Major Section: ACL2 Documentation
Bob Boyer and Warren Hunt have developed a canonical representation for ACL2 data objects and a function memoization mechanism to facilitate reuse of previously computed results. This facility includes procedures to read and print ACL2 expressions in such a way that repetition of some ACL2 objects is eliminated, thereby permitting a kind of on-the-fly file compression. The implementation does not alter the semantics of ACL2 except to add a handful of definitions.
Much of the documentation for this topic is taken from the paper ``Function Memoization and Unique Object Representation for ACL2 Functions'' by Robert S. Boyer and Warren A. Hunt, Jr., which has appeared in the Sixth International Workshop on the ACL2 Theorem Prover and Its Applications, ACM Digital Library, 2006.
You can build an experimental version of ACL2 that includes hash
cons, function memoization, and fast association lists (applicative
hash tables). An easy way is to include the following with a
ACL2_HONS=hSo for example, to make an executable image and also documentation (which will appear in subdirectories
make large DOC ACL2_HONS=h
(cons-subtrees x nil)builds a fast alist that associates each subtree
of X with T, without duplication.
(fast-alist-free alist)throws away the hash table associated with a fast
(fast-alist-len alist)counts the number of unique keys in a fast alist.
(fast-alist-summary)prints some basic statistics about any current fast
(hons-acons key val alist)is the main way to create or extend
(hons-acons! key val alist)is an alternative to
produces normed, fast alists.
(hons-assoc-equal key alist)is not fast; it serves as the logical
(hons-clear gc)is a drastic garbage collection mechanism that clears out
the underlying Hons Space.
(hons-copy-persistent x)returns a normed object that is equal to X
and which will be re-normed after any calls to
(hons-equal x y)is a recursive equality check that optimizes when parts of
its arguments are normed.
(hons-equal-lite x y)is a non-recursive equality check that optimizes if
its arguments are normed.
(hons-get key alist)is the efficient lookup operation for
(hons-resize ...)can be used to manually adjust the sizes of the hash
tables that govern which ACL2 Objects are considered normed.
(hons-shrink-alist alist ans)can be used to eliminate "shadowed pairs"
from an alist or to copy fast-alists.
(hons-shrink-alist! alist ans)is an alternative to
that produces a normed result.
(hons-summary)prints basic information about the sizes of the tables in
the current Hons Space.
(make-fast-alist alist)creates a fast-alist from the input alist,
alistitself or, in some cases, a new object equal to it.
(number-subtrees x)returns the number of distinct subtrees of X, in the
fast-alistsare used inefficiently
In the implementation of the ACL2 logic, ACL2 data objects are
represented by Common Lisp objects of the same type, and the ACL2
pairing operation is internally implemented by the Common Lisp
cons function. In Common Lisp,
cons is guaranteed to
provide a new pair, distinct from any previously created pair. We
have defined a new ACL2 function
HONS that is logically
identical to the ACL2
cons function, but whose implementation
usually reuses an existing pair if its components are identical to
the components of an existing pair. A record of ACL2 HONS objects
is kept, and when an ACL2 function calls
hons ACL2 searches for
an existing identical pair before allocating a new pair; this
operation been called ``hash consing''.
It appears that hash consing was first conceived by A. P. Ershov in 1957, to speed up the recognition of common subexpressions. Ershov showed how to collapse trees to minimal DAGs by traversing trees bottom up, and he used hashing to eliminate the re-evaluation of common subexpressions. Later, Eiichi Goto implemented a Lisp system with a built-in hash consing operation: his h-CONS cells were rewrite protected and free of duplicate copies, and Goto used this hash consing operation to facilitate the implementation of a symbolic algebra system he developed.
Memoizing functions also has a long history. In 1967, Donald Michie proposed using memoized functions to improve the performance of machine learning. Rote learning was improved by a learning function not forgetting what it had previously learned; this information was stored as memoized function values.
The use of hash consing has appeared many times. For instance, Henry Baker used hash consing to improve the performance of the well-known Boyer rewriting benchmark. Baker used both hash consing and function memoization to improve the speed of the Takeuchi function, exactly in the spirit of our implementation, but without the automated, system-wide integration we report here.
The ACL2 implementation permits memoization of user-defined functions. During execution a user may enable or disable function memoization on an individual function basis, may clear memoization tables, and may even keep a stack of memoization tables. This facility takes advantage of our implementation where we keep one copy of each distinct ACL2 data object. Due to the functional nature of ACL2, it is sufficient to have at most one copy of any data structure; thus, a user may arrange to keep data canonicalized. This implementation extends to the entire ACL2 system the benefits enjoyed by BDDs: canonicalization, memoization, and fast equality check.
We have defined various algorithms using these features, and we have observed, in some cases, substantial performance increases. For instance, we have implemented unordered set intersection and union operations that operate in time roughly linear in the size of the sets. Without using arrays, we defined a canonical representation for Boolean functions using ACL2 objects. We have investigated the performance of rewriting and tree consensus algorithms to good effect, and we believe function memoization offers interesting opportunities to simplify algorithm definition while simultaneously providing performance improvements.
We recommend that users focus at first on the logical definitions of
hons and other primitives rather than their underlying Common
Lisp implementations. Integrated with this memoization system is a
performance monitoring system, which can provide real-time feedback
on the operation and usefulness of
hons and function
memoization. For a more detailed description of these tools, please
see the ACL2 2006 workshop paper mentioned above.
The Fibonacci function is a small example that demonstrates the utility of function memoization. The following definition exhibits a runtime that is exponential in its input argument.
(defun fib (x) (declare (xargs :guard (natp x))) (mbe :logic (cond ((zp x) 0) ((= x 1) 1) (t (+ (fib (- x 1)) (fib (- x 2))))) :exec (if (< x 2) x (+ (fib (- x 1)) (fib (- x 2))))))
Below we show how the ACL2
time$ utility can measure the time
to execute a call to the
fib function (with some editing to
avoid noise from the underlying Common Lisp implementation).
Memoize is actually an ACL2 macro that expands to a call of
the ACL2 function used to identify a function for memoization;
see memoize. This function also accepts a well-formed term that
must be true in order for the system to memoize a call of the
memoized function; to ensure that an instance of the term is safe
for evaluation in Common Lisp, a check is performed that if the
guard of the memoized function is satisfied, then this instance
will execute without error.
ACL2 !>(time$ (fib 40)) ; (EV-REC *RETURN-LAST-ARG3* ...) took ; 0.99 seconds realtime, 0.98 seconds runtime ; (1,296 bytes allocated). 102334155 ACL2 !>(memoize 'fib) Summary Form: ( TABLE MEMOIZE-TABLE ...) Rules: NIL Time: 0.01 seconds (prove: 0.00, print: 0.00, other: 0.01) Summary Form: ( PROGN (TABLE MEMOIZE-TABLE ...) ...) Rules: NIL Time: 0.01 seconds (prove: 0.00, print: 0.00, other: 0.01) FIB ACL2 !>(time$ (fib 40)) ; (EV-REC *RETURN-LAST-ARG3* ...) took ; 0.00 seconds realtime, 0.00 seconds runtime ; (2,864 bytes allocated). 102334155 ACL2 !>(time$ (fib 100)) ; (EV-REC *RETURN-LAST-ARG3* ...) took ; 0.00 seconds realtime, 0.00 seconds runtime ; (7,024 bytes allocated). 354224848179261915075 ACL2 !>(unmemoize 'fib)
We see that once the function
fib is identified as a function
for which function calls should be memoized, the execution times are
substantially reduced. Finally, we can prevent
fib from being
further memoized; in fact,
unmemoize erases the previously
The implementation of hash consing, memoization, and applicative hash tables involves several facets: canonical representation of ACL2 data, function memoization, and the use of Lisp hash tables to improve the performance of association list operations. We discuss each of these in turn, and we mention some subtle interrelationships. Although it is not necessary to understand the discussion in this section, it may permit some users to better use this implementation. This discussion may be confusing for some ACL2 users as it makes references to Lisp implementation features.
The ACL2 system is actually implemented as a Lisp program that is layered on top of a Common Lisp system implementation. ACL2 data constants are implemented with their corresponding counterparts in Common Lisp; that is, ACL2 cons pairs, strings, characters, numbers, and symbols are implemented with their specific Common Lisp counterparts. This choice permits a number of ACL2 primitive functions to be implemented with their corresponding Common Lisp functions, but there are indeed differences. ACL2 is a logic, and as such, it does not specify anything to do with physical storage or execution performance. When ACL2 is used, the knowledgeable user can write functions to facilitate the reuse of some previously computed values.
Recall the three mechanisms under discussion: hash consing, function
memoization, and fast association list operations. The function
memoization mechanism takes advantage of the canonical
representation of data objects provided by the
as does the fast association list operation mechanism. Each time
hons is invoked, its arguments are themselves converted, if
necessary, to uniquely represented objects.
The ACL2 universe is recursively closed under the
operation and the atoms. Hash consing (
hons) is logically
cons, but a set of tables is used to record each
hons pair. When a
hons pair is requested, the
implementation checks, in the current set of tables, whether the
requested pair already exists. If not, a new pair is created and a
record of that pair is made; otherwise, a reference to the
previously created pair is returned. Thus, any data object created
hons has a unique representation, as does every
subcomponent. We also arrange for strings to have a unique
representation -- only one copy of each different string is kept
-- and when any previously unseen string is an argument to
hons, we add the string to a unique table of strings. A
system-wide benefit of having a canonical representation for data is
that equality comparisons between any two data objects can be done
in constant time.
The definition of
hons in no way changes the operation of
hons creates a
cons pair. When asked to
hons, the implementation checks to see if there is a
cons with the same
cdr as the
requested. Thus, the only difference between the results of a
hons call and a corresponding
cons call is a notation in
some invisible (to the ACL2 logic) tables where we record which
cons elements are also
hons elements. Since a
nothing but a
cons, the operation of
unchanged. In fact, the implementation is designed so that at any
time it is safe to clear the table identifying which
elements are also considered
User-defined functions with defined and verified guards can be
memoized. When a function is memoized, a user-supplied condition
restricts the domain when memoization is attempted. When the
condition is satisfied, memoization is attempted (assuming that
memoization for the function is presently enabled); otherwise, the
function is just evaluated. Each memoized function has a hash table
that is used to keep track of a unique list of function arguments
paired with their values. If appropriate, for each function the
corresponding table is checked to see if a previous call with
exactly the same arguments already exists in the table: if so, then
the associated value is returned; if not, then the function is
evaluated and a new key-value pair is added to the table of memoized
values so that some future call will benefit from the memoization.
With ACL2 user functions memoization can be dynamically enabled and
disabled. There is an ACL2 function that clears a specific
memoization table. And finally, just as with the
hons table, a
stack of these function memoization tables is maintained; that is,
it is possible to memoize a function, use it a bit, set the memoized
values aside, start a new table, use it, and then return to the
We next discuss the fast lookup operation for association lists.
When a pair is added to an association list using the functions
hons-acons!, the system also records the
key-value pair in an associated hash table. As long as a user only
used these two functions when placing key-value pairs on an
association list, the key-value pairs in the corresponding hash
table will be synchronized with the key-value pairs in the
association list. Later, if
hons-get is used to look up a key,
then instead of performing a linear search of the association list
we consult the associated hash table. If a user does not strictly
follow this discipline, then a linear search may be required. In
some sense, these association lists are much like ACL2 arrays, but
without the burden of explicitly naming the arrays. The ACL2 array
compress1 function is analogous to the functions
hons-shrink-alist!. There are
user-level ACL2 functions that allow the associated hash tables to
be cleared and removed.
Finally, we would advise anyone who is using CCL in a research
environment to stay plugged into the ``trunk'' or ``bleeding edge''
of CCL development. This is very easy to do by typing a few
commands to a shell, for example standing above the target directory
as follows, provided one has
For linux: rm -rf ccl svn co http://svn.clozure.com/publicsvn/openmcl/trunk/linuxx8664/ccl For an x86 Macintosh running the Darwin OS: svn co http://svn.clozure.com/publicsvn/openmcl/trunk/darwinx8664/ccl To keep up to date, you may find it sufficient to do: cd ccl svn update Whether obtaining a fresh CCL or just updating, finally issue these commands. ./lx86cl64 (rebuild-ccl :full t) (quit) ./lx86cl64 (rebuild-ccl :full t) (quit)
Baker, Henry G., The Boyer Benchmark at Warp Speed. ACM Lisp Pointers V,3 (Jul-Sep 1992), pages 13-14.
Baker, Henry G., A Tachy 'TAK'. ACM Lisp Pointers Volume 3, July-September, 1992, pages 22-23.
Robert S. Boyer and Warren A. Hunt, Jr., Function Memoization and Unique Object Representation for ACL2 Functions, in the Sixth International Workshop on the ACL2 Theorem Prover and Its Applications, ACM Digital Library, 2006.
A. P. Ershov. On Programming of Arithmetic Operations. In the Communications of the ACM, Volume 118, Number 3, August, 1958, pages 427-430.
Eiichi Goto, Monocopy and Associative Algorithms in Extended Lisp, TR-74-03, University of Toyko, 1974.
Richard P. Gabriel. Performance and Evaluation of Lisp Systems. MIT Press, 1985.
Donald Michie. Memo functions: a Language Feature with Rote Learning Properties. Technical Report MIP-R-29, Department of Artificial Intelligence, University of Edinburgh, Scotland, 1967.
Donald Michie. Memo Functions and Machine Learning. In Nature, Volumne 218, 1968, pages 19-22.