Bi-rewrite Systems. Universitat Politecnica de Catalunya. Consejo Superior de Investigaciones Cientcas. Abstract

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1 Bi-rewrite Systems Jordi Levy z and Jaume Agust x z Departament de Llenguatges i Sistemes nformatics Universitat Politecnica de Catalunya x nstitut d'nvestigacio en ntelligencia Articial Consejo Superior de nvestigaciones Cientcas Abstract n this article we propose an extension of term rewriting techniques to automate the deduction in monotone pre-order theories. To prove an inclusion a b from a given set of them, we generate from, using a completion procedure, a bi-rewrite system h; i, that is, a pair of rewrite relations! and!, and seek a common term c such that a! c and b! c. Each component of the bi-rewrite system! and! is allowed to be a subset of the corresponding inclusion relation or dened by the theory of. n order to assure the decidability and completeness of such proof procedure we study the termination and commutation of! and!. The proof of the commutation property is based on a critical pair lemma, using an extended denition of critical pair. We also extend the existing techniques of rewriting modulo equalities to bi-rewriting modulo a set of inclusions. Although we center our attention on the completion process a la Knuth-Bendix, the same notion of extended critical pair is suitable of being applied to the so called unfailing completion procedures. The completion process is illustrated by means of an example corresponding to the theory of the union operator. We show that conuence of extended critical pairs may be ensured adding rule schemes. Such rule schemes contain variables denoting schemes of expressions, instead of expressions. We propose the use of the linear second-order typed -calculus to codify these expression schemes. Although the general second-order unication problem is only semi-decidable, the second-order unication problems we need to solve during the completion process are decidable. 1 ntroduction ewrite systems are usually associated with rewriting on equivalence classes of terms, dened by a set of equations. However term rewriting techniques may be used to compute other relations than congruence. Particularly interesting are non-symmetric relations like pre-orders. n this article we will show the applicability of rewrite techniques to monotonic pre-order relations on terms, that is the deduction of inequalities here we call them inclusions from a given set of them. This work was partially supported by the project DSCO (TC C02-01) funded by the CCYT. 1

2 The idea of applying rewrite techniques to the deduction of inclusions between terms, like a b, is very simple. We compute by repeatedly replacing both 1) subterms of a by \bigger" terms using the axioms and 2) subterms of b by \smaller" terms using the same axioms until a path is found between a and b. Evidently there are many paths starting from a in the direction! and from b in the direction! (see gure 1). Many of them are blind alleys and others are not terminating. t is essential that the search procedure avoids innite sequences of rewrite steps with innitely many dierent terms (innite paths due to cycles can be avoided if we control the introduction of repeated terms). Obviously innitely many dierent rewrite steps would prevent the termination of the procedure. The solution to non-termination is, like in term rewriting systems, to orient the axioms using a well founded ordering on terms. Because the relation is non-symmetric, the orientation results in a pair of rewrite systems h ; i, i.e. we get what we call a bi-rewrite system. We introduce the denitions of Church-osser and quasi-terminating bi-rewrite system in order to assure the soundness, completeness and termination of the search procedure. That is, given a set of axioms, if we can orient and complete them obtaining a quasi-terminating and Church-osser bi-rewrite system, then we will have a decision algorithm to test a b. a (a [ b) [ a b [ a a [ (a (b [ a) [ a -b [ (a [ J] J J a HY H a a [ (b [ H (a [ a) [ [ (b [ c) b J c b [ c 8 < X [ Y! Y [ X = (X [ Y ) [ Z! X [ (Y [ Z) a [ c : X [ X! X = X [ Y! X [ Y! X Y Figure 1: A graphical representation of the bi-rewrite algorithm Most of the notions of rewriting developed for the equational case can be extended to birewriting and the development of the article follows the same pattern as equational rewriting: the Church-osser property is proved by means of a critical pair lemma, and we use a completion process to ensure the conuence of the critical pairs (Knuth and Bendix, 1970; Huet, 1980; Klop, 1987; Dershowitz and Jouannaud, 1990). However there are also some dierences. Equational rewriting is in essence a theory of normal forms, while bi-rewriting disregards this notion. Bi-rewriting can also be seen as a generalization of equational rewriting: equations can be translated to pairs of inclusions and then we can reproduce the equational case. One of the costs of this generalization is that bi-rewriting is based on a search procedure, which is avoided in canonical rewrite systems thanks to the existence of unique normal forms. Another cost is that now critical pairs must be computed considering variable overlapping, producing possibly innitely many of them, which are represented as critical pair schemes. This article proceeds as follows. n section 2 we present a version of the critical pair lemma for bi-rewrite systems using 2

3 an extended denition of critical pairs. We also give a counter-example that invalidates this lemma stated only in terms of standard critical pairs. n section 3 we generalize the results of section 2 to bi-rewrite systems modulo a set of (non-orientable) inclusions. We have divided this section in two subsections, the rst devoted to abstract bi-rewrite properties and the second to term dependent properties. n section 4 we present an example of canonical bi-rewrite system for the theory of nondistributive lattices. We show that although in general extended critical pairs could be intractable, there exist for this theory, and possibly for others, practical ways to handle them. We also show in section 5 some of the disadvantages of using equations to model inclusions in lattice theories. Unfortunately, the set of extended critical pairs is in general innite. Although there exists canonical bi-rewrite systems for many inclusion theories, the standard completion procedures are of little practical help to automatically complete a bi-rewrite system. n section 6 we show how these innitely many extended critical pairs can be made conuent introducing rule schemes, where these rule schemes can be implemented using second-order rules. However, the use of the simply typed second-order -calculus for rewriting purposes introduces some problems, stated in subsection 6.1. Because of that, we dene a restricted second-order language called linear second-order -calculus, which is described in section 7. There we also describe a unication procedure for the linear second-order typed -calculus. Then the new critical pair lemma for second-order bi-rewrite systems is proved in section 8. We illustrate how the Knuth-Bendix completion procedure could be implemented for second-order bi-rewrite systems by means of an example in section 9. n section 10 we present related work and in section 11 we conclude summarizing present and further work. 2 nclusions and Bi-rewrite Systems f nothing is said, we follow the notation and the standard denitions used in (Huet, 1980; Klop, 1987; Dershowitz and Jouannaud, 1990). We are concerned with rst-order terms T (F; X ) over a nonempty signature F = S n2nf n of function symbols, and a denumerable set X of variables. 1 The set of variables of a term t is denoted by FV(t). A position p is a sequence of positive integers. Given two positions, p 1 p 2 denotes their concatenation. We write p 1 p 2 when p 1 is a prex of p 2 and p 1 jp 2 when they are disjoint. The occurrence of a subexpression at a position p of a term t is denoted by tj p. The expression t[u] p denotes the result of replacing in t the occurrence of tj p by u. A context F [] p is an expression with a hole [] at a distinguished position p. 2 A substitution = [X 1 7! t 1 ; : : :; X n 7! t n ] is a mapping from a nite set fx 1 ; : : :; X n g X of variables to T (F; X ), extended as a morphism to T (F; X )! T (F; X ). The set Dom() def = fx 1 ; : : :; X n g is called the domain of the substitution. 1 As we will see later, in most cases we also require the niteness of F. We suppose that F n are disjoint sets. The set T (F; X ) is dened as the smallest set containing X such that if f 2 F n and t i 2 T (F; X ) for i = 1; : : : ; n then f(t 1; : : : ; t n) 2 T (F; X ). 2 We write p 1 p 2 when there exists a sequence q such that p 2 = p 1 q, and p 1jp 2 when p 1 6 p 2 def and p 2 6 p 1. f p is an empty sequence then tj p is dened by tj <> = t otherwise it is dened inductively by f(t 1; : : : ; t n)j <i1 ;i 2 ;:::;ir> = t i1 j <i2 ;:::;ir>. f p is the empty sequence then t[u] <> = u, otherwise def def f(t 1; : : : ; t n)[u] <i1 ;:::;im> def = f(t 1; : : : ; t i1 [u] <i2 ;:::;im>; : : : ; t n). 3

4 We use the relational logic notation (dekogel, 1992; Baumer, 1992) to present the abstract bi-rewriting properties. The inverse of the relation is denoted by?1, its reexive-transitive closure by, the transitive composition by 1 2, the union by 1 [ 2, and the intersection by 1 \ 2. Notation + is a shorthand for. A relation is said to be terminating if + is a well-founded ordering, quasi-terminating if the set fu j t ug is nite for any value t; and nitely branching if fu j t ug is nite for any t. A binary relation on terms is said to be closed under substitutions if t u implies (t) (u), for any substitution and pair of terms t and u; monotonic if t u implies F [t] p F [u] p, for any context F [] p ; and a rewrite relation if it is closed under substitutions and monotonic. We denote by! the rewrite relation dened by the set of rules. 3 Notation is a shorthand for (! )?1. An inclusion is a pair of terms s; t 2 T (F; X ) written s t. Given a nite set of inclusions Ax and a pair of terms s and t, we say that s Ax t i Ax `POL s t, where POL stands for Partial Order Logic and `POL is the entailment relation dened by the following inference rules `POL s t `POL t u ; s t `POL s t `POL s s `POL s u `POL s t `POL (s) (t) `POL s t `POL u[s] p u[t] p where is a substitution, p a position in u, i.e. u[] p is a context, and is a nite set of inclusions. Meseguer (1990,1992) has studied widely the logic of conditional inequalities, which he names rewriting logic, and its models. The set of inclusions s t that can be inferred from Ax using `POL forms an inclusion theory, noted by T h(ax). Notice that, in rst-order logic, T h(ax) is a denumerable set and the deduction problem Ax `POL s t is semi-decidable. n the following we will propose sucient conditions to have a decision algorithm for Ax `POL s t based on rewrite techniques. Given an inclusion s t of Ax, we can orient it obtaining either a term rewriting rule s! t or a rule t! s. Thus, the orientation, for rewriting purposes, of a nite set of inclusions Ax results in two sets of rewrite rules, with rules like s! t and with rules like s! t. The pair h ; i is called a bi-rewrite system. Denition 1 A (term) bi-rewriting system is a pair h ; i of nite sets of (term) rewriting rules = fs 1! t 1 ; : : :; s n! t n g = fu 1! v 1 ; : : :; u m! v m g Given a bi-rewrite system h ; i, its corresponding inclusion theory is dened by the set of axioms Ax = fs t j s! t 2 _ t! s 2 g. The orientation criteria is based, like in rewrite systems, on a well-founded ordering on terms (noted as ) (Dershowitz, 1987). n this section we suppose that each inclusion s t in Ax may be oriented, putting either s! t in if s t, or t! s in if t s. n the next section we will consider the case of inclusions which can not be oriented because s 6 t and t 6 s. For example, inclusions dening the inclusion theory of the union may be oriented using a simplication ordering as it is shown in gure 2. 3 The minimal rewrite relation satisfying s! t for any rule s! t 2. 4

5 8 >< Ax = >: X [ X X X X [ Y Y X [ Y n = = ( r 1 : X [ X! X r 2 : X [ Y! X r 3 : X [ Y! Y Figure 2: Orientation of the inclusion theory of the union. Given a bi-rewrite system h ; i the monotonic and substitution closure of each one of its components and results in a pair of rewrite relations, noted by! and! respectively, dened as follows. Denition 2 We say that s -rewrites to t, written s! [p;;l!r] t, or simply s! t when there is no confusion, if there exist a rule l! r 2, a position p in s, and a substitution, such that sj p = (l) and t = s[(r)] p. f sj p = (l) then we say that sj p and l match. Notice that if FV(r) FV(l) then the substitution in the previous denition, with its domain restricted to Dom() FV(l), is unique. A variant of the theorem of Birkho (Birkho, 1935) allows to prove the following lemma. Lemma 3 Given a bi-rewrite system h ; i and its corresponding inclusion theory Ax, for any pair of terms s, t we have s (! ) t if, and only if, Ax `POL s t. [ However, the relation (! [ ) is in general not computable, i.e. given two terms s and t there does not exist a decision algorithm for s (! [ ) t. We are interested in reducing the previous relation into the subrelation!, which we will show is computable. Based on the bi-rewrite system h ; i a deduction procedure for its corresponding inclusion theory T h(ax) can be easily dened (see gure 1). To prove Ax `POL s t the procedure enumerates recursively the nodes of two trees T 1 and T 2, dened by roott 1 = s, roott 2 = t, brancht 1 (s 1 ) = fs 2 j s 1! s 2 g and brancht 2 (t 1 ) = ft 2 j t 1! t 2 g, avoiding repeated nodes. f the procedure nds a common node in both trees then it stops and answers true, otherwise if both sets of nodes are nite then it stops and answers false or else it does not stop. Notice that the nodes of both trees are always recursively enumerable, although the trees may be innitely branching. We say that a tree is innitely branching if it contains a node with innitely many branches. The following denition states sucient conditions for the soundness and completeness, and for the termination of this procedure. Notice that the soundness and completeness properties are based on the equivalence of the relation! computed by the algorithm and the relation (! [ ) implementing the inclusion relation dened by the theory. The termination property is based on the niteness of both search trees. Denition 4 A bi-rewrite system h ; i is said to be (i). terminating i (! [! ) is a well-founded ordering; 5

6 (ii). quasi-terminating or globally nite i the sets fu j t! both nite for any term t; and (iii). Church-osser i (! [ )!. ug and fv j t! vg are n previous versions of this work (Levy and Agust, 1993; Levy, 1994), a bi-rewrite system h ; i is said to be terminating i both! and! are well-founded orderings. This is a weaker condition and clearly it is not enough to prove later the equivalence between the Church-osser and the local bi-conuence properties. This error was communicated to the authors by Professor Harald Ganzinger. We can prove the following results for the decision procedure based on a bi-rewrite system, and the Ax `POL t u deduction problem of its corresponding inclusion theory. Lemma 5 f the bi-rewrite system h ; i is Church-osser then the decision procedure based on it is sound and complete, i.e. Ax `POL t u holds if, and only if, the procedure terminates and answers true. f the bi-rewrite system is Church-osser and quasi-terminating then the decision procedure is sound, complete and terminates, therefore the satisability problem is decidable. We only need to require the quasi-termination property of the bi-rewrite system which is (strictly) weaker than the termination property in order to prove the termination of the procedure; whereas in the equational case, the termination property of the rewrite system is needed to prove the termination of a procedure based on the computation of the normal form. Lemma 6 Any terminating term bi-rewriting system is quasi-terminating. Proof: f (! [! ) is terminating then both! and! are terminating, and the problem is reduced to prove that any terminating term rewrite system is quasi-terminating. First we prove that any terminating term rewriting relation is nitely branching. f! is terminating then any rewrite rule l! r in satises FV(r) FV(l). Now, to rewrite a term we have nitely many ways to select a rule l! r and a subterm tj p. Once we have xed them, if it exists, there is a unique substitution satisfying Dom() FV(l) and tj p = (l). Finally, if FV(r) FV(l), such substitution determines the result of the rewrite step. Second to prove that any nitely branching and terminating relation is quasi-terminating is a straightforward application of the Koenig's lemma. n order to test automatically the Church-osser property we extend the standard procedure used in term rewriting to bi-rewriting. So we reduce the Church-osser property to three simpler properties, namely bi-conuence (or commutativity), local bi-conuence and critical pair bi-conuence. Denition 7 A bi-rewrite system h ; i is said to be (i). bi-conuent i!! (ii). locally bi-conuent i!! A pair of terms hs; ti is said to be bi-conuent i s! t. 6

7 A variant of the Newman's lemma (Newman, 1942; Huet, 1980) proves the following result for bi-rewrite systems. f fact the statement is implied by lemma 1.2 in (Bachmair and Dershowitz, 1986a). Lemma 8 A terminating bi-rewrite system is Church-osser i it is locally bi-conuent. Proof: The only if implication is trivially proved since! (! [ ). The proof for the if implication is done by noetherian induction. We prove that property u P (t) def = 8u; v : u v 0 local t! v ) u! holds for any term t by noetherian induction, using the well-founded ordering (! [ u induc. t 0 t = induc.?? =!! ) +.? The base cases t = u or t = v satised. The induction case follows directly from the induction hypothesis P (u 0 ) and P (v 0 ) using the diagram on the left. Notice that in the previous lemma we require the union of both rewrite relations to be wellfounded, and it is not sucient if both relations are well-founded separately. The following counter-example was communicated to the authors by Professor Harald Ganzinger to show this fact. This counter-example invalidates the corresponding previous results in (Levy and Agust, 1993; Levy, 1994). The bi-rewrite system dened by = fb! c; c! dg and = fc! b; b! ag is locally bi-conuent and both rewrite relations! and! are terminating, not their union. However, the bi-rewrite system is not Church-osser. A simple adaptation of the standard critical pair denition (Knuth and Bendix, 1970) can be given for bi-rewrite systems. However, as we will see, it is not sucient to prove the critical pair lemma. This simple denition of critical pair arises from the most general non-variable overlap between the left hand side of a rule in and a sub-term of the left hand side of a rule in, (or vice versa). Given a pair of rules l! r and s! t, a position p of a non-variable subterm of s, and the most general unier of l and sj p, the pair (t) (s[r] p ) is a (standard) critical pair between and ; and similarly for critical pairs between and. Unfortunately, in the presence of non-left-linear rules, 4 the critical pair lemma stated in terms of such standard critical pairs can not be proved because the conuence of variable overlaps is no longer possible. The same fact has already been discussed in (Bachmair, 1991). Here is a simple counter-example to the validity of this lemma. Counter-example 9 The following bi-rewrite system = f(x; X)! X = a! b is terminating and has no standard critical pairs, however the divergence f(a; b) f(a; a)! a 4 A rule l! r is left- (right-) linear i any variable in l (in r) occurs at most once in l (in r). 7

8 does not satisfy the Church-osser property (the pair f(a; b) a is not bi-conuent). This problem would be avoided if a! b 2, but then the inclusion theory corresponding to the bi-rewrite system would be dierent. Non-left-linear rules also invalidate the bi-rewrite parallel of Toyama's theorem (Toyama, 1987) as the following counter-example shows. Counter-example 10 The following bi-rewrite system 8 >< = >: X [ X! X X [ Y! X [ (Y [ Z)! Y [ X (X [ Y ) [ Z = ( X [ Y! X [ Y! X Y is Church-osser and quasi-terminating, if we consider a signature containing uniquely constants and the binary union operator, i.e. F 2 = f[g and F i = ; for i 62 f0; 2g. However, if we introduce a new 1-ary symbol in the signature f 2 F 1 then we have the following divergence which is not bi-conuent. f(x) [ f(y ) f(x) [ f(x [ Y ) f(x [ Y ) [ f(x [ Y )! f(x [ Y ) This means that many properties of bi-rewrite systems depend not only on the axioms of the theory but also on the signature. Using the standard denition of critical pairs, the critical pair lemma is only true for left-linear systems: a terminating and left-linear bi-rewrite system is Church-osser i all standard critical pairs are bi-conuent. n order to keep this lemma for non-left-linear birewrite systems, we have to enlarge the set of critical pairs to be considered as follows. Denition 11 f l! r 2 and s! t 2 are two rewrite rules (with variables distinct) and p a position in s, then (i). if sj p is a non-variable subterm and is the most general unier of sj p and l then (t) (s[r] p ) is a (standard) critical pair of ECP ( ; ) (ii). if sj p = x is a repeated variable in s, F is a term not sharing variables with s! that Fj q = l, and l! r does not hold, 5 then t such t[x 7! F ] (s[x 7! F ])[r] pq is an (extended) critical pair of ECP ( ; ). Similarly for critical pairs between and, written ECP ( ; ). 5 f this condition is satised then we can make the pair resulting from the variable overlapping conuent like in the equational case. 8

9 The set of (extended) critical pairs of the previous denition is in general innite, t[x 7! F ] (s[x 7! F ])[r] pq is really a critical pair scheme because we do not impose any restriction on the context F [] q (notice that the only condition imposed to F is Fj q = l). n section 4 we will see an example where we use such kind of schemes. So the critical pair lemma even if true with this denition of critical pairs, will be of little practical help to test bi-conuence. Then the conditions of bi-conuence have to be studied in each case taking into account the particular shape of the non-left-linear rules. n section 6 we face the problem of testing bi-conuence automatically by codifying extended critical pairs using the linear second-order typed -calculus. Theorem 12 (Extended Critical Pair Lemma) A terminating bi-rewrite system h ; i is Church-osser i any (standard or extended) critical pair s t in ECP ( ; ) or s t in ECP ( ; ) is bi-conuent, i.e. it satises s! t. Proof: For the if part, see the proof of theorem 24, which states a more general result, taking = ;. For the only if part, extended critical pairs are sound deductions, therefore if s t is an extended critical pair, then s(! [ ) t holds. Now, if the bi-rewrite system is Church-osser, then s! t. This theorem, lemma 21 and theorem 24 could be considered as instances of the general critical pair theorem proved by Geser in his thesis (Geser, 1990). Nevertheless, we think it is worthy to face the critical pair problem directly for our case. The extended critical pair theorem generalizes the critical pair lemma (Knuth and Bendix, 1970) for bi-rewrite systems. However, we require the bi-conuence of not only the standard critical pairs, but also of the extended critical pairs. Nevertheless, if all rules come from the translation of an equational theory E, then any equation a = b with a b results in two bi-rewrite rules a! b in and a! b in and both bi-rewrite relations! =! are equal. Then we only obtain standard critical pairs because the condition l! r in the denition 11 of extended critical pair is always satised. So we recover the old results for the equational case. 3 Bi-rewriting Modulo a Set of nclusions Like in equational rewriting, in bi-rewriting it is not always possible to orient all inclusions of a theory presentation in two terminating rewrite relations, as was assumed in the previous section. Frequently enough, we must handle three rewrite relations, the terminating relations! and! resulting from the inclusions and oriented to the right and to the left respectively, and the non-terminating relation?! resulting from the non-oriented inclusions. Then we say to have a h ; i bi-rewrite system modulo. Although we use the word modulo, it does not mean that?! is a congruence, be aware it is a non-symmetric relation (monotonic pre-order). Figure 3 in section 4 shows an example of these bi-rewrite systems. The inverse of the relation is noted?!. The Birkho's theorem is stated then as Ax `POL t u i t (! [?! [ ) u. 9

10 3.1 From Church-osser to Local Bi-Conuence The simplest way to have a complete and decidable proof procedure based on the h ; i bi-rewrite system modulo is reducing it to the bi-rewrite system h [ ; [ i and, using the results of the previous section, requiring of it the following properties: 1. The relations! [?! and! [?! are both quasi-terminating, and 2. they satisfy the (weak) Church-osser property (! [?! [ ) (! [?! ) ( [?! However, as we have seen in the previous section the quasi-termination of! [?! and! [?! is not enough to reduce the (weak) Church-osser property to the local bi-conuence property ( [?! )(! [?! ) (! [ ) ( [?! ) using lemma 8. To do this we would need the termination of! [?! [! [?!, which of course never holds, because the relation?! [?! is cycling. The solution to this problem comes from requiring the termination of?!! [?!!. Using this termination property, the weak Church-osser property can be reduced to a local bi-conuence property. is terminating, then the following proper- Lemma 13 f the relation?! ties! [?!! (! [?! [ ) (! [?! ) ( [?!?!! (?!! )?! (?! are equivalent. ) ) (weak) Church-osser ) (weak) local bi-conuence Proof: Using the equalities (A [ B) = (A B) A = A (BA ) we prove that right hand sides of both inclusions are equal. Now?!! (! [?! [ ) shows that Church-osser implies local bi-conuence. For the converse we use (A [ B) A B, B A A B to prove the equivalence between the Church-osser property and the following one.?! (?! ) (?!! )?! (?!! )?! (?! Now, if?!! [?!! is terminating we can prove by noetherian induction that this property is equivalent to the local bi-conuence property. The base cases?! (?! ) n (?!! ) m?! with n = 0 or m = 0 trivially hold. The following diagram shows a sketch of the proof for n > 0 and m > 0.?@ A bi-conf. A - - induc.?? =?!! =?!!? - =?! ) 10

11 f?! is symmetric (?! =?! ) the above termination property becomes similar to the termination property required in rewriting modulo a set of equations (Bachmair and Dershowitz, 1989a). That is,?! symmetric means we can dene equivalence classes ([s]! [t] i s?!!?! t) and, the termination of?!! [?!! is ensured by the existence of a well-founded -compatible quasi-ordering, i.e. by the existence of a well-founded, reexive and transitive relation satisfying!,! and?! =?!, where the equivalence relation is the intersection of and and the strict ordering is the dierence of and. The quasi-termination property of?! means that each?! -class of equivalence is nite. However, like in the equational case, rewriting by?!! is inecient, and the local commutativity of?!! and?!! can not be reduced to the bi-conuence of a nite set of critical pairs. Therefore we will approximate them by two weaker, but more practical rewrite relations, named n and n respectively by similarity to the corresponding equational denitions. Notice that although we use the notation! n, it does not means that this relation is the monotonic and substitution closure of a set of rules. n the following, we prove the abstract properties of these relations. We will suppose that they satisfy:!?! n?!!!?! n?!! leaving their denition for the next subsection. We require these new rewrite relations to satisfy what we call a strong Church-osser modulo property, dened as follows. Denition 14 The bi-rewrite system h ; i modulo is (strong) Church-osser i (! [?! [ )?! n?!? The following lemma states sucient conditions to dene a search decision procedure for Ax `POL t u based on the relations n and n. Lemma 15 f the relations?! n are both computable 6 and quasi-terminating, the relation?! is decidable, and h ; i is strong Church-osser modulo, then there exists a decision procedure for the inclusion relation dened by these relations. n and?! Proof: Like in the simpler case of the previous section, given two terms s and t, the algorithm generates the sets fs 0 j s?! n s 0 g and ft 0 j t?! n t 0 g and seek for a term s 0 from the rst set and a term t 0 from the second one such that s 0?! t 0. f relations?! n and?! n satisfy the above inclusions and the Church-osser property then (! [?! [ ) =?! n?!? n. Now, it is easy to prove that the algorithm is a decision procedure for the relation?! n?!? n and Ax `POL s t is equivalent to s(! [?! [ ) t. n The solutions we propose of reducing the strong Church-osser property to a local biconuence property are inspired mainly by the solutions known for the equational case. n the following we consider how they can be adapted to bi-rewriting. 6 We say that a relation! is computable i for any term t, the set fu j t! ug is calculable. We say that a relation! is decidable i for any pair of terms t and u, it is decidable when t! u holds or not. 11

12 Huet (1980) proved that given a set of rules and equations E such that?! E! is terminating, is strong Church-osser modulo E i all peaks and clis are conuent:!!?! E and?! E!!?! E. Notice that these are sucient and, what is also important, necessary conditions. Besides, the niteness of the E-equivalence classes is not required. However, these conuence properties are too strong and can not be reduced to the conuence of critical pairs unless the rules are left-linear. To overcome this limitation for non-left-linear systems Peterson and Stickel (1981) propose the use of a new rewrite relation En satisfying!?! En?! E!. They prove that when this relation is E-compatible, that is when?! E!?! En?! E (?! E ), and terminating, then the Church-osser property becomes equivalent to the conuence of peaks of the form En??! En?! En?! E? En. They also study how a rewrite relation can be extended to obtain a E-compatible rewrite relation En when E is an associative and commutative theory. However, in this case the problem is that the set of critical pairs of the form t En? u?! En v is in general innite. Jouannaud and Kirchner (1986, theorem 5) and Kirchner (1985, theorem 4, chapter 2) generalize the Peterson and Stickel's concept of E-compatibility to coherence, and prove that when?! E! is terminating ( is E-terminating) then the following three conditions are equivalent 7 1. is En-Church-osser modulo E (! [?! E [ )?! En?! E? En 2. En is conuent and coherent modulo E En??! En?! En?! E? En?! E! En?! En?! E? En 3. En is locally conuent and locally coherent modulo E?! En?! En?! E? En?! E! En?! En?! E? En (global) peak (global) cli local peak local cli Then local conuence and coherence can be reduced to critical pair conuence and to extended rules respectively. Here we call these properties conuence of peaks and conuence of clis respectivelly, following the notation of Dershowitz and Jouannaud (1990). Jouannaud and Kirchner also notice that this result is false if we require termination of?! En instead of that for?! E!. As a counter-example 8 we can take the rewrite system = En = fb! a; a! dg with E = fa = b; b = cg. t satises local conuence properties and termination of?! En, but it is not Church-osser. However, termination of?! En is enough to prove the equivalence between Church-osser property and \global" conuence properties (see (Huet, 1980) for a similar proof). Evidently, this termination property is not enough to prove equivalence between local and global conuence properties. For our purposes, we are more interested on global conuence properties than on the local ones, therefore we use them in the following lemma. 7 They dene coherence as?! E! En! + En?! E En, however, as they notice, if is E terminating, both denitions are equivalent. 8 This counter-example is, in fact, equivalent to one given by Huet (1980). 12

13 n be two rewrite relations satisfying!?! n. f their union?! n [?! n is terminating then the following three global conuence properties Lemma 16 Let?! n and?!?!! and!?! n?!!?!?! n?! n?!? n n??!?! n?!? n 9 >= >; clis n??! n?! n?!? n peaks and the strong Church-osser property are equivalent. (! [?! [ )?! n?!? Proof: t is evident that the Church-osser property implies the three local bi-conuence properties, so we will prove the opposite implication. Such proof is based on the ideas of proof transformation and proof ordering proposed by Bachmair in his thesis (Bachmair, 1991) and in (Bachmair et al., 1986b). Given a sequence of terms hv 1 ; : : :v n i, we say that it is a proof of s t i v 1 = s, v n = t, and for any i 2 [1::n?1] we have v i?! n v i+1 or v i n? v i+1 or v i?! + v i+1. Notice that we allow to concentrate one or more?! rewrite steps in a single proof step. Evidently, t u has a proof i t(! [?! [ ) u. n the following we dene a set of transformations on the proofs of an inclusion. Given a proof transformation rule hs; t; ui ) hs; v; ui, we can use it to transform hw 1 ; s; t; u; w 2 i ) hw 1 ; s; v; u; w 2 i. To prove the termination of such transformation relation we associate a multiset S(hv 1 ; : : :; v n i) of terms to each proof hv 1 ; : : :; v n i dened as follows. S(hvi) = ; S(hv 1 ; : : :; v n i) = S(hv 1 ; : : :; v n?1 i) [ 8 >< >: n fv n?1 ; v n g if v n?1?! n v n or v n?! n v n?1 fv 2 n?1 ; ng v2 if v n?1?! + where [ denotes the multiset union operator and superscripts denote the number of occurrences of an element in a multiset. We dene a well-founded ordering on these term multisets as the multiset extension of the order relation?! + n [?! + n which we have supposed terminating. This ordering on associated multisets denes a well-founded ordering on proofs. Notice that this ordering is monotonic, i.e. if S(hs; t; ui) S(hs; v; ui), then S(hw 1 ; s; t; u; w 2 i) S(hw 1 ; s; v; u; w 2 i). This is a key point to prove that if any proof transformation rule hs; t; ui ) hs; v; ui satises S(hs; t; ui) S(hs; v; ui) then the proof transformation relation is terminating. f clis are bi-conuent, then for any cli s?! + t?! u we have n s?! n v 1 v p?1?! n v p?! w q n? w q?1 w 1? n v n u 13

14 and we can apply one of the following proof transformations rules to eliminate it hs; t; ui ) hs; v 1 ; : : :; v p ; w q ; : : :; w 1 ; ui if v p?! + w q hs; t; ui ) hs; v 1 ; : : :; v p?1 ; w q ; : : :; w 1 ; ui if s?! + n v p = w q + hs; t; ui ) hs; w q?1 ; : : :; w 1 ; ui if s = v p = w q n? hs; t; ui ) hsi if s = v p = w q = u where p; q 0, except in the second rule where p 1, and the third rule where q 1. Now, taking into account that s v 1 v p and t u w 1 w q, we can prove that the multiset associated to the left part of the rules S(hs; t; ui) = fs 2 ; t 3 ; ug is strictly greater than the multisets associated to the right part of the rules, which are respectively: S(hs; v 1 ; : : :; v p ; w q ; : : :; w 1 ; ui) = fs; v 2 1 ; : : :; v2 p ; w2 q ; w2 2 ; : : :; w2 1 ; ug [ 9 fv p ; w q g S(hs; v 1 ; : : :; v p?1 ; w q ; : : :; w 1 ; ui) = fs; v 2 1 ; : : :; v2 p?1 ; w2 q ; : : :; w2 1 ; ug S(hs; w q?1 ; : : :; w 1 ; ui) = fs; w 2 q?1 ; : : :; w2 1 ; ug S(hsi) = ; Similarly, if peaks are bi-conuent, then we can also apply the same proof transformations rule to any peak s n? t?! n u. And, taking into account that now t s v 1 v p and t u w 1 w q, we can also prove that the multiset associated to the left part of the rule, now S(hs; t; ui) = fs; t 2 ; ug is also strictly greater than the multisets associated to the corresponding right parts of the rules. Evidently, if we iterate this process, the resulting canonical (normal) proof will not contain any clis nor peaks. Therefore it will be of the form?! n?!? n. The process can not be applied innitely, because the transformation relation is terminating. We conclude that if s t has a proof, then it has a canonical proof of the form s?! n?!? n t. Therefore, the Church-osser property holds for these rewrite relations. Now, the logical process would be to reduce the bi-conuence of peaks of the form n??! n to the bi-conuence of peaks of the form n?! or?! n, as Jouannaud and Kirchner did for the equational case. However, as the following counterexample shows, not any denition of?! n satisfying!?! n?!! permits such reduction, unless we require termination of (?! [?! ) (! [! ). Counter-example 17 Consider the rewrite relations dened by the following sets of rules. = fa 1?! b; b?! a 2 g = fa 1! b; a 2! c c 2 g 1 a 1 - b - a 2 - c 2 = fa 2! b; a 1! c 1 g def def f we dene?! n =! [?!! and?! n =! [?!!, we will obtain two rewrite relations such that?! n [?! n is terminating and the properties!?! n?!! hold. However, although any peak of the form n?! or?! n and any cli is bi-conuent, there is a peak c 1 n? b?! n c 2 which is not bi-conuent. Notice that in this counter-example?!! [?!! is also terminating, not so (?! [?! ) (! [! ). cases. 9 Notice that in this case we can have s = v p, u = w q or both together. With such union we capture four u 14

15 emark 18 The reader may prove that, when (?! [?! ) (! [! ) is terminating, then (strong) Church-osser property and \local" conuence properties are equivalent. This result is closer to the theorem proved by Jouannaud and Kirchner (1986, theorem 5), and its proof is left to the reader. Alternately, if we only require the termination of?! n [?! n, then the method of rule extensions and the concrete denition of the relation?! n ensures that, if inclusions in are linear, then?! and?! n commute, i.e.?!?! n?! n?!. This property is stronger than the conuence of clis, and permits the desired reduction. However, such method takes into account the structure of terms, so we will describe it in the next subsection. 3.2 From Local Bi-Conuence to (Extended) Critical Pairs Till now, we have studied Church-osser, termination and bi-conuence properties in the framework of relational algebra (Baumer, 1992). All proofs were done without any reference to the structure of terms. n the following, we will consider the term structure in order to reduce the bi-conuence properties to the bi-conuence of critical pairs and rule extensions. We begin dening the rewrite relations n and n that were only axiomatically characterized by!?! n?!! in the previous subsection. The choice of such denition is motivated, as in the equational case, by the fact that local bi-conuence of peaks?! n and n?! can be reduced to the bi-conuence of a selected set of critical pairs. t, i there (l) n [p;;l!r] t when we want to make explicit the position, Denition 19 We say that s -rewrites to t modulo, written s?! n exists a rule l! r in, an occurrence p in s, and a substitution such that sj p?! and t = s[(r)] p. We write s?! substitution and rule involved in the rewrite step. Similarly for s -rewrites to t modulo, written s?! n t. With this denition?! n really veries!?! n?!! although in general?!! 6?! n. Notice that in a?! n rewrite step, the?! rewrite steps take place bellow the! rewrite step. We say that the! rewrite step covers such?! rewrite steps. We will use the notions of E-matching and E-unication from (Peterson and Stickel, 1981) but adapted to bi-rewriting. Given two terms s and t, we say that s -matches t i there exists a substitution such that s?! (t), and s?1 -matches t i there exists a substitution such that s?! (t). We say that s -unies with t i there exists a substitution such that (s)?! (t). Notice that, since?! is not necessarily symmetric, -matching and?1 -matching are in general dierent non-symmetric relations, and -unication also is a non-symmetric relation. We will suppose in the following that -unication and and?1 - matching are decidable. We have to be careful dening minimum uniers since denition of critical pairs is based on them. Given two terms s and t, we say that M is a complete set of minimum uniers i for any -unier of s and t, there exists a minimum unier 2 M and a substitution such that (x)(?! \?! )((x)) for any x 2 Dom(). We will suppose in the following that a nite and complete set of minimum uniers exists for our relation?!. As in the equational case (to prove bi-conuence of clis or E-compatibility), we will prove the commutativity properties by means of the rule extension and the extensionally closed property dened as follows. 15

16 Denition 20 Given an inclusion l r in, and a rule s! s, being a minimum unier, and rj p is neither a variable nor equal to r, then we say that (l)! (r[t] p ) is a right--extended rule of. t in, if rj p -unies with Given a set of rules and inclusions, is said to be right--extensionally closed n?! r. i any right--extended rule l! r of satises l?! We dene left--extended rule and left--extensionally closed similarly changing by and \rj p -unies with s" by \s -unies with rj p ". Notice that in the previous denition, to consider a bi-rewrite system extensionally closed, we require any rule extension l! r to satisfy l?! n?! r. t is not enough to require the pair l r to be bi-conuent. Since?! may be non-symmetric, we have had to distinguish between right- and leftextensionality in the previous denition. We will use a completion procedure to ensure that the nal bi-rewrite system satises that is right--extensionally closed, and that is left--extensionally closed. The following lemma states that, if all inclusions in are linear, then the extensionally closed property ensures the commutativity of?! and?! n. Notice that this property is stronger than the bi-conuence of clis required in the previous subsection. Lemma 21 (Critical Cli Lemma) f all inclusions in are linear, and is right- -extensionally closed, then?! and?! n commute, i.e. they satisfy?!?! n?!. Moreover, we also have?! Similarly for?!?! n?! n?! [?! and?! n. if the later is left--extensionally closed.?! Proof: The conclusion of the lemma is equivalent to?!?! n?! n?! [?!. Suppose that a?! [p1 ;;st] b?! n [p2 ;;l!r] c where p 1 and p 2 are positions, is a substitution (assume that FV(t) \ FV(l) = ;), s t is an inclusion of and l! r a rule of. We have to consider the following four cases. case p 1 jp 2 n f p 1 and p 2 are disjoint occurrences then both rewrite steps trivially commute. case p 1 p 2 Let v satisfy p 2 = p 1 v. We have (t)j v?! (l). There are two possibilities: variable overlapping There exist two occurrences v 1 and v 2 satisfying p 1 v 1 v 2 = p 2 and being tj v1 = x a variable. f all inclusions in are right-linear then tj v1 is the only occurrence of x in t, moreover if all inclusions are left-linear then x occurs at most once in s. Let v 0 1 be this occurrence of x in s, if there is one. First, we have aj p1 v 0 1 v 2?! (l) and therefore a?! n [p1 v 0 1 v a[(r)] 2;;l!r] p1 v 0 1 v. Second, 2 we prove that a[(r)]?! p1 v 0 1 v 2 [p 1 ; 0 ;st] c where 0 is dened as 0 (y) = (y) for any y 6= x, and 0 (x) = (x)[(r)] v2. Otherwise, if x does not occur in s then we have a?! [p1 ; 0 ;st]; c, where 0 is dened as above. Notice that it is in this case, with variable overlapping, when we have to require both left- and right-linearity of s t. 16

17 strict overlapping f v is a position in t, and tj v is not a variable, we are in the conditions of denition 20, i.e. tj v -unies with l being a minimum unier, and we can generate an extensional rule l 0! r 0 def = (s)! (t[r] v ) between s t and l! r. Now, using our concrete denition of minimum -unier, a variant of the E-critical pair lemma (Jouannaud, 1983) ensures that aj p1 (?! \?! )(l 0 ) and cj p1 (?! \?! )(r 0 ) where =. n particular, we have aj p1?! (l 0 ) and (r 0 )?! cj p1. n the equational case (Jouannaud and Kirchner, 1986) we would need to require the termination of the subterm relation modulo. However, the stronger condition required in the denition of extensional closure allows us to disregard this requirement. f is -extensionally closed, then l 0?! n?! r 0. The aj p1?! (l 0 ) rewrite steps are \covered" by the?! n rewrite step, obtaining aj p1?! n?! (r 0 )?! cj p1. Notice that the proof aj p1?! n?! (r 0 )?! cj p1 is normal, whereas if we only require extended rules to be bi-conuent, we would obtain aj p1?! n?!? n (r 0 )?! cj p1. The later of course is not a normal proof and we would need to require the well-foundness of the strict subterm modulo relation to prove that it is smaller than the original proof. case p 1 p 2 The a?! b rewrite step is covered by the?! n rewrite step. Let v be the occurrence such that p 2 v = p 1. We prove aj p2?! [v;;st] bj p2, so aj p2?! bj p2?! (l) and we have a?! n [p2 ;;l!r] c. Notice that like in (Peterson and Stickel, 1981), and dierently from (Jouannaud and Kirchner, 1986), the inclusions in are required to be (both left- and right-) linear. However, thanks to the stronger condition required to extended rules, we can disregard the well-founded condition on the strict subterm modulo relation. emark 22 The attentive reader will notice that our assumptions dier from those assumed in the equational case by Jouannaud and Kirchner (1986). Everywhere we have tried to require the weakest termination condition. Another option, similar to the one developed by Struth (1996), 10 starts from requiring the termination of the strict subterm relation modulo?! [?! and the termination of the relation (?! [?! ) (! [! ), (notice that these conditions are stronger than the termination of?! n [?! n assumed in this paper). They would allow us to relax the condition on extended rules l r, requiring them to be bi-conuence l?! n?!? n r instead of l?! n?! r. Moreover, if we also applied the notion of extended critical pair to extended critical cli, requiring them to be bi-conuent, then we could drop the requirement on the linearity of the inclusions dening?!. These assumptions would also allow us to reproduce the classical results when any inclusion comes from an equality, and we have?! =?! and! =!. The reader is invited to reproduce the proofs of Jouannaud and Kirchner (1986) for bi-rewrite systems. 10 Struth (1996) requires the termination of?!!?! [?!!?!, which is a weaker condition than the termination of (?! [?! ) (! [! ). However, then he has to use a condition for the local peaks and clis stronger than bi-conuence. 17

18 The conclusion of the critical cli lemma, not only ensures the bi-conuence of clis, but also allows to reduce the bi-conuence of peaks of the form n??! n to the conuence of the peaks of the form?! n or n?! using the following sequence of inclusions n??! n?!?! n : : : (?! n?! [?! ) if clis commute?! n?!? n?! if peaks are bi-conuent?! n?!? if clis commute n For the bi-conuence of peaks we use a denition of (extended) critical pairs similar to the one introduced in the previous section. Denition 23 f l! r 2 and s! t 2 are two rewrite rules normalized apart, and p is a position in s, then (i). if sj p is not a variable and sj p -unies with l being the minimum -unier, then (t) (s[r] p ) is a (standard) critical pair of ECP (n ; ) (ii). if sj p = x is a repeated variable in s, F is a term not sharing variables with s! that Fj q?! (l) for some, and l?! n?! r does not hold, then t such t[x 7! F ] (s[x 7! F ])[(r)] pq is an (extended) critical pair of ECP (n ; ). Moreover, if?! is symmetric, then we can restrict extended critical pairs to those which satisfy Fj q = l, like in denition 11. The set ECP (n ; ) can be dened similarly. Again we have had to introduce critical pair schemes which may generate innitely many critical pairs. Unlike denition 11 of previous section, here, if?! is non-symmetric, then the whole term F is undetermined, not only the context F [] q. The only restriction on Fj q is Fj q?! (l) for some substitution. Using this extended denition of critical pairs and the denition of extensionally closed bi-rewrite system we can prove the following theorem which characterizes the strong Church- osser property of a h ; i bi-rewrite system modulo. Theorem 24 (Critical Pair Theorem) Given two sets of rules and and a set of inclusions, if?! n [?! n is terminating,?! n is right--extensionally closed,?! n is left--extensionally closed, all inclusions in are linear, and all standard and extended critical pairs ECP (n ; ) and ECP ( ; n ) are bi-conuent, then h ; i is (strongly) Church-osser modulo. Proof: We use lemma 16 of previous sub-section to prove the Church-osser property. We are in the conditions of lemma 21, therefore we can suppose that clis commute. As we 18

19 have commented, this condition is stronger than the bi-conuence of global clis required by lemma 16. Let's concentrate on the bi-conuence of global peaks. Assume that a n? [p 1 ;;s!t] b?! n [p 2 ;;l!r] c f p 1 jp 2, as in the commutativity case, both rewrite steps trivially commute and we can reduce a and c to the same term b[(t)] p1 [(r)] p2 = b[(r)] p2 [(t)] p1. Otherwise, we can suppose, without lose of generality that p 1 p 2. We have: a? [p 1 ;;s!t] a 0?! b?! n [p 2 ;;l!r] c where all the?! rewrite steps between a 0 and b takes place bellow the position p 1 (notice that they are covered by the?! n rewrite step). Now, lemma 21 allow us to commute these?! steps and the?! step. We will have either or n a? [p 1 ;;s!t] a 0?! a? [p 1 ;;s!t] a 0?! n [p 0 2 ;;l!r] c 0?! where p 0 is a position bellow or equal to 2 p 1 (notice that the original steps were all them bellow p 1, therefore, after commuting them, the resulting steps will also be bellow p 1 ). Taking into account that we can also commute?! steps and?! n steps, we only have to prove that local peaks of the form: c t 1? [p 1 ;;s!t] t 2?! n [p 2 ;;l!r] t 3 where p 1 p 2, are bi-conuent. Let v be the occurrence such that p 2 = p 1 v. There are three possibilities: Strict overlapping Position v is a non-variable occurrence of s. This case works as case strict overlapping in the proof of critical cli lemma 21. That is, there exists a standard critical pair l 0 r 0, obtained -unifying sj v and l, such that t 1 j p1?! (l 0 ) and (r 0 )?! t 3 j p1. f standard critical pairs are conuent then (l 0 )?! n?!? n (r 0 ). Finally, if lemma 21 holds, then pair t 1 t 3 is bi-conuent. Non-repeated variable overlapping Subterm sj v is a or is bellow a non-repeated variable, i.e. there exist two positions such that v = v 1 v 2 and sj v1 = x is a non-repeated variable of s. This case works similarly to case variable overlapping of lemma 21. That is, we can rewrite t 1 and t 3 into a common term in the following way. We apply the rewrite step (x) =?! n [v 2 ;;l!r] (x)[(r)] v2 to any occurrence of x in t, i.e. to some sub-terms (x) of t 1. On the other side, we apply the rule s! t to the position p 1 of t 3, but using the substitution 0 dened as 0 (y) = (y) for any y 6= x and 0 (x) = (x)[(r)] v2 instead of. t can be proved that in both cases we obtain the same result. epeated variable overlapping Subterm sj v is a or is below a repeated variable x of s, i.e. there exists a pair of positions v 1 v 2 = v such that sj v1 is a repeated variable x of s. c 19