Canonical Sets of Horn Clauses. Nachum Dershowitz. University of Illinois West Springeld Avenue. Urbana, IL 61801, U.S.A.

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1 Caical Sets f Hr Clauses Nachum Dershwitz Departmet f Cmputer Sciece Uiversity f Illiis 1304 West Sprigeld Aveue Urbaa, IL 61801, U.S.A. achum@cs.uiuc.edu 1 Backgrud Rewrite rules are rieted equatis used t replace equals-by-equals i the specied directi. Iput terms are repeatedly rewritte accrdig t the rules. Whe ad if rule applies, the resultat rmal frm is csidered the value f the iitial term. If iite sequeces f rewrites is pssible, a rewrite system is said t have the termiati prperty. Cuece f a rewrite system is a prperty that esures that term has mre tha e rmal frm. A cverget rewrite system is e with bth the cuece ad termiati prperties. Let T be a set f (rst-rder) terms, with variables take frm a set X, ad G be its subset f grud (variable-free) terms. If t is a term i T, by tj we sigify the subterm f t rted at psiti ad by t[s] (r simply t[s]) we dete t with its subterm tj replaced by a term s. We use the fllwig tatis fr equatial deducti: s ' t stads fr the usual sese f equality i lgical systems; s $ e t (r just s $ t) detes e step f replacemet f equals fr equals (usig equati e); s! R t (r just s! t) stads fr e replacemet accrdig t the rietati f a rewrite rule (i R); s! t, fr ay umber (icludig zer) f rewrites; s $ t als stads fr e rewrite step i either directi. Tw terms s ad t are said t be jiable if there is a term v such that s! v t, r s # t fr shrt. Fr cverget R, a idetity s ' t hlds i the thery deed by R (each rule viewed as a equati) if ad ly if the rmal frms f s ad t are idetical. Thus, validity f equatis is decidable fr ite cverget R, sice the jiability (#) relati is decidable. A grud cverget rewrite system is e that termiates ad dees uique rmal frms fr all grud terms. Grud cverget systems ca be used t decide validity by sklemizig s ad t ad reducig t rmal frm. Fr a survey f the thery f rewritig, see [Dershwitz ad Juaaud, 1990]. A cditial equati is a uiversally-quatied (deite) Hr clause i which the ly predicate symbl is equality. We write such a clause i the frm e 1 ^ ^ e ) s ' t ( 0), meaig that equality s ' t hlds wheever all the atecedet equatis e i, hld. The term s will be called the left-had side; t is the right-had side; ad the e i are the cditis. If = 0, the the (psitive uit) clause will be called a ucditial equati. Cditial equatis are imprtat fr specifyig abstract data types ad expressig lgic prgrams with equatis. A cditial (rewrite) rule is a equatial implicati i which the equati i the csequet (s ' t) is rieted. T give peratial sematics t such a system, the cditis uder which a rewrite may be perfrmed eed t made precise. The mst ppular cveti (see [Dershwitz ad Okada, 1990]) fr cditial rewritig is that the terms i each cditi be This research supprted i part by the Natial Sciece Fudati uder Grat CCR

2 jiable. Thus, a rule u 1 ' v 1 ^ ^ u ' v ) l! r, is applied t a term t ctaiig a istace l f the left-had side, if u i # v i fr each cditi u i ' v i, i which case t[l]! t[r]. We call sets f such rules (stadard) cditial rewrite systems; they prvide a applicative prgrammig laguage with especially clea sytax ad sematics, ad ca be exteded t lgic prgrammig paradigms. The grud rmal frms f grud cverget cditial systems frm a iitial algebra fr the uderlyig system f cditial equatis. I fact, a rst-rder thery admits iitial term mdels if ad ly if it is a uiversal Hr thery (see [Makwsky, 1985]). I this sese, (grud cverget) cditial rewritig implemets the iitial-algebra sematics fr peratis cstraied by cditial equatis. Oe f the basic results i (ucditial) rewritig is the Critical Pair Lemma [Kuth ad Bedix, 1970], which states that cuece f (ite) termiatig systems ca be eectively tested by checkig jiability f a ite set f equatis, called \critical pairs", frmed by verlappig left-had sides. I the cditial case, we prpse the fllwig deiti: Let be sme partial rderig grud terms with the \replacemet" prperty, s t implies u[s] u[t] fr all ctexts u[]. We write s t fr grud terms if s t fr all grud substitutis ad s 6 t if s 6 t fr sme. If p ) l ' r ad q ) u ' v are cditial equatis, the the cditial equati p ^ q ) u[r] ' v is a (rdered) cditial critical pair if l uies via mst geeral uier (mgu) with a variable subterm uj f u, u 6 v; u[r], ad als u 6 p; q (meaig that u is t smaller tha ay side f a istatiated cditi i p r q). It was shw i [Dershwitz et al., 1987] that there exists a termiatig cditial rewrite system all critical pairs f which are jiable, but which is t cuet. O the ther had, whe the cditial system is such that recursively evaluatig the cditis als termiates, the critical pair cditi suces. Mre precisely, we say that a cditial system is decreasig if there exists a well-fuded extesi f! (i ther wrds, rewritig always reduces terms i the rderig) with tw additial requiremets: has the \subterm" prperty (each term is greater uder tha its prper subterms) ad cditis fr rule applicati are smaller tha the term that gets rewritte (fr each rule c ) l! r ad substituti, u[l] c). Fr such decreasig systems, all the basic tis are decidable, i.e., the rewrite relati, jiability relati, ad rmal frm attribute are all recursive. Decreasig systems geeralize the ccept f \hierarchy" i [Remy ad Zhag, 1984], ad are slightly mre geeral tha the \simplifyig" ad \reductive" systems csidered i [Kapla, 1987] ad [Juaaud ad Waldma, 1986], respectively. I fact, it ca be shw [Dershwitz ad Okada, 1990] that decreasig systems exactly capture the iteess f recursive evaluati f terms. Thus, they are ideal fr mst cmputatial purpses. It is well kw that ay cditial equatial thery is expressible as a set f uiversally quatied Hr clauses (i which the equality symbl is uiterpreted), sice the axims f equality are themselves Hr. Thus, psitive-uit resluti, r ay ther variati f resluti that is cmplete fr Hr clauses, culd be used t prve therems i equatial Hr theries, but the cst f treatig equality axims like ay ther clause is prhibitively high. Fr this reas, special iferece mechaisms fr equality, tably paramdulati [Rbis ad Ws, 1969], have bee devised. I recet years, term rderigs have bee prpsed as a apprpriate tl with which t restrict paramdulati. O the ip side, ay Hr thery ca be expressed as a ucditial equatial thery. Sme f the implicatis f this crrespdece are explred i Secti 2. I [Kuth ad Bedix, 1970], it was suggested that a cuet ucditial system be \cmpleted" by addig ew rules (accrdig t sme user-supplied partial rderig) wheever a critical pair fails the jiability test. Whe this prcess succeeds, a ite set f equatis is btaied frm which all therems fllw by rewritig. Cmpleti, as deed i [Kuth ad Bedix, 1970] ad studied i [Huet, 1981], fails whe a critical pair, after its tw sides have

3 bee reduced t rmal frm, is either trivial r rietable by the rderig supplied t the prcedure fr this purpse. Cmpleti was rst exteded t cditial equatis by Kapla [1987]. Equatis are tured it rules ly if they satisfy a decreasigess cditi. The prblem is that the critical pair f tw decreasig rules ca easily be decreasig. Gaziger [1987] suggested arrwig the cditis f decreasig rules. Like stadard cmpleti, bth these methds may fail accut f iability t frm ew rules. I Secti 3, we exted these methds aalgus t ufailig rdered cmpleti (as described i [Bachmair et al., 1989]) t prvide a rderig-based therem-prvig methd. As i [Hsiag ad Rusiwitch, 1986; Kualis ad Rusiwitch, 1987; Zhag ad Kapur, 1988; Rusiwitch, 1989; Bachmair ad Gaziger, 1990; Nieuwehuis ad Orejas, 1990; Dershwitz, 1991], ur gal i develpig therem prvig prcedures is t miimize the amut f paramdulati, while maximizig the amut f simplicati, withut threateig cmpleteess thereby. Orderigs are used t chse which literals participate i a paramdulati step, ad which side f a equality literal t use. Our methd als allws fr (almst urestricted) simplicati (demdulati) by directed decreasig equatis. It requires less paramdulati ad ers mre simplicati tha [Kualis ad Rusiwitch, 1987], fr example. Fr ur cmpleteess prfs, we adapt the prf-rderig methd f [Bachmair et al., 1986; Bachmair ad Dershwitz, t appear] t cditial prfs (usig a rderig that is much simpler tha the e i [Gaziger, 1987]). A reduced rewrite system is e such that each right-had side is i rmal frm, as are prper geeralizatis ad prper subterms f all left-had sides. (Fr cverget systems, this is equivalet t the deiti i [Huet, 1981] which requires that left-had sides t be rewritable by ther rules.) Reduced cverget systems are called caical i [Dershwitz ad Juaaud, 1990]. If tw caical systems have the same equatial thery ad are ctaied i the same well-fuded rderig, the they must be literally similar (i.e. the same except fr variable reamigs). This imprtat result was rst metied i [Butler ad Lakfrd, 1980]. It meas that all implemetatis f (stadard) cmpleti must yield the same system, give the same iputs E ad, prvided they use the ecmpassmet relati [Dershwitz ad Juaaud, 1990] fr simplicati f rules. I ur view, simplicati i cmpleti is itimately related t reducti: by strivig t d the uique reduced cverget systems, ecessary simplicatis are illumed. T guide the chice f simplicati strategies fr cditial cmpleti, we develp, i Secti 4, a ti f reduced cditial system, ad lk fr a apprpriate uiqueess result. Oly i circumstaces that esure that a cverget system will be fud wheever there is e, d we csider it reasable t refer t a cditial iferece mechaism as \cmpleti", rather tha \therem prvig". 2 Hr theries We begi ur discussi with Hr clauses t ctaiig (iterpreted) equality symbls. Ay Hr clause p 1 ^ ^ p ) q is lgically equivalet t (the equivalece) p 1 ^ ^ p ^ q p 1 ^ ^ p. Sice the left-had side is lger tha the right, we view this as a termiatig ucditial rewrite rule p 1 ^ ^ p ^ q! p 1 ^ ^ p, with the rder f cjucts left itact. Let H be a set f Hr clauses ad! be the crrespdig rewrite relati. The cmpleteess f selected psitive-uit (SPU) resluti meas, i this framewrk, that, fr a arbitrary cjucti P f atms, H ` P by rst-rder reasig if ad ly if P! (T ^ ^T ) ca be derived frm rules geerated i the fllwig maer:

4 Frm p ^ s ^ q! p ^ s ad p 0! T, where atms p ad p 0 are uiable with mst geeral uier, q is ay atm, ad s is ay cjucti f atms, ifer s^q! s. Whe s is empty, this is q! T. Fr example, frm the tw Hr clauses p(a)! T p(x) ^ p(f(x))! p(x) all p(f i (a))! T are geerated, e after the ther. The abve iferece rule is suciet fr cmpleteess, but ur gal is t allw as much simplicati as pssible. I particular, give a rule p ^ q! q, r eve p ^ r! r, we are tempted t simplify a clause like p ^ q ^ r p ^ r t q ^ r p ^ r. The prblem is that the latter has sides f equal legth, ad cat, i geeral, be rieted it a rule (e.g. if p is x < y ad q is y < x). Hece, addig simplicati wuld lead t icmpleteess f this iferece mechaism. T recver cmpleteess, we eed ifereces that apply t mre geeral equivaleces betwee cjuctis. The idea is t apply the rdered cmpleti methd fr ucditial equatis i [Hsiag ad Rusiwitch, 1987; Bachmair et al., 1989; Dershwitz ad Juaaud, 1990] t these equivaleces. There is eed t use Blea idetities (hece eed fr assciativecmmutative uicati), sice reasig equatially with these equivaleces suces; the ly Blea rule eeded is T ^ x! x. This results i a better methd tha the e i [Bachmair et al., 1989] fr Hr clauses, sice mre simplicati is pssible. Additial ptial simplicati strategies may be icrprated i this therem prvig strategy, just as lg as they are sud ad d t make mre cmplex prfs ecessary (cf. Secti 3). A ttal rderig > grud terms G is called a cmplete simplicati rderig if it has (a) the replacemet prperty, s > t implies u[s] > u[t] fr all ctexts u[], ad (b) the subterm prperty, t tj fr all subterms tj f t. Such a grud-term rderig must be a wellrderig [Dershwitz, 1982]. A cmpletable simplicati rderig all terms T (cf. [Hsiag ad Rusiwitch, 1987]) is a well-fuded partial rderig that (c) ca be exteded t a cmplete simplicati rderig > grud terms, such that (d) s t implies that s > t fr all grud substitutis. Furthermre, we will assume that (e) the (truth) cstats T ad F are miimal i. Of curse, the empty rderig is cmpletable, as are the plymial ad path rderigs cmmly used i rewrite-based therem prvers (see [Dershwitz, 1987]). By results i [Bachmair et al., 1989], prvidig rdered cmpleti with a cmpletable simplicati rderig is guarateed t succeed i dig a caical system fr the give thery, if e exists that is cmpatible with the give rderig. Ordered cmpleti with simplicati is likewise guarateed t derive a ctradicti frm Hr clauses H ad the Sklemized egati f a atmic frmula p such that H ` p. The pit is that the ly Blea equati used (implicitly) i the abve SPU-mimickig iferece rule is T ^ x ' x, frm which it fllws that the equatial represetati f H (plus this Blea equati) prvides a equatial prf f p ' T. The cmpleteess f rdered cmpleti fr equatial reasig [Bachmair et al., 1989] meas that the ctradicti F ' T will be geerated frm these equatis plus ^p ' F, where ^p is p with its variables replaced by Sklem cstats. Rather tha give the geeral case (which is dieret frm rdiary rdered cmpleti except that assciative-cmmutative matchig ca, but eed t, be used whe simplifyig e rule via ather), we shw here hw simplicati prvides, i the prpsitial case, a quadratic algrithm t cvert a set f grud Hr clauses t a uique represetati i the frm f a (ucditial) caical rewrite system. Give ay well-rderig f atms, dee a

5 well-rderig > cjuctis uder which lger cjuctis are bigger, ad equal-legth es are cmpared lexicgraphically. The algrithm perates Hr clauses expressed as equivaleces: Repeat the fllwig, util lger pssible: Chse the equivalece p q (r q p) that has t yet bee csidered such that q is miimal amg all sides vis-a-vis the ttal rderig >. If all the atms i p ccur tgether e side r f ay ther equivalece r s (r s r), remve them frm r ad merge what is left i r with the atms i q. Delete duplicate atms ad ccurreces f the cstat T (uless T is the ly atm) frm all equivaleces. Discard equivaleces with idetical sides ad duplicate equivaleces. Fr example, the rst clause f p ^ q p; p ^ q q; p ^ q ^ r p ^ q rewrites the thers (assumig p < q < r) t q p ad p ^ r p. The, the rst becmes p ^ p p ad is deleted, leavig the Hr clauses p ) q, q ) p, ad p ) r. This algrithm is based the cmpleti-based cgruece clsure methd i [Lakfrd, 1975], shw t be dable with lw plymial time cmplexity i [Gallier et al., 1988]. By the therem i [Lakfrd ad Ballatye, 1983] fr uiqueess f caical assciative-cmmutative rewritig systems, it results i a uique set f equivaleces, determied by the rderig >. The resultat system ca be used t decide satisability i the give prpsitial Hr thery, thugh t as fast as i [Dwlig ad Gallier, 1984]. The equivaleces ca ptially be cverted back t Hr frm. 3 Cmpleti I this secti, we tur t Hr clauses with equality, that is, t cditial equatial theries. Fr eciecy, it is ureasable t just add axims f equality ad use Hr-clause therem-prvig methds. Istead, we develp a ufailig cmpleti prcedure fr cditial equatis, based the icmplete methd i [Gaziger, 1987]. (Cmplete, rderigbased therem-prvig methds fr such theries iclude [Kualis ad Rusiwitch, 1987; Dershwitz, 1991].) The allwable ifereces are a striget restricti f paramdulati. A user-supplied rderig is used t guide the iferece mechaism, s that ly maximal terms are used i ay iferece step. Whe is the empty rderig, the methd reduces t \special" paramdulati, i which the fuctial-reexive axims are t eeded ad paramdulati it variables is t perfrmed (see [Lakfrd, 1975]). Mst imprtat, a empty rderig allws cditial equatis that are simpliable t be replaced withut cmprmisig (refutatial) cmpleteess. Hece, the pwer f the methd, bth i miimizig pssible ifereces ad maximizig ptetial simplicatis, is brught t bear by emplyig rderigs that are mre cmplete tha the empty e. The methd is like the e i [Bertlig, 1990], but we give a specic strategy fr simplicati. Give a set E f cditial equatis, a prf i E f a equati u ' v is a sequece f terms u = t 1 $ t 2 $ $ t = v ( 1), each step t k $ t k+1 f which is justied by a apprpriate cditial equati i E, psiti i t k, substituti fr variables i the equati, ad subprfs fr each f its cditis. Steps emplyig a ucditial equati d t have subprfs as part f their justicati. Ay equati s ' t that is valid fr E is ameable t such a equatial prf. Nte that a cditial equati e 1 ^ ^ e ) s ' t is valid fr E if

6 ad ly if s ' t is valid fr fe 1 ; ; e g. Hece, prvig validity f cditial equatis reduces t prvig validity f ucditial es. We write u! e v (with respect t a partial rderig ), if u $ e v usig a istace p ) s ' t f e, ad u v; p (by which we mea that u is bigger tha v ad bigger tha bth sides f each cditi i p). A cditial equati may have sme grud istaces that are decreasig i the cmplete rderig (if s > t; p), ad thers that are t decreasig. The Critical Pair Lemma f [Kapla, 1987] fr decreasig systems ca be adapted t grud cuece f decreasig systems: Let E be a set f cditial equatis ad a cmpletable simplicati rderig. If all grud istaces f rdered cditial critical pairs rewrite, uder! E, t the idetical term, the the system is grud cuet. Hwever, a cuterexample i [Dershwitz et al., 1987] shws that this critical pair cditi is isuciet whe rewritig by decreasig istaces f equatis is icluded. We frmulate ur therem-prvig prcedure as a iferece system peratig a set f cditial equatis, ad parameterized by a cmpletable rderig. The rules may be classi- ed it three \expasi" rules ad fur \ctracti" rules. The ctracti rules f stadard cmpleti sigicatly reduce its space requiremets, but they make prfs f cmpleteess much mre subtle. The rst expasi rule geerates critical pairs frm clauses that may have decreasig istaces: ( ) 8 >< >: p ) l ' r; q ) u ' v p ) l ' r; q ) u ' v; p ^ q ) u[r] ' v 9 >= >; if 8 >< >: uj 62 X = mgu(uj ; l) u 6 p; q; v; u[r] Superpsiti (i.e. rieted paramdulati f psitive equatial literals) is perfrmed ly at variable psitis (uj 62 X ). Oly psitive equatis are used i this rule, ad ly i a decreasig directi (u 6 p; q). Of curse, if the relati 6 is t kw precisely, e must be cservative ad apply the iferece wheever it cat be guarateed that all grud istaces f u are larger tha the crrespdig istaces f p, q, v, ad u[r]. Either side f a equati may be used fr superpsiti, but ly if, i the ctext f the paramdulati, it is (believed t be) ptetially the largest term ivlved (u 6 v; u[r] ). (This ca prbably be stregtheed a bit t require l 6 p istead f u 6 p.) Nw ad hecefrth, whe a rule refers t a clause f the frm q ) u ' v, a ucditial equati (u ' v) is als iteded. We eed, additially, a rule that applies decreasig equatis t egative literals: 8 >< >: ( p ) l ' r; q ^ s ' t ) u ' v p ) l ' r; q ^ s ' t ) u ' v; p ^ q ^ s[r] ' t ) u ' v ) 9 >= >; if 8 >< >: sj 62 X = mgu(sj ; l) s 6 p; q; t; s[r] Wheever this r subsequet rules refer t a cditial equati like q ^ s ' t ) u ' v, the itet is that s ' t is ay e f the cditis ad s is either side f it. (A) (B)

7 The last expasi rule i eect reslves a maximal egative literal with reexivity f equals (x ' x): ( q ^ s ' t ) u ' v = mgu(s; t) ( ) if q ^ s ' t ) u ' v; s 6 q (C) q ) u ' v The fur ctracti rules all simplify the set f cditial equatis. The rst tw elimiate trivial equatis: q ) u ' u E q ^ s ' s ) u ' v q ) u ' v The last tw use decreasig clauses t simplify ther clauses. Oe simplies cditis; the ther applies t the equati part. I bth cases, the rigial clause is replaced by a versi that is equivalet but strictly smaller uder. p ) u ' v q ) u ' v (D) (E) if p! E q (F) I simplifyig equatis, we utilize a extesi f the ecmpassmet rderig > frm terms t clauses (i which terms are larger tha prper subterms ad smaller tha prper istaces): Terms are cmpared with ; equatis are cmpared by cmparig the multiset f their tw terms i the multiset extesi mul f the term rderig (see [Dershwitz ad Maa, 1979]); t cmpare terms with equatis we make s bigger tha u ' v if s u; v; ally, q ) u ' v > p ) l ' r if q mul p, r q = mul p (i.e. q = p as multisets) ad u > l i the ecmpassmet rderig, r q = mul p, u = l, ad v r. q ) u ' v q ) w ' v if ( u! e w; e 2 E q u _ v u _ (q ) u ' v) > e Here q u meas that there is always e side f e cditi i q that is bigger tha u; hece, the clause is decreasig. As a simple example, csider the fllwig three clauses: (G) 0 < c(0) ' T (1) c(y) < c(z) ' y < z (2) c(c(0)) < x ' T ) c(0) < x ' T (3) The rst tw are decreasig; the third is t. We emply a straightfrward rderig (e.g. leftt-right lexicgraphic path rderig [Kami ad Levy, 1980; Dershwitz, 1987]). Expasi iferece (A) des t apply betwee (1) ad (2), sice 0 ad c(y) d t uify. By the same tke, (B) des t apply betwee (1) ad the cditi i (3). Applyig (B) betwee (2) ad (3) yields c(0) < x ' T ) c(0) < c(x) ' T which ctracts, usig (G) ad (2), t ather decreasig clause c(0) < x ' T ) 0 < x ' T (4)

8 Applyig (B) t it yields a decreasig clause 0 < x ' T ) 0 < c(x) ' T (5) The critical pair btaied by superpsig (1) 0 < c(x) gives a trivial equati (ctractable by (D)). Sice (5) des t uify with the cditis f (3) ad (4), we are de. Nte that the decreasig rules (1,2,5) reduce ay term c i (0) < c j (0), such that i < j, t T. A valley prf s ' t is e i which the steps take the frm s # t. We dee a rmal-frm prf f s ' t t be a valley prf i which each subprf is als i rmal frm ad each term i a subprf is smaller tha the larger f s ad t. Ay -rmal-frm prf has a peak made frm decreasig istaces with rmal-frm subprfs, r has a decreasig step with rmal-frm subprfs, r has a trivial step. We say that a sequece f ifereces is fair if expasis f all persistet cditial equatis have bee csidered. Frmally, that may be expressed as exp(e 1 ) [E i, where E 0, E 1,... is the sequece f cditial equatis geerated, E 1 = lim sup E j is the set f cditial equatis that each persist frm sme E j, ad exp(e 1 ) is the set f cditial equatis that may be iferred i e expasi step frm persistig equatis. Fr a methd based these rules t be cmplete, we eed t shw that with eugh ifereces, ay grud therem evetually has a rmal-frm prf. Precisely stated: If a iferece sequece is fair, the fr ay prf f s ' t i the iitial set E 0 f cditial equatis, there is a rmal-frm prf f s ' t i the limit E 1. This is a csequece f the fllwig bservatis: If E 0 ca be iferred frm E, the fr ay prf i E there exists a prf i E 0 f equal r lesser cmplexity, ad, furthermre, that there are always ifereces that ca decrease the cmplexity f -rmal prfs. Cmplexity may be measured by assigig t each step s $ t i a grud prf r its subprfs the weight hfq 1 ;...; q ; sg; ei, where e is the cditial equati q 1 ^ ^ q ) l ' r justifyig the step, is the substituti, ad s is the larger f s ad t (i the cmplete simplicati rderig > extedig ). Steps are cmpared i the lexicgraphic rderig f these pairs. The rst cmpets f pairs are cmpared i the multiset extesi f the rderig equatis ad terms described abve. Secd cmpets are cmpared usig >. Prfs are cmpared i the well-fuded multiset extesi f the lexicgraphic rderig steps. We use t dete this prf rderig. It ca be shw by stadard argumets [Dershwitz ad Maa, 1979] that is well-fuded. By iducti with respect t, the evetual existece f a rmal-frm prf fllws: If P is a -rmal-frm prf i E, there exists a prf P 0, usig equatis i E ad exp(e), such that the cmplexity f P is strictly greater (i the prf rderig ) that that f P 0. I particular, trivial steps ca be elimiated, reducig cmplexity by remvig elemets frm the multiset f prf steps. Peaks betwee decreasig steps will have smaller prfs accut f iferece rule (A) ad the Critical Pair Lemma. A decreasig step s! e t with a decreasig step ut f its largest cditi p breaks dw it tw cases: If the decreasig step p! q is i the substituti part f p, the a applicati f rule (B) supplies a ew equati that ca be used i a step f smaller cmplexity (sice p, which was the largest elemet f the rst cmpet f the cmplexity f the step s! e t, is replaced by q). If the decreasig step takes place i the substituti part, the there is a alterative prf s! s 0! e t 0 t, where s ad t are rewritte by the same decreasig equati. The ew e step is smaller that the ld e sice its cditis are. Ay ew steps itrduced are smaller tha the elimiated cst f p! q, sice they apply t terms smaller tha p. Lastly, a decreasig step with trivial subprfs ca be replaced after geeratig a ew equati usig (C).

9 The ctracti rules were als desiged t decrease prf cmplexity. Thus, ay fair sequece f ifereces must allw fr a simpler prf, ad evetually a rmal-frm prf must persist. A alterative cmpleti prcedure may be based the fact [Dershwitz et al., 1987] that a system is cuet if all its critical pairs are jiable ad are frmed frm verlaps betwee left-had sides at their tpmst psiti. I such a prcedure, ay -rt critical pair wuld be elimiated by pullig ut subterms. Fr example, the rules a! b ad h(f(a))! c verlap, but t at the tp. T get arud that, the secd rule ca be replaced by the mre pwerful x ' a ) h(f(x))! c, elimiatig the edig pair. Nte that iterpretig Hr clauses as cditial rules (rewritig predicates t T ) gives a system satisfyig the abve cstrait, because predicate symbls are ever ested i the head f a clause. Furthermre, all critical pairs are jiable, sice all right-had sides are the same. This als applies t patter-directed fuctial laguages i which deed fuctis are t ested left-had sides. 4 Uiqueess f Systems I ur view, there is a qualitative dierece betwee therem prvig ad cmpleti. As pited ut i [Huet ad Oppe, 1980], cmpleti is a cmpilati-like prcess; the gal is t d a cverget system that ca later be used t prve (a certai class f) therems eectively. A therem prver is, accrdigly, deemed \cmplete" if it ca prve ay prvable therem (i the class f therems uder csiderati); a cmpleti prcedure, the ther had, is \cmplete" (i the sese f [Dershwitz, 1989]) if it will d a cverget system wheever there is e (fr the give rderig). The uit methd f [Dershwitz, 1991], fr example, shuld t qualify as a cmpleti prcedure fr cditial rules, sice it may g prducig (perhaps iitely) may ucditial rules, eve whe e cditial rule suces. Fr example, give the cuet system fh(f(x))! h(x); h(x) = h(a) ) g(x)! cg, it will prceed t add superuus csequeces g(f i (a))! c (amg thers). Suppse R is a cverget (cditial) rewrite system fr sme thery E ad the rewrite relati! R is ctaied i a partial rderig. The, the rmal frm f ay term t is the elemet i the E-cgruece class f t that is miimal vis-a-vis [Avehaus, 1986]. Hece, if R ad S are tw such systems fr the same thery E ad rderig, the R ad S have the same rmal frms ad the same reducibility relati. We say that a term t is reduced, with respect t thery E ad rderig, if, f all elemets i its E-cgruece class, it is miimal with respect t. A ucditial rewrite system is said t be reduced if, fr each f its rules, l! r, the right-had side r is reduced ad all terms s less tha l i the ecmpassmet rderig are als reduced. The ctracti iferece rules fr ucditial systems (see [Bachmair et al., 1986]) are themselves \cuet", implyig that the same reduced system is btaied regardless f the rder i which they are appplied t a give cuet system. Reduced ucditial systems are uique with respect t i the strger sese that if R ad S are caical, have the same thery, ad their rewrite relatis are bth ctaied i, the R ad S are (essetially) idetical (cf. [Butler ad Lakfrd, 1980; Metivier, 1983]). We saw i Secti 2 hw this gives a uique represetati fr equatial Hr clauses. But fr cditial systems, reducti is clearly isuciet. Fr example, the tw equivalet rules, a ' b ) f(a)! c ad a ' b ) f(b)! c, are each cverget ad reduced i the abve sese. Applyig the ctracti rules f Secti 3 t a set f cditial equatis (a prcess that will f ecessity termiate) is t eugh fr this purpse. Oe eeds, rst f all, sme srt f \ctextual rewritig" (a la [Zhag ad Remy, 1985]) t reduce the left-had side f the rule s that it ctais the smaller f the hypthesized-equal terms a ad b. This suggests a additial

10 iferece rule like: q ) r q ) s if r! e s; e 2 T h( q) (H) where T h( q) is the set f equatial csequeces f E ad q. But eve this ieective rule is isuciet, as ca be see frm the fllwig alteratives: a ' b ^ a ' c ) f(b)! c vs. a ' b ^ a ' c ) f(c)! c. If a b; c, we still eed t chse betwee the miimal terms b ad c. Let p ) l! r be a cditial rule e i a system R. It is deemed reduced if r ad all terms smaller tha l i the ecmpassmet rderig are reduced with respect t E^p, l itself is reduced with respect t R [ p? feg, ad there is lgically weaker cditi such that l is reducible. Let R ad S be tw reduced cverget cditial systems fr E ad. If p ) l! r is i R, the l ' r has a prf i S [ p. Eve if we culd shw that l must be a left-had side f a rule e 0 i S which must have right-had side r (a questi we leave pe), the cditis i e ad e 0 may dier, ad additial cmpleti ad simplicati are required t preclude that. Imagie a rule p ) l! r. T get true uiqueess, e must cmplete the equatis p (mdul ay ther equatis) t d a uique represetati (that is, the ite caical system) fr p (if e exists at all). Als, e wuld wat t elimiate cditis f the frm x ' t; therwise, the cditial rule x ' a ) f(x)! b culd be preferred ver f(a)! b. Ackwledgmet I thak Subrata Mitra ad the referees fr their cmmets. Refereces [Avehaus, 1986] Jurge Avehaus. O the descriptive pwer f term rewritig systems. J. Symblic Cmputati, 2:109{122, [Bachmair ad Dershwitz, t appear] Le Bachmair ad Nachum Dershwitz. Equatial iferece, caical prfs, ad prf rderigs. J. f the Assciati fr Cmputig Machiery, t appear. [Bachmair ad Gaziger, 1990] Le Bachmair ad Harald Gaziger. Cmpleti f rstrder clauses with equality. I M. Okada, editr, Prceedigs f the Secd Iteratial Wrkshp Cditial ad Typed Rewritig Systems, Mtreal, Caada, Jue Lecture Ntes i Cmputer Sciece, Spriger, Berli. [Bachmair et al., 1986] Le Bachmair, Nachum Dershwitz, ad Jieh Hsiag. Orderigs fr equatial prfs. I Prceedigs f the IEEE Sympsium Lgic i Cmputer Sciece, pages 346{357, Cambridge, MA, Jue [Bachmair et al., 1989] Le Bachmair, Nachum Dershwitz, ad David A. Plaisted. Cmpleti withut failure. I H. At-Kaci ad M. Nivat, editrs, Resluti f Equatis i Algebraic Structures 2: Rewritig Techiques, chapter 1, pages 1{30. Academic Press, New Yrk, [Bertlig, 1990] Hubert Bertlig. Kuth-Bedix cmpleti f Hr clause prgrams fr restricted liear resluti ad paramdulati. I S. Kapla ad M. Okada, editrs, Exteded Abstracts f the Secd Iteratial Wrkshp Cditial ad Typed Rewritig Systems, pages 89{95, Mtreal, Caada, Jue Revised versi t appear i Lecture Ntes i Cmputer Sciece, Spriger, Berli. [Butler ad Lakfrd, 1980] Gerge Butler ad Dallas S. Lakfrd. Experimets with cmputer implemetatis f prcedures which fte derive decisi algrithms fr the wrd prblem

11 i abstract algebras. Mem MTP-7, Departmet f Mathematics, Luisiaa Tech. Uiversity, Rust, LA, August [Dershwitz ad Juaaud, 1990] Nachum Dershwitz ad Jea-Pierre Juaaud. Rewrite systems. I J. va Leeuwe, editr, Hadbk f Theretical Cmputer Sciece B: Frmal Methds ad Sematics, chapter 6, pages 243{320. Nrth-Hllad, Amsterdam, [Dershwitz ad Maa, 1979] Nachum Dershwitz ad Zhar Maa. Prvig termiati with multiset rderigs. Cmmuicatis f the ACM, 22(8):465{476, August [Dershwitz ad Okada, 1990] Nachum Dershwitz ad Mitsuhir Okada. A ratiale fr cditial equatial prgrammig. Theretical Cmputer Sciece, 75:111{138, [Dershwitz et al., 1987] Nachum Dershwitz, Mitsuhir Okada, ad G. Sivakumar. Cuece f cditial rewrite systems. I S. Kapla ad J.-P. Juaaud, editrs, Prceedigs f the First Iteratial Wrkshp Cditial Term Rewritig Systems, pages 31{44, Orsay, Frace, July Vl. 308 f Lecture Ntes i Cmputer Sciece, Spriger, Berli (1988). [Dershwitz, 1982] Nachum Dershwitz. Orderigs fr term-rewritig systems. Theretical Cmputer Sciece, 17(3):279{301, March [Dershwitz, 1987] Nachum Dershwitz. Termiati f rewritig. J. f Symblic Cmputati, 3(1&2):69{115, February/April Crrigedum: 4, 3 (December 1987), 409{410. [Dershwitz, 1989] Nachum Dershwitz. Cmpleti ad its applicatis. I H. At-Kaci ad M. Nivat, editrs, Resluti f Equatis i Algebraic Structures 2: Rewritig Techiques, chapter 2, pages 31{86. Academic Press, New Yrk, [Dershwitz, 1991] Nachum Dershwitz. Orderig-based strategies fr Hr clauses. I Prceedigs f the 12th Iteratial Jit Cferece Articial Itelligece, Sydey, Australia, August T appear. [Dwlig ad Gallier, 1984] William F. Dwlig ad Jea H. Gallier. Liear-time algrithms fr testig the satisability f prpsitial Hr frmulae. J. f Lgic Prgrammig, 1(3):267{284, [Gallier et al., 1988] Jea Gallier, Paliath Naredra, David Plaisted, Sta Raatz, ad Waye Syder. Fidig caical rewritig systems equivalet t a ite set f grud equatis i plymial time. I E. Lusk ad R. Overbeek, editrs, Prceedigs f the Nith Iteratial Cferece Autmated Deducti, pages 182{196, Arge, Illiis, May Vl. 310 f Lecture Ntes i Cmputer Sciece, Spriger, Berli. [Gaziger, 1987] Harald Gaziger. A cmpleti prcedure fr cditial equatis. I S. Kapla ad J.-P. Juaaud, editrs, Prceedigs f the First Iteratial Wrkshp Cditial Term Rewritig Systems, pages 62{83, Orsay, Frace, July Vl. 308 f Lecture Ntes i Cmputer Sciece, Spriger, Berli (1988). [Hsiag ad Rusiwitch, 1986] Jieh Hsiag ad Michael Rusiwitch. A ew methd fr establishig refutatial cmpleteess i therem prvig. I J. H. Siekma, editr, Prceedigs f the Eighth Iteratial Cferece Autmated Deducti, pages 141{152, Oxfrd, Eglad, July Vl. 230 f Lecture Ntes i Cmputer Sciece, Spriger, Berli. [Hsiag ad Rusiwitch, 1987] Jieh Hsiag ad Michael Rusiwitch. O wrd prblems i equatial theries. I T. Ottma, editr, Prceedigs f the Furteeth EATCS Iteratial Cferece Autmata, Laguages ad Prgrammig, pages 54{71, Karlsruhe, West Germay, July Vl. 267 f Lecture Ntes i Cmputer Sciece, Spriger, Berli. [Huet ad Oppe, 1980] Gerard Huet ad Derek C. Oppe. Equatis ad rewrite rules: A survey. I R. Bk, editr, Frmal Laguage Thery: Perspectives ad Ope Prblems, pages 349{405. Academic Press, New Yrk, [Huet, 1981] Gerard Huet. A cmplete prf f crrectess f the Kuth-Bedix cmpleti algrithm. J. Cmputer ad System Scieces, 23(1):11{21, 1981.

12 [Juaaud ad Waldma, 1986] Jea-Pierre Juaaud ad Berard Waldma. Reductive cditial term rewritig systems. I Prceedigs f the Third IFIP Wrkig Cferece Frmal Descripti f Prgrammig Ccepts, Ebberup, Demark, [Kami ad Levy, 1980] Sam Kami ad Jea-Jacques Levy. Tw geeralizatis f the recursive path rderig. Upublished te, Departmet f Cmputer Sciece, Uiversity f Illiis, Urbaa, IL, February [Kapla, 1987] Stephae Kapla. Simplifyig cditial term rewritig systems: Uicati, termiati ad cuece. J. Symblic Cmputati, 4(3):295{334, December [Kuth ad Bedix, 1970] Dald E. Kuth ad P. B. Bedix. Simple wrd prblems i uiversal algebras. I J. Leech, editr, Cmputatial Prblems i Abstract Algebra, pages 263{ 297. Pergam Press, Oxfrd, U. K., Reprited i Autmati f Reasig 2, Spriger, Berli, pp. 342{376 (1983). [Kualis ad Rusiwitch, 1987] Emmauel Kualis ad Michael Rusiwitch. O wrd prblems i Hr theries. I S. Kapla ad J.-P. Juaaud, editrs, Prceedigs f the First Iteratial Wrkshp Cditial Term Rewritig Systems, pages 144{160, Orsay, Frace, July Vl. 308 f Lecture Ntes i Cmputer Sciece, Spriger, Berli (1988). [Lakfrd ad Ballatye, 1983] Dallas S. Lakfrd ad A. Michael Ballatye. O the uiqueess f term rewritig systems. Upublished te, Departmet f Mathematics, Luisiaa Tech. Uiversity, Rust, LA, December [Lakfrd, 1975] Dallas S. Lakfrd. Caical iferece. Mem ATP-32, Autmatic Therem Prvig Prject, Uiversity f Texas, Austi, TX, December [Makwsky, 1985] J. A. Makwsky. Why Hr frmulas matter i cmputer sciece: Iitial structures ad geeric examples. I Mathematical Fudatis f Sftware Develpmet, pages 374{385, Vl. 185 f Lecture Ntes i Cmputer Sciece, Spriger, Berli. [Metivier, 1983] Yves Metivier. Abut the rewritig systems prduced by the Kuth-Bedix cmpleti algrithm. Ifrmati Prcessig Letters, 16(1):31{34, Jauary [Nieuwehuis ad Orejas, 1990] Rbert Nieuwehuis ad Ferad Orejas. Clausal rewritig. I S. Kapla ad M. Okada, editrs, Exteded Abstracts f the Secd Iteratial Wrkshp Cditial ad Typed Rewritig Systems, pages 81{88, Mtreal, Caada, Jue Ccrdia Uiversity. Revised versi t appear i Lecture Ntes i Cmputer Sciece, Spriger, Berli. [Remy ad Zhag, 1984] Jea-Luc Remy ad Hata Zhag. REVEUR4: A system fr validatig cditial algebraic specicatis f abstract data types. I Prceedigs f the Sixth Eurpea Cferece Articial Itelligece, pages 563{572, Pisa, Italy, [Rbis ad Ws, 1969] G. Rbis ad L. Ws. Paramdulati ad therem-prvig i rst rder theries with equality. I B. Meltzer ad D. Michie, editrs, Machie Itelligece 4, pages 135{150. Ediburgh Uiversity Press, Ediburgh, Sctlad, [Rusiwitch, 1989] Michael Rusiwitch. Demstrati Autmatique: Techiques de reecriture. IterEditis, Paris, Frace, [Zhag ad Kapur, 1988] Hata Zhag ad Deepak Kapur. First-rder therem prvig usig cditial equatis. I E. Lusk ad R. Overbeek, editrs, Prceedigs f the Nith Iteratial Cferece Autmated Deducti, pages 1{20, Arge, Illiis, May Vl. 310 f Lecture Ntes i Cmputer Sciece, Spriger, Berli. [Zhag ad Remy, 1985] Hata Zhag ad Jea-Luc Remy. Ctextual rewritig. I Prceedigs f the First Iteratial Cferece Rewritig Techiques ad Applicatis, pages 46{62, Dij, Frace, May Vl. 202 f Lecture Ntes i Cmputer Sciece, Spriger, Berli (September 1985).