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Expressiveness of Full First Order Constraints in the Algebra of Finite or Infinite Trees
Alain Colmerauer, Thi-Bich-Hanh Dao
To cite this version:
Alain Colmerauer, Thi-Bich-Hanh Dao. Expressiveness of Full First Order Constraints in the Algebra
of Finite or Infinite Trees. Principles and Practice of Constraint Programming - CP 2000, 2000,
Singapore, Singapore. pp.172-186. �hal-00144924�
in the algebra of nite or innite trees
AlainColmerauerandThi-Bih-HanhDao
Laboratoired'InformatiquedeMarseille,CNRS,
UniversitésdelaMéditerranéeetdeProvene
Abstrat. Weareinterestedintheexpressivenessofonstraintsrepre-
sentedbygeneralrstorderformulae,withequalityasuniquerelational
symbolandfuntionalsymbolstakenfromaninnitesetF.Thehosen
domainisthe setof treeswhosenodes,inpossiblyinnitenumber,are
labeledbyelementsofF.Theoperationlinkedto eahelementf of F
isthe mapping(a1;:::;an)7!b,where bisthetree whoseinitial node
islabeledf andwhosesequeneofdaughtersisa
1
;:::;a
n .
Werstonsideronstraintsinvolvinglongalternatedsequenesofquan-
tiers9898::: .Weshowhowtoexpresswinningpositionsoftwo-partners
gameswithsuhonstraintsandapplyourresultstotwoexamples.
Wethen onstruta familyofstrongly expressiveonstraints, inspired
byaonstrutiveproofofaomplexityresultbyPawelMielnizuk.This
familyinvolvesthehugenumber(k),obtainedbyevaluatingtopdown
apowertowerof2's,ofheight k.Withelementsofthis family,ofsizes
at most proportional to k, we dene a nite tree having (k) nodes,
and we expressthe resultof aProlog mahine exeuting atmost (k)
instrutions.
By replaing the Prolog mahine by a Turing mahine we redisover
thefollowingresultofSergeiVorobyov:theomplexityofanalgorithm,
deidingwhether aonstraint withoutfreevariables istrue, annotbe
boundedabovebyafuntionobtainedbyniteompositionofelemen-
taryfuntionsinludingexponentiation.
Finally,takingadvantageofthefatthatwehaveatourdisposalanalgo-
rithmforsolvingsuhonstraintsinalltheirgenerality,weprodueaset
ofbenhmarksforseparating feasibleexamplesfrompurelyspeulative
ones.Amongotherswesolveonstraintsinvolvingalternatedsequenes
ofmorethan160quantiers.
1 Introdution
The algebra of (possibly) innite trees plays a fundamental at in omputer
siene:itisamodelfordatastrutures,programshemesandprogramexeu-
tions.Asearlyas1976,GérardHuetproposedanalgorithmforunifyinginnite
terms,thatissolvingequationsinthatalgebra[11℄.BrunoCourellehasstudied
thepropertiesof innitetreesin thesopeofreursiveprogramshemes[8,9℄.
AlainColmerauerhasdesribedtheexeutionofPrologII,IIIandIVprograms
trees[12℄.Amongothers,hehasshownthatinthistheory,andthusinthealgebra
ofinnitetrees, anyrstorder formulaisequivalenttoaBooleanombination
ofonjuntionsofequations(partiallyortotally)existentiallyquantied.Sergei
Vorobyovhas shown that the omplexity of an algorithm, deiding whether a
formulawithoutfreevariablesistrueinthattheory,annotbeboundedabove,
byafuntionobtainedbyniteomposition ofelementaryfuntions,inluding
exponentiation[14℄.PawelMielnizukhasshown asimilar resultin thetheory
offeaturetrees,but withamoreonstrutivemethod, whihhasinspiredsome
ofourexamples[13℄.
Wehavereentlydevelopedanalgorithmforsolvinggeneralrstorder on-
straints in the algebra of innite trees [10℄. The purpose of this paper is not
thepresentationofthisalgorithm,butofexamples,rstimaginedastests,then
extended to show the expressiveness of suh general onstrains. The paper is
organizedasfollows.
(1)Weendthisrstsetionbymakinglearthenotionsoftreealgebraand
rstorderonstraintsin thatalgebra.
(2) Inthe seond setion we onsider onstraintsinvolvinglongalternated
sequenesofquantiers9898:::. Weshowhowto expresswinning positions of
two-partnersgameswithsuhonstraintsandapplyourresultstotwoexamples.
(3) In the third setion, we investigate the most expressive family of on-
straintsweknow.Itinvolvesthetrulyhugenumber(k),obtainedbyevaluating
topdown atowerofpowersof2's,ofheightk.With elementsof thisfamily, of
sizes at mostproportionalto k, we denea nitetree having(k) nodes, and
weexpresstheresultofaPrologmahine exeutingatmost(k).Byreplaing
thePrologmahinebyaTuringmahineweredisovertheomplexityresultof
SergeiVorobyovmentionedatthebeginningofthissetion.This parthasbeen
stronglyinuenedbythework ofPawelMielnizuk[13℄.
(4)We onludebydisussions andbenhmarks separatingthefeasibleex-
amplesfromthepurelyspeulativeones.
1.1 The algebra ofinnite trees
Treesare well known objets in theomputer sieneworld. Here aresome of
them:
. . . . . .
0 1
0 1
0
0 0 0
ff s
f f
f f
f f
f f s s s s s
0 1 0 s
takenfromasetF offuntionalsymbols,whihweassumeto beinnite.Note
thatthersttreeistheonlyonehavinganitesetofnodes,butthattheseond
onehasstillanitesetof(patternsof)subtrees.WedenotebyAthesetofall
trees 1
onstrutedonF.
Weintroduein Aa set ofonstrutionoperations 2
, oneforeah element
f 2F whihisthemappings(a
1
;:::;a
n
)7!b,wherenisthearityoff andbthe
treewhoseinitialnodeislabeledf andthesequeneofdaughtersis(a
1
;:::;a
n )
andwhihbeshematizedas
1 1
. . . . . .
f
a a
a an
n
Wethusobtainthealgebraof innite treesonstrutedonF,whihwedenote
by(A ;F).
1.2 Treeonstraints
We are interested in the expressiveness of onstraints represented by general
rst order formulae, with equality as unique relational symbol and funtional
symbolstakenfrom aninnitesetF.These tree onstraintsare ofoneofthe9
forms:
s=t; true; false; :(p); (p^q); (p_q); (p!q); 9xp; 8xp;
where p andq are shorter treeonstraints, x avariable taken from aninnite
set ands;tterms,that areexpressionsofoneoftheforms
x; ft
1 :::t
n
wheren0,f 2F,witharityn,andthet
i
'sareshorterterms.
ThevariablesrepresentelementsofthesetAoftreesonstrutedonF and
thefuntionalsymbolsf areinterpretedasonstrutionoperationsinthealgebra
ofinnitetrees(A ;F).Thusaonstraintwithoutfreevariablesiseithertrueor
false anda onstraint p(x
1
;:::;x
n
) with nfree variablesx
i
establishan n-ary
relationinthesetoftrees.
1
Morepreiselywedenerstanodetobeawordonstrutedonthesetofstritly
positive integers. A tree a, onstruted onF, is thena mapping of type E !F,
where E isanon-emptysetofnodes,eahonei1:::i
k
(with k0) satisfyingthe
twoonditions:(1)if k>0then i
1 :::i
k1
2E,(2)if thearity ofa(i
1 :::i
k ) is n,
thenthesetofnodesofE ofthe formi1:::ikik+
1
is obtainedbygiving toik
+1 the
values1;:::;n.
2
Infat,theonstrutionoperationlinkedtothen-arysymbolfofF isthemapping
(a1;:::;an) 7!b,where the ai's are any trees and bis the treedened as follows
fromtheai's andtheirsetofnodesEi's:thesetE ofnodesof ais f"g[fixjx2
Ei andi21::ng and,foreahx2E,ifx=",thena(x)=f andifxisoftheform
iy,withibeinganinteger,a(x)=a(y).
We rst introdue the notions of k-winning and k-losing position in any two-
partners gamesandin twoexamples.Weshowhowto express,in any domain,
thesetofk-winningpositionsbyaonstraint.Weendthesetionbyexpressing
the k-winning positions of the two examples by tree onstraints involving an
alternatedembeddingof2k quantiers.
2.1 Winning positionsin a two-partners game
Let(V;E)beadiretedgraph,withV asetofvertiesandEV V asetof
edges. Thesets V and E maybe empty andthe elements ofV are alsoalled
positions. Weonsideratwo-partnersgamewhih,given aninitialpositionx
0 ,
onsists,oneafteranother,inhoosingapositionx
1
suhthat(x
0
;x
1
)2E,then
apositionx
2
suh that(x
1
;x
2
)2E,thenapositionx
3
suhthat (x
2
;x
3 )2E
andsoon... Therstonewhoannotplayanymorehaslostandtheotherone
has won. For example thetwo following innitegraphs orrespond to the two
followinggames:
1
0 2 3 4 5 6
0,0 1,0 2,0 3,0
0,1 1,1 2,1 3,1
0,2 1,2 2,2 3,2
0,3 1,3 2,3 3,3
Game 1Anon-negativein-
tegeriisgivenand,oneafter
another, eah partner sub-
trats1or2fromi,butkeep-
ing i non-negative.Therst
person whoannot play any
morehaslost.
Game 2 An ordered pair (i;j) of non-negative
integers is given and, one after another, eah
partner hoosesoneoftheintegersi;j.Depend-
ing onthefat that thehosenintegeruisodd
oreven,hetheninreasesordereasestheother
integerv by1,but keepingv non-negative.The
rstpersonwhoannotplayanymorehaslost.
Letx2V beanyvertexofthedireted graph(V;E)andsupposethat itis
theturnofpersonAtoplay.Thepositionxissaidtobek-winningif,nomatter
thewaytheotherperson B plays,itis alwayspossiblefor Ato winin making
at mostkmoves.Thepositionx is saidtobek-losingif, nomatterthewayA
plays,B analwaysforeAtoloseand toplayatmostkmoves.
Considerthe twopreeding graphs and mark with +k the positions whih
are k-winningand with k thepositions whih are k-losing, witheah time k
being as small aspossible. Vertex 0of the rst graphand vertex (0;0) of the
ndsuessivelythesetofvertiestobemarkedby+1,then 1,then+2,then
2,then+3, then 3,andsoon.Weget
-0 +1 +1 -1 +2 +2 -2
-0 +1 -1 +2 -2 +3
+1 +2 +3
+3 +3 +3 +3 -2 +2 -1 +2
-2 -3
-3 -2
andonvineourselvesthattheset ofk-winningpositionsofgame1is
fi2Nji<3kandimod36=0g
andofgame2
f(i;j))2N 2
ji+j<2kand(i+j)mod2=1}:
whereNisthesetofnon-negativeintegers.
2.2 Expressingk-winningpositions bya onstraint
LetD beadomain, that isanon-emptysetand letG=(V;E)thegraphofa
two-partnersgame, withV D.We willexpress thek-winningpositions ofG
byaonstraintinDinvolvinganembedding989:::of2kalternatedquantiers.
Letus introdueinD thepropertiesmove,winning
k
et losing
k
,dened by
move(x;y) $(x;y)2E;
winning
k
(x)$x isak-winningpositionofG;
losing
k
(x) $x isak-losingpositionofG:
(1)
InDwethenhavetheequivalenes,forallk0:
winning
0
(x) $false;
winning
k +1
(x)$9ymove(x;y)^losing
k (y);
losing
k
(x) $8ymove(x;y)!winning
k (y):
(2)
Contrarytowhat wemaybelieve,itfollowsthatwehave:
winning
k
(x)!winning
k+1
(x); losing
k
(x)!losing
k+1 (x):
Indeed, from the rst and the last equivalene of (2) we onlude that these
impliationsholdfork=0and,ifweassumethattheyholdforaertaink0,
k
winning
k (x) $
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
9ymove(x;y)^:(
9xmove(y;x)^:(
9ymove(x;y)^:(
9xmove(y;x)^:(
:::
9ymove(x;y)^:(
9xmove(y;x)^:(
false ):::)
|{z}
2k 3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(3)
whereofourseallthequantiersapplyonelementsofD.Bymovingdownthe
negations,wethusgetanembeddingof2k alternatedquantiers.
Inequivalene (3) it is possible to use a moregeneral denition of move
than the one given in (1). We rst remark, that for any non-negative k, the
followingpropertyholds:
Property 1 Let three direted graphs be of the form G
1
= (V
1
;E
1 ), G
2
=
(V
2
;E
2
)andG=(V
1 [V
2
;E
1 [E
2
).ThegraphsG
1
andGhavethesamesetof
k-winningpositions, ifboth:
1. thesetsof vertiesV
1 andV
2
aredisjoint,
2. forallx2V
2
, thereexistsy2V
2
with(x;y)2E
2 .
Indeed,from therst onditionitfollowsthatE
1 andE
2
aredisjointandthus
that the set of k-winning positions of G is the union of the set of k-winning
positionsofG
1
withthesetofk-winningpositionsofG
2
.Thislastsetisempty
beauseoftheseondondition.
Itfollowsthat:
Property 2(Generalizedmove relation) Equivalene(3)holdsalsoforany
move relationobeyingtothe twoonditions:
1. forallx2V andy2V, move(x;y) $ (x;y)2E,
2. forallx2D V there existsy2D V suhthat move(x;y).
2.3 Formalizinggame 1in the algebraof innitetrees
Wenowreonsidergame1introduedin setion2.1.AsdomainD wetakethe
set A of trees onstruted on a set F of funtional symbolsinluding among
others thesymbols0;s, of respetivearities 0;1.We odethe vertiesi of the
gamegraphbythetrees 3
s i
(0).LetG=(V;E)bethegraphobtainedthisway.
As generalized relation move we then antake in the algebra of innite
trees:
move(x;y) def
= x=s(y)_x=s(s(y))_(:(x=0)^:(9ux=s(u))^x=y)
3 0 i+1 i
ofsolutionsinxof theonstraintwinning
k
(x) denedin(3).
Forexample,withk=1theonstraintwinning
k
(x) isequivalentto
x=s(0)_x=s(s(0))
andwithk=2to
x=s(0)_x=s(s(0))_x=s(s(s(s(0))))_x=s(s(s(s(s(0)))))
2.4 Formalizinggame 2in the algebraof innitetrees
Wealsoreonsidergame2introdued insetion2.1. AsdomainDwetakethe
set A of trees onstruted on a set F of funtional symbolsinluding among
others thesymbols0;f;g;,of respetivearities0;1;1;2.Weodethe verties
(i;j)of thegame graphby thetrees (i;j)with i= (fg) i
2
(0) ifi is even, and
i=g(i 1)ifiisodd 4
. LetG=(V;E)bethegraphobtainedthis way.
Theperspiaiousreaderwill onvinehimself that, asgeneralizedrelation
move,weantakeinthealgebraofinnitetrees:
move(x;y) def
= transition(x;y)_(:(9u9vx=(u;v))^x=y)
with
transition(x;y) def
= 2
6
6
6
6
6
6
4
9u9v9w
(x=(u;v)^y=(u;w))_
(x=(v;u)^y=(w;u))
^
(9iu=g(i)^su(v;w))_
(:(9iu=g(i))^pred(v;w))
3
7
7
7
7
7
7
5
su(v;w)
def
=
((9jv=g(j))^w=f(v))_
(:(9jv=g(j))^w=g(v))
pred(v;w) def
= 2
6
6
6
6
6
4
(9jv=f(j)^
(9kj=g(k)^w=j)_
(:(9kj=g(k))^w=v)
)_
(9jv=g(j)^
(9kj=g(k)^w=v)_
(:(9kj=g(k))^w=j)
)_
(:(9jv=f(j))^:(9jv=g(j))^:(v=0)^w=v) 3
7
7
7
7
7
5
Aordingto property2,theset of k-winningpositions ofgame 2is theset of
solutionsinx oftheonstraintwinning
k
(x)dened in(3).
Forexample,withk=1theonstraintwinning
k
(x) isequivalentto
x=(g(0);0)_x=(0;g(0)))
andwithk=2to
x=(0;g(0))_x=(g(0);0)_x=(0;g(f(g(0))))_
x=(g(0);f(g(0)))_x=(f(g(0));g(0))_x=(g(f(g(0)));0)
4
0 i+1 i
Afterallthesequantiers,wemoveto onstraints,whih aresoexpressivethat
theirsolvingbeomesquasi-undeidable.
3.1 Dening a huge nitetreeby a onstraint
Weset(k)=2 2
: :
: 2
,withk ourrenesof2.Morepreiselywetake
(0)=1; (k+1)=2 (k )
;
withk0.Thefuntioninreasesinastunningway,sine(0)=1,(1)=2,
(2) = 4, (3) = 16, (4) = 65536 and (5) = 2 65536
. Thus (5) is greater
than10 20000
,anumberprobablymuhgreaterthanthenumberofatomsofthe
universeorthenumberofnanoseondswhihelapsedsineitsreation!
Wesuppose that the set Aof trees isonstruted ona set F of funtional
symbolsinludingamong others thesymbols0;1;2;3;s;f,of respetivearities
0;0;0;0;1;4.Fork0letusintroduetheonstraint:
huge
k (x)
def
= 9z triangle
k
(3;x;z;0)
withstillfork0,
triangle
0
(t;x;z;y) def
= z=x^z=y
triangle
k +1
(t;x;z;y) def
= 2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4 [9u
1 9u
2
z=f(x;u
1
;u
2
;y)℄
^
2
6
6
6
6
6
6
6
6
6
6
4 8t
0
8y 0
8z 0
(t 0
=1_t 0
=2)^
triangle
k (t
0
;z;z 0
;y 0
)
!
2
6
6
6
6
4 (t
0
=1^form1(y 0
))_
(t 0
=2^ 2
6
6
4
9u9v form2(u;y 0
;v)^
(t=1!trans1(u;v))^
(t=2!trans2(u;v))^
(t=3!trans3(u;v)) 3
7
7
5 3
7
7
7
7
5 3
7
7
7
7
7
7
7
7
7
7
5 3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5 (4)
and
form1(x) def
= 9u
1 :::9u
4
x=f(u
1
;f(u
2
;u
2
;u
2
;u
2 );f(u
3
;u
3
;u
3
;u
3 );u
4 )
form2(x;z;y) def
= 9u
1 :::9u
6 z=f(u
1
;f(u
1
;u
2
;u
3
;x);f(y;u
4
;u
5
;u
6 );u
6 )
trans1(x;y) def
= 9u
1 :::9u
4 x=f(u
1
;u
2
;u
3
;u
4
)^(y=u
2
_y=u
3 )
trans2(x;y) def
= trans1(x;y)_x=y
def
