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last edited 14 years ago by Bill Page |
Edit detail for SandBoxCachedFunction revision 5 of 10
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Editor: Bill Page
Time: 2009/10/29 10:00:22 GMT-7 |
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| Note: Caching a function with more than one argument | ||
changed: -gcd2cached:CachedFunction(Pair,Integer):=new gcd2 gcd2cached:CachedFunction(Pair,Integer):=cachedFunction gcd2 changed: -gcd2cached:CachedFunction(Pair,Integer):=new gcd2 gcd2cached:CachedFunction(Pair,Integer):=cachedFunction gcd2
A generic way to define function that cache their values.\begin{spad} )abbrev domain CACHEDF CachedFunction ++ Author: Ralf Hemmecke ++ Date Created: 2009 ++ Date Last Updated: Oct 27, 2009 ++ Basic Operations: ++ Related Domains: Table, AssociationList ++ Also See: ++ AMS Classifications: ++ Keywords: ++ Examples: ++ References: ++ Description: ++ This domain is a domain of functions with associated cache. CachedFunction(A: SetCategory, B: SetCategory): Exports == Implementation where Exports ==> with function: % -> (A -> B) cachedFunction: (A -> B) -> % apply: (%, A) -> B recursiveDefine: (%, f: A -> B) -> %
Implementation ==> add
Rep := Record(cache: Table(A, B), fun: A -> B);
function(x: %): A -> B == x.fun
cachedFunction(f: A -> B): % == [empty()$Table(A, B), f]
apply(x: %, a: A): B == c := x.cache u: Union(B, "failed") := search(a, c) if u case B then u :: B else f: A -> B := x.fun c.a := f(a)
recursiveDefine(x: %, f: A -> B): % == x.fun := f; x \end{spad}
\begin{axiom} )set message time on I := Integer f(n:I): I ==1 fib: CachedFunction(I,I) := cachedFunction(f) recursiveDefine(fib,(n:I):I+->if n<2 then 1 else fib(n-1) + fib(n-2)) fib 40 g(n:I): I == if n<2 then 1 else g(n-1)+g(n-2) g 40 h: I -> I := function fib h 40 \end{axiom}
Caching a function with more than one argument
We need to combine more than one domain into a single domain in a universal manner like a cross-product. There are seveal ways to do this in Axiom including
List,Record,ProductandDirectProduct. For example: \begin{axiom} -- combine two domain in one using Record Pair:=Record(x1:Integer,x2:Integer) pair(x:Integer,y:Integer):Pair == [x,y] -- gcd2(arg:Pair):Integer == gcd(arg.x1,arg.x2) gcd2cached:CachedFunction(Pair,Integer):=cachedFunction gcd2 gcdCached(x1:Integer,x2:Integer):Integer == gcd2cached pair(x1,x2) )set message time on for i in 1..1000 repeat for j in 1..100 repeat gcd(i,j) for i in 1..1000 repeat for j in 1..100 repeat gcdCached(i,j) )set message time off testhash() == for i in 1..100 repeat for j in 1..100 repeat if gcdCached(i,j)~=gcd(i,j) then error "bad hash!" "good" testhash() \end{axiom} I was rather disappointed with the performance of the cache! So I took a look at the source code for Table and it seems thatTableonly uses an efficient hashing method ifhashable(Key)$Lispis true, whereKeyis the first domain ofTable(Key,Entry). This is true forIntegerandhashable(String)$Lispbut notRecord(x1:Integer,x2:Integer)or evenList(Integer). \begin{axiom} hashable(Integer)$Lisp hashable(String)$Lisp hashable(Record(x1:Integer,x2:Integer))$Lisp hashable(List Integer)$Lisp \end{axiom} But as far as I can see the implementation ofListandRecordin Axiom do satisfy the requirements for the use of HashTable::++ This creates a \spadtype{HashTable} if equal for the Key ++ domain is consistent with Lisp EQUAL otherwise an ++ \spadtype{AssociationList}
if their component domains do. So I think it would be a good idea to make the
hashablefunction in 'spad.lisp':(defun |knownEqualPred| (dom) (let ((fun (|compiledLookup| '= '((|Boolean|) $ $) dom))) (if fun (get (bpiname (car fun)) '|SPADreplace|) nil))) (defun |hashable| (dom) (memq (|knownEqualPred| dom) #-Lucid '(EQ EQL EQUAL) #+Lucid '(EQ EQL EQUAL EQUALP) ))a little smarter. This test only seems to check if the compiler was able to optimize equality to a lisp operator. I suppose we could improve the compiler but even failing that, there should be a way to recognize it at this level. I think it's something like having the
Canonicalproperty "all the way down".We can test this for now by using HashTable? explicitly and just assuming that that domain
Asatisfies the requirement: \begin{spad} )abbrev domain CACHEDF CachedFunction? CachedFunction?(A: SetCategory?, B: SetCategory?): Exports == Implementation where Exports ==> with function: % -> (A -> B) cachedFunction: (A -> B) -> % apply: (%, A) -> B recursiveDefine: (%, f: A -> B) -> %Implementation ==> add
Rep := Record(cache: HashTable?(A, B, "UEQUAL"), fun: A -> B);
function(x: %): A -> B == x.fun
cachedFunction(f: A -> B): % == [empty()$HashTable(A, B, "UEQUAL"), f]?
apply(x: %, a: A): B == c := x.cache u: Union(B, "failed") := search(a, c) if u case B then u :: B else f: A -> B := x.fun c.a := f(a)
recursiveDefine(x: %, f: A -> B): % == x.fun := f; x \end{spad}
Now let's try that again. \begin{axiom} -- combine two domain in one Pair:=Record(x1:Integer,x2:Integer) pair(x:Integer,y:Integer):Pair == [x,y]? -- gcd2(arg:Pair):Integer == gcd(arg.x1,arg.x2) gcd2cached:CachedFunction?(Pair,Integer):=cachedFunction gcd2 gcdCached(x1:Integer,x2:Integer):Integer == gcd2cached pair(x1,x2) )set message time on for i in 1..1000 repeat for j in 1..100 repeat gcd(i,j) for i in 1..1000 repeat for j in 1..100 repeat gcdCached(i,j) )set message time off testhash() == for i in 1..100 repeat for j in 1..100 repeat if gcdCached(i,j)~=gcd(i,j) then error "bad hash!" "good" testhash() \end{axiom}
perhaps recursiveDefine not necessary --Bill Page, Wed, 28 Oct 2009 22:06:18 -0700 reply\begin{axiom} )set message time on )lib CACHEDF fib2: CachedFunction?(INT,INT):= cachedFunction (n+-> if n<2 then 1 else fib2(n-1)+fib2(n-2) ) fib2 40 \end{axiom} Looks like the time difference (if any) is not significant.
Some or all expressions may not have rendered properly, because Axiom returned the following error:Error: export AXIOM=/usr/local/lib/fricas/target/x86_64-unknown-linux; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 240; $AXIOM/bin/AXIOMsys < /var/zope2/var/LatexWiki/5132263765150484496-25px.axmChecking for foreign routines AXIOM="/usr/local/lib/fricas/target/x86_64-unknown-linux" spad-lib="/usr/local/lib/fricas/target/x86_64-unknown-linux/lib/libspad.so" foreign routines found openServer result -2 FriCAS (AXIOM fork) Computer Algebra System Version: FriCAS 2009-10-26 Timestamp: Tuesday October 27, 2009 at 17:02:14 ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave FriCAS and return to shell. -----------------------------------------------------------------------------
(1) -> (1) -> (1) -> (1) -> (1) -> <spad> )abbrev domain CACHEDF CachedFunction ++ Author: Ralf Hemmecke ++ Date Created: 2009 ++ Date Last Updated: Oct 27, 2009 ++ Basic Operations: ++ Related Domains: Table, AssociationList ++ Also See: ++ AMS Classifications: ++ Keywords: ++ Examples: ++ References: ++ Description: ++ This domain is a domain of functions with associated cache. CachedFunction(A: SetCategory, B: SetCategory): Exports == Implementation where Exports ==> with function: % -> (A -> B) cachedFunction: (A -> B) -> % apply: (%, A) -> B recursiveDefine: (%, f: A -> B) -> %
Implementation ==> add
Rep := Record(cache: Table(A, B), fun: A -> B);
function(x: %): A -> B == x.fun
cachedFunction(f: A -> B): % == [empty()$Table(A, B), f]
apply(x: %, a: A): B == c := x.cache u: Union(B, "failed") := search(a, c) if u case B then u :: B else f: A -> B := x.fun c.a := f(a)
recursiveDefine(x: %, f: A -> B): % == x.fun := f; x</spad> Compiling FriCAS source code from file /var/zope2/var/LatexWiki/8303643491408128431-25px001.spad using old system compiler. CACHEDF abbreviates domain CachedFunction processing macro definition Exports ==> -- the constructor category processing macro definition Implementation ==> -- the constructor capsule ------------------------------------------------------------------------ initializing NRLIB CACHEDF for CachedFunction compiling into NRLIB CACHEDF compiling exported function : $ -> A -> B CACHEDF;function;$M;1 is replaced by QCDR Time: 0.05 SEC.
compiling exported cachedFunction : A -> B -> $ Time: 0 SEC.
compiling exported apply : ($,A) -> B Time: 0.01 SEC.
compiling exported recursiveDefine : ($,A -> B) -> $ Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |CachedFunction| REDEFINED
;;; *** |CachedFunction| REDEFINED Time: 0.01 SEC.
Cumulative Statistics for Constructor CachedFunction Time: 0.07 seconds
finalizing NRLIB CACHEDF Processing CachedFunction for Browser database: --->-->CachedFunction((function ((Mapping B A) %))): Not documented!!!! --->-->CachedFunction((cachedFunction (% (Mapping B A)))): Not documented!!!! --->-->CachedFunction((apply (B % A))): Not documented!!!! --->-->CachedFunction((recursiveDefine (% % (Mapping B A)))): Not documented!!!! --------constructor--------- ; compiling file "/var/zope2/var/LatexWiki/CACHEDF.NRLIB/CACHEDF.lsp" (written 29 OCT 2009 09:55:58 AM): ; compiling (/VERSIONCHECK 2) ; compiling (PUT (QUOTE |CACHEDF;function;$M;1|) ...) ; compiling (DEFUN |CACHEDF;function;$M;1| ...) ; compiling (DEFUN |CACHEDF;cachedFunction;M$;2| ...) ; compiling (DEFUN |CACHEDF;apply;$AB;3| ...) ; compiling (DEFUN |CACHEDF;recursiveDefine;$M$;4| ...) ; compiling (DEFUN |CachedFunction| ...) ; compiling (DEFUN |CachedFunction;| ...) ; compiling (MAKEPROP (QUOTE |CachedFunction|) ...)
; /var/zope2/var/LatexWiki/CACHEDF.NRLIB/CACHEDF.fasl written ; compilation finished in 0:00:00.072 ------------------------------------------------------------------------ CachedFunction is now explicitly exposed in frame initial CachedFunction will be automatically loaded when needed from /var/zope2/var/LatexWiki/CACHEDF.NRLIB/CACHEDF
(1) -> )set message time on
I := Integer
$$ Integer \leqno(1) $$
Type: Domain Time: 0.06 (OT) = 0.06 sec f(n:I): I ==1
Function declaration f : Integer -> Integer has been added to workspace. Type: Void Time: 0.01 (IN) = 0.01 sec fib: CachedFunction(I,I) := cachedFunction(f)
Compiling function f with type Integer -> Integer
LISP output: (#<HASH-TABLE :TEST EQUAL :COUNT 0 {1005F3E391}> #<FUNCTION |*1;f;1;initial|>) Type: CachedFunction(Integer,Integer) Time: 0.01 (IN) + 0.02 (OT) = 0.03 sec recursiveDefine(fib,(n:I):I+->if n<2 then 1 else fib(n-1) + fib(n-2))
LISP output: (#<HASH-TABLE :TEST EQUAL :COUNT 0 {1005F3E391}> #<FUNCTION |*1;anonymousFunction;0;initial;internal|>) Type: CachedFunction(Integer,Integer) Time: 0.02 (IN) + 0.01 (OT) = 0.03 sec fib 40
$$ 165580141 \leqno(5) $$
Type: PositiveInteger Time: 0.01 (IN) = 0.01 sec g(n:I): I == if n<2 then 1 else g(n-1)+g(n-2)
Function declaration g : Integer -> Integer has been added to workspace. Type: Void Time: 0 sec g 40
Compiling function g with type Integer -> Integer
$$ 165580141 \leqno(7) $$
Type: PositiveInteger Time: 11.48 (EV) + 0.02 (OT) = 11.50 sec h: I -> I := function fib
$$ \mbox{theMap(...)} \leqno(8) $$
Type: (Integer -> Integer) Time: 0 sec h 40
$$ 165580141 \leqno(9) $$
Type: PositiveInteger Time: 0 sec (10) -> -- combine two domain in one using Record Pair:=Record(x1:Integer,x2:Integer)
$$ \mbox{\rm Record(x1: Integer,x2: Integer)} \leqno(10) $$
Type: Domain Time: 0 sec pair(x:Integer,y:Integer):Pair == [x,y]
Function declaration pair : (Integer,Integer) -> Record(x1: Integer, x2: Integer) has been added to workspace. Type: Void Time: 0 sec -- gcd2(arg:Pair):Integer == gcd(arg.x1,arg.x2)
Function declaration gcd2 : Record(x1: Integer,x2: Integer) -> Integer has been added to workspace. Type: Void Time: 0.01 (IN) = 0.01 sec gcd2cached:CachedFunction(Pair,Integer):=cachedFunction gcd2
Compiling function gcd2 with type Record(x1: Integer,x2: Integer) -> Integer
LISP output: (() #<FUNCTION |*1;gcd2;1;initial|>) Type: CachedFunction(Record(x1: Integer,x2: Integer),Integer) Time: 0.01 (OT) = 0.01 sec gcdCached(x1:Integer,x2:Integer):Integer == gcd2cached pair(x1,x2)
Function declaration gcdCached : (Integer,Integer) -> Integer has been added to workspace. Type: Void Time: 0.01 (IN) = 0.01 sec )set message time on
for i in 1..1000 repeat for j in 1..100 repeat gcd(i,j)
Type: Void Time: 3.78 (EV) = 3.78 sec for i in 1..1000 repeat for j in 1..100 repeat gcdCached(i,j)
Compiling function pair with type (Integer,Integer) -> Record(x1: Integer,x2: Integer) Compiling function gcdCached with type (Integer,Integer) -> Integer
Some or all expressions may not have rendered properly, because Latex returned the following error:
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