Complete Sets of Unifiers and Matchers in Equational Theories 1986 Theoretical Computer Science

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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Complete sets of unifiers and marchers

193

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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Complete sets of unifiers and marchers

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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196

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-~


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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~


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Complete sets of unifiers and marchers 199

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Documents

Complete Sets of Unifiers and Matchers in Equational Theories 1986 Theoretical Computer Science