_{i_0\mapsto 2}= i_0\mapsto 4,i_1\mapsto 2$, is indeed a constant indexing, even if the domain of the original indexing is not covered by the constant one.}.
     476The definition is extended to indexed labels by $\alpha\la I\ra|_C\ass \alpha\la I|_C\ra$.
     477
     478Constant indexings will be used to keep track of the exact iterations of the original code that the emitted labels belong to.
     479We thus define two basic actions to update constant indexings: $C[i_k{\uparrow}]$ increments the value of $i_k$ by one, and $C[i_k{\downarrow}0]$ resets it to $0$.
     480
     481We are ready to update the definition of the operational semantics of indexed labelled \imp.
     482The emitted cost labels will now be ones indexed by constant indexings.
     483We add a special indexed loop construct for continuations that keeps track of active indexed loop indices:
     484$$
     485K,\ldots  \gramm \cdots | i_k:\sop{while} b \sbin {do} S \sbin{then}  K
     486$$
     487The difference between the regular stack concatenation $i_k:\sop{while}b\sbin{do}S\cdot K$ and the new constructor is that the latter indicates the loop is the active one in which we already are, while the former is a loop that still needs to be started\footnote{In the presence of \s{continue} and \s{break} statements active loops need to be kept track of in any case.}.
     488The \s{find} function is updated accordingly with the case
     489$$
     490\s{find}(\ell, i_k:\sop{while}b\sbin{do}S, K) \ass \s{find}(\ell, S, i_k: \sop{while}b\sbin{do}S\sbin{then}K)
     491$$
     492The state will now be a 4-tuple $(S,K,s,C)$ which adds a constant indexing to the triple of the regular semantics.
     493The small-step rules for all statements remain the same, without touching the $C$ parameter (in particular unindexed loops behave the same as usual), apart from the ones regarding cost-labels and indexed loops.
     494The remaining cases are:
     495$$\begin{aligned}
     496   (\alphab : S,K,s,C) &\to[\alphab|_C]_P (S,K,s,C)\\
     497   (i_k:\sop{while}b\sbin{do}S,K,C) &\to[\varepsilon]_P
     498    \begin{cases}
     499     (S,i_k:\sop{while}b\sbin{do}S\sbin{then} K,s,C[i_k{\downarrow}0])
     500     \\\hskip 125pt \text{if $(b,s)\eval v\neq 0$,}\\
     501     \rlap{(\s{skip}, K, s, C)}\hskip 125pt \text{otherwise}
     502    \end{cases}\\
     503   (\s{skip}, i_k:\sop{while}b\sbin{do}S\sbin{then}K,C) &\to[\varepsilon]_P
     504    \begin{cases}
     505     (S,i_k:\sop{while}b\sbin{do}S\sbin{then} K,s,C[i_k{\uparrow}])
     506      \\\hskip 125pt \text{if $(b,s)\eval v\neq 0$,}\\
     507     \rlap{(\s{skip}, K, s, C)} \hskip 125pt \text{otherwise}
     508    \end{cases}
     509  \end{aligned}$$
     510Some explanations are in order:
     511\begin{itemize}
     512\item
     513Emitting a label always instantiates it with the current indexing.
     514\item
     515Hitting an indexed loop the first time initializes the corresponding index to 0; continuing the same loop increments the index as expected.
     516\item
     517The \s{find} function ignores the current indexing: this is correct under the assumption that all indexed loops are single entry, so that when we land inside an indexed loop with a \s{goto}, we are sure that its current index is right.
     518\item
     519The starting state with store $s$ for a program $P$ is $(P,\s{halt},s,(i_0\mapsto 0,\dots,i_{n-1}\mapsto 0)$ where $i_0,\ldots,i_{n-1}$ cover all loop indices of $P$\footnote{For a program which is the indexed labelling of an \imp{} one this corresponds to the maximum nesting of single-entry loops.
     520We can also avoid computing this value in advance if we define $C[i{\downarrow}0]$ to extend $C$'s domain as needed, so that the starting constant indexing can be the empty one.}.
     521\end{itemize}
     522
     523\paragraph{Compilation}
     524Further down the compilation chain the loop structure is usually partially or completely lost.
     525We cannot rely on it anymore to keep track of the original source code iterations.
     526We therefore add, alongside the \s{emit} instruction, two other sequential instructions $\sop{ind_reset}k$ and $\sop{ind_inc}k$ whose sole effect is to reset to 0 (resp.\ increment by 1) the loop index $i_k$, as kept track of in a constant indexing accompanying the state.
     527
     528The first step of compilation from $\iota\ell\imp$ consists of prefixing the translation of an indexed loop $i_k:\s{while}\ b\ \s{do}\ S$ with $\sop{ind_reset}k$ and postfixing the translation of its body $S$ with $\sop{ind_inc}k$.
     529Later in the compilation chain we must propagate the instructions dealing with cost labels.
     530
     531We would like to stress the fact that this machinery is only needed to give a suitable semantics of observables on which preservation proofs can be done.
     532By no means are the added instructions and the constant indexing in the state meant to change the actual (let us say denotational) semantics of the programs.
     533In this regard the two new instruction have a similar role as the \s{emit} one.
     534A forgetful mapping of everything (syntax, states, operational semantics rules) can be defined erasing all occurrences of cost labels and loop indices, and the result will always be a regular version of the language considered.
     535
     536\paragraph{Stating the preservation of semantics}
     537In fact, the statement of preservation of semantics does not change at all, if not for considering traces of evaluated indexed cost labels rather than traces of plain ones.
     538
     539\subsection{Dependent costs in the source code}
     540\label{ssec:depcosts}
     541The task of producing dependent costs from constant costs induced by indexed labels is quite technical.
     542Before presenting it here, we would like to point out that the annotations produced by the procedure described in this Subsection, even if correct, can be enormous and unreadable.
     543In Section~\ref{sec:conc}, where we detail the actual implementation, we will also sketch how we mitigated this problem.
     544
     545Having the result of compiling the indexed labelling $\Ell^\iota(P)$ of an \imp{} program $P$, we may still suppose that a cost mapping can be computed, but from indexed labels to naturals.
     546We want to annotate the source code, so we need a way to express and compute the costs of cost labels, \ie group the costs of indexed labels to ones of their atoms.
     547In order to do so we introduce \emph{dependent costs}.
     548Let us suppose that for the sole purpose of annotation, we have available in the language ternary expressions of the form
     549$$\tern e {f_1}{f_2},$$
     550and that we have access to common operators on integers such as equality, order and modulus.
     551
     552\paragraph{Simple conditions}
     553
     554First, we need to shift from \emph{transformations} of loop indices to \emph{conditions} on them.
     555We identify a set of conditions on natural numbers which are able to express the image of any composition of simple expressions.
     556\emph{Simple conditions} are of three possible forms:
     557\begin{itemize}
     558\item
     559Equality $i_k=n$ for some natural $n$.
     560\item
     561Inequality $i_k\ge n$ for some natural $n$.
     562\item
     563Modular equality together with inequality $i_k\bmod a = b\wedge i_k\ge n$ for naturals $a, b, n$.
     564\end{itemize}
     565The `always true' simple condition is given by $i_k\ge 0$.
     566We write $i_k\bmod a = b$ as a simple condition for $i_k\bmod a = b\wedge i_k\ge 0$.
     567
     568Given a simple condition $p$ and a constant indexing $C$ we can easily define when $p$ holds for $C$ (written $p\circ C$).
     569A \emph{dependent cost expression} is an expression built solely out of integer constants and ternary expressions with simple conditions at their head.
     570Given a dependent cost expression $e$ where all of the loop indices appearing in it are in the domain of a constant indexing $C$, we can define the value $e\circ C\in \mathbb N$ by:
     571$$n\circ C\ass n,\qquad (\tern p e f)\circ C\ass
     572\begin{cases}
     573  e\circ C& \text{if $p\circ C$,}\\
     574  f\circ C& \text{otherwise.}
     575\end{cases}$$
     576
     577\paragraph{From indexed costs to dependent ones}
     578Every simple expression $e$ corresponds to a simple condition $p(e)$ which expresses the set of values that $e$ can take.
     579Following is the definition of such a relation.
     580We recall that in this development, loop indices are always mapped to simple expressions over the same index.
     581If it was not the case, the condition obtained from an expression should be on the mapped index, not the indeterminate of the simple expression.
     582We leave all generalisations of what we present here for further work:
     583$$
     584p(a*i_k+b)\ass
     585\begin{cases}
     586i_k = b & \text{if $a = 0$,}\\
     587i_k \ge b & \text{if $a = 1$,}\\
     588i_k\bmod a = b' \wedge i_k \ge b & \text{otherwise, where $b' = b\bmod a$}.
     589\end{cases}
     590$$
     591Now, suppose we are given a mapping $\kappa$ from indexed labels to natural numbers.
     592We will transform it in a mapping (identified, via abuse of notation, with the same symbol $\kappa$) from atoms to \imp{} expressions built with ternary expressions which depend solely on loop indices.
     593To that end we define an auxiliary function $\kappa^\alpha_L$, parameterized by atoms and words of simple expressions, and defined on \emph{sets} of $n$-uples of simple expressions (with $n$ constant across each such set, \ie each set is made of words all with the same length).
     594
     595We will employ a bijection between words of simple expressions and indexings, given by:\footnote{Lists of simple expressions are in fact how indexings are -represented in CerCo's current implementation of the compiler.}
     596$$
     597i_0\mapsto e_0,\ldots,i_{k-1}\mapsto e_{k-1} \cong e_0\cdots e_{k-1}.
     598$$
     599As usual, $\varepsilon$ denotes the empty word/indexing, and juxtaposition is used to denote word concatenation.
     600
     601For every set $s$ of $n$-uples of simple expressions, we are in one of the following three exclusive cases:
     602\begin{itemize}
     603\item
     604$S=\emptyset$.
     605\item
     606$S=\{\varepsilon\}$.
     607\item
     608There is a simple expression $e$ such that $S$ can be decomposed in $eS'+S''$, with $S'\neq \emptyset$ and none of the words in $S''$ starting with $e$.
     609\end{itemize}
     610Here $eS'$ denotes prepending $e$ to all elements of $S'$ and $+$ is disjoint union.
     611This classification can serve as the basis of a definition by recursion on $n+\card S$ where $n$ is the size of tuples in $S$ and $\card S$ is its cardinality.
     612Indeed in the third case in $S'$ the size of tuples decreases strictly (and cardinality does not increase) while for $S''$ the size of tuples remains the same but cardinality strictly decreases.
     613The expression $e$ of the third case will be chosen as minimal for some total order\footnote{The specific order used does not change the correctness of the procedure, but different orders can give more or less readable results. A ``good'' order is the lexicographic one, with $a*i_k+b \le a'*i_k+b'$ if $a<a'$ or $a=a'$ and $b\le b'$.}.
     614
     615Following is the definition of the auxiliary function $\kappa^\alpha_L$, which follows the recursion scheme presented above:
     616$$
     617\begin{aligned}
     618\kappa^\alpha_L(\emptyset) &\ass 0\\
     619\kappa^\alpha_L(\{\varepsilon\}) &\ass \kappa(\alpha\la L\ra) \\
     620\kappa^\alpha_L(eS'+S'') &\ass \tern{p(e)}{\kappa^\alpha_{Le}(S')}{\kappa^\alpha_L(S'')}
     621\end{aligned}
     622$$
     623\noindent
     624Finally, the wanted dependent cost mapping is defined by
     625$$
     626\kappa(\alpha)\ass\kappa^\alpha_\varepsilon(\{\,L\mid \alpha\la L\ra \text{ appears in the compiled code}\,\})
     627$$
     628
     629\paragraph{Indexed instrumentation}
     630The \emph{indexed instrumentation} generalises the instrumentation presented in~\ref{sec:labelling}.
     631We described above how cost atoms can be mapped to dependent costs.
     632The instrumentation must also insert code dealing with the loop indices.
     633As instrumentation is done on the code produced by the labelling phase, all cost labels are indexed by identity indexings.
     634The relevant cases of the recursive definition (supposing $c$ is the cost variable) are then:
     635$$
     636\begin{aligned}
     637\mathcal I^\iota(\alpha\la Id_k\ra:S) &= c\ass c + \kappa(\alpha);\mathcal I^\iota(S)\\
     638\mathcal I^\iota(i_k : \sop{while}b\sbin{do}S) &=
     639  i_k \ass 0; \sop{while}b\sbin{do}(\mathcal I^\iota (S); i_k \ass i_k + 1)
     640\end{aligned}
     641$$
     642
     643\subsection{A detailed example}\label{ssec:detailedex}
     644Take the program in \autoref{fig:example1}. Its initial labelling will be:
     645$$\begin{array}{l}
     646\alpha\la\ra : s\ass 0;\\
     647i\ass 0;\\
     648i_0:\sop{while}i<n\sbin{do}\\
     649\inde \beta\la i_0\ra : p\ass 1;\\
     650\inde j\ass 1;\\
     651\inde i_1:\sop{while}j \le i\sbin{do}\\
     652\inde \inde \gamma\la i_0, i_1\ra : p\ass j*p\\
     653\inde \inde j\ass j+1;\\
     654\inde \delta\la i_0\ra : s\ass s+p;\\
     655\inde i\ass i+1;\\
     656\epsilon\la\ra:\s{skip}
     657\end{array}
     658$$
     659(a single \s{skip} after the $\delta$ label has been suppressed, and we are using the identification
     660between indexings and tuples of simple expressions explained in \autoref{ssec:depcosts}).
     661Supposing for example, $n=3$
     662the trace of the program will be
     663$$\alpha\la\ra\,\beta\la 0 \ra\, \delta\la 0\ra\,\beta\la 1\ra\,\gamma\la 1,0\ra\,
     664\delta\la 1\ra\,\beta\la 2\ra\,\gamma\la 2,0\ra\,\gamma\la 2, 1\ra\,\delta\la 2\ra\,
     665\epsilon\la\ra$$
     666Now let as apply the transformations of \autoref{fig:example1} with the additional
     667information detailed in \autoref{ssec:looptrans}. The result is shown in
     668\autoref{fig:example2}.
     669\begin{figureplr}
     670$$
     671\begin{array}{l}
     672\mbox{\color{blue}\boldmath$\alpha\la\ra $}:s\ass 0;\\
     673i\ass 0;\\
     674\tikztargetm{a}{\s{if}}\ i<n\sbin{then}\\
     675\inde \mbox{\color{blue}\boldmath$\beta\la0\ra $}:p\ass 1;\\
     676\inde j\ass 1;\\
     677\inde i_1:\sop{while}j \le i\sbin{do}\\
     678\inde \inde \mbox{\color{blue}\boldmath$\gamma\la 0, i_1\ra $}:p\ass j*p\\
     679\inde \inde j\ass j+1;\\
     680\inde \mbox{\color{blue}\boldmath$\delta\la 0\ra $}: s\ass s+p;\\
     681\inde i\ass i+1;\\
     682\inde i_0:\tikztargetm{d}{\s{while}}\ i<n\sbin{do}\\
     683\inde \inde \mbox{\color{blue}\boldmath$\beta\la 2*i_0+1\ra $}:p\ass 1;\\
     684\inde \inde j\ass 1;\\
     685\inde \inde \tikztargetm{b}{\s{if}}\ j \le i\sbin{then}\\
     686\inde \inde \inde \mbox{\color{blue}\boldmath$\gamma\la  2*i_0+1, 0\ra $}:p\ass j*p\\
     687\inde \inde \inde j\ass j+1;\\
     688\inde \inde \inde \sop{if}j \le i\sbin{then}\\
     689\inde \inde \inde \inde \mbox{\color{blue}\boldmath$\gamma\la  2*i_0+1, 1\ra $}: p\ass j*p\\
     690\inde \inde \inde \inde j\ass j+1;\\
     691\inde \inde \inde \inde i_1:\s{while}\ j \le i\sbin{do}\\
     692\inde \inde \inde \inde \inde \mbox{\color{blue}\boldmath$\gamma\la  2*i_0+1, 2*i_1 + 2 \ra $}:p\ass j*p\\
     693\inde \inde \inde \inde \inde j\ass j+1;\\
     694\inde \inde \inde \inde \inde \sop{if}j \le i\sbin{then}\\
     695\inde \inde \inde \inde \inde \inde \mbox{\color{blue}\boldmath$\gamma\la  2*i_0+1, 2*i_1 + 3\ra $}:p\ass j*p\\
     696\inde \inde \inde \inde \inde \inde \tikztargetm{c}j\ass j+1;\\
     697\inde \inde \mbox{\color{blue}\boldmath$\delta\la 2*i_0+1\ra$}: s\ass s+p;\\
     698\inde \inde i\ass i+1;\\
     699\inde \inde \sop{if}i<n\sbin{then}\\
     700\inde \inde \inde \mbox{\color{blue}\boldmath$\beta\la 2*i_0+2\ra $}:p\ass 1;\\
     701\inde \inde \inde j\ass 1;\\
     702\inde \inde \inde i_1:\tikztargetm{e}{\s{while}}\ j < i\sbin{do}\\
     703\inde \inde \inde \inde \mbox{\color{blue}\boldmath$\gamma\la 2*i_0+2, 2*i_1\ra$}: p\ass j*p\\
     704\inde \inde \inde \inde j\ass j+1;\\
     705\inde \inde \inde \inde \s{if}\ j < i\sbin{then}\\
     706\inde \inde \inde \inde \inde \mbox{\color{blue}\boldmath$\gamma\la2*i_0+2, 2*i_1+1\ra$}: p\ass j*p\\
     707\inde \inde \inde \inde \inde \tikztargetm{f}j\ass j+1;\\
     708\inde \inde \inde \mbox{\color{blue}\boldmath$\delta\la 2*i_0+2\ra $}: s\ass s+p;\\
     709\inde \inde \inde i\ass i+1\tikztargetm{g};{}\\
     710\mbox{\color{blue}\boldmath$\epsilon\la\ra $}:\s{skip}
     711\end{array}$$
     712\caption{The result of applying reindexing loop transformations on the
     713program in \autoref{fig:example1}.}\label{fig:example2}
     714\end{figureplr}
     715One can check that the transformed code leaves the same trace when executed.
     716
     717Now let us compute the dependent cost of $\gamma$, supposing no other loop transformations
     718are done. Ordering its indexings we
     719have the following list:
     720\begin{equation}
     721\label{eq:inds}
     722\begin{aligned}
     723  &0, i_1\\
     724  &2*i_0+1, 0\\
     725  &2*i_0+1, 1\\
     726  &2*i_0+1, 2*i_1+2\\
     727  &2*i_0+1, 2*i_1+3\\
     728  &2*i_0+2, 2*i_1\\
     729  &2*i_0+2, 2*i_1+1
     730  \end{aligned}
     731\end{equation}
     732
     733The resulting dependent cost will then be
     734\def\indetern#1#2#3{\begin{tabular}[t]{nL}(#1)\mathrel ?{}\\\hskip 15pt #2:{}\\\hskip 15pt #3\end{tabular}}
     735\def\tern#1#2#3{(#1)\mathrel ? #2:#3}
     736\begin{equation}\label{eq:ex}
     737\kappa^\iota(\gamma)=
     738\indetern{i_0 = 0}
     739  {\tern{i_1\ge 0}a0}
     740  {\indetern{i_0\bmod 2 = 1 \wedge i_0\ge 1}
     741    {\indetern{i_1=0}
     742      b
     743      {\indetern{i_1 = 1}
     744        c
     745        {\indetern{i_1\bmod 2 = 0 \wedge i_1\ge 2}
     746          d
     747          {\tern{i_1\bmod 2 = 1 \wedge i_1\ge 3}e0}
     748        }
     749      }
     750    }
     751    {\indetern{i_0\bmod 2 = 0 \wedge i_0\ge 2}
     752      {\indetern{i_1 \bmod 2 = 0 \wedge i_1 \ge 0}
     753        f
     754        {\tern{i_1 \bmod 2 = 1 \wedge i_1 \ge 1}g0}
     755      }
     756      0
     757    }
     758  }
     759\end{equation}
     760We will see later on \autopageref{pag:continued} how such an expression can be simplified.
     761\section{Notes on the implementation and further work}
     762\label{sec:conc}
     763Implementing the indexed label approach in CerCo's untrusted Ocaml prototype does not introduce many new challenges beyond what has already been presented for the toy language, \imp{} with \s{goto}s.
     764\s{Clight}, the \s{C} fragment source language of CerCo's compilation chain~\cite{D2.1}, has several more fetaures, but few demand changes in the indexed labelled approach.
     765
     766\paragraph{Indexed loops \emph{vs}. index update instructions}
     767In our presentation we have indexed loops in $\iota\ell\imp$, while we hinted that later languages in the compilation chain would have specific index update instructions.
     768In CerCo's actual compilation chain from \s{Clight} to 8051 assembly, indexed loops are only in \s{Clight}, while from \s{Cminor} onward all languages have the same three cost-involving instructions: label emitting, index resetting and index incrementing.
     769
     770\paragraph{Loop transformations in the front end}
     771We decided to implement the two loop transformations in the front end, namely in \s{Clight}.
     772This decision is due to user readability concerns: if costs are to be presented to the programmer, they should depend on structures written by the programmer himself.
     773If loop transformation were performed later it would be harder to create a correspondence between loops in the control flow graph and actual loops written in the source code.
     774However, another solution would be to index loops in the source code and then use these indices later in the compilation chain to pinpoint explicit loops of the source code: loop indices can be used to preserve such information, just like cost labels.
     775
     776\paragraph{Break and continue statements}
     777\s{Clight}'s loop flow control statements for breaking and continuing a loop are equivalent to appropriate \s{goto} statements.
     778The only difference is that we are assured that they cannot cause loops to be multi-entry, and that when a transformation such as loop peeling is complete, they need to be replaced by actual \s{goto}s (which happens further down the compilation chain anyway).
     779
     780\paragraph{Function calls}
     781Every internal function definition has its own space of loop indices.
     782Executable semantics must thus take into account saving and resetting the constant indexing of current loops upon hitting a function call, and restoring it upon return of control.
     783A peculiarity is that this cannot be attached to actions that save and restore frames: namely in the case of tail calls the constant indexing needs to be saved whereas the frame does not.
     784
     785\paragraph{Cost-labelled expressions}
     786In labelled \s{Clight}, expressions also get cost labels, due to the presence of ternary conditional expressions (and lazy logical operators, which get translated to ternary expressions too).
     787Adapting the indexed labelled approach to cost-labelled expressions does not pose any particular problems.
     788
     789\paragraph{Simplification of dependent costs}
     790As previously mentioned, the na\"{i}ve application of the procedure described in~\ref{ssec:depcosts} produces unwieldy cost annotations.
     791In our implementation several transformations are used to simplify such complex dependent costs.
     792
     793Disjunctions of simple conditions are closed under all logical operations, and it can be computed whether such a disjunction implies a simple condition or its negation.
     794This can be used to eliminate useless branches of dependent costs, to merge branches that share the same value, and possibly to simplify the third case of simple condition.
     795Examples of the three transformations are respectively:
     796\begin{itemize}
     797\item $
     798\verb+(_i_0 == 0)?+x\verb+:(_i_0 >= 1)?+y\verb+:+z
     799\mapsto
     800\verb+(_i_0 == 0)?+x\verb+:+y,
     801$
     802\item $
     803c\texttt{?}x\verb+:(+d\texttt{?}x\texttt{:}y\verb+)+
     804\mapsto
     805\texttt{(}c\texttt{ || }d\texttt{)?}x\texttt{:}y,
     806$
     807\item \begin{tabular}[t]{np{\linewidth}n}
     808$\verb+(_i_0 == 0)?+x\verb+:(_i_0 % 2 == 0 && _i_0 >= 2)?+y\verb+:+z
     809\mapsto$ \\\hfill
     810$\verb+(_i_0 == 0)?+x\verb+:(_i_0 % 2 == 0)?+y\verb+:+z.
     811$\end{tabular}
     812\end{itemize}
     813The second transformation tends to accumulate disjunctions, to the detriment of readability.
     814A further transformation swaps two branches of the ternary expression if the negation of the condition can be expressed with fewer clauses.
     815For example:
     816$$ \verb+(_i_0 % 3 == 0 || _i_0 % 3 == 1)?+x\verb+:+y \mapsto
     817\verb+(_i_0 % 3 == 2)?+y\verb+:+x.
     818$$
     819Picking up again the example depicted in \autoref{ssec:detailedex}, \label{pag:continued}
     820we can see that the cost in \eqref{eq:ex} can be simplified to the following,
     821using some of the transformation described above:
     822$$
     823\kappa^\iota(\gamma)=
     824\indetern{i_0 = 0}
     825  a
     826  {\indetern{i_0\bmod 2 = 1}
     827    {\indetern{i_1=0}
     828      b
     829      {\indetern{i_1 = 1}
     830        c
     831        {\indetern{i_1\bmod 2 = 0}
     832          de
     833        }
     834      }
     835    }
     836    {\indetern{i_1 \bmod 2 = 0}
     837      fg
     838    }
     839  }
     840$$
     841One should keep in mind that the example was wilfully complicated, in practice
     842the cost expressions produced have rarely more clauses
     843than the number of nested loops containing the annotation.
     844\paragraph{Updates to the frama-C cost plugin}
     845Cerco's frama-C~\cite{framac} cost plugin\todo{is there a reference for this?}{} has been updated to take into account our new notion of dependent costs.
     846The frama-c framework expands ternary expressions to branch statements, introducing temporaries along the way.
     847This makes the task of analyzing ternary cost expressions rather daunting.
     848It was deemed necessary to provide an option in the compiler to use actual branch statements for cost annotations rather than ternary expressions, so that at least frama-C's use of temporaries in cost annotation could be avoided.
     849The cost analysis carried out by the plugin now takes into account such dependent costs.
     850
     851The only limitation (which actually simplifies the code) is that, within a dependent cost, simple conditions with modulus on the same loop index should not be modulo different numbers.
     852This corresponds to a reasonable limitation on the number of times loop unrolling may be applied to the same loop: at most once.
     853
     854\paragraph{Further work}
     855For the time being, indexed labels are only implemented in the untrusted Ocaml compiler, while they are not present yet in the Matita code.
     856Porting them should pose no significant problem.
     857Once ported, the task of proving properties about them in Matita can begin.
     858
     859Because most of the executable operational semantics of the languages across the frontend and the backend are oblivious to cost labels, it should be expected that the bulk of the semantic preservation proofs that still needs to be done will not get any harder because of indexed labels.
     860The only trickier point that we foresee would be in the translation of \s{Clight} to \s{Cminor}, where we pass from structured indexed loops to atomic instructions on loop indices.
     861
     862An invariant which should probably be proved and provably preserved along the compilation chain is the non-overlap of indexings for the same atom.
     863Then, supposing cost correctness for the unindexed approach, the indexed one will just need to amend the proof that
     864$$\forall C\text{ constant indexing}.\forall \alpha\la I\ra\text{ appearing in the compiled code}.
     865  \kappa(\alpha)\circ (I\circ C) = \kappa(\alpha\la I \ra).
     866$$
     867Here, $C$ represents a snapshot of loop indices in the compiled code, while $I\circ C$ is the corresponding snapshot in the source code.
     868Semantics preservation will ensure that when, with snapshot $C$, we emit $\alpha\la I\ra$ (that is, we have $\alpha\la I\circ C\ra$ in the trace), $\alpha$ must also be emitted in the source code with indexing $I\circ C$, so the cost $\kappa(\alpha)\circ (I\circ C)$ applies.
     869
     870Aside from carrying over the proofs, we would like to extend the approach to more loop transformations.
     871Important examples are loop inversion (where a for loop is reversed, usually to make iterations appear to be truly independent) or loop interchange (where two nested loops are swapped, usually to have more loop invariants or to enhance strength reduction).
     872This introduces interesting changes to the approach, where we would have indexings such as:
     873$$i_0\mapsto n - i_0\quad\text{or}\quad i_0\mapsto i_1, i_1\mapsto i_0.$$
     874In particular dependency over actual variables of the code would enter the frame, as indexings would depend on the number of iterations of a well-behaving guarded loop (the $n$ in the first example).
     875
     876Finally, as stated in the introduction, the approach should allow some integration of techniques for cache analysis, a possibility that for now has been put aside as the standard 8051 target architecture for the CerCo project lacks a cache.
     877Two possible developments for this line of work present themselves:
     878\begin{enumerate}
     879\item
     880One could extend the development to some 8051 variants, of which some have been produced with a cache.
     881\item
     882One could make the compiler implement its own cache: this cannot apply to \textsc{ram} accesses of the standard 8051 architecture, as the difference in cost of accessing the two types of \textsc{ram} is only one clock cycle, which makes any implementation of cache counterproductive.
     883So for this proposal, we could either artificially change the accessing cost of \textsc{ram} of the model just for the sake of possible future adaptations to other architectures, or otherwise model access to an external memory by means of the serial port.
     884\end{enumerate}
    7885
    8886\section{Compiler proof}
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