Complete Monotonic Semantic Path Orderings 1 ... - CiteSeerX
Complete Monotonic Semantic Path Orderings Cristina Borralleras Maria Ferreira Albert Rubio [email protected] [email protected] [email protected] Dept. L.S.I. Dept. Informatica Dept. L.S.I., Univ. Univ. de Vic Univ. Nova de Lisboa Politecnica de Catalunya C/ de Miramarges, s/n 2825-114 Caparica, C/Jordi Girona, 1-3 08500 Vic Spain Portugal 08034 Barcelona Spain
Abstract
Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination, i.e., a TRS is terminating if and only if its rules are included in some MSPO. Hence MSPO is a complete termination method. On the practical side, it can be easily automated using as ingredients standard interpretations and general-purpose orderings like RPO. This is shown to be a suciently powerful way to handle several non-trivial examples and to obtain methods like dummy elimination or dependency pairs as particular cases. Finally, we obtain some positive modularity results for termination based on MSPO.
1 Introduction Rewrite systems are sets of rules (directed equations) used to compute by repeatedly replacing parts of a given formula with equal ones until the simplest possible form is obtained. Depending on the kind of objects that are rewritten there are di erent kinds of rewrite systems, like string rewrite systems (Thue or semi-Thue systems) or term rewrite systems (TRS; see [DJ90, Klo92] for detailed surveys). Termination is a fundamental property for most applications of rewrite systems. Termination of a TRS is, in general, an undecidable property, even if it consists of a single rule. Termination of TRS can be proved by showing that the induced rewrite relation is included in a well-founded ordering on terms. If the ordering is also monotonic and stable under substitutions, i.e., a reduction ordering , then it suces to check that l r for every rule l ! r in the system. Monotonic orderings including the subterm relation are called simpli cation orderings, and its well-foundedness follows from Kruskal's theorem [Kru60]. Inside this class, path orderings, and in particular the recursive path ordering 1
(RPO) [Der82], have received a special attention (see [Der87, DJ90]). Unfortunately, although these orderings are simple and easy to use, they turn out, in many cases, to be a weak termination proving tool, as there are many TRS that are terminating but are not contained in any simpli cation ordering, i.e. they are not simply terminating. To avoid this problem, di erent transformation methods have been developed, e.g. [BD86, BL90, Zan94, FZ93, Ste95, Xi98]. By transforming the TRS into a set of ordering constraints, the dependency pairs method [AG97] has become a successful general technique for proving termination of (non-simply terminating) TRS. As an alternative to transformation methods, more powerful term orderings can be used. Due to its simplicity, the Semantic Path Ordering (SPO) ([KL80]) becomes a potential well-known candidate: in SPO the scheme of RPO is generalized by replacing the precedence on function symbols by any (well-founded) underlying (quasi-)ordering involving the whole term and not only its head symbol. Although the simplicity of the presentation is kept, this makes the ordering much more powerful. In fact, for every terminating TRS there is some SPO that includes its rewrite relation. Unfortunately, SPO is not so useful in practice. Due to the generalization, the monotonicity property is lost, even if the underlying ordering is monotonic. Hence, it is not sucient to check that the rules are included in the ordering to ensure termination, since this does not imply that each rewrite step is inside the ordering. In order to ensure termination of a TRS R, the user is responsible for proving (by hand) monotonicity restricted to all terms s and t such that s rewrites to t with R in one step. In this paper we present a monotonic version of the SPO. On the one hand, this monotonic semantic path ordering (MSPO) is still very powerful for proving termination of TRS: on the theoretical side, it also characterizes termination, i.e. a TRS is terminating if and only if the rules are included in some MSPO; and on the practical side, it generalizes most of the (automatable) termination proof methods. On the other hand, termination can be automatically checked once the ingredients, i.e. the underlying (base) quasi-orderings, of the MSPO are provided. The rst and only other, as far as we know, monotonic version of SPO is due to Geser [Ges92]. On the one hand, this proposal is not as general as ours (in fact, as we will show, it does not characterize termination), and on the other hand, it is less suitable for practical implementations. In section 5.3 a detailed comparison with this work is provided. We are not only interested in using MSPO for checking termination, but also for proving termination automatically. Since we cannot expect to automatically generate an adequate MSPO for a given TRS whenever it exists, we have studied particular classes of underlying quasi-orderings, which can be automatically generated. As a hint of the power of the resulting family of MSPO's, it is shown that some known methods like dummy elimination [FZ93] or dependency pairs [AG97] (without the dependency graph re nement) are particular instances. 2
Using these classes of underlying quasi-orderings the termination proofs of several non-simply terminating, but terminating, term rewriting systems can be fully automated. A rst system based on MSPO has been developed by which many examples, including the ones in the paper, have been checked. The software and examples are available at www.lsi.upc.es/albert. Additionally, applying known abstract sucient conditions ensuring modularity of termination [Ohl94, Gra94], modularity results for termination based on MSPO are obtained. In particular, for TRS proved terminating using MSPO with the aforementioned classes of underlying quasi-orderings, termination is proved to be modular for disjoint systems and for nite constructor-sharing systems. Note that these modularity properties are crucial for many practical applications of automatic termination proof systems. Formal de nitions and basic tools are introduced in section 2. In section 3 we present and study the monotonic semantic path ordering. Section 4 is devoted to examples. Other termination methods are analyzed in section 5. In section 6 modularity results are presented. Some conclusions are given in section 7.
2 Preliminaries In the following we consider that F is a set of function symbols, X a set of variables and T (F ; X ) is the set of terms built from F and X . Let s and t be arbitrary terms in T (F ; X ), let f be a function symbol in F and let be a substitution. A (strict partial) ordering is a transitive irre exive relation. It is monotonic if s t implies f(: : :s : : :) f(: : :t : : :), and stable under substitution if s t implies s t. Monotonic orderings that are stable under substitutions are called rewrite orderings. A reduction ordering is a rewrite ordering that is well-founded : there are no in nite sequences t1 t2 : :: The re exive-transitive closure of a binary relation ! is denoted by ! and +. the transitive closure by ! A term rewrite system (TRS) is a (possibly in nite) set of rules l ! r where l and r are terms. Given a TRS R, s rewrites to t with R, denoted by s !R t, if there is some rule l ! r in R, sjp = l for some position p and substitution and t = s[r]p . A TRS R is terminating if there exists no in nite sequence + of any terminating TRS is a t1 !R t2 !R : : : Thus, transitive closure ! R reduction ordering. Furthermore, reduction orderings characterize termination of TRS. Theorem 2.1. A rewrite system R is terminating if and only if all rules are contained in a reduction ordering , i.e., l r for every l ! r 2 R. Another interesting property of reduction orderings is that they can be combined with the subterm relation without loosing well-foundedness. However, note that in general, monotonicity will be lost, since we only add the subterm relation, and not its monotonic closure.
Proposition 2.2. If is a reduction ordering then [ is well-founded. 3
Given a relation , the multiset extension of on nite multisets, denoted by , is de ned as the smallest transitive relation containing X [ fsg X [ ft1; : : : ; tng if s ti for all i 2 f1 : : :ng If is a well-founded ordering on terms then is a well-founded ordering on nite multisets of terms. A quasi-ordering is a transitive and re exive binary relation. Its inverse is denoted by . Its strict part is the strict ordering n (i.e, s t i s t and s 6 t). Its equivalence is \ . Note that is the disjoint union of and , and that if = denotes syntactic equality then [ = is a quasi-ordering whose strict part is . Notation: In the remainder of this paper, (possibly with subscripts) will always denote a quasi-ordering. The following de nitions for will be used: 1. is monotonic if is. 2. is well-founded if is. 3. is stable under substitutions if is and s t whenever s t. 4. is quasi-monotonic if f(: : : ; s; : : :) f(: : : ; t; : : :) whenever s t. 5. is a quasi-reduction quasi-ordering if it ful lls the above properties 2, 3 and 4. Note that if is quasi-monotonic then is not necessarily monotonic. The lexicographic combination of 1 ; : : : ; n , denoted by (1 ; : : : ; n )lex , is de ned as usual as, s(1 ; : : : ; n)lex t i either s i t for some i and s j t for all j < i, or s i t for all i. If all i are well-founded (stable under substitutions) then its lexicographic combination also is. A precedence F is a well-founded quasi-ordering on F . Given a precedence F , the recursive path ordering (RPO), denoted by rpo , is de ned recursively as follows: s = f(s1 ; : : : ; sm ) rpo t i 1. si rpo t, for some i = 1; : : : ; m, or 2. t = g(t1 ; : : : ; tn) with f F g and s rpo ti for all i = 1 : : :n or 3. t = g(t1 ; : : : ; tn) with f F g and fs1; : : : ; sm g rpo ft1; : : : ; tng.
3 A monotonic Semantic Path Ordering Let us now recall the de nition of the semantic path ordering. Then we will analyze an example of non-monotonicity, which will provide the intuition behind the monotonic version we will propose. 4
De nition 3.1. Given Q , called the underlying (or base) quasi-ordering, the semantic path ordering (SPO) [KL80], denoted as Qspo , is de ned as
s = f(s1 ; : : : ; sm ) Qspo t i 1. si Qspo t, for some i = 1; : : : ; m, or 2. s Q t = g(t1 ; : : : ; tn) and s Qspo ti for all i = 1; : : : ; n, or 3. s Q t = g(t1 ; : : : ; tn) and fs1 ; : : : ; sm g Qspo ft1; : : : ; tng, The semantic path ordering ful lls the following property.
Lemma 3.1 ([KL80]). If Q is well-founded and stable under substitutions then Qspo is well-founded and stable under substitutions.
But, as said before, the semantic path ordering is, in general, non-monotonic, even when Q is quasi-monotonic (in fact, the same problem appears if Q is monotonic). This is shown in the following example.
Example 3.2. Consider the following quasi-ordering Q de ned for all terms
s and t as: (i) f(s) Q g(t); (ii) f(s) Q f(t) i s Q t; and (iii) g(s) Q g(t),
and re exive in variables and constants. This quasi-ordering is well-founded, since its strict part is u[f(s)] Q u[g(t)] for all terms s and t and contexts u containing only the symbol f (note that the length of any decreasing sequence is at most the number of f 's above the rst g symbol of the initial term in the sequence); it is stable under substitutions and quasi-monotonic. However, the induced SPO is not monotonic: by case 1 of SPO we have g(f(a)) Qspo f(a), but by adding the context f([]) onto both terms we have f(g(f(a))) 6 Qspo f(f(a)) (in fact, in this case we can even prove f(f(a)) Qspo f(g(f(a)))).
Analyzing this example we can observe that, even if Q is quasi-monotonic (or monotonic), since in case 1 of SPO we do not require s Q t, it may happen that by adding some context u, if we do not have u[s] Q u[t], we cannot apply any of the cases. Then, in order to ensure monotonicity of Qspo , we need to be sure that u[s] Q u[t] for any context. A way to obtain this is to require always s Q t, provided that Q is quasi-monotonic, that is s t i and only if s Q t and s Qspo t. However, requiring always s Q t can be a bit to strong and, in fact, from the example above we only need to have u[s] Q u[t] for every context u. Hence instead of requiring s Q t, we will ask for something weaker that ensures u[s] Q u[t]. De nition 3.2. We say that I is quasi-monotonic on Q (or Q is quasimonotonic wrt. I ) if s I t implies f(: : :s : : :) Q f(: : : t : : :) for all terms s and t and function symbols f. 5
De nition 3.3. A pair hI ; Qi is called a quasi-reduction pair if I is quasi-
monotonic, Q is well-founded, both are stable under substitutions and I is quasi-monotonic on Q . Now we de ne the monotonic semantic path ordering (MSPO): De nition 3.4. Let hI ; Qi be a quasi-reduction pair. The corresponding monotonic semantic path ordering, denoted by mspo , is de ned as: s mspo t if and only if s I t and s Qspo t for all terms s and t.
Theorem 3.3.
mspo
is a reduction ordering.
and Qspo is well-founded. Stability under substitutions follows from the stability of I and Q spo . For monotonicity we have to show that s mspo t implies f(: : : ; s; : : :) mspo f(: : : ; t; : : :), i.e. f(: : : ; s; : : :) I f(: : : ; t; : : :) and also f(: : : ; s; : : :) Qspo f(: : : ; t; : : :), for all terms s and t and function symbols f. By de nition of mspo , s mspo t implies s I t. Hence, by the quasimonotonicity of I , we have f(: : : ; s; : : :) I f(: : : ; t; : : :). On the other hand, by quasi-monotonicity of I on Q , we have f(: : : ; s; : : :) Q f(: : : ; t; : : :). Then, by de nition of mspo , s mspo t implies s Qspo t and, hence f: : :s : : : g Q Q spo f: : :t : : : g, which implies f(: : : ; s; : : :) spo f(: : : ; t; : : :) by case (3). The previous theorem shows that mspo provides a correct method for proving termination of term rewriting systems. The following result shows that it is also a complete method, i.e., for any terminating term rewriting there is a monotonic semantic path ordering that includes its rules. Therefore mspo characterizes termination. Theorem 3.4. A rewrite system R is terminating if and only if there exists some quasi-reduction pair hI ; Q i, s.t. l mspo r for every rule l ! r in R. Proof. The right to left implication follows from Theorem 3.3. For the left to right, we will build an appropriate pair hI ; Q i for any terminating TRS R. Let I be !R and let Q be (!R [) . By de nition of rewriting I is quasi-monotonic and both I and Q are stable under substitutions. By termination of R and Property 2.2, the strict part of Q , i.e. (!R [)+ , is well-founded. Finally, since I Q , and I is quasi-monotonic, we have that I is quasi-monotonic on Q . Therefore, since hI ; Q i is a quasi-reduction pair, we only have to prove that l mspo r for every rule l ! r in R. By de nition, we have l I r and l Q r, and, moreover, for every subterm r0 of r we have l Q r0, since l !R r r0 . Therefore, we have l I r and l Qspo r, by repeatedly applying case 2, which implies l mspo r.
Proof. Well-foundedness follows from the fact that mspo
6
Q spo
The quasi-reduction pair condition is quite tight: as shown in the following example, if Q is required to be a quasi-reduction quasi-ordering then even SPO does not characterize termination, i.e. there are terminating sets of rules that are not included in any SPO with an underlying quasi-reduction quasi-ordering.
Example 3.5. There is no quasi-reduction quasi-ordering following TRS is included in the generated SPO.
a f(b) h(a)
! ! !
Q
such that the
b g(a) h(f(b))
First, note that the TRS is terminating (it will be shown using MSPO in example 4.1). Now we show that these rules cannot be included in SPO with a quasi-reduction quasi-ordering Q . To this end it is important to remark that in SPO to have c Qspo t for some constant c and term t we need c Q t0 for all t0 subterm of t. By de nition we need a Q b to be able to include the rst rule. Then to include the second one we need f(b) Q g(a) and f(b) Q a, in order to conclude by case 2 rst and then by case 2 or 3 for the recursive call with f(b) Qspo a, since any other possibility requires b Q a, which contradicts the rst assumption. Now we have (i) a Q b, (ii) f(b) Q g(a) and (iii) f(b) Q a. We proceed with the third rule. If a Qspo h(f(b)) we need a Q f(b), which contradicts assumption (iii). Otherwise we need h(a) Q h(f(b)). By assumption (iii) and quasi-monotonicity of Q , we have h(f(b)) Q h(a), which implies h(a) 6 Q h(f(b)). Therefore we have to apply case 3 of SPO, and hence we need a Qspo f(b), which requires a Q f(b) contradicting assumption (iii).
Theorem 3.4 shows the theoretical power of the ordering as a termination proof method, but in order to make it more useful in practice in the following two sections we will show general methods to obtain quasi-reduction pairs hI ; Q i.
3.1 Building
I
We consider I obtained by combining an interpretation I on terms with some quasi-reduction quasi-ordering B , called the basic quasi-ordering, which can be obtained by well-known practical general-purpose methods like the path orderings or polynomial interpretations. For the interpretation I, as a general property, we require the preservation of the quasi-monotonicity and stability under substitutions of the basic quasi-ordering. Below some particular such interpretations which are suitable for practical applications are provided. These interpretations are not original; they have been used in more ad-hoc ways in many di erent termination methods, but here they are all placed in the same framework. We will consider interpretations as mappings from terms to terms I : T (F ; X ) ! T (F 0 ; X 0), although, of course, we can also consider interpretation from terms to multisets of terms or any other domain provided that the required properties are ful lled. 7
From now on we consider that I is de ned as follows s I t if and only if I(s) B I(t) Note that this does not imply any loss of generality since I can be the identity mapping. Additionally, it follows that s I t if and only if I(s) B I(t). For any I, if B is a quasi-ordering then I is a quasi-ordering which is wellfounded if B also is. For quasi-monotonicity and stability under substitutions this is not always the case. Let us give some examples of interpretations that do preserve this properties. Each symbol f can be interpreted either by a projection on a single argument, denoted by the pair (f(x1 ; : : : ; xn); xi), or else by a function symbol fI applied to an arbitrary sequence obtained from the arguments of f, denoted by the pair (f(x1 ; : : : ; xn); fI (xi1 ; : : : ; xi )), for some k 0 and i1 ; : : : ; ik 2 f1; : : : ; ng. Additionally we consider I to be the identity for variables (although it can be any bijection). We assume that there is only one pair for each symbol. Usually the identity pairs will be omitted. Thus the interpretation I is recursively de ned from these pairs as, I(x) = x and I(f(t1 ; : : : ; tn)) is I(ti ) if we have the pair (f(x1 ; : : : ; xn); xi), or fI (I(ti1 ); : : : ; I(ti )) if we have the pair (f(x1 ; : : : ; xn); fI (xi1 ; : : : ; xi )). It is easy to show that these interpretations preserve quasi-monotonicity and stability under substitutions (recall that the stability of quasi-orderings requires also the stability of its strict part). Proposition 3.6. Let I be an interpretation as de ned above. k
k
k
1. If B is quasi-monotonic then I is quasi-monotonic. 2. If B is stable under substitutions then I is stable under substitutions.
3.2 Building h i I;
Q
In this section we show how quasi-reduction pairs hI ; Q i can be obtained. First, as basic cases, we present two possible quasi-orderings Q ful lling the quasi-monotonicity requirement for a given I .
Proposition 3.7. Let I be a quasi-reduction quasi-ordering. Then
1. hI ; I i is a quasi-reduction pair. 2. Let F be a precedence on F and let FT be de ned as s FT t i top(s) F top(t). Then hI ; FT i is a quasi-reduction pair.
Now we show how to obtain new quasi-reduction pairs from one or several given quasi-reduction pairs. Hence, we can start by pairs as in the proposition above and then (repeatedly) apply the following properties to obtain more suitable quasi-reduction pairs. First we de ne what a renaming quasi-ordering is. 8
De nition 3.5. Let N be a mapping from F to F , called a renaming in F ,
and let fN denote N(f). We extend N to terms, obtaining a head renaming map, in the following way N(f(t1 ; : : : ; tm )) = fN (t1 ; : : : ; tm ) for every symbol f in F . Given and a renaming map in F , the renaming quasi-ordering N is de ned as s N t if and only if N(s) N(t) Note that the renaming is only applied to the head symbol of the term, and not to the arguments, and that it is not de ned for variables. Proposition 3.8. hI ; Qi is a quasi-reduction pair if 1. hI ; Q0 i is a quasi-reduction pair and Q is well-founded and stable under substitutions and Q0 Q . 2. hI ; Q0 i is a quasi-reduction pair and Q is NQ0 for some renaming map N in F . 3. hI ; Q i is a quasi-reduction pair for all i 2 f1 : : :ng and Q is (Q1 ; : : : ; Q )lex . i
n
Example 3.9. Let I be a quasi-reduction quasi-ordering, let F be a precedence on F and let N be a renaming map in F . By proposition 3.7.1, hI ; I i is a quasi-reduction pair and, by proposition 3.8.2, hI ; NI i is a quasi-reduction pair as well. Now since, by proposition 3.7.2, hI ; FT i is a quasi-reduction pair, we can conclude by proposition 3.8.3, that hI ; (NI ; FT )lex i is a quasi-reduction pair.
4 Examples In the examples we will always provide the quasi-reduction pair hI ; Q i and in some cases the details of the checking of mspo will be included. First we give the de nition of Q using I and then the de nition of I . In all cases we use the methods described in sections 3.1 and 3.2. Since for the basic quasi-ordering B of I we always use RPO, to avoid confusion, its precedence will be denoted by P (note that we can use other precedences to build Q ), and for simplicity we will directly give its strict part P .
Example 4.1. The following TRS comes from example 3.5 a f(b) h(a)
! ! !
b g(a) h(f(b))
In this case for Q we use NI with the renaming map N which is the identity except N(f) = F . For I we use the interpretation I generated by the pairs: (f(x); b) and (g(x); b); and RPO with the precedence h P F , F P g, F P a, a P b. Note that we have added to the signature the symbol F .
9
For the rst rule we have a I b since I(a) = a rpo b = I(b) and since N(a) = a and N(b) = b, we have a Q b, which implies a Qspo b by case 2 of SPO, and hence a mspo b. For the second rule I(f(b)) = b = I(g(a)), and hence f(b) I g(a). To prove f(b) Qspo g(a), since I(N(f(b))) = F(b) rpo b = I(N(g(a))), which implies f(b) Q g(a), by case 2 of SPO we only need to check f(b) Qspo a, which follows
again by case 2. For the third rule I(h(a)) = h(a) rpo h(b) = I(h(f(b))) and hence h(a) I h(f(b)). To prove h(a) Qspo h(f(b)), since I(N(h(a))) = h(a) rpo h(b) = I(N(h(f(b)))), which implies h(a) Q h(f(b)), by case 2 of SPO we only need to check h(a) Qspo f(b). This follows again by case 2, since I(N(h(a))) = h(a) rpo F(b) = I(N(f(b))), and hence h(a) Q f(b), and h(a) Qspo b follows as well by case 2. Example 4.2. In the following example of nested recursion (from [FZ93]) we use a precedence F as rst component in Q .
f(g(x)) f(h(x))
! !
g(f(f(x))) h(g(x))
We can take as Q the lexicographic combination (FT ; NI )lex with the precedence f F g and f F h, and the renaming map N which is the identity except N(f) = F. For I , the interpretation is given by (f(x); x) and (h(x); a); and the basic quasi-ordering is RPO generated by the empty precedence. Note that we have added to the signature the function symbol F and the constant a. We only show here that f(g(x)) mspo g(f(f(x))). First, we have f(g(x)) I g(f(f(x))), since I(f(g(x))) = g(x) = I(g(f(f(x)))). To prove f(g(x)) Qspo g(f(f(x))) we apply case 2, since f F g and hence f(g(x)) Q g(f(f(x))). For the recursive call f(g(x)) Qspo f(f(x)) we apply case 2 as well, since I(N(f(g(x)))) = F(g(x)) rpo F(x) = I(N(f(f(x)))) and hence f(g(x)) Q g(f(f(x))). For the recursive call f(g(x)) Qspo f(x) we apply again case 2, since I(N(f(g(x)))) = F(g(x)) rpo F(x) = I(N(f(x))), and for the recursive call f(g(x)) Qspo x we apply twice case 1.
Example 4.3. In the following non-simply terminating example (from [AG97]) we use a precedence F as second component in Q . le(0; y) le(s(x); 0) le(s(x); s(y)) minus(0; y) minus(s(x); y) if(true; s(x); y) if(false; s(x); y) quot(0; s(y)) quot(s(x); s(y))
! ! ! ! ! ! ! ! !
true false le(x; y) 0 if(le(s(x); y); s(x); y) 0 s(minus(x; y)) 0 s(quot(minus(x; y); s(y))) 10
We can take as Q the lexicographic combination (I ; FT )lex with the precedence le F true, le F false, minus F if and if F s. For I , the interpretation is given by (le(x; y); b), (true; b), (false; b), (if(x; y; z); y) and (minus(x; y); x); and the basic quasi-ordering is RPO generated by the precedence quot P s, s P b and s P 0. We only show here that minus(s(x); y) mspo if(le(s(x); y); s(x); y). First minus(s(x); y) I if(le(s(x); y); s(x); y) hlods since I(minus(s(x); y)) = s(x) = I(if(le(s(x); y); s(x); y)). To prove minus(s(x); y) Qspo if(le(s(x); y); s(x); y), we have minus(s(x); y) Q if(le(s(x); y); s(x); y), since minus F if. Then by case 2 of SPO we only need to check the recursive call minus(s(x); y) Qspo le(s(x); y), minus(s(x); y) Qspo s(x) and minus(s(x); y) Qspo y. The last two follow from case 1 of SPO. For the rst one we have minus(s(x); y) Q le(s(x); y), since I(minus(s(x); y) = s(x) rpo b = I(le(s(x); y)), and as before we can conclude by case 2.
5 Generalizing other termination proof methods As an application of the provided methods to generate suitable quasi-orderings we will show how two known termination proof methods, namely dummy elimination [FZ93] and the dependency pairs [AG97], can be seen as a particular instance of the monotonic semantic path ordering. Note that as a side e ect, this provides a new simple proof of their correctness. Finally we study Geser's proposal and show that it is strictly weaker than ours.
Q ,
5.1 Dummy elimination
Dummy elimination consists in a transformation which eliminates function symbols from a signature replacing them by a constant ( in our notation); terms and rewrite rules are transformed accordingly. The soundness result states that a TRS R, de ned over T (F [ Fa ; X ), where Fa contains symbols of arity 1 which are to be eliminated, is terminating if the transformed TRS E(R), de ned over T (F [ fg; X ), is terminating. Let Ia be the interpretation de ned by the pairs (g(x1 ; : : : ; xn); g ) for every symbol g to be eliminated (and the identity pair for all other symbols). The system E(R) is given by E(R) = fcap(l) ! u ju 2 fcap(r)g [ dec(r); l ! r 2 Rg where cap(s) = Ia (s) and dec(s) contains Ia (s0 ) for all s0 subterm of s just below a function symbol to be eliminated (for details see [FZ93]). For example, R = ff(f(x)) ! f(g(f(x)))g is transformed, via the elimination of g, in the system E(R) = ff(f(x)) ! f(); f(f(x)) ! f(x)g, where is the constant replacing g. 11
In the following we show that whenever termination of R can be shown using dummy elimination, we can nd a simple quasi-reduction pair hI ; Q i s.t. R is contained in MSPO. Let be the reduction ordering containing E(R), and be [ =. Then we de ne I by means of the interpretation Ia de ned above and as basic quasi-ordering, i.e. s I t i Ia (s) Ia (t). On the other hand, we de ne Q as s Q t i Ia (s)( [) Ia (t). Note that if is a simpli cation ordering like RPO, the subterm relation is already included. Now we show that hI ; Q i is a quasi-reduction pair. Since is a reduction ordering, is a quasi-reduction quasi-ordering, and, by proposition 3.6, I also is. As already said, since is a reduction ordering, we have that ( [) is a well-founded quasi-ordering stable under substitutions, and since I Q , by propositions 3.7.1 and 3.8.1, we conclude that hI ; Q i is a quasi-reduction pair. Finally we show that if E(R) is contained in then R is contained in mspo . Since cap(l) ! cap(r) 2 E(R) we have Ia (l) = cap(l) cap(r) = Ia (r) and hence l I r. To show that l Qspo r we prove that l Qspo r0 for all r0 subterm of r, by induction on jr0j. We have that u cap(r0) for some u 2 fcap(r)g [ dec(r). By de nition, cap(l) ! u 2 E(R), and hence cap(l) u. Consequently, since the strict part of ( [) includes ( [)+ , we have Ia (l) = cap(l)( 0 [)+ cap(r0) = Ia (r0 ) and thus, l Q r0. By induction hypothesis l Q spo ri for 0 0 Q 0 all arguments ri of r , and therefore, by case 2, we have l spo r .
5.2 Dependency Pairs
We consider here the plain dependency pair method, i.e. the method without using what is called the dependency graph and the usable rules restriction (see [AG97] for details). Note that theses re nements of the method are based on analyzing (operationally) the actual rewrite relation that one is considering. In this method for a given TRS R the signature F is split into two sets: the constructor symbols set C and the de ned symbols set D. De ned symbols are those heading the left hand side of a rule in R and constructor symbols are all others. Let N be a renaming map in F de ned as N(f) = F for all symbols f 2 D and the identity for the others. The dependency pairs of a rule l ! r in R is the set of pairs hN(l); N(r0)i for every subterm r0 of r headed by a symbol in D. For example, the rule f(f(x)) ! f(g(f(x))) has the dependency pairs hF(f(x)); F(g(f(x)))i and hF(f(x)); F(x)i. Then R is terminating if there is a quasi-reduction quasi-ordering such that for every rule l ! r we have l r and for every dependency pair hs; ti of l ! r we have s t. In the following we show that whenever termination of R can be shown using dependency pairs, we can a nd a simple quasi-reduction pair hI ; Q i s.t. R is contained in MSPO. Let F be a precedence on F where f F g i f 2 D (hence f F g i f 2 D and g 2 C). Then we take I as and Q as (FT ; NI )lex . By 12
propositions 3.7 and 3.8 ( 2 and 3), we have that hI ; Q i is a quasi-reduction pair. Finally we show that R is contained in mspo . We have directly that l I r since l r in the dependency pairs proof. To show that l Qspo r we prove that l Qspo r0 for all r0 subterm of r, by induction on jr0j. If top(r0) 2 C then top(l) F top(r0) and hence l Q r0 . Otherwise top(l) F top(r0) and there is a dependency pair hN(l); N(r0)i, s.t. N(l) N(r0 ), which implies l NI r0 , and hence l Q r0 . Since, by induction hypothesis l Qspo ri0 for all arguments ri0 of r0 , by case 2 of SPO, we have l Qspo r0 . Note that the precedence we have used to build Q is quite weak in its strict part. We believe that by using better precedences as a rst component of Q , e.g. adding strict comparisons between the de ned symbols, or adding another (di erent) precedence after NI in the lexicographic combination, we can capture easily part of the power of the dependency graph.
5.3 Geser's monotonic semantic path ordering
Now we analyze Geser's proposal [Ges92] for a monotonic SPO. We give here the strict part of his de nition. De nition 5.1. where Let Q be a quasi-reduction quasi-ordering. s G t i s Qspo t and f(: : : ; s; : : :) Q f(: : : ; s; : : :) for all f 2 F Although this version is an important step in the right direction, it has two main weaknesses. First of all, the requirement of Q to be a quasi-reduction quasi-ordering is too strong. As shown in example 3.5, this makes SPO loose completeness, and hence Geser's proposal cannot be complete. Furthermore, in a more practical view, with such a Q neither the dummy elimination technique nor the dependency pair technique can be included (note that, for instance, the renaming mapping does not preserve quasi-monotonicity). On the other hand, with respect to eciency of an implementation of the method, a termination proof requires a huge number of comparisons with Q , since for every rule l ! r, every symbol f and every argument position of this symbol f(: : : ; l; : : :) Q f(: : : ; r; : : :) has to be checked. Note that instead we only have to check l I r. Now we show that G is included in mspo .
Lemma 5.1. Let Q be a quasi-reduction quasi-ordering. There exists some s.t. hI ; Q i is a quasi-reduction pair and G mspo . Proof. We take I as s I t i f(: : : ; s; : : :) Q f(: : : ; t; : : :) for all f 2 F . It
I ,
is obvious that in this case G mspo . Then, we have to prove that hI ; Q i is a quasi-reduction pair, that is, I is quasi-monotonic on Q , which follows by de nition of I , and I is a quasi-monotonic quasi-ordering stable under substitutions, which follows directly from the properties of Q . 13
6 Modularity In this section we present some modularity results for MSPO which are obtained applying known abstract sucient conditions ensuring modularity of termination. We consider disjoint unions of TRS's, i.e., systems that do not share any symbol, and constructor sharing unions of TRS's, i.e., systems which share only constructors. A TRS R is called terminating under non-deterministic collapses, denoted C" -terminating, if R [ fG(x; y) ! x; G(x; y) ! yg terminates for some new symbol G. For C"-termination we have the following results:
[Ohl94] C" -termination is a modular property for disjoint unions of TRS. [Gra94] C"-termination is a modular property for constructor-sharing unions of nite TRS.
Lemma 6.1. If B and R is included in mspo then R is C"-terminating. Note that since B includes the subterm relation and G is a new symbol, we have I(G(x; y)) = G(x; y) and hence G(x; y) I x and G(x; y) I y. Therefore, since by case 1 both rules are included in SPO, we can conclude that G(x; y) mspo x and G(x; y) mspo y.
Corollary 6.2. Let 1mspo and 2mspo be two MSPO whose basic orderings include the subterm relation and let R1 and R2 be TRS's that are included in and 2mspo respectively.
1mspo
If R1 and R2 are disjoint then R1 [ R2 is C" -terminating, and thus terminating. If R1 and R2 share only constructors and are nite then R1 [ R2 is C" terminating, and thus terminating.
7 Conclusion In this paper we have described a new ordering-based general method for proving termination of TRS's. MSPO is based on the well-known SPO, but unlike SPO it is monotonic, which makes it useful in practice. It is a complete method, i.e., it characterizes termination. The method generalizes, in a simple way, many known methods based on transformations. In the case of the dependency pairs method, which is by now one of the most successful general methods applied in practice, we can only show that we generalize it without the \usable rules" and \dependency graph" re nements. These kind of \operational" re nements do not t so well in our framework, but by using more elaborated quasi-orderings I and Q inside MSPO we believe that we can obtain similar results. In order to study the behavior of MSPO, we are developing a termination system based on this method. Currently this system can only check termination once the 14
ingredients for the MSPO are provided, but our aim is to fully automate the termination proofs, by using some heuristics for selecting ingredients to be tried. Besides its application to rst-order term rewriting, the fact that the method is de ned by means of orderings, for which the properties to be ful lled are wellknown, opens the door to other important classes like AC-rewriting (i.e., rewriting modulo associativity and commutativity axioms) or higher-order rewriting, for which the lack of general methods is more important. Finally, apart from the presented modularity results, we are studying other kinds of combinations, in particular the so called hierarchical combinations , since reusing termination proofs or proving the termination of the TRS by splitting it in di erent parts may be crucial in practice. Acknowledgments: We want to thank Pilar Nivela, Roberto Nieuwenhuis and Enno Ohlebusch for their help in the development of this work.
References [AG97] T. Arts and J. Giesl. Automatically proving termination where simpli cation orderings fail. In TAPSOFT: 7th International Joint Conference on Theory and Practice of Software Development (in CAAP), LNCS 1214, 1997. [BD86] L. Bachmair and N. Dershowitz. Commutation, transformation, and termination. 8th International Conference on Automated Deduction, LNCS 230, pag. 5{20, Oxford, England, 1986. [BL90] F. Bellegarde and P. Lescanne. Termination by completion. Applicable Algebra in Engineering, Communication and Computing, 1:79{96, 1990. [Der82] N. Dershowitz. Orderings for term-rewriting systems. Theoretical Computer Science, 17(3):279{301, 1982. [Der87] N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69{116, 1987. [DJ90] N. Dershowitz and J.-P. Jouannaud. Rewrite systems. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B: Formal Models and Semantics, chap. 6, pag. 244{320. Elsevier Science Publishers B.V., 1990. [FZ93] M. Ferreira and H. Zantema. Dummy elimination: making termination easier. 10th Int. Conf. on Fundamentals of Computation Theory, LNCS 965, pag. 243{252, 1995. [Ges92] A. Geser. On a monotonic semantic path ordering. Technical Report 9213, Ulmer Informatik-Berichte, Universitat Ulm, Ulm, Germany, 1992. 15
[Gra94] B. Gramlich. Generalized sucient conditions for modular termination of rewriting. Applicable Algebra in Engineering, Communication and Computing, 5:131{158, 1994. [KL80] S. Kamin and J.-J. Levy. Two generalizations of the recursive path ordering. Unpublished note, Dept. of Computer Science, Univ. of Illinois, Urbana, IL, 1980. [Klo92] J.W. Klop. Term rewriting systems. In S. Abramsky, D.M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, pages 1{116. Oxford University Press, 1992. [Kru60] J.B. Kruskal. Well-quasi-ordering, the Tree Theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210{225,1960. [Ohl94] E. Ohlebusch. On modularity of termination of term rewriting systems. Theoretical Computer Science, 136(2):333{360, 1994. [Ste95] J. Steinbach. Automatic termination proofs with transformation orderings. 6th Int. Conf. on Rewriting Techniques and Applications, LNCS 914, 11{25, Kaiserslautern, Germany, 1995. [Xi98] H. Xi. Towards automated termination proofs through freezing. In 9th Int. Conf. on Rewriting Techniques and Applications, LNCS 1379, pag. 271{285, Tsukuba, Japan, 1998. [Zan94] H. Zantema. Termination of term rewriting: interpretation and type elimination. Journal of Symbolic Computation, 17:23{50, 1994.
16
Abstract
Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination, i.e., a TRS is terminating if and only if its rules are included in some MSPO. Hence MSPO is a complete termination method. On the practical side, it can be easily automated using as ingredients standard interpretations and general-purpose orderings like RPO. This is shown to be a suciently powerful way to handle several non-trivial examples and to obtain methods like dummy elimination or dependency pairs as particular cases. Finally, we obtain some positive modularity results for termination based on MSPO.
1 Introduction Rewrite systems are sets of rules (directed equations) used to compute by repeatedly replacing parts of a given formula with equal ones until the simplest possible form is obtained. Depending on the kind of objects that are rewritten there are di erent kinds of rewrite systems, like string rewrite systems (Thue or semi-Thue systems) or term rewrite systems (TRS; see [DJ90, Klo92] for detailed surveys). Termination is a fundamental property for most applications of rewrite systems. Termination of a TRS is, in general, an undecidable property, even if it consists of a single rule. Termination of TRS can be proved by showing that the induced rewrite relation is included in a well-founded ordering on terms. If the ordering is also monotonic and stable under substitutions, i.e., a reduction ordering , then it suces to check that l r for every rule l ! r in the system. Monotonic orderings including the subterm relation are called simpli cation orderings, and its well-foundedness follows from Kruskal's theorem [Kru60]. Inside this class, path orderings, and in particular the recursive path ordering 1
(RPO) [Der82], have received a special attention (see [Der87, DJ90]). Unfortunately, although these orderings are simple and easy to use, they turn out, in many cases, to be a weak termination proving tool, as there are many TRS that are terminating but are not contained in any simpli cation ordering, i.e. they are not simply terminating. To avoid this problem, di erent transformation methods have been developed, e.g. [BD86, BL90, Zan94, FZ93, Ste95, Xi98]. By transforming the TRS into a set of ordering constraints, the dependency pairs method [AG97] has become a successful general technique for proving termination of (non-simply terminating) TRS. As an alternative to transformation methods, more powerful term orderings can be used. Due to its simplicity, the Semantic Path Ordering (SPO) ([KL80]) becomes a potential well-known candidate: in SPO the scheme of RPO is generalized by replacing the precedence on function symbols by any (well-founded) underlying (quasi-)ordering involving the whole term and not only its head symbol. Although the simplicity of the presentation is kept, this makes the ordering much more powerful. In fact, for every terminating TRS there is some SPO that includes its rewrite relation. Unfortunately, SPO is not so useful in practice. Due to the generalization, the monotonicity property is lost, even if the underlying ordering is monotonic. Hence, it is not sucient to check that the rules are included in the ordering to ensure termination, since this does not imply that each rewrite step is inside the ordering. In order to ensure termination of a TRS R, the user is responsible for proving (by hand) monotonicity restricted to all terms s and t such that s rewrites to t with R in one step. In this paper we present a monotonic version of the SPO. On the one hand, this monotonic semantic path ordering (MSPO) is still very powerful for proving termination of TRS: on the theoretical side, it also characterizes termination, i.e. a TRS is terminating if and only if the rules are included in some MSPO; and on the practical side, it generalizes most of the (automatable) termination proof methods. On the other hand, termination can be automatically checked once the ingredients, i.e. the underlying (base) quasi-orderings, of the MSPO are provided. The rst and only other, as far as we know, monotonic version of SPO is due to Geser [Ges92]. On the one hand, this proposal is not as general as ours (in fact, as we will show, it does not characterize termination), and on the other hand, it is less suitable for practical implementations. In section 5.3 a detailed comparison with this work is provided. We are not only interested in using MSPO for checking termination, but also for proving termination automatically. Since we cannot expect to automatically generate an adequate MSPO for a given TRS whenever it exists, we have studied particular classes of underlying quasi-orderings, which can be automatically generated. As a hint of the power of the resulting family of MSPO's, it is shown that some known methods like dummy elimination [FZ93] or dependency pairs [AG97] (without the dependency graph re nement) are particular instances. 2
Using these classes of underlying quasi-orderings the termination proofs of several non-simply terminating, but terminating, term rewriting systems can be fully automated. A rst system based on MSPO has been developed by which many examples, including the ones in the paper, have been checked. The software and examples are available at www.lsi.upc.es/albert. Additionally, applying known abstract sucient conditions ensuring modularity of termination [Ohl94, Gra94], modularity results for termination based on MSPO are obtained. In particular, for TRS proved terminating using MSPO with the aforementioned classes of underlying quasi-orderings, termination is proved to be modular for disjoint systems and for nite constructor-sharing systems. Note that these modularity properties are crucial for many practical applications of automatic termination proof systems. Formal de nitions and basic tools are introduced in section 2. In section 3 we present and study the monotonic semantic path ordering. Section 4 is devoted to examples. Other termination methods are analyzed in section 5. In section 6 modularity results are presented. Some conclusions are given in section 7.
2 Preliminaries In the following we consider that F is a set of function symbols, X a set of variables and T (F ; X ) is the set of terms built from F and X . Let s and t be arbitrary terms in T (F ; X ), let f be a function symbol in F and let be a substitution. A (strict partial) ordering is a transitive irre exive relation. It is monotonic if s t implies f(: : :s : : :) f(: : :t : : :), and stable under substitution if s t implies s t. Monotonic orderings that are stable under substitutions are called rewrite orderings. A reduction ordering is a rewrite ordering that is well-founded : there are no in nite sequences t1 t2 : :: The re exive-transitive closure of a binary relation ! is denoted by ! and +. the transitive closure by ! A term rewrite system (TRS) is a (possibly in nite) set of rules l ! r where l and r are terms. Given a TRS R, s rewrites to t with R, denoted by s !R t, if there is some rule l ! r in R, sjp = l for some position p and substitution and t = s[r]p . A TRS R is terminating if there exists no in nite sequence + of any terminating TRS is a t1 !R t2 !R : : : Thus, transitive closure ! R reduction ordering. Furthermore, reduction orderings characterize termination of TRS. Theorem 2.1. A rewrite system R is terminating if and only if all rules are contained in a reduction ordering , i.e., l r for every l ! r 2 R. Another interesting property of reduction orderings is that they can be combined with the subterm relation without loosing well-foundedness. However, note that in general, monotonicity will be lost, since we only add the subterm relation, and not its monotonic closure.
Proposition 2.2. If is a reduction ordering then [ is well-founded. 3
Given a relation , the multiset extension of on nite multisets, denoted by , is de ned as the smallest transitive relation containing X [ fsg X [ ft1; : : : ; tng if s ti for all i 2 f1 : : :ng If is a well-founded ordering on terms then is a well-founded ordering on nite multisets of terms. A quasi-ordering is a transitive and re exive binary relation. Its inverse is denoted by . Its strict part is the strict ordering n (i.e, s t i s t and s 6 t). Its equivalence is \ . Note that is the disjoint union of and , and that if = denotes syntactic equality then [ = is a quasi-ordering whose strict part is . Notation: In the remainder of this paper, (possibly with subscripts) will always denote a quasi-ordering. The following de nitions for will be used: 1. is monotonic if is. 2. is well-founded if is. 3. is stable under substitutions if is and s t whenever s t. 4. is quasi-monotonic if f(: : : ; s; : : :) f(: : : ; t; : : :) whenever s t. 5. is a quasi-reduction quasi-ordering if it ful lls the above properties 2, 3 and 4. Note that if is quasi-monotonic then is not necessarily monotonic. The lexicographic combination of 1 ; : : : ; n , denoted by (1 ; : : : ; n )lex , is de ned as usual as, s(1 ; : : : ; n)lex t i either s i t for some i and s j t for all j < i, or s i t for all i. If all i are well-founded (stable under substitutions) then its lexicographic combination also is. A precedence F is a well-founded quasi-ordering on F . Given a precedence F , the recursive path ordering (RPO), denoted by rpo , is de ned recursively as follows: s = f(s1 ; : : : ; sm ) rpo t i 1. si rpo t, for some i = 1; : : : ; m, or 2. t = g(t1 ; : : : ; tn) with f F g and s rpo ti for all i = 1 : : :n or 3. t = g(t1 ; : : : ; tn) with f F g and fs1; : : : ; sm g rpo ft1; : : : ; tng.
3 A monotonic Semantic Path Ordering Let us now recall the de nition of the semantic path ordering. Then we will analyze an example of non-monotonicity, which will provide the intuition behind the monotonic version we will propose. 4
De nition 3.1. Given Q , called the underlying (or base) quasi-ordering, the semantic path ordering (SPO) [KL80], denoted as Qspo , is de ned as
s = f(s1 ; : : : ; sm ) Qspo t i 1. si Qspo t, for some i = 1; : : : ; m, or 2. s Q t = g(t1 ; : : : ; tn) and s Qspo ti for all i = 1; : : : ; n, or 3. s Q t = g(t1 ; : : : ; tn) and fs1 ; : : : ; sm g Qspo ft1; : : : ; tng, The semantic path ordering ful lls the following property.
Lemma 3.1 ([KL80]). If Q is well-founded and stable under substitutions then Qspo is well-founded and stable under substitutions.
But, as said before, the semantic path ordering is, in general, non-monotonic, even when Q is quasi-monotonic (in fact, the same problem appears if Q is monotonic). This is shown in the following example.
Example 3.2. Consider the following quasi-ordering Q de ned for all terms
s and t as: (i) f(s) Q g(t); (ii) f(s) Q f(t) i s Q t; and (iii) g(s) Q g(t),
and re exive in variables and constants. This quasi-ordering is well-founded, since its strict part is u[f(s)] Q u[g(t)] for all terms s and t and contexts u containing only the symbol f (note that the length of any decreasing sequence is at most the number of f 's above the rst g symbol of the initial term in the sequence); it is stable under substitutions and quasi-monotonic. However, the induced SPO is not monotonic: by case 1 of SPO we have g(f(a)) Qspo f(a), but by adding the context f([]) onto both terms we have f(g(f(a))) 6 Qspo f(f(a)) (in fact, in this case we can even prove f(f(a)) Qspo f(g(f(a)))).
Analyzing this example we can observe that, even if Q is quasi-monotonic (or monotonic), since in case 1 of SPO we do not require s Q t, it may happen that by adding some context u, if we do not have u[s] Q u[t], we cannot apply any of the cases. Then, in order to ensure monotonicity of Qspo , we need to be sure that u[s] Q u[t] for any context. A way to obtain this is to require always s Q t, provided that Q is quasi-monotonic, that is s t i and only if s Q t and s Qspo t. However, requiring always s Q t can be a bit to strong and, in fact, from the example above we only need to have u[s] Q u[t] for every context u. Hence instead of requiring s Q t, we will ask for something weaker that ensures u[s] Q u[t]. De nition 3.2. We say that I is quasi-monotonic on Q (or Q is quasimonotonic wrt. I ) if s I t implies f(: : :s : : :) Q f(: : : t : : :) for all terms s and t and function symbols f. 5
De nition 3.3. A pair hI ; Qi is called a quasi-reduction pair if I is quasi-
monotonic, Q is well-founded, both are stable under substitutions and I is quasi-monotonic on Q . Now we de ne the monotonic semantic path ordering (MSPO): De nition 3.4. Let hI ; Qi be a quasi-reduction pair. The corresponding monotonic semantic path ordering, denoted by mspo , is de ned as: s mspo t if and only if s I t and s Qspo t for all terms s and t.
Theorem 3.3.
mspo
is a reduction ordering.
and Qspo is well-founded. Stability under substitutions follows from the stability of I and Q spo . For monotonicity we have to show that s mspo t implies f(: : : ; s; : : :) mspo f(: : : ; t; : : :), i.e. f(: : : ; s; : : :) I f(: : : ; t; : : :) and also f(: : : ; s; : : :) Qspo f(: : : ; t; : : :), for all terms s and t and function symbols f. By de nition of mspo , s mspo t implies s I t. Hence, by the quasimonotonicity of I , we have f(: : : ; s; : : :) I f(: : : ; t; : : :). On the other hand, by quasi-monotonicity of I on Q , we have f(: : : ; s; : : :) Q f(: : : ; t; : : :). Then, by de nition of mspo , s mspo t implies s Qspo t and, hence f: : :s : : : g Q Q spo f: : :t : : : g, which implies f(: : : ; s; : : :) spo f(: : : ; t; : : :) by case (3). The previous theorem shows that mspo provides a correct method for proving termination of term rewriting systems. The following result shows that it is also a complete method, i.e., for any terminating term rewriting there is a monotonic semantic path ordering that includes its rules. Therefore mspo characterizes termination. Theorem 3.4. A rewrite system R is terminating if and only if there exists some quasi-reduction pair hI ; Q i, s.t. l mspo r for every rule l ! r in R. Proof. The right to left implication follows from Theorem 3.3. For the left to right, we will build an appropriate pair hI ; Q i for any terminating TRS R. Let I be !R and let Q be (!R [) . By de nition of rewriting I is quasi-monotonic and both I and Q are stable under substitutions. By termination of R and Property 2.2, the strict part of Q , i.e. (!R [)+ , is well-founded. Finally, since I Q , and I is quasi-monotonic, we have that I is quasi-monotonic on Q . Therefore, since hI ; Q i is a quasi-reduction pair, we only have to prove that l mspo r for every rule l ! r in R. By de nition, we have l I r and l Q r, and, moreover, for every subterm r0 of r we have l Q r0, since l !R r r0 . Therefore, we have l I r and l Qspo r, by repeatedly applying case 2, which implies l mspo r.
Proof. Well-foundedness follows from the fact that mspo
6
Q spo
The quasi-reduction pair condition is quite tight: as shown in the following example, if Q is required to be a quasi-reduction quasi-ordering then even SPO does not characterize termination, i.e. there are terminating sets of rules that are not included in any SPO with an underlying quasi-reduction quasi-ordering.
Example 3.5. There is no quasi-reduction quasi-ordering following TRS is included in the generated SPO.
a f(b) h(a)
! ! !
Q
such that the
b g(a) h(f(b))
First, note that the TRS is terminating (it will be shown using MSPO in example 4.1). Now we show that these rules cannot be included in SPO with a quasi-reduction quasi-ordering Q . To this end it is important to remark that in SPO to have c Qspo t for some constant c and term t we need c Q t0 for all t0 subterm of t. By de nition we need a Q b to be able to include the rst rule. Then to include the second one we need f(b) Q g(a) and f(b) Q a, in order to conclude by case 2 rst and then by case 2 or 3 for the recursive call with f(b) Qspo a, since any other possibility requires b Q a, which contradicts the rst assumption. Now we have (i) a Q b, (ii) f(b) Q g(a) and (iii) f(b) Q a. We proceed with the third rule. If a Qspo h(f(b)) we need a Q f(b), which contradicts assumption (iii). Otherwise we need h(a) Q h(f(b)). By assumption (iii) and quasi-monotonicity of Q , we have h(f(b)) Q h(a), which implies h(a) 6 Q h(f(b)). Therefore we have to apply case 3 of SPO, and hence we need a Qspo f(b), which requires a Q f(b) contradicting assumption (iii).
Theorem 3.4 shows the theoretical power of the ordering as a termination proof method, but in order to make it more useful in practice in the following two sections we will show general methods to obtain quasi-reduction pairs hI ; Q i.
3.1 Building
I
We consider I obtained by combining an interpretation I on terms with some quasi-reduction quasi-ordering B , called the basic quasi-ordering, which can be obtained by well-known practical general-purpose methods like the path orderings or polynomial interpretations. For the interpretation I, as a general property, we require the preservation of the quasi-monotonicity and stability under substitutions of the basic quasi-ordering. Below some particular such interpretations which are suitable for practical applications are provided. These interpretations are not original; they have been used in more ad-hoc ways in many di erent termination methods, but here they are all placed in the same framework. We will consider interpretations as mappings from terms to terms I : T (F ; X ) ! T (F 0 ; X 0), although, of course, we can also consider interpretation from terms to multisets of terms or any other domain provided that the required properties are ful lled. 7
From now on we consider that I is de ned as follows s I t if and only if I(s) B I(t) Note that this does not imply any loss of generality since I can be the identity mapping. Additionally, it follows that s I t if and only if I(s) B I(t). For any I, if B is a quasi-ordering then I is a quasi-ordering which is wellfounded if B also is. For quasi-monotonicity and stability under substitutions this is not always the case. Let us give some examples of interpretations that do preserve this properties. Each symbol f can be interpreted either by a projection on a single argument, denoted by the pair (f(x1 ; : : : ; xn); xi), or else by a function symbol fI applied to an arbitrary sequence obtained from the arguments of f, denoted by the pair (f(x1 ; : : : ; xn); fI (xi1 ; : : : ; xi )), for some k 0 and i1 ; : : : ; ik 2 f1; : : : ; ng. Additionally we consider I to be the identity for variables (although it can be any bijection). We assume that there is only one pair for each symbol. Usually the identity pairs will be omitted. Thus the interpretation I is recursively de ned from these pairs as, I(x) = x and I(f(t1 ; : : : ; tn)) is I(ti ) if we have the pair (f(x1 ; : : : ; xn); xi), or fI (I(ti1 ); : : : ; I(ti )) if we have the pair (f(x1 ; : : : ; xn); fI (xi1 ; : : : ; xi )). It is easy to show that these interpretations preserve quasi-monotonicity and stability under substitutions (recall that the stability of quasi-orderings requires also the stability of its strict part). Proposition 3.6. Let I be an interpretation as de ned above. k
k
k
1. If B is quasi-monotonic then I is quasi-monotonic. 2. If B is stable under substitutions then I is stable under substitutions.
3.2 Building h i I;
Q
In this section we show how quasi-reduction pairs hI ; Q i can be obtained. First, as basic cases, we present two possible quasi-orderings Q ful lling the quasi-monotonicity requirement for a given I .
Proposition 3.7. Let I be a quasi-reduction quasi-ordering. Then
1. hI ; I i is a quasi-reduction pair. 2. Let F be a precedence on F and let FT be de ned as s FT t i top(s) F top(t). Then hI ; FT i is a quasi-reduction pair.
Now we show how to obtain new quasi-reduction pairs from one or several given quasi-reduction pairs. Hence, we can start by pairs as in the proposition above and then (repeatedly) apply the following properties to obtain more suitable quasi-reduction pairs. First we de ne what a renaming quasi-ordering is. 8
De nition 3.5. Let N be a mapping from F to F , called a renaming in F ,
and let fN denote N(f). We extend N to terms, obtaining a head renaming map, in the following way N(f(t1 ; : : : ; tm )) = fN (t1 ; : : : ; tm ) for every symbol f in F . Given and a renaming map in F , the renaming quasi-ordering N is de ned as s N t if and only if N(s) N(t) Note that the renaming is only applied to the head symbol of the term, and not to the arguments, and that it is not de ned for variables. Proposition 3.8. hI ; Qi is a quasi-reduction pair if 1. hI ; Q0 i is a quasi-reduction pair and Q is well-founded and stable under substitutions and Q0 Q . 2. hI ; Q0 i is a quasi-reduction pair and Q is NQ0 for some renaming map N in F . 3. hI ; Q i is a quasi-reduction pair for all i 2 f1 : : :ng and Q is (Q1 ; : : : ; Q )lex . i
n
Example 3.9. Let I be a quasi-reduction quasi-ordering, let F be a precedence on F and let N be a renaming map in F . By proposition 3.7.1, hI ; I i is a quasi-reduction pair and, by proposition 3.8.2, hI ; NI i is a quasi-reduction pair as well. Now since, by proposition 3.7.2, hI ; FT i is a quasi-reduction pair, we can conclude by proposition 3.8.3, that hI ; (NI ; FT )lex i is a quasi-reduction pair.
4 Examples In the examples we will always provide the quasi-reduction pair hI ; Q i and in some cases the details of the checking of mspo will be included. First we give the de nition of Q using I and then the de nition of I . In all cases we use the methods described in sections 3.1 and 3.2. Since for the basic quasi-ordering B of I we always use RPO, to avoid confusion, its precedence will be denoted by P (note that we can use other precedences to build Q ), and for simplicity we will directly give its strict part P .
Example 4.1. The following TRS comes from example 3.5 a f(b) h(a)
! ! !
b g(a) h(f(b))
In this case for Q we use NI with the renaming map N which is the identity except N(f) = F . For I we use the interpretation I generated by the pairs: (f(x); b) and (g(x); b); and RPO with the precedence h P F , F P g, F P a, a P b. Note that we have added to the signature the symbol F .
9
For the rst rule we have a I b since I(a) = a rpo b = I(b) and since N(a) = a and N(b) = b, we have a Q b, which implies a Qspo b by case 2 of SPO, and hence a mspo b. For the second rule I(f(b)) = b = I(g(a)), and hence f(b) I g(a). To prove f(b) Qspo g(a), since I(N(f(b))) = F(b) rpo b = I(N(g(a))), which implies f(b) Q g(a), by case 2 of SPO we only need to check f(b) Qspo a, which follows
again by case 2. For the third rule I(h(a)) = h(a) rpo h(b) = I(h(f(b))) and hence h(a) I h(f(b)). To prove h(a) Qspo h(f(b)), since I(N(h(a))) = h(a) rpo h(b) = I(N(h(f(b)))), which implies h(a) Q h(f(b)), by case 2 of SPO we only need to check h(a) Qspo f(b). This follows again by case 2, since I(N(h(a))) = h(a) rpo F(b) = I(N(f(b))), and hence h(a) Q f(b), and h(a) Qspo b follows as well by case 2. Example 4.2. In the following example of nested recursion (from [FZ93]) we use a precedence F as rst component in Q .
f(g(x)) f(h(x))
! !
g(f(f(x))) h(g(x))
We can take as Q the lexicographic combination (FT ; NI )lex with the precedence f F g and f F h, and the renaming map N which is the identity except N(f) = F. For I , the interpretation is given by (f(x); x) and (h(x); a); and the basic quasi-ordering is RPO generated by the empty precedence. Note that we have added to the signature the function symbol F and the constant a. We only show here that f(g(x)) mspo g(f(f(x))). First, we have f(g(x)) I g(f(f(x))), since I(f(g(x))) = g(x) = I(g(f(f(x)))). To prove f(g(x)) Qspo g(f(f(x))) we apply case 2, since f F g and hence f(g(x)) Q g(f(f(x))). For the recursive call f(g(x)) Qspo f(f(x)) we apply case 2 as well, since I(N(f(g(x)))) = F(g(x)) rpo F(x) = I(N(f(f(x)))) and hence f(g(x)) Q g(f(f(x))). For the recursive call f(g(x)) Qspo f(x) we apply again case 2, since I(N(f(g(x)))) = F(g(x)) rpo F(x) = I(N(f(x))), and for the recursive call f(g(x)) Qspo x we apply twice case 1.
Example 4.3. In the following non-simply terminating example (from [AG97]) we use a precedence F as second component in Q . le(0; y) le(s(x); 0) le(s(x); s(y)) minus(0; y) minus(s(x); y) if(true; s(x); y) if(false; s(x); y) quot(0; s(y)) quot(s(x); s(y))
! ! ! ! ! ! ! ! !
true false le(x; y) 0 if(le(s(x); y); s(x); y) 0 s(minus(x; y)) 0 s(quot(minus(x; y); s(y))) 10
We can take as Q the lexicographic combination (I ; FT )lex with the precedence le F true, le F false, minus F if and if F s. For I , the interpretation is given by (le(x; y); b), (true; b), (false; b), (if(x; y; z); y) and (minus(x; y); x); and the basic quasi-ordering is RPO generated by the precedence quot P s, s P b and s P 0. We only show here that minus(s(x); y) mspo if(le(s(x); y); s(x); y). First minus(s(x); y) I if(le(s(x); y); s(x); y) hlods since I(minus(s(x); y)) = s(x) = I(if(le(s(x); y); s(x); y)). To prove minus(s(x); y) Qspo if(le(s(x); y); s(x); y), we have minus(s(x); y) Q if(le(s(x); y); s(x); y), since minus F if. Then by case 2 of SPO we only need to check the recursive call minus(s(x); y) Qspo le(s(x); y), minus(s(x); y) Qspo s(x) and minus(s(x); y) Qspo y. The last two follow from case 1 of SPO. For the rst one we have minus(s(x); y) Q le(s(x); y), since I(minus(s(x); y) = s(x) rpo b = I(le(s(x); y)), and as before we can conclude by case 2.
5 Generalizing other termination proof methods As an application of the provided methods to generate suitable quasi-orderings we will show how two known termination proof methods, namely dummy elimination [FZ93] and the dependency pairs [AG97], can be seen as a particular instance of the monotonic semantic path ordering. Note that as a side e ect, this provides a new simple proof of their correctness. Finally we study Geser's proposal and show that it is strictly weaker than ours.
Q ,
5.1 Dummy elimination
Dummy elimination consists in a transformation which eliminates function symbols from a signature replacing them by a constant ( in our notation); terms and rewrite rules are transformed accordingly. The soundness result states that a TRS R, de ned over T (F [ Fa ; X ), where Fa contains symbols of arity 1 which are to be eliminated, is terminating if the transformed TRS E(R), de ned over T (F [ fg; X ), is terminating. Let Ia be the interpretation de ned by the pairs (g(x1 ; : : : ; xn); g ) for every symbol g to be eliminated (and the identity pair for all other symbols). The system E(R) is given by E(R) = fcap(l) ! u ju 2 fcap(r)g [ dec(r); l ! r 2 Rg where cap(s) = Ia (s) and dec(s) contains Ia (s0 ) for all s0 subterm of s just below a function symbol to be eliminated (for details see [FZ93]). For example, R = ff(f(x)) ! f(g(f(x)))g is transformed, via the elimination of g, in the system E(R) = ff(f(x)) ! f(); f(f(x)) ! f(x)g, where is the constant replacing g. 11
In the following we show that whenever termination of R can be shown using dummy elimination, we can nd a simple quasi-reduction pair hI ; Q i s.t. R is contained in MSPO. Let be the reduction ordering containing E(R), and be [ =. Then we de ne I by means of the interpretation Ia de ned above and as basic quasi-ordering, i.e. s I t i Ia (s) Ia (t). On the other hand, we de ne Q as s Q t i Ia (s)( [) Ia (t). Note that if is a simpli cation ordering like RPO, the subterm relation is already included. Now we show that hI ; Q i is a quasi-reduction pair. Since is a reduction ordering, is a quasi-reduction quasi-ordering, and, by proposition 3.6, I also is. As already said, since is a reduction ordering, we have that ( [) is a well-founded quasi-ordering stable under substitutions, and since I Q , by propositions 3.7.1 and 3.8.1, we conclude that hI ; Q i is a quasi-reduction pair. Finally we show that if E(R) is contained in then R is contained in mspo . Since cap(l) ! cap(r) 2 E(R) we have Ia (l) = cap(l) cap(r) = Ia (r) and hence l I r. To show that l Qspo r we prove that l Qspo r0 for all r0 subterm of r, by induction on jr0j. We have that u cap(r0) for some u 2 fcap(r)g [ dec(r). By de nition, cap(l) ! u 2 E(R), and hence cap(l) u. Consequently, since the strict part of ( [) includes ( [)+ , we have Ia (l) = cap(l)( 0 [)+ cap(r0) = Ia (r0 ) and thus, l Q r0. By induction hypothesis l Q spo ri for 0 0 Q 0 all arguments ri of r , and therefore, by case 2, we have l spo r .
5.2 Dependency Pairs
We consider here the plain dependency pair method, i.e. the method without using what is called the dependency graph and the usable rules restriction (see [AG97] for details). Note that theses re nements of the method are based on analyzing (operationally) the actual rewrite relation that one is considering. In this method for a given TRS R the signature F is split into two sets: the constructor symbols set C and the de ned symbols set D. De ned symbols are those heading the left hand side of a rule in R and constructor symbols are all others. Let N be a renaming map in F de ned as N(f) = F for all symbols f 2 D and the identity for the others. The dependency pairs of a rule l ! r in R is the set of pairs hN(l); N(r0)i for every subterm r0 of r headed by a symbol in D. For example, the rule f(f(x)) ! f(g(f(x))) has the dependency pairs hF(f(x)); F(g(f(x)))i and hF(f(x)); F(x)i. Then R is terminating if there is a quasi-reduction quasi-ordering such that for every rule l ! r we have l r and for every dependency pair hs; ti of l ! r we have s t. In the following we show that whenever termination of R can be shown using dependency pairs, we can a nd a simple quasi-reduction pair hI ; Q i s.t. R is contained in MSPO. Let F be a precedence on F where f F g i f 2 D (hence f F g i f 2 D and g 2 C). Then we take I as and Q as (FT ; NI )lex . By 12
propositions 3.7 and 3.8 ( 2 and 3), we have that hI ; Q i is a quasi-reduction pair. Finally we show that R is contained in mspo . We have directly that l I r since l r in the dependency pairs proof. To show that l Qspo r we prove that l Qspo r0 for all r0 subterm of r, by induction on jr0j. If top(r0) 2 C then top(l) F top(r0) and hence l Q r0 . Otherwise top(l) F top(r0) and there is a dependency pair hN(l); N(r0)i, s.t. N(l) N(r0 ), which implies l NI r0 , and hence l Q r0 . Since, by induction hypothesis l Qspo ri0 for all arguments ri0 of r0 , by case 2 of SPO, we have l Qspo r0 . Note that the precedence we have used to build Q is quite weak in its strict part. We believe that by using better precedences as a rst component of Q , e.g. adding strict comparisons between the de ned symbols, or adding another (di erent) precedence after NI in the lexicographic combination, we can capture easily part of the power of the dependency graph.
5.3 Geser's monotonic semantic path ordering
Now we analyze Geser's proposal [Ges92] for a monotonic SPO. We give here the strict part of his de nition. De nition 5.1. where Let Q be a quasi-reduction quasi-ordering. s G t i s Qspo t and f(: : : ; s; : : :) Q f(: : : ; s; : : :) for all f 2 F Although this version is an important step in the right direction, it has two main weaknesses. First of all, the requirement of Q to be a quasi-reduction quasi-ordering is too strong. As shown in example 3.5, this makes SPO loose completeness, and hence Geser's proposal cannot be complete. Furthermore, in a more practical view, with such a Q neither the dummy elimination technique nor the dependency pair technique can be included (note that, for instance, the renaming mapping does not preserve quasi-monotonicity). On the other hand, with respect to eciency of an implementation of the method, a termination proof requires a huge number of comparisons with Q , since for every rule l ! r, every symbol f and every argument position of this symbol f(: : : ; l; : : :) Q f(: : : ; r; : : :) has to be checked. Note that instead we only have to check l I r. Now we show that G is included in mspo .
Lemma 5.1. Let Q be a quasi-reduction quasi-ordering. There exists some s.t. hI ; Q i is a quasi-reduction pair and G mspo . Proof. We take I as s I t i f(: : : ; s; : : :) Q f(: : : ; t; : : :) for all f 2 F . It
I ,
is obvious that in this case G mspo . Then, we have to prove that hI ; Q i is a quasi-reduction pair, that is, I is quasi-monotonic on Q , which follows by de nition of I , and I is a quasi-monotonic quasi-ordering stable under substitutions, which follows directly from the properties of Q . 13
6 Modularity In this section we present some modularity results for MSPO which are obtained applying known abstract sucient conditions ensuring modularity of termination. We consider disjoint unions of TRS's, i.e., systems that do not share any symbol, and constructor sharing unions of TRS's, i.e., systems which share only constructors. A TRS R is called terminating under non-deterministic collapses, denoted C" -terminating, if R [ fG(x; y) ! x; G(x; y) ! yg terminates for some new symbol G. For C"-termination we have the following results:
[Ohl94] C" -termination is a modular property for disjoint unions of TRS. [Gra94] C"-termination is a modular property for constructor-sharing unions of nite TRS.
Lemma 6.1. If B and R is included in mspo then R is C"-terminating. Note that since B includes the subterm relation and G is a new symbol, we have I(G(x; y)) = G(x; y) and hence G(x; y) I x and G(x; y) I y. Therefore, since by case 1 both rules are included in SPO, we can conclude that G(x; y) mspo x and G(x; y) mspo y.
Corollary 6.2. Let 1mspo and 2mspo be two MSPO whose basic orderings include the subterm relation and let R1 and R2 be TRS's that are included in and 2mspo respectively.
1mspo
If R1 and R2 are disjoint then R1 [ R2 is C" -terminating, and thus terminating. If R1 and R2 share only constructors and are nite then R1 [ R2 is C" terminating, and thus terminating.
7 Conclusion In this paper we have described a new ordering-based general method for proving termination of TRS's. MSPO is based on the well-known SPO, but unlike SPO it is monotonic, which makes it useful in practice. It is a complete method, i.e., it characterizes termination. The method generalizes, in a simple way, many known methods based on transformations. In the case of the dependency pairs method, which is by now one of the most successful general methods applied in practice, we can only show that we generalize it without the \usable rules" and \dependency graph" re nements. These kind of \operational" re nements do not t so well in our framework, but by using more elaborated quasi-orderings I and Q inside MSPO we believe that we can obtain similar results. In order to study the behavior of MSPO, we are developing a termination system based on this method. Currently this system can only check termination once the 14
ingredients for the MSPO are provided, but our aim is to fully automate the termination proofs, by using some heuristics for selecting ingredients to be tried. Besides its application to rst-order term rewriting, the fact that the method is de ned by means of orderings, for which the properties to be ful lled are wellknown, opens the door to other important classes like AC-rewriting (i.e., rewriting modulo associativity and commutativity axioms) or higher-order rewriting, for which the lack of general methods is more important. Finally, apart from the presented modularity results, we are studying other kinds of combinations, in particular the so called hierarchical combinations , since reusing termination proofs or proving the termination of the TRS by splitting it in di erent parts may be crucial in practice. Acknowledgments: We want to thank Pilar Nivela, Roberto Nieuwenhuis and Enno Ohlebusch for their help in the development of this work.
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