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ESEARCH ARTICLE Adv. Sci. Lett. X, XXXXXX, 2015

Copyright 2015 Aeric!" Scie"ti#ic $%&'i(her( Adv!"ced Scie"ce Letter(

A'' right( re(erved )o'.X, XXX*XXX, 2015

$ri"ted i" the +"ited St!te( o# Aeric!

Parallel Theorem Proving Algorithm

Based on Semi-Extension Rule

Lii"g h!"g1,2, -!"to"g %y!"g1,2

1Schoo' o# Cop%ter Scie"ce !"d Tech"o'ogy, /i'i" +"iver(ity, Ch!"gch%" 10012, Chi"!2ey L!&or!tory o# Sy&o'ic Cop%t!tio" !"d "o'edge E"gi"eeri"g o# 3i"i(try o# Ed%c!tio",

/i'i" +"iver(ity, Ch!"gch%" 10012, Chi"!

A#ter ! deep i"ve(tig!tio" o" the !4i% ter( (p!ce o# the c'!%(e (et, the co"cept o# the p!rti!' !4i% ter( (p!ce o#

the c'!%(e (et, hich the !4i% ter( o# the c'!%(e (et decopo(ed, i( &ro%ght #or!rd. y i"ve(tig!ti"g the e4te"(io"

r%'e, thi( p!per i"trod%ce( the co"cept o# the (!ti(#i!&i'ity !"d the %"(!ti(#i!&i'ity o# the p!rti!' !4i% ter( (p!ce, !"d

give( !" !'gorith deteri"i"g the (!ti(#i!&i'ity o# ! p!rti!' (p!ce o# the !4i% ter( * !'gorith $SER 6$!rti!' Sei*

E4te"(io" R%'e7. The", the T$ pro&'e i( decopo(ed i"to (ever!' (%&*pro&'e( i"depe"de"t o# e!ch other, hich c!" &e

(o'ved &y the give" p!r!''e' cop%ti"g ethod $$SER 6$!r!''e' $!rti!' Sei*E4te"(io" R%'e7.

Keywords: Theore $rovi"g, $!r!''e' A'gorit, E4te"(io" R%'e.

1. !TR"#$%T"!

The c'!((ic!' 8$*cop'ete pro&'e o# T$ h!( (ee"%ch i"tere(t i" "ot 9%(t the theoretic!' cop%ter (cie"ceco%"ity, &%t !'(o i" !re!( here pr!ctic!' (o'%tio"( tothi( pro&'e hich e"!&'e (ig"i#ic!"t pr!ctic!'

!pp'ic!tio"(1

. Hoever, 8$*Cop'ete"e(( doe( "ote4c'%de the po((i&i'ity o# #i"di"g !'gorith( th!t !ree##icie"t e"o%gh #or (o'vi"g !"y i"tere(ti"g (!ti(#i!&i'ityi"(t!"ce(. The(e i"(t!"ce( !ri(e #ro !"y diver(e !re!( *!"y pr!ctic!' pro&'e( i" AI p'!""i"g2*5, circ%itte(ti"g:,;!"d veri#ic!tio"orthe ge"er!tio" o# the t!rget '!"g%!ge &!(ed o" the?"o'edge copi'!tio", !"d !chieved good re(%'t(.

@E!i' Addre(( 'ii"gBh!"g,o%yd9'%.ed%.c"

e(ide(, !"y re(e!rcher( !pp'ied the e4te"(io" r%'e tothe ode' co%"ti"g pro&'e1i"!''y, o%r

or? o# thi( p!per i( (%!riBed i" the '!(t (ectio".

1


2/4

&. E'TE!S"!-R$(E BASE# T)E"RE*

PR"+!, *ET)"#

e &egi" &y (peci#yi"g the "ot!tio" th!t i'' &e %(ed

i" the re(t o# thi( p!per. e %(e F to de"ote ! (et o#

c'!%(e( i" co"9%"ctive "or!' #or 6C8>7, C to de"ote !

(i"g'e c'!%(e, !"d 3 to de"ote the (et o# !'' the !to( th!t!ppe!r i"F. The e4te"(io" r%'e i( de#i"ed !( #o''o(.

-E>I8ITI8 115.Give" ! c'!%(e C, CF , D

CA, CA J KA i( !" !to,AM, KA !"d KA doe("ot !ppe!r i" CM, e c!'' the ded%ctio" proce((

proceedi"g #ro C to Dthe e4te"(io" r%'e o# o" C, !"d

c!''Dthe re(%'t o# !pp'yi"g the e4te"(io" r%'e o# o" C.

THERE3 115.A c'!%(e Ci( 'ogic!''y eN%iv!'e"t to

the re(%'t o# the e4te"(io" r%'eD.

Thi( theore e"(%red the eN%iv!'e"ce &etee" the

origi"!' c'!%(e (et !"d the e4p!"ded c'!%(e (et, th%(

e4te"(io" r%'e c!" &e reg!rded !( !" i"#ere"ce r%'e.

-E>I8ITI8 222.A "o"*t!%to'ogy c'!%(e i( !!4i% ter o" ! (etMi## it co"t!i"( !'' the !to( i"

Mi" either po(itive #or or "eg!tive #or.

THERE3 215.Give" ! (et o# c'!%(e( F, 'et M &e

the (et o# !'' the !to( i" it 6JMJ m7. I# !'' the c'!%(e( i"

F !re !4i% ter( o" M, the" the c'!%(e (et F i(

%"(!ti(O!&'e i## it co"t!i"( 2mc'!%(e(.

App!re"t'y the (et o# !'' the !4i% ter( co"(i(t

o# !to( i( (%re'y co"t!i"( 2m !4i% ter(.

There#ore, it i( o"'y "eed to cop%te the "%&er o#

di(ti"ct !4i% ter( c!" &e ded%ced #ro the c'!%(e

(et th!t e c!" deteri"e it( (!ti(#i!&i'ity. I" !dditio",

he" co%"ti"g the "%&er o# the !4i% ter( th!t

c!" &e ded%ced #ro the c'!%(e (et, e c!" %(e the

i"c'%(io"*e4c'%(io" pri"cip'e pre(e"ted &e'o.

THERE3 22.6I"c'%(io"e4c'%(io" pri"cip'e7 The

e'ee"t "%&er o# the %"io" o# (et( o# (etA1, A2,P,Anc!" &e cop%te %(i"g the #or%'! &e'o

A1A2PAn= 1J J

n

i

i

A

=

1

J Ji j

i j n

A A

<

+P+ (*

1)nQ1A1A2 PAn.THERE3 =15.The i"ter(ectio" o# the (et( th!t

co"(i(t o# the !4i% ter( e4p!"ded &y to c'!%(e(

re(pective'y i'' &e epty i## the(e to c'!%(e( co"t!i"cop'ee"t!ry 'iter!'(.

Give" ! (et o# c'!%(e( F={C1, C2, P , Cn}, 'etM&ethe (et o# !to( th!t !ppe!r i" it 6JMJ m7.LetPi&e the

(et( o# !'' the !4i% ter( e c!" get #ro Ci &y

%(i"g the e4te"(io" r%'e, !"d 'et S &e the "%&er o#

di(ti"ct !4i% ter( e c!" get #ro F. y %(i"g the

e4te"(io" r%'e, e i'' h!ve S=P1P2PP".

. PARA((E( PR"+!, A(,"RT)* BASE# "!SE*-E'TE!S"! R$(E

The ide! o# the p!r!''e'ed (ei*e4te"(io" r%'e &!(ed

!'gorith i( !( #o''o(. >ir(t'y, the !'gorith decopo(e(

the !4i% ter( (p!ce o# the c'!%(e (et i"to (ever!'

p!rti!' !4i% ter( (p!ce(, hich co"vert the SAT

pro&'e o# the c'!%(e (et i"to the SAT pro&'e o# the

p!rti!' !4i% ter( (p!ce(. I# there i( ! cert!i" p!rti!'

!4i% ter( (p!ce th!t i( (!ti(#i!&'e, the" the c'!%(e

(et i( (!ti(#i!&'e. I# !'' the p!rti!' !4i% ter( (p!ce(

!re %"(!ti(#i!&'e, the" the c'!%(e (et i( %"(!ti(#i!&'e. I"

other ord(, the c'!%(e (et i( (!ti(#i!&'e. I" the #o''oi"g,

the co"cept o# the p!rti!' !4i% ter( (p!ce i'' &egive".

-E>I8ITI8 . >or the (etML1,L2,PLmM, the 2m

!4i% ter( corre(po"di"g to M i( L1L2PLm*1Lm, L1L2PLm*1Lm,P,L1L2PLm*1Lm, L1L2PLmM, !"d e "%&er e!ch!4i% ter !( i607,i617,P,i62m*27,i62m*17.

-E>I8ITI8 =.Give" ! c'!%(e (et F={C1, C2,P,Cn}, 'etM&e the (et o# it( 'iter!'(, !"dM= m. e c!''!4i% ter( (p!ce o# 3 !( 3I6M7. A((%i"g th!t

12k2m,i# e o%'d 'i?e to decopo(e the !4i%

ter( (p!ce i"to 2

k

(p!ce(, the" e!ch (p!ce i( o# thi( #or3IS6j7i6j7Jj i62m6j*17 2k7, i62m6j*172kQ17,P,

i62m6j7 2k*17MM, 1j2k.

-E>I8ITI8 5. >or the p!rti!' !4i% ter(

(p!ce 3IS6j7, 1j2k2m. I# !'' the !4i% ter( i" it

c!" &e e4p!"ded &y the c'!%(e( o# the c'!%(e (et, the"

3IS6j7 i( (!id to &e %"(!ti(#i!&'e. I# there e4i(t ! cert!i"

!4i% ter th!t c!""ot &e e4p!"ded &y !"y c'!%(e o#

the c'!%(e (et, the" 3IS6j7 i( (!id to &e (!ti(#i!&'e.

THERE3 5.I# every p!rti!' !4i% ter( (p!ce

i( %"(!ti(#i!&'e, the" the c'!%(e (et i( %"(!ti(#i!&'e. I#

there i( ! cert!i" p!rti!' !4i% ter( (p!ce th!t i(

(!ti(#i!&'e, the" the c'!%(e (et i( (!ti(#i!&'e.I" the #o''oi"g, the !'gorith $SER hich

deteri"e( the (!ti(#i!&i'ity o# the p!rti!' !4i% ter(

(p!ce i'' &e give"

-E>I8ITI8 :. 3L1,L2,P,LmM,mJMJ. Let

c'!%(e C LiPLjPLd, 1ijdm, hich d i(

re#erred !( the degree o# c'!%(e C. FCLk, CLkJdUkmM, e c!'' the oper!tio" proceedi"g #ro Cto the

e'ee"t( o# F the (ei*e4te"(io" r%'e, !"d the e'ee"t(

o# F the re(%'t o# the (ei*e4te"(io" r%'e.

$R$SIT8 1.Accordi"g to de#i"itio" :, he"

!pp'yi"g the (ei*e4te"(io" r%'e o" C, the re!i"i"g *d

!to( co%'d &e po(itive or "eg!tive, there#ore Cc!" (ei*

e4p!"d 2m*dc'!%(e(.

$R$SITI8 2.Let d1 !"d d2 &e the degree( o#

c'!%(e C1 !"d C2 re(pective'y, hi'e d1Ud2 !"d C1C2.

Accordi"g to propo(itio" 1, the c'!%(e th!t C1 or C2 c!"

(ei*e4p!"d i( o&t!i"ed &y copo(e the m*d1 !to( or

m*d2 !to( i" po(itive #or or i" "eg!tive #or.

There#ore, c'!%(e( th!t C2c!" (ei*e4p!"d !re ! (%&(et o#

the c'!%(e( th!t C1c!" e4p!"d.

Accordi"g to propo(itio" 2, he" deteri"i"g

hether the !4i% ter( c'!%(e c!" &e e4p!"ded &y

the c'!%(e(, e (ho%'d deteri"e hether it c!" &ee4p!"ded &y the c'!%(e( o# (!''er degree #ir(t. I" the

#o''oi"g, the !'gorith deteri"i"g the (!ti(#i!&i'ity o#

the p!rti!' !4i% ter( (p!ce i'' &e give".


3/4

>%"ctio" $SER6C8>F, I8Tstarti, I8Tendi7

1 EGI8

2 iVstartiW F-egreeSort6F7W

hi'e6iU endi7

= EGI8

5 TVget3!4Ter6i7W

: I# E4p!"d6F, T7#!'(e; The" Ret%r" SATW

< E'(e I# 6lenUm7 iL!(t3i6C,M7

D iQQW

10

E8-

11

Ret%r" +8SATW

12 E8-

I" the #o''oi"g, the re'!ted theore !"d !'gorith

o# E4p!"d i'' &e give".

THERE3 :21.Give" ! c'!%(e (etFC1,C2,P,CnM,

LetM&e the (et o# it( 'iter!'(, !"d JMJm. A !4i%

ter TL1L2PLmo" 3 c!" &e e4p!"ded &y c'!%(e

C LiPL9PLd, 1ijdm, i## Li,P,Lj,P,Ld ML1,L2,P,LmM.

I" the !&ove, e g!ve the (o'vi"g ethod o# p!rti!'

!4i% ter( (p!ce !"d the !'gorith deteri"i"g it(

(!ti(#i!&i'ity. The !4i% ter( (p!ce o# ! c'!%(e (et

c!" &e decopo(ed i"to (ever!' p!rti!' !4i% ter(

(p!ce(. I" doi"g (o, the SAT pro&'e o# ! c'!%(e (et i(

co"verted i"to the SAT pro&'e( o# (ever!' p!rti!'

!4i% ter( (p!ce(. I# there i( ! p!rti!' !4i%

ter( (p!ce th!t c!""ot &e e4p!"ded, the" the c'!%(e (et i(

(!ti(#i!&'e. r e'(e, i# !'' the p!rti!' !4i% ter( (p!ce

i( %"(!ti(#i!&'e, the" the c'!%(e (et i( %"(!ti(#i!&'e. I" the

#o''oi"g, the p!r!''e' T$ !'gorith &!(ed o" (ei*

e4te"(io" r%'e i'' &e give".

>%"ctio" $$SER6 C8> F, I8T

thre!d"%7

1 hi'e i U thre!d"% do

2 EGI8

tid cre!tthre!d 67W

= I# 6 tid 0 7

5 EGI8

: Re(%'tiY $SER6F,

(t!rt6i7, e"d6i77W

; e4it607W< E8-

D iQQ

10 E8-

11 hi'e 617 do

12 EGI8

1 i"t Count0W

1= hi'e jU threadnum do

15 EGI8

1: i# 6Re(%'tjYSAT7

1; ret%r" SATW

1< i#

6Re(%'tjY+8SAT7

1D CountQQW

20 E8-

21 i# 6Countn7 ret%r"

+8SATW

22 E8-

The co"crete #'o o# the !'gorith i( !( #o''o( The

p!re"t proce(( di(tri&%te( Thre!d"% (%&*thre!d(, !"d

the" the(e (%&*thre!d( !re !rr!"ged to !"y core( o# the

proce((or &y the per!ti"g Sy(te, re(pective'y. E!ch(%&*thre!d c!''( the #%"ctio" $SER, !"d record( the

corre(po"di"g ret%r"ed re(%'t( %(i"g Re(%'tjY, hi'e the

p!re"t proce(( o"itor( r%""i"g re(%'t o# e!ch (%&*thre!d.

I# there i( ! (%&*thre!d th!t it( Re(%'tjY i( SAT, the" the

!'gorith ret%r"( SAT. I# the Re(%'tjY o# every (%&*thre!d

i( +8SAT, the" the !'gorith ret%r"( +8SAT.

E'PER*E!TA( RES$(T

" the &!(i( o# 'iter!t%re 21, i" thi( (ectio", e

cop!re o%r !'gorith $$SER ith !'gorith 8ER,

!'gorith IER !"d -irectio"!' Re(o'%tio" !'gorith 12propo(ed &y -echter !"d et !', re(pective'y. The

e4perie"t( !re c!rried o%t o" ! -e'' -ie"(io" C521,

A3- Ath'o"6t7 := X2 -%!' Core $roce((or :00Q,

1.DGHB 10223 RA3 ith i"do( X$. Thi(

e4perie"t %(e( +"i#or R!"do**SAT &e"ch!r? 2

'!y( o" ph!(e ch!"ge Bo"e !"d (t!"d!rd te(t c!(e( o# #r&

2= !( te(t c!(e(. >or the e"tire 1000 i((%e( o# %#20*D0,

t!&'e 1 o"'y (ho( the e4perie"t!' re(%'t( o# 10 i((%e(

r!"do'y (e'ected. The #ir(t co'%" o# the t!&'e (ho(

the (!p'e "!e, the '!tter co'%"( (ho( the r%"tie

o# three !'gorith( corre(po"di"g to e!ch i((%e hich the

%"it o# d!t! i( Seco"d( 6(7. >or e!ch te(t c!(e, e i'' te(t10 tie( !"d t!?e the e!" v!'%e !( the e4perie"t!'

re(%'t(. A( e c!" (ee #ro the te(t, o%r !'gorith $$SER

h!( ! (ig"i#ic!"t !dv!"t!ge o" e##icie"cy cop!red ith

the origi"!' !'gorith, hich i(


4/4

the #i"!' re(%'t. The e4perie"t!' re(%'t i( i" >ig%re 1 !"d

>ig%re 2.

>ig.1. UN, 20, 10Z

>ig.2. UN, 20, 10Z

y te(ti"g the R!"do SAT pro&'e(, e c!" (ee#ro #ig%re 1 !"d #ig%re 2, th!t o%r !'gorith $$SER h!(!" o&vio%( !dv!"t!ge o" e##icie"cy, hich i( :*15 tie(higher th!" the re'!tive'y #!(t !'gorith 8ER. 3oreover,he" the "%&er o# c'!%(e( i"cre!(ed to 10 !&ove, thecop%ti"g tie o# o%r !'gorith $$SER i( i"cre!(ed

ge"t'y.

/. %"!%($S"!S

Thi( ethod decopo(e the !4i% ter( (p!ceo# the c'!%(e (et i"to (ever!' p!rti!' !4i% ter((p!ce(, !"d deteri"e( the (!ti(#i!&i'ity o# e!ch p!rti!'!4i% ter( (p!ce. I# !'' the p!rti!' !4i% ter((p!ce !re %"(!ti(#i!&'e, the c'!%(e (et i( %"(!ti(#i!&'e. r

e'(e, i# there i( ! cert!i" p!rti!' !4i% ter( (p!ce th!ti( (!ti(#i!&'e, the" the c'!%(e (et i( (!ti(#i!&'e. e(ide(,he" deteri"i"g the (!ti(#i!&i'ity o# the p!rti!'!4i% ter( (p!ce, i# e #o%"d th!t ! !4i% terc!" &e e4p!"ded &y ! cert!i" c'!%(e, the" !'' the!4i% ter( (ei*e4p!"ded &!(ed o" thi( c'!%(e h!ve"o "eed to deteri"e their e4p!"d!&i'ity, th%( e##ective'yred%ced the "%&er o# the !4i% ter( to &edeteri"ed. The e4perie"t re(%'t( (ho th!t o%r!'gorith $$SER h!( ! re!(o"!&'e e4ec%tio" e##icie"cy,hich i( (%perior to !'gorith 8ER, -R, IER !"d et !'.

A%K!"0(E#,*E!TS

Thi( or? !( (%pported i" p!rt &y 8S>C %"der

Gr!"t 8o(. :0D;0 o%((e!rt, > Heery, C Leco%tre. R!"do co"(tr!i"t

(!ti(#!ctio" e!(y ge"er!tio" o# h!rd 6(!ti(#i!&'e7 i"(t!"ce( ,

Arti#ici!' I"te''ige"ce, 1;1 6200;7 51=*5=.

Received 22 Septe&er 2010. Accepted 1< cto&er 2010

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