Classical Recursion Theory - The Theory of Functions and Sets of

344 Pages·2002·13.22 MB·English

by  Odifreddi, Piergiorgio

STUDIES IN LOGIC CLASSICAL
AND RECURSION THEORY

HL EOCNDATIONS OF MATHEMATICS

VOLUME 125

The Theory of Functions and
Sets of Natural Numbers

Piergiorgie ODIFREDDI

BSE
NE

bs
Sarat tou ico sabe

‘To Lidia

Wr ts Lance wel ean he a

‘Wis Una owt well all no et rom shew

‘Whit The lcs well hp ve haitage
{Perm Cantos, OC}
Foreword

kif ee writers dllightal pe elely wootiae on roemion thoor:
‘Wate cle cam one re nous mnezie ect Tle boo eotmiree hie aces
to tho central quemiou of reeusion they: Wnt la mason tleey? His
nsw, Tam please 10 ncte. is Wlnjieraie. Menlo nurs eden
lerset idm, exaenpte Keine salyory theecan, ste anayeeol Bisratey
le pvteati Hevarty, 200 Uke anion of ablermioatences, To my mind
sa uedervtactng of reenrion thecty even at the lvel of Turing desneet and
romsively ecumcexbie cts, is incomplete unt che coaneecion Ce ighe Ie
ia mado via st thoy

“Trecoeon thpory i about anyon, they Ue ali Je. Lewaeealls
only a tothon of eile. Lede ug computations, woes
sroouul of fire lype ob et, or 0 in'Tokew's version of eecursive Sanction
Df ovdinnls, ermit a deeper insight int te tate uf wtaptiaicn, Tin bs
bom on by the wore if Sinan ard elwrs or Tragonats f mites
polytomal redial, in which hes Tov high cere ple hoe

‘Lacanrer's wee of eva? in fo sitlexpailly eopy iesl,
bo date the morale overs, Tle eottaalates ou the osc caye of ec¥rson
hwy. Peres Uoeo were the slr lays. Pers only thm ely rena l

‘The wurhor mnce the set thio nmaeticn lucas psa aly
hese, Let us hope he oes Vi et olan on tedeen eure as, bs
swing Bret olume prove ins worth of Ue Lk,

G8 Sch
Morvan rivers aad MULT:
onesie 19ST
Preface

1 ign of oxi 30 bark ts Girt is ago ty os te
ty scueae on a cry UBSL ae Rov later naib Le me, ts gill
‘underdescloped, thought | could lara Kecuraion‘Theeny by writing "here
‘were at tho fine a tex testhoots, potted mirigy To Ken [162 sd
Regen 1997], bor T soe una ivi ih The Tene apes Tiler
inn progretions af fortal ayaa, -eraniygle ree ey eels st
hey aid ast vee. ‘This Ts doe, sac 4 Io, sod wrote fal veri
Aahon, Fortunately, 1 did nox publi i.

‘atatile, had geet in fet th sone rusian Thre nd deri
2 wohl go te Tita Slay to study soume aaee, Tho Iuiae Cooler
of Hewanties (CUR) provided sopra xu 1076 Flaca Urbano
Clanpaina, ober the sora meus up Ue we, Tlouad thao a vey aesiire
‘sed at loachee, Coe) Jockaeth, who baceh, aie ia oce pleasant year mote
than T could have tought muself in ferme. And [found a fend in Dik
stein, ftom eho F Tere ow is write nuaderkine. Tham, Fasc al
sow of their papers, (wena the (OTA, ple for ty al Peal
Tried thie pater with ny ins sions Thece Toru whol Thus of
Sez Theory aud Generalized Novursioa ‘Theor, (eaazh the teaching and hep
of Tey Mictin, Yaunia Moschoenkia and John Steeh Hack ir aly, | sewtore
he whee book, chs ine in Reales

In the sean T hel prow avi of tho fact that. mathematics was aot
ke universal scivue ua! Tog once thousht i waa: not outy pecsonal, bot
‘leo social nisl hideraa intucnsce shape the wate of the researchers. More
‘specially, Thal lenennd tone the Sores mere ing, in Racoon Teh,
sretic chat He Wines di net few anc a td Ue eli ere
Incgcly onaware of hal people did fn The Wes I hod Usa ons ain
ns decided L nov ps tothe Soviee Laka tie vw go, hast in
Ieapmdecge. ‘The Sabian al Soviet State Deaton en al 1
toyed ia Noveeibite for Bal yous, i LIG2-ER. gain eri
Shand metbcasetes ad ure tee. Ta eaticulae sre lp a poe
bby Maral Arson, Sergei Deane, Vari Resoe. sat ietor Selene. Dreyer
x Preface

some difiuline Kher shih oat a 9 smringeauiong otlwe Hing Letie
back wich nore perio, ad the sone wae Ys

final umnapero uel was ae be Anil Nevo ind Richer Shove,
ssho breil ms tm Carnal forse jer HR, and ia th fling sume,
(Wh hen sarin fa as) wey ffi ellen, ply Faced ya
Joint WSC 9.0. rane. Te prtientay Richa! Share bn elena raed
Moeroma that woven Mal epee tie bok. Ty Cotoel T nla met Baie
artnabisy wh hanger any preepetns in Crnupcaiy Por

addition tn all he peuple enti nb, De gretly hala Ly thee
‘she Ive rex, ml cowed som, mttautil parts of he ayaauea ps
‘ae faughe rn ine Ung, Serlncing Kise AvcaweSpis, [ne Catt
tg Alesande Degter, Leo Tarington, Coorg Kreonl Govtgh Mabie. Moni
“erring, Ju Ligln, Cabecie Loli, Flav Verve, Mash Seqwcn ual BOS
Snare, Many elter people ave provided tale hiv uf ep anch etce-
‘Hone, in putralarthoee wha silences al. meninars nero pars
ofthe aol in “ovine aed Sn (Hay), Cela CHA wad Corael {Tae
States, Novoribiek and Kaman (Soviet Uwion). Ze st ale tao ih pace
fn retin Uno all be to evplily ge iy incre hank.

Since Cmtenbarg, Looks inne nay doen witen to be pete. Ta mp
vase thie wae made puaslie ly Solemn Fels oud Richard Shur, wip
inleunced tho buck lu dire, edicrs, Michas! Rlaley conve me (bat
‘old lye it mgs BT, su ie when Yd wel een oe
ss voupter on, wal im wal Sail Rerede help acrworda eth te axehnace,
in me aye. Wile € tw preparing the typeset, the amar Lill Casein
ard Riskaud Slr provid over! carnations St rae. ally, nope
‘he espe ngs provided by le C.K, Thon le a of them, cm

Filan amt were epoca thane pr va Lidia. Sle we there beter ia
se The inn tala Sbape, se Had about Ae more then anvlen’y Uso. She
[lowe me an ay iets, aod Dewi perceive how wre tal tha ae
Uoking om er only he iL Ragen ae, She i "we amore tore he
off ach Jr enact 28h mont ef time nace Coen hee
dand devwraé lo tis wank» pelle sespotyite hr hee shoe. Ro double
eas a top is, ats nie fyurtcen Years he Sethe bck: devoted to
hres part. ne coupes for what nh gwerved. anal 6 ERRDIE to

Nevo - Urbana fas Angola
‘Nevosbirsk -Iekaea
Tae iges

Preface to the
Second Edition

Altea ts ts aracianin 1980, Casson! Resnorion Theory haseeoo a socond
printing sw paperback edition in (9%, lM wt With the appearance of
Volume Tin 199. he une haa come te peblil «onl edition of Vole 1
rer nd corscied of al che argprnes md macakes Chal ave bot nied
Uucel the years. seize this opeorlusity to Ube all the ready rhe have
inal eiata exe in tonne et, a ling eto

“Recursion Tloary bas soon mon sxciting delyarvnlin rh eed,
bot most of tne cone one ane which ste eel in Vans Te. Sire
Voom 1 hse Ine [et Inte antonchal hy ase svloyuneria, V haan we
anieyted acy major roerting af the rival lest. Tralend, ve users fn
ic two volun ile of Carne are hale ite fo pag, so
Ane capper lo wk the whole book the rane, flowing the wew
‘Gaal af Bee Be,

"ori, Fone 198.
Contents

Foreword by G.E, Sacks

Preface

Preface

to the Second Ftisn

Introdaction
Whe i Clas
‘Whet iin co

Aplentions of Rararaion ees

lise Cae she Bok,
‘Natalia ard Ganeontons

1 RECURSIVENFSS AND COMPUTTSBILITY

"

1a

Teetnetion.
eiyitions hr incaetiowe
rock Uy induction
Rene
ister onls of Bac
Fusunal Arithmetic n

Some prmive veiw Gancticnn al praia

Cadivgs of Ce ple

Baivetien of peinbive eect

Systems of Equator:
‘Tho formtiya of equacions

etnatiiy 97 aye of epuons
Decibiliy trom systems of equations
8 logie pregrunmting latins ©
‘Arithiastical Foemal Syetete

Notions of reprmunaitity

on vy =
18

Laaal svete tagesertig te resi fnaiors
Iwai daly»

inability of funtion

‘Turing Machines

‘Vavatone nf the“Lorig msec model

Praseel Ting aowhine «

nite antean «

aring maehine comqutlity

Macaine-capundet peugrantng language =
Plembaxts

estemvured progremming languages

Vanlinied egiler, random nest chines +
lowes ecnaplabi

Stewotored prom soming agungse +

Progen lr price reagan »

Pete ns»

Guciions as Rules

Dewveuhis

(Onher fortnulbogs of tae Acalenh &
enable

Fein yrogramenng gages +
‘AslthaetinaLion

Estero sensi «

[Raxezrca ol fr auheretation
‘The Bormal Kortn Theres

Equtenco of the meius ampmuiches to retreive
‘The lic cel ofthe fotos of Reenesnm Theres
Claurei’s Thesis +

Ieaeuaootion to Chane eis

stein veicks

Copter and pyri
Chie esbnies
Probable ubles
Compe le
The bisia
Construcivinas
Conclusion

Conia

Contents

1 BASIC RECURSION THEORY
WL Partial Recureive Funetions
The auton of poeta Function
Patil recuniee Goncione
Univeral Torin orachines onl omens
Teaeivey eusmioedberar:
og ata Flin of Rexursion Toone =
A pregennaving Wangs host on 6 >
112 Dingonelteation
The eeu of Cingotaliation
Retunsine uecidebtily ou
imitans of mech 8
Pownlait Yheocem
Linton of feta ©

Selective ©

Selerepducion wad ele wats ©
Vd Porliat Recursive Fuuetiumale
Onvele wagons aah Lec gear
“The notion of functional
Pala rerutlve fenetonals
cen These
“Topological digression
Tteeaton and te pore»
Monte nf alta get 1)
Torok wots af merce owas
La Eseetive Operations
[Eec:ivo operatiore on partial cursive louctions
ifecsve apertions on tal nse Pnetions
Bieccivw apersia i geal +
eeureive aval +
145 Indices and Enmnerations +
‘Acceyttleeslonns of chee
S.vonlig Reeurian Thea 4
Models of -aleul (par ID «
Thee far resi and fit ata
Eneimaeaies of hoses of la
‘he Thaury af Boratiens
16 Retracesble and Regreselve Sete «
‘Rewcuceabe pera weneive
wi Contents Contents wi

UIPOSTS PHOBUPAL AND STHONG REDUCIBILINES 261 Wwinierancitis Anny weas neDUcED MATES 0
TIL Beat's Problem. Be 11 The Avitioneles! Hlocarchy 1
Osiginn of Port's Probleen + 253 The dedisition of truth» ccs

12 Shap a ne ra ir in Per ite is
Simple Sat ed Me. Oue Decree 2 ‘he ketal Mane es
Effeerively: aitoyshe ts + 263 AY sete a

LW Llyperalmple Sts and ‘Teuth-Table egroes 267 Aelativiations & vat
vrai Fa 12th anal Hira *

Ui Rypectypertaple Set Q.Degee ao Toc oan erty a
Q-rodueibiticy ear Alf sets jal
Tharsis to inte a
Mosimel sets ake [iescrial ier et Theory # av
Is a’Boinon vo Yow Prob my Prin Te wt
Semin mt Poets Tren wu usd ey © i
Blectiely nowseunive set he ‘Truth in Set Theory 307

arr Recuccompbee ne ns 1 aa he Avenel Pian ie
thontenaad Lieve oo ‘Ruolenes ee dy a ty ni
Reeursive ieuinorphesm types: out Adtrabnitle sols 42d
Revue ens gre wa a 1 ea ic eee oH

tue Varin of aia oi acy & o “Te otras Hien te
‘Benalal twullivalile dogrons: sat The ewes of the Conslruetible |inrereky: 422
tte wotiaes of eekucity + he Conscenetibve sule uf walural nemsers, ag
THLS ‘Thr World of Complete Sete + HL Sewn “ on

Forwol wenn ond wt ae a0 Telatiszationn « 1
TURING NFGREFS.
VL The Language of agree Theory
The in mperoee
‘The jump aperot
ust propel of grees
"The Asis of Deceeminary «|
V2 ‘The Hinite Exeenslon Methead|
Inconreiy degaes
Reseda ity resis
Te icing acho
owing ss jump
Wa Ren Cntegery ©
Thales on ctl ftom

Someager ste
Saige Category sac Degg heury
Menge tts of dog
Mnaure Teor aa Lege Pheory »
VA The Colufinile Extension Methoa
Tract pairs aids
‘Sree leer tote nd ews upper bamnds
Excensiona of ols
Vib ‘the Tree Meta
Hypsimunns fre drgroce
Minin agro
Misia uppre hon ©
in’ ani td ean ©
Compe exteusoas ef Pena ratte »
6 Tita! Segments +
niform trees
Sinina degra yp cecusive eats extensions
he ubeerciens chin
‘The ini gmat oF the ngrers ¢
WF Global Pmywertlos
Dafibiy from parauieten,
"The complerity ofthe tinory of dogs
Abwolate defaalee
Des seeily
Avononbiins
WH Degree ‘Hhenry mith Jump =

Contenta

ar
ai8
as
110)
al
rc
46
sr
0
408
‘07
#2
ans
an
a8
481
4a
185
88
490
463
495
198
6
sy
16
sin
500
a2

0
56
540
a

0

Contents

VIMANY-ONE AND OTHER DRGREES
M11 Distrtbativily
Dateibalive uppersowibation
Ideals of disribatie uppers,
V12 Connteble Taltial Serments
vite intial gnc
Cae ai sits
VEY aemontit: Tait] Beet
Sern rine en
CCapwntaste Rorae erry

Corannta tial sees
VIA Global Properties
(Chaoacleiticn of Ue ateucice of mmm depen
Urfinahtiy, hangers, ad sulancep hon
"The enmplaity ef the fcnrs of ange Ag A8
“VIS Comparicon nf Dogren theories =
Lalgesee
“Teh Unble degen a ele rachel Ages
temettory inquire
“Vio Btructune Incite Dogrece =
Celindene
Ine nny ore degra
Inch eutletanie dogs
Tide Tsing degra

Vibtogeaphy
Notation Tndex

Sublet Index

on
Introduction

‘Claasleal Recupsion ‘Phanny te the sburly of real vammbers or, egulva-
ently, functions over Hue mathral mimhare Aamich thre Tory hivany
fend a mmr of tie sul raat that Wate egal proved Ge illo
Tnlc2 and for diferent parpinsn we incrparsled, wed nd extended 8
pone dy She ae thing. ern, for cmon, of Hae dilleret ei
‘lon dfniione of ra autor, of Cantor» tore thal Uke eal eben
{xo unonmiable, of iekTe cae uf surat oe watabere 4620.99 A
Ub choi ste tow pert of Hrnueson Thecry aod uf our stuly, but the theory
tou prove 2o3 too: of ty oon, the origins of which cae be trace back
Tein [J885|- Le iacedaced the esd of fovetions define over the wat
22 of the natal ans Ly aveurcse ang she wl deve septs
‘whence the nn Racursian Then.

"The pone of socuvion we a tal ft Aang Felons nae azul i
ela Ly Sader [0029], Poa [1994) sal Lille ane Herma [1994], bot
Ferlito were al usa yt, Gedo tbe collective werk af owt
[is], Chet [P33], Gite [1884 Cer 1996), mal Tag IOI, Ie vo the
Hlebieaien ul vie tal yster ere The roeusiow piace wd at
wo not eal roenzive functions, In 8 Bok phigh sittin Chase
[1086] repro ls Weve wake 0 effete compen Lime wu" of
hatiral wamlars with th of resi Furetion, chon providing olin of
hvolnteners lv the alii, Wh Met [IAL Recursion ‘Theory eon ot
Indepencen eich of tenses, lind Fr i om Ske

What is ‘Classical’

In mote seca aves Revssion Thoey hae bow seul in vious os
fn Aifeent douivea, cians Diggs Uso 2, Fusctanade of lites oder, a>
eeiostecte, AD Urea seta beng be hin: oo ull Geueeellze Recursion
"Thoory. We use Ue word ‘chewiel a andi the ac nat we coins wat
ueetavcu te the win eng, wi we wD ed LL olin v2 Cela