Complete Inference Rules f o r t h e Cancellation L a w s
—Extended
1.
Abstract—
Jieh Hsiang
Michael Rusinowitch
Department of Computer Science
CRIN
ICOT
Ko
Sakai
SUNY at Stony Brook
B.P. 239
M i t a Kokusai Bldg, 2 I F
Stony Brook, NY 11794
54506 Vandoeuvre-les-Nancy
1-4-28 M i t a , M i n a t o - k u
U.S.A.
France
108 T o k y o , Japan
Introduction
In
many
cases,
specialiied
inference
rules
which
incorporate certain axioms into the inference mechanism can
produce
fewer redundant consequences and
proofs.
The
most
notable
example
more efficient
is
paramodulation.
Inference rules for inequalities, partial orerings, special binary
relations have also been found ([BIH80], [S1N73], [MaW85]).
Recently
there
has
also
been
considerable
inference rules for the cancellation law.
interests
in
xy =yx
in
ring
inference rule called
consequences
theory.
[W0M86]
introduced
negative paramodulation to
resulting from
methods provide only
cancellation.
These
t w o inference rules, w i t h resolution and paramodulation, form
i t , w i t h the Knuth-Bendix method, to prove t h a t
implies
The soundness of the rules is quite immediate.
Stickel ((Sti84j) used
an
find
useful
However,
these
a complete theorem proving strategy for first order predicate
calculus
with
cancellable.
equality
The
only
when
some
extra axiom
operators
needed
is
ad hoc treatments of cancellation.
are
right
reflexivity,
In this
\
\
paper we present some complete sets of inference rules, which
1
can replace the cancellation axioms. Due to space limitation,
1
These inference rules are not complete and cannot eliminate
the cancellation axiom from the input set of clauses.
we only
outline
the inference
theorem w i t h o u t proofs.
illustrate how
rules and
state the
main
Only simple examples are given to
the rules are used.
The proof and
more
2.2.
Cancellation with Identity
Similar inference rules can be designed for other types of
examples will be given in the full paper.
cancellation laws.
an identity t
2. Inference Rules for Cancellation Laws
We present inference rules for three types of cancellation
with
For example, in an algebraic theory w i t h
for an operator /, there is the cancellation law
identity:
Although this law
laws, the basic cancellation, cancellation w i t h identity, and
is a consequence of right cancellation and left identity, it may
cancellation except the null element.
also be given as an independent axiom without the other two.
2.1.
([Sti84j) has demonstrated the power of incorporating this
The operator + in ring theory satisfies this law, and Stickel
Right Cancellation
A function / is right cancellable if it satisfies the right
cancellation
law
convenience
we
For
only
consider
right
cancellation is handled symmetrically.
cancellation.
axiom intoinference rules.
W i t h this axiom, we modify rule Cl into:
Left
Although cancellation
has only one axiom, it may lead to many resolvents and
paramodulants in a resolution based theorem prover if simply
treated as a clause.
Replacing it w i t h inference rules ensures
It is interesting to note t h a t this inference rule alone suffices
In
to secure completeness, i.e., resolution and paramodulation,
the following we give a "rough-cut" version of the inference
w i t h C I l for each operator satisfying this axiom, form a
t h a t only those " r e l e v a n t " ones are generated and kept.
rules.
A
more
efficient
orderings is given later.
version
involving
We say two lists
simplification
and
complete strategy for first order logic w i t h equality when
some operators are right cancellable w i t h identity.
The rule C2 can also be modified into a rule
unifier such t h a t s, a = f t a for all 1.
990
REASONING
Although
the
reduction
rule
is
similar
to
([WRC67|), there are some subtle differences.
demodulation
Reductions are
based on a well-founded ordering, while demodulation may
not be.
Each reduction step reduces a clause to a "smaller"
one. Reductions can be applied whenever applicable, as much
as possible, without the risk of falling into an infinite loop.
Paramodulation can utilize this ordering on terms and be
improved to the oriented paramodulation inference rule:
3.
Inference
Simplification
Rules
with
Ground-Linear
Orderings
where
The inference rules introduced in the above section can
r
is
be improved substantially (with completeness preserved) and
restriction of
with more powerful equational inference rules such as oriented
other
paramodulatton and
paramodulants.
demodulation, if a notion of simplification
orderings ([Der82]) on the term structure is introduced.
Let
non-variable
is
different
subterm
from
of
D .
Oriented
paramodulation
in
the
is not smaller than any
literal
Using
T(F , X ) be the set of terms and .
a
paramodulation
in
the
C.
This
ground-linear
eliminates
many
simplification
potential
ordering
to
be
compare literals w i t h i n a clause also restricts the cancellation
the set of atoms in the language. Also let T (F ) be the set of
inference rules to be performed only on the literals among the
ground
largest in a clause. The rules C 1 and C 2 become:
terms
Herbrand base.
(Herbrand
universe)
We also consider s — t
same atom.
An ordering
simplification
ordering
if
h
e
n
a
n
d
is
a
ordering
and t — s
partial
has
the
the
as the
is
ordering if it is also total on
a
ordering
monotonic
subterm
superterm of a term is larger than
simplification
I
is a
it
preserves the substitution property, is
t
and
which
(if t <s
property
the term
ground-linear
itself).
(any
A
simplification
The most
important property of these orderings is that they are wellfounded.
Variations of simplification orderings have appeared in
It is interesting to note t h a t if a pre dic ate-first type ground-
A comprehensive discussion is in
linear simplification ordering is used on the term structure
[Der85]. A more detailed description of the version presented
(that is, atoms are compared lexicographically first by their
[Der82], [Pla78], [Pet83],
Hsiang, Rusinowitch, and Sakai
991
predicate symbols, w i t h = as the smallest predicate) then no
cancellation inference need be performed on any clause which
contains
a non-equality literal, since all equality
literals are
smaller than any literal w i t h non-equality predicate.
Even the resolution inference rule can be restricted by
the ordering (into oriented resolution).
For simplicity we give
the binary resolution version here.
[WoM86]
introduced
handling cancellation.
negative
paramodulation
for
This inference rule cannot completely
eliminate the cancellation axiom.
A simple counterexample is
the unsatisfiable set
is right cancellable.
Neither negative paramodulation nor
paramodulation/resolution is applicable to any of the clauses.
However,
negative paramodulation is compatible w i t h the
cancellation inference rules introduced here, and it provides
for backward chaining.
The inclusion of such an inference rule
should further improve the efficiency of a refutational proving
system involving the cancellation axioms.
Oriented resolution is different from ordered resolution or
ordered predicate resolution (see [ChL73]) in t h a t the ordering
5.
on literals are given in a natural way.
[B1H80]
For example, there is
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[ChL73]
Theorem
W.
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[MaW85]
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Discussion
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The cancellation
Proving
inference rules can also be used to
Ordering
([Sti84]).
682-688.
In a prototype implementation these inference rules
[Sti84]
14 critical pairs rather than the usual 17.
[KnB70j
another
way
of dealing
with
the
basic
each cancellable operator, is given.
This method can be
extended easily to first order theory.
The completeness of
their
of
theorem.
is
an
easy
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our
However, their approach seems to produce more
REASONING
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Prover
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Partial
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[WRC67]
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992
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