Theorem Proving with Transitive Relations
from a Practical Point of View
W. Marco Schorlemmer
Jaume Agust
Institut d'Investigacio en Intelligencia Articial
Consejo Superior de Investigaciones Cientcas
Campus Universitat Autonoma de Barcelona
08193 Bellaterra, Catalonia (Spain)
E-mail: fmarco,[email protected]
Abstract
Rewrite techniques have been typically applied to reason with the equality relation and have
turned out to be among the more successful approaches to equational theorem proving. In fact, it
is not only in reasoning with the equality relation where these techniques naturally apply, but in
reasoning with arbitrary, probably non-symmetric, transitive relations, being the equality relation
just a special case of monotone transitive relation, which is also symmetric. The work done so far in
applying rewrite techniques to arbitrary transitive relations showed several important dierences
with the equational case. Although most equational results can be extended to non-symmetric
relations, new problems appear which must be solved in a quite dierent way. In this paper we
review the use of rewrite techniques for reasoning with arbitrary possibly non-symmetric transitive
relations and we analyze the reasons why an ecient treatment of this generalization to nonsymmetric transitive relations is not yet achieved. We also point to the future work which needs
to be done in order to come up with these diculties.
1 Introduction
Rewrite techniques have been typically applied to reason with the equality relation and have turned
out to be among the more successful approaches to equational theorem proving. They implicitly
capture the transitivity and congruence properties of the equality relation in a natural way, and avoid
the explicit use of the equality axioms, which pose severe problems in the design of ecient automated
theorem provers. Recently an eort has been made to apply these techniques in theorem proving for
rst-order clauses with equality.
But, since rewrite rules rewrite terms in one direction, it is not only in reasoning with the equality relation where these techniques naturally apply, but in reasoning with arbitrary, probably nonsymmetric, transitive relations. Symmetry is not a property captured by rewriting. The equality
relation is just a special case of monotone transitive relation, which is also symmetric. So the generalization of rewrite systems is not considering rewriting on equivalence classes of terms, as presented
by Huet [Hue80], but even further considering rewrite systems as a logic itself, as pointed out by
This work was supported by the project DISCOR (TIC 94-0847-C02-01) funded by the CICYT, and a research
fellowship funded by the CSIC
1
Meseguer [Mes92], or considering them as a deduction mechanism for theories with non-symmetric
relations, the so called bi-rewriting systems of Levy and Agust [LA93]. Further work in this direction
has been done by Bachmair and Ganzinger [BG94c], who extended the notions of the bi-rewriting
systems to the more general chaining calculus of rst-order clauses with transitive relations.
The work done so far in applying rewrite techniques to arbitrary transitive relations showed several
important dierences with the equational case, due to the lack of symmetry, which dicults notably
the deduction mechanism in theories with this kind of relations. New problems appear which must
be solved in a quite dierent way. In this report we review the use of rewrite techniques for reasoning
with arbitrary possibly non-symmetric transitive relations, presenting reasoning with the equality
relation as a special case, and we analyze the reasons why an ecient treatment of this generalization
to non-symmetric transitive relations is not yet achieved. We also point to the future work which
needs to be done in order to come up with these diculties. The report is organized as follows: First
of all, section 2 introduces to some preliminary notions used throughout the paper. Section 3 gives
a historical overview of the use of rewrite techniques in automated theorem proving to reason with
the equality relation. In Section 4 we introduce rewrite techniques, putting the emphasis on their
application to arbitrary transitive relations. The application of rewrite techniques to theorem proving
for full rst-order clauses with transitive relations is analyzed in section 5. The description in section
6 of important eciency criteria and their use in the Saturate theorem prover developed by Nivela
and Nieuwenhuis [NN93] allows us to determine the requirements for an ecient implementation of
these techniques, and to expose the drawbacks that appear when generalizing these eciency criteria
for arbitrary transitive relations. Finally, section 7 concludes with some of the future perspectives in
order to come up with the analyzed problems.
2 Preliminaries
If nothing is said, we follow the notation and the standard denitions used in [DJ90]. We are concerned
with rst-order terms T (F ; X ) over a nonempty signature F of function symbols and a denumerable
set X of variables. The expression t[u] denotes that u occurs as subterm in term t. If the occurrence
of this subterm u is replaced by term v, we will denote this ambiguously with t[v]. We refer to t[ ]
as the context in which the replacement takes place. A substitution = [X1 7! t1 ; : : : ; Xn 7! tn ] is
a mapping from a nite set fX1 ; : : : ; Xn g X of variables to T (F ; X ), extended as a morphism to
a mapping from T (F ; X ) ?! T (F ; X ). We will use substitutions in postx notation, so we denote
substitution applied to a term t with t.
A binary relation ?! is called a rewrite relation if s ?! t implies u[s] ?! u[t], for all terms s,
+ the transitive closure and with ?!
the reexivet, and u, and substitution . We denote with ?!
transitive closure of rewrite relation ?!. If the rewrite relation is also symmetric then we will denote
it with !, and with +! and ! its transitive and its reexive-transitive closure, respectively.
An ordering is an irreexive, transitive binary relation. Given a rewrite relation ?!, its transitive
+ is said to be a reduction ordering, if it is well-founded, i.e. no endless chains of rewrites
closure ?!
can be build. A reduction ordering captures the notion of simplication of terms which is essential in
the rewrite approach of theorem proving.
3 Equational Reasoning
Most of the eort made in order to deal eciently with special predicates in the context of resolutionbased theorem proving, has focused mainly on the equality relation, due to its importance in a large
number of domains. The use of rewrite techniques in resolution-based theorem proving appears to be
one of the most promising approaches developed so far.
2
The use of rewrite techniques in automated theorem proving has its origins in the completion
procedure stated by Knuth and Bendix [KB70] for solving the word problem in equational theories.
Equations are used as one-way rewrite rules, namely in the direction they simplify terms. The completion process adds to the initial set of equations E suciently many new ones, by computing critical
pairs, in order to obtain a convergent (i.e. terminating and Church-Rosser) rewrite system. Convergent rewrite systems, when they can be obtained, provide a decision procedure for the word problem
in E , based on the computation of normal forms, since they exist and are unique in each equivalence
class of the congruence dened by the set of equations E . If a set of equations E can be completed
into a nite and convergent rewrite system R, then this system R is a way to nitely encode the
whole congruence closure dened by E , which is innite in general: all possible proofs of equational
consequences of E can be represented as rewrite proofs. Furthermore, if rewrite rules are kept as
interreduced as possible, having in R the minimal number of rewrite rules required to represent the
congruence closure, then the rewrite system R is canonical, and therefore unique up to renaming of
variables [DMT88].
But since the word problem in equational theories is not decidable in general, a convergent rewrite
system can not always be obtained. On one hand the completion procedure may fail, due to the
impossibility to orient an equation as a simplifying rewrite rule. On the other hand it may not
terminate. Failure can be avoided by ordered (or unfailing) completion (cf. [Lan75, HR87, BDP89]).
Due to the \unfailing" property of ordered completion we can see it as a refutational deduction process.
During this process we may distinguish two main kinds of deductions, those who are based on the
generation of new equations by computation of critical pairs, and which will participate in the process
of refutation, and those which simplify equations, in order to avoid deduction of rules which would be
unnecessary for the purpose of refutation (in the spirit of building a canonical rewrite system). This
distinction between productive and simplication deduction is a key aspect in theorem proving and
hence also in its rewrite approach, as we will see along the paper.
The eort to extend completion to general rst-order clauses has introduced the use of well-founded
orderings to resolution-based theorem proving. In the theorem proving context, the desire to encode
the equality predicate into the logical language led to the paramodulation inference rule stated by
Robinson and Wos [RW69]:
0 =v
Paramodulation: ??!; ; s!= t ;;!u[t;]u[s=] v
where is a most general unier of s and s0 . Paramodulation, together with resolution and factoring,
was proved to be complete, whenever the reexivity and functional-reexivity axioms were added
to the initial set of clauses. The paramodulation rule was signicantly improved by Brand [Bra75]
and Peterson [Pet83] by proving that the functional-reexivity axioms were unnecessary, and that
paramodulation through variables was not required. But even under this restrictions, paramodulation
is dicult to control, because it generates the complete congruence closure of equality. Therefore,
since convergent rewrite systems avoid the generation of the congruence closure, the extension of
rewrite techniques to paramodulation of rst-order clauses led to the superposition calculus which is in
essence a generalization of the critical pair generation of the Knuth-Bendix completion procedure. This
extension was done on completion procedures for conditional equations (cf. [Kap84, KR91, Gan91]).
Hsiang and Rusinowitch [HR91] explicitly put some ordering restrictions on paramodulation.
The generalization of completion to full rst-order clauses required the development of more powerful techniques for the completeness proofs. Hsiang and Rusinowitch [HR91] proved the refutational completeness of ordered paramodulations using the transnite semantic tree method. Bachmair
[Bac91] used proof orderings in order to obtain similar results. But it was with the model construction method of Bachmair and Ganzinger, when the refutational completeness of an inference system
based on strict superposition was proved [BG90, BG94b]. At the same time the model construction
3
method provides a powerful abstract redundancy notion which generalizes the simplication methods
of Knuth-Bendix completion. Model construction also allowed one to prove refutational completeness
of inference calculi based on strict superposition with more restrictions like the use of basic strategies
[NR92a, BGLS92] and ordering constrained clauses [NR92b], which may reduce the search space to
be explored. The process of deduction of non-redundant (productive deduction) clauses based on
superposition is known as saturation, and it generalizes the process of completion towards canonical
rewrite systems of equational theories.
All these theoretical results have been put into practice in the Saturate theorem prover, developed
by Nivela and Nieuwenhuis [NN93], and its study will show us the importance of implementing powerful redundancy provers in order to guarantee the eciency of the whole theorem prover. Through
redundancy we avoid the generation of clauses which will not participate in the process of saturation.
4 Reasoning with arbitrary transitive relations
Most of the research done so far has focused mainly on the equality relation, but there have been
also some attempts to include transitive relations other than equality into the logical language. Slagle
[Sla72] was the rst to propose an inference system for theories with equality, orderings and sets,
based on the chaining inference rule, which essentially is the equivalent of the paramodulation rule
for arbitrary transitive relations:
; u < s ! ; t < v
Chaining: ??!
; ! ; ; u < v
where is a most general unier of s and t.
But for the completeness of such inference systems the functional-reexive axioms and chaining
through variables were required. By investigating special theories like dense total orderings without
endpoints, Bledsoe and Hines [BH80] have developed techniques for eliminating certain occurrences
of variables from formulas, avoiding in this manner the prolic chaining through variables. Bledsoe,
Kunen and Shostak [BKS85] and Hines [Hin92] achieved also completeness results for particular such
systems of restricted chaining. Subterm chaining methods for general clauses in the presence of
(anti)monotonicity have been proposed by Manna and Waldinger [MW86], but their calculus was
proved to be incomplete [MW92].
But in the same manner as ordering restrictions on the paramodulation rule have led to the
superposition calculus which avoids the generation of the whole congruence closure of equality, ordering
restrictions on the chaining inference rule also avoid the generation of such transitive closure. Therefore
rewrite systems may also be used in order to provide decision procedures for the word problems in
theories expressed with arbitrary transitive relations. This has been done for the rst time by Levy
and Agust [LA93], who developed the so called bi-rewriting systems, which we will introduce in the
next subsection. The generalization of bi-rewriting systems to theorem proving with full rst-order
clauses with arbitrary transitive relations has been made by Bachmair and Ganzinger [BG94c], who
introduced the ordered chaining calculus, which generalizes the superposition calculus to arbitrary
transitive relations.
4.1 Bi-rewriting systems
Let denote a reexive, monotonic, but non-symmetric transitive relation, which we will refer to as
inclusion. Given a nite set of inclusions I , we say that s ?!I t if there exist an inclusion u v 2 I
and a substitution such that u is a subterm
of s, and t is the term resulting from replacing u with
t if we are able to obtain term t from s by replacing terms
v in term s. Intuitively, we say that s ?!
I
with \bigger" terms, using instances of inclusions in I . The same can be done in the inverse direction,
4
from \bigger" to \smaller" terms1 . Given two terms s and t, the word problem of a theory I is to
decide whether I j= s t. By the equivalent of Birkho's theorem for arbitrary transitive relations
it is known that I j= s t if and only if s ?!
I t and in this case we say that the inclusion s t
is provable in I . But it is an arduous task to automatically prove the validity of an inclusion s t
by application of all possible replacements on subterms of s until term t is obtained. It is impractical
to build a decision procedure based on this methodology, due to the huge search space that may be
generated. Even further, due to the existence of innite branches that may be explored, if I 6j= s t,
the procedure would not terminate.
Bi-rewriting systems avoid all potentially innite branches by using inclusions to replace subterms
by smaller ones with respect to an ordering on terms, i.e. they are used as rewrite rules. Given a
nite set of rewrite rules R, we say that a term s is reduced to t using rule l ?! r 2 R, and denote
it by s ?!R t, if a subterm u of s, which is an instance of the left-hand side l, is replaced by the
corresponding instance of the right-hand side r, obtaining so a smaller term t. If t is a term such that
no rewrite rule of R can be applied, then we say that it is an irreducible
term with respect to R, and
s a reachable term from s with
call it a normal from. Given a term t, we call a term s, such that t ?!
?!R .
Notice that if we orient a set of inclusions, following an ordering on terms, we get two independent
rewrite systems, one containing those rewrite rules which rewrite terms into \bigger" ones (with
respect to the inclusion relation), and the other one containing rewrite rules which rewrite terms into
\smaller" ones2 . We will distinguish the two separate rewrite relations by denoting the rst one with
and the second one with ?!
. Consider for example an inclusion theory dened by the following
?!
axioms:
f (a; x) x
f (x; c) x
b f (a; c)
If we orient these inclusions, following an ordering on terms, we obtain the following two rewrite
systems:
8
x
<
R1 = : f (a; x) ?!
x
f (x; c) ?!
n
R2 = f (a; c) ?!
b
We say that these two rewrite systems form a bi-rewriting system B = hR1 ; R2 i. In such a system a
bi-rewrite proof consists of two paths, one using rules of R1 and the other using rules of R2 , which join
together in a common term:
?!
u ? ? t
s ?!
There will be a bi-rewrite proof if the set of reachable terms from s with ?!R1 and the set of reachable
terms from t with ?!R2 intersect in an nonempty set.
In order to have a decision procedure for the word problem of a theory I we need the bi-rewriting
t
system B to satisfy two properties: First, whenever we have two terms s and t such that s ?!
I
then a term u should exist such that s ?!R1 u and t ?!R2 u. We denote the existence of such a
term u by s #B t. This property is referred to as the Church-Rosser property. Second, and in order to
avoid reducing terms s or t innitely many times, we require that no innite branching and no innite
sequences of rewrites with rule in R1 (or R2 ) can be build. In this case we say that the bi-rewriting
1
2
\bigger" and \smaller" refer to the inclusion relation, not to the ordering on terms
Here we suppose that all inclusions can be oriented. The case with unorientable inclusions will be treated later.
5
system B is terminating3 . Bi-rewriting system fullling termination and the Church-Rosser property
are said to be convergent.
t we
The decision procedure for the word problem is then straightforward: To check if s ?!
I
reduce s and t applying rewrite rules of each rewrite system until a common term is reached. Since
the bi-rewriting system is terminating, no innite branch will be explored:
Z}Z >
ZZ
*
ZZ
. J] -Z
.
HH
J
HHH
ZZ . . . . JJ9 j
~. . )
H
ZZ
In the same manner as convergent rewrite systems are a nite representation of the congruence
closure of equality, convergent bi-rewriting systems nitely encode the reexive, transitive and monotone closure of . With convergent bi-rewriting systems all possible consequences of a set of inclusions
I using transitivity, reexivity and monotonicity can be represented by a bi-rewrite proof.
An arbitrary bi-rewriting system, obtained by orienting the inclusions of a inclusional theory
I is non-convergent in general. But, like in the equational case, there exist necessary and sucient
conditions for a terminating bi-rewriting systems to be Church-Rosser, which were stated by Levy and
Agust adapting the original results of Knuth and Bendix [KB70]. First of all we give two denitions
and then the theorem which summarizes this result (cf. [LA93] for further details):
Denition 4.1 Given a bi-rewriting system B = hR1; R2 i and two rules l1 ?! r1 2 R1 , l2[u] ?!
r2 2 R2 , where u is not a variable. If is a most general unier of l1 and u, then hl2 [r1 ]; r2 i is
called a critical pair.
Denition 4.2 Given a bi-rewriting system B = hR1; R2 i and a rule l1 ?! r1 of R1 and an instance
l2 ?! r2 of a rewrite rule l2 ?! r2 of R2 , where is such that, for some term v and some variable
x that appears more than once in l2, x = v[l1 ] and y = y, if y =
6 x. The critical pair hl2 [v[r1 ]]; r2 i
is called a variable instance pair4.
A critical pair or a variable instance pair hl; ri is said to be convergent if l #B r, and divergent
otherwise.
Theorem 4.1 A terminating bi-rewriting system B is Church-Rosser (and thus, convergent) if and
only if there are no divergent critical pairs or variable instance pairs between the rules of R1 and the
rules of R2 .
Following the same ideas proposed by Knuth and Bendix [KB70], one can attempt to complete
a non-convergent terminating bi-rewriting system, by means of adding divergent critical pairs and
variable instance pairs as new rewrite rules to the systems R1 or R2 . Notice that the number of
critical pairs among the rewrite rules of sets R1 and R2 is always nite. But from the denition of
variable instance pairs, we can observe that the overlap of term l1 on l2 is done below a variable position
of l2 , and therefore unication always succeeds. Furthermore, term v is arbitrary, which means that
if a variable instance pair exists between two rewrite rules then there are an innite number of them.
As we will see later, this is one of the major drawbacks for the tractability of the generalization of
actually only quasi-termination is required (cf. [LA93])
The denition of variable instance pairs also appears in the context of rewrite systems modulo a congruence (cf.
[BDH86])
3
4
6
rewrite techniques to arbitrary transitive relations, because a completion procedure which attempts
to add variable instance pairs as new rewrite rules is impossible to manage in general.
The example bi-rewriting system stated above is non-convergent since, for example rules f (a; x) ?!
b yield a divergent critical pair:
x and f (a; c) ?!
f (a; c)
b
?
?
@@R
c
Suppose that in our ordering on terms, c b. Since b c, we obtain the following rule:
b
c ?!
A Knuth-Bendix like completion procedure would add the critical pair hb; ci as a new rewrite rule to
the system. In this example a convergent system can be build, because during the whole completion
process no variable instance pairs among two rules appear. In fact, for this inclusional theory the
completion procedure would generate the following convergent bi-rewriting system:
8
x
>> f (a; x) ?!
>< f (x; c) ?!
x
R =
1
>> b
>: f (x; b)
8
<
R2 = : f (a; c)
c
a
?!
x
?!
b
?!
b
?!
Now we can prove the validity of inclusion f (f (a; b); f (b; b)) f (c; f (a; c)) in the example inclusional
theory with the following bi-rewrite proof:
f (f (a; b); f (b; b))
@
f (c; f (a; c))
?
R
@
?
f (b; f (b; b))
f (b; f (a; c))
@@R
?
?
f (b; b)
b. In fact, f (b; b)
Notice that f (b; b) is not an irreducible term with respect to R1 , since f (b; b) ?!
is the only term reachable from f (f (a; b); f (b; b)) with ?!R1 which is also reachable from f (c; f (a; c))
with ?!R2 . This shows us, in disadvantage to the equational case, that normal forms do not play any
role in this search for a bi-rewrite proof.
Let us now come back to some notions about termination. Recall that theorem 4.1, on which
Knuth-Bendix completion is based, requires a bi-rewriting system to be terminating. The following
theorem puts the conditions for it:
+
Theorem 4.2 A bi-rewriting system B = hR1 ; R2i is terminating if the rewrite orderings ?!
R1 and
+
?!R2 dened by R1 and R2 respectively are contained in a unique reduction ordering.
7
Since during the process of completion new critical pairs or variable instance pairs are generated,
which need to be oriented in order to add them as new rewrite rules to the bi-rewriting system, a
suitable reduction ordering between terms must be dened in such a way as to orient all these possible
pairs.
Obviously, like in equational theories, not for every given inclusional theory, and for every reduction
ordering we could dene, the completion procedure terminates successfully providing a convergent birewriting system for the inclusional theory. There may be theories for which the completion procedure
will keep generating new critical pairs and therefore will not stop. But completion also can terminate
unsuccessfully, i.e. it may abort without nding a convergent bi-rewriting system, for a given inclusional theory, though one exists. This happens when it nds a critical pair which cannot be oriented
with the given reduction ordering. Sometimes this can be solved by using a more powerful reduction
ordering (one that orients more pairs), but there are also cases of pairs, which are non-orientable by
any reduction ordering. A typical example of an inclusion which causes the failure of the completion
procedure is the commutativity-like axiom:
f (x; y) f (y; x)
It is obvious that no orientation of the inclusion preserves the termination property, because endless
sequences of reductions can always be build:
f (y; x) ?!
f (x; y) ?!
f (x; y) ?!
Dierent methods have been developed to overcome this problem. In the equational case Lankford and Ballantyne [LB77] extended rewriting to rewriting modulo a set of equations. These set of
equations used to be structural axioms like the commutativity and the associativity axiom. Peterson
and Stickel [PS81] extended the matching mechanism used in the generation of critical pairs to a more
general matching modulo associativity and commutativity, which Jouannaud and Kirchner [JK86] extended to the general case. Both approaches treat the troublesome equations implicitly, not as rewrite
rules. These ideas also can be generalized to bi-rewriting systems (cf. [LA95]).
4.2 The role of symmetry in bi-rewriting systems
If the arbitrary transitive relation we need to reason with is, besides reexive and monotonic, also
symmetric, then we are dealing with a congruence relation. Bi-rewriting systems based on equality
appear simply to be rewrite systems as known from the equational case, because when orienting
equations following a given reduction ordering no two distinct rewrite systems are obtained. For
example, the theory of groups is dened by the following set of equations:
ex = x
e
(x y) z = x (y z )
x?1 x =
These equations can be oriented leading to the following rewrite system:
ex
?! x
x?1 x ?! e
(x y) z ?! x (y z )
The symmetry of the equality relation relation makes an equality to be equivalent to two inclusions
(since the relation dened by the two inclusions is also an equivalence relation) and therefore we can
8
also dene the theory of groups by the following set of inclusions:
ex
ex
x?1 x
x?1 x
(x y) z
(x y) z
x
x
e
e
x (y z)
x (y z)
By orienting them with respect to a given reduction ordering we obtain the following bi-rewriting
system:
8
>< e x
R1 = > x?1 x
: (x y) z
8
>< e x
R2 = > x?1 x
: (x y) z
?!
?!
?!
?!
?!
?!
x
e
x (y z)
x
e
x (y z)
Notice that each former equation appears as a rewrite rule in both rewrite systems. Therefore the
generation of critical pairs, which is done by looking for overlaps between two rules, one of each rewrite
system, is equivalent to looking for overlaps among the rules of one unique rewrite system with rules
that rewrite equations, since every overlap can be done twice, leading each time to two new inclusions.
The result can be resumed in the standard denition of a critical pair:
Denition 4.3 Given a term rewrite system R and two rules l1 ?! r1 , l2[u] ?! r2 in R where u is
not a variable. If is a most general unier of l1 and u, than hl2 [r1 ]; r2 i is called a critical pair.
Notice that due to symmetry no variable instance pairs need to be generated, i.e. no overlap on
and below variable positions is needed, because now all of them are convergent.
Symmetry plays an important role, because when reasoning with equivalence relations, we can deal
with the notion of equivalence class. Since we do not have two separate rewrite systems any more,
critical pairs are computed by overlapping left-sides of rules of one unique rewrite system, which means
that, if the rewrite system is convergent, each term not only has an irreducible term, the normal form,
but this normal form is also unique for each term. Rewriting is done within an equivalence class, and
all the members of this class share the same normal form. A decision procedure for the word problem
in equational theories, based on convergent rewriting systems, is much simpler than with arbitrary
transitive relations, since no backtracking must be performed (cf. section 4.1). Just the normal forms
of the two terms of the equation we want to validate are computed and checked for identity. The
substantial dierence of bi-rewriting systems with respect to rewrite systems for equational theories
lies in the fact that rewriting is not done within equivalence classes, and therefore the notion of normal
form becomes meaningless.
For a more exhaustive study of rewrite systems refer to the surveys of Dershowitz and Jouannaud
[DJ90], Huet [HO80], Plaisted [Pla93] and Klop [Klo92].
5 Theorem proving with rewrite techniques
As mentioned before in section 3, in order to avoid failure during the completion process, completion
itself could be used as a deduction process for proving theorems, by unfailing completion. The same
9
principle can be generalized for the unfailing completion of bi-rewriting systems. But let us emphasize
these notions by studying the equational case.
It was Huet [Hue81] who suggested the use of the completion procedure not only to obtain convergent rewrite systems but also for theorem proving in equational theories. He showed, that whenever
the completion process does not fail, i.e. all newly generated equations are orientable, it can be used
as a semi-decision procedure for the word problem of equational theories:
Let E be a set of equations, and R the set of rewrite rules obtained by orienting the equations of
E . Suppose we want to check if s !E t. We rst reduce s and t to their normal forms with respect
to R obtaining s0 and t0 respectively. If they are identical, s = t is provable in E . Otherwise, we run
the completion procedure on R, following a suitable fairness criterion5 , nding divergent critical pairs
and adding them as new rewrite rules to R until a new system R0 is obtained, for which s0 or t0 are
reducible. Again the normal forms of s and t (now with respect to R0 ) are computed and checked for
identity. This process is repeated until two identical normal forms are reached, and thus the equation
s = t is provable in E , or no more divergent critical pairs are found, and therefore a convergent rewrite
system is obtained for which the normal forms of s and t are not identical, which means that s = t
is not provable in E . Since the word problem is semi-decidable it may also happen that s = t is
not provable in E , but the completion process never ends, without providing us a convergent rewrite
system. Notice that the idea is to progressively make the divergent critical pairs, that may appear in
the proof of the equation, convergent, until a rewrite proof is obtained. The same principle can be
generalized for arbitrary transitive relations, since we also would like to progressively make convergent
the critical pairs or variable instance pairs in order to nally obtain a bi-rewrite proof.
Unfortunately the completion procedure may fail to orient a critical pair as a rewrite rule and the
process is prone to abort. This makes standard Knuth-Bendix completion unappropriated for theorem
proving purposes. Bachmair, Dershowitz and Plaisted [BDP89] overcame this situation with a variant
of completion, the so called unfailing completion, which is refutationally complete for equational
theories.
5.1 Unfailing completion
In unfailing completion rst of all we attempt to complete a set of equations rather then a set of
rewrite rules, because we want to manage unorientable equations. Even if there is no convergent
rewrite system for a given equational theory, it may still be possible to construct a system of equations
which possesses a weaker Church-Rosser property, namely a Church-Rosser property only on ground
(variable free) terms. Such a system denes unique ground normal forms, which is sucient for most
purposes, including refutational theorem proving, as we will see later.
In this approach we rene the notion of rewriting:
Denition 5.1 Given a reduction ordering , s is rewritten to t and we denote it by s ?!E t, if
s !E t by applying some equation u = v in E with a substitution for which u v. We call
u ?! v an orientable instance of the equation u = v.
Notice that rewriting is performed now by the use of equations rather than rewrite rules. Since
we want the system to be ground Church-Rosser the unfailing completion process needs to add to the
system of equations those ones, which may have as instance a divergent critical pair. These divergent
critical pairs must be generated by the overlap of orientable instances of equations.
Let s = t be the equations obtained by replacing each variable in s and t with a unique Skolem
constant. If the given reduction ordering is complete on T (F [ K; V ), where K is the set of Skolem
constants occurring in s and t, then unfailing completion is refutationally complete: If s = t is provable
5
the fairness criterion guarantees that all possible critical pairs will sooner or later be generated
10
in E , then s = t is also provable in E , and hence unfailing completion can be used as a semi-decision
procedure to nd a ground rewrite proof of s = t. Conversely if s = t is provable in some E 0 obtained
during the unfailing completion process, then s = t is provable in E .
The idea of completion as a refutationally complete deduction process, can also be generalized to
full rst-order clauses, as we show in the next subsection.
5.2 Ordered chaining
Like paramodulation in the rst-order theories with equality (cf. section 3) there have been attempts
to use additional inferences to resolution-based theorem provers in order to avoid the transitivity axiom
of arbitrary transitive relations. As mentioned in section 4, Slagle [Sla72] introduced for these purposes
the chaining inference rule. Chaining can be seen as the generalization of paramodulation for arbitrary
transitive relations, but in this case without chaining on subterms, because the transitive relation
needs not to be monotonic. As paramodulation, chaining has the drawback that it explicitly generates
the transitive closure of the binary relation. As mentioned previously there is an extra cost in this
generalization, since chaining through variables is now needed in general, unlike in paramodulation.
But like ordering restrictions on the paramodulation inference have led to the superposition calculus
(which in essence generalizes the critical pair computation of the unfailing variant of the Knuth-Bendix
completion procedure), so ordering restrictions on the chaining inference rule take the advantages of
rewrite techniques resulting from bi-rewriting systems, and avoid generating the whole closure using
bi-rewrite proofs to prove the validity of a transitive relation. So ordered chaining, introduced by
Bachmair and Ganzinger [BG94c], is chaining with the additional restrictions that u 6 s, v 6 t,
u < s is strictly maximal with respect to ? ! , and v < t is strictly maximal with respect
to ! (cf. the denition of chaining in section 4).
In order to use ordering restrictions in inference calculi we need to dene a simplication ordering
over rst-order terms and we must also be able to extend it to an ordering over literals and clauses.
The latter is done extending orderings to multisets. For a survey about orderings see Dershowitz
[Der87].
Recall the use of unfailing completion as a semi-decision procedure for theorem proving. Like in the
equational case, the generation of new inclusions from critical pairs or variable instance pairs can be
seen as an ordered version of the chaining inference rule. In this context, as in the equational case, the
process of completion, is known as saturation. In analogy to a completion procedure which attempts
to produce a convergent bi-rewriting system, in which all critical pairs (and variable instance pairs,
if the transitive relation is also monotonic) are convergent (have a bi-rewrite proof), the saturation
process attempts to provide us a set of clauses in which all inferences are redundant. We say that
the set of clauses is saturated, i.e. closed up to redundancy. Notice that this is the criterion in order
to nish the process of completion, or saturation respectively. In the same manner as during the
completion process rewrite rules are kept as interreduced as possible, during saturation redundant
clauses are deleted, and redundant inferences avoided. Although from now on we will often come back
to the notion of redundancy, we will dene this notion in more detail later, in the context of putting
all these ideas into practice.
As we will see next, Bachmair and Ganzinger [BG94c] proved the refutational completeness of
several inference calculi based on ordered chaining for Horn clauses and for general rst-order clauses
with arbitrary transitive relations by means of bi-rewriting systems.
5.3 Bi-rewriting systems and the completeness of ordered chaining calculi
As explained in section 4.1, Levy and Agust [LA93] gave the sucient and necessary conditions to
identify those critical pairs that need to converge in order to build a Knuth-Bendix like completion pro11
cedure for inclusional theories. We have seen that by completion we may get a convergent bi-rewriting
system which is a nite representation of the transitive closure of the binary relation. Bachmair and
Ganzinger used these techniques at a more abstract level in the context of their \model construction"
method.
Bachmair and Ganziger [BG90] introduced this method, which is very useful for proving refutational completeness of inference systems based on saturation and which also gives us a very elegant
way to dene the notion of redundancy, as we will see later. It consists in constructing a Herbrand
interpretation I from a saturated set of clauses N in such a way that, whenever N is satisable, I is
a model of N . But whenever N is unsatisable, the minimal counter-example showing that I is not
a model of N is the empty clause. This means that in some step of saturation, the empty clause is
added to the set of clauses. Note that the minimal counter-example is the smallest clause in N (with
respect to the extension to an ordering on clauses of the reduction ordering ) which is false in I .
The construction of this Herbrand interpretation is made inductively over a given total reduction
ordering on ground clauses. For simplifying purposes here we consider clauses without transitive
relations: Starting with the empty interpretation I = ;, we repeat the following steps: First we take
the smallest possible instance C of a clause of N which is false in I . Then we take a positive literal A
of C , which meets some \desired" conditions6 and add it to I . A clause C with such a literal is called
a productive clause. Obviously, C is true in the new interpretation I [ fAg. The result of repeating
these steps is an interpretation with the properties explained above.
For example [BG94d], consider the following ground clauses, which are statements about some
property P of the naturals in Peano notation, and which is a saturated set, in the sense explained
before. The clauses are listed in increasing order:
P (0)
P (0) _ P (succ(0))
P (succ(0)) _ P (succ(succ(0)))
:::
P (succ2n?1 (0)) _ P (succ2n (0))
P (succ2n (0)) _ P (succ2n+1 (0))
:::
This is how the interpretation would be constructed:
Clause set
Interpretation
P (0)
P (0) _ P (succ(0))
P (succ(0)) _ P (succ(succ(0)))
:::
P (succ2n?1 (0)) _ P (succ2n (0))
P (succ2n (0)) _ P (succ2n+1 (0))
:::
P (0)
P (succ(succ(0)))
:::
P (succ2n (0))
:::
The framed literals represent literals of productive clauses which meet the desired condition (in this
simple example it must be the strictly maximal literal of the clause). The constructed interpretation is
the union of all literals appearing under the column Interpretation and is a model of this set of clauses.
Notice, that clause P (0) _ P (succ(0)) is not productive, because it is true in the interpretation fP(0)g
constructed so far.
6
for further details on these conditions cf. [BG90]
12
In the case of full rst-order clauses with arbitrary transitive relations, productive literals will be
ordered inequalities, i.e. rewrite rules of a bi-rewriting system. Therefore the constructed interpretation will be represented by a convergent bi-rewriting system, which nitely encodes the transitive
closure of the binary relation.
5.4 Redundancy
The model construction method gives a very simple way to dene the notion of redundancy, introduced
before: A ground clause C is redundant in the saturated set N of clauses if it is non-productive, and
therefore it does not contribute to the model construction. This means that other clauses produce
the necessary literals, in order that clause C is true in the nal interpretation. A non-ground clause
is redundant if all its ground instances are. Under this model, it is easy to give a sucient condition
for redundancy: A ground clause C is redundant in N if there exist ground instances D1 ; : : : ; Dn
of clauses in N , such that D1 ; : : : ; Dn j= C , and C Di for all i = 1 : : : n. With this denition,
tautologies, subsumed clauses and condensed clauses, for example, are redundant.
A ground inference is redundant when either one of its premises is redundant or else the conclusion
doesn't contribute to the model construction. A non-ground inference is redundant if all its ground
instances are. As before we can give a sucient condition for redundancy: A ground inference with
premises C and D and conclusion B is redundant (C being the maximal premise), whenever there
exist ground instances D1 ; : : : ; Dn such that D1 ; : : : ; Dn j= B , and C Di for all i = 1 : : : n. This
redundancy notion generalizes the idea of conuent critical pairs, i.e. those pairs that have already a
rewrite proof and need not to be added as a new equations to the set of equations.
For example [BG94d], with this notion it is easy to prove the redundancy of superposition on
variables: If we have an ordering on terms such that ground terms u v and f (u; u) g(u), then the
superposition inference
u = v f (x; x) = g(x)
f (u; v) = g(u)
is redundant. It is obvious that the only possible ground instance of this inference, namely
u = v f (u; u) = g(u)
f (u; v) = g(u)
has as maximal premise f (u; u) = g(u). Now we have that u = v and f (v; v) = g(v) j= f (u; v) = g(u),
and f (u; u) = g(u) u = v and f (u; u) = g(u) f (v; v) = g(v).
For further details refer to the work of Bachmair and Ganzinger [BG94b].
6 Putting it into practice
Theorem provers typically consist of two components: On one hand we require a deductive inference
system that is used to generate new formulas, such as the chaining inference rule. On the other hand
we need also techniques for simplifying formulas and eliminating redundancies, because we would like
a theorem prover to be as ecient as possible, avoiding it to generate superuous clauses, or to explore
blind allays of the search space.
In the context of the saturation of a set of full rst-order clauses the theorem prover needs to have
implemented powerful redundancy provers, which allow to drastically cut down the search space of
the theorem prover, by maintaining only the minimum number of necessary clauses, keeping them as
small as possible with respect to the ordering on clauses, and by avoiding those inferences which do
not contribute to the saturation of the set.
These redundancy provers are supposed to detect for example, tautologies, subsumption and condensment within clauses, and they simplify clauses as much as possible, for example by demodulation.
13
Redundancy provers also should notice when a set is yet saturated and no more inferences need to
be performed. Without the use of powerful redundancy provers the saturation of a set of clauses
is practically impossible. We will see that the most important drawbacks when handling arbitrary
transitive relations appear at this point.
6.1 The Saturate Theorem Prover
For the equational case, all these techniques, which have been treated in an abstract level, have been
put into practice in the Saturate theorem prover for rst-order clauses with equality, developed by
Nivela and Nieuwenhuis [NN93], with some additional constraints introduced by Nieuwenhuis and
Rubio to the superposition inference rule, like the use of basic strategies [NR92a] and of constrained
clauses [NR92b], which prune even more the required search space.
Special emphasis was put on the implementation of powerful redundancy provers, since on them lies
the eciency of the theorem prover. So Saturate checks for tautology, subsumption, condensment. It
also simplies clauses by demodulation and makes case analysis by constrained rewriting. Redundant
inferences are checked by clausal rewriting. All these redundancy provers are based on the notion of
redundancy introduced by Bachmair and Ganzinger, and therefore they are based on rewrite proofs.
So for example tautology of a clause is detected as follows:
Proposition 6.1 Clause ? ! is a tautology i S (?) j= S (), where S is a Skolem function
assigning dierent new constants to the variables occurring in the clause.
Saturate completes the set of ground equations S (?) getting a rewrite system R, and checks
whether some equation of S (?) can be proved by rewriting in R.
6.2 Problems for the saturation of general clauses with arbitrary (monotonic)
transitive relations
We have pointed out several dierences which appear when applying rewrite techniques to arbitrary
transitive relations rather than equality. So instead of dealing with a single rewrite relation we have
to manage a bi-rewriting system. No rewriting within equivalence classes is done, which invalidates
the notion of unique normal form on which equational rewrite systems are based. The order of the
application of the rewrite rules is now signicant for a rewrite proof to be found. We also pointed out
in section 4.1 that for the completion of bi-rewriting systems variable instance pairs need to be found
by computing critical pairs between instances of rules.
Though Levy and Agust, and Bachmair and Ganzinger proved the completeness of their respective
inference systems, and Ganzinger has put them into practice extending the Saturate theorem prover to
arbitrary transitive relations by generalizing superposition to ordered chaining, all the dierences with
respect to the equational case appear to be problematic for an ecient application of these rewrite
techniques as we will see in the following subsections.
6.2.1 Redundancy Provers and Rewrite Proofs
Though the model construction method provides a powerful abstract notion of redundancy, which is
indispensable for the implementation of ecient theorem provers, without a corresponding ecient
decision procedure which gives a bi-rewrite proof for transitive relations, we will not be able to have
ecient redundancy provers.
The notion of redundancy which derives from the model construction method is also applicable to
clauses and inferences with arbitrary transitive relations. But recall from section 6.1 that all existing
techniques for redundancy proving in the equational case are based on rewrite proofs. Since in order
14
to nd bi-rewrite proofs based on a bi-rewriting system we need to search among all possible paths
which can be build starting from either term, this is an important drawback for ecient redundancy
provers and hence for ecient theorem proving in these kind of theories. It appears to be necessary
to nd an ecient treatment of this search space rst.
Unfortunately, the techniques of rewriting applied to theories with arbitrary transitive relations
have been treated only in an abstract level, without giving a procedure which provides such bi-rewrite
proofs. In the equational case, once a convergent rewrite system is given, the validity of an equation
can be determined eciently. But, how can this done in the arbitrary case?
Furthermore, in the equality case, theorem provers make use of very powerful redundancy provers
like demodulation, which in practice drastically reduces the number of clauses that need to be managed.
Such simplication techniques have no equivalent for theories with arbitrary transitive relations, which
means that for the refutation of clauses with such relations much more clauses need to be inferred.
We need new redundancy proving techniques to come up with this situation.
6.2.2 Variable Chaining
Brand [Bra75] made an important renement to the paramodulation inference by proving that no
paramodulation through variable positions is required for completeness. We have seen in section 5.4
that this is captured in the redundancy notion based on the model construction, since inferences
through variable position are redundant. Unfortunately this is not yet true with chaining. Chaining
through variables is necessary for completeness. This is an important drawback, because unication
with variables always succeeds, and therefore the inference is very prolic. So in order to make
the theorem proving derivation ecient, we have to avoid as much variable chainings as possible.
Fortunately the ordering restrictions of ordered chaining avoid many of them: The variable x must be
a maximal term in the premise, and therefore x cannot occur in a negative literal and also must be
unshielded in the clause. We say that a variable is unshielded if it only occurs as an argument of <,
but not as an argument of any other predicate or function symbol. In addition the redundancy notion
makes inferences redundant, when the variable x is linear, that is, it only occurs once in the premise.
In these cases the conclusion is subsumed by one of the premises. For further details refer to the work
of Bachmair and Ganzinger [BG94c].
A way to restrict the prolic variable chaining is by studying special theories, for which these kind
of chainings are redundant. In dense total orderings without endpoints, for example, Bledsoe and Hines
[BH80] developed a technique to eliminate variables from formulas. Bachmair and Ganzinger [BG94a]
showed that the variable elimination technique in these theories is a simplication rule captured by
the notion of redundancy, in the sense explained above. So after applying variable elimination to a
clause, it becomes redundant, and can be eliminated. In this way variable chaining through variables
can be completely avoided.
6.2.3 Chaining below Variables
A quite problematic case, in the sense of practicability, is when we handle functions which are monotonic or antimonotonic with respect to the transitive relation, but are not symmetric, i.e. if for some
function f , a < b ?! f (a) < f (b) or a < b ?! f (a) > f (b). In these cases variable subterm chaining
is necessary which leads to a lot of problems if completeness results are wanted.
In section 4.1 we pointed out that computing the critical pairs between rules of each rewrite
relation, as in the Knuth-Bendix completion, was not sucient in order to guarantee the bi-rewriting
system to be Church-Rosser. Levy and Agust [LA93] proved that extended critical pairs need to be
computed. The denition of extended critical pairs is slightly dierent and more restrictive as the one
of variable instance pairs of section 4.1:
15
Denition 6.1 Given a bi-rewriting system B = hR1; R2 i and rules l1 ?! r1 of R1 and l2 ?! r2 of
R2 , such that x is a variable that occurs more than once in l2, and v is an arbitrary term such that l1
is subterm of v and x does not occur in v, and l1 ?!R2 r1 does not hold. If is such that x = v[l1 ]
and y = y, if y =
6 x, then hl2 [v[r1 ]]; r2 i is called an extended critical pair.
Levy and Agust [LA93] extended the notion Church-Rosser known of equational rewrite systems
to bi-rewriting systems intended in order to give a completion procedure for bi-rewriting systems
based
on theories of reexive and monotonic transitive relations (inclusions). Notice that if l1 ?!
R2 r1 holds,
then the extended critical pair converges, as happens when the transitive relations is also symmetric
(cf. section 4.2).
For example with rewrite rules:
x
rule2 : f (x; x) ?!
b
rule1 : a ?!
where x occurs twice in l2 . If v is, for example, the term g(a), and substitutes x with g(a), then
hf (g(b); g(a)); g(a)i is an extended critical pair between rule1 and rule2.
Notice that v may be any term build with function symbols of the signature whenever it has a as
subterm. Therefore, there are innite many extended critical pairs between rules rule1 and rule2 , and
c[a]),
hence innite many new rewrite rules, which can be represented by a rule schema: f (c[b]; c[a]) ?!
where c[ ] can be seen as an arbitrary context.
Generation of extended critical pairs can also be seen as the conclusion of an inference which chains
below variables:
b a f (x; x) x
f (c[b]; c[a]) c[a]
Note that a is unied with some term below variable x, and therefore context c[ ] may be arbitrary.
Computation of extended critical pairs can be avoided by using only left-linear rewrite system (cf.
[Hue80]), that is rewrite system where all left-sides of rules have only single occurrences of variables,
but this is a too strong restriction. Though it is impossible in practice to add innite many new rewrite
rules to a rewrite systems, Levy and Agust [LA93] attempt to build a completion procedure based
on rule schemes, which encode the innite number of new rewrite rules. In this way they were able
to give canonical bi-rewriting systems for the inclusion theory of the union, and of non-distributive
and distributive lattices. Based on this idea of handling extended critical pairs with rule schemes,
Levy proposed in [Lev93] to attack this problem by means of using second-order bi-rewriting systems
in order to handle the contexts mentioned above as second-order variables. In this case the inference
requires the unication of second-order terms. Levy studied the unication of linear second-order
terms (these are the only kind of terms which appear in his Calculus of Renements [Lev94]) and
proposes a semi-decision procedure for it. The decidability of the unication of linear second-order
terms remains an open problem. Additionally, the unication of second-order terms with the procedure
proposed by Levy gives us not only the critical pairs we require for building rewrite proofs, but also
all the pairs which are yet convergent, and hence redundant. This means that the search space is
importantly enlarged if no powerful redundancy provers for second-order terms are developed.
7 Conclusions
As we have seen, rewrite techniques are used for dealing with the equality relation, but naturally
generalize to arbitrary transitive relations and can be used to automate the deduction in theories of
rst-order clauses with arbitrary transitive relations. But unfortunately, for relations which do not
16
fulll the symmetry property, several important problems arise, which put diculties on an ecient
treatment of these kind of relations. So a rewrite proof of an inequality is only obtained by applying
all possible rewrite rules to both terms of the inequality, i.e. all possible branches of the rewrite tree
need to be explored. And since redundancy provers are build on rewrite proofs, no ecient technique
for proving redundancy is known. Furthermore, such indispensable simplication techniques like
demodulation do not exist for arbitrary transitive relations, which means that much more clauses
need to be treated during the inference process than with equality. But it is well known, that theorem
provers without an ecient treatment of redundancy degenerate in a huge search space, making an
automated deduction inference impossible.
In addition the inference rules for reasoning with arbitrary transitive relations themself are much
more prolic as in the equational case. So chaining through variables is required for arbitrary nonsymmetric transitive relations. If the function are monotonic with respect to this relations, chaining
below variables is also needed, which leads to an innite number of new non-redundant clauses. To
manage these clauses second-order terms are needed, with the additional need of redundancy provers
for second-order clauses and inferences.
But we have seen that, though the need of dealing with contexts forces us to handle with secondorder terms, in some theories this can be overcome by the use of some kind of rules schemes. Levy
and Agust [LA93] gave canonical bi-rewriting systems for the inclusion theory of the union, and for
distributive and non-distributive lattices. Therefore it seams that by studying special theories the
second-order case may be avoided.
Also it may be reasonable to focus special attention on the generation of rewrite proofs for transitive
relations in an ecient way, by studying techniques for exploring the rewrite tree. Finally, it is also
necessary to nd new simplication techniques which restrict the search space in a similar way as
demodulation does for theorem provers with equality.
Acknowledgments
We specially thank Robert Nieuwenhuis and Jordi Levy for their valuable comments and helpful
suggestions on previous versions of this paper.
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