Mechanized systems for equational inference often produce many terms that are permutations of one another. We propose to gain efficiency by dealing with such sets of terms in a uniform manner, by the use of efficient general algorithms on permutation groups. We show how permutation groups arise naturally in equational ...
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... 2001. 2001 Kluwer Academic Publishers. Printed in the Netherlands. 223General
Algorithms for Permutations in EquationalInference ?J RGEN AVENHAUSUniversit t Kaiserslautern, Fachbereich Informatik, D-67663 ... September 1998)Abstract. Mechanized systems for equational inference often produce many
terms that are permu-tations of one another. We propose to gain ... propose to gain efficiency by dealing with such sets of
terms in a uniformmanner, by the use of efficient
general algorithms on permutation groups. We show how permutationgroups arise naturally in ... problems, and study some of their properties. We alsostudy some
general algorithms for processing permutations and permutation groups, and considertheir application to ... and permutation groups, and considertheir application to equational reasoning and
term- -rewriting systems. Finally, we show how thesetechniques can be incorproated ... be incorproated into resolution theorem-proving strategies.Key words: equational inference, permutations,
term rewriting.1. IntroductionA common problem in applications of mechanized reasoning is ... E |= s = t . Even in the more
general context of first-order the-orem proving, deduction using equations plays an ... role. A problem thatfrequently arises in such theorem-proving applications of
term- -rewriting systemsand equational reasoning is that many permutations of a ... systemsand equational reasoning is that many permutations of a given
term or equationare often produced. For example, we may obtain many ... a renaming of the variables, and s is an arbitrary
term. .? This research was partially supported by the National Science ... + are associative and commutative. There are threeproducts of three
terms; ; each product can independently be permuted in 6 ways,leading ... the same polynomial equation.Of course, for polynomials having many more
terms and products, this number caneasily become astronomical. This is unsatisfactory, ... methods to deal with this problem is to usespecialized unification
algorithms for various equational theories, such as AC-theories. This has led ... This has led to a large number of specialized unification
algorithms forvarious theories, and a large number of results concerning how ... unificationalgorithms can be combined. However, we would like to develop
general methodsfor dealing directly with subterm permutations in
term- -rewriting theorem proving,regardless of which set of equations produces them, ...
... other than theRobbins problem involving AC operators and concluded thatGENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 225although incomplete, the superset limit ... also states thatAC unification has two distinct effects when making
terms identical: it per-mutes arguments and introduces new variables. We believe ... approachthat we have not tried would be to develop a
general E-unification theory for equa-tions that produce subterm permutations. For discussions ... for equa-tions that produce subterm permutations. For discussions of the
general area ofequational reasoning, see [DJ90, Pla93, Klo92]. An early approach ... is given in [Chr89].Our goal, then, is to handle E-equivalent
terms efficiently, where E is a setof equations. The idea of ... our approach is not to deal with entire E-equivalenceclasses of
terms, , as is done in classical E-unification, but rather to ... but their processing may be faster. Also, our method is
general, , apply-ing to all permutations, and so may be able ... to all permutations, and so may be able to exploit
algorithmic efficiencies notcovered by known equational unification algorithms.One difference between our ... E explicitly at all, since they are incorporatedinto a unification
algorithm. . For us, this is not always possible, but we ... is a flat permutative theory and s is a ground
term ofthe form f (a1, a2, . . . , an) ... elements exponentialin n. This motivates the search for other efficient
algorithms relating to subtermpermutations and
term rewriting. The
algorithm for Theorem 1.1 is based on thefollowing result, from [FHL80]:THEOREM ... the problem of deciding whether g is in the group
generated by g1, g2, . . . , gnis solvable in ... permutation group membership problem.1.1. TERMINOLOGYWe first develop some terminology. A
term is a well-formed expression composedof function symbols, constant symbols, and ... be a set of variables, and T(F,X) be theset of
terms over F and X. A ground
term is a
term containing no variables, such asf (a, g(a, b)). The top-level ... asf (a, g(a, b)). The top-level function symbol of a
term f (t1 . . . tn) is the functionsymbol f ... tn) is the functionsymbol f . If t is a
term and 2 is a substitution, then t2 denotes the application ...
... or disjoint. We write r ? sto indicate that the
terms r and s are identical. If ? is a position ... identical. If ? is a position and t is a
term, , wewrite t|? or ?(t) for the subterm of t ... , tm)) ? ?(ti). Often it is easier to speakin
terms of contexts instead of positions. A context is a
term with occurrences of2 in it. For example, f (2, g(a,2)) ... context. Ifn is an integer, then an n-context is a
term with n occurrences of 2. If t is an n-context ... t with the leftmost m occurrencesof 2 replaced by the
terms t1, . . . , tm, respectively. If r is ... r is an n-context and m ? n and the
terms si are0-contexts, then r[s1, . . . , sm] ? ... . . , sm] ? r[s1][s2] . . . [sm].Two
terms r and s are variants if they are instances of ... yi ? yj , and all occurrences of the variablesGENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 227xi and yi are explicitly ... EQUATIONAL INFERENCE 227xi and yi are explicitly listed. Thus the
terms f (x, f (y, z)) and f (y, f (x,w)) ... usual, a superscript of ?indicates reflexive and transitive closure. A
term t is linear if each variable appearsat most once in ... permutation or is leafpermutative if r and s are linear
terms that have the same set of variables and arevariants of ... = f (v, u).This set is not uniform because the
terms f (x, f (y, f (z,w))) and f (u, v) ... : e ? E}. Also, ?5[E]? is the permutation group
generated by 5[E]; this228 J RGEN AVENHAUS AND DAVID A. PLAISTEDis the ...
... (1) and (2) in sequence,we obtain the permutation ??. In
general, , arbitrary compositions of permutationsmay be obtained in this way. ... define the application of an equation r1 = r2to a
term t to be the replacement of an instance of r1 ... in E, which is so iff ? ? ?5[E]?. 2GENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 229COROLLARY 1.5. There is a ... IN EQUATIONAL INFERENCE 229COROLLARY 1.5. There is a polynomial time
algorithm to test whether E |= e,where E ? {e} is ... ? E2 is uniform, then there is a polynomial time
algorithm to test whetherE1 |= E2.Proof. E1 |= E2 if for ... E2, E1 |= e. 2The fact that such polynomial time
algorithms exist in these special cases mo-tivates us to develop systematic ... of handling leaf permutative equations inequational theorem proving and in
general first-order theorem proving. It is oftenthe case that many leaf ... to do this, we need to have methodsfor dealing with
terms in which the permutations can be applied at more than ... To this end, we introduce subterm permutation groups and stratified
terms. .2. Subterm Permutation GroupsA subterm permutation group G is an ... AND DAVID A. PLAISTEDRecall that T(F,X) is the set of
terms whose function symbols are in the set Fand whose variables ... definitions are consid-ered to be over the set T(F,X) of
terms. . Thus we can define a set E of equationsover ... over T(F,X), and so on. We nowextend this set of
terms by allowing “labeled function symbols,” as follows: If theequations of ... superscripts of elements of F′(X). Let T′(F,X)be the set of
terms ... in X. Thus T′(F,X) = T(F′(X) ? F,X). Foran arbitrary
term f (s1 . . . sn), with f ? F ... sn), with f ? F and si ? T′(F,X), the
term f G(s1 . . . sn)is also written as f ... f ({f (x,y)=f (y,x)},1)(a, b) is an exampleof a labeled
term in T′(F,X)whose top-level function symbol is f ({f (x,y)=f(y,x)},1).Two labeled ... is a labeled term.A labeled redex for G is a
term of the form tG in T′(F,X), where t ? T(F,X)is ... (y, z)) = f (x, f (z, y))}, then the
term f G(x, f (y, z)) is a labeled redexfor G, ... (x, f (y, z)) is a redex for G.A stratified
term t ? T′(F,X) is a
term t such that for any labeled subterm vGof t , ... distinct occurrences of functionsymbols in t are distinct. Note that
terms in T(F,X) are also stratified
terms, , bydefault. When we say t is a stratified
term, , we imply that t ? T′(F,X).
GENERAL ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 231EXAMPLE 2.5. Let G1 = ... and h(x, y, z) is a redex forG2. Therefore the
term gG1(a, b, hG2(d, e, f )) is astratified
term. . This is because gG1(a, b, hG2(d, e, f )) ... f (z,w))) is a labeled redex for G3. How-ever, the
term f G3(a, f (b, f G4(c, f (d, f (e, ... G4(c, f (d, f (e, e′))))) is not a stratified
term, , sincef G3(a, f (b, f G4(c, f (d, f ... of f G3(x, f (y, f (z,w))).We note that stratified
terms require that the labeled function symbols be sep-arated from one ... of theterm “stratified.” We will see later that when these
terms are rewritten, this levelstructure is preserved.Typically we use the symbols ... and substitutions, r, s, t, u, v,w(possibly with subscripts) for
terms, , f, g, h (possibly with subscripts) for functionsymbols or ... with subscripts) for variables, i, j, k for integers,R for
term- -rewriting systems over T(F,X) or T′(F,X), and u and w ... a fixed single set of equations as in typical E-unification
algorithms, , weenvision the sets Ei being
generated as the theorem-proving process progresses.Therefore we cannot specify in advance ...
... kept and used in inferences. Thisguarantees completeness even when stratified
terms do not completely capture theE-equivalence class of a
term. . Despite the fact that equations in Ei are kept, ... PLAISTED2.1. STRATIFIED REWRITINGWe now return to the discussion of stratified
terms. . A stratified
term in T′(F,X) isa representation for a set of
terms in T(F,X). This set of
terms is defined using aterm-rewriting system associated with the
term. . We now define this
term- -rewritingsystem.DEFINITION 2.7. If t is a stratified
term, , then the stratified
term- -rewriting systemR[t] is {t1G ? t2G : G labels some ... z)}, and R2[t] is {gG2(x, y) ? gG2(y, x)}. In
general, ,Ri[t] will have as many rewrite rules as Ei has ... r1 are referred to.Let us again consider the preceding example
term t ? f G1(a, b, gG2(c, d)),where E1 is {f ... a)?s2 f G1(b, gG2(c, d), a). We observe that all
terms pro-duced from t by stratified rewriting are also stratified
terms with the same set oflabels. We also observe that applying ... R[t] to an arbitrary termenough times will produce the original
term back again. Furthermore, the “levelstructure" of
terms is preserved by stratified rewriting; we note that G1 occursabove ... rewriting; we note that G1 occursabove G2 in all the
terms obtained from t by stratified rewriting.In
general, , a stratified rewrite rule f G(. . .)? f ... f G(. . .) over T′(F,X) can applyonly to a
term in T′(F,X) of the form f G(. . .), since ... to restrict theapplication of the rewrite rules.Note that R[t] in
general is not terminating, but it is left and right linear ... rules. We now show that R[t] is confluent for all
terms t .THEOREM 2.9. R[t] is confluent.GENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 233Proof. We first note that ... in Ri[t], then applying this rule enough times to a
term will give the originalterm back again (since there are only ... that n applications of therule r ? s to a
term u will produce u back again. Let v be the ... will produce u back again. Let v be the second
term in thissequence of
terms, , that is, u? v. Then v?? u by n? ... easy consequence. 2To use the definition of stratified rewriting in
general, ... , we need to know that ifr1 is a stratified
term, , so is r2, and has the same set of ... of G′ is 2.THEOREM 2.11. If r1 is a stratified
term and r1 ?s r2, then r2 is a stratifiedterm, and ... then r2 is a stratifiedterm, and Id[r2] = Id[r1].Proof. The
term r2 is obtained by permuting some of the subterms of ... Id[r2] = Id[r1].To show that r2 is also a stratified
term, , we need to show that any labeled sub-
term of r2 is an instance of a labeled redex. Let ...
... is a labeled subterm of r1. Sincer1 is a stratified
term, , w[u1] is an instance of a labeled redex. Suppose ... subterms w appear in r1 as well, which is astratified
term. . Therefore such subterms w are all instances of labeled ... of?si , as usual.DEFINITION 2.12. If t is a stratified
term in T′(F,X), then [t]s is the set of t ′ ... terms.We will later show that [t]s is finite.So, a stratified
term t is a convenient way of representing the set [t]s ... of representing the set [t]s of terms.This set [t]s of
terms has properties that make it possible to treat it more ... in theorem proving applications.We now give some examples of stratified
terms t and their associated sets [t]sof
terms. . Then we will prove some properties of stratified rewriting ... properties of stratified rewriting and stratifiedsets.EXAMPLE 2.13. Let the stratified
term t be f (E1,1)(a, f (b, f (c, d))), where ... f (x, f (y,f (w, z)))}. Then [t]s contains all
terms of the form f (E1,1)(u1, f (u2, f (u3, u4))),where ... u4) is a permutation of (a, b, c, d), 24
terms in all.EXAMPLE 2.14. Let t be f (E2,2)(a, b, c, ... = f (x, y, z, v,w, u)}. Then [t]scontains all
terms of the form f (E2,2)(u1, u2, u3, u4, u5, u6), ... u6) is a cyclic permutation of (d, e, f ),nine
terms in all.EXAMPLE 2.15. Let t be f (E3,3)(a, b, c, ... (y, z, x, v,w, u)}. Then [t]s contains the three
terms {f (E3,3)(a, b, c, d,e, f ), f (E3,3)(b, c, ... all possible ways.We now give an example showing how stratified
terms relate to associative andcommutative operators. The associative and commutative equations ... x),f (x, f (y, z)) = f (f (x, y), z).
GENERAL ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 235We note that if we ... (y, z)) =f (y, f (x, z)), and we can
generate all equations of the following form:f (x, f (y, f ... f (z,w))),where the equations of G consist of the following
generators of the permutationgroup on four elements:f (x, f (y, f ...
... these examples wouldnot necessarily be handled by a single unification
algorithm. . However, they all canbe handled in a uniform manner ... possessed by special unification algorithms.We have defined stratified sets of
terms. . What we are really interested in is theset [t]s ... which we now define.DEFINITION 2.17. If t is a stratified
term in T′(F,X), then um[t] ? T(F,X)(“unmark” t) is t with ... other labelsremaining.DEFINITION 2.18. If t ? T′(F,X) is a stratified
term, , then S[t] ? T(F,X) is{um[u] : u ? [t]s}. ... We also refer to S[t] as a stratified set of
terms. . But note thatthe
terms in S[t] are ordinary (unlabeled)
terms. . If X is a set of stratified
terms, ,then S[X] is ?{S[t] : t ? X}.Note that if ... factorof 3 from the equation in E4.We now discuss some
general properties of stratified rewriting and stratifiedsets.THEOREM 2.24. If r ?i ... and i and j are distinct, then there is a
term usuch that r ?j u?i t .
GENERAL ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 237Proof. This follows by reasoning ... 2.24.COROLLARY 2.25. If r ?? r ′, then there are
terms r1r2 . . . rn such that r ??1r1 ??2 ... we get a bound on sizes.LEMMA 2.26. The set of
terms r2 such that r1 ?s,?i r2 is finite.Proof. Let uG[x1 ... si . Repeatedapplications of rules of G will only produce
terms of the same form. Since thereare only finitely many permutations ... that r1 ?s,?i r2is finite. 2THEOREM 2.27. The set of
terms r ′ such that r ?s,? r ′ is finite.Proof. ... That is, if r ?s,? r ′, then there are
terms r1r2 rn suchthat r ??1 r1 ??2 ... obtain the result. 2COROLLARY 2.28. If t is a stratified
term, , then [t]s and hence [t]s,i are finite.DEFINITION 2.29. If ... and hence [t]s,i are finite.DEFINITION 2.29. If t is a
term, , then M[t], the multiset of t , is the ...
... lower indices never appear farther from the root of the
term, , withoutloss of
generality. . We want to show that if ri ??i ri+1 ... right, since eachvariable and constant appears only once in the
term. . Further rewriting can changeonly the order of the variables ... rewriting and equational reason-ing.THEOREM 2.32. Let r be a stratified
term, , and let E be the union of all thelabeled ... (5)However, the system R′ has the property that for any
terms u and v, u ??R′ viff v??R′ u. This implies ... implies that(r ??R′ r ′) ? (r ??R′ r ′). (6)
GENERAL ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 239From (5) and (6), we ... interested in is equational conse-quences involving unlabeled equations and the
term r with the labels removed.DEFINITION 2.33. A stratified
term r is consistently labeled if for all distinctlabeled subterms f ... same set of equations as G. A set of stratified
terms is con-sistently labeled if each
term in the set is consistently labeled and, in addition,the labelings ... all distinct labeledsubterms f G(r1 . . . rm) of
terms in the set, all occurrences of f in the set ... the same set of equations as G.For example, a stratified
term f G(a, f (b, c)) is not consistently labeled be-cause ... have the same set of equations,but not otherwise.Note that some
terms cannot be consistently labeled (except with the emptylabeling). For example, ... having a redex of f G(x, f (y, z)),then the
term f (u, f (v, f (w, x))) cannot be consistently ... them. If we label all of them, then weobtain the
term f G(u, f G(v, f G(w, x))), which is not ... where the si are either vari-ables or constants. Then any
term can be consistently labeled using these subtermpermutation groups.THEOREM 2.35. Let ... subtermpermutation groups.THEOREM 2.35. Let r be a consistently labeled stratified
term, , and let E bethe union of all the equations ...
... subterms t of [r]. This willbe useful for describing an
algorithm to test membership in S[r].DEFINITION 2.36. If f G(r1, . ... f G(r1, . . . , rn) is a stratified
term and the index of G is i,then top[f G(r1, . ... set E of equations.DEFINITION 2.38. If t is a stratified
term whose top-level function symbol islabeled with G and u is ... z))},then the top context of t is f G′(2, f (2,2)).
GENERAL ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 241The following results follow easily ... following results follow easily from the definition of [t]s by
term rewriting(Definition 2.12):PROPOSITION 2.40. If the top-level function symbol f of ... 2.12):PROPOSITION 2.40. If the top-level function symbol f of stratified
term f (t1 . . . tn)is not labeled, then [f ... 2PROPOSITION 2.42. If r is a top context for the
term t = [t1 . . . tn], then [r[t1 . ... .tn]]s = {r[w1 . . . wn]: there exists a
term r[u1 . . . un] ? [t]top[t ] such that ... rewriting on the subterms. Rewriting using the top group gives
terms ofthe form r[u1 . . . un], where u1 . ... . . tn, and then rewritingon the subterms ui gives
terms of the form r[w1 . . . wn] for wi ... of operationson stratified sets.DEFINITION 2.43. If r is a stratified
term of the form f (r1 . . . rn) and ... isunlabeled, then Split[r] = {r}. If r is a stratified
term of the form f G(r1 . . . rn),then Split[r] ... f (gG2(a, b), x)}.PROPOSITION 2.45. If r is a stratified
term, , then S[r] =?{S[t] : t ? Split[r]},and no
term of Split[r] has a label at the top-level function symbol.3. ... is a predicate symbolin P, followed by a list of
terms ... per-mutation groups over T(P ? F,X), and in which all
terms in the clause are stratifiedterms (in T′(F,X)). Let C(P,F,X) be ... clauses over T(P,F,X). For purposes ofalgorithms and definitions involving stratified
terms, , we can consider the predi-cate symbol to be just ... 3.3. Stratified rewriting??,s on stratified clauses is defined as forstratified
terms. . Note that if C is a stratified clause, and ... (a, x)) ?Q(f (y, b)), and P(f (a, x)) ?Q(f (b, y)).
GENERAL ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 243Now, for any operation on ...
... ? S[Cn] ? S[D], T |= S[D]T ? {D} expansion.GENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 245Both operations deletion and expansion ... be applied together. For example, suppose a clause C[t]contains the
term t ? g(a, f (b, c, d)) and the equations ... thatif f is transformed into f G1 , then the
term g(a, f G1(b, c, d)) is no longer aninstance of ... which paramodulation refinement is used, buthave suggested ordered paramodulation. In
general, , any complete refinement canbe used, and then the lifted ...
... used. Manycomplete resolution-paramodulation strategies permit such rewriting. Suppose Ris a
term- -rewriting system such that for every rule r ? s ... this purpose,we introduce some definitions.DEFINITION 3.13. If t is a
term and u is a context obtained by replacing onesubterm of ... are unrelated.DEFINITION 3.14. Suppose v ? u[rG] is a labeled
term and s is the contextof G. Suppose ti are
terms ... or contexts, and all but one of them is a
term, , and theremaining ti is a 1-context. Then u[s[t1, . ... inside the redex of G in v represent posi-tions of
term structure that is altered by the permutations in G. Contexts ... Contexts that areoutside (below) the redex ofG in v represent
term structure that may be moved to adifferent position by the ... C is a clausecontaining a literal L[r1?], R is a
term- -rewriting system, and > is a simplificationordering:T ? {C[L[r1?]]}, r1 ... ? S′iT ? {C[umi(L[r2? ])]} ? {C[umi(L′)] : L′ ? S′i}.
GENERAL ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 247In words, [L[r1?]]s,i are the ... applying permutations to L[r1?] may cause the structure of the
term r1?to be changed, so that it may become impossible to ...
... of r1? by r2? . This cannotbe done on lifted
terms, , since r1? may contain labels. In order to apply ... group redex, we also have to ensure that for all
terms s1 in S[r1?] ands2 in S[r2?], s1 > s2.We now ... = f G1(x, f (y, f (z,w))),where E1 has the
generators of the full permutation group on four elements.If we have ... E into T are not necessary (they do not produceGENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 249anything new), because of the ...
... for paramodulation intothe equations of E or into other equations
generated from them, because of thelabeling restriction. 2Thus we see that ... equationsthat could appear in the labels of the stratified clauses.4.
Algorithms for DeductionWe now present some
algorithms for operating on stratified
terms ... r and t , be-ginning with a membership test. These
algorithms ... could all be done by explicitlyenumerating S[r] and S[t] for
terms r and t and then performing operations on ele-ments of ... in efficiency. Instead,250 J RGEN AVENHAUS AND DAVID A. PLAISTEDthe following
algorithms are written to take as much advantage as possible of ... It should be clear upon a little reflection that these
algorithms can bemuch more efficient than a direct enumeration of S[r] ... oftheir complexity, there may be errors in some of these
algorithms; ; if so, we hopethat others will point them out.We ... out.We can say something precise about the efficiency of these
algorithms, , too.Recalling that the size of a stratified set is ... exponential in the termsize, one can show that the following
algorithms all have complexity of the sameorder, that is, single exponential. ... that the time bound is basedon a linear representation for
terms, , and unifying such
terms can increase the termsize by an exponential amount. This implies ... termsize by an exponential amount. This implies that our unification
algorithm can beperformed in roughly single exponential time, but the results, ... roughly single exponential time, but the results, when represented aslinear
terms, , will take possibly exponential time each to write out. ... exponential, but successive operations could take muchlonger on these large
terms. . There may be a way to represent common subtermsonly ... be a way to represent common subtermsonly once in stratified
terms and avoid this exponential size blowup, but we havenot investigated ... choose stratifications thatwould respect the repeated subterm structure of a
term, , and thereby enable a moreefficient representation.For theorem-proving applications, r ... we refer to them simply as terms.We first present two
algorithms to test for membership in S[r]. Given a
term tand a stratified
term r, the
algorithms test if t is a member of S[r]. This could ... some sense t is superfluous and canbe deleted. Therefore these
algorithms have applications to deduction. The firstalgorithm is simpler, but the ... is more efficient in some cases.In a number of these
algorithms, , we use a construct of the form return(A) whereA ... else false should be returned.procedure Member(t, r)[[test if t in S[r]]]
GENERAL ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 251if the multisets M[um[r]] and ...
... : r2 ? Split[r]})end Member;procedure Member2(t, r)[[ we know neither
term is a variable and neither top-level functionsymbol has a label ... call does not have to be executed more than once.This
algorithm requires us to iterate through all elements of Split[r], whichcan ... test or the unequal top-level function symbol test.Here is another
algorithm to test if t ? S[r], which is more efficient ... desired.For example, suppose that we want to test whether the
term g(f (a, b), f (y, x),f (c, d)) is a ... 2}. This set B is found by a perfect matching
algorithm. . B ′ equalsB, since this permutation is in ?5[E1]?. ... and for each one, eight combinations of permutations of the
terms rj , for 48elements of S[r] altogether to test for ... For larger examples, the effect becomes much more pronounced.The following
algorithm tests whether t is a subterm of some element of ... a subtermof S[r] is done, but we present the simpler
algorithm here in which t must beidentical to a subterm of ... r);[[test if t is a proper subterm of some elementGENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 253of S[r]. We know r ... ? Subterm(t, rn)end Subterm2;It is straightforward to extend these
algorithms to test whether t unifies with anelement of S[r] or ... someelement of S[r], and so forth.We now present some other
algorithms that are useful in a
general theoremproving context. These
algorithms require another definition, however. The generalidea is that some
algorithms may be done more efficiently if the topmost subtermpermutation groups ... done more efficiently if the topmost subtermpermutation groups of two
terms permute the same or similar sets of subterms. Inorder to ...
their topmost groups more similar so that some
algorithms can bemore efficient. This mechanism is based on conformability and ... restrictive than this, because it gives conditions guaranteeingthat such a
term r ′ will exist.DEFINITION 4.1. Suppose stratified
term t (? T′(F,X)) has a top label and topcontext u. ... (? T′(F,X)) has a top label and topcontext u. A
term r ? T′(F,X) is conformable to
term t if there is a context u′ ofr such that(a) ... . . . r?(n)] for somepermutation ? and(c) for all
terms s1 . . . sn, u′[s1 . . . sn] ... Theterm “extension” below is motivated by the use of this
term in the AC unificationalgorithm of Stickel [Sti81].DEFINITION 4.2. If
term r in T′(F,X) is conformable to t and t has ... has top contextu, then the t-extension of r is the
term r? in T′(F,X)whose top context is um[u′] as254 J RGEN AVENHAUS ... is not t-conformable.(The E2 redex goes too deep in the
term, , outside the top context of t .)THEOREM 4.6. S[r?] ... 4.6. S[r?] = S[r].Proof. By the definition of S[r] in
terms of stratified rewriting and the definitionof r?. 2Note. If no ... present a procedure to test the subset condition for stratified
terms. . Firstwe need a notation for representatives of equivalence classes.DEFINITION ... t isnot needed and can be deleted. Therefore the subset
algorithm has an applicationto deduction. First we give a very simple ... applicationto deduction. First we give a very simple but inefficient
algorithm, , followed by amore complex but in some cases much ... elsereturn((?t ′ ? Split[t])(?r ′ ? Split[r])Subset2(t ′, r ′))end Subset;
GENERAL ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 255procedure Subset2(t ′, r ′)[[a ...
... be done, when r1 and r2 have no labels. TheGENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 265reasoning is similar to that ... C;return (? literals M1 . . .Mn ? D such thatMore-
General- -
Term( (f (L1 . . . Ln), f (M1 . . ... . . .Mn)))end Subsumes;Finally, we can give a more systematic
algorithm for performing the expansionoperation of Section 3. Let LL be ... been derived at some point in the proof. For a
term t , let Et be the leafpermutative equations in LL ... Ei of equations may notbe known in advance but are
generated as the proof search progresses.266 J RGEN AVENHAUS AND DAVID A. ... chosen for the input clauses.5. DiscussionWe have presented a fully
general method of dealing with arbitrary permutations inthe context of equational ... deduction and theorem proving. This method takes advan-tage of efficient
algorithms for permutation groups, but it is not exactly a unificationalgorithm ... but not instantiation (narrowing) in the definition of stratified setsof
terms. . However, in that our method considers
general sets of permutations thatcannot be handled by AC unification (such ... AC unification (such as cyclic permutations, for example),it is more
general than AC unification. The fact that polynomial time algorithmsexist for ... stimulate additional work in equational and first-order theorem proving. The
algorithms presented here can readily be implementedand tested, and we encourage ... to do so. In fact, the efficiencies of some ofthe
algorithms can probably be improved beyond what we have done by ...
... permutation ? must satisfy the property that there is some
term t ′ in [t]top[t ] andsome subterm r ′ in ... of each such equivalence class. This equivalence relationis defined in
terms of certain transpositions being in the permutation groups oftop[t] and ... |= u[x1 . . . xn] =u[x?(1) . . . x?(n)]
GENERAL ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 259?(?? ? ??(B/ ?)) (Etop[r] ... a procedure to unify two stratified sets, that is, given
terms r and t , tofind all most
general substitutions 2 such that S[r]2 and S[t]2 have a nonemptyintersection. ... pairwise ordinary unification problems, each having at mostone solution. This
algorithm has obvious applications to deduction.Let ? be the identity substitution.First ... : (?x ?S)(y ? x)}.procedure Unify(r, t);[[ return the most
general substitutions in {mgu(r ′, t ′) : r ′ ? ... t ′ ? Split[t]})end Unify;procedure Unify2(r, t)[[ we know neither
term is a variable and neither top-level functionsymbol has a label ...
... efficient version of Unify follows:procedure Unify(r, t):[[ return the most
general substitutions in {mgu(r ′, t ′) : r ′ ? ... ] thenreturn Prune(?{Unify2(r ′, t) : r ′ ? Split[r]})elseGENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 261return Prune(?{Unify2(r ′, t ′) ... save a lot of work. Wealso note that the unification
algorithm in
general can be much more efficient thana direct treatment of the ... more efficient thana direct treatment of the underlying sets of
terms. . If the top groups of r and tare identical ... 120 elements, then we need only to permute one ofthe
terms r and t during unification to avoid repeated unification results, ... (x, y) = f (y, x), if we unify the
terms f G(x, y) and f G(y, x), we obtainthe identity ... b, y ? a}. Suppose now that G is a sub-
term permutation group that contains the equations f (x, f (y, ... = f (y, f (z, f (w,f (u, x)))). These
generate the full permutation group on 5 elements, which has 120elements. ... is unlabeled even if f is com-mutative, since the unification
algorithm won’t add any labels. But the instanceg(f E1(a, b), f ... to be a lifting of ordinaryunification to stratified clauses.The unification
algorithm made use of a procedure Prune which has not yetbeen ...
... Match_Lists;For pruning, we want to say that t1 is more
general than t2 if S[t2] is a subset ofS[t1], or if ... if t2 is an instance of t1. The following test
generalizes both of these butstill seems to be reasonably efficient computationally.procedure ... both of these butstill seems to be reasonably efficient computationally.procedure More-
General- -
Term( (t, r)[[ sufficient test for
term t to be more
general than
term r ]]let ? be Match(t, r);if ? is false then ... r);if ? is false then return false;return Subset(r, t? )end;procedure More-
General( (?, ?)[[check if ? is more
general than ?]]let x1 . . . xn be all variables ... z? 6= zlet f be a new n-ary function symbolGENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 263return More-
General- -
Term( (f (x1 . . . xn)?, f (x1 . . ... . . . xn)?, f (x1 . . . xn)?)end More-
General; ;We can now give a Prune procedure for use with ... (?? ′ ? T )(? ′ and ? are distinctand More-
General( (? ′, ?))})end Prune;THEOREM 4.8. The procedure Unify given above ... a lifting of unification be-tween two terms.Proof. By induction on
term depth and case analysis. 24.2. FASTER OPERATIONS AND CACHINGSuppose we ... the procedure P ; we assume P takes the stratified
terms r1and r2 as input and returns P (r1, r2). The ... operations (modulothe time to compute P on each pair of
terms) ). This can result in an efficiencyimprovement, but it can ... operations. For this, we assume that Unify is extended from
terms to literals264 J RGEN AVENHAUS AND DAVID A. PLAISTEDin the straightforward ...
... of each such equivalence class. This equivalence relation isdefined in
terms of certain transpositions being in the permutation groups of top[t]and ... u[t? (1) . . . t? (n)] and there are
terms wi in[t? (i)]s such that w is u[w1 . . ... S[r] as claimed.Here is an example. Suppose t is the
term gE1(f (a, b), f (x, y), f (c, d)) andr ... b), f (x, y), f (c, d)) andr is the
term gE1(f E4(c, d), f E2(a, b), f E3(x, y)). Suppose ... = {f (x, y) = f (y, x)}. Then BGENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 257would be the permutation {1 ... true. This example should give an idea of how thefollowing
algorithms operate, as well.The following procedure tests whether two stratified sets ... test for S[t ′] ? S[r ′] 6= {}, assumingneither
term is a variable and both have the same top levelfunction ...
... Subsetand Intersect. Ordinary Knuth–Bendix completion can be lifted to stratified
terms, ,since it essentially performs a sequence of paramodulations, and we ... paramodulations, and we have shownhow to lift paramodulations to stratified
terms. . We feel that this lifting of Knuth–Bendix completion is ... in efficiency withoutthe high complexity associated with traditional E-unification algorithms.GENERAL
ALGORITHMS FOR PERMUTATIONS IN EQUATIONAL INFERENCE 267AcknowledgmentsWe thank the referees for ... (1992), 39–44.[FHL80] Furst, M., Hopcroft, J. and Luks, E.: Polynomial-time
algorithms for permutationgroups, in Proceedings of the 21st IEEE Symposium on ... method, J. Assoc. Comput. Mach. 38(3) (1991),559–587.[Klo92] Klop, J. W.:
Term rewriting systems, in S. Abramsky, D. M. Gabbay and T. ... J. Automated Reasoning 19(3)(1997), 263–276.[Pla93] Plaisted, D.: Equational reasoning and
term rewriting systems, in D. Gabbay, C. Hog-ger, J. A. Robinson ... Comput. Mach. 27(4) (1980), 772–796.[Sti81] Stickel, M. E.: A unification
algorithm
Abstract:
<p>Mechanized systems for equational inference often produce many
terms that are permutations of one another. We propose to gain ... propose to gain efficiency by dealing with such sets of
terms in a uniform manner, by the use of efficient
general algorithms on permutation groups. We show how permutation groups arise naturally ... and study some of their properties. We also study some
general algorithms for processing permutations and permutation groups, and consider their application ... permutation groups, and consider their application to equational reasoning and
term- -rewriting systems. Finally, we show how these techniques can be ...
References:
{FHL80} Furst, M., Hopcroft, J. and Luks, E.: Polynomial-time
algorithms for permutation groups, in
Proceedings of the 21st IEEE Symposium on Foundations of Computer Science, Syracuse, New York, 1980, pp. 36-41.
{Klo92} Klop, J. W.:
Term rewriting systems, in S. Abramsky, D. M. Gabbay and T. S. E. Maibaum (eds),
Handbook of Logic in Computer Science, Vol. 2, Oxford University Press, Oxford, 1992, Ch. 1, pp. 1-117.
{LBB84} Lankford, D., Butler, G. and Ballantyne, A.: A progresss report on new decision
algorithms for finitely presented Abelian groups, in
Proceedings of the 7th International Conference on Automated Deduction, Lecture Notes in Comput. Sci. 170, Springer-Verlag, 1984, pp. 128-141.
{Pla93} Plaisted, D.: Equational reasoning and
term rewriting systems, in D. Gabbay, C. Hogger, J. A. Robinson and J. Siekmann (eds),
Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 1, Oxford University Press, 1993, pp. 273-364.
{Sti81} Stickel, M. E.: A unification
algorithm for associative-commutative functions,
J. Assoc. Comput. Mach. <b>28</b> (1981), 423-434.
Keywords:
term rewriting
Title:
General Algorithms for Permutations in Equational Inference
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