Download One Sound and Complete R-Calculus with Pseudo

Journal of Computer and Communications, 2013, 1, 20-25
http://dx.doi.org/10.4236/jcc.2013.15004 Published Online October 2013 (http://www.scirp.org/journal/jcc)
One Sound and Complete R-Calculus with
Pseudo-Subtheory Minimal Change Property*
Wei Li1, Yuefei Sui2
1
State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing, China;
Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Sciences, Beijing,
China.
Email: [email protected], [email protected]
2
Received July 2013
ABSTRACT
The AGM axiom system is for the belief revision (revision by a single belief), and the DP axiom system is for the
iterated revision (revision by a finite sequence of beliefs). Li [1] gave an R-calculus for R-configurations ∆ | Γ, where
∆ is a set of atomic formulas or the negations of atomic formulas, and Γ is a finite set of formulas. In propositional
logic programs, one R-calculus N will be given in this paper, such that N is sound and complete with respect to operator
s (∆, t ) , where s (∆, t ) is a pseudo-theory minimal change of t by ∆ .
Keywords: Belief Revision; R-Calculus; Soundness and Completeness of a Calculus; Pseudo-Subtheory
1. Introduction
The AGM axiom system is for the belief revision
(revision by a single belief) [2-5], and the DP axiom
system is for the iterated revision (revision by a finite
sequence of beliefs) [6,7]. These postulates list some
basic requirements a revision operator Γ  Φ (a result
of theory Γ revised by Φ ) should satisfy.
The R -calculus ([1]) gave a Gentzen-type deduction
system to deduce a consistent one Γ′ ∪ ∆ from an inconsistent theory Γ ∪ ∆, where Γ′ ∪ ∆ should be a
maximal consistent subtheory of Γ ∪ ∆ which includes
∆ as a subset, where ∆ | Γ is an R-configuration, Γ
is a consistent set of formulas, and ∆ is a consistent
sets of atomic formulas or the negation of atomic formulas. It was proved that if ∆ | Γ ⇒ ∆ | Γ′ is deducible
and ∆ | Γ′ is an R-termination, i.e., there is no R-rule to
reduce ∆ | Γ′ to another R-configuration ∆ | Γ′′, then
∆ ∪ Γ′ is a contraction of Γ by ∆.
The R -calculus is set-inclusion, that is, Γ, ∆ are taken as belief bases, not as belief sets [8-11]. In the following we shall take ∆, Γ as belief bases, not belief sets.
We shall define an operator s (∆, t ), where ∆ is a
set of theories and t is a theory in propositional logic
programs, such that
*
This work was supported by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE2010KF-06, Beijing University of Aeronautics and Astronautics, and
by the National Basic Research Program of China (973 Program) under
Grant No. 2005CB321901.
Copyright © 2013 SciRes.
•
•
∆, s (∆, t ) is consistent;
s (∆, t ) is a pseudo-subtheory of t ;
• s (∆, t ) is maximal such that ∆, s(∆, t ) is consistent,
and for any pseudo-subtheory η of t , if s (∆, t ) is
a pseudo-subtheory of η and η is not a pseudosubtheory of η then either ∆,η  s (∆, t ) and
∆, s (∆, t )  η , or ∆,η is inconsistent.
Then, we give one R -calculus N such that N is sound
and complete with respect to operator s (∆, t ), where
 N is sound with respect to operator s (∆, t ), if
∆ | t ⇒ ∆, s being provable implies s = s (∆, t ), and
 N is complete with respect to operator s (∆, t ), if
∆ | t ⇒ ∆, s (∆, t ) is provable.
Let  be the pseudo-subtheory relation, P (t ) be
the set of all the pseudo-subtheories of t , and ≡ ∆ be
an equivalence relation on P (t ) such that for any
η1 ,η2 ∈ P(t ),η1 ≡ ∆ η2 iff ∆,η1  ∆,η2 . Given a
pseudo-subtheory η  t , let [η ] be the equivalence
class of r with respect to ≡ ∆ .
About the minimal change, we prove that [ s (∆, t )] is
 -maximal in P(t )/ ≡ ∆ such that ∆, s (∆, t ) is consistent, that is,
♦ ∆, s (∆, t ) is consistent; and
♦ for any η such that [ s (∆, t )]  [η ]  [t ], either
[η ]  [ s (∆, t )] or ∆,η is inconsistent.
[ s (∆, t )] being  -maximal implies that s (∆, t ) is a
minimal change of t by ∆ in the syntactical sense, not
in the set-theoretic sense, i.e., s (∆, t ) is a minimal
change of t by ∆ in the theoretic form such that s (∆, t )
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One Sound and Complete R-Calculus with Pseudo-Subtheory Minimal Change Property
is consistent with ∆.
The paper is organized as follows: the next section
gives the basic elements of the R-calculus and the definition of subtheories and pseudo-subtheories; the third
section defines the R-calculus N; the fourth section proves that N is sound and complete with respect to the
operator s (∆, t ); the fifth section discusses the logical
properties of t and s (∆, t ), and the last section concludes the whole paper.
2. The R-Calculus
The R-calculus ([1]) is defined on a first-order logical
language. Let L′ be a logical language of the first-order
logic; ϕ ,ψ formulas and Γ, ∆ sets of formulas (theories), where ∆ is a set of atomic formulas or the negations of atomic formulas.
Given two theories Γ and ∆, let ∆ | Γ be an Rconfiguration.
The R-calculus consists of the following axiom and
inference rules:
( A ¬ ) ∆¬
, ϕ | Γϕ , ⇒∆ϕ , Γ|
Γ , ϕ  ψ ϕ  T ψ Γ 2 ,ψ  χ ∆ | χ , Γ 2 ⇒ ∆ | Γ 2
(R cut ) 1
∆ | ϕ , Γ1 , Γ 2 ⇒ ∆ | Γ1 , Γ 2
∆ | ϕ, Γ ⇒ ∆ | Γ
∆∧| ϕ Γ
ψ , ⇒∆ Γ|
∆ | ϕ, Γ ⇒ ∆ | Γ ∆ |ψ , Γ ⇒ ∆ | Γ
(R ∨ )
∆∨| ϕ Γ
ψ , ⇒∆ Γ|
∆¬| Γϕ , ⇒∆ Γ| ∆ Γ| ψ , ⇒∆ Γ|
(R → )
∆ | ϕ →ψ , Γ ⇒ ∆ | Γ
∆ | ϕ[t / x], Γ ⇒ ∆ | Γ
(R ∀ )
∆ | ∀xϕ , Γ ⇒ ∆ | Γ
(R ∧ )
where in R cut , ϕ  T ψ means that ϕ occurs in the
proof tree T of ψ from Γ1 and ϕ ; and in R ∀ , t is
a term, and is free in ϕ for x .
Definition 2.1. ∆ | Γ ⇒ ∆ ′ | Γ′ is an R-theorem,
denoted by R ∆ | Γ ⇒ ∆ ′ | Γ′, if there is a sequence
{(∆ i , Γi , ∆ i ′ , Γi ′ ) : i ≤ n} such that
(i) ∆ | Γ ⇒ ∆ ′ | Γ′ = ∆ n | Γ n ⇒ ∆ n′ | Γ n′ ,
(ii) for each 1 ≤ i ≤ n, either ∆ i | Γi ⇒ ∆ i′ | Γi′ is an
axiom, or ∆ i | Γi ⇒ ∆ i′ | Γi′ is deduced by some R-rule
∆ | Γ ⇒ ∆ i′−1 | Γi′−1
of form i −1 i −1
.
∆ i | Γi ⇒ ∆ i′ | Γi′
Definition 2.2. ∆ | Γ ⇒ ∆ | Γ′ is valid, denoted by
 ∆ | Γ ⇒ ∆ | Γ′, if for any contraction Θ of Γ′ by
∆, Θ is a contraction of Γ by ∆.
Theorem 2.3(The soundness and completeness theorem of the R-calculus). For any theories Γ, Γ′ and ∆,
 ∆ | Γ ⇒ ∆ | Γ′
if and only if
Copyright © 2013 SciRes.
21
 ∆ | Γ ⇒ ∆ | Γ′.
Theorem 2.4. The R-rules preserve the strong validity.
Let L be the logical language of the propositional
logic. A literal l is an atomic formula or the negation of
an atomic formula; a clause c is the disjunction of
finitely many literals, and a theory t is the conjunction
of finitely many clauses.
Definition 2.5. Given a theory t , a theory s is a
sub-theory of t , denoted by s  t , if either t = s, or
(i) if t = ¬t1 then s  t1 ;
(ii) if t = t1 ∧ t 2 then either s  t1 or s  t2 ; and
(iii) if t = c1 ∨ c2 then either s  c1 or s  c2 .
Let t = ( p ∨ q ) ∧ ( p ′ ∨ q ′). Then,
p ∨ q, p ′ ∨ q ′  t ;
and
p ∧ p ′, q ∧ p ′, p ∧ ( p ′ ∨ q ′)  t.
Definition 2.6. Given a theory t[ s1 ,..., sn ], where s1
is an occurrence of s1 in t , a theory
s = t[λ ,..., λ ] = t[ s1 / λ ,..., sn / λ ], where the occurrence
si is replaced by the empty theory λ , is called a pseudo-subtheory of t , denoted by s  t.
Let t = ( p ∨ q ) ∧ ( p ′ ∨ q ′). Then,
p ∨ q, p ′ ∨ q ′, p ∧ p ′, q ∧ p ′, p ∧ ( p ′ ∨ q ′)  t.
Proposition 2.7. For any theories t1 , t2 , s1 and s2 ,
(i) s1  t1 implies s1  t1 ∨ t2 and s1  t1 ∧ t2 ;
(ii) s1  t1 and s2  t2 imply
¬s1  ¬t1 , s1 ∨ s2  t1 ∨ t2 and s1 ∧ s2  t1 ∧ t2 .
Proposition 2.8. For any theories t and s, if
s  t then s  t.
Proof. By the induction on the structure of t.
Proposition 2.9.  and  are partial orderings on
the set of all the theories.
Given a theory t , let P (t ) be the set of all the pseudo-subtheories of t. Each s ∈ P(t ) is determined by a
set τ ( s ) = {[ p1 ],...,[ pn ]}, where each [ pi ] is an
occurrence of pi in t , such that
s = t ([ p1 ] / λ ,...,[ pn ] / λ ).
Given any s1 , s2 ∈ P (t ), define
s1  s2 = max{s : s  s1 , s  s2 };
s1  s2 = min{s : s  s1 , s  s2 }.
Proposition 2.10. For any pseudo-subtheories
s1 , s2 ∈ P (t ), s1  s2 and s1  s2 exist.
Let P (t ) = ( P (t ), , , t , λ ) be the lattice with the
greatest element t and the least element λ.
Proposition 2.11. For any pseudo-subtheories
s1 , s2 ∈ P (t ), s1  s2 if and only if τ ( s1 ) ⊇ τ ( s2 ). Moreover,
τ ( s1  s2 ) = τ ( s1 )∪τ ( s2 );
τ ( s1  s2 ) = τ ( s1 ) ∩τ ( s2 ).
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One Sound and Complete R-Calculus with Pseudo-Subtheory Minimal Change Property
22
¬l1 | l1
⇒ ¬l1
¬l1 | l2
⇒ ¬l1 , l2
¬l1 | l1 ∨ l2
⇒ ¬l1 , λ ∨ l2 ≡ ¬l1 , l2
¬l1 , l2 | l1
⇒ ¬l1 , l2
∆, s1 | t2 ⇒ ∆, s1 , s2 . Therefore, ∆ | t1 ∧ t2 ⇒ ∆, s1 , s2
and s = s1 ∧ s2 .
If t = c1 ∨ c2 then by the induction assumption, there
are theories s1 , s2 such that ∆ | c1 ⇒ ∆, s1 and
∆ | c2 ⇒ ∆, s2 . Therefore, ∆ | c1 ∨ c2 ⇒ ∆, s1 ∨ s2 and
s = s1 ∨ s2 .
Proposition 3.3. If ∆ | t ⇒ ∆, s is N-provable then
s  t.
Proof. We prove the proposition by the induction on
the length of the proof of ∆ | t ⇒ ∆, s.
If the last rule used is ( N1a ) then t = l , and
s = l  t = l;
If the last rule used is ( N 2a ) then t = l , and
s = λ  t = l;
If the last rule used is ( N ∧ ) then ∆ | t1 ⇒ ∆, s1 and
∆, s1 | t2 ⇒ ∆, s1 , s2 . By the induction assumption,
s1  t1 and s2  t2 . Hence, s1 ∧ s2  t1 ∧ t2 = t ;
If the last rule used is ( N ∧ ) then ∆ | c1 ⇒ ∆, s1 and
∆ | c2 ⇒ ∆, s2 . By the induction assumption, s1  t1
and s2  t2 . Hence, s1 ∨ s2  c1 ∨ c2 = t.
Proposition 3.4. If ∆ | t ⇒ ∆, s is N-provable then
∆∪{s} is consistent.
Proof. We prove the proposition by the induction on
the length of the proof of ∆ | t ⇒ ∆, s.
If the last rule used is ( N1a ) then ∆  ¬l , and
∆ | l ⇒ ∆, l. Then, ∆∪{l} is consistent;
If the last rule used is ( N 2a ) then ∆  ¬l , and
∆ | l ⇒ ∆, λ. Then, ∆∪{λ} is consistent;
If the last rule used is ( N ∧ ) then ∆ | t1 ⇒ ∆, s1 and
∆, s1 | t2 ⇒ ∆, s1 , s2 . By the induction assumption,
∆∪{s1} and ∆∪{s1 , s2 } is consistent, and so is
∆∪∪
{s1 ∧ s2 } = ∆ {s};
If the last rule used is ( N ∨ ) then ∆ | c1 ⇒ ∆, s1 and
∆ | c2 ⇒ ∆, s2 . By the induction assumption, ∆∪{s1}
and ∆∪{s2 } is consistent, and so is
∆∪∪
{s1 ∨ s2 } = ∆ {s}.
¬l1 , l2 | ¬l2
⇒ ¬l1 , l2
4. The Completeness of the R-Calculus N
3. The R-Calculus N
The deduction system N:
∆  ¬l
∆ | l ⇒ ∆, l
∆ | t1 ⇒ ∆, s1
(N ∧ )
∆ | t1 ∧ t2 ⇒ ∆, s1 | t2
( N1a )
(N ∨ )
( N 2a )
∆  ¬l
∆ | l ⇒ ∆, λ
∆ | c1 ⇒ ∆, d1 ∆ | c2 ⇒ ∆, d 2
∆ | c1 ∨ c2 ⇒ ∆, d1 ∨ d 2
where ∆,t denotes a theory ∆∪{t}; λ is the empty
string, and if s is consistent then
λ∨s ≡ s∨λ ≡ s
λ ∧s ≡ s∧λ ≡ s
∆, λ ≡ ∆
and if s is inconsistent then
λ∨s ≡ s∨λ ≡ λ
λ ∧s ≡ s∧λ ≡ λ
Definition 3.1. ∆ | t ⇒ ∆, s is N-provable if there is a
statement sequence {∆ i | t i ⇒ ∆ i , si : 1 ≤ i ≤ n} such that
∆ | t ⇒ ∆, s = ∆ n | tn ⇒ ∆ n , sn ,
and for each i ≤ n, ∆ i | ti ⇒ ∆ i , si is either by an N a rule or by an N ∧ -,or N ∨ -rule.
An example is the following deduction for
¬l1 | l1 ∨ l2 , l1 ∨ ¬l2 :
¬∨
l1 , l2 | l1 ¬⇒¬
l2
l1 , l2
¬l1 | l1 ∨∨l2 , l1 ¬⇒
l2 ¬∨ l1 ,¬l2 | l1
l2
⇒ ¬l1 , l2 .
Notice that ¬l1 | l1 ⇒ ¬l1 and
l1 ≡ (l1 ∨ l2 ) ∧ (l1 ∨ ¬l2 ).
Theorem 3.2. For any theory set ∆ and theory t ,
there is a theory s such that ∆ | t ⇒ ∆, s is N-provable.
Proof. We prove the theorem by the induction on the
structure of t.
If t = l is a literal then either ∆  ¬l or ∆  ¬l.
If ∆  ¬l then ∆ | l ⇒ ∆, λ and s = λ ; if ∆  ¬l
then ∆ | l ⇒ ∆, l and s = l ;
If t = t1 ∧ t2 then by the induction assumption, there
are theories s1, s2 such that ∆ | t1 ⇒ ∆, s1 and
Copyright © 2013 SciRes.
For any theory t , define s (∆, t ) as follows:
if t = l and ∆  ¬ l
λ

if t = l and ∆  ¬ l
l

s (∆, t ) =  s (∆, t1 ) ∧ s (∆∪{s (∆, t1 )}, t2 )

if t = t1 ∧ t2

 s (∆, t ) ∨ s (∆, t ) if t = t ∨ t

1
2
1
2
About the inconsistence, we have the following facts:
• if ∆  ¬l then ∆∪{l} is inconsistent;
• ∆∪{t1 ∧ t2 } is inconsistent if and only if either
∆∪{t1} is inconsistent or ∆∪{t1 , t2 } is in- consistent;
• ∆∪{c1 ∨ c2 } is inconsistent if and only if both
∆∪{c1} and ∆∪{c2 } are inconsistent.
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One Sound and Complete R-Calculus with Pseudo-Subtheory Minimal Change Property
Proposition 4.1. For any consistent theory set ∆ and
a theory t , ∆∪{s (∆, t )} is consistent.
Proof. We prove the proposition by the induction on
the structure of t.
If t = l and l is consistent with ∆ then s (∆, l ) = l
is consistent with ∆; if t = l and l is inconsistent
with ∆ then s (∆, l ) = λ is consistent with ∆;
If t = t1 ∧ t2 then by the induction assumption,
∆∪{s (∆, t1 )} and ∆∪∪
{s (∆, t1 ), s (∆ {s (∆, t1 )}, t2 )} are
consistent, so ∆∪{s (∆, t1 ∧ t2 )} is consistent;
If t = c1 ∨ c2 then by the induction assumption,
∆∪{s (∆, c1 )} and ∆∪{s (∆, c2 )} are consistent, so
∆∪∪
{s (∆, c1 ∨ c2 )} = ∆ {s (∆, c1 ) ∨ s (∆, c2 )} is consistent.
About the consistence, we have the following facts:
• if ∆  ¬l then ∆∪{l} is consistent;
• ∆∪{t1 ∧ t2 } is consistent if and only if ∆∪{t1} is
consistent and ∆∪{t1 , t2 } is consistent;
• ∆∪{c1 ∨ c2 } is consistent if and only if either
∆∪{c1} or ∆∪{c2 } is consistent.
Theorem 4.2. If ∆∪{t} is consistent then
∆, t  s (∆, t ) and ∆, s (∆, t )  t.
Proof. We prove the theorem by the induction on the
structure of t.
If t = l and l is consistent with ∆ then
s (∆, l ) = l , and the theorem holds for l ;
If t = t1 ∧ t2 then ∆∪{t1} and ∆∪{t1 , t2 } is consistent, and by the induction assumption,
∆, t1
 s (∆, t1 )
∆, s (∆, t1 )
 t1 ;
∆, s1 , t2
 s (∆∪{s1}, t2 )
∆, s (∆∪{s1}, t2 )  t2 ,
where s1 = s (∆, t1 ). Hence,
∆, t1 ∧ t2
 s (∆, t1 ) ∧ s (∆∪{s1}, t2 )
∆, s (∆, t1 ) ∧ s (∆∪{s1}, t2 )
 t1 ∧ t2 .
If t = c1 ∨ c2 then either ∆∪{c1} or ∆∪{c1 , c2 } is
consistent, and by the induction assumption, either
∆, c1
 s (∆, c1 )
∆, s (∆, c1 )  c1 ;
or
∆, c2
 s (∆, c2 )
∆, s (∆, c2 )  c2 .
Hence, we have
∆, c1 ∨ c2
 s (∆, c1 ) ∨ s (∆, c2 )
∆, s (∆, c1 ) ∨ s (∆, c2 )  c1 ∨ c2 .
Theorem 4.3. ∆ | t ⇒ ∆, s is N-provable if and only
if s = s (∆, t ).
Proof. (⇒) Assume that ∆ | t ⇒ ∆, s is N-provable.
We assume that for any i < n, the claim holds.
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23
If t = l and the last rule is ( N1a ) then ∆  ¬l and
∆ | l ⇒ ∆, l. It is clear that s = l = s (∆, l );
If t = l and the last rule is ( N 2a ) then ∆  ¬l and
∆ | l ⇒ ∆, λ. It is clear that s = λ = s (∆, l );
If t = t1 ∧ t2 and the last rule is ( N ∧ ) then
∆ | t1 ⇒ ∆, s1 and ∆ | t1 ∧ t2 ⇒ ∆, s1 | t2 ⇒ ∆, s1 , s2 . By
the induction assumption, s (∆, t1 ) = s1 and
s (∆∪{s1}, t2 ) = s2 . Then,
s = s1 ∧ s2 = s (∆, t1 ) ∧ s (∆∪{s1}, t2 ) = s (∆, t1 ∧ t2 );
If t = c1 ∨ c2 and the last rule is ( N ∨ ) then
∆ | c1 ⇒ ∆, s1 and ∆ | c2 ⇒ ∆, s2 . By the induction
assumption, s1 = s (∆, c1 ), s2 = s (∆, c2 ), and
s = s1 ∨ s2 = s (∆, c1 ) ∨ s (∆, c2 ) = s (∆, c1 ∨ c2 ).
(⇐) Let s = s (∆, t ). We prove that ∆ | t ⇒ ∆, s is
N-provable by the induction on the structure of t.
If t = l and ∆  ¬l then s (∆, l ) = λ , and
∆ | l ⇒ ∆, λ , i.e., ∆ | l ⇒ ∆, s;
If t = l and ∆  ¬l then s (∆, l ) = l , and
∆ | l ⇒ ∆, l , i.e., ∆ | l ⇒ ∆, s;
If t = t1 ∧ t2 then
s (∆, t1 ∧ t2 ) = s (∆, t1 ) ∧ s (∆∪{s (∆, t1 )}, t2 ). By the induction assumption, ∆ | t1 ⇒ ∆, s (∆, t1 ) and
∆, s1 | t2 ⇒ ∆, s1 , s (∆∪{s (∆, t1 )}, t2 ). Therefore,
∆ | t1 ∧ t2 ⇒ ∆, s1 , s (∆∪{s (∆, t1 )}, t2 );
If t = c1 ∨ c2 then
s (∆, c1 ∨ c2 ) = s (∆, c1 ) ∨ s (∆∪{s (∆, c1 )}, c2 ). By the induction assumption, ∆ | c1 ⇒ ∆, s (∆, c1 ) and
∆ | c2 ⇒ ∆, s (∆, c2 ). Therefore,
∆ | c1 ∨ c2 ⇒ ∆, s (∆, c1 ) ∨ s (∆, c2 ).
5. The Logical Properties of t and s (∆ , t )
It is clear that we have the following
Proposition 5.1. For any theory set ∆ and theory t ,
ξ (∆, t )  s (∆, t ).
Theorem 5.2. For any theory set ∆ and theory t ,
∆, ξ (∆, t )  s (∆, t );
∆, s (∆, t )  ξ (∆, t ).
Proof. By the definitions of s (∆, ξ ), ξ (∆, t ) and the
induction on the structure of t.
Proposition 5.3. (i) If ∆, s (∆, t )  t then ∆,t is inconsistent;
(ii) If ∆, s (∆, t )  t then ∆,t is consistent.
Define
Ct∆ = {s ∈ P(t ) : ∆∪{s} is consistent};
I t∆ = {s ∈ P(t ) : ∆∪{s} is inconsistent}.
Then, Ct∆ ∪I t∆ = P(t ) and Ct∆ ∩ I t∆ = ∅.
Define an equivalence relation ≡ ∆ on P(t ) such
that for any s1 , s2 ∈ P (t ),
s1 ≡ ∆ s2 iff ∆, s1  ∆, s2 .
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One Sound and Complete R-Calculus with Pseudo-Subtheory Minimal Change Property
Given a pseudo-subtheory s ∈ P(t ), let [r ] be the
equivalence class of s. Then, we have that
[ s (∆, t )],[ξ (∆, t )] ⊆ Ct∆ .
Proposition 5.4. [ s (∆, t )] = [ξ (∆, t )].
Define a relation  on P (t ) such that for any s1
and s2 ∈ P(t ), s1  s2 iff
if s1 = l1 and s2 = l2
l1 = l2
c = c & c = c o c r = c & c = c
22
12
21
11
21
12
22
 11
if
s
=
c
∨
c
and
s
=
c

1
11
12
2
21 ∨ c22
s = s & s = s o s r = s & s = s
22
12
21
11
21
12
22
 11
if s1 = s11 ∧ s12 and s2 = s21 ∧ s22

Proposition 5.5.  is an equivalence relation on
P (t ), and for any s1 , s2 ∈ P (t ), if s1  s2 then
s1  s2 .
Theorem 5.6. If ∆ | t ⇒ ∆, s is provable then for any
η with s  η  t , ∆ | η ⇒ ∆, s is provable.
Proof. We prove the theorem by the induction on the
structure of t.
If t = l and ∆  ¬l then s = λ , and for any η
with s  η  t ,η = λ , and ∆ | η ⇒ ∆, λ is provable;
If t = l and ∆  ¬l then s = l , and for any η
with s  η  t ,η = l , and ∆ | η ⇒ ∆, s is provable;
If t = t1 ∧ t2 and the theorem holds for both t1 and
t2 then s = s1 ∧ s2 , and for any η with s  η  t ,
there are η1 and η2 such that s1  η1  t1 and
s2  η2  t2 . By the induction assumption,
∆ | η1 ⇒ ∆, s1 , ∆, s1 | η2 ⇒ ∆, s1 , s2 , and by ( N ∧ ) ,
∆ | η1 ∧ η2 ⇒ ∆, s1 , s2 ≡ ∆, s1 ∧ s2 ;
If t = c1 ∨ c2 and the theorem holds for both c1 and
c2 then s = s1 ∨ s2 , and for any η with s  η  t ,
there are η1 and η2 such that s1  η1  c1 and
s2  η2  c2 . By the induction assumption,
∆ | η1 ⇒ ∆, s1 ; ∆ | η2 ⇒ ∆, s2 , and by ( N ∨ ) ,
∆ | η1 ∨ η2 ⇒ ∆, s1 ∨ s2 .
Theorem 5.7. For any η with s  η  t , if ∆,η
is consistent then ∆,η  ∆, s, and hence, [η ] = [ s ];
and if ∆,η is inconsistent then ∆,η  ∆, t , and
hence, [η ] = [t ].
Proof. If ∆,η is consistent then by Theorem 6.6,
∆ | η ⇒ ∆, s, and we prove by the induction on the
structure of t that ∆, t  ∆, s.
If t = l and ∆  ¬l then s = l , and ∆, t  ∆, s;
If t = t1 ∧ t2 and the claim holds for both t1 and t2
then s = s1 ∧ s2 , ∆, t1  ∆, s1 and ∆, t2  ∆, s2 . Therefore, ∆, t1 ∧ t2  ∆, s1 ∧ s2 .
If t = c1 ∨ c2 and the theorem holds for both c1 and
c2 then d = d1 ∨ d 2 , and there are three cases:
Case 1. ∆, c1 and ∆, c2 are consistent. By the induction assumption, we have that
∆, c1  ∆, d1 , ∆, c2  ∆, d 2 , and hence,
∆, c1 ∨ c2  ∆, d1 ∨ d 2 ;
Copyright © 2013 SciRes.
Case 2. ∆, c1 is consistent and ∆, c2 is inconsistent.
By the induction assumption, we have that
∆, c1  ∆, d1 , and ∆ | c2 ⇒ ∆. Then,
∆, d1 ≡ ∆, d1 ∨ d 2
 c1
 c1 ∨ c2 ;
and
∆, c1 ∨ c2 ≡ (∆ ∧ c1 ) ∨ (∆ ∧ c2 )
≡ ∆ ∧ c1
≡ ∆, c1  d1  d1 ∨ d 2 ,
where d 2 = λ.
Case 3. Similar to Case 2.
Corollary 5.8. For any η with s  η  t , either
[η ] = [ s ] or [η ] = [t ]. Therefore, [ s ] is  -maximal
such that ∆, s is consistent.
6. Conclusions and Further Works
We defined an R-calculus N in propositional logic programs such that N is sound and complete with respect to
the operator s (∆, t ).
The following axiom is one of the AGM postulates:
Extensionality : if p  q then K  p = K  q
It is satisfied, because we have the following
Proposition 7.1. If t1  t2 ; t1 | s ⇒ t1 , s1 and
t2 | s ⇒ t2 , s2 then s1  s2 .
It is not true in N that
(*) if s1  s2 ; t | s1 ⇒ t , s1′ and t | s2 ⇒ t , s2′ then
s1′  s2′ .
A further work is to give an R-calculus having the
property (∗) .
A simplified form of (∗) is
(**) if s1  s2 ; t | s1 ⇒ t , s1′ and t | s2 ⇒ t , s2′ then
s1′  s2′ , which is not true in N either.
Another further work is to give an R-calculus having
the property (∗∗) and having not the property (∗) .
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