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8/10/2019 Complete Sets of Unifiers and Matchers in Equational Theories 1986 Theoretical Computer Science
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Theoretica l Comput er Science 43 (1986) 189-200
North-Holland
189
COMP LETE SETS OF UNIFI ERS AND MATCHER S IN
E Q U A T I O N A L T H E O R I E S *
Franfois FAGES and Grrard HUET
CNRS LITP), INR IA Rocquencourt), 78153 Le Chesnay Cedex, France
Communicated by M. Nivat
Received October 1985
Abstract. We propose an abstract framework to present unification and matching prob,lems. We
argue about the necessity of a somewhat complicated definition of a basis of unifiers. In
particular, we prove the nonexistence of complete sets of minimal unifiers (and matehers) in some
equational theories, even regular.
1 . E q u a t i o n a l t h e o r i e s
We assume to be well known the concept of an algebra A = (A, F) with A a set
of elements (the card er o f A) and F a fami ly of operators given with their arities.
More generally, we may consider heterogeneous algebras over some set of sorts,
but all the not ions considered here carry over to sorted algebras without difficulty
and so we will forget sorts and even arities for simplici ty of notat ion. With this
provision, all our definitions are consistent with [22].
We denote by T ( F ) the set of (ground) terms over F. We assume that there is at
least one c onstant (ope rator of arity 0) in F so that this set is not empty. We also
assume the existence of a denumerable set of variables V, disjoint from F, and
denote by T(F, V) the set of terms with variables over F and V. When F and V
are clear from the context, we abbreviate T(F , V) as T and T ( F ) as G (for ground).
We denote terms l~y M, N , . . . , and write V ( M ) for the set of variables appearing
in M.
We denote by T (respectively G) the algebra with carrier T(respectively G) and
with operators the term constructors corresponding to each operator of F
Th e substitutions are all mappings fro m V to T, exte nded to T, as endomorph isms
of T. We de note by S the set of all substitutions. I f or ~ S and M e T, we denote by
trM the application of cr to M. Since we are only interested in substitutions for
the ir effect on terms, we shall general ly assume that o'x =x , except on a finite set
of variables D(cr) which we call the domain of tr by abuse of notation. Such
substitutions can then be represented by the finite set of pairs {x ~ crxlx ~ D(tr)}.
* A preliminary version of this paper was pres ented in March 1983 at CAAP'83.
0304-3975/86/ 3.50 1986, Elsevier Science Publisher s B.V. (Nor th-Ho lland)
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190 E Fa ges , G . H u e t
T h e e m p t y s u b s t i t u ti o n ( i d e n t i ty ) i s d e n o t e d b y I d . W e d e f i n e t h e se t I ( cr ) o f variables
in t roduced by t r a s
I ( c r ) = U V( o - x ) .
xeD (o ' )
W e s a y t h a t o - i s
g r o u n d
i ff I ( o ' ) = 0 . T h e c o m p o s i t i o n o f s u b s t i t u t i o n s i s t h e u s u a l
c o m p o s i t i o n o f m a p p i n g s : (o - o
p) x = c r( px ) .
A n d w e s a y t h a t o i s
m or e g ener a l t han
p- o -
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Com plete sets of unifiers an d marchers
191
A x iom a t i c equa t iona l t heor i e s a re sem idec idab le ( e .g . , by enum era t ing a l l poss ib l e
p roof s o f equ a l i ty o f tw o t e rm s) , so U E is a lw ays r ecur s ive ly enum erab le ( e.g ., by
enum era t ing a l l subs t i t u t ions and check ing in pa ra l l e l w he the r t hey a re un i f i e r s o r
no t ) , bu t , o f cour se , w e a re m os t ly in te res t ed in a g ene ra t ing se t o f the E -un i f ie r s
( ca l l ed 'C om ple t e S e t o f E -U ni f i e r s ' i n [39] and deno ted by C S U ~) , f rom w hich
w e can gene ra t e U ~ by ins t an t i a tions s ince o-M = E o - N ~ V , ( , o c r )M = E ( , o c r)N.
O r be t t e r , by a bas is o f U E (ca ll ed 'C om ple t e S e t o f M in im a l U ni f i e r s ' and deno ted
b y # C S U e ) s at is fy in g th e m i n im a l it y c o n d it io n s 0 4 ~ ' ~ c r ~ v o r', w h e re V =
V ( M ) u V ( N ) .
H ence , w e sha l l m ake the d i f f e rence be tw een un i f i ca t ion procedures, which enu-
m era te a C S U e ( the exhaus t ive enum era t ion p rocedure in sem idec idab le theor i e s
enum era te s U e com ple t e ly ) , un i f ica t ion algorithms, w hich a lw ays t e rm ina te w i th a
f in it e C S U e , em pty i f t e rm s a re no t un i f iab le , and minimal un i f i ca tion p rocedures
o r a l g o r i t h m s w h i c h c o m p u t e a # C S U e .
Un if ica t ion was f i rs t s tudie d in f i r s t-order langua ges ( the case E = ~J) by H erbran d
in [ 16]. In h is thes is , he gave an expl ic i t a lgo r i thm to c om pute a mos t gen era l uni fier.
H ow ever , t he no ta t ion o f un i f ica t ion r ea lly g rew ou t o f t he w ork o f t he r e sea rche rs
in au tom at i c t heo rem -prov ing s ince the un i f i ca t ion a lgor i thm i s t he bas i c m echan i sm
need ed to exp la in the m utu a l i n t e rac t ion o f in fe rence ru l e s. R ob inson [41] gave the
a lgor i thm in connec t ion w i th the r e so lu t ion ru l e and p roved tha t i t indeed-com putes
a most genera l uni f ier , tha t i s , a ~CSU~ equal to a s ingle ton whose exis tence i s a
fun dam enta l p ro pe r ty o f f i r st -o rde r l anguages. Inde pend en t ly , G ua rd [15] p resen ted
unif ic a t ion in var ious sys tems o f logic . Un if ica t ion i s a lso cent ra l in the t rea tm ent
o f equ a l i ty [29 , 42 ]. Im plem en ta t ion and com plex i ty ana lys i s o f un i f ica t ion i s d is -
cussed in [1 , 20, 25, 37, 50, 53] and Paterson an d W egm an give a l inear a lgor i thm
to compute a most genera l uni f ier .
F i rs t order uni f ica t ion was extended to inf in i te ( regular ) t rees by Huet [20] , who
show ed tha t a s ing le m os t gene ra l un i f i e r ex i s t s fo r t h i s c l a s s , com putab le by an
a lm os t l i nea r a lgor i thm . T h i s p rob lem i s r e l evan t to the im plem e nta t ion o f P R O L O G -
l ike pr og ram min g langu ages [4 , 5 , 6 , 9] .
In the con tex t o f h ighe r -o rde r l ogic, the p ro b lem of un i f i ca tion w as s tud ied by
G o uld [14] , w ho d e f ined ' gene ra l m a tch ing se t s ' o f t e rm s , a w eak er no t ion than
tha t o f C S U . T h e ex i s t ence o f un i fi e r i s show n to be u ndec idab le in th i rd -o rde r
l anguag es in [18] , and in second-orde r i n [13]. T he gene ra l t heory o f C S U ' s and
p C S U ' s in the con tex t o f h ighe r o rde r l og ic i s s tud ied in [20 , 24 ].
Unif ica t ion in equat ional theor ies was f i rs t s tudied by P lotkin [39] in the context
o f r e so lu t ion theorem prover s t o bu i ld up the under ly ing equa t iona l t heory in to the
ru le s o f i n fe rence . In th i s pape r , P lo tk in con jec tu red tha t t he re ex i s t ed an equa t iona l
theory E w here a / zC S U ~ d id no t a lw ays ex i s t . T heorem 2 .1 in the nex t sec t ion
proves th is conjecture .
F ur the r i n t e res t i n un i f i ca t ion in equa t iona l t heor i e s a rose f rom the p rob lem of
im plem ent ing p rogram m ing l anguages w i th ' ca l l by pa t t e rns ' , such a s Q A 4 [43] .
A ssoc ia t ive un i f i ca t ion ( f ind ing so lu t ions to w ord equa t ions ) i s a pa r t i cu la r ly ha rd
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192 F. Fages, G. Huet
prob lem . P lo tk in [39] g ives a p rocedure to enum era te a / zC S U A (poss ib ly in f in i t e ) ,
and M akan in [34] show s tha t t he w ord equa t ion p rob lem i s dec idab le . S t i cke l
[47 , 49] and , i ndepen den t ly , L ivesey and S iekm ann [32 , 33] g ive an a lgor i thm fo r
un i f i ca t ion in the p resence o f a s soc ia t ive -com m uta t ive ope ra to r s , t he t e rm ina t ion
of w hich has b een p roved in the gene ra l case by F ages [9 , 10 ] . T h i s r e su lt o f
t e rm ina t ion has been gene ra l i zed r ecen t ly to the com bina t ion o f un i f ica t ion
a lgor i thm s fo r t heor i e s w i th d i s jo in t s e t s o f sym bol s by K i rchner [28] , T iden [51]
and Yel l ick [52] . S iekmann [44] s tudied the genera l problem in his Ph.D. Thesis ,
e spec ia l ly the ex tens ion o f t he A C -un i f i ca tion a lgor i thm to idem potence and iden ti ty .
L ank ford [30 , 31 ] gave the ex tens ion to a un i f i ca t ion p roced ure in A be l i an g roup
theory , fo r w hich T iden [51] r ecen t ly go t a p roof o f t e rm ina t ion .
T h e c o m p l e x i ty o f A C - u n if i ca t io n i s u n k n o w n . T h e c o m p l e x i ty o f A C - m a t c h i n g
( i .e . , f inding one subst i tu t ion o such that crM = A c N ) has been show n to be
N P - c o m p l e t e b y C h a n d r a a n d K a n n e l a ki s ( u n p u b l i s h e d ) a n d i n d e p e n d e n t ly b y
K a pur e t a l. [26 ]. T he co m plex i ty o f A C -equ iva lence i s li nea r.
In the c l a s s o f equa t iona l t heor i e s fo r w hich the re ex i s ts a canon ica l t e rm rew r i ti ng
sys t em ( see [22] ) , F ay [12] g ives a un ive r sa l p roced ure to enum era te a C S U e . I t i s
based on the no t ion o f ' na r row ing ' a s de f ined in [46] . H uU ot [23] g ives a s im i l a r
pro ced ure and a suff ic ient term inat ion cr i ter ion, fur th er genera l ized in [25]. S ick-
m an n a nd S zabo [33] inves t iga t e the dom ain o f r egu la r canon ica l t e rm rew r i ti ng
sys t em s in o rde r t o f ind gene ra l m in im a l un i f i ca t ion p rocedures , bu t w e sha l l show
here tha t even in th i s f r am ew ork /zC S U e m ay no t ex i s t (T heorem 4 .2 ) .
T e rm ina t ion o r m in im a l i ty o f un if i ca tion p rocedu res is m uch ha rde r to ob ta in
than com ple t eness . H ow ever , t he m a in app l i ca t ions o f un i fi ca t ion in equa t iona l
theor i e s t o the g ene ra l iza t ions o f the K nuth and B end ix a lgor i thm , such a s in [ 17, 38 ] ,
a re cove red by the a s soc ia t ive -com m uta t ive un i f i ca t ion a lgor i thm .
2.2. Definitions
Let M , N ~ T, V = V ( M ) u V ( N ) , and W be a f in it e se t o f 'p ro t ec t ed va r i ab les '
not ap pe ar in g in M or N , W n V = ~. S is a comp lete set o f E-unif iers o f M and N
away f rom W
i f a n d o n l y i f
( a ) V t r s S D ( t r ) _ _ _ V a n d
I ( t r ) c ~ ( W u D ( t r ) ) = ~ )
(pur i ty) ,
(b) S ~ U n (M , N ) (correctness) ,
( c ) V p ~
U (M , N ) 3 c r ~ _ S t r ~ v p
( com ple t eness ) .
F ur the rm ore , S i s a comple te se t of minimal E -un i f i e r s o f M and N aw ay f rom
W i f, add i t iona l ly ,
(d ) V tr , t r ' ~ S cr ~ c r ' ~ t r ~ v tr (m in im a l i ty ) .
T he r eason to cons ide r W n onem pty i s t ha t i n equa t ion a l t heor i e s in gene ra l
som e un i f i e r s m us t i n t roduce new va r i ab les and in m any a lgor i thm s , un i f i ca t ion i s
pe r fo rm ed on sub te rm s , so i t i s necessa ry to sepa ra t e the va r i ab les in t roduced by
un i f i ca t ion f rom the va r iab les o f t he con tex t no t app ea r ing in M and N . T h i s is t he
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c a s e , f o r i n s t a n c e , f o r re s o l u t io n i n e q u a t i o n a l t h e o r ie s [ 3 9 ] a n d f o r t h e g e n e r a l i z a ti o n
o f t h e K n u t h - B e n d i x c o m p l e t i o n p r o c e d u r e i n c o n g r u e n c e c l as s e s o f t e r m s [ 38 ] . I f
W w a s n o t t a k e n d i s jo i n t f r o m V , t h e n t h e v a r ia b l e s in c o m m o n s h o u l d b e r e n a m e d
by the un i f i e rs , e .g . , w i th W = V = {x, y} , t he sub s t i t u t i on {x ~ -z , y ~ z} i s a un i f i e r
o f x a n d y w h i c h s a t is f i es c o n d i t i o n ( a ) , b u t { x ~ y } o r { y ~ x } a r e n o t . B y t a k i n g
W c ~ V = ~ , v a r i a b l e r e n a m i n g i s n o t n e c e ss a r y . T h e c o n d i t i o n D ( c r ) c~ l ( c r ) = O i s
e q u i v a l e n t t o idempotence: c r o ~r = o- an d can a lw ays be sa t i s f i ed b y a un i f i e r [8 ] ;
t h e r e f o r e , i t i s e a s y to s h o w t h a t t h e r e a l w a y s e x is ts a C S U n a w a y f r o m W , b y t a k i n g
a l l E - u n i f i e r s s a t i s f y i n g ( a ) .
H o w e v e r , w e c a n n o t p u t i d e m p o t e n c e i n to t h e g e n e r a l d e f in i ti o n o f s u b s t i tu t i o n s
s i n c e , i n o r d e r t o c o m p a r e t w o u n i f i e r s c r a n d p w i t h t h e p r e o r d e r
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194 F. Fages, (3. Huet
i s a c a n o n i c a l t e r m r e w r i t in g s y s t e m f o r E . W e d e n o t e b y -~ o n e s t e p o f r e d u c t io n
b y R , a s u s u a l , b y - > a d e r i v a t i o n o f n r e d u c t i o n s t e p s , a n d b y ~ , [ M ] t h e n o r m a l
f o r m o f t e r m M i n s y s t e m R . W e h a v e M = E N if f ~ [ M ] = ~ [ N ] . T h e s e t o f n o r m a l
t e r m s d e f i n e s a m o d e l o f E i n t h e u s u a l w a y . L e t
0.0 = { x 0 } , V = { x } , W = 0 .
F i r st w e p r o v e th a t S in a C S U n o f M a n d N a w a y f r o m W .
(1) Pu r i t y : V i i > 0 D(o- i ) - ' {x} a n d I (0 . i ) n {x} = 0 .
( 2 ) C o r r e c t n e s s : V i ~ > 0 o - i g ( x ) = g ( f ( x , f ( x ~ _ ~ , . . . , f ( x ~ , 0 ) . . . ) ) ) - -> ~g (0 ), s o
0 iM = E N .
( 3 )
C o m p l e t e n e s s : L e t 0 ~
U E ( M , N )
a n d A = ~ [ o - x ] , w e h a v e
g ( A ) = ~ g ( O ) .
W e
s h o w t h e c o m p l e t e n e s s o f S b y p r o v i n g =l i ~> 0 0 i x ~< A b y s t r u c t u r a l i n d u c t i o n o n A .
I f A i s a v a r i a b l e o r a c o n s t a n t , t h e n g ( A ) i s i r r e d u c i b l e , s o g ( A ) = E g (O ) o n l y
i f A = 0 . W e t a k e i = 0.
I f A = g ( A ' ) , t h e n g ( A ) = g ( g ( A ') ) i s a l s o i n R - n o r m a l f o r m , s o t h i s c a s e d o e s
n o t a r is e s in c e t h e ( u n i q u e ) n o r m a l f o r m i s g ( 0 ) .
I f A = f ( A ' , A ) , h e n g ( A ) - > g ( A ) , s o g ( A ) = e g ( O ) . B y s t r u c t u r a l i n d u c t i o n ,
w e g e t a j s u c h t h a t o )x 1 , l e t p i = {x l * 0} , we ha v e
p~0 ~x= f ( 0 , 0 H x) --> 0 ~-1x,
he nc e , 0 ~
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1 9 5
V a ' e
U 2 3 p 'e U 1 p ' ~< v a ' .
We define ~(a ') as one such substitution p'. Therefore,
V~reU~/,(~o(a))~
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F. Fages, G. Huet
P r o o f .
F o r t h e n o n t r i v i a l w a y , a s s u m e S i s n o t m i n i m a l , i. e. , 3 t r , t r ' e S or ~ t r' a n d
3 p p ar = v tr '. S i n c e E i s r e g u l a r a n d N i s g r o u n d , w e h a v e I ( t r ) = I ( t r ' ) = ~. H e n c e ,
V x
~ V
V ( o -x ) = ~ , s o p o x = o -x a n d a x = ~ t r 'x , l e a d i n g t o a c o n t r a d i c t i o n . [ ]
A g a i n , h o w e v e r , a / z C S U ~ m a y n o t e x i st in a r e g u l a r t h e o r y , f o r i t m a y s t il l b e
n e c e s s a r y to i n t r o d u c e n e w v a r i a b l e s t o e x p r e s s m o s t g e n e r a l E - u n i f i e r s .
T h e o r e m
4.2.
I n s o m e r e g u l a r t h e o r y E , t h e r e e x i s ts E - u n i f i a b l e t e r m s f o r w h i c h th e r e
i s n o / ~ C S U ~ .
P r o o f . L e t E b e t h e e q u a t i o n a l t h e o r y d e f i n e d b y t h e f u n c t i o n s y m b o l s 0 , a , f , g o f
a r i t y 0 , 0 , 2 , 1, r e s p e c t i v e l y , a n d R b e t h e c a n o n i c a l t e r m r e w r i t i n g s y s t e m :
: (O , x ) x ,
f ( x , O) - , x ,
R = , g ( f ( x , y ) ) - > f ( g ( x ) , g ( y ) ) ,
g(0 ) -> 0,
f ( f ( g ( x ) , y ) , z )-> f ( g ( x ) , f ( y , z ) ) .
T h e p r o o f o f c a n o n i c i t y h a s b e e n c h e c k e d o n t h e K B s y s te m [ 1 1 ] , a n d is le f t h e r e
t o t h e r e a d e r ' s c o m p u t e r . W e d e n o t e b y --> o n e s t e p o f r e d u c t i o n b y R , a n d b y ~ [ M ]
t h e R - n o r m a l f o r m o f t e r m M . F i rs t, w e s t a te a n o r m a l f o r m l e m m a .
L e m m a 4 . 3 .
L e t P a n d Q b e t w o t e r m s i n R - n o r m a l f o r m a n d d i f f e r e n t f r o m O . T h e n
w e h a v e ~ , [ f ( P , Q ) ] = f ( P l , f ( P 2 , f ( P , , , , Q ) . . . )) f o r s o m e m > I 1 a n d t e r m s P 1 , . . . , P , ,
i n R - n o r m a l fo r m . M o r e o v e r , P = f ( P l , . . . , f ( P , , - 1 , P r o ) . . . ) .
T h e p r o o f b y s t ru c t u r a l i n d u c t i o n o n P i s o m i t t e d .
P r o o f o f T h e o r e m 4 . 2 ( c o n t i n u e d ) . L e t M = g ( x ) a n d N = f ( y , g ( a ) ) , w e s h a l l s h o w
t h a t t h e r e d o e s n o t e xi st a / z C S U E o f M a n d N . L e t
t r o = { X ~ a ,y < - O } , t r l = { x ~ f ( x l , a ) , y ~ g ( x t ) } ,
o '2 = { x ~ f ( x 2 , f ( x l , a ) ) , y ~ f ( g ( x 2 ) , g ( x l ) ) } , ,
tr~ = { x ~ f ( x ~ , t r y _ ix ) , y ~ f ( g ( x , ) ,
t r ~ _ l y ) } , . .
a n d S = { t r ~ [ i ~ O } , V = { x , y } , W = ~ .
F i r s t w e s h o w t h a t S i s a C S U ~ o f M a n d N a w a y f r o m W .
( 1 ) P u r i t y : V i ~ > 0 D ( t r~ ) = { x , y } a n d
I ( ~ r ~ ) c ~ { x , y } = l ~ .
( 2 ) C o r r e c t n e s s : V i ~ > 0
$ [ t r i M ] = f ( g ( x ~ ) , f ( g ( x ~ _ ~ ) , . . . , f ( g ( x l ) , g ( a ) ) . . . ) )
b y i
a p p l i c a t i o n s o f t h e t h i r d r u l e o f R . I n t h e s a m e w a y w e h a v e $ [ t r o N ] = g ( a ) i f i = 0 ,
a n d i f i > 0 , w e h a v e $ [ t r i N ] = ~ [ t r J ( y , g ( a ) ) ] = $ [ t r g ( x ) ] b y i - 1 a p p l i c a t i o n s o f
t h e l a s t r u l e o f R , h e n c e , tr ~M - - ~ o '~ N.
( 3) C o m p l e t e n e s s : L e t
t r ~ U n ( M , N )
a n d A = ~ [ t r x ] ,
B = ~ [ o 3 ' ] ,
w e h a v e
g ( A ) = n f ( g ( B ) , g ( a )) .
I f B # 0 , b y L e m m a 4 . 3 , w e h a v e
$ [ 0 - N] =
~ [ f ( B , g ( a ) ) ] = f ( B ~ , f ( B 2 , . . . , f ( B m , g ( a ) ) . . . ))
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Complete sets of unifiers and matchers 197
a n d
B = f ( B I , f ( B 2 , . . . , f ( B n , _ 1 , B m ) . . . ) ) f o r s o m e m / > 1 .
W e s h o w t h e c o m p l e t e n e s s o f S b y p r o v i n g : l i t> 0 o -ix ~< A a n d o -iy 2 o' i_~y, he n ce , o-~
8/10/2019 Complete Sets of Unifiers and Matchers in Equational Theories 1986 Theoretical Computer Science
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198 F. Fages , G. Huet
L e t R b e a n y C S U n o f M a n d N . S i n c e S i s c o m p l e t e , w e h a v e V p ~ R 3~r~
S cr~
8/10/2019 Complete Sets of Unifiers and Matchers in Equational Theories 1986 Theoretical Computer Science
11/12
Complete sets of unifiers and marchers 199
[22] G . H ue t an d D . Opp en , E qua t ions and rewr i t e ru l e s: A survey , i n : R .V . Book, ed . , Formal L anguages :
Perspectives and Open Problems
(Acad emic P res s , New Y ork /L ondo n, 1980).
[23] J .M. H ul lo t , C omp i l a t ion de form es canoniqu es dan s l e s theor i e s 6qua t ionne l le s , Th~se de 3~me
cycle , Univ. de Paris Sud, 1980.
[24] D . J ensen and T . P ie t rzykowski , Mech aniz ing to -orde r type theory th rough un i f i ca t ion , Theoret.
Comput. Sci . 3 (1977) 123-171.
[25] J .P . Jou an nau d a nd C. K i rchner , Inc rem enta l cons t ruc t ion of un i f ica t ion a lgor i thms in equa t iona l
theories ,
Proc. lOth ICALP,
1983.
[26] D . Ka put e t a l ., Com plex i ty o f ma tch ing prob lem, Proc. Rewrite Techniques and Applications, Di jon ,
1985, Lecture Notes in Computer Science 202 (Springer , Berl in, 1985) 417-429.
[27] H. K irch ner an d C . Kirchn er , Con tr ibut io n ~t la r~solut ion d '~quat ions dan s les a lg~bres l ibres e t
les vari r t~s 6quat ionnel les d 'a lg~bres , Th~se de 3~me cycle , Univ. de Nancy, 1982.
[28] C. K i rchner , M ~thodes e t ou t il s de concep t ion sys t~mat ique d 'a lgor i thmes d 'un i f i ca t ion dans l e s
theor i e s ~qua t ionne l le s , Th~se d ' r t a t , Univ . de Nan cy , 1985 .
[29] D . Kn uth a nd P . Bendix , S imple word prob lems in un ive rsa l a lgebras , i n: J . Leech , ed . , Computat ional
Problems in Abstract Algebra (Pe rga mo n P res s, Ox ford /Ne w Y ork , 1970) 263-297 .
[30] D .S. L ankfo rd , A u nf i ca tion a lgor i thm for Ab e l i an group theory , Rept . MTP-1 , M ath . Dept . ,
Loui s i ana Tec hn . U niv ., Rus ton , LA, 1979 .
[31] D .S. L ankfo rd , G . But l e r and B. Brady , Abe l i an group un i f ica t ion a lgori thms for e l emen ta ry t e rms ,
Math . D ept . , Lo ui s i ana Techn . Univ . , Rus ton LA, 1983.
[32] M. Livesey and J . S iekman n, Unif icat ion of bags an d sets, Int . Rept . 3/76, Ins t i tut f i i r Info rm at ik
I , Univ. Karls ruhe, 1976.
[33] M. L ivesey , J . S i ekmann, P . Szabo and E . Unver i ch t , Uni f i ca tion prob lems for combina t ions of
as soc ia t iv i ty , com muta t iv i ty , d i s t ribu t iv ity an d idem potenc e ax ioms , 4 th Workshop on Au toma ted
Deduction,
Aust in, TX (1979) 161-167.
[34] G .S. M akan in , Th e prob lem o f so lvab i l it y o f equa t ions in a f ree semigroup ,
Akad . Nauk . SSSI~
T O M
23(2) (1977) 287-290.
[35] A. M arte l l i and U. M ontan ari , An eff ic ient unif icat ion a lgori thm,
A C M T O P L A S .
4(2) (1982)
258-282.
[36] Y . Mat iyasev ich , D iophant ine represen ta t ion o f recursive ly enum erab le p red ica te s , P ro ,
2nd Scan-
dinavian Logic Symposium
(Nor th-Hol l and , Ams te rdam, 1970) .
[37] M.S . Pa te r son and M .N. Wegm an, L inea r un i f i ca tion ,
J. Comput. System ScL
16 (1978) 158-167.
[38] G .E . P e te r son an d M.E . S t ickel , Com ple te s e t s o f reduc t ion for equa t iona l t heor i e s w i th comple te
unif icat ion a lgori thms, J .
A C M
28(2) (1981) 233-264.
[ 3 9 ] G .
Plotkin, Bui lding-in equat ional theories ,
Ma chine Intelligence
7 (1972) 73-90.
[40] P . Raulefs , J . S iekmann, P . Szabo and E. Unvericht , A short survey on the s ta te of the ar t in
match ing and un i f i ca t ion prob lems ,
S igsam Bu l l 1979
13 (1979).
[41] J .A . Robinson , A m achine -or i en ted log ic based o n the re so lu t ion pr inc ip le ,
J. A C M
12 (1965) 32-41.
[42] G .A . Rob inson and L .T. Wos, Pa ram odu la t ion and theorem proving in fi r s t-o rde r theor i e s w i th
equal i ty,
Machine Intell igence 4 ( 1 969)
135-150.
[43] J .F . Rul i fson , J .A. ,De rksen and R.J . Waldin ger , QA4: A proced ural calculus for intui t ive reasonin g,
Tech. Note 73, A.I . Center , SRI, Menlo Park, 1972,
[44] J . S i ekman n, Uni f i ca t ion and ma tch ing prob lems , Ph .D . Thes i s , Mem o CSM-4-78 , Univ . o f Es sex ,
1978.
[45] J . S i ekm ann an d P . Szabo , Unive rsa l un i f i ca t ion in regula r equa tiona l ACFM theor i e s,
6 th C A D E ,
New York , 1982 .
[46] J .R~ Slagle , Auto m ated theor em -prov ing for theor ies wi th simpl if iers, co mm utat ivi ty an d associa t iv-
ity, J.
A C M
21 (1974) 622-642.
[47] M.E. St ickel , A com plete unif icat ion a lgori thm fo r associa t ive-comm utat ive funct ion s ,
4th lnternat.
Joint Co nf. on Arti f icial Intelligence, Tbilisi , U.S.S.R., 1975.
[48] M.E. St ickel , Unif icat ion a lgori thms for ar t if ic ia l inte l l igence languages , Ph .D . T hes is , Carn egie-
Mellon Univers i ty, Pi t tsburgh, PA, 1976.
[49] M .E. St ickel , A com plete unif icat ion a lgori thm fo r associa t ive-com mutat ive funct ion s , J . A C M
28(3) (1981) 423-434.
[50] M. Ven turini Zi l l i, Com plexi ty of the un if icat ion a lgori thm for f i rs t -order express ions , Calcolo XII
(IV) (1975) 361-372.
8/10/2019 Complete Sets of Unifiers and Matchers in Equational Theories 1986 Theoretical Computer Science
12/12
200 F. Fages, G. Huet
[51] E. Tiden, Unif ication in com binations of theo ries with disjoint se ts of function symbo ls, Royal Inst .
of Techno logy, Dept. of Com pute r Science, S-100 44, Stockholm , 1985.
[52] K. Yellick, C om bining unif ication a lgorithms for confined equatio nal theories, M.I .T. , Camb ridge,
MA, 1985.
[53] J .A. R obinson , Com putatio nal logic : The un if ication comp utation , in: B. Meltzer and D. Michie,
eds. , Machine Intelligence VoL 6 (Am erican Elsevier , New Y ork, 1971) .