The theory of total unary rpo is decidable. State University of New York at Albany, Albany, NY 12222, USA.

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1 The theory of total unary rpo is deidale Paliath Narendran 1 and Mihael Rusinowith 2 1 Institute of Programming and Logis, Department of Computer Siene, State University of New York at Alany, Alany, NY 12222, USA dran@s.alany.edu 2 LORIA-INRIA Lorraine, 615, rue du jardin otanique, BP 101, Villers les Nany edex, Frane rusi@loria.fr The Reursive Path Ordering (rpo) is a syntati ordering on terms that has een widely used for proving termination of term-rewriting systems [7, 20]. How to omine term-rewriting with ordered resolution and paramodulation is now well-understood and it has een suessfully applied in many theorem-proving systems [11,16, 21]. In this setting an ordering suh as rpo is used oth to orient rewrite rules and to selet maximal literals to perform inferenes on. In order to further prune the searh spae the ordering requirements on onditional inferenes are etter handled when they are treated as onstraints [12, 18]. Typially a nonorientale equation s = t will e split as two onstrained rewrite rules: s! t j s > t and t! s j t > s. Suh onstrained rules are useless when the onstraint is unsatisale. Therefore it is important for the eieny of automated reasoning systems to investigate deision proedures for the theory of terms with ordering prediates. Other types of onstraints an e introdued too suh as disuniation onstraints [1]. It is often the ase that they an e expressed with ordering onstraints (although this might e ineient). We prove that the rst-order theory of the reursive path ordering is deidale in the ase of unary signatures with total preedene. This solves a prolem that was mentioned as open in [6]. The result has to e ontrasted with the undeidaility results of the lexiographi path ordering [6] for the ase of symols with arity 2 and total preedene and for the ase of unary signatures with partial preedene. We reall that lexiographi path ordering (lpo) and the reursive path ordering and many other orderings suh as [13, 10] oinide in the unary ase. Among the positive results it is known that the existential theory of total lpo is deidale [3, 17]. The same result holds for the ase of total rpo [8,15]. The proof tehnique we use for our deidaility result might e interesting y itself. It relies on enoding of words as trees and then on uilding a tree automaton to reognize the rpo relation. Key words: Reursive path ordering, rst-order theory, ground reduiility, tree automata, ordered rewriting. 1 The reursive path ordering on words We assume that A is a nite alphaet, A the set of words on A and is the empty word. We shall often identify a letter with the orresponding word of length one. There is a orrespondane etween words and terms on a unary alphaet: every letter an e onsidered as a unary funtion and every word a 1 ; a 2 ; : : : ; a n an e onsidered as a term a 1 (a 2 (: : :a n (x) : : :)) where x is an element of the set of variales X. The Reursive Path Ordering (rpo) originally introdued y Dershowitz [7] is dened as follows on words: Given a nite total preedene A on A, if and only if one of the following holds: s A rpo t

2 1. s 6= and t = 2. s = as 0 and t = t 0 and either (a) a A and s A rpo t0 or () a = and s 0 A rpo t 0 or () A a and s 0 A rpo t where s 0 A rpo t is an areviation for s 0 A rpo t or s 0 = t. The following properties of rpo are well-known and their proofs an e found in the literature (e.g. [7]). We omit the supersript A when it is lear from ontext. Proposition 1. The relation rpo is antireexive, transitive, monotoni (i.e. aw rpo aw 0 if w rpo w 0 ), total, well-founded and has the suterm property (i.e. w rpo w 0 when w 0 is a proper suterm of w). When the preedene is not total, all properties ut totality of rpo remain valid. We now give a few properties that will e useful in the sequel. Lemma 1. If s rpo t then for all w 2 A, ws rpo wt. Proof: y indution on the length of w and y the monotoniity property. 2 Lemma 2. If a 2 A and w rpo w 0 then for all w 00 2 A suh that a rpo w 00, we have aw rpo w 00 aw 0. Proof: y indution on the length of w 00. If w 00 is empty it is ovious. If w 00 = u where 2 A, then y denition of rpo we must have a A and a rpo u. Sine u is shorter than w 00, y indution hypothesis we have aw rpo uaw 0. By the denition of rpo, we also have aw rpo uaw 0 and the result follows. 2 The next lemma shows that when terms s; t have the same numer of maximal symols a then the omparison an e done y onsidering only the rightmost (innermost) ourrene of a whose arguments are dierent in s and t. Lemma 3. Let w 1 ; w 2 ; : : : ; w k, u 1 ; u 2 ; : : : ; u k e two sequenes of words suh that eah letter in eah word is stritly smaller than a 2 A. If w rpo w 0 then we have: Proof: By indution hypothesis we an assume that: y the previous lemma we have: y the suterm property and transitivity we have: aw 1 aw 2 : : : aw k aw rpo au 1 au 2 : : : au k aw 0 aw 2 : : : aw k aw rpo au 2 : : :au k aw 0 aw 2 : : :aw k aw rpo u 1 au 2 : : : au k aw 0 w 1 aw 2 : : : aw k aw rpo u 1 au 2 : : : au k aw 0 The result follows y monotoniity. 2 We denote y max(w; A) the maximal letter of A that ours in word w and y mul(w; A) the numer of ourrenes of this letter in w. As a onsequene of the previous lemmas we get: Lemma 4. w rpo w 0 i one of the following holds: 1. max(w; A) max(w 0 ; A)

3 2. max(w; A) = max(w 0 ; A) and mul(w; A) > mul(w 0 ; A) 3. a = max(w; A) = max(w 0 ; A), mul(w; A) = mul(w 0 ; A), w = w 0 aw 1 aw 2 : : :aw k, w 0 = u 0 au 1 au 2 : : :au k and there exists 0 i k suh that w i rpo u i and for all j > i we have w j = u j. A rst idea would e to try to use word automata in order to reognize the relation rpo. However this is not possile. Consider A = fa; g with a. We introdue the produt alphaet A 2 = f(u; v) j u; v 2 A [ f?gg, where? is a new symol. Classially we an assoiate to every ouple of words U = u 1 :u 2 : : : u m 2 A and V = v 1 :v 2 : : :v n 2 A a word U V = (u 1 ; v 1 ); (u 2 ; v 2 ); : : : on A 2 y ompleting the shortest word (if any) y some?'s in order to get two words of the same length. Assume that the relation R = fu V j U; V 2 A and U V g is reognizale on A 2 y an automaton A with N states. Then N a N+1 a N N+1 elongs to R. By a pigeon-hole argument the automaton will enter twie the same state when reading the seond half of the word. Then y pumping etween the orresponding positions, A should aept also N a m a N m, with m N. Sine a N m N a m, this raises a ontradition. 2 Coding words as trees We shall now dene a tree representation of words so that omparison of words an e performed y an automaton. Our goal is to represent a word w = w 1 aw 2 : : :aw k aw k+1 y a inary tree a(t k+1 ; a(t k ; : : : ; a(t 1 ; ) : : :)), where t i represents w i and a 2 A is maximal in w w.r.t. A. First we introdue a new signature F that ontains a inary symol for eah element of the alphaet A. The inary symol assoiated with a 2 A will e denoted also a, y ause of notation. We shall introdue also a onstant symol a for eah a 2 A. We shall denote y a 0 the suessor of a with respet to the total preedene A of A. We denote y o (resp. m) the minimal (resp. maximal) element of A. The translation funtion from A to T (F ) is dened using auxiliary funtions a, a 2 A. Now we dene (w) = m (w). a 0(w 1 a 0 w 2 ) = a 0 ( a (w 2 ); a 0(w 1 )) when a 0 62 w 2 a 0(w) = a 0 ( a (w); a 0) when a 0 62 w o (w:o) = o( o ; o (w)) o () = o( o ; o ) Example 1. Assume A = fa; ; g with a A A. Then () = a((( ; ); ); a ). Example 2. Assume A = fa; ; g with a A A. Then (aa) = a((( ; ); (( ; ( ; ); ))); a((( ; ); ); a((( ; ( ; )); ); a ))) Note that is a real enoding. In other terms, two dierent words are never oded as the same tree. Lemma 5. The funtion is injetive. Proof: Let I a = f A ag. We prove y indution on a that a is injetive on Ia. Sine I m = A this will imply the result. If a = o then o (o k ) = o (o j ) learly entails j = k. Now we assume y indution hypothesis that I a is injetive. Consider two words w 1 ; w 2 2 I a 0 suh that a 0(w 1 ) = a 0(w 2 ). If neither of w 1 ; w 2 ontains a 0 we have a 0 ( a (w 1 ); a 0) = a 0 ( a (w 2 ); a 0) and therefore a (w 1 ) = a (w 2 ) whih implies w 1 = w 2 y the indution hypothesis. The ase when w 1 = u 1 a 0 v 1 (for some words u 1 2 I a 0; v 1 2 I a ) and a 0 does not our in w 2 is impossile sine a 0(w 1 ) should ontain in that ase more a 0 that a 0(w 2 ). Now assume that w 1 = u 1 a 0 v 1 and w 2 = u 2 a 0 v 2 with u 1 ; u 2 2 I a 0; v 1 ; v 2 2 I a. We prove y indution on the sum of the lengths of w 1 and w 2 that a 0(w 1 ) = a 0(w 2 ) implies w 1 = w 2. The ase ase is trivial. a 0 ( a (v 1 ); a 0(u 1 )) = a 0 ( a (v 2 ); a 0(u 2 ))

4 a a a a whih implies y deomposition a (v 1 ) = a (v 2 ) and a 0(u 1 ) = a 0(u 2 ). From the indution hypothesis we have v 1 = v 2, u 1 = u 2 and therefore w 1 = w 2. 2 A nite tree automaton over a signature F is a tuple A = (Q; Q f ; ) where Q is a nite set of states, Q f Q is a suset of aepting states and is a set of transition rules of type: f(q 1 ; : : : ; q n )! q where n 0, f is symol of F of arity n and q; q 1 ; : : : ; q n 2 Q. We onsider here ottom-up tree automata: they are applied to ground terms indutively from the leaves to the root. A ground term t is aepted y A if t! q for some state q 2 Q f. For more details aout tree automata a reent referene is [4]. Lemma 6. The set of trees f(w)j w 2 A g is reognizale y a tree automaton C. Proof: The following automaton does the work. Let Q = fq a j q 2 Ag [ fqs o g, Q f = fq m g where m is the maximal element in the preedene. The transitions are, for all a 2 A: a! q a ; o! qs o o(qs o ; qs o )! q o o(qs o ; q o )! q o a(q a ; q a 0)! q a 0 a 6= o 2 3 Tree automata We dene now a tree automaton for omparing two words. First let us extend F with a onstant?. By ause of notation we denote y fg a new funtion symol assoiated with the ouple (f; g) 2 (F [ f?g) (F [ f?g). We denote y F 2 the signature ffg j f; g 2 F [ f?gg (produt alphaet), where the arity of fg is equal to the maximum of the arities of f and g (? has arity 0). The automaton will traverse in a ottom-up way the tree otained y gluing together the tree strutures assoiated to the two words. The oding of a ouple of trees (t 1 ; t 2 ) 2 T (F ) 2 as a tree t 1 t 2 on the produt alphaet F 2 is dened reursively as follows:

5 f(s 1 ; s 2 ) g(r 1 ; r 2 ) = fg(s 1 r 1 ; s 2 r 2 ) f(s 1 ; s 2 ) a = f a (s 1?; s 2?) a f(s 1 ; s 2 ) = a f(? s 1 ;? s 2 ) f(s 1 ; s 2 )? = f?(s 1?; s 2?)? f(s 1 ; s 2 ) =?f(? s 1 ;? s 2 ) Lemma 7. The set of trees f(w) (v)j w; v 2 A g is reognizale y a tree automaton. Proof: We just need to take the produt of two opies of the automaton in Lemma 6. 2 Theorem 1. The set of trees f(w) (v) j w rpo vg is reognizale y a tree automaton. Proof: let Q = fq a ; + qa ;? qa > ; qa < ; qa j a 2 Ag e the set of states. The set of aepting states is Q = f = fq m ; + qm > g where m is the maximum element of the alphaet. The meaning of q a (resp. + qa?) is that on the right ranh we have enountered more (resp. less) a's in (w) than in (v). The meaning of q> a (resp. q< a, resp. q a = ) is that the numer of a's is the same ut some a-free suword tips the alane in favour of w (resp. v, resp. equality). The transitions of the ottom-up tree automaton inlude, for all a; 2 A with a = 0 : aa( ; q a +)! q a + aa( ; q a?)! q a? a a ( ; )! q a + a a( ; )! q a??a( ; )! q a? a?( ; )! q a + a a! q a =? a! q a? a?! q a + aa(q =; q a =)! q a = aa(q =; q a >)! q a > aa(q =; q a <)! q a < aa(q >; q a =)! q a > aa(q >; q a <)! q a > aa(q >; q a >)! q a > aa(q +; q a =)! q a > aa(q + ; qa < )! qa > aa(q +; q a >)! q a > aa(q?; q a =)! q a < aa(q?; q a <)! q a < aa(q?; q a >)! q a < aa(q < ; qa = )! qa < aa(q < ; qa < )! qa < aa(q <; q a >)! q a < In order to omplete the automaton we may add a failure state and transitions for deteting trees that do not stand for a ouple of words. We all A the automaton we get nally. We now show that if w rpo u then (w) (u) is aepted y A. For this we show the more general result: Claim: if a is the maximal letter in w then a (w) a (u)! q 2 fq a +; q a >g. We proeed y indution on a (w.r.t. A ). The ase ase when w 2 fog is left to the reader. Assume that the result is true for all a. Consider now the words w; u and a 2 A suh that a = max(w; A) and w rpo u. Let t e a (w) a (u). By Lemma 4 there are three ways to get w rpo u: If a = max(w; A) d = max(u; A) then a (t) = aa(t 1 ; a a (t 2 ; t 3 )) for some terms t 1 ; t 2 ; t 3. It an e heked that t! q a +. The ase when a = max(w; A) = max(u; A) and mul(w; A) > mul(u; A) is similar. Assume now that w = w 0 aw 1 aw 2 : : : aw k aw k, u = u 0 au 1 au 2 : : :au k au k and a is larger than every symol in w i ; u j where i; j = 0; : : : ; k. Let e the predeessor of a: 0 = a. By Lemma 4, w rpo u i there exists j suh that w j rpo u j and l > j implies w l = u l. Let us denote (w j ) (u j ) y s j.

6 Then t = aa(s k ; aa(s k ; aa(: : : ; aa(s j ; : : : ; aa(s 0 ; a ) : : :)). Running the automaton on t one an get, after some steps, aa(s k ; : : :aa(q 0 ; q) : : :) for some q 2 fq a ; = qa < ; qa > g. By the indution hypothesis (sine a) s j! q0. where q 0 is either q + or q>. Hene applying one more rule of the automaton we also have: aa(q 0 ; q)! q>. a Sine s l odes a ouple of idential words, we have s l! q =, for l > j. Applying several times the rule aa(q=; q>) a! q> a allows one to onlude. By symmetry we an show that (w) (u)! q 2 fq m? ; q< m g whenever w rpo u. For the onverse one needs to show that whenever (w) (u)! q 2 Q f we have w rpo u. Sine the ordering rpo is total either w rpo u or w rpo u. In the rst ase q 2 fq m? ; q< m g and in the seond, q 2 fq m + ; q> m g = Q f. Now sine the automaton A is deterministi if q 2 Q f then neessarily w rpo u. 2 The rst-order theory we will show deidale has oth inequality interpreted as rpo and equality interpreted as identity. When a 2 A we shall redue the satisaility of a formula au rpo v (resp. u rpo av) to the satisaility of w rpo v ^ w = au (resp.u rpo w ^ w = au). This motivates the next theorem. Note that the set fav vjv 2 A g is reognizale y a word automaton. However we have now adopted a tree representation for the words to handle inequalities and we annot mix dierent representations of the same word. Hene we have to prove: Theorem 2. The set of trees f(av) (v) j v 2 A g (where a 2 A) is reognizale y a tree automaton. Proof: Note that adding a letter a to a word w amounts to replaing a leaf y a node of the tree representation of w. There is a unique position where this node an e inserted: it is at the position of the rightmost ourrene of a. Let Q = fq 0 ; q 1 ; q 2 g e the set of states. The state q 1 reognizes the trees on F 2 otained as the produt of two idential trees. The aepting state is q 0. The transitions of the ottom-up tree automaton inlude, for all 2 A: a?! q 0! q 1 (q 1 ; q 1 )! q 1 a a (q 1 ; q 0 )! q 0 (q 1 ; q 0 )! q RPO theory An RPO formula is a rst-order formula onstruted from terms on the unary signature A and the inary prediate symols \" and \=". The formula is interpreted in T (A), with as rpo and = as identity. We denote y (x 1 ; : : : ; x n ) an RPO formula with free variales x 1 ; : : : ; x n. A at formula is an RPO formula whose atoms are of type: x = y; x = ay; x = a; x y; a x or x a, where x; y are variales. Hene terms in at formulas are restrited to types x; ax; a, where a 2 A; x 2 X. Using transformations that preserve solutions we an redue the deision of RPO formulas to the deision of at formulas. These transformations inlude the following astration rules, where a; 2 A: au rpo v ` 9u 0 (u 0 rpo v ^ u 0 = au) (1) u rpo av ` 9v 0 (u rpo v 0 ^ v 0 = av) (2) au = v ` 9u 0 (au 0 = v ^ u 0 = u) (3) u = av ` 9v 0 (av 0 = v ^ v 0 = v) (4) These rules are ompleted y the deomposition rules derived from the denition of rpo, and the deomposition and lash rules for the prediate = in T (A).

7 Theorem 3. Given an RPO formula (x 1 ; : : : ; x n ) there exists an automaton that reognizes fu 1 : : : u n j u 1 ; : : : ; u n is a solution of g: Proof: We an assume the formula is at. The proof will e y indution on the struture of. The tehnique is lassial ([4]). Let U e an automaton that reognizes all terms. We rst remark that given two automata A 1 ; A 2 for the solutions of 1 (xz) and 2 (zy), where x; y; z are disjoint sets of variales, the set of solutions of 1 (xz) ^ 2 (zy) is the intersetion of the regular languages reognized y A 1 ( jyj U) and ( jxj U) A 2. Base ase: assume that (x 1 ; : : : ; x n ) is atomi. Its satisaility is equivalent to the onjuntion of at atomi formulas otained y replaing (repeatedly) every strit maximal suterm t in every atom y a fresh variale x t. Automata reognizing the solutions of formula of type x y (resp. x = ay) an e onstruted thanks to Theorem 1 (resp. 2). For the other ases (x = y; x = a; a x; x a) automata are easily otained too. Step ase: i) Suppose (x 1 ; : : : ; x n ) = : (x 1 ; : : : ; x n ). By taking the intersetion of the omplement automaton for (x 1 ; : : : ; x n ) with n C (produt of n opies of C) we get an automaton for. ii) If (x 1 ; : : : ; x n ) = 9x 1 (x 1 ; : : : ; x n ). then y projetion (i.e. forgetting rst omponent) one get an automaton for. 2 We an now onlude with the main result whih is a diret onsequene of the previous theorem: Theorem 4. The rst-order theory of rpo is deidale when the signature is uilt from a nite set of onstants and unary symols. 5 On normal forms and ordered rewriting 5.1 The reursive path ordering on terms We assume that F is a nite set of funtion symols given with their arity. T (F; V ) is the set of nite terms uilt on F and an alphaet V of (rst-order) variale symols. denotes syntati equality of terms. T (F ) is the set of terms whih do not ontain any variales. A multiset over a set X is a funtion M from X to the natural numers. Any ordering > on X an e extended to an ordering >> on nite multisets over X as follows: M >> N if a) M 6= N and ) whenever N(x) > M(x) then M(y) > N(y) for some y > x. Note that if > is well-founded so is >>. Given a nite total preedene F on funtions, if and only if one of the following holds: s = f(s 1 ; : : : ; s m ) F rpo g(t 1 ; : : : ; t n ) = t 1. f F g and s F rpo t i for all 1 i n 2. f = g; m = n and fs 1 ; : : : ; s m g F rpo ft 1 ; : : : ; t m g 3. There exists a j, 1 j m, suh that s j F rpo t or s j rpo t. where rpo is dened as permutatively equal, or, in other words, the terms are treated as unordered trees. The set t= rpo is the equivalene lass of t for rpo. The multiset extension of rpo used aove, namely rpo, is dened in terms of this equivalene. 5.2 Ordered rewriting An ordered term rewrite system (TRS) is a pair (E; ), where E is a set of equations, and is an ordering on terms. The ordered rewriting relation dened y (E; ) is the smallest monotoni inary relation! E; on terms suh that s! E; t whenever s = t 2 E and s t.

8 Let us reall that a term t is ground reduile w.r.t. a rewrite system R i all instanes t 2 T (F ) of t are reduile y R. This denition extends to ordered rewriting, replaing R with! E;> when E is a nite set of (unonstrained) equations. Ground reduiility is deidale for aritrary nite term rewriting systems [19]. It is undeidale for nite sets of equations [9]. We show it is deidale in the speial ase where all symols ourring in the set of equations E have arity 0 or 1, (in that ase we say that the equations are unary) and the ordering is rpo with a total preedene. In fat, we an state a slightly more general result. An equation is said to e semiground if at least one memer is a ground term. Theorem 5. Given a term t, and system of equations E suh eah element of E is either unary or semiground it is deidale whether t is ground reduile y! E; rpo. Given set of terms G 1 ; : : : ; G n and a funtion symol f of arity n we denote y f(g 1 ; : : : ; G n ) the set of terms ff(g 1 ; : : : ; g n ) j g 1 2 G 1 ; : : : ; g n 2 G n g. Lemma 8. fu 2 T (F ) j s(u) rpo t(u)g, for unary terms s; t is either empty or T (F ). Proof: Applying the rpo denition we have that s(x) rpo t(x) is either equivalent to v(x) rpo x or x rpo v(x) for some term v. 2 Lemma 9. fw 2 T (F ) j 9v : s(w) rpo t(v)g is either equal to T (F ) or to fx 2 T (F ) j x rpo ug, where u is a ground term. Proof: s(x) rpo t(y) for some y i s(x) rpo t(?) where? is the minimal onstant in the signature. By deomposition we get the result. 2 Lemma 10. fu 2 T (F ) j u rpo tg where t is a ground term, is reognizale y a tree automaton. Proof: It is y simple indution on t w.r.t. rpo. If t is? this is trivial. Otherwise assume y the indution hypothesis that for all t 0 rpo t, L t 0 = fu 2 T (F ) j u rpo t 0 g is a regular tree language. Let t = f(t 1 ; : : : ; t n ). Assume w.l.o.g. that t 1 rpo t 2 ; : : : ; rpo t n. Then x rpo f(t 1 ; : : : ; t n ) i either 1. x = g(x 1 ; : : : ; x m ) and f g and for some i: x i rpo t. 2. x = g(x 1 ; : : : ; x m ) and f g and for all j: x rpo t j. 3. x = f(x 1 ; : : : ; x n ) and fx 1 ; : : : ; x n g rpo ft 1 ; : : : ; t n g The language L t is regular sine it an e dened as a omponent of the least solution of the following system of equations: L t = L 1 [ L 2 [ L 3 [ ( L t if i = j; L 1 = g(h 1 ; : : : ; h i ; : : : ; h m ) where h i = gf; [ T (F ) otherwise 1jm L 2 = g(t \ (F ); : : : ; T (F )) \ L ti fg L 3 = [ 1in 1jn; 2Sn f(l j (1) ; : : : ; lj (n) ) where lj k = 8 > < > : t k = rpo if k < j; L tj if k = j; T (F ) if k > j: and where S n is the permutation group of f1; : : : ; ng. Note that L 2 and L 3 are regular y indution hypothesis sine they are otained y omposition of languages of type L u with u rpo t. 2 Sine fu 2 T (F ) j t rpo ug is the omplement in T (F ) of the language fu 2 T (F ) j u rpo tg[ftg whih is regular we also have:

9 Lemma 11. fu 2 T (F ) j t rpo ug where t is a ground term, is reognizale y a tree automaton. Now given a set of equations E satisfying the hypothesis of Theorem 5 it an e deomposed as a set of orientale unary equations E 1 union a set E 2 of non-orientale unary ones union E 3 a set of semiground equations. Hene y Lemma 8 and Lemma 9 the set of ground and reduile terms for E is the set of ground terms that have a suterm in the set: fl(t) j l(x) = r(x) 2 E 1 ; l(x) rpo r(x); t 2 T (F )g [ fl(t) j l(x) = r(y) 2 E 2 ; l(t) rpo r(?); t 2 T (F )g [ fs j s = t 2 E 3 ; s rpo t; t 2 T (F )g [ fs j s = t 2 E 3 ; s rpo t; s 2 T (F )g (5) By Lemma 10 and Lemma 11 this set is a regular tree language. We denote it y RED E. Given a term t of T (F; X) it an e deided whether it is ground reduile y the ordered rewrite system E sine in this speial ase it amounts to heking whether t elongs to a regular tree language. This may e otained as a onsequene of stronger results [2]. We give here a simple diret proof. Let A = (Q; Q f ; ) e the automaton for RED E. We may assume that all states are reahale. And Q f = fq f g. Assume that t has m variales x 1 ; : : : ; x m with possily repeated ourrenes. Let t(q 1 ; : : : ; q m ) e the term (on an extended signature F [ Q) otained y replaing every variale x i with a state q i. We an ompute the result of applying the automaton to it. Let SUC = fhq 1 ; : : : ; q m i j t(q 1 ; : : : ; q m )! q fg. Now it sues to show that for every m-tuples of ground terms ht 1 ; : : : ; t m i we have ht 1 ; : : : ; t m i! m hq 1; : : : ; q m i 2 SUC This is a reahaility prolem on the produt automaton m and therefore is deidale. Remark: The result also holds for lexiographi path orderings. Conlusion Using a non-standard oding of words as trees and tree automata tehniques we have een ale to show deidaility of the theory of total unary rpo. A question that remains open is whether the existential theory of rpo with partial preedene is deidale. Aknowledgements: We thank Florent Jaquemard and Pierre Lesanne for their omments. Referenes 1. R. Caferra and N. Peltier. Disinferene rules, model uilding and adution. Logi at work: Essays dediated to the memory of Helena Rasiowa (Part 5: Logi in Computer Siene, Chap. 20). Physia-Verlag, A-C. Caron, J-L. Coquide, and M. Dauhet. Enompassment properties and automata with onstraints. In C. Kirhner, editor, Proeedings 5th Conferene on Rewriting Tehniques and Appliations, Montreal (Canada), volume 690 of Leture Notes in Computer Siene, pages 328{342. Springer-Verlag, H. Comon. Solving inequations in term algeras. In Pro. 5th IEEE Symposium on Logi in Computer Siene (LICS), Philadelphia, June H. Comon, M. Dauhet, R. Gilleron, F. Jaquemard, D. Lugiez, S. Tison, M. Tommasi. Tree Automata Tehniques and Appliations H. Comon, P. Narendran, R. Nieuwenhuis, and M. Rusinowith. Deision prolems in ordered rewriting. In Pro. 13th IEEE Symp. Logi in Computer Siene (LICS'98), Indianapolis, IN, USA, June 1998, pages , H. Comon and R. Treinen. The rst-order theory of lexiographi path orderings is undeidale. Theoretial Computer Siene 176, April 1997.

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