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\title{Demand-Driven Indexing of Prolog Clauses}
\titlerunning{Demand-Driven Indexing of Prolog Clauses}

\author{V\'{\i}tor Santos Costa\inst{1} \and Konstantinos
  Sagonas\inst{2} \and Ricardo Lopes\inst{1}}
\authorrunning{V. Santos Costa, K. Sagonas and R. Lopes}

\institute{
  University of Porto, Portugal
  \and
  National Technical University of Athens, Greece
}

\begin{document}
\maketitle

\begin{abstract}
  As logic programming applications grow in size, Prolog systems need
  to efficiently access larger and larger data sets and the need for
  any- and multi-argument indexing becomes more and more profound.
  Static generation of multi-argument indexing is one alternative, but
  applications often rely on features that are inherently dynamic
  (e.g., generating hypotheses for ILP data sets during runtime) which
  makes static techniques inapplicable or inaccurate. Another
  alternative, which has not been investigated so far, is to employ
  dynamic schemes for flexible demand-driven indexing of Prolog
  clauses. We propose such schemes and discuss issues that need to be
  addressed for their efficient implementation in the context of
  WAM-based Prolog systems. We have implemented demand-driven indexing
  in two different Prolog systems and have been able to obtain
  non-negligible performance speedups: from a few percent up to orders
  of magnitude. Given these results, we see very little reason for
  Prolog systems not to incorporate some form of dynamic indexing
  based on actual demand. In fact, we see demand-driven indexing as
  the first step towards effective runtime optimization of Prolog
  programs.
\end{abstract}


\section{Introduction}
%=====================
The WAM~\cite{Warren83} has mostly been a blessing but occasionally
also a curse for Prolog systems. Its ingenious design has allowed
implementors to get byte code compilers with decent performance --- it
is not a fluke that most Prolog systems are still based on the WAM. On
the other hand, \emph{because} the WAM gives good performance in many
cases, implementors have not incorporated in their systems many
features that drastically depart from WAM's basic characteristics.
%
For example, first argument indexing is sufficient for many Prolog
applications. However, it is clearly sub-optimal for applications
accessing large databases; for a long time now, the database community
has recognized that good indexing is the basis for fast query
processing.

As logic programming applications grow in size, Prolog systems need to
efficiently access larger and larger data sets and the need for any-
and multi-argument indexing becomes more and more profound. Static
generation of multi-argument indexing is one alternative. The problem
is that this alternative is often unattractive because it may
drastically increase the size of the generated byte code and do so
unnecessarily. Static analysis can partly address this concern, but in
applications that rely on features which are inherently dynamic (e.g.,
generating hypotheses for inductive logic programming data sets during
runtime) static analysis is inapplicable or grossly inaccurate.
Another alternative, which has not been investigated so far, is to do
flexible indexing on demand during program execution.

This is precisely what we advocate with this paper. More specifically,
we present a small extension to the WAM that allows for flexible
indexing of Prolog clauses during runtime based on actual demand. For
static predicates, the scheme we propose is partly guided by the
compiler; for dynamic code, besides being demand-driven by queries,
the method needs to cater for code updates during runtime. Where our
schemes radically depart from current practice is that they generate
new byte code during runtime, in effect doing a form of just-in-time
compilation. In our experience these schemes pay off. We have
implemented \JITI in two different Prolog systems (Yap and XXX) and
have obtained non-trivial speedups, ranging from a few percent to
orders of magnitude, across a wide range of applications. Given these
results, we see very little reason for Prolog systems not to
incorporate some form of indexing based on actual demand from queries.
In fact, we see \JITI as only the first step towards effective runtime
optimization of Prolog programs.

This paper is structured as follows. After commenting on the state of
the art and related work concerning indexing in Prolog systems
(Sect.~\ref{sec:related}) we briefly review indexing in the WAM
(Sect.~\ref{sec:prelims}). We then present \JITI schemes for static
(Sect.~\ref{sec:static}) and dynamic (Sect.~\ref{sec:dynamic})
predicates, and discuss their implementation in two Prolog systems and
the performance benefits they bring (Sect.~\ref{sec:perf}). The paper
ends with some concluding remarks.


\section{State of the Art and Related Work} \label{sec:related}
%==============================================================
% Indexing in Prolog systems:
Even nowadays, some Prolog systems are still influenced by the WAM and
only support indexing on the main functor symbol of the first
argument. Some others, like YAP~\cite{YAP}, can look inside compound
terms. SICStus Prolog supports \emph{shallow
backtracking}~\cite{ShallowBacktracking@ICLP-89}; choice points are
fully populated only when it is certain that execution will enter the
clause body. While shallow backtracking avoids some of the performance
problems of unnecessary choice point creation, it does not offer the
full benefits that indexing can provide. Other systems like
BIM-Prolog~\cite{IndexingProlog@NACLP-89}, SWI-Prolog~\cite{SWI} and
XSB~\cite{XSB} allow for user-controlled multi-argument indexing (via
an \code{:-~index} directive). Notably, ilProlog~\cite{ilProlog} uses
compile-time heuristics and generates code for multi-argument indexing
automatically. In all these systems, this support comes with various
implementation restrictions. For example, in SWI-Prolog at most four
arguments can be indexed; in XSB the compiler does not offer
multi-argument indexing and the predicates need to be asserted
instead; we know of no system where multi-argument indexing looks
inside compound terms. More importantly, requiring users to specify
arguments to index on is neither user-friendly nor guarantees good
performance results.

% Trees, tries and unification factoring:
Recognizing the need for better indexing, researchers have proposed
more flexible index mechanisms for Prolog. For example, Hickey and
Mudambi proposed \emph{switching trees}~\cite{HickeyMudambi@JLP-89},
which rely on the presence of mode information. Similar proposals were
put forward by Van Roy, Demoen and Willems who investigated indexing
on several arguments in the form of a \emph{selection tree}~\cite{VRDW87}
and by Zhou et al.\ who implemented a \emph{matching tree} oriented
abstract machine for Prolog~\cite{TOAM@ICLP-90}. For static
predicates, the XSB compiler offers support for \emph{unification
factoring}~\cite{UnifFact@POPL-95}; for asserted code, XSB can
represent databases of facts using \emph{tries}~\cite{Tries@JLP-99}
which provide left-to-right multi-argument indexing. However, in XSB
none of these mechanisms is used automatically; instead the user has
to specify appropriate directives.

% Comparison with static analysis techniques and Mercury:
Long ago, Kliger and Shapiro argued that such tree-based indexing
schemes are not cost effective for the compilation of Prolog
programs~\cite{KligerShapiro@ICLP-88}. Some of their arguments make
sense for certain applications, but in general we disagree with their
conclusion because they underestimate the benefits of indexing on
large datasets. Nevertheless, it is true that unless the modes of
predicates are known we run the risk of doing indexing on output
arguments, whose only effect is an unnecessary increase in compilation
times and, more importantly, in code size. In a programming language
like Mercury~\cite{Mercury@JLP-96} where modes are known the compiler
can of course avoid this risk; indeed in Mercury modes (and types) are
used to guide the compiler generate good indexing tables. However, the
situation is different for a language like Prolog. Getting accurate
information about the set of all possible modes of predicates requires
a global static analyzer in the compiler --- and most Prolog systems
do not come with one. More importantly, it requires a lot of
discipline from the programmer (e.g., that applications use the module
system religiously and never bypass it). As a result, most Prolog
systems currently do not provide the type of indexing that
applications require. Even in systems like Ciao~\cite{Ciao@SCP-05},
which do come with built-in static analysis and more or less force
such a discipline on the programmer, mode information is not used for
multi-argument indexing!

% The grand finale:
The situation is actually worse for certain types of Prolog
applications. For example, consider applications in the area of
inductive logic programming. These applications on the one hand have
big demands for effective indexing since they need to efficiently
access big datasets and on the other they are very unfit for static
analysis since queries are often ad hoc and generated only during
runtime as new hypotheses are formed or refined.
%
Our thesis is that the Prolog abstract machine should be able to adapt
automatically to the runtime requirements of such or, even better, of
all applications by employing increasingly aggressive forms of dynamic
compilation. As a concrete example of what this means in practice, in
this paper we will attack the problem of providing effective indexing
during runtime. Naturally, we will base our technique on the existing
support for indexing that the WAM provides, but we will extend this
support with the technique of \JITI that we describe in the next
sections.


\section{Indexing in the WAM} \label{sec:prelims}
%================================================
To make the paper relatively self-contained we briefly review the
indexing instructions of the WAM and their use. In the WAM, the first
level of dispatching involves a test on the type of the argument. The
\switchONterm instruction checks the tag of the dereferenced value in
the first argument register and implements a four-way branch where one
branch is for the dereferenced register being an unbound variable, one
for being atomic, one for (non-empty) list, and one for structure. In
any case, control goes to a (possibly empty) bucket of clauses. In the
buckets for constants and structures the second level of dispatching
involves the value of the register. The \switchONconstant and
\switchONstructure instructions implement this dispatching: typically
with a \fail instruction when the bucket is empty, with a \jump
instruction for only one clause, with a sequential scan when the
number of clauses is small, and with a hash lookup when the number of
clauses exceeds a threshold. For this reason the \switchONconstant and
\switchONstructure instructions take as arguments the hash table
\instr{T} and the number of clauses \instr{N} the table contains (or
equivalently, \instr{N} is the size of the hash table). In each bucket
of this hash table and also in the bucket for the variable case of
\switchONterm the code performs a sequential backtracking search of
the clauses using a \TryRetryTrust chain of instructions. The \try
instruction sets up a choice point, the \retry instructions (if any)
update certain fields of this choice point, and the \trust instruction
removes it.

The WAM has additional indexing instructions (\instr{try\_me\_else}
and friends) that allow indexing to be interspersed with the code of
clauses. For simplicity we will not consider them here. This is not a
problem since the above scheme handles all cases. Also, we will feel
free to do some minor modifications and optimizations when this
simplifies things.

We present an example. Consider the Prolog code shown in
Fig.~\ref{fig:carc:facts}. It is a fragment of the well-known machine
learning dataset \textit{Carcinogenesis}~\cite{Carcinogenesis@ILP-97}.
The five clauses get compiled to the WAM code shown in
Fig.~\ref{fig:carc:clauses}. The first argument indexing indexing code
that a Prolog compiler generates is shown in
Fig.~\ref{fig:carc:index}. This code is typically placed before the
code for the clauses and the \switchONconstant instruction is the
entry point of predicate. Note that compared with vanilla WAM this
instruction has an extra argument: the register on the value of which
we will index ($r_1$). The extra argument will allow us to go beyond
first argument indexing. Another departure from the WAM is that if
this argument register contains an unbound variable instead of a
constant then execution will continue with the next instruction; in
effect we have merged part of the functionality of \switchONterm into
the \switchONconstant instruction. This small change in the behavior
of \switchONconstant will allow us to get \JITI. Let's see how.

%------------------------------------------------------------------------------
\begin{figure}[t]
\centering
\subfigure[Some Prolog clauses\label{fig:carc:facts}]{%
  \begin{ScriptProg}
    has\_property(d1,salmonella,p).\\
    has\_property(d1,salmonella\_n,p).\\
    has\_property(d2,salmonella,p). \\
    has\_property(d2,cytogen\_ca,n).\\
    has\_property(d3,cytogen\_ca,p).
  \end{ScriptProg}
}%
\subfigure[WAM indexing\label{fig:carc:index}]{%
  \begin{sf}
    \begin{\wamcodesize}
      \begin{tabular}[b]{l}
        \switchONconstant $r_1$ 5 $T_1$  \\
        \try   $L_1$ \\
        \retry $L_2$ \\
        \retry $L_3$ \\
        \retry $L_4$ \\
        \trust $L_5$ \\
	\\
	\begin{tabular}[b]{r|c@{\ }|l|}
	  \Cline
	  $T_1$: & \multicolumn{2}{c|}{Hash Table Info}\\ \Cline\Cline
	  \      & d1 & \try   $L_1$ \\
	  \      &    & \trust $L_2$ \\ \Cline
          \      & d2 & \try   $L_3$ \\
	  \      &    & \trust $L_4$ \\ \Cline
	  \      & d3 & \jump  $L_5$ \\
	  \Cline
	\end{tabular}
      \end{tabular}
    \end{\wamcodesize}
  \end{sf}
}%
\subfigure[Code for the clauses\label{fig:carc:clauses}]{%
  \begin{sf}
    \begin{\wamcodesize}
      \begin{tabular}[b]{rl}
	$L_1$: & \getcon $r_1$ d1            \\
	\      & \getcon $r_2$ salmonella    \\
	\      & \getcon $r_3$ p             \\
        \      & \proceed                    \\
	$L_2$: & \getcon $r_1$ d1            \\
        \      & \getcon $r_2$ salmonella\_n \\
        \      & \getcon $r_3$ p             \\
        \      & \proceed                    \\
	$L_3$: & \getcon $r_1$ d2            \\
        \      & \getcon $r_2$ salmonella    \\
        \      & \getcon $r_3$ p             \\
        \      & \proceed                    \\
	$L_4$: & \getcon $r_1$ d2            \\
	\      & \getcon $r_2$ cytogen\_ca   \\
	\      & \getcon $r_3$ n             \\
	\      & \proceed                    \\
	$L_5$: & \getcon $r_1$ d3            \\
	\      & \getcon $r_2$ cytogen\_ca   \\
	\      & \getcon $r_3$ p             \\
	\      & \proceed
      \end{tabular}
    \end{\wamcodesize}
  \end{sf}
}%
\subfigure[Any arg indexing\label{fig:carc:jiti_single:before}]{%
  \begin{sf}
    \begin{\wamcodesize}
      \begin{tabular}[b]{l}
        \switchONconstant $r_1$ 5 $T_1$  \\
        \jitiONconstant $r_2$   5 3    \\
        \jitiONconstant $r_3$   5 3    \\
        \try   $L_1$ \\
        \retry $L_2$ \\
        \retry $L_3$ \\
        \retry $L_4$ \\
        \trust $L_5$ \\
	\\
	\begin{tabular}[b]{r|c@{\ }|l|}
	  \Cline
	  $T_1$: & \multicolumn{2}{c|}{Hash Table Info}\\ \Cline\Cline
	  \      & \code{d1} & \try   $L_1$ \\
	  \      &           & \trust $L_2$ \\ \Cline
          \      & \code{d2} & \try   $L_3$ \\
	  \      &           & \trust $L_4$ \\ \Cline
	  \      & \code{d3} & \jump  $L_5$ \\
	  \Cline
	\end{tabular}
      \end{tabular}
    \end{\wamcodesize}
  \end{sf}
}%
\caption{Part of the Carcinogenesis dataset and WAM code that a byte
  code compiler generates}
\label{fig:carc}
\end{figure}
%------------------------------------------------------------------------------


\section{Demand-Driven Indexing of Static Predicates} \label{sec:static}
%=======================================================================
For static predicates the compiler has complete information about all
clauses and shapes of their head arguments. It is both desirable and
possible to take advantage of this information at compile time and so
we treat the case of static predicates separately.
%
We will do so with schemes of increasing effectiveness and
implementation complexity.

\subsection{A simple WAM extension for any argument indexing}
%------------------------------------------------------------
Let us initially consider the case where the predicates to index
consist only of Datalog facts. This is commonly the case for all
extensional database predicates where indexing is most effective and
called for.

Refer to the example in Fig.~\ref{fig:carc}.
%
The indexing code of Fig.~\ref{fig:carc:index} incurs a small cost for
a call where the first argument is a variable (namely, executing the
\switchONconstant instruction) but the instruction pays off for calls
where the first argument is bound. On the other hand, for calls where
the first argument is a free variable and some other argument is
bound, a choice point will be created, the \TryRetryTrust chain will
be used, and execution will go through the code of all clauses. This
is clearly inefficient, more so for larger data sets.
%
We can do much better with the relatively simple scheme shown in
Fig.~\ref{fig:carc:jiti_single:before}. Immediately after the
\switchONconstant instruction, we can statically generate
\jitiONconstant (demand indexing) instructions, one for each remaining
argument. Recall that the entry point of the predicate is the
\switchONconstant instruction. The \jitiONconstant $r_i$ \instr{N A}
instruction works as follows:
\begin{itemize}
\item if the argument register $r_i$ is a free variable, then
  execution continues with the next instruction;
\item otherwise, \JITI kicks in as follows. The abstract machine will
  scan the WAM code of the clauses and create an index table for the
  values of the corresponding argument. It can do so because the
  instruction takes as arguments the number of clauses \instr{N} to
  index and the arity \instr{A} of the predicate. (In our example, the
  numbers 5 and 3.) For Datalog facts, this information is sufficient.
  Also, because the WAM byte code for the clauses has a very regular
  structure, the index table can be created very quickly. Upon its
  creation, the \jitiONconstant instruction will get transformed to a
  \switchONconstant. Again this is straightforward because of the two
  instructions have similar layouts in memory. Execution of the
  abstract machine will continue with the \switchONconstant
  instruction.
\end{itemize}
Figure~\ref{fig:carg:jiti_single:after} shows the index table $T_2$
which is created for our example and how the indexing code looks after
the execution of a call with mode \code{(out,in,?)}. Note that the
\jitiONconstant instruction for argument register $r_2$ has been
appropriately patched. The call that triggered \JITI and subsequent
calls of the same mode will use table $T_2$. The index for the second
argument has been created.
%------------------------------------------------------------------------------
\begin{figure}
  \centering
  \begin{sf}
    \begin{\wamcodesize}
      \begin{tabular}{c@{\hspace*{2em}}c@{\hspace*{2em}}c}
	\begin{tabular}{l}
          \switchONconstant $r_1$ 5 $T_1$ \\
          \switchONconstant $r_2$ 5 $T_2$ \\
          \jitiONconstant $r_3$   5 3     \\
          \try $L_1$   \\
          \retry $L_2$ \\
          \retry $L_3$ \\
          \retry $L_4$ \\
          \trust $L_5$ \\
	\end{tabular}
	&
	\begin{tabular}{r|c@{\ }|l|}
	  \Cline
	  $T_1$: & \multicolumn{2}{c|}{Hash Table Info}\\ \Cline\Cline
	  \      & \code{d1} & \try   $L_1$ \\
	  \      &           & \trust $L_2$ \\ \Cline
          \      & \code{d2} & \try   $L_3$ \\
	  \      &           & \trust $L_4$ \\ \Cline
	  \      & \code{d3} & \jump  $L_5$ \\
	  \Cline
	\end{tabular}
	&
	\begin{tabular}{r|c@{\ }|l|}
	  \Cline
	  $T_2$: & \multicolumn{2}{|c|}{Hash Table Info}\\ \Cline\Cline
	  \      & \code{salmonella}    & \try $L_1$   \\
	  \      &                      & \trust $L_3$ \\ \Cline
	  \      & \code{salmonella\_n} & \jump $L_2$  \\ \Cline
	  \      & \code{cytrogen\_ca}  & \try $L_4$   \\
	  \      &                      & \trust $L_5$ \\
	  \Cline
	\end{tabular}
      \end{tabular}
    \end{\wamcodesize}
  \end{sf}
  \caption{WAM code after demand-driven indexing for argument 2;
    table $T_2$ is generated dynamically}
  \label{fig:carg:jiti_single:after}
\end{figure}
%------------------------------------------------------------------------------

The main advantage of this scheme is its simplicity. The compiled code
(Fig.~\ref{fig:carc:jiti_single:before}) is not significantly bigger
than the code which a WAM-based compiler would generate
(Fig.~\ref{fig:carc:index}) and, even if \JITI turns out unnecessary
during runtime (e.g. execution encounters only open calls or with only
the first argument bound), the extra overhead is minimal: the
execution of some \jitiONconstant instructions for the open call only.
%
In short, this is a simple scheme that allows for \JITI on \emph{any
single} argument. At least for big sets of Datalog facts, we see
little reason not to use this indexing scheme.

\paragraph*{Optimizations.}
Because we are dealing with static code, there are opportunities for
some easy optimizations. Suppose we statically determine that there
will never be any calls with \code{in} mode for some arguments or that
these arguments are not discriminating enough.\footnote{In our example,
suppose the third argument of \code{has\_property/3} had the atom
\code{p} as value throughout.} Then we can avoid generating
\jitiONconstant instructions for them. Also, suppose we detect or
heuristically decide that some arguments are most likely than others
to be used in the \code{in} mode. Then we can simply place the
\jitiONconstant instructions for these arguments \emph{before} the
instructions for other arguments. This is possible since all indexing
instructions take the argument register number as an argument.

\subsection{From any argument indexing to multi-argument indexing}
%-----------------------------------------------------------------
The scheme of the previous section gives us only single argument
indexing. However, all the infrastructure we need is already in place.
We can use it to obtain (fixed-order) multi-argument \JITI in a
straightforward way.

Note that the compiler knows exactly the set of clauses that need to
be tried for each query with a specific symbol in the first argument.
This information is needed in order to construct, at compile time, the
hash table $T_1$ of Fig.~\ref{fig:carc:index}. For multi-argument
\JITI, instead of generating for each hash bucket only \TryRetryTrust
instructions, the compiler can prepend appropriate demand indexing
instructions. We illustrate this on our running example. The table
$T_1$ contains four \jitiONconstant instructions: two for each of the
remaining two arguments of hash buckets with more than one
alternative. For hash buckets with none or only one alternative (e.g.,
for \code{d3}'s bucket) there is obviously no need to resort to \JITI
for the remaining arguments. Figure~\ref{fig:carc:jiti_multi} shows
the state of the hash tables after the execution of queries
\code{has\_property(C,salmonella,T)}, which creates table $T_2$, and
\code{has\_property(d2,P,n)} which creates the $T_3$ table and
transforms the \jitiONconstant instruction for \code{d2} and register
$r_3$ to the appropriate \switchONconstant instruction.

%------------------------------------------------------------------------------
\begin{figure}[t]
  \centering
  \begin{sf}
    \begin{\wamcodesize}
      \begin{tabular}{@{}cccc@{}}
	\begin{tabular}{l}
          \switchONconstant $r_1$ 5 $T_1$ \\
          \switchONconstant $r_2$ 5 $T_2$ \\
          \jitiONconstant $r_3$   5 3     \\
          \try $L_1$   \\
          \retry $L_2$ \\
          \retry $L_3$ \\
          \retry $L_4$ \\
          \trust $L_5$ \\
	\end{tabular}
	&
	\begin{tabular}{r|c@{\ }|l|}
	  \Cline
	  $T_1$: & \multicolumn{2}{c|}{Hash Table Info}\\ \Cline\Cline
	  \      & \code{d1} & \jitiONconstant $r_2$ 2 3 \\
	  \      &           & \jitiONconstant $r_3$ 2 3 \\
	  \      &           & \try   $L_1$ \\
	  \      &           & \trust $L_2$ \\ \Cline
          \      & \code{d2} & \jitiONconstant $r_2$ 2 3 \\
	  \      &           & \switchONconstant $r_3$ 2 $T_3$ \\
	  \      &           & \try   $L_3$ \\
	  \      &           & \trust $L_4$ \\ \Cline
	  \      & \code{d3} & \jump  $L_5$ \\
	  \Cline
	\end{tabular}
	&
	\begin{tabular}{r|c@{\ }|l|}
	  \Cline
	  $T_2$: & \multicolumn{2}{|c|}{Hash Table Info}\\ \Cline\Cline
	  \      & \code{salmonella}    & \jitiONconstant $r_3$ 2 3 \\
	  \      &                      & \try $L_1$   \\
	  \      &                      & \trust $L_3$ \\ \Cline
	  \      & \code{salmonella\_n} & \jump $L_2$  \\ \Cline
	  \      & \code{cytrogen\_ca}  & \jitiONconstant $r_3$ 2 3 \\
	  \      &                      & \try $L_4$   \\
	  \      &                      & \trust $L_5$ \\
	  \Cline
	\end{tabular}
	&
	\begin{tabular}{r|c@{\ }|l|}
	  \Cline
	  $T_3$: & \multicolumn{2}{|c|}{Hash Table Info}\\ \Cline\Cline
	  \      & \code{p} & \jump $L_3$ \\ \Cline
	  \      & \code{n} & \jump $L_4$ \\
	  \Cline
	\end{tabular}
      \end{tabular}
    \end{\wamcodesize}
  \end{sf}
  \caption{\JITI for all argument combinations;
    table $T_1$ is static; $T_2$ and $T_3$ are generated dynamically}
  \label{fig:carc:jiti_multi}
\end{figure}
%------------------------------------------------------------------------------

\paragraph{Implementation issues.}
In the \jitiONconstant instructions of Fig.~\ref{fig:carc:jiti_multi}
notice the integer 2 which denotes the number of clauses that the
instruction will index. Using this number an index table of
appropriate size will be created, such as $T_3$. To fill this table we
need information about the clauses to index and the symbols to hash
on. The clauses can be obtained by scanning the labels of the
\TryRetryTrust instructions following \jitiONconstant; the symbols by
looking at appropriate byte code offsets (based on the argument
register number) from these labels. In our running example, the
symbols can be obtained by looking at the second argument of the
\getcon instruction whose argument register is $r_2$. In the loaded
bytecode, assuming the argument register is represented in one byte,
these symbols are found $sizeof(\getcon) + sizeof(opcode) + 1$ bytes
away from the clause label; see Fig.~\ref{fig:carc:clauses}. Thus,
multi-argument \JITI is easy to get and the creation of index tables
can be extremely fast when indexing Datalog facts.

\subsection{Beyond Datalog and other implementation issues}
%----------------------------------------------------------
Indexing on demand clauses with function symbols is not significantly
more difficult. The scheme we have described is applicable but
requires the following extensions:
\begin{enumerate}
\item Besides \jitiONconstant we also need \jitiONterm and
  \jitiONstructure instructions. These are the \JITI counterparts of
  the WAM's \switchONterm and \switchONstructure.
\item Because the byte code for the clause heads does not necessarily
  have a regular structure, the abstract machine needs to be able to
  ``walk'' the byte code instructions and recover the symbols on which
  indexing will be based. Writing such a code walking procedure is not
  hard.\footnote{In many Prolog systems, a procedure with similar
  functionality often exists for the disassembler, the debugger, etc.}
\item Indexing on an argument that contains unconstrained variables
  for some clauses is tricky. The WAM needs to group clauses in this
  case and without special treatment creates two choice points for
  this argument (one for the variables and one per each group of
  clauses). However, this issue and how to deal with it is well-known
  by now. Possible solutions to it are described in a 1987 paper by
  Carlsson~\cite{FreezeIndexing@ICLP-87} and can be readily adapted to
  \JITI. Alternatively, in a simple implementation, we can skip \JITI
  for arguments with variables in some clauses.
\end{enumerate}
Before describing \JITI more formally, we remark on the following
design decisions whose rationale may not be immediately obvious:
\begin{itemize}
\item By default, only table $T_1$ is generated at compile time (as in
  the WAM) and the additional index tables $T_2, T_3, \ldots$ are
  generated dynamically. This is because we do not want to increase
  compiled code size unnecessarily (i.e., when there is no demand for
  these indices).
\item On the other hand, we generate \jitiSTAR instructions at compile
  time for the head arguments.\footnote{The \jitiSTAR instructions for
  the $T_1$ table can be generated either by the compiler or by the
  loader.} This does not noticeably increase the generated byte code
  but it greatly simplifies code loading. Notice that a nice property
  of the scheme we have described is that the loaded byte code can be
  patched \emph{without} the need to move any instructions.
% The indexing tables are typically not intersperced with the byte code.
\item Finally, one may wonder why the \jitiSTAR instructions create
  the dynamic index tables with an additional code walking pass
  instead of piggy-backing on the pass which examines all clauses via
  the main \TryRetryTrust chain. Main reasons are: 1) in many cases
  the code walking can be selective and guided by offsets and 2) by
  first creating the index table and then using it we speed up the
  execution of the queries encountered during runtime and often avoid
  unnecessary choice point creations.
\end{itemize}
This is \JITI as we have implemented it.
% in one of our Prolog systems.
However, we note that these decisions are orthogonal to the main idea
and are under compiler control. If, for example, analysis determines
that some argument sequences will never demand indexing we can simply
avoid generation of \jitiSTAR instructions for these. Similarly, if we
determine that some argument sequences will definitely demand indexing
we can speed up execution by generating the appropriate index tables
at compile time instead of at runtime.

\subsection{Demand-driven index construction and its properties}
%---------------------------------------------------------------
The idea behind \JITI can be captured in a single sentence: \emph{we
can generate every index we need during program execution when this
index is demanded}. Subsequent uses of these indices can speed up
execution considerably more than the time it takes to construct them
(more on this below) so this runtime action makes sense.\footnote{In
fact, because choice points are expensive in the WAM, \JITI can speed
up even the execution of the query that triggers the process, not only
subsequent queries.}
%
We describe the process of demand-driven index construction.

% \subsubsection{Demand-driven index construction}
%-------------------------------------------------
Let $p/k$ be a predicate with $n$ clauses.
%
At a high level, its indices form a tree whose root is the entry point
of the predicate. For simplicity, we assume that the root node of the
tree and the interior nodes corresponding to the index table for the
first argument have been constructed at compile time. Leaves of this
tree are the nodes containing the code for the clauses of the
predicate and each clause is identified by a unique label \mbox{$L_i,
1 \leq i \leq n$}. Execution always starts at the first instruction of
the root node and follows Algorithm~\ref{alg:construction}. The
algorithm might look complicated but is actually quite simple.
%
Each non-leaf node contains a sequence of byte code instructions with
groups of the form \mbox{$\langle I_1, \ldots, I_m, T_1, \ldots, T_l
\rangle, 0 \leq m \leq k, 1 \leq l \leq n$} where each of the $I$
instructions, if any, is either a \switchSTAR or a \jitiSTAR
instruction and the $T$ instructions are either a sequence of
\TryRetryTrust instructions (if $l > 1$) or a \jump instruction (if
\mbox{$l = 1$}). Step~2.2 dynamically constructs an index table $\cal
T$ whose buckets are the newly created interior nodes in the tree.
Each bucket associated with a single clause contains a \jump
instruction to the label of that clause. Each bucket associated with
many clauses starts with the $I$ instructions which are yet to be
visited and continues with a \TryRetryTrust chain pointing to the
clauses. When the index construction is done, the instruction mutates
to a \switchSTAR WAM instruction.
%-------------------------------------------------------------------------
\begin{Algorithm}[t]
  \caption{Actions of the abstract machine with \JITI}
  \label{alg:construction}
  \begin{enumerate}
  \item if the current instruction $I$ is a \switchSTAR, \try, \retry,
    \trust or \jump, the action is an in the WAM;
  \item if the current instruction $I$ is a \jitiSTAR with arguments $r,
    l$, and $k$ where $r$ is a register then
    \begin{enumerate}
    \item[2.1] if register $r$ contains a variable, the action is simply to
      \instr{goto} the next instruction in the node;
    \item[2.2] if register $r$ contains a value $v$, the action is to
      dynamically construct the index as follows:
      \begin{itemize}
      \item[2.2.1] collect the subsequent instructions in a list $\cal I$
	until the next instruction is a \try;\footnote{Note that there
	will always be a \try following a \jitiSTAR instruction.}
      \item[2.2.2] for each label $L$ in the \TryRetryTrust chain
	inspect the code of the clause with label $L$ to find the
	symbol~$c$ associated with register $r$ in the clause; (This
	step creates a list of $\langle c, L \rangle$ pairs.)
      \item[2.2.3] create an index table $\cal T$ out of these pairs as
	follows:
	\begin{itemize}
	\item if $I$ is a \jitiONconstant or a \jitiONstructure then
	  create an index table for the symbols in the list of pairs;
	  each entry of the table is identified by a symbol $c$ and
	  contains:
	  \begin{itemize}
	  \item the instruction \jump $L_c$ if $L_c$ is the only label
	    associated with $c$;
	  \item the sequence of instructions obtained by appending to
	    $\cal I$ a \TryRetryTrust chain for the sequence of labels
	    $L'_1, \ldots, L'_l$ that are associated with $c$
	  \end{itemize}
	\item if $I$ is a \jitiONterm then
	  \begin{itemize}
	  \item partition the sequence of labels $\cal L$ in the list
	    of pairs into sequences of labels ${\cal L}_c, {\cal L}_l$
	    and ${\cal L}_s$ for constants, lists and structures,
	    respectively;
	  \item for each of the four sequences ${\cal L}, {\cal L}_c,
	    {\cal L}_l, {\cal L}_s$ of labels create code as follows:
	    \begin{itemize}
	    \item the instruction \fail if the sequence is empty;
	    \item the instruction \jump $L$ if $L$ is the only label in
	      the sequence;
	    \item the sequence of instructions obtained by appending to
	      $\cal I$ a \TryRetryTrust chain for the current sequence
	      of labels;
	    \end{itemize}
	  \end{itemize}
	\end{itemize}
      \item[2.2.4] transform the \jitiSTAR $r, l, k$ instruction to
	a \switchSTAR $r, l, \&{\cal T}$ instruction; and
      \item[2.2.5] continue execution with this instruction.
      \end{itemize}
    \end{enumerate}
  \end{enumerate}
\end{Algorithm}
%-------------------------------------------------------------------------

\paragraph*{Complexity properties.}
Complexity-wise, dynamic index construction does not add any overhead
to program execution. First, note that each demanded index table will
be constructed at most once. Also, a \jitiSTAR instruction will be
encountered only in cases where execution would examine all clauses in
the \TryRetryTrust chain.\footnote{This statement is possibly not
valid the presence of Prolog cuts.} The construction visits these
clauses \emph{once} and then creates the index table in time linear in
the number of clauses. One pass over the list of $\langle c, L
\rangle$ pairs suffices. After index construction, execution will
visit only a subset of these clauses as the index table will be
consulted.
%% Finally, note that the maximum number of \jitiSTAR instructions
%% that will be visited for each query is bounded by the maximum
%% number of index positions (symbols) in the clause heads of the
%% predicate.
Thus, in cases where \JITI is not effective, execution of a query will
at most double due to dynamic index construction. In fact, this worst
case is extremely unlikely in practice. On the other hand, \JITI can
change the complexity of evaluating a predicate call from $O(n)$ to
$O(1)$ where $n$ is the number of clauses.

\subsection{More implementation choices}
%---------------------------------------
The observant reader has no doubt noticed that
Algorithm~\ref{alg:construction} provides multi-argument indexing but
only for the main functor symbol of arguments. For clauses with
compound terms that require indexing in their sub-terms we can either
employ a program transformation like \emph{unification
factoring}~\cite{UnifFact@POPL-95} at compile time or modify the
algorithm to consider index positions inside compound terms. This is
relatively easy to do but requires support from the register allocator
(passing the sub-terms of compound terms in appropriate argument
registers) and/or a new set of instructions. Due to space limitations
we omit further details.

Algorithm~\ref{alg:construction} relies on a procedure that inspects
the code of a clause and collects the symbols associated with some
particular index position (step~2.2.2). If we are satisfied with
looking only at clause heads, this procedure only needs to understand
the structure of \instr{get} and \instr{unify} instructions. Thus, it
is easy to write. At the cost of increased implementation complexity,
this step can of course take into account other information that may
exist in the body of the clause (e.g., type tests such as
\code{var(X)}, \code{atom(X)}, aliasing constraints such as \code{X =
Y}, numeric constraints such as \code{X > 0}, etc).

A reasonable concern for \JITI is increased memory consumption during
runtime due to the index tables. In our experience, this does not seem
to be a problem in practice since most applications do not have demand
for indexing on all argument combinations. In applications where it
becomes a problem or when running in an environment where memory is
limited, we can easily put a bound on the size of index tables, either
globally or for each predicate separately. The \jitiSTAR instructions
can either become inactive when this limit is reached, or better yet
we can recover the space of some tables. To do so, we can employ any
standard recycling algorithm (e.g., least recently used) and reclaim
the space for some tables that are no longer in use. This is easy to
do by reverting the corresponding \jitiSTAR instructions back to
\switchSTAR instructions. If the indices are needed again, they can
simply be regenerated.


\section{Demand-Driven Indexing of Dynamic Predicates} \label{sec:dynamic}
%=========================================================================
We have so far lived in the comfortable world of static predicates,
where the set of clauses to index is fixed beforehand and the compiler
can take advantage of this knowledge. Dynamic code introduces several
complications. However, note that most Prolog systems do provide
indexing for dynamic predicates. In effect, they already deal with
these issues.


\section{Performance Evaluation} \label{sec:perf}
%================================================


\section{Concluding Remarks}
%===========================
\begin{itemize}
\item Mention the non-trivial speedups in actual applications; also
  that it is important to realize that certain applications have ad
  hoc query patterns (e.g., ILP) are not amenable to static analyses
\end{itemize}

%==============================================================================
\bibliographystyle{splncs}
\bibliography{lp}
%==============================================================================

\end{document}
