Selected Title s i n Thi s Serie s

192 Le v Beklemishev , Mat i Pentus , an d Nikola i Vereshchagin , Provability ,

Complexity, Grammar s

191 A . Yu . Morozo v an d M . A . Olshanetsky , Editors , Mosco w Semina r i n

Mathematical Physic s

190 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knot s an d Curve s

189 V . Buslaev , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d

Spectral Theor y (M . Sh . Birman' s 70t h anniversar y collection )

188 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume I V

187 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y

186 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l Equations

185 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e

Invariants (Mathematic s a t th e Independen t Universit y o f Moscow )

184 Pete r Kuchmen t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s

(Dedicated t o Seli m Krein )

183 K . Nomizu , Editor , Selecte d Paper s o n Harmoni c Analysis , Groups , an d Invariant s

182 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e

181 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y

180 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y

Theory

179 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology :

On th e Crossroa d

178 V . Kreinovic h an d G . Mints , Editors , Problem s o f Reducin g th e Exhaustiv e Searc h

177 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors ,

Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

176 E . V . Shikin , Editor , Som e Question s o f Differentia l Geometr y i n th e Larg e

175 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume )

174 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Net sve taev , Editors , Mathematic s

in St . Petersbur g

173 V . Kharlamov , A . Korchagin , G . Polotovskit , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s

172 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theor y an d Algebrai c Geometr y

171 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pesin , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s

170 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s

169 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B .

Dynkin's Semina r

168 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s

167 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l

Equations

166 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume II I

165 Yu . Ilyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m

164 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s

163 L . A . Bokut' , M . Hazewinkel , an d Yu . G . Reshetnyak , Editors , Thir d Siberia n

School 'Algebr a an d Analysis "

162 S . G . Gindikin , Editor , Applie d Problem s o f Rado n Transfor m

(Continued in the back of this publication)

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Provability, Complexity, Grammars

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 19 2

Provability, Complexity, Grammars Lev Beklemishe v Mati Pentu s Nikolai Vereshchagi n

A* America n Mathematical Societ y /•? Providence , Rhode Island

http://dx.doi.org/10.1090/trans2/192

AMS Subcommitte e Robert D . MacPherso n

Grigorii A . Marguli s James D . Stashef F (Chair )

ASL Subcommitte e Steffe n Lemp p (Chair ) IMS Subcommitte e Mar k I . Freidli n (Chair )

1991 Mathematics Subject Classification. Primar y 68Q15 , 68S05, 03B45; Secondary 03B65 , 03F40.

ABSTRACT. Thi s boo k i s a collectio n o f thre e outstandin g dissertation s i n mathematica l logi c an d complexity theory .

The stud y o f moda l logic s axiomatizin g provabilit y trace s bac k t o Godel' s paper s o f th e earl y 1930's. Sinc e the n severa l infinit e serie s o f provabilit y logic s hav e bee n found . Th e Ph.D . dis -sertation b y Le v Beklemishe v establishe s tha t n o othe r suc h logic s exist , thu s completin g thei r classification. Fo r thi s pape r Dr . Beklemishe v receive d th e Mosco w Mathematica l Societ y Awar d in 1994 .

The Ph.D . dissertatio n b y Mat i Pentu s prove s th e Chomsk y conjectur e tha t establishe s th e equivalence o f tw o competin g approache s t o definin g forma l languages : th e Chomsk y hierarch y and th e Lambe k grammars . Fo r thi s resul t an d relate d paper s Dr . Pentu s wo n th e internationa l research priz e fro m th e Europea n Associatio n fo r Logic , Languag e an d Informatio n i n 1994 , an d the Mosco w Mathematica l Societ y Awar d i n 1998 .

In hi s Docto r o f Science s dissertatio n Nikola i Vereshchagi n propose s a genera l framewor k fo r the criteri a o f relativizabilit y i n complexit y theory .

The boo k i s usefu l fo r researcher s an d graduat e student s workin g i n mathematica l logi c an d complexity theory .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Beklemishev, Le v Dmitrievich , 1967 -Provability, complexity , grammar s / Le v Beklemishev , Mat i Pentus , Nikola i Vereshchagin .

p. cm . — (America n Mathematica l Societ y translations , ISS N 0065-929 0 ; ser. 2 , v . 192 ) Contains thre e doctora l dissertation s i n mathematica l logic , mathematica l linguistics , an d

complexity theory , translate d fro m th e Russian . Includes bibliographica l references . ISBN 0-8218-1078- 2 (handcove r : alk . paper ) 1. Modalit y (Logic ) 2 . Proo f theory , 3 . Computationa l complexity . 4 . Mathematica l lin -

guistics. I . Pentus , Mat i Reinovich , 1967 - . II . Vereshchagin , Nikola i Konstantinovich , 1958 - . III. Title . IV . Series . QA3.A572 ser . 2 , vol . 19 2 [QA9.46] 510 s—dc2 1 [511.3] 99-2017 7

CIP

© 199 9 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

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Information o n copyin g an d reprintin g ca n b e foun d i n th e bac k o f thi s volume . Visit th e AM S hom e pag e a t URL : ht tp: / /www.ams.org /

10 9 8 7 6 5 4 3 2 1 0 4 0 3 0 2 0 1 0 0 9 9

Contents

Preface i x

Classification o f Propositiona l Provabilit y Logic s L. D . Beklemishe v 1

Introduction 1

1. Preliminaries 7 2. Semantic s fo r S , D , an d A 1 4 3. Trac e classificatio n o f provability logic s 2 1

4. Prim e A-model s an d thei r characteristi c formula s 3 0 5. Provabilit y logic s containing D 3 5 6. Provability logic s containing A 3 8 7. Main result s 4 3

8. Examples , comments , an d relate d result s 4 8 References 5 5

Lambek Calculu s an d Forma l Grammar s Mati Pentu s 5 7

Introduction 5 7

1. Preliminaries 6 0 2. Fre e grou p interpretatio n 6 3

3. Thi n sequent s 6 5 4. Interpolatio n 6 6

5. Mai n theore m 6 9 6. Interpolation i n fragment s 7 3

7. Constructio n o f a context-fre e gramma r fo r a product-fre e Lambe k

grammar 7 9

8. Conjoinabl e type s i n th e Lambe k calculu s 8 0

9. Multiplicativ e cycli c linea r logi c 8 1

References 8 6

Relativizability i n Complexit y Theor y Nikolai K . Vershchagi n 8 7

Notation 8 7

1. Introduction 8 7

V l l l CONTENTS

2. A uniform wa y t o defin e complexit y classe s 8 9 3. Genera l criteri a 9 2 4. Relativizabl e inclusion s betwee n particula r complexit y classe s 10 4 5. Turing reducibilit y betwee n particula r complexit y classe s 11 4 6. Complet e language s i n particula r complexit y classe s 12 1 7. Perceptrons an d oracl e separatio n o f AM f l co-AM fro m P P 12 6 8. The universu m metho d 13 2 9. Relations betwee n complexit y classe s relativize d wit h a rando m oracl e 15 9 References

Preface

This book consist s o f English translation s o f three outstandin g dissertation s i n mathematical logic , mathematica l linguistics , an d complexit y theory .

The area of modal logics axiomatizing provability traces back to Godel' s discov-ery i n the earl y 1930' s o f the incompletenes s phenomeno n i n formal theories . Sinc e then severa l infinit e serie s o f provabilit y logic s hav e bee n found . Th e Ph.D . dis -sertation "Classificatio n o f Propositiona l Provabilit y Logics " b y Le v Beklemishe v establishes tha t n o other suc h logics exist. Thi s result ha s completed th e effort s o f a number o f researchers i n a classical are a of mathematical logic . Fo r it , Beklemishe v received th e Mosco w Mathematica l Societ y Awar d i n 1994 .

The Ph.D . dissertatio n "Lambe k Calculu s an d Forma l Grammars " b y Mat i Pentus prove s th e well-know n Chomsk y conjectur e o f th e earl y 1960' s tha t al l formal language s generate d b y Lambe k grammar s ar e context-free . Th e founda -tional significance o f this resul t fo r mathematica l linguistic s consist s i n establishin g the equivalenc e o f tw o competin g approache s t o definin g forma l languages : th e Chomsky hierarch y an d th e Lambe k categoria l grammars . Th e proo f i s an elegan t combination o f algebraic , logical , an d combinatoria l methods . Fo r thi s resul t an d related work , Pentu s wo n th e awar d "Fo r th e bes t ide a o f th e year " fro m th e Eu -ropean Associatio n fo r Logic , Languag e an d Informatio n i n 1994 , an d th e Mosco w Mathematical Societ y Awar d i n 1998 .

In 197 5 Baker, Gill and Solova y showe d tha t th e P = NP proble m i n complex -ity theor y relativize d t o differen t oracle s ha s opposit e solutions . I n hi s Docto r o f Sciences dissertatio n "Relativizabilit y i n Complexit y Theory " Nikola i Vereschagi n proposes a genera l framewor k fo r formulatin g th e relativizabilit y criteri a an d fo r analyzing thei r limits . Th e autho r finds al l relativizabl e inclusion s fo r som e o f th e known complexit y classes .

Lev Beklemishev i s now an Alexander von Humboldt Researc h Fellow at th e In-stitut fu r Mathematisch e Logi k un d Grundlagenforschung , Westfalisch e Wilhelms -Universitat Minister , an d a senio r researc h fello w a t th e Russia n Academ y o f Sci -ences, Moscow . Mat i Pentu s i s a n associat e professor a t Mosco w Stat e University . Nikolai Vereschagi n i s a professo r a t Mosco w Stat e University .

Sergei Artemo v

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Selected Title s i n Thi s Serie s (Continued from the front of this publication)

161 K . Nomizu , Editor , Selecte d Paper s o n Analysis , Probability , an d Statistic s

160 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theory , Algebrai c Geometry , an d

Differential Geometr y

159 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume I I

158 A . K . Kelmans , Editor , Selecte d Topic s i n Discret e Mathematics : Proceeding s o f th e

Moscow Discret e Mathematic s Semina r 1972-199 0

157 M . Sh . Birman , Editor , Wav e Propagation . Scatterin g Theor y

156 V . N . Gerasimov , N . G . Nesterenko , an d A . I . Valitskas , Thre e Paper s o n

Algebras an d Thei r Representation s

155 O . A . Ladyzhenskay a an d A . M . Vershik , Editors , Proceeding s o f th e St .

Petersburg Mathematica l Society , Volum e I

154 V . A . Artamono v e t al. , Selecte d Paper s i n if-Theor y

153 S . G . Gindikin , Editor , Singularit y Theor y an d Som e Problem s o f Functiona l Analysi s

152 H . Draskovicov a e t al. , Ordere d Set s an d Lattice s I I

151 I . A . Aleksandrov , L . A . Bokut' , an d Yu . G . Reshetnyak , Editors , Secon d

Siberian Winte r Schoo l "Algebr a an d Analysis "

150 S . G . Gindikin , Editor , Spectra l Theor y o f Operator s

149 V . S . Afratmovic h e t al. , Thirtee n Paper s i n Algebra , Functiona l Analysis , Topology ,

and Probability , Translate d fro m th e Russia n

148 A . D . Aleksandrov , O . V . Belegradek , L . A . Bokut' , an d Yu . L . Ershov ,

Editors, Firs t Siberia n Winte r Schoo l 'Algebr a an d Analysis "

147 I . G . Bashmakov a e t al. , Nin e Paper s fro m th e Internationa l Congres s o f

Mathematicians, 198 6

146 L . A . Aizenber g e t al. , Fiftee n Paper s i n Comple x Analysi s

145 S . G . Dalalya n e t al. , Eigh t Paper s Translate d fro m th e Russia n

144 S . D . B e r m a n e t al. , Thirtee n Paper s Translate d fro m th e Russia n

143 V . A . Belonogo v e t al. , Eigh t Paper s Translate d fro m th e Russia n

142 M . B . Abalovic h e t al. , Te n Paper s Translate d fro m th e Russia n

141 H . Draskovicov a e t al. , Ordere d Set s an d Lattice s

140 V . I . Berni k e t al. , Eleve n Paper s Translate d fro m th e Russia n

139 A . Ya . AYzenshta t e t al. , Ninetee n Paper s o n Algebrai c Semigroup s

138 I . V . Kovalishin a an d V . P . Potapov , Seve n Paper s Translate d fro m th e Russia n

137 V . I . A r n o l d e t al. , Fourtee n Paper s Translate d fro m th e Russia n

136 L . A . Aksent'e v e t al. , Fourtee n Paper s Translate d fro m th e Russia n

135 S . N . Ar temo v e t al. , Si x Paper s i n Logi c

134 A . Ya . AYzenshta t e t al. , Fourtee n Paper s Translate d fro m th e Russia n

133 R . R . Sunchelee v e t al. , Thirtee n Paper s i n Analysi s

132 I . G . Dmitr ie v e t al. , Thirtee n Paper s i n Algebr a

131 V . A . Zmorovic h e t al. , Te n Paper s i n Analysi s

130 M . M . Lavrent'ev , K . G . Reznitskaya , an d V . G . Yakhno , One-dimensiona l Invers e

Problems o f Mathematica l Phvsic s

129 S . Ya . Khavinson , Tw o Paper s o n Extrema l Problem s i n Comple x Analysi s

128 I . K . Zhu k e t al. , Thirtee n Paper s i n Algebr a an d Numbe r Theor y

127 P . L . Shabali n e t al. , Eleve n Paper s i n Analysi s

126 S . A . A k h m e d o v e t al. , Eleve n Paper s o n Differentia l Equation s

125 D . V . Anoso v e t al. , Seve n Paper s i n Applie d Mathematic s

124 B . P . Allakhverdie v e t al. , Fiftee n Paper s o n Functiona l Analysi s (See th e AM S catalo g fo r earlie r titles )

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