Selected Title s i n Thi s Serie s

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 , Topic s i n

Statistical 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 , Contemporar y

Mathematical Physic s (F . A . Berezi n Memoria l Volume )

174 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Netsvetaev , Editors , Mathematic s i n

St. Petersbur g

173 V . Kharlamov , A . Korchagin , G . Polotovskii , an d 0 . Viro , Editors , Topolog y o f Rea l

Algebraic 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 Semina r o n

Dynamical System s

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

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

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 Schoo l

"Algebra an d Analysis "

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

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 Algebra s an d

Their Representation s

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

Mathematical Society , Volum e I

154 V . A . Artamono v e t al. , Selecte d Paper s i n X-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 Siberia n Winte r

School "Algebr a an d Analysis "

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

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

Probability, Translate d fro m th e Russia n

148 A . D . Aleksandrov , O . V . Belegradek , L . A . Bokut 7, an d Yu . L . Ershov , Editors , First Siberia n Winte r Schoo l "Algebr a an d Analysis "

(Continued in the back of this publication)

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 17 8

Problems o f Reducin g the Exhaustiv e Searc h

V. Kreinovic h G. Mint s Editors

American Mathematical Societ y Providence, Rhode Island

http://dx.doi.org/10.1090/trans2/178

EDITORIAL COMMITTE E

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 . Preidli n (Chair )

V. Kreinovic h an d G . Mint s (editors )

n P O B J I E M b l COKPAHIEHHf l I IEPEBOP A

Hay^HBiM CoBe T n o KoMnjieKCHoi i IIpo6jieM e «Kn6epHeTHKa»

AKaAeMKH Hay K C C C P MocKBa, 198 7

Transla ted b y V . Minachi n

1991 Mathematics Subject Classification. P r imar y 68Q15 ; Secondar y 03D15 , 68Q60 .

ABSTRACT. Thi s collectio n contain s translation s o f paper s o n propositiona l satisfiabilit y an d re -lated logica l problems . A significan t rol e i n th e collectio n i s playe d b y paper s relate d t o Maslov' s iterative metho d o f searc h reduction . Th e boo k i s usefu l fo r researcher s an d graduat e student s working i n mathematica l logic , theor y o f algorithms , an d i n relate d area s o f compute r science .

L i b r a r y o f C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a

Problemy sokrashchenii a perebora . English . Problems o f reducin g th e exhaustiv e searc h / V . Kreinovich , G . Mints , editors ,

p. cm . — (America n Mathematica l Societ y translation s ; ser. 2 , v . 178 ) Includes bibliographica l references . ISBN 0-8218-0386- 7 (alk . paper ) 1. Computationa l complexity . I . Kreinovich , Vladik . II . Mints , G . E . III . Series .

QA3A572 ser . 2 . vol . 17 8 [QA267.7] 510 s—dc20 [511.3] 96-2757 7

CIP

C o p y i n g a n d r e p r i n t i n g . Materia l i n this book may be reproduced b y any means for educationa l and scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y service s that collec t fee s fo r deliver y o f documents an d provide d tha t th e customar y acknowledgmen t o f th e source i s given. Thi s consen t doe s no t exten d t o othe r kind s o f copying fo r genera l distribution , fo r advertising o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercia l us e o f material shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematica l Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permissiori@ams. org .

Excluded fro m thes e provision s i s materia l i n article s fo r whic h th e autho r hold s copyright . I n such cases , request s fo r permissio n t o us e o r reprin t shoul d b e addresse d directl y t o th e author(s) . (Copyright ownershi p i s indicate d i n th e notic e i n th e lowe r right-han d corne r o f th e firs t pag e o f each article. )

© 199 7 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 .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

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

To the Memory of

Sergei Maslov and Nina Maslova

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Contents

Foreword t o th e Englis h Translatio n i x

Algorithmics o f TVP-Har d Problem s R. I . FREIDZO N 1

Algorithmics o f Propositiona l Satisfiabilit y Problem s E. YA . DANTSI N 5

Semantics o f S . Yu. Maslov' s Iterativ e Metho d V. YA . KREINOVIC H 2 3

Ergodic Propertie s o f Maslov' s Iterativ e Metho d M. I . ZAKHAREVIC H 5 3

Anomalous Propertie s o f Maslov' s Iterativ e Metho d I. V . MELICHE V 6 5

Possible Nontraditiona l Method s o f Establishin g Unsatisfiabilit y o f Propositional Formula s Yu. V . MATIYASEVIC H 7 5

Dual Algorithm s i n Discret e Optimizatio n G. V . DAVYDO V an d I . M . DAVYDOV A 7 9

Models, Methods , an d Mode s fo r th e Synthesi s o f Progra m Scheme s A. YA . DIKOVSK Y 10 3

Effective Calcul i a s a Techniqu e fo r Searc h Reductio n M. I . KANOVIC H 13 3

Lower Bound s o f Combinatoria l Complexit y fo r Exponentia l Searc h Reduction N. K . KOSSOVSKI I 14 9

On a Clas s o f Polynomia l System s o f Equation s Followin g fro m th e Formula fo r Tota l Probabilit y an d Possibilitie s fo r Eliminatin g Searc h in Solvin g The m AN. A . MUCHNI K 16 1

S. Maslov' s Iterativ e Method : 1 5 Years Late r (Freedo m o f Choice , Neura l Networks, Numerica l Optimization , Uncertaint y Reasoning , an d Chemical Computing ) V. KREINOVIC H 17 5

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Foreword t o th e Englis h Translatio n

This collectio n contain s translation s o f paper s o n propositiona l satisfiabilit y and relate d problem s i n logic which appeare d i n a book "Problem y Sokrashcheniy a Perebora," Vopros y Kibernetiki no . 133 , published i n Russian i n 198 7 by the Scien -tific Counci l "Cybernetics " o f the USSR Academy of Sciences. Thes e problems for m the nucleus of an intensively developin g area of investigation. A general descriptio n of this are a an d o f th e separat e paper s i s given i n a shor t introductor y not e b y R . Preidzon. Som e of the late r development s ar e described i n a supplement writte n b y V. Kreinovich fo r this translation. Relate d recen t investigation s involvin g ideas an d results o f th e presen t volum e concer n suc h divers e topic s a s biologica l computin g [1], combinatorial optimizatio n [2] , decision problems fo r linea r logi c [5] , improving upper bound s fo r satisfiabilit y algorithm s [6] , and applie d logi c programming [3] .

This translatio n i s dedicated t o th e memor y o f two remarkabl e Russia n math -ematicians, Serge i Maslo v [9 ] and hi s wife Nin a Maslova .

S. Maslov is known as the author o f the inverse method i n automated deductio n [7], which was discovered a t th e same time a s the resolution metho d o f J. A . Robin-son an d ha s approximatel y th e sam e rang e o f applications . I n 1981 , Maslov pro -posed a n iterativ e algorith m fo r propositiona l satisfiabilit y base d o n som e genera l search idea s describe d i n detai l i n hi s boo k [8] , published posthumously . Whe n i n 1982 Maslov wa s killed a t th e ag e o f 43 in a ca r accident , a seminar wa s started t o follow throug h o n hi s ideas . Th e resul t o f this collectiv e researc h form s th e bul k of the presen t book .

Nina Maslov a wa s a mathematicia n i n he r ow n right . Sh e wa s awarde d th e National Priz e i n Mathematics , on e o f the highes t distinction s fo r mathematic s i n the former Sovie t Union . Whe n she died in 1993 at th e age of 54, the St . Petersbur g Mathematical Societ y convene d a specia l meetin g i n he r honor .

The intellectua l influenc e o f th e Maslo v famil y wa s no t restricte d howeve r t o their scientifi c achievements . Thei r hom e i n Leningra d (no w St . Petersburg ) wa s a meetin g plac e o f a semina r wher e talk s o n socia l an d scientifi c problem s wer e presented. On e ha s t o fee l th e gravit y o f th e ideologica l pressur e o f a totalitaria n state t o appreciat e th e importanc e o f suc h a fre e forum . Th e emergenc e o f suc h seminars seem s t o b e characteristi c fo r intellectua l lif e unde r oppressiv e regimes : recall Zilsel's seminar in Vienna where Godel presented in January 193 8 his overview [4] o f possibilitie s fo r continuin g Hilbert' s program . Anothe r foru m fo r dissiden t thought i n th e USS R wa s provide d b y a Samizda t (unofficiall y published ) journa l

ix

x FOREWOR D T O TH E ENGLIS H TRANSLATIO N

"Summa" edite d b y S . Maslo v [10] , whic h wa s designe d a s a revie w journa l fo r Samizdat publications .

One ma y hop e tha t thi s boo k wil l contribut e t o furthe r progres s o f knowledg e and understanding , whic h wa s th e goa l o f Nina Maslov a an d Serge i Maslov .

Grigori Mint s Stanford , Jun e 199 6

References

1. L . Adleman , Molecular computation of solutions to combinatorial problems, Scienc e 26 6 (1994), 1021-1024 .

2. G . Davydo v an d I . Davydova , Tautologies and positive solvability of linear homogeneous systems, Ann . Pur e Appl . Logi c 5 7 (1992) , 27-43 .

3. A . Dikovsky , On computational complexity of Prolog programs, Theoret . Comput . Sci . 11 9 (1993), 63-102 .

4. K . Godel , Lecture at ZilseVs, Collecte d Work s o f Kur t Gode l (S . Feferma n e t al. , eds.) , Vol . Ill , Oxfor d Universit y Press , 1995 , pp . 87-113 .

5. M . Kanovich , The complexity of neutrals in linear logic, Proc . 10-t h Annua l IEE E Symposiu m on Logi c i n Compute r Scienc e (Sa n Diego , California , 1995) , pp . 486-495 .

6. O . Kullman n an d H . Luckhardt , Various complexity upper bounds for decisions on proposi-tional tautology, Inform , an d Comput . (1996 ) (t o appear) .

7. S . Yu . Maslov , An inverse method of establishing deducibility in the classical predicate calcu-lus, Dokl . Akad . Nau k SSS R 15 9 (1964) , 17-20 ; Englis h transl. , Sovie t Math . Dokl . 5 (1964) , 1420-1424.

8. , Theory of Deductive Sytems and Its Applications, "Radi o i Svyaz" , Moscow , 1986 ; English transl. , MI T Press , Cambridge , MA , 1987 .

9. G . Davydov , Yu . Matiyasevich , G . Mints , V . Orevkov , A . Slisenko , an d N . Shanin , Sergei Yuryevich Maslov. Obituary, Uspekh i Mat . Nau k 3 9 (1984) , no . 2 (236) , 129-130 ; Englis h

transl., Russia n Math . Survey s 3 9 (1984) , no . 2 , 133-135 . 10. A . Vershik , Concealed digest of stagnation times, Zvezd a 1991 , no . 11 . (Russian )

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

147 I . G. Bashmakova et al. , Nin e Paper s fro m th e Internationa l Congres s o f Mathematicians , 1986

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. Berma 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 . Draskovicova 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. Aizenshta 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 . Arnol' d e t al. , Fourtee n Paper s Translate d fro m th e Russia n

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

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

134 A . Ya . Aizenshta 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 . Dmitrie v e t al. , Thirtee n Paper s i n Algebr a

131 V . A . Zmorovic h et 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 Physic 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 et al. , Thirtee n Paper s i n Algebr a an d Numbe r Theor y

127 P . L . Shabalin et al. , Eleve n Paper s i n Analysi s

126 S . A . Akhmedo 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

123 V . G . Maz'y a e t al. , Ellipti c Boundar y Valu e Problem s

122 N . U . Arakelya n e t al. , Te n Paper s o n Comple x Analysi s

121 V . D . Mazurov , Yu . I . Merzlyakov , an d V. A . Churkin , Editors , Th e Kourovk a Notebook :

Unsolved Problem s i n Grou p Theor y

120 M . G . Krei n and V. A . Jakubovic , Fou r Paper s o n Ordinar y Differentia l Equation s

119 V . A . Dem'janenk o e t al. , Twelv e Paper s i n Algebr a

118 Ju . V . Egoro v e t al. , Sixtee n Paper s o n Differentia l Equation s

117 S . V . Bockare v e t al. , Eigh t Lecture s Delivere d a t th e Internationa l Congres s o f

Mathematicians i n Helsinki , 197 8

116 A . G . Kusnirenko, A . B . Katok, an d V. M. Alekseev, Thre e Paper s o n Dynamica l System s

115 I . S . Belov e t al. , Twelv e Paper s i n Analysi s

114 M . S . Birma n an d M . Z . Solomjak , Quantitativ e Analysi s i n Sobole v Imbeddin g

Theorems an d Application s t o Spectra l Theor y

113 A . F . Lavri k e t al. , Twelv e Paper s i n Logi c an d Algebr a

112 D . A . Gudko v an d G . A . Utkin , Nin e Paper s o n Hilbert' s 16t h Proble m

111 V . M . Adamja n et al. , Nin e Paper s o n Analysi s

110 M . S . Budjan u e t al. , Nin e Paper s o n Analysi s

109 D . V . Anoso v e t al. , Twent y Lecture s Delivere d a t th e Internationa l Congres s o f Mathematicians i n Vancouver , 197 4

(See th e AM S catalo g fo r earlie r titles )