Computational problems in infinite groups -8mm - CARMA · PDF fileComputational problems in...

Computationalproblemsininfinitegroups

Laura Volker Murray Andrew BuksCiobanu Diekert Elder Rechnitzer vanRensburg

MathematicsandComputation, CARMA June2015

PartI:Equationsinfreegroups

Let A = a1, a−11 , . . . , ad, a

−1d .

Let RA ⊂ A∗ bethesetofallwordsthatdonotcontainany aia−1i or

a−1i ai pair. Wecallsuchwords reduced.

The freegroup on A, denoted FA, isthesetofallreducedwordswiththeoperationof concatenatethenreduce.

Let Ω = X1, X−11 , . . . , Xs, X−1

s beanotherset.

An equationin FA isanexpression U = V where U, V ∈ (A ∪ Ω)∗.

...Eg: a−1X = Yb aXXb = YYbX

A solution totheequationisamap Xj 7→ uj, X−1j 7→ u−1

j with uj ∈ RAwhichmakes U = V truein FA.

Let A = a1, a−11 , . . . , ad, a

−1d .

Let Ω = X1, X−11 , . . . , Xs, X−1

s beanotherset.

Let A = a1, a−11 , . . . , ad, a

−1d .

Let Ω = X1, X−11 , . . . , Xs, X−1

s beanotherset.

Let A = a1, a−11 , . . . , ad, a

−1d .

Let Ω = X1, X−11 , . . . , Xs, X−1

s beanotherset.

Equationsinfreegroups

Eg: aXXb = YYbX

Y→ aY aXXb = aYaYbX

XXb = YaYbX

X→ Xb−1 Xb−1Xb−1b = YaYbXb−1

Xb−1X = YaYbXb−1

History

1982Makanin: algorithmtodecideifanequationinafreegrouphasasolution

Algorithmextremelyinefficient, timecomplexity DTIME

222poly(n)

)1987Razborov: algorithmicdescriptionofallsolutions(usingMakanin)

1990KoscielskiandPacholski: Makanin’sschemenotprimitiverecursive

1998Plandowski: newapproachtosolvingequationsover freemonoidsinPSPACE using datacompression

2000Gutierrez: solvingequationsoverfreegroupsinPSPACE

2013Jez: new recompression technique–simpliedallpreviousproofs

2014Diekert, JezandPlandowski: NSPACE(n2) algorithm

History

222poly(n)

1987Razborov: algorithmicdescriptionofallsolutions(usingMakanin)

History

222poly(n)

History

222poly(n)

History

222poly(n)

History

222poly(n)

History

222poly(n)

History

222poly(n)

Howcomplicatedaresolutionsets?

Wewereinterestedinunderstandingthesetofsolutionstoanequationasaformallanguage

u1# . . .#us | uj ∈ RA, Xj 7→ uj solves U = V.

Weaskedhowcomplicatedsolutionssetscanbe, intermsofformallanguages.

Itisclearthatthesetofsolutionstoanyequationisrecognisedbyalinearboundedautomatonsosolutionsarecontext-sensitive.

Buttheyareevensimpler:

Theorem(Ciobanu, Diekert, E 2015)

ThesetofsolutionsinreducedwordstoanequationinafreegroupisEDT0L.

Moreprecisely, weprovethatthesetofsolutionsisequalto

h(#) | h ∈ Rwhere R isaregularlanguageofendomorphismsover C ⊇ A ∪ #,

theNFA accepting R canbeconstructedinNSPACE(n log n),

andtheequationhaszero/finitelymanysolutionsifftheNFA hasnoacceptstate/nocycle.

Thisisasurprisingresult–beforeMakaninitwasthoughttheproblemcouldbeundecidable,

andtryingtoobtainactualsolutionsbyfollowingMakanin-likeschemes(eg. using Makanin-Razborovdiagrams)seemsprettyhopeless.

Ourresultsaysyoucandescribethesetofallsolutionsinaparticularlyeasyway.

Outlineoftheproof

1. Reducetotheproblemoffindingallsolutionstoanequationoverafreemonoidwithinvolution over A = ai, ai.

2. Constructafinitegraphwithverticeslabeledby extendedequations, andedgeswhichenablethefollowingmovesbetweenthem:

– popvariables X→ aX, X→ Xa or X, X→ 1 (equationlengthgrows)

– compresspairsofconstants ab→ c (equationlengthshrinks,

– compressblocksofconstants aa . . . a→ aℓ alphabetofconstantsgrows)

Wefixanenlargedsetofconstantswithinvolution C ⊃ A ofsizeO(|UV|+ |A|) andrestricttoextendedequationsover C ∪ Ω oflengthatmostafixedboundin O(|UV|+ |A|).

Thisguaranteesthegraphisfinite. Wemustprovethatthegraphencodesallsolutionswiththeserestrictions.

Outlineoftheproof

Provingthegraphcontainssolutions

Wedefinean initialvertex tobelabeledbyanextendedequationwithU = V, anda finalvertex tobelabeledbyanextendedequationwithP = P where P ∈ C∗ (ie. novariables).

Edgesaredefinedsothatasolutiontothetargetextendedequationimpliesasolutiontothesourceextendedequation.

Itfollowsthatapathfrominitialtofinalverticesencodesasequenceofmovesontheequation U = V convertingittoanequation P = P withnovariables. Since P = P hasasolution, itcanbecarriedbacktoasolutionfor U = V.

Provingthegraphcontains all solutions

Supposewe know asolution Xj → ui for U = V.

Sinceweknowwhatthesolutionis, wecouldsimplyfollow X→ aX,X→ 1 edgesuntilallvariablesdisappear, andarriveatafinalvertex.

Butthiswouldtakeusoutofthegraph.

Notethataswepop, thenumberofvariablesintheequationneverincreases, andthesubstringsofconstantsinbetweenthemgrowinlength.

Applyingthecompressionmovesshrinkstheequationbackdown, sowecancontinuepoppingtofindasolution.

Hereistheprocedure. Put n = |UV|+ |A|.

– Assumeequationlengthisatmost29n (trueforaninitialvertex).

– Popthefirstandlastletterfromeachvariable. Nowtheequationlengthisatmost31n.

– Runasubroutine(blockcompression)toreplacemaximalblocksaa . . . a by a. Nowtheequationhasno aa factors, andlengthatmost31n.

– Runasubroutine(paircompression)whichappliesthemovesab→ c inacarefulwaytoensurethattheequationlengthreducestoatmost29n. Repeat.

Eachroundreducesthe weight ofthesolution(definedasthesumofthelengthofwordssubstitutedforeachvariable)sotheprocedureterminatesatafinalvertex.

MoredetailsinourpapertoappearinICALP2015proceedings, longerversiontoappearIJAC.

PartII:Cogrowth

Let G beagroupwithpresentation ⟨S | R⟩, so G ∼= FS/⟨⟨R⟩⟩.

...Eg: Z2 = ⟨a, b | aba−1b−1⟩ Fa,b,a−1,b−1 = ⟨a, b | −⟩

Define cn = # wordsin ⟨⟨R⟩⟩ oflength n. Thefunction n 7→ cn iscalledthe cogrowthfunction for (G, S).

cn ≤ (2|S|)(2|S| − 1)n−1 so lim sup c1/nn ≤ 2|S| − 1

Theorem(Grigorchuk/Cohen)

LetG beanon-freegroup. G is amenable iff lim sup c1/nn = 2|S| − 1.

PartII:Cogrowth

Cogrowth

Theformalpowerseries f(z) =∑

cnzn isthe cogrowthseries.

lim sup c1/nn isthereciprocaloftheradiusofconvergenceof f(z). Wecall

thisnumberthe cogrowth.

Example

G = ⟨a | a2⟩. Weconsider a, a−1 asdistinctformalsymbols, so:

c2n+1 = 0 sincetherelatorhasevenlength

c0 = 1 ϵc2 = 2 aa, a−1a−1

c4 = 2 aaaa, a−1a−1a−1a−1

f(z) = 1+∞∑n=1

2z2n = 1+ 2

1− z2− 1

1− z2

(radiusofconvergenceis1)

Theorem(Kouksov)

LetG beanon-freegroup. ThecogrowthseriesisrationaliffG isfinite.

Example

c0 = 1 ϵc2 = 2 aa, a−1a−1

c4 = 2 aaaa, a−1a−1a−1a−1

f(z) = 1+∞∑n=1

2z2n = 1+ 2

1− z2− 1

1− z2

Theorem(Kouksov)

Example

c0 = 1 ϵc2 = 2 aa, a−1a−1

c4 = 2 aaaa, a−1a−1a−1a−1

f(z) = 1+∞∑n=1

2z2n = 1+ 2

1− z2− 1

1− z2

Theorem(Kouksov)

Example

c0 = 1 ϵc2 = 2 aa, a−1a−1

c4 = 2 aaaa, a−1a−1a−1a−1

f(z) = 1+∞∑n=1

2z2n = 1+ 2

1− z2− 1

1− z2

Theorem(Kouksov)

Alternativedn = # all wordsin (S ∪ S−1)∗ equaltoid

Thereareformulastoswitchbetween dn and cn atthelevelofgeneratingfunctions

G isamenableiff lim sup d1/nn = 2|S|.

Z2 = ⟨a, b | aba−1b−1⟩

.a a b−1 b a b a−1 a b a−1 b a−1 a−1 b−1 a−1 b−1 b−1 a+ + − + + + − + + − + − − − − − − ++ + + − + − − + − − − − − + − + + +

d2n =(2nn

)d2n+1 = 0

Z2 = ⟨a, b | aba−1b−1⟩

.a a b−1 b a b a−1 a b a−1 b a−1 a−1 b−1 a−1 b−1 b−1 a+ + − + + + − + + − + − − − − − − ++ + + − + − − + − − − − − + − + + +

d2n =(2nn

)d2n+1 = 0

Openproblemforamenability

A famousopenproblemiswhetheroneparticular(infamous)exampleisamenable:

RichardThompson’sgroup F = ⟨a, b | [ab−1, a−1ba], [ab−1, a−2ba2]⟩

Severalauthorshavetriedcomputationalapproachestotheproblem:

– Burillo, Cleary, Weist2007

– Arzhantseva, Guba, Lustig, Préaux2008

– E,Rechnitzer, Wong2012

– Haagarup, Haagarup, Ramirez-Solano2015

Openproblemforamenability

A famousopenproblemiswhetheroneparticular(infamous)exampleisamenable:

RichardThompson’sgroup F = ⟨a, b | [ab−1, a−1ba], [ab−1, a−2ba2]⟩

Severalauthorshavetriedcomputationalapproachestotheproblem:

– Burillo, Cleary, Weist2007

– Arzhantseva, Guba, Lustig, Préaux2008

– E,Rechnitzer, Wong2012

– Haagarup, Haagarup, Ramirez-Solano2015

Newmethod: “Randomwalkonthesetoftrivialwords”

G = ⟨S | R⟩

X = ⟨⟨R⟩⟩

R = allfreelyreducedcyclicpermutationsofwordsin R ∪ R−1

P : R → [0, 1] aprobdistst P(r) > 0 and P(r) = P(r−1) forall r ∈ R

Wedefinea Markovchain withstates X andtransitionsbetweenstates(trivialwords)withprescribedprobabilities

sothatmovesare“uniquelyreversible”andthereisapositiveprobabilityofreachingeverystatefromanygivenstate.

G = ⟨S | R⟩

X = ⟨⟨R⟩⟩

G = ⟨S | R⟩

X = ⟨⟨R⟩⟩

Moveson w ∈ X1. Conjugation by x ∈ S ∪ S−1:

– write xwx−1 andfreelyreducetoobtain w′.

2. Left-insertion by r ∈ R atposition m ∈ [0, |w|]:– write w = uv with |v| = m– freelyreduce ur toobtain u′

– freelyreduce u′v toobtain w′

– ifanypartof v iscancelled, set w′ = w (rejectthemove)

Eachmoveis uniquelyreversible inthefollowingsense:conjugationby x←→ conjugationby x−1

left-insertionof r at m←→ left-insertionof r−1 at m

MetropolisMCMC algorithm

Next, wedefinethetransitionprobabilitiesforourMarkovchain. Letpc, β ∈ (0, 1) and α ∈ R beparameters.

Let wn bethecurrentword. Constructthenextword wn+1 asfollows:

1. Withprobability pc, choosetodoaconjugation, else(1− pc)chooseaninsertion.

2. Ifconjugation, choose s ∈ S ∪ S−1 withprobability 12|S| , and

conjugatetoobtain w′.

Withprobability min

(|w′|+ 1)1+α

(|w|+ 1)1+α· β

|w′|

put wn+1 = w′.

Elseput wn+1 = wn (rejectthemove)

3. Ifleft-insertion, choose r ∈ R withprobability P(r) andm ∈ [0, |w|] withuniformprobability. Left-inserttoobtain w′.

Withprobability min

(|w′|+ 1)α

(|w|+ 1)α· β

|w′|

put wn+1 = w′.

Eg: G = ⟨a | a2⟩. Thestatespace X is

:::: a−4 :::: a−2 :::: ϵ :::: a2 :::: a4 ::::Startingat w0 = a2k

S conjugation: nochange

S left-insert a±2: moveleft/right(ornochangeifrejected)

3. Ifleft-insertion, choose r ∈ R withprobability P(r) andm ∈ [0, |w|] withuniformprobability. Left-inserttoobtain w′.

Withprobability min

(|w′|+ 1)α

(|w|+ 1)α· β

|w′|

put wn+1 = w′.

Eg: G = ⟨a | a2⟩. Thestatespace X is

:::: a−4 :::: a−2 :::: ϵ :::: a2 :::: a4 ::::Startingat w0 = a2k

S conjugation: nochange

S left-insert a±2: moveleft/right(ornochangeifrejected)

Theprobabilitiesarechosenspecificallysothatwecanprovethat:

thereisaprobabilitydistribution π on X suchthattheprobabilitythatthealgorithmreachesstate w afterN stepsconvergesto π(w).

If Pr(u→ v) istheprobabilityofmovingfrom u to v inonestep, adistributionis stationary fortheMC if π(u) =

∑v Pr(v→ u)π(v).

Theorem(E,Rechnitzer, vanRensburg)

π(w) =(|w|+ 1)1+αβ|w|

where Z isanormalisingconstantistheuniquestationarydistributionforthealgorithm.

i.e. theprobabilitythatthealgorithmreachesstate w after N stepsconvergesto π(w).

π(w) =(|w|+ 1)1+αβ|w|

Relationtocogrowth

Since π isaprobabilitydistributionwehave∑w∈X

π(w) = 1

sosince π(w) =(|w|+ 1)1+αβ|w|

Zweget

Z =∑w∈X

(|w|+ 1)1+αβ|w| =∑n

cn(n+ 1)1+αβn

whichconvergeswhen β islessthantheradiusofconvergenceofthecogrowthseries (= recipofcogrowthrate)

Relationtocogrowth

Themeanlengthofawordsampledbyrunningthealgorithmis

E(|w|) =∑u∈X|u|π(u) =

∑u∈X|u|(|u|+ 1)1+αβ|u|

n(n+ 1)1+αβncnZ

∑n n(n+ 1)1+αβncn∑n (n+ 1)1+αβncn

As β → recipofcogrowthrate(frombelow), themeanlength→ +∞

Z2 = ⟨a, b | aba−1b−1⟩

0.25 0.275 0.3 0.325

〈n〉

Themeanlengthofsampledwordsplottedagainst β for ⟨a, b | aba−1b−1⟩ withα = 1. Thecrossesindicatedataobtainedfromanimplementationofthealgorithmwhilethecurveindicatestheexpectationderivedfromtheexactcogrowthseriesforthegroup. Theverticallineindicates βc = 1/3.

Freeproduct ⟨a, b, c|a2, b2, c2⟩

0.15 0.2

〈n〉

Meanlengthofsampledwordsvs. β for ⟨a, b, c|a2, b2, c2⟩ sampledwith α = 1.Thecrossesindicatedataobtainedfromthealgorithm, whilethecurvesindicatestheexpectationderivedfromtheexactcogrowthseries(foundbyKouksov).Verticallinesat 1/5 and0.2192752634(thereciprocalofthecogrowth).

Baumslag-Solitar(3,3)

0.2 0.3 0.4

〈n〉

Meanlengthofsampledwordsvs. β for BS(3, 3) = ⟨a, b | ba3b−1a−3⟩ withα = 1. Thecrossesindicatedataobtainedfromthealgorithm, whilethecurvesindicatestheexpectationderivedfromthe(known)cogrowthseriesforthegroup(foundbyE,Rechnitzer, vanRensburgandWong). Verticallinesat 1/3 and0.417525628(thereciprocalofthecogrowth).

Surfacegroup ⟨a, b, c, d | [a, b][c, d]⟩

0.1 0.2 0.3 0.4

〈n〉

Meanlengthsampledwordsvs. β forthepresentation ⟨a, b, c, d | [a, b][c, d]⟩for α = 1. Bluelinesindicateboundsupperandlowerbounds0.35473, 0.3547provenbyNagnibedaandGouëzel.

Thompson’sgroup F

Let F = ⟨a, b | [ab−1, a−1ba], [ab−1, a−2ba2]⟩

|S| = 2 soif F isamenable, lim sup c1/nn = 3

andwewouldexpectthemeanlengthofwordssampledbytheMCalgorithmtoblowupat 1/3.

Let’srunthealgorithm . . .

0.25 0.3 0.35 0.4

〈n〉

Thompson’sgroup F

Let F = ⟨a, b | [ab−1, a−1ba], [ab−1, a−2ba2]⟩

|S| = 2 soif F isamenable, lim sup c1/nn = 3

andwewouldexpectthemeanlengthofwordssampledbytheMCalgorithmtoblowupat 1/3.

Let’srunthealgorithm . . .

0.25 0.3 0.35 0.4

〈n〉

Thompson’sgroup F

Anotherpresentation⟨a, b, c, d | c = a−1ba, d = a−1ca, [ab−1, c], [ab−1, d]⟩ forF

0.1 0.15

〈n〉

Willthismethodevergiveaproof?

ThereareveryfewexamplesofMarkovchainalgorithmsforwhichonecan prove rateofconvergencetothestationarydistribution.

Eg: shuffleadeckofcardsbyselectingacardatrandomandplacingitatthebottomofthedeck

Startwith onthebottomofthedeck, when isontop, stop–uniformdistribution.

CameronRogersisexploringpathalogicalcasesthatbreakthealgorithm

Eg: ⟨a, b | abab−1a−1b−1, anb⟩

aswellasconnectionsbetweenconvergenceofthewalkandcomplexityoftheFølnerfunction.

MoredetailsinourpapertoappearinExperimentalMathematics.

Computational problems in infinite groups -8mm - CARMA · PDF fileComputational problems in...

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