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Journal ArticleDOI

Paramodulation with Well-founded Orderings

01 Apr 2009-Journal of Logic and Computation (Oxford University Press)-Vol. 19, Iss: 2, pp 263-302
TL;DR: By using a new restricted form of rewriting, a completeness proof of ordered paramodulation for Horn clauses with equality is obtained, where well-foundedness of the ordering suffices.
Abstract: For many years, all existing completeness results for Knuth–Bendix completion and ordered paramodulation required the term ordering ≻ to be well-founded, monotonic and total(izable) on ground terms. Then, it was shown that well-foundedness and the subterm property were enough for ensuring completeness of ordered paramodulation. Here we show that the subterm property is not necessary either. By using a new restricted form of rewriting, we obtain a completeness proof of ordered paramodulation for Horn clauses with equality, where well-foundedness of the ordering suffices. Apart from the theoretical significance of this result, some potential applications motivating the interest of dropping the subterm property are given. The proof of the results included in this article, being still technical in some parts, is pretty much shorter and easier to read than the one we have in the preliminary version of this work presented at the CADE, 2002 conference (Bofill, and Rubio, 2002, CADE, Vol. 2392 of LNAI, pp. 456–470).

Summary (3 min read)

1 Introduction

  • Equations are ubiquitous in mathematics, logic, and many areas of computer science.
  • If the authors take the opposite orientation, i.e., ffg(x)→ fgfg(x) the system is already closed, i.e., standard Knuth-Bendix completion would generate only this additional rule.
  • Another typical situation is in deduction modulo built-in equational theories E. And, even worse, for many theories E such orderings cannot exist.
  • Another (less obvious) potential application of their results may be in goaloriented deduction since, in some cases, a goal-oriented paramodulation proof can only be obtained if the ordering contradicts the subterm property.

1.1 About the proof technique

  • The authors proof follows the lines of Bachmair and Ganzinger’s model generation proof method of [3] and its variation used in [7] for the non-monotonic case.
  • In order to express these blockings the authors use abstracted terms, which have variables in the place of blocked subterms.
  • In fact, most of this paper is devoted to the proof of confluence of the generated rewrite system R. Once confluence is proved, the authors can easily show that there exists a B-rewrite proof with R for every logical consequence of the set of equations from which R has been generated.
  • In Section 5, using the results obtained for the equational case, the authors prove refutation completeness of ordered paramodulation with well-founded orderings for the case of Horn clauses with equality.

2 Basic notions

  • The authors use the standard definitions and notation of [10,17].
  • Note that constants and variables have size 1 (since n = 0).
  • By Newman’s lemma, terminating locally-confluent relations are confluent.
  • (It is commonly assumed that lefthand sides of rules are not variables, and that all variables occurring in a righthand side also occur in the corresponding lefthand side.).
  • Every well-founded ordering can be totalized [20], and hence every well-founded ordering satisfying the subterm property can be extended to a west-ordering.

3 Blocked rewriting

  • Here the authors define a special kind of rewrite relation, called blocked rewriting (Brewriting for short), which is an essential ingredient for their completeness proof.
  • This relation restricts standard rewriting by mainly forbidding rewrite steps below righthand sides introduced in previous rewrite steps.
  • This makes this relation terminating by definition.
  • In order to express these blockings, the authors use abstracted terms, which have variables in the place of the blocked subterms.
  • Two abstracted terms s·γ and u·δ are called equivalent, denoted by s·γ ≡ t·δ, if s and t have the same variable positions and sγ ≡ uδ.

4 Paramodulation with equations

  • This section is devoted to the pure equational case.
  • The authors will also assume that all equations at hand have disjoint variables (if necessary, they can be renamed).

4.1 The inference system

  • Note that this situation is possible, together with lθσ ≻ rθσ, as the ordering ≻ is allowed to contradict the subterm property.
  • The authors must note that the need of paramodulations from variables into variables at top positions has to do with their proof technique.
  • At present, the authors have not found any counterexample showing incompleteness when paramodulation into variables is completely disallowed.
  • Let ≻ be the well-founded ordering used in the ordered paramodulation inference rule E.

4.2 The rewrite system

  • In order to take into account the fact that RE will be used applying Brewriting, and hence the righthand sides will become blocked after each rewrite step, in the construction of RE the authors will not require irreducibility at those positions that are below subterms belonging to Right(RE).
  • The main difference is that left- and righthand sides of rules can be reducible at positions below subterms in Right(RE).
  • For technical reasons, many of the results are proved for R◦E .

4.3 Some relevant properties

  • Recall that the generated RE can be overlapping and non-terminating and, hence, confluence does not follow easily.
  • This property generalizes the notion in [7] of irreducibility for the rules of the ground TRS that defines the model.
  • Altogether, this lets us conclude by the induction hypothesis.
  • Then there are no two rules in RE with the same lefthand side.

4.4 The Return Property

  • This strong property is crucial to prove that their generated RE , in spite of being potentially overlapping and non-terminating, is confluent: after the Return Property is proved, confluence follows from the fact that the lefthand sides only overlap below subterms corresponding to righthand sides.
  • The following lemma states that, in a sequence by B-rewriting with a TRS R, if some step has taken place at the border, w.r.t. Right(R), of the initial term, then it is not possible to get back to the initial term until a normal form has been reached.
  • Now the authors prove the main lemma in which the Return Property is based.
  • Let σ′′ be the substitution of that instance.

4.5 Rewrite proofs for the equational case

  • The Return Property is the cornerstone for proving local confluence.
  • So the diamond closes with a final step with the same rule.
  • The authors proceed by induction on size of the terms without counting the variables, i.e., |u1|v + |u2|v. Lemma 30.
  • Note that rewriting with yRE also provides a rewrite sequence with RE .

6 Conclusion

  • By showing that wellfoundedness of the term ordering is sufficient for completeness of ordered paramodulation for Horn clauses with equality.the authors.
  • The authors hope that the shorter and easier to read proof that they have presented here can open the door to extend their completeness results to deduction modulo any theory E with a decidable E-unification problem.
  • Moreover, like in [7], similar problems concerning the use of demodulation, i.e., simplification by rewriting, arise.
  • The problem of providing concrete redundancy notions for paramodulation with non-monotonic orderings has been addressed in [9], where results on restricted forms of demodulation for paramodulation with well-founded orderings fulfilling the subterm property, are given.
  • The problem is that, within the model generation proof technique, the completeness for general clauses is usually based on the fact that, at most, one positive equation is satisfied in the generated model.

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Paramodulation with well-founded orderings
Miquel Bofill
1
and Albert Rubio
2
1
Universitat de Girona, Dept. IMA, Girona, Spain
mbofill@ima.udg.edu
2
Universitat Polit`ecnica de Catalu nya, Dept. LSI, Barcelona, Spain
rubio@lsi.upc.edu
Abstract For many years, all existing completeness results for Knuth-
Bendix completion and ordered paramodulation required the term order-
ing to be well-founded, monotonic and total(izable) on ground terms.
Then, it was shown that well-foundedness and the subterm property were
enough for ensuring completeness of ordered paramodulation.
Here we show that the subterm property is not necessary either. By us-
ing a new restricted form of rewriting, we obtain a completeness proof
of ordered paramodulation for Horn clauses with equality, where well-
foundedness of the ordering suffices. Apart from the theoretical signifi-
cance of th is result, some potential applications motivating the interest
of dropping the subterm property are given.
The proof of the results included in this paper, being still technical in
some parts, is pretty much shorter and easier to read than the one we
have in the preliminary version of this work presented at the CAD E 2002
conference [8].
Key words: term rewriting, equational reasoning, theorem proving,
paramodulation, Knuth-Bendix completion
1 Introduction
Equations are ubiquitous in mathematics, logic , and many areas of com-
puter science. Knuth-Bendix-like completion techniques and their extensions to
ordered paramodulation for first-order clauses are among the most successful
methods for automated deduction with equality [4,17]. At the present time most,
if not all, state-of-the-art theorem provers for first-order logic with equality are
based on variations o f the ordered paramodulation inference r ule.
These completion and deduction procedures are parameterized by an or-
dering on terms. For many years, all known completeness results for Knuth-
Bendix completion and ordered paramodulation r equired the term ordering
to be well-founded, monotonic and total (or extendable to a total ordering) on
ground terms [11,2,1,3]. Under these strong requirements, powerful abstract re-
dundancy notions (covering many practical re dundancy elimination techniques
such as tautology deletion, subsumption and demodulation) can be applied with-
out losing completeness. However, under ce rtain circumstances, imposing these
strong properties on the ordering can be too restrictive, and hence weakening

them has been a well-known res e arch challenge. For instance, frequently it is
not clear what is a good ordering fo r a particular problem. The following is an
intere sting e xample for that. It shows that sometimes Knuth-B e ndix completion
terminates only if an unusual orientation is chosen for certain equations:
Example 1.
3
Consider the closure under standard Knuth-Bendix completion of
the following two equations:
h(x) f g(x)
hfg(x) ffg(x)
Between these two rules there is a single critical pair since hfg(z) can be rewrit-
ten into both ffg(z) and fgfg(z). Therefore in a Knuth-Bendix completion the
equation
ffg(z) fgf g(z)
should be added. If we work with a well-founded and monotonic total ordering
then the equation is necessarily oriented into
fgfg(z) ffg(z)
(otherwise we will contradict either well-foundedness or monotonicity or totality
on ground terms —see commentaries below—). Unfortunately, with this orienta-
tion standard Knuth-Bendix completion would generate an infinite set of rules:
fgf
n
g(x) f
n+1
g(x) with n 1.
However, if we take the opposite orientation, i.e.,
ffg(x) fgfg(x)
the system is already closed, i.e., standard Knuth-Bendix completion would gen-
erate only this additional rule.
It is important to remark that with this orientation, i.e., with ffg(x)
fgfg(x), the ordering cannot be extended to a total one, since f g(x) and
gfg(x) must be incomparable in any well-founded and monotonic extension of
:
if fg(x) gf g(x) then it contradicts the subterm relation and then by mono-
tonicity we can obtain an infinite decreasing sequence f g(x) gfg(x)
ggfg(x) . . .
if gfg(x) fg(x) then by monotonicity we have fgf g(x) ffg(x) leading
to reflexivity and hence to the existence o f an infinite decreasing sequence.
Notice that, even if we compute inferences with lefthand sides into righthand
sides (in the sense of ordered paramodulation) the system would be clo sed after
adding hfg(x) fgfg(x) to the set.
3
We thank Christopher Lynch for providing us this example.
2

Another typical situatio n is in deduction modulo built-in equational theo-
ries E. There, apart from replacing unification by E-unification, the ordering
needs to be total up to E-equal ground terms and E- c ompatible as well as well-
founded and monotonic (se e e.g. [17]). Unfortunately having all these properties
can be a too strong requirement. For example, the existence of such an ordering
for the case where E consists of associativity and commutativity (AC) prop-
erties for some symbols remained o pen for a long time.
4
And, even worse, for
many theories E such orderings cannot exist. For example, when E contains
an idempotency axiom f(x, x) x, then, if s t, by monotonicity one should
have f (s, s) f (s, t), which by E-compatibility implies s f(s, t) and, hence,
well-foundedness is lost.
The first results on o rdered paramodulation and Knuth-Bendix completion
with weaker orderings were given in [6,7]. There, the monotonicity requirement
was dropped and well-foundedness and the subterm property were s hown to be
sufficient for completeness. Note that any such ordering can be to talized without
losing these two properties. After this, the fundamental question arises whether
more requirements can be dropped.
In this paper we prove that, for the case of Horn clauses with equality, the
subterm property is not necessary either, that is, equations may be ordered
from subterms to superterms such as in a f(a), and ordered paramodulation
remains refutation complete. The only requirement le ft for the ordering is being
well-founded (note that a f(a) ca n be oriented in the well-founded ordering
induced by f
m
(a) f
n
(a) a f(a) with m > n > 1).
This is a new important step in the theory of paramodulation, which shows
the power of ordered paramodulation regardless of the properties of the ordering
that is used, and leaves as the last question whether even well-foundedness is
necessary.
Apart from its theoretical value, this result has its most significant po-
tential application in deduction modulo built-in e quational theories E, since
the requirement o f a (only) well-founded E-compatible ordering does not ex-
clude any theory. Note that, like any denumerable set, the set of E-congruence
classes admits a well-founded ordering and, hence, there is a well-founded E-
compatible ordering for any theory E. Therefore, if our results are extended to
deduction modulo E, the only remaining condition to obtain refutation complete
E-paramodulation calculi relates to the need of computing complete s e ts of E-
unifiers in the paramodulation steps. Moreover, we believe that our results can
be shown to be compatible with basic strategies [5,15]. Then, in fact, as shown
in [1 9,16], if basic strategies a re applied, only decidability of the E-unification
problems is nec e ssary.
Another (less obvious) potential application of our results may be in goal-
oriented deduction since, in some cases, a goal-oriented (ordered) par amodula-
tion pr oof can only be obtained if the o rdering contradicts the subterm property.
Let us illustrate this with a simple example:
4
Once it was found [13,18], it immediately led to a number of follow-up results, such
as the decidability of the ground AC-word and -unification problems [12,14].
3

Example 2. Consider the set of equations
y
log
y
x
x
log
y
y
x
x
x
y+z
x
y
· x
z
and the goal
log
a
b + log
a
c log
a
(b · c).
With the following well-fo unded order ing
y
log
y
x
x
x log
y
y
x
if x is not headed by log
log
y
y
x
x if x is headed by log
x
y+z
x
y
· x
z
which contr adicts the subterm property, we have a goal-oriented proof by ordered
paramodulation (note that some variable renaming have been applied):
x log
y
y
x
log
a
b + log
a
c 6≃ log
a
(b · c)
x
y
1
+z
1
1
x
y
1
1
· x
z
1
1
log
y
y
log
a
b+log
a
c
6≃ log
a
(b · c)
y
log
y
1
x
1
1
x
1
log
y
( y
log
a
b
· y
log
a
c
) 6≃ log
a
(b · c)
y
log
y
1
x
1
1
x
1
log
a
(b · a
log
a
c
) 6≃ log
a
(b · c)
log
a
(b · c) 6≃ log
a
(b · c)
2
Note that there is no such fully goal-oriented proof with respect to a well-
founded ordering including the subterm relation, because no paramodulation
step can then be applied on the goal: for the first two equations, due to the
subterm property, only the term on the left c an be used and it does not unify
with any subterm of the goal; for the last equation neither side overlaps with
the goal.
This simple e xample illustra tes tha t certain expansions of terms, which are
possible only if the ordering contradicts the subterm property, can help in ob-
taining goal-oriented (ordered) paramodulation proofs. Observe, however, tha t
4

the unusual orientation x log
y
y
x
is only taken if x is not headed by log.
Therefore, we are no t suggesting that it is reasonable to have rules which always
have a variable as lefthand side, but having the variable as lefthand side only
for some substitutions. Roughly speaking, this approach can be seen as a kind
of definition unfolding. The idea is that we can apply such kind of expansion
only finitely many times since, although the ordering can contradict the sub-
term property, it must be well-founded.
1.1 About the proof technique
Our proof follows the lines of Bachmair and Ganzinger’s model generation
proof method of [3] and its variation used in [7] for the non-monotonic case.
In this method for proving completeness o f or dered paramodulation ca lculi, a
(possibly infinite) ground rewrite system is generated from a clo sed (or saturated,
i.e., closed up to redundancy) set of clauses. This rewrite sy stem R defines a n
equality Herbrand interpretation
R
, that is shown to be a model of the set
of clauses whenever the empty clause is not contained in it (hence the name
of model generation”). The rules in the rewrite system are oriented by the
ordering that parameterizes the ordered paramodulation inference rule, and
confluence and termination are crucial properties for the completeness proof. In
the original proof of [3], confluence is e nsured by the fact that the rewrite system
is non-overlapping, while ter mination is due to the fa c t that is a well-founded
and monotonic ordering. In the variatio n us e d in [7], confluence is also ensured
by the fact that the rewrite system is non-overlapping, but termination is due to
the fact the righthand sides are irreducible. In our proof, the rewrite system is
neither non-overla pping nor terminating (note that rules of the form a f (a)
necessarily cause non-termination of standard rewriting).
To avoid non-termination and keep the simplicity o f the original proof, we
define a new restricted form of rewriting, called blocked rewriting (B-rewriting for
short) which, roughly, blocks the introduced righthand side for future rewriting
steps. More precisely, when a rewrite step takes place at some position p, then
all positions below p —that is, p and all positions strictly below p are marked
as blo cked, and further rewrite steps at blocked pos itions are disallowed.
In order to express these blockings we use abstracted terms, which have vari-
ables in the place of blocked subterms. Blocked subterms are stored in a sub-
stitution. More precisely, an abstracted term t · γ is a term split into two parts:
the abstraction (or term) part t and the substitution part γ, such that is a
ground ter m. Variable positions in the term part t correspond to positions at
which a rewrite step ha s taken place, while the s ubstitution part γ stores the
righthand sides tha t have b e en introduced at these positions. That is, we write
t · γ y t[x]
p
· (γ {x 7→ r}), where x is a fresh variable, if |
p
is l, l r
is a (ground) rewrite rule, and p is a non-variable position of t. Note that this
relation is well-founded, since p is required to be a non-variable position of t
and at every step the number of non-var iable symbols of the term part t strictly
decreases. Figure 1 shows a rewrite sequence by B-rewriting.
5

Citations
More filters
Journal ArticleDOI
TL;DR: A complete ordered paramodulation calculus for non-monotonic orderings which is compatible with powerful redundancy notions including demodulation is presented, hence strictly improving the previous results and making the calculus more likely to be used in practice.
Abstract: Ordered paramodulation and Knuth-Bendix completion are known to remain complete when using non-monotonic orderings. However, these results do not imply the compatibility of the calculus with essential redundancy elimination techniques such as demodulation, i.e., simplification by rewriting, which constitute the primary mode of computation in most successful automated theorem provers. In this paper we present a complete ordered paramodulation calculus for non-monotonic orderings which is compatible with powerful redundancy notions including demodulation, hence strictly improving the previous results and making the calculus more likely to be used in practice. As a side effect, we obtain a Knuth-Bendix completion procedure compatible with simplification techniques, which can be used for finding, whenever it exists, a convergent term rewrite system for a given set of equations and a (possibly non-totalizable) reduction ordering.

8 citations


Cites background from "Paramodulation with Well-founded Or..."

  • ...And in [9, 11] it was shown that well-foundedness of the ordering suffices for completeness of ordered paramodulation for Horn clauses, i....

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References
More filters
Book ChapterDOI
01 Jan 1990
TL;DR: In this paper, the authors focus on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained.
Abstract: Publisher Summary This chapter focuses on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained. As a formalism, rewrite systems have the full power of Turing machines and may be thought of as nondeterministic Markov algorithms over terms rather than strings. The theory of rewriting is in essence a theory of normal forms. To some extent, it is an outgrowth of the study of A. Church's Lambda Calculus and H. B. Curry's Combinatory Logic. The chapter discusses the syntax and semantics of equations from the algebraic, logical, and operational points of view. To use a rewrite system as a decision procedure, it must be convergent. The chapter describes this fundamental concept as an abstract property of binary relations. To use a rewrite system for computation or as a decision procedure for validity of identities, the termination property is crucial. The chapter presents the basic methods for proving termination. The chapter discusses the question of satisfiability of equations and the convergence property applied to rewriting.

1,551 citations

Book
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TL;DR: In this article, the authors focus on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained.
Abstract: Publisher Summary This chapter focuses on rewrite systems, which are directed equations used to compute by repeatedly replacing sub-terms of a given formula with equal terms until the simplest form possible is obtained. As a formalism, rewrite systems have the full power of Turing machines and may be thought of as nondeterministic Markov algorithms over terms rather than strings. The theory of rewriting is in essence a theory of normal forms. To some extent, it is an outgrowth of the study of A. Church's Lambda Calculus and H. B. Curry's Combinatory Logic. The chapter discusses the syntax and semantics of equations from the algebraic, logical, and operational points of view. To use a rewrite system as a decision procedure, it must be convergent. The chapter describes this fundamental concept as an abstract property of binary relations. To use a rewrite system for computation or as a decision procedure for validity of identities, the termination property is crucial. The chapter presents the basic methods for proving termination. The chapter discusses the question of satisfiability of equations and the convergence property applied to rewriting.

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TL;DR: This paper forms an abstract notion of redundancy and shows that the deletion of redundant clauses during the theorem proving process preserves refutation completeness, and presents various refutationally complete calculi for first-order clauses with equality that allow for arbitrary selection of negative atoms in clauses.
Abstract: We present various refutationally complete calculi for first-order clauses with equality that allow for arbitrary selection of negative atoms in clauses. Refutation completeness is established via the use of well-founded orderings on clauses for defining a Herbrand model for a consistent set of clauses. We also formulate an abstract notion of redundancy and show that the deletion of redundant clauses during the theorem proving process preserves refutation completeness. It is often possible to compute the closure of nontrivial sets of clauses under application of non-redundant inferences. The refutation of goals for such complete sets of clauses is simpler than for arbitrary sets of clauses, in particular one can restrict attention to proofs that have support from the goals without compromising refutation completeness. Additional syntactic properties allow to restrict the search space even further, as we demonstrate for so-called quasiHorn clauses. The results in this paper contain as special cases or generalize many known results about Knuth-Bendix-like completion procedures (for equations, Horn clauses, and Horn clauses over built-in Booleans), completion of first-order clauses by clausal rewriting, and inductive theorem proving for Horn clauses.

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324 citations

Frequently Asked Questions (2)
Q1. What are the contributions in "Paramodulation with well-founded orderings" ?

Here the authors show that the subterm property is not necessary either. The proof of the results included in this paper, being still technical in some parts, is pretty much shorter and easier to read than the one the authors have in the preliminary version of this work presented at the CADE 2002 conference [ 8 ]. Apart from the theoretical significance of this result, some potential applications motivating the interest of dropping the subterm property are given. 

The authors hope that the shorter and easier to read proof that they have presented here can open the door to extend their completeness results to deduction modulo any theory E with a decidable E-unification problem. Although these potential applications are interesting, additional techniques for redundancy elimination, like simplification by rewriting, should be investigated in this framework to make the calculus useful in practice. However, the authors have found no counterexample showing that subsumption can cause incompleteness. B-rewriting is closely related to basic strategies ( see [ 5,15 ] ), which the authors believe can be compatible with their results.