1 L -Semantics as a Fibring over a Combination of Temporal and Kripke Frames
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231
– (σ, σ ) ∈ Rns6 if and only if σ = τ Rl and σ = τ t for some l ∈ {1, 2, . . . , N }
and t ∈ N, where char(σ) = char(τ ) = n;
– (σ, σ ) ∈ Rn7 if and only if σ = σRn t for some t ∈ N;
– (σ, σ ) ∈ Rns7 if and only if σ = σ Rn t for some t ∈ N.
Since the N -preﬁxes σ and σRl will be interpreted to represent the same
object, if we want the accessibility relation to be reﬂexive, then R1 will not be
suﬃcient for the purpose and we shall require R2 and R3 as well. For a similar
reason, we will require Rn5 , Rn6 , Rn7 in addition to Rn4 to give the accessibility
relation for the class of dynamic K spaces. Rns4 , Rns5 , Rns6 , Rns7 will be used to
give symmetry.
The following deﬁnition gives the accessibility relations for diﬀerent classes
of dynamic spaces using the relations deﬁned above.
Deﬁnition 53. Let I ∈ {K, K4, T, B, S4, KT B, KB4, S5} and n ≤ N . We
deﬁne the I-accessibility relation for characteristic n, RnI , on the set P (N ) \ {ε}
of N -preﬁxes as follows. Let us write R∗ to denote the transitive closure of the
relation R.
–
–
–
–
–
–
–
–
RnK := Rn4 ∪ Rn5 ∪ Rn6 ∪ Rn7 .
RnT := R1 ∪ R2 ∪ R3 ∪ RnK .
RnK4 := (RnK )∗ .
RnB := Rn4 ∪ Rn5 ∪ Rn6 ∪ Rn7 ∪ Rns4 ∪ Rns5 ∪ Rns6 ∪ Rns7 .
RnKB4 := (RnB )∗ .
RnS4 := (RnT )∗ .
RnKT B := R1 ∪ R2 ∪ R3 ∪ RnB .
RnS5 := (RnKT B )∗ .
Now, we are in a position to give the tableau extension rules for each of the
classes of dynamic I spaces, I ∈ {K, K4, T, B, S4, KT B, KB4, S5}. Figures 3–4
give the rules for these classes corresponding to the N -preﬁxed wﬀs. We call
these (I, N )-tableau extension rules. Except one, the rules for the Since operator
are not listed, as these can be given in the lines of those for the Until operator.
The rules involving temporal operators, that is, ⊕, , U, S-rules, will be called
the temporal rules.
In order to understand the tableau extension rules, let us consider a wﬀ ⊕α.
To evaluate ⊕α at the time point t at the object w, we need to evaluate the
wﬀ α at the same object w, but at the time point t + 1. To capture this fact,
the ⊕ rule is designed such that if the branch contains σm ⊕ α, m ∈ N, then
using the ⊕(b)-rule, we introduce the wﬀ σmRl+1 α (assuming char(σm) = l).
The N -preﬁx in the latter is used to indicate the same object referred to by σm
(recall the comments following Deﬁnition 49), along with the fact that we have
shifted focus to the time point l + 1. Similarly, one may explain the rule U(d).
Note that in a model M with number of time points N , for any w ∈ U and
n < N , we have M, n, w |= ¬(XUY ), if and only if either M, k, w |= ¬Y for all
k with n ≤ k ≤ N , or there exists s with n ≤ s < N such that M, s, w |= ¬X
and for all k such that n ≤ k ≤ s, M, k, w |= ¬Y . Observe that the wﬀ XUY
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Md.A. Khan
1st
σα
σα1
σα2
2nd
σβ
σβ1
α-rule
σRn ⊕, n < N
∗
σn+1
⊕0
σβ2
β-rule
σRn t1 t2 · · · tr ⊕, n < N
∗
σn+1
⊕0
⊕(a)-rule
σRn , n > 1
∗
σn−1
0
σRn t1 t2 · · · tr , n > 1
∗
σn−1
0
⊕(b)-rule
(a)-rule
(b)-rule
σν
σ ν0
σπ
σ π0
I
, σ is
where (σ, σ ) ∈ Rn
used on the branch and
char(σ) = char(σ ) = n.
σRn F p
σF p
σ is an unrestricted simple p is propositional variable.
extension of σ.
ν-rule
π-rule
σF (XUY ), char(σ) = N
σF Y
U end point-rule
σRn T p
σT p
Propositional-rule
σF (XSY ), char(σ) = 1
σF Y
S ﬁrst point-rule
Fig. 3. (I, N )-tableau extension rules.
can be made false at (n, w) in N − n + 1 possible ways. Accordingly, the rule
U(d) for τ F (XUY ), n < N , where τ = σRn t1 t2 · · · tr , asks for the introduction
∗
in the
of N − n + 1 branches. We again note that all N -preﬁxes of the kind τm
rule indicate the same object referred to by τ , along with the fact that we have
shifted focus to the time point m. The tableau extension rules for the other wﬀs
may be understood in a similar manner.
Given a N -preﬁx σ, we would like to investigate next what would be the form
of any N -preﬁx μ related to σ by the relation RnI . We would then be able to see
the possible forms of an accessible N -preﬁx – this is crucial for the application of
the ν rule. Moreover, the information about the form of such N -preﬁxes will also
be used for obtaining important results related with the soundness, completeness
Multiple-Source Approximation Systems, Evolving Information Systems
Fig. 4. (I, N )-tableau extension rules, continued.
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Md.A. Khan
and termination of the proof procedures. The form of μ depends on I as well
as on the form of σ itself. Here we will only give the propositions determining
the form for I = S5, but one can obtain similar results for other Is. In fact, the
argument will be simpler for all the other choices of Is.
Let us ﬁrst consider the N -preﬁx of the form τ := τ Rn t1 t2 · · · ts , s ≥ 1 and
the relation RnS5 . We investigate the form of μ for which we have (τ, μ) ∈ RnS5 .
Since RnS5 is reﬂexive, we should have (τ, τ ) ∈ RnS5 . Moreover, due to symmetry
and transitivity of RnS5 , we also expect to have (τ, τ Rn d1 d2 · · · dk ) ∈ RnS5 . Furthermore, as mentioned above, σ and σRl will be interpreted to represent the
same object, and thus we should also have (τ, τ Rl ) and (τ, τ Rn d1 d2 · · · dk Rl ) ∈
RnS5 . In fact, the following proposition shows that these are the only possibilities
for μ.
Proposition 67. Let τ = τ Rn t1 t2 · · · ts , s ≥ 1 and (τ, μ) ∈ RnS5 . Then μ must
be in either of the following forms.
(a)
(b)
(c)
(d)
μ
μ
μ
μ
is
is
is
is
τ
τ
τ
τ
.
Rl , where Rl ∈ {R1 , R2 , . . . , RN }.
Rn d1 d2 · · · dk , where di ∈ N.
Rn d1 d2 · · · dk Rl , where Rl ∈ {R1 , R2 , . . . , RN } and di ∈ N.
Proof. The proposition is proved by showing that if
τ RnKT B σ1 RnKT B σ2 · · · RnKT B σj ,
then σj = τ or σj is in either of the above-mentioned forms. We use induction
on j. Note that since τ Rn t1 t2 · · · ts ∈ P (N ), we have char(τ ) = n.
Basis case: Let j = 1.
Since (τ, σ1 ) ∈ RnKT B and RnKT B = R1 ∪ R2 ∪ R3 ∪ Rn4 ∪ Rn5 ∪ Rn6 ∪ Rn7 ∪ Rns4
∪Rns5 ∪Rns6 ∪Rns7 , we must have (τ, σ1 ) ∈ R for some R ∈ {R1, R2, R3, Rn4 , Rn5 ,
Rn6 , Rn7 , Rns4 , Rns5 , Rns6 , Rns7 }. But note that (τ, σ1 ) ∈ R for R ∈ {R2, Rn5 , Rn6 , Rn7 }.
Therefore, σ1 = τ (in the case when (τ, σ1 ) ∈ R1), or σ1 must be in one of the
following forms:
–
–
–
–
–
–
–
τ Rl (when (τ, σ1 ) ∈ R3), or
τ Rn t1 t2 · · · ts ts+1 (when (τ, σ1 ) ∈ Rn4 ), or
τ Rn , where s = 1 (when (τ, σ1 ) ∈ Rns4 ), or
τ Rn t1 t2 · · · ts−1 , where s ≥ 2 (when (τ, σ1 ) ∈ Rns4 ), or
τ Rl , where s = 1 (when (τ, σ1 ) ∈ Rns5 ), or
τ Rn t1 t2 · · · ts−1 Rl , where s ≥ 2 (when (τ, σ1 ) ∈ Rns6 ), or
τ , where s = 1 (when (τ, σ1 ) ∈ Rns7 ).
Thus in each case, we obtain σj in the desired form.
Induction case: Suppose (a) σj = τ , or σj is in one of the following forms:
(b) τ Rl , or
(c) τ Rn d1 d2 · · · dk , or
(d) τ Rn d1 d2 · · · dk Rl
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235
Case (a): σj = τ
Since char(τ ) = n and (σj , σj+1 ) ∈ RnKT B , σj+1 = τ (when (σj , σj+1 ) ∈ R1)
or σj+1 must be in either of the following forms:
– τ Rl (when (σj , σj+1 ) ∈ R3), or
– τ Rn d1 (when (σj , σj+1 ) ∈ Rn7 ).
Case (b): σj is of the form τ Rl .
In this case, σj+1 = τ (when (σj , σj+1 ) ∈ R2), or σj+1 must be in either of the
following forms:
– τ Rl (when (σj , σj+1 ) ∈ R1), or
– τ Rl d1 , l = n (when (σj , σj+1 ) ∈ Rn4 ), or
– τ Rn d1 , (when (σj , σj+1 ) ∈ Rn5 ).
Case (c): σj is of the form τ Rn d1 d2 · · · dk .
Then, σj+1 = τ and k = 1 (when (σj , σj+1 ) ∈ Rns7 ), or σj+1 must be in one of
the following forms:
–
–
–
–
–
–
–
τ
τ
τ
τ
τ
τ
τ
Rn d1 d2 · · · dk (when (σj , σj+1 ) ∈ R1), or
Rn d1 d2 · · · dk Rl (when (σj , σj+1 ) ∈ R3), or
Rn d1 d2 · · · dk dk+1 (when (σj , σj+1 ) ∈ Rn4 ), or
Rn d1 d2 · · · dk−1 and k ≥ 2 (when (σj , σj+1 ) ∈ Rns4 ), or
Rn and k = 1 (when (σj , σj+1 ) ∈ Rns4 ), or
Rl and k = 1 (when (σj , σj+1 ) ∈ Rns5 ), or
Rn d1 d2 · · · dk−1 Rl and k ≥ 2 (when (σj , σj+1 ) ∈ Rns6 ).
Case (d): σj is of the form τ Rn d1 d2 · · · dk Rl .
Then, σj+1 must be in either of the following forms:
– τ Rn d1 d2 · · · dk Rl (when (σj , σj+1 ) ∈ R1), or
– τ Rn d1 d2 · · · dk (when (σj , σj+1 ) ∈ R2), or
– τ Rn d1 d2 · · · dk dk+1 (when (σj , σj+1 ) ∈ Rn6 ).
Thus, in each case, we obtain σj+1 in the desired form. This completes the
proof.
As a direct consequence of Proposition 67, we obtain the following corollary.
Corollary 15. Let μ, τ ∈ P (N ) be such that (i) τ = τ Rn t1 t2 · · · ts , s ≥ 1, (ii)
(τ, μ) ∈ RnS5 and (iii) char(μ) = n. Then μ must be either τ Rn , or of the form
τ Rn d1 d2 · · · dk , di ∈ N.
We end this section with the two following propositions, similar to
Proposition 67.
Proposition 68. Let τ = τ Rh t1 t2 · · · ts Rn ∈ P (N ), (so, s ≥ 1, h = n) and
(τ, μ) ∈ RnS5 . Then μ = τ Rh t1 t2 · · · ts or μ must be in either of the form (a)
τ Rh t1 t2 · · · ts Rl , or (b) τ d1 d2 · · · dr , or (c) τ d1 d2 · · · dr Rl .
Proposition 69. Let τ = τ Rh t1 t2 · · · ts , s ≥ 1, h = n and (τ, μ) ∈ RnS5 . Then
μ = τ or μ must be in either of the form (a) τ Rl , or (b) τ Rn d1 d2 · · · dr , or (c)
τ R n d1 d2 · · · dr R l .
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Md.A. Khan
10.1
Soundness
In this section, we shall prove the soundness of the tableau-based proof procedures proposed in Sect. 10. We begin with the following deﬁnitions. Recall the
relations RnI given by Deﬁnition 53.
Deﬁnition 54.
(i) Let S be a set of N -preﬁxed wﬀs and F := F1 , F2 , . . . FN be a dynamic I
space, where Fi = (W, Pi ). By an I-interpretation of S in F, we mean a
mapping Int from the set of N -preﬁxes that occur in S to W such that the
following conditions are satisﬁed:
(a) If (σ, σ ) ∈ RnI then (Int(σ), Int(σ )) ∈ Pn , n = 1, 2, . . . , N .
(b) Int(σ) = Int(σRl ), for all l ∈ N.
(ii) A set S of N -preﬁxed wﬀs is said to be (I, N )-satisﬁable if there exists some
dynamic I space F := F1 , F2 , . . . , FN of cardinality N , an I-interpretation
Int of S in F, and a valuation function V : P → 2W such that for each
σZ ∈ S,
M, n, Int(σ) |= Z
where char(σ) = n and M = (F, V ).
(iii) A branch of an (I, N )-tableau is said to be (I, N )-satisﬁable if the set of
N -preﬁxed wﬀs on it is (I, N )-satisﬁable.
(iv) An (I, N )-tableau is said to be (I, N )-satisﬁable if some branch of it is
(I, N )-satisﬁable.
Observe that the condition (b) in the deﬁnition of interpretation signiﬁes that
the N -preﬁxes σ and σRl represent the same object.
Example 16. Let us consider a (S5, 2)-tableau for the wﬀ F r → ♦p ∨ q,
p, q, r ∈ P V , given in Fig. 5.
Let S be the set of all 2-preﬁxed wﬀs occurring on the right branch of the
tableau of Fig. 5. Note that {R1 1, R1 11, R1 1R2 } is set of all 2-preﬁxes that occur
in S. Let us consider the dynamic S5 space F := F1 , F2 , where Fi := (W, Pi ),
W := {x, y}, W/P1 := W × W and W/P2 := {{x}, {y}}. Let Int be the function
from the set of 2-preﬁxes occurring in S to W deﬁned as
Int(R1 1) = Int(R1 1R2 ) = x;
Int(R1 11) = y.
Note that Int is an interpretation of S in F as it satisﬁes both the deﬁning
conditions (a), and (b) of an I-interpretation (cf. Deﬁnition 54). In fact, condition
/ R2S5 ,
(a) is a direct consequence of the fact that (R1 1, R1 11), (R1 1R2 , R1 11) ∈
which, in turn, follows from Proposition 67. Further, using Int and the valuation
V which maps p to ∅, q and r to {x}, one can show that the right branch of
the above tableau is (S5, 2)-satisﬁable. For instance, for the wﬀ R1 1F (F r →
♦p∨ q) lying on the right branch, we have M, 1, Int(R1 1) |= F (F r → ♦p∨ q)
as M, 2, x |= r, but M, 1, x |= ♦p and M, 1, x |= q, M := (F, V ).