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doc/aaron_comp_II/comp_II.tex
rce76eb9 re394618 37 37 \setlength{\headsep}{0.25in} 38 38 39 \usepackage{caption} 40 \usepackage{subcaption} 41 39 42 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 40 43 … … 62 65 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 63 66 64 \newcommand{\bigO}[1]{O\ left( #1 \right)}67 \newcommand{\bigO}[1]{O\!\left( #1 \right)} 65 68 66 69 \begin{document} … … 404 407 If crossargument resolution dependencies cannot be completely eliminated, effective caching strategies to reduce duplicated work between equivalent argumentparameter matches in different combinations may mitigate the asymptotic defecits of the wholecombination matching approach. 405 408 The final area of investigation is heuristics and algorithmic approaches to reduce the number of argument interpretations considered in the common case; if argumentparameter matches cannot be made independent, even small reductions in $i$ should yield significant reductions in the $i^{p+1}$ resolver runtime factor. 409 406 410 The discussion below presents a number of largely orthagonal axes for expression resolution algorithm design to be investigated, noting prior work where applicable. 411 Though some of the proposed improvements to the expression resolution algorithm are based on heuristics rather than asymptoticly superior algorithms, it should be noted that user programmers often employ idioms and other programming patterns to reduce the mental burden of producing correct code, and if these patterns can be identified and exploited by the compiler then the significant reduction in expression resolution time for common, idiomatic expressions should result in lower total compilation time even for code including difficulttoresolve expressions that push the expression resolver to its theoretical worst case. 407 412 408 413 \subsection{ArgumentParameter Matching} 409 The first axis we consider is argumentparameter matching  whether the type matching for a candidate function to a set of candidate arguments is directed by the argument types or the parameter types. 410 411 \subsubsection{Argumentdirected (``Bottomup'')} 412 Baker's algorithm for expression resolution\cite{Baker82} precomputes argument candidates, from the leaves of the expression tree up. 414 The first axis for consideration is argumentparameter matching direction  whether the type matching for a candidate function to a set of candidate arguments is directed by the argument types or the parameter types. 415 All expression resolution algorithms form a DAG of interpretations, some explicitly, some implicitly; in this DAG, arcs point from functioncall interpretations to argument interpretations, as below: 416 \begin{figure}[h] 417 \centering 418 \begin{subfigure}[h]{2in} 419 \begin{lstlisting} 420 int *p; // $p_i$ 421 char *p; // $p_c$ 422 423 double *f(int*, int*); // $f_d$ 424 char *f(char*, char*); // $f_c$ 425 426 f( f( p, p ), p ); 427 \end{lstlisting} 428 \end{subfigure}~\begin{subfigure}[h]{2in} 429 \includegraphics{resolution_dag} 430 \end{subfigure} 431 \caption{Resolution DAG for a simple expression. Functions that do not have a valid argument matching are covered with an \textsf{X}.}\label{fig:res_dag} 432 \end{figure} 433 434 Note that some interpretations may be part of more than one superinterpretation, as with $p_i$ in the bottom row, while some valid subexpression interpretations, like $f_d$ in the middle row, are not used in any interpretation of their containing expression. 435 436 \subsubsection{Argumentdirected (Bottomup)} 437 Baker's algorithm for expression resolution~\cite{Baker82} precomputes argument candidates, from the leaves of the expression tree up. 413 438 For each candidate function, Baker attempts to match argument types to parameter types in sequence, failing if any parameter cannot be matched. 414 439 415 Bilson \cite{Bilson03} similarly precomputes argument candidates in the original \CFA compiler, but then explicitly enumerates all possible argument combinations for a multiparameter function; these argument combinations are matched to the parameter types of the candidate function as a unit rather than individual arguments.416 This is less efficient than Baker's approach, as the same argument may be compared to the same parameter many times, but allows a more straightforward handling of polymorphic type binding and multiple returntypes.417 It is possible the efficiency losses here relative to Baker could be significantly reduced by application of memoization to the argumentparameter type comparisons.418 419 \subsubsection{Parameterdirected ( ``Topdown'')}420 Unlike Baker and Bilson, Cormack's algorithm \cite{Cormack81} requests argument candidates whichmatch the type of each parameter of each candidate function, from the toplevel expression down; memoization of these requests is presented as an optimization.440 Bilson~\cite{Bilson03} similarly precomputes argument candidates in the original \CFA compiler, but then explicitly enumerates all possible argument combinations for a multiparameter function; these argument combinations are matched to the parameter types of the candidate function as a unit rather than individual arguments. 441 This approach is less efficient than Baker's approach, as the same argument may be compared to the same parameter many times, but allows a more straightforward handling of polymorphic typebinding and multiple returntypes. 442 It is possible the efficiency losses here relative to Baker could be significantly reduced by keeping a memoized cache of argumentparameter type comparisons and reading previouslyseen argumentparameter matches from this cache rather than recomputing them. 443 444 \subsubsection{Parameterdirected (Topdown)} 445 Unlike Baker and Bilson, Cormack's algorithm~\cite{Cormack81} requests argument candidates that match the type of each parameter of each candidate function, from the toplevel expression down; memoization of these requests is presented as an optimization. 421 446 As presented, this algorithm requires the result of the expression to have a known type, though an algorithm based on Cormack's could reasonably request a candidate set of any return type, though such a set may be quite large. 422 447 423 448 \subsubsection{Hybrid} 424 449 This proposal includes the investigation of hybrid topdown/bottomup argumentparameter matching. 425 A reasonable hybrid approach might be to take a topdown approach when the expression to be matched is known to have a fixed type, and a bottomup approach in untyped contexts. 426 This may include switches from one type to another at different levels of the expression tree, for instance: 450 A reasonable hybrid approach might take a topdown approach when the expression to be matched has a fixed type, and a bottomup approach in untyped contexts. 451 This approach may involve switching from one type to another at different levels of the expression tree. 452 For instance: 427 453 \begin{lstlisting} 428 454 forall(otype T) … … 433 459 int x = f( f( '!' ) ); 434 460 \end{lstlisting} 435 Here, the outer call to ©f© must have a return type that is (implicitly convertable to) ©int©, so a topdown approach could be used to select \textit{(1)} as the proper interpretation of ©f©. \textit{(1)}'s parameter ©x© here, however, is an unbound type variable, and can thus take a value of any complete type, providing no guidance for the choice of candidate for the inner ©f©. The leaf expression ©'!'©, however, gives us a zerocost interpretation of the inner ©f© as \textit{(2)}, providing a minimalcost expression resolution where ©T© is bound to ©void*©. 436 437 Deciding when to switch between bottomup and topdown resolution in a hybrid algorithm is a necessarily heuristic process, and though finding good heuristics for it is an open question, one reasonable approach might be to switch from topdown to bottomup when the number of candidate functions exceeds some threshold. 461 The outer call to ©f© must have a return type that is (implicitly convertable to) ©int©, so a topdown approach is used to select \textit{(1)} as the proper interpretation of ©f©. \textit{(1)}'s parameter ©x©, however, is an unbound type variable, and can thus take a value of any complete type, providing no guidance for the choice of candidate for the inner call to ©f©. The leaf expression ©'!'©, however, determines a zerocost interpretation of the inner ©f© as \textit{(2)}, providing a minimalcost expression resolution where ©T© is bound to ©void*©. 462 463 Deciding when to switch between bottomup and topdown resolution to minimize wasted work in a hybrid algorithm is a necessarily heuristic process, and though finding good heuristics for which subexpressions to swich matching strategies on is an open question, one reasonable approach might be to set a threshold $t$ for the number of candidate functions, and to use topdown resolution for any subexpression with fewer than $t$ candidate functions, to minimize the number of unmatchable argument interpretations computed, but to use bottomup resolution for any subexpression with at least $t$ candidate functions, to reduce duplication in argument interpretation computation between the different candidate functions. 464 465 \subsubsection{Common Subexpression Caching} 466 With any of these argumentparameter approaches, it may be a useful optimization to cache the resolution results for common subexpressions; in Figure~\ref{fig:res_dag} this optimization would result in the list of interpretations $[p_c, p_i]$ for ©p© only being calculated once, and reused for each of the three instances of ©p©. 438 467 439 468 \subsection{Implicit Conversion Application} 440 Baker's \cite{Baker82} and Cormack's\cite{Cormack81}algorithms do not account for implicit conversions\footnote{Baker does briefly comment on an approach for handling implicit conversions.}; both assume that there is at most one valid interpretation of a given expression for each distinct type.469 Baker's and Cormack's algorithms do not account for implicit conversions\footnote{Baker does briefly comment on an approach for handling implicit conversions.}; both assume that there is at most one valid interpretation of a given expression for each distinct type. 441 470 Integrating implicit conversion handling into their algorithms provides some choice of implementation approach. 442 471
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