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 UIUCDCS-R-79-986 
 
 A Note on Simplification Orderings 
 
 by 
 
 Nachum Dershowitz 
 
 UILU-ENG 79 1736 
 
 '<N 
 
 
 April 1979 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/noteonsimplifica986ders 
 
A Note on Simplification Orderings 11 
 
 Nachum Dershowitz 
 
 Department of Computer Science 
 
 University of Illinois at Urbana-Champaign 
 
 Urbana, Illinois 61801 
 
 April 1979 
 
 Keywords: simplification orderings, term-rewriting, termination, 
 well-founded orderings. 
 
 *Research supported under NSF Grant MCS77-22830. 
 
1. A TERMINATION THEOREM 
 
 In this note we present a powerful method for proving the 
 termination of term-rewriting systems based on the following notion 
 of a simplification ordering. 
 
 Definition : A transitive and irreflexive relation ^ is a 
 simplification ordering on a set of terms T if for any terms 
 t,t',f(...t...),f(...t\..)eT 
 
 1) t > t" implies f(...t...) > f(...t'...) 
 and 2) £(...t...) > t. 
 
 Note that this definition does not require that ^ be well-founded (cf. [5,7]). 
 We shall assume throughout this note that all function symbols have fixed 
 arity. 
 
 A term-rewriting system P over a set of terms T is a finite 
 
 set of rewrite rules of the form £.(a) -*■ r.(a), where the a are variables 
 
 i i 
 
 ranging over T. Such a rule is applied to a term t e T in the following 
 
 manner: if t contains a subterm £.(a), i.e. the variables a are instantiated 
 
 l ' 
 
 with terms a, then replace that subterm with the corresponding term r.(a), 
 
 thereby obtaining t 1 . The choice of rule and subterm is nondeterministic. 
 
 We write t =* t f to indicate that the term t' can be derived from the term t 
 
 by a single application of a rule in P to one of the subterms of t. (The 
 
 variables appearing in r, must be a subset of those in I. . ) For example, 
 
 the system consisting of the one rule (a*3)*Y "* ot*(£*Y) reparenthesizes a product by 
 
 associating to the right. Applying that rule twice to the term t = (a»b) • ((c*d) »e) , 
 
 -1- 
 
-2- 
 
 we get t ■* a-(b*((cd)«e)) ■* a* (b* (c (d«e))) or, alternatively, 
 
 t =* (a«b) • (c* (d*e)) =* a* (b* (c* (d*e)) ) . In either case, no further 
 
 applications of the rule are possible. We say that a term-rewriting 
 
 system P terminates , if there exist no infinite sequences of terms t. e T 
 
 such that t =* t => t =*• . . . . In general, It is undecidable whether a system 
 
 terminates [21. 
 
 The following theorem gives a sufficient criterion for proving 
 that a term-rewriting system terminates for all inputs. 
 
 Termination Theorem : 
 
 A term-rewriting system P = {Z . -> r . } . , over a set of 
 
 11 i=l 
 
 terms T terminates if there exists a simplification ordering > 
 over T such that 
 
 £ i ^ r i ' i=1 '-'- >P > 
 for any assignment of terms in T to the variables of £ . . 
 
 The proof of this theorem is based on the following: 
 
 Tree Theorem [4]: 
 
 In any infinite sequence t-,t_,... of terms over a finite set 
 of function symbols, there exists a pair of terms t. and t., i < j, 
 such that t. is homeomorphically embedded in t. (when t. and t. 
 are viewed as trees) . 
 
 We shall denote this embedding relation by <. We have 
 
 s = f(s ,s ,...,s ) < g(t ,t ,...,t ) = t , 
 1 z m — 1 z n 
 
-3- 
 
 if and only if 
 
 (a) f = g and s,, < t . for all i, l<i<m=n 
 & ^ _ x — — 
 
 or else (b) s < t . for some i, 1 <_ i j< n. 
 
 Proof of Tree Theorem [8]: 
 
 Assume that the theorem is false. Then there exist one or more 
 infinite "counterexample" sequences of terms such that no element can 
 be embedded in a subsequent one. We construct a "minimal" counterexample 
 
 sequence t = (t.) . in the following manner: if the elements 
 t 1 ,t.,...,t. . (i _> 1) have already been chosen, then let t. be a 
 
 minimal size (that is, number of function symbols) i-th element 
 
 of a counterexample sequence beginning with the elements already chosen. 
 
 We need the following observation: Let s = (s ) . be any 
 
 infinite sequence whose elements are proper subterms of successive 
 
 elements of some subsequence of t. By the assumption of minimality of 
 
 t, if s 1 is a subterm of some t, , then the sequence t- 9 t , . . . ,t, . ,s. ,s«. . . 
 
 must contain an embedding pair s. < s . (i < j). Moreover, there must 
 
 exist an infinite chain s. _<s. _<s. _< . . . of subterms (i < i_ < i„ < ...), 
 
 x x i 2 i 3 1 l 5 
 
 each embedded in the next. (Otherwise, there would be an infinite number 
 of maximal length, but finite, chains, each beginning beyond where the 
 previous chain ended. But then the last elements of those chains form an 
 infinite sequence of subterms which must also contain an embedding pair, 
 implying that some chain was not maximal.) 
 
-4- 
 
 Now, by the pigeon-hole principle, there exists an infinite 
 
 11 1 2 2 2 
 subsequence f(s ,s ...,s ), f(s s ...,s ),... of t, all elements of 
 
 LA m 1 z m 
 
 which have the same outermost function symbol, f . Furthermore, by the above 
 observation, the sequence (s ) . of first arguments contains an infinite embedding 
 
 chain s < s < ... . Consider the sequence of terms f(s, ,s , . . . ,s ) 
 1 — 1 — 1 L m 
 
 h i 2 H 
 
 f(s ,s , . . . ,s ),... . Repeating the above procedure m-1 times for 
 the remaining arguments s , . . . ,s yields an infinite subsequence of terms 
 of t, every argument of which can be embedded in the corresponding 
 argument of the subsequent term. Consequently, each term in the 
 subsequence can be embedded in the next; hence, t could not have been 
 a counterexample. rj 
 
 We shall need the following: 
 
 Lemma : Let s and t be terms in T. If s < t, then s 4 t in any 
 simplification ordering > over T. 
 
 Proof of Lemma : 
 
 The proof is by induction on the size of t. Assume that 
 s' js t' implies s' < t 1 for any t' smaller than t. By the definition 
 of < y if s = f(s ,s 2 ,.. .s m ) < g(t-,t 2 ,... ,t ) = t (m or n may be 0), 
 then either (a) f = g and s. < t. for all l_<i_<m = n, in which case 
 s. < t. and therefore s <t (property (1) of simplification orderings); 
 or else (b) s j t, for some i, in which case s j(t, ^g(...t....) = t 
 (property (2)). 
 
 We are ready for the 
 
-5- 
 
 Proof of Termination Theorem : 
 
 Assume that P does not terminate. Then there exists an infinite 
 sequence of terms t.. * t~ =*■ . . . . Note that there can be only a finite 
 number of function symbols appearing in the sequence (those in t, and in P). 
 If for a simplification ordering > we have I. ^ r., then it follows 
 (using property (1)) that t-, ^ t~ ^ ...» and by transitivity that 
 t. > t. for all i < i . But, by the Tree Theorem t. <l t. for some i < j, 
 and by the lemma t. ^ t.. This contradicts the asymmetry of > 
 (asymmetry follows from transitivity and irref lexivity) . D 
 
 This result may be used to simplify proofs of well-foundedness 
 and termination, e.g. those in [1, 3, 6, 9, 10]. 
 
 2. SOME EXAMPLES 
 
 Consider the one-rule term-rewriting system 
 
 (or3) *Y "*" «• (3*y) 
 and the following recursively defined ordering: t ^ t' for two terms t and 
 t' , if (i) |t| > |t'|, where |t| denotes the number of function symbols in t, 
 or else t and t 1 are products of the forms a*3 and ct'^B 1 , respectively, and 
 (ii) |t| = | t 1 | and a > a' or else (iii) |t| = |t' | , a = a 1 , and 3 > g' 
 (cf. the ordering in [3]). To see that this is a simplification ordering, 
 note that t > t' implies 1 1| _> 1 1' | . Thus, if a > a 1 , then a* 3 > a' •£ 
 (by (i) or (ii)) and 3*a^ B'a' (by (i) or (iii)). Furthermore, 
 | ex • 3 1 > |a|,|3| and therefore a*3 ^ a, 3 (by (i)). To prove termination, we 
 need only verify that (a*3)*Y ^ a*(3*y) in this ordering. This follows 
 (by (ii)) from the fact that | (a* B) * Y | = | a* (6 • y) | and | ex * 3 | > |ot|. 
 
-6- 
 
 As a second example, we prove that the system 
 a- (8+7) + (a»8)+(a«Y) 
 (S+7)'a ■*- (B*a)+(ya), 
 for distributing multiplication over addition, terminates. Let the function 
 t map terms into multisets recursively: (i) for any sum a+8, 
 x(a+8) = T(a)Ux(8), where U denotes union of multisets, (ii) for any product 
 a«8, x (a»0) = (T(a)Ux(8) } , and (iii) for any atomic term u, x(u) = {0}, where 
 denotes the empty multiset. For example, x( ((a+b) • (c+d) )+e) = {{0,0,0,0} ,0} . 
 
 We use the following simplification ordering: t > t' , for two 
 terms t and t' , if x(t) » x(t') in the nested multiset ordering ». 
 In this ordering (see [1]), XUZ » YUZ, for multisets X ^ 0, Y, and Z, if 
 for each element y e Y there is some x e X such that x » y. For example, 
 {{0,0,0,0},0} » {{0,0,0},{0,0,0},0h since {0,0,0,0} » {0,0,0}. It 
 should be obvious from this definition of » that XUZ » YUZ if X»Y and that 
 XUZ» Z if X 4 0. It is also easy to prove that {...X...} » X for any 
 multiset X, by induction on the nested depth of X. With these facts in mind, 
 it is straightforward to verify that ^ is a simplification ordering on terms. 
 
 To prove termination it remains to show that a* (8+7) ^ (a*0)+(a , y) 
 and (B+y)*a^ (8 # a)+(ya), i.e. we must show x(a*(8+y)) = x((0+7)*a) = 
 {x(a)U T (B)U T ( Y )} » x((a-8)+(a- Y )) = t ((B-a)+(ya)) = {x(a)U T (8), x(a)U T (y)}. 
 This in turn follows from the fact that the multisets x(0) and x(y) cannot be 
 empty and therefore x (a)U T (8)U T (y) » x(a)U T (8), x(a)U T (y). 
 
 3. AN APPLICATION 
 
 As an application of the Termination Theorem, consider the following 
 class of orderings: With each function symbol f of arity n >^ associate 
 
-7- 
 
 some real polynomial F(x-,...,x ) of n variables. Then extend this relation 
 
 i n 
 
 to a morphism on terms, i.e. f(t.,...,t ) = F(t,,...,t ), where t is the real 
 
 1 n l n 
 
 expression associated with the term t. The set of terms T 
 constructed from those function symbols may be ordered according to the 
 real values of the associated expressions, i.e. t^ t' , for two terms t 
 and t' in T, if and only if t > t' . Thus, to prove that a term- rewriting 
 system P over T terminates, one must show that the polynomials F(x , ...,x ) 
 satisfy the two conditions for simplification orderings (1) x > x' implies 
 F(...x...) > F(...x'...) and (2) F(r..x...) > x, and that FT > r~" for 
 each rule I. -*■ r . in P. Conveniently, these are all decidable properties 
 for polynomials over the reals [11]. By the same token, it is decidable if 
 there exists any polynomial of degree lees than any given n that 
 demonstrates termination. In this manner, the undecidability for 
 polynomials over the natural numbers, encountered in the method of [6], is 
 circumvented. 
 
 Finally, we note that the Termination Theorem provides sufficient 
 but not necessary conditions for termination. To see this, consider, for 
 example, the one-rule system ffa -> fgfa, where f and g are unary function 
 symbols. This system always terminates, since each application of the rule 
 decreases the number of adjacent f's. On the other hand, ffa <_ fgfa and 
 therefore ffa 4 fgfa in any simplification ordering. Consequently, there 
 is no simplification ordering ^ under which ffa ^ fgfa. Moreover, there can 
 be no well-ordering ^ of all the terms constructable from f and g - that satisfies 
 the monotonicity property (1) - under which ffa ^ fgfa. (Since ^ cannot 
 be a implif ication ordering, for some term a it must be that a^ ha, where 
 h is f or g; consequently, a^ha^hha^... is an infinite descending sequence and 
 ^ cannot be a well-ordering. ) 
 
-8- 
 
 REFERENCES 
 
 [1] N. Dershowitz and Z. Manna [Aug. 19 79], Proving termination with 
 multiset orderings , Comm. ACM, vol. 22, no. 8, pp. 465-476. 
 
 [2] G. Huet and D. S. Lankford [1978], On the uniform halting problem for 
 term rewriting systems , Report 2 83, IRIA, Le Chesney, France. 
 
 [3] D. E. Knuth and P. B. Bendix [1969], Simple word problems in universal 
 algebras , Computational Problems in Universal Algebras (J. Leech, ed.), 
 Pergamon Press, Oxford, pp. 263-297. 
 
 [4] J. B. Kruskal [May 1960], Well-quasi-ordering, the tree theorem, and 
 Vazsonyi's conjecture , Trans. Amer. Math. Soc. , vol. 95, pp. 210-225. 
 
 [5] D. S. Lankford [May 19 75], Canonical algebraic simplification in 
 
 computational logic , Memo ATP-25, Automatic Theorem Proving Project, 
 Univ. of Texas, Austin, TX. 
 
 [6] D. S. Lankford [May 19 79], On proving term rewriting systems are Noetherian , 
 Memo MTP-3, Mathematics Dept. , Louisiana Tech. Univ. , Ruston, LA. 
 
 [7] Z. Manna and S. Ness [Jan. 19 70], On the termination of Markov algorithms , 
 
 Proc. Third Hawaii Intl. Conf. on System Sciences, Honolulu, HI, pp. 789-792 
 
 [8] C. St. J. A. Nash-Williams [1963], On well-quasi-ordering finite trees , 
 Proc. Cambridge Philos. Soc, vol. 59, pp. 833-835. 
 
 [9] D. Plaisted [July 1978], Well-founded orderings for proving the termination 
 of rewrite rules, Report R-78-932, Dept. of Computer Science, Univ. of 
 Illinois, Urbana, IL. 
 
 [10] D. Plaisted [Sept. 1978], A recursively defined ordering for proving 
 
 termination of term rewriting systems , Report R-78-943, Dept. of Computer 
 Science, Univ. of Illinois, Urbana, IL. 
 
 11] A. Tarski [1951], A decision method for elementary algebra and geometry , 
 Univ. of California Press, Berkeley, CA. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R-79-986 
 
 2. 
 
 3. Recipient's Accession No. 
 
 4. Title and Subtitle 
 
 A Note on Simplification Orderings 
 
 5. Report Date 
 
 April 19 79 
 
 6. 
 
 7. Author(s) 
 
 Nachum Dershowitz 
 
 8. Performing Organization Rept. 
 
 No. 
 
 9. Performing Organization Name and Address 
 
 Dept. of Computer Science 
 University of Illinois 
 Urbana, IL 61801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 NSF MCS 77-22830 
 
 12. Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, D.C. 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 A new method is given for proving that term- rewriting systems 
 terminate. Examples of its use are given. In particular, the method allows 
 the use of polynomials over the reals in termination proofs. 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 simplification orderings, term- rewriting, termination, well-founded orderings. 
 
 17b. Identifiers/Open-Ended Terms 
 17c. COSATI Field/Group 
 
 18. Availability Statement 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 11 
 
 
 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 22. Price 
 
 FORM NTIS-38 (10-70) 
 
 USCOMM-DC 40329-P7I 
 

FEB 
 
 20