Expressiveness of Full First-Order Constraints in the Algebra of Finite or Infinite Trees

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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. Constraints, Springer Verlag, 2003, 8 (3), pp.283-302. �hal-00144931�

in the algebra of nite or innite trees

Alain Colmerauer Thi-Bih-Hanh Dao

Marh5, 2003

Abstrat

We are interested inthe expressiveness of onstraints represented by general rstorder

formulae,withequalityasuniquerelationsymbolandfuntionsymbolstakenfromaninnite

set F. The hosen domainis the setof trees whose nodes, inpossibly innite number, are

labelled by elements of F. The operation linked to eah element f of F is the mapping

(a

1

;:::;a

n

) 7!b,where bis thetreewhose initial nodeis labelledf andwhose sequeneof

daughtersisa1;:::;an.

Werstonsidertreeonstraintsinvolvinglongalternatedsequenesofquantiers9898:::.

Weshowhowtoexpresswinningpositionsoftwo-persongameswithsuhonstraintsandapply

ourresultstotwoexamples.

Wethenonstrutafamilyofstronglyexpressivetreeonstraints,inspiredbyaonstru-

tiveproofofaomplexityresultbyPawelMielnizuk. Thisfamilyinvolvesthehugenumber

(k), obtained by top downevaluating a power tower of 2's, of height k. By a tree on-

straintofsizeproportionaltok,itisthenpossibletodeneatreehavingexatly(k)nodes

or to expressthe multipliationtable omputedby aProlog mahine exeuting up to (k)

instrutions.

ByreplaingthePrologmahinewithaTuringmahineweshowthequasi-universalityof

treeonstraints,thatistosay,theabilitytooniselydesribetreeswhihthemostpowerful

mahine will neverhave timeto ompute. We alsoredisover thefollowing resultof Sergei

Vorobyov: the omplexity of analgorithm, deidingwhether atree onstraint without free

variablesistrue,annotbeboundedabovebyafuntionobtainedfromniteompositionof

simplefuntionsinludingexponentiation.

Finally,takingadvantageofthefatthatwehaveatourdisposalanalgorithmforsolving

suhonstraintsinalltheirgeneralities,weprodueasetofbenhmarksforseparatingfeasible

examplesfrompurelyspeulativeones. Amongotherswenotiethatitispossibletosolvea

onstraintof5000symbolsinvolving160alternatingquantiers.

1 Introdution

Thealgebraof(possibly)innitetreesplaysafundamentalroleinomputersiene: itisamodel

for data strutures, program shemes and program exeutions. As early as 1976, Gerard Huet

proposed an algorithm for unifying inniteterms, that is solvingequations in that algebra[12℄.

Bruno Courelle has studied the properties of innite trees in the sope of reursive program

shemes[8, 9℄. Alain Colmerauerhasdesribedthe exeutionof Prolog II, III and IV programs

in terms of solving equations and disequations in that algebra [4, 5, 6, 1℄. Mihael Maher has

introdued andjustiedaomplete theoryofthealgebraofinnitetrees [13℄. Amongothers,he

hasshown thatin this theory,and thusin thealgebraof innitetrees,anyrstorder formulais

equivalenttoaBooleanombinationofonjuntionsofequations(partiallyortotally)existentially

quantied. SergeiVorobyovhas shown that the omplexityof an algorithm, deiding whether a

formula without free variables is true in that theory, annot be bounded above, by a funtion

obtainedfrom nite omposition of simplefuntions, inludingexponentiation [16℄. Pawel Miel-

nizuk[14℄ hasshownasimilarresultinthetheoryoffeaturetrees,butwithamoreonstrutive

method,whihhasinspiredalargepartof theworkpresentedhere.

ofinnitetrees[10, 11℄. Thepurpose ofthis paperisnotthepresentationof thisalgorithm,but

of examples, rst imagined as tests, then extended to show the expressiveness of suh general

onstraints. Thepaperisorganizedasfollows:

We end this rst setion bymaking lear thenotions of innitetrees algebraand rst-order

onstraintsinthat algebra.

Intheseondsetion,weusetwo-partnergamesfordeningonstraintsinvolvinglongsequenes

ofquantiers9898:::.

Inthethirdsetion,weintrodueaompositiononstraintwhihrepeatsthesameonstraint

atremendouslylargenumberoftimes. A longpartofthesetion isdevotedto provingitsmain

property.

Atsetion four, wemoveon to themost expressiveonstraintswe know. Theyare obtained

byhangingthe nature of the repeated onstraint. We produe several examples,among whih

aonstraintdening ahugenite treeand analmost perfet multipliation onstraint. Then by

simulating a Turing mahine, we show the quasi-universality of tree onstraint, that is to say,

theability to onisely desribetrees whih the most powerfulmahine will neverhave time to

ompute. ThisalsoallowsustogiveanotherproofoftheomplexityresultofSergeiVorobyov.

Weonludebydisussions andbenhmarksseparatingthefeasibleexamplesfromthepurely

speulativeones.

1.1 The algebra of innite trees

Asusual,afuntionsymbolisasymboltogetherwithanon-negativeinteger,itsarity. Trees,with

nodeslabelled byfuntion symbols,are well known objetsin theomputersieneworld. Here

aresomeofthem:

. . . ^{. . .}

f f

s

f f

f f

f f

s f

s

s s

f

s s

a b a b

a

a b

a

a

a

a

with, ofourse, the funtion symbolsa;b;s;f having respetivelythe arities0;0;1;2. Note that

thersttreeistheonlyonehavinganitesetof nodes,butthat theseondonehasstill anite

set of (patterns of) subtrees. Wedenote by A theset of all trees built on theinnite set F of

funtionsymbols.

1

Weequip Awithaset ofonstrutionoperations,oneforeahelementf 2F,whihare the

mappings(a

1

;:::;a

n

)7!b,where nisthearityof f andb thetreewhose initialnode islabelled

1

Morepreiselywedenerstanodetobeawordonthesetofstritlypositiveintegers. Atree,builtonF,is

thenamappinga:E!F,whereE isanon-emptysetofnodes,eahonei1:::i

k

(withk0)satisfyingthetwo

onditions:(1)ifk>0theni

1 :::i

k1

2E,(2)ifthearityofa(i

1 :::i

k

)isn,thenthesetofnodesofEoftheform

i

1 :::i

k i

k+1

isobtainedbygivingtoi

k+1

thevalues1;:::;n.

f andwhose sequeneof daughtersis(a

1

;:::;a

n ):

. . . . . .

a a

a a

f

1 n

1 n

The set A, together with the onstrution operations, is the algebra of innite trees and is

denotedby(A;F).

1.2 Tree onstraints

Weareinterestedin theexpressivenessof onstraintsrepresentedbygeneralrstorder formulae,

withequalityasuniquerelationsymbolandfuntion symbolstakenfromaninniteset F. These

treeonstraintsareofoneofthe10forms:

s=t; true; false; :('); ('^ ); ('_ ); ('! ); ('$ ); 9x'; 8x';

where'and are shortertreeonstraints,x avariable takenfromaninnitesetands;tterms,

thatistosayexpressionsofoneoftheforms:

x; ft

1 :::t

n

;

wheren0,f 2F,witharityn,andthet

i

'sareshorterterms.

The variables represent elements of the set A of trees built on F, the equality symbol is

interpretedas equalityand thefuntion symbols f are interpreted asonstrutionoperationsin

thealgebraofinnitetrees(A;F). Thusaonstraintwithoutfreevariablesiseither trueorfalse

andaonstraint'(x

1

;:::;x

n

)with nfreevariables x

i

establishes ann-ary relationin theset of

trees. Forexamplethetreeonstraint

(u;v;w;x;y) def

= 9z

z=f(x;f(y;z))^

z=f(u;f(v;f(w;z)))

isequivalentto

x=u^ y=v ^x=w ^ y=u ^x=v ^ y=w:

Thusitexpressesthatthetreesu;v;w;x;y areequal.

2 Long nesting of alternated quantiers

Toshow theexpressive powerof our onstraints,we startexamining deeply embedded quanti-

ations. We onsider two-partnergames and haraterize the positions from whih it is always

possibletowin inatmostkmoves. Inthismannerweobtainonstraintsinvolvinganalternated

sequeneof2k quantiers.

2.1 Winning positions in a two-partner game

Let(V;E)beadiretedgraph,withV asetofvertiesandEVV asetofedges. ThesetsV

andE maybe inniteand theelementsof E arealsoalled positions. Weonsiderthefollowing

two-partnergame: startingfromaninitialpositionx

0

,eahpartnerinturn,hoosesapositionx

1

suh that (x

0

;x

1

) 2 E, then aposition x

2

suh that (x

1

;x

2

) 2 E, then aposition x

3

suh that

2

Infat,theonstrutionoperationlinkedtothe n-arysymbolf ofF isthemapping(a1;:::;an)7!b,where

thea

i

'sareanytrees andbisthe treedenedas followsfromthea

i

'sandtheirsetsofnodesE

i

's: the setE of

nodesofbis f"g[fixjx2E

i

andi21::ng and,foreahx2E,ifx=",thenb(x)=fandifxisoftheformiy,

withibeinganinteger,b(x)=a

i (y).

2 3

followinginnitegraphsorrespondtothetwofollowinggames:

1

0 2 3 4 5 6

Game 1 A non-negative integer i

is given and eah partner, in turn,

subtrats1or2fromi,butkeeping

inon-negative. Therstpersonwho

annotplayanymorehaslost.

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 2An orderedpair (i;j) ofnon-negativeintegersis

givenandeahpartner,inturn,hoosesoneoftheintegers

i;j. Depending onwhether thehosenintegeruis oddor

even,hetheninreasesordereasestheotherintegerv by

1,butkeepingvnon-negative. Therstpersonwhoannot

playanymorehaslost.

Letx2V beanyvertexofthediretedgraph(V;E)andsupposethatitistheturn ofperson

Atoplay. Theposition xissaidtobek-winningif,nomatterthewaytheotherpersonB plays,

itisalwayspossibleforAtowin,afterhavingmadeatmostkmoves. Thepositionxissaidtobe

k-losingif,nomatterthewayAplays,B analwaysforeAtoloseandtoplayatmostkmoves.

3

2.2 Expressing k-winning positions by a rst-order onstraint

Insteadoftreeonstraints,onsidergeneralonstraints,whiharerst-orderformulaewhosefun-

tionand relation symbols are interpreted in a given struture D, that is a set D together with

operations and relations. Let G = (V;E) be the graph representing a two-partner game, with

V D,andletmove(x;y)beaonstraintinDsuhthat:

4

foreaha2V andb2V, move(a;b) i (a;b)2E: (1)

We want to build onstraints winning

k

(x) and losing

k

(x) in D, suh that, for k 0 and eah

a2V,

winning

k

(a) i aisak-winningpositionofG;

losing

k

(a) i aisak-losingpositionofG:

>From the denition of k-winning and k-losingpositions in the preeding setion, we inferrst

that,foreveryk0:

winning

0

(x) $ false;

winning

k +1

(x) $ 9y move(x;y)^losing

k (y);

(2)

losing

k

(x) $ 8y move(x;y)!winning

k

(y): (3)

3

Fortherstgame,itanbeshownthatthesetofk-winningpositionsisthesetofnon-negativeintegersisuh

thati<3k andiisnot divisibleby3. For theseondgame,it isthe set oforderedpairs(i;j)of non-negative

integers,suhthati+j<2kandi+j isodd.

4

If'(x

1

;:::;x

n

)denotesaonstraintwhosefreevariablesareamongthex

i

's,andifthea

i

'sareelementsofD,

then'(a

1

;:::;a

n

)istheinterpretationoftheonstraintinD,wherethefreeourrenesofthex

i

'sareinterpreted

bytheorrespondingai's.

bereplaedby

winning

k (x) $

9ymove(x;y)^:(

9xmove(y;x)^:(

9ymove(x;y)^:(

9xmove(y;x)^:(

:::

9ymove(x;y)^:(

9xmove(y;x)^:(

false ):::)

|{z}

2k

(4)

Thusequivalenes(4)and(3)provideanexpliitwayforbuildingtheonstraintsweneed. Notie

that,bymovingthenegationsdownin(4), wegetanestingof2k alternatedquantiers.

5

Itispossibleto keepequivalene(4),whileweakeningondition(1). Werstremark,that for

anynon-negativek,thefollowingpropertyholds:

Property 1 Let three direted graphs be of the form G

1

= (V

1

;E

1 ), G

2

= (V

2

;E

2

) and G =

(V

1 [V

2

;E

1 [E

2

). ThegraphsG

1

andGhavethe sameset ofk-winningpositions, if both:

1. the setsof verties V

1 andV

2

are disjoint,

2. for eahx2V

2

,thereexistsy2V

2

with(x;y)2E

2 .

Indeed,fromtherstonditionitfollowsthatE

1 and E

2

aredisjointand thus thattheset ofk-

winningpositionsofGistheunionofthesetofk-winningpositionsofG

1

withthesetofk-winning

positionsofG

2

. Butthis lastsetisemptybeauseoftheseondondition.

Itfollowsthat:

Property 2(Generalized move) Equivalene(4)holdsalsoforanyonstraintmove(x;y)obey-

ingthe threeonditions:

1. for eaha2V andb2V,if(a;b)2E thenmove(a;b),

2. for noa2D V andnob2V wehave move(a;b),

3. for eaha2D V,thereexistsb2D V suhthatmove(a;b).

2.3 Example : tree onstraint for game 1

Wenowreonsidergame 1introduedin setion2.1. Asstruture Dwe takethe algebra(A ;F)

ofinnite treesonstruted on aset F of funtion symbols inludingamong others thesymbols

0;s, of respetivearities0;1. We ode thevertiesi of thegame graphby thetrees s i

(0).

6

Let

G=(V;E)bethegraphobtainedthisway.

Then,asgeneralizedmove(x;y)onstraintweantake:

move(x;y) def

= x=s(y)_x=s(s(y))_(:(x=0)^:(9ux=s(u))^x=y)

andaordingtoproperty2,thesetofk-winningpositionsofgame1isthesetofsolutionsinxof

theonstraintwinning

k

(x) denedby(4).

5

Fromthethreeequivalenesin(2)and(3)itfollowsthat8xwinning

k

(x)!winning

k+1

(x)and8xlosing

k (x)!

losing

k+1

(x),foranyk0. Indeed,fromthe rstandthelastequivaleneweonludethattheimpliationshold

fork=0and,ifweassumethattheyholdforaertaink0,fromthelasttwoequivalenesweonludethatthey

alsoholdfork+1.

6

Ofourse,s 0

(0)=0ands i+1

(0)=s(s i

(0)).

k

x=s(0)_x=s(s(0))

and,fork=2,to

x=s(0)_x=s(s(0))_x=s(s(s(s(0))))_x=s(s(s(s(s(0))))):

2.4 Example : tree onstraint for game 2

Wealsoreonsider game2introduedin setion 2.1. As strutureD wetakethe algebra(A ;F)

ofinnite treesonstruted on aset F offuntion symbols inludingamong others thesymbols

0;f;g;,of respetivearities0;1;1;2. We ode theverties(i;j) ofthegame graphbythe trees

(i;j)with i=(fg) i

2

(0), if iis even, and i=g(i 1), ifi isodd.

7

Let G=(V;E) be thegraph

obtainedthisway.

Theperspiaiousreader will onvinehimself that, as generalizedonstraintmove(x;y), we

antake:

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

Aordingtoproperty2,thesetofk-winningpositionsofgame2isthesetofsolutionsinxofthe

onstraintwinning

k

(x)denedin (4).

Forexample,fork=1theonstraintwinning

k

(x)isequivalentto

x=(g(0);0)_x=(0;g(0))

and,fork=2,to

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)

3 Composition of onstraints

We now move on to a systemati way of ompressing a onjuntion of a very large number of

onstraintsintoasmall onstraint.

7

Ofourse,(fg) 0

(x)=xand(fg) i+1

(x)=f(g((fg) i

(x))).