Turing machines application to the rate of decrease in logical depth for general Logical depth for reversible Turing machines with an Theoretical Computer Science

Theoretical Computer Science 778 (2019) 78–80

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Theoretical Computer Science

www.elsevier.com/locate/tcs

Note

Logical depth for reversible Turing machines with an

application to the rate of decrease in logical depth for general Turing machines

Paul M.B. Vitányi

^a

^,

^b

^,∗

aCWI,theNetherlands

bUniversityofAmsterdam,theNetherlands

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received16January2019 Accepted17January2019 Availableonline25January2019 CommunicatedbyP.G.Spirakis

Keywords:

Logicaldepth Kolmogorovcomplexity Compression

The logical depth of a reversible Turing machine equals the shortest running time of a shortest program for it. This is applied to show that the result in [1] is valid notwithstandingtheerrornotedinCorrigendum [7].

©^{2019ElsevierB.V.Allrightsreserved.}

1. Introduction

Abookonnumbertheoryisdifficult,or‘deep.’Thebooklistsanumberofdifficulttheoremsofnumbertheory.However, it hasvery low Kolmogorov complexity,sinceall theorems arederivable fromtheinitial few definitions.Ourestimate of thedifficulty,or‘depth,’ofthebookisbasedonthefactthat ittakesalongtime toreproducethebookfrompartofthe informationinit.Theexistenceofadeepbookisitselfevidenceofsomelongevolutionprecedingit.

Thelogicaldepthofa(finite)stringisrelatedtocomplexitywithboundedresourcesandmeasuresthetradeoffbetween program sizesandrunningtimes.Computinga stringxfromoneofitsshortestprogramsmaytakea verylongtime,but computingthesamestringfroma simple“print(x)” programoflengthabout|^x|^{bits (thelengthofx) takesverylittle} time.

Logical depth asdefinedin [4] for a string comes in two versions: one based on thecompressibility of programs of prefixTuringmachinesandtheotherusingtheratiobetweenalgorithmicprobabilitieswithandwithouttimelimits.Since both areapproximatelythesame([5,Theorem7.7.1] basedon[4,Lemma3])itisnorestrictiontousethecompressibility version.

Theusednotionsofcomputability,resource-boundedcomputation time,self-delimitingstrings,big-Onotation,andKol- mogorovcomplexityarewell-knownandtheproperties,notations,aretreatedin[5].

*

Correspondenceto:CWI,SciencePark123,1098XGAmsterdam,theNetherlands.

E-mailaddress:Paul.Vitanyi@cwi.nl.

https://doi.org/10.1016/j.tcs.2019.01.031

0304-3975/©^{2019ElsevierB.V.Allrightsreserved.}

P.M.B. Vitányi / Theoretical Computer Science 778 (2019) 78–80 79

2. Preliminaries

AllTuringmachinesinthispaperareprefixTuringmachines.AprefixTuringmachineisaTuringmachinewithaone-way read-onlyprogramtape,anauxiliarytape,oneormoreworktapesandanoutputtape.Alltapesarelinearone-wayinfinite anddividedintocellscapableofcontaining onesymbolout ofa finiteset.Initially theprogramtapeisinscribedwithan infinitesequenceof0’sand1’sandtheheadisscanningtheleftmostcell.Whenthecomputationterminatesthesequence ofbitsscanned ontheinputtape istheprogram.Foreveryfixed finitecontentsoftheauxiliary tapethesetofprograms forsuch a machine isa prefix code(no program isa proper prefix of another program). Let T₀

,

T₁

, . . .

be the standard enumeration ofprefix Turingmachines. A universalprefixTuringmachine simulates every prefix Turing machine givenits indexnumber.Wealsorequireittobeoptimalwhichmeansthatthesimulationprogramisasshortaspossible.Wechoose areferenceoptimaluniversalprefixTuringmachineandcallitU.

Theprefix Kolmogorovcomplexity isbasedontheprefixTuringmachinesimilarto the(plain)Kolmogorov complexity based on the(plain) Turing machine.Let x

,

y be finite binary strings. The prefixKolmogorovcomplexity K

(

x|^y

)

of x with auxiliary y isdefinedby

K

(

x

|

^y

) =

^min

p

{|

^p

| :

^U

(

p

,

y

) =

^x

}.

Ifxis abinary stringoflengthnthen K

(

x|^y

)

≤ⁿ+^O

(

logn

)

.Restrictingthe computationtime resourceisindicated bya superscriptgivingthe allowed numberofsteps, usually denotedby d.The notation U^d

(

p

,

y

)

=^{x meansthat U}

(

p

,

y

)

=^x withindsteps.Iftheauxiliarystringy istheemptystring ,thenweusuallydropit.Similarly,wewriteU

(

p

)

forU

(

p

, )

. Thestringx^∗ isashortestprogramforxifU

(

x^∗

)

=^xandK

(

x

)

= |^x∗|^.Astringxisb-incompressibleif|^x∗|≥ |^x|−^b.

3. ReversibleTuringmachines

ATuringmachinebehavesaccordingtoafinitelistofrules.Theserulesdetermine,fromthecurrentstateofthefinite controlandthesymbolcontainedinthecellunderscan,theoperationtobeperformednextandthestatetoenteratthe endofthenextoperationexecution.

Thedevice is(forward) deterministic.Notevery possiblecombinationofthe firsttwoelements hastobe intheset;in thiswaywe permitthedevice toperformnooperation. Inthiscasewesaythat thedevicehalts.Hence,wecandefine a Turingmachinebyatransitionfunction.

Definition1.A reversible Turing machine[3,2] is a Turing machinethat is forwarddeterministic (any Turing machine as definedis)butalsobackwarddeterministic,thatis,thetransitionfunctionhasasingle-valuedinverse.Thedetailsofthefor- maldefinitionareintricate[3,2] andneednotconcernushere.Thisdefinitionextendsintheobviousmannertomultitape Turingmachines.

In[3] forevery 1-tapeordinaryTuringmachine T a3-tapereversibleTuringmachineT_revisconstructedthatemulates T inlineartimesuchthatwithinput ptheoutputisT_rev

(

p

)

=

(

p

,

T

(

p

))

.ThereversibleTuringmachinethatemulatesU is calledU_rev.

Definition2.Letxbeastringandbanonnegativeinteger.Thelogicaldepthofx atsignificancelevel bis depth_b

(

x

) =

^min

d

:

^p

∈ {

⁰

,

1

}

^∗

∧

^Ud

(

p

) =

^x

∧ |

^p

| ≤

^K

(

p

) +

^b

,

theleastnumberofstepstocompute xbyab-incompressibleprogram.

Theorem1.Thelogicaldepthofastringx atsignificancelevelb∈N ^{forreversibleTuringmachinesisequalto} depth_b^rev

(

x

) =

^min

d

:

^p

∈ {

⁰

,

1

}

^∗

∧

^Urev^d

(

p

) = (

p

,

x

) ∧ |

^p

| ≤

^K

(

x

) +

^b

,

theleastnumberofstepstocompute

(

p

,

x

)

byU_revfromaprogramoflengthatmostK

(

x

)

+^b.

Proof. SinceareversibleTuringmachineisbackwardsdeterministic,andan incompressibleprogram cannotbecomputed froma shorter program,the length ofan incompressibleprogram withb=^{0 for xcan only be the lengthof a shortest} programforx.Thelogicaldepthatsignificanceb isthentheleastnumberofstepstocomputexbyUrevfromaprogramp oflengthatmostK

(

x

)

+^b. 2

80 P.M.B. Vitányi / Theoretical Computer Science 778 (2019) 78–80

4. Therateofdecreaseoflogicaldepth

InCorrigendum [7] thefollowingisobserved:In[1,Section 4] itisassumedthat,forallx∈ {⁰

,

1}^{∗,thestringx∗ isthe} only incompressiblestringsuch thatU

(

x^∗

)

=^{x.Thatis,logicaldepth resemblingtheone accordingto Theorem1isused.}

However, thisassumptioniswrongforgeneralTuringmachinesinthat formanyxtheremaybeanincompressiblestring pwith|^x|≥ |^p|

>

|^x∗|^suchthatU

(

p

)

=^x.ThecomputationofU

(

p

)

=^{xmaybefasterthanthatofU}

(

x^∗

)

=^{x.Forexample,} the functionfromx∈ {⁰

,

1}^{∗ totheleastnumberofsteps ina}computation U

(

p

)

=^xforan incompressiblestring pmay be computable. The argument in the paper is, however, correct forthe set ofreversible Turing machines. These Turing machinesareasubsetofthesetofallTuringmachines[3,2] andemulatetheminlineartime.Thisimpliesthecorrectness of[1,Theorem2] asweshallshow.

Lemma1.Let

ψ

bedefinedby

ψ (

n

) =

max

|x|=nmin

d

{

d

:

U^d_rev

(

x^∗

) = (

x^∗

,

x

)}.

Then

ψ

isnotcomputableandgrowsfasterthananycomputablefunction.

Proof. Ifa function

ψ

asinthe lemma were computable,then foran x oflength n we couldrun U_rev emulating U [3]

forwardfor

ψ(

n

)

stepsonallprogramsoflengthn+^O

(

logn

)

.Amongthoseprogramsthathaltwithin

ψ(

n

)

steps,wecould selectthe programs p whichoutput

(

p

,

x

)

.Subsequently,we could selectfromthat setaprogram p ofminimumlength, say x^∗.Sucha programx^∗ haslength K

(

x

)

sinceU_rev isemulating U.Thiswouldimplythat K wouldbecomputable.But thefunction K isincomputable[6,5]:contradiction.Therefore

ψ

cannotbecomputable.Sincethisholdsforeveryfunction majoring

ψ

,thefunction

ψ

mustgrowfasterthananycomputablefunction. 2

Corollary1.ThesetofreversibleTuringmachinesisasubsetofthesetofallTuringmachines.TheemulationofU

(

p

)

byU_rev

(

p

)

is lineartimeforallbinaryinputsp by[3].Therefore,replacinginthelemmaUrevbyU changes

ψ(

n

)

to

φ (

n

)

=

(ψ(

n

))

.Hencethe lemmaholdswith

ψ

replacedby

φ

andUrevbyU .Thisgivesus[1,Lemma1] andtherefore[1,Theorem2](theBusyBeaverupper boundisprovedasitisin[1]):

Theorem2.Thefunction f

(

n

) =

^max

|^x|=ⁿ,0≤^b≤ⁿ

{

^depthb

(

x

) −

^depthb+¹

(

x

)}

growsfasterthananycomputablefunctionbutnotasfastastheBusyBeaverfunction.

(Thestatementof[1,Theorem2] containsatypo“x:^{”whichshouldbedeletedandthe}superscript“

(

2

)

”is(canbe)omittedhere.) References

[1]L.F.Antunes,A.Souto,P.M.B.Vitányi,Ontherateofdecreaseinlogicaldepth,Theoret.Comput.Sci.702(2017)60–64.

[2]H.B.Axelsen,R.Glück,AsimpleandefficientuniversalreversibleTuringmachine,in:Proc.5thInt.Conf.LanguageandAutomataTheoryandApplica- tions,in:LectureNotesinComputerScience,vol. 6638,Springer,2011,pp. 117–128.

[3]C.H.Bennett,Logicalreversibilityofcomputation,IBMJ.Res.Develop.17 (6)(1973)525–532.

[4]C.H.Bennett,Logicaldepthandphysicalcomplexity,in:R.Herken(Ed.),TheUniversalTuringMachineaHalf-CenturySurvey,OxfordUniversityPress, 1988,pp. 227–257.

[5]M.Li,P.M.B.Vitányi,AnIntroductiontoKolmogorovComplexityandItsApplications,Springer,NewYork,2008.

[6]A.N.Kolmogorov,Threeapproachestothequantitativedefinitionofinformation,Probl.Inf.Transm.1 (1)(1965)1–7.

[7]P.M.B.Vitányi,Corrigendumto“Ontherateofdecreaseinlogicaldepth”byL.F.Antunes,A.Souto,andP.M.B.Vitányi[Theoret.Comput.Sci.702(2017) 60–64],Theoret.Comput.Sci.770(2019)101.