V. V. Rimatskiy1
FINITE BASES OF ADMISSIBLE INFERENCE RULES
FOR MODAL LOGICS OF WIDTH 2
1. Introduction
Presented article is devoted to study of inference rules admissible for strong
modal logics. We investigate bases for admissible rules. This problem has
algebraic analog: if some corresponding quasi-variety has finite basis for
its quasi-identity. Such basis of inference rules allows to increase deductive
ability of logic and solve various connected problems. The main result is
the following
Theorem. Any tabular modal logic of width 2 has finite basis of admissible
inference rules.
We begin with recalling certain necessary facts for further explanations.
2. Admissible Inference Rules
All definitions and theorems (and notation) reviewed here can be found in
[1].
An inference rule
r = {α1 (x1 , . . . , xn ), . . . , αk (x1 , . . . , xn )/β(x1 , . . . , xn )}
is called admissible in a logic λ, if for any formulas δ1 , . . . , δn the assertion
(∀j(αj (δ1 , . . . , δn ) ∈ λ) =⇒ β(δ1 , . . . , δn ) ∈ λ) holds. A frame F := hF, Ri
is a pair, where F is a nonempty set and R is a binary relation on F . Basis
set and a frame itself are below often denoted by the same letter.
1 The
investigation was supported by RFFI grant 96-01-00228
126
Any subset C of a frame F which consists of all mutually R–comparable
elements of F is called a cluster of the frame F . A cluster C is proper if
|C| > 1. The maximal number of clusters in chains of clusters generated
by an element or a cluster is said to be a depth of the element or of the
cluster in the frame (or the model) F. For any transitive frame F (or transitive Kripke model M ) n-slice Sn (F ) (Sn (M )) is the set of all elements
of depth n from F (M ). S≤n (F ) is the set of all elements from F with
depth not greater than n. We put cR := {x|∃y ∈ {c} : yRx} ∪ {c}, where
{c} is a cluster of a frame F consisting of element c. For any subset X,
X R is defined similarly. Element c ∈ F is co-covering for a set X ⊆ F , if
cR \{c} = X R and {c} is the cluster. Co-covering cluster for a subset X is
defined similarly.
Admissible inference rules have following algebraic description:
Statement 1. An inference rule
r = {α1 (x1 , . . . , xn ), . . . , αk (x1 , . . . , xn )/β(x1 , . . . , xn )}
is admissible in a logic λ iff quasi-identity
r∗ = {α1 (x1 , . . . , xn ) = 1 & . . . & αk (x1 , . . . , xn ) = 1 =⇒ β(x1 , . . . , xn ) = 1}
is true on free algebra of countable rank Fw (λ) from the variety V ar(λ)
generated by λ.
A set Ad∗ (λ) of admissible rules of logic λ is called a basis of admissible inference rules iff any admissible inference rule r is derivable from
Ad∗ (λ) in λ.
Statement 2. {r1 , . . . , rk } is a basis of admissible inference rules in λ
iff {r1∗ , . . . , rk∗ } is a basis of quasi-identities of Fw (λ).
Statement 3. Fw (λ) |= q ⇐⇒ ∀n Fn |= q.
A Kripke model hWn , R, V i, where V : Wn → 2Wn ,Wn = {p1 , . . . , pn },
is n- characterizing for a logic λ iff for any formula α which is built up from
p1 , . . . , pn , α ∈ λ iff hWn , R, V i |= α.
A mapping φ : hT, Ri → hT1 , R1 i is a p-morphism, if
i) xRy =⇒ φ(x)R1 φ(y);
ii) φ(x)R1 y =⇒ ∃z(y = φ(z) & xRz).
Let λ be a tabular logic over K4.
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Statement 4. ( see [3]) Fw (λ) does not have a finite basis for quasiidentities iff there exists a sequence of algebras Bn ∈ V ar(λ) (where indexes
n form strictly ascending sequence) such that:
1) for any n, Bn is n+1-generated;
2) for any n, Bn 6∈ FwQ (λ);
3) any n -generated subalgebra of algebra Bn is proper and belongs to FwQ (λ),
where FwQ is quasi-variety generated by free algebra of countable rank Fw (λ)
from V ar(λ).
3. Certain semantic results
Let λ be a tabular logic over K4 with fmp which does not have finite bases
of admissible inference rules. Then according to Statement 4 there is a
sequence of algebras Bn ∈ V ar(λ) (where indexes n form strictly ascending
sequence) such that:
1) for any n, Bn is n + 1-generated;
2) for any n, Bn 6∈ FwQ (λ);
3) any n-generated subalgebra of algebra Bn is proper and belongs to
FwQ (λ).
We need to describe n-characterizing models for λ. The method of
constructing and general properties of n-characterizing for λ model Cn (λ)
are contained in Rybakov [1]. All we need here is: S1 (Fn+ ) consists of all
n+1
possible λ-clusters with not more than 22
elements; any antichain of
clusters has all possible co-covering ones if they generate λ-frames.
Theorem 1. (see [3]) A finite algebra A generated by the frame F belongs
to quasi-variety FwQ (λ) iff there exists p-morphism from Fn+1 t. . .tFn+k onto
F for some n1 , . . . nk ∈ N , where Fn+ is a frame generating free algebra of
rank n from V ar(λ).
Pick any algebra Bn satisfying the conditions of Statement 4.
Lemma 1. Let Bn 6∈ FwQ (λ). Then there exists some antichain A of clusters
Bn+ such that there is no co-covering cluster for A, but at the same time
a frame K0R = AR ∪ {K0 } generated by ascribing a cluster K0 to a frame
AR is λ-frame.
Proof. Assume that for any antichain of clusters A ∈ Bn+ , there exists a
one-element co-covering cluster {a} if the frame {a}R = AR ∪{a} generated
128
by ascribing a cluster {a} to a frame AR is a λ-frame. We will show that
+
a frame Bn+ is a p-morphic image of a frame Fn+1
.
We begin with the following: for any n ≥ 3 there exists a cluster
K0 ∈ S1 (Bn+ ) such that |K0 | = 1. Indeed, for suppose that any cluster
Ki ∈ S1 (Bn+ ), i ∈ I is proper. Then the following S
valuation V can be
introduced on Bn+ : V (a) = ai , i ∈ I, V (b) = Bn+ \ i {ai }, where ai is
arbitrary element of Ki ∈ S1 (Bn+ ), i ∈ I. Elements V (a), V (b) generate
2-generated subalgebra A of algebra Bn+ . It is not difficult to check that
all clusters of S1 (A+ ) are proper. Hence by Theorem 1, A 6∈ FwQ (λ) and at
the same time A ∈ FwQ (λ) by the condition 3) for Bn+ . A contradiction.
Let B0 ∈ S1 (Bn+ ) be such improper cluster. We define a p-morphism
+
from S1 (Fn+1
) onto S1 (Bn+ ) as follows: for any cluster Ki ∈ S1 (Bn+ ), i ∈
+
I, there is a cluster Ci ∈ S1 (Fn+1
) such that Ki ∼
= Ci (as models) by
+
+
the
S construction of Fn+1 . Then define f1 (Ci ) = Ki , i ∈ I; f1 (S+1 (Fn+1 \
i {Ci })) := B0 . Note that for any open subspace V ⊆ S1 (Bn ), there
+
exists W ⊆ S1 (Fn+1
) with W ∼
= V ; f1 (W ) = V .
Let f1 , . . . fk be constructed and assume that for any open subspace
+
V ⊆ S≤k (Bn+ ), there exists a subspace W ⊆ S≤k (Fn+1
) with W ∼
= V ; fk ◦
. . . ◦ f1 (W ) = V .
+
Now we define a p-morphism fk+1 from Sk+1 (Fn+1
) onto Sk+1 (Bn+ ).
+
Pick any cluster K0 ∈ Sk+1 (Bn ) and suppose that clusters K1 , . . . Kl ∈
S≤k+1 (Bn+ ) are seen from K0 immediately. By IH any Ki , i ≤ l has Ci ∈
+
+
S≤k (Fn+1
), i ≤ k such that KiR (⊆ Bn+ ) ∼
); fk ◦ . . . ◦ f1 (CiR ) =
= CiR (⊆ Fn+1
R
Ki .
+
) C0 ∈
As KiR is a λ-frame so there exists (by construction of Fn+1
+
Sk+1 (Fn+1
) such that C0R ∼
= K0R , C0 RCi , i ≤ l. Then we define fk+1 (C0 )
:= K0 .
So defined fk+1 is a partial p-morphism which satisfy IH. Now we
+
complete a definition of fk+1 on the remaining part of Sk+1 (Fn+1
). If any
+
cluster C0 ∈ Sk+1 (Fn+1 ) sees immediately some antichain C1 , . . . Cm ∈
+
S≤k+1 (Fn+1
), then, for each Ci , i ≤ m there exists a cluster Ki = fk ◦
. . . ◦ f1 (CiR ), i ≤ m such that CiR ∼
= KiR . By our hypothesis for antichain
K1 , . . . Km there is one-element co-covering cluster K0 ∈ Bn+ . Then we put
+
fk+1 (C0 ) := K0 . In the case, when from the cluster C0 ∈ Sk+1 (Fn+1
) only
a cluster C1 of depth k is seen immediately, we pick cluster K1 = fk (C1 ) ∈
Sk (Bn+ ). Then define fk+1 (C0 ) := K0 provided that from K0 ∈ Sk+1 (Bn+ )
only K1 ∈ Sk (Bn+ ) is immediately seen else fk+1 (C0 ) := K1 .
129
+
Going on this construction until R-minimal clusters of Fn+1
(existing
+
by finiteness of this frame) we define a p-morphism from Fn+1 onto Bn+ .
Then Bn is a subalgebra of Fn , i.e. Bn ∈ FwQ (λ), which contradicts the
condition 2) for Bn and proves our lemma.
Let a λ-frame F satisfies the conditions:
1) S1 (F ) = {a}, i.e. first slice is one-element cluster only;
2) for any clusters K1 , K2 ∈ F of the depth greather than 2,
if K1R \ {K1 } = K2R \ {K2 }, then K1 = K2 .
Statement 5. Let ϕ be a p-morphism from Fn+1 t. . . t Fn+k onto F . Then,
for any open subspace V ⊆ F , for any ni , i ≤ k, there exists a subspace
W ⊆ Fn+i such that W ∼
= V, ϕ(W ) = V .
This statement can be easily proved by induction on the depth of V .
Lemma 2. Assume that there exists an antichain A of clusters of F such
that there is no co-covering cluster for A, but at the same time a frame
K0R = AR ∪ {K0 } generated by ascribing a cluster K0 to a frame AR is a
λ-frame. Then ∀n1 , . . . nk ∈ N there is no a p-morphism from Fn+1 t. . .tFn+k
onto F .
Proof. Let for some n1 , . . . nk ∈ N a mapping ϕ : Fn+1 t. . .tFn+k →onto F
be a p-morphism. According to Statement 5, for any subspace AR which
is generated by antichain A, for any ni there is a subspace W ⊆ Fni such
that ϕ(W ) = AR , W ∼
= AR .
Since the frame AR is a λ-frame, then by construction of Fn+ , for any
ni , there exists a co-covering cluster Ci ∈ Fni for R-minimal antichain of
W (isomorphic to A) which is isomorphic to the cluster K0 . Since ϕ is a
p-morphism, for all i, ϕ(Ci ) is co-covering cluster in F for A. But such a
cluster does not exist in the frame F by our hypothesis, a contradiction.
Let us return to a sequence of algebras Bn from Statement 4. According to the statement, for any n there exists some antichain of clusters
{K1 , K2 } such that there is no co-covering cluster for it in Bn+ , but at the
same time a frame K0R = K1R ∪ K2R ∪ {K0 } generated by a cluster K0 is a
λ-frame.
Let us examine the first case when both clusters {K1 , K2 } ∈ Bn+ have
a depth greater than 1. We fix them and transform the frame Bn+ as follows:
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1) all elements of Bn+ of depth not greater than the depth of the clusters
K1 , K2 are contracted into one-element cluster of the first slice;
2) all elements of Bn+ which do not see neither K1 nor K2 are contracted into the same one-element cluster of the first slice;
3) if clusters K1 , K2 are R-minimal then transformation is completed;
4) if only one from fixed clusters K1 , K2 is R-minimal then all remaining elements of Bn+ are contracted into one-element cluster of depth
3;
5) if the fixed clusters K1 , K2 have immediate predecessors (but no
co-covering!) of depth 3 then:
a) all elements of Bn+ of depth greater than 2 which see the cluster K1 or K2 are contracted into improper cluster K13 or K23 of depth 3
correspondingly;
b) all elements of Bn+ of depth greater than 3 which immediately see
cluster K1 or K2 are contracted into improper cluster K14 or K24 of depth
4 – immediate predecessors of K13 or K23 correspondingly;
c) all remaining elements of Bn+ are contracted into improper cocovering cluster of K14 , K24 or K13 , K23 .
It is not difficult to check that composition of all transformations 1)
– 5) described above is a p-morphism from Bn+ onto some λ-frame B ∗ .
Directly from 1) – 5) the following properties of B ∗ follow:
(i) S1 (B ∗ ) = {a} is one-element cluster;
(ii) S2 (B ∗ ) = {K1 , K2 } and fixed K1 , K2 does not have co-covering
cluster;
(iii) a frame B ∗ has the depth not greater than 5;
(iv) for i ≤ 5, Si (B ∗ ) consists of no more than two distinguished
one-element clusters;
(v) for any clusters C1 , C2 ∈ B ∗ of the depth greater than 2, if C1 6=
C2 , then C1R \ {C1 } 6= C2R \ {C2 }.
Provided that no one from 1)– 5) is carried out, i.e. Bn+ = B ∗ , by
(iii) – (v) algebra Bn is not more than 10-generated which contradicts the
condition 1) for Bn .
Else p-morphism from Bn+ onto B ∗ is proper, hence algebra B ∗ generated by the frame B ∗ belongs to quasi-variety FwQ (λ) by condition 3) for
Bn+ . At the same time from (i), (ii), (v) it follows that the frame B ∗ satisfies
the assumption of Lemma 2. Hence by this lemma algebra Bn generated
by B ∗ does not belong to quasi-variety FwQ (λ), a contradiction.
131
Now let us examine the case when one or both fixed clusters K1 , K2
have depth 1. Transform a frame Bn+ as it was described in 1) – 5).
Obviously, the fixed antichain from Lemma 1 either has common predecessor or does not have. If it does not have such a predecessor in B ∗ ,
then it does not have in Bn+ too. In such case, without loss of generality, we can consider that any antichain A ∈ B ∗ has co-covering improper
R
R
cluster KA ∈ B ∗ provided that it generates a λ-frame KA
= AR ∪ {KA
}.
+
So the set of clusters of Bn is divided into the subsets L1 =
{K|KRK1 }; L2 = {K|KRK2 }; L3 = {K|¬KRK1 , ¬KRK2 }, and we can
+
+
generate LR
i = {x ∈ Bn |∃y ∈ Li , yRx}, i ≤ 3. Note that the frame Bn is
R
a p-morphic image of the disjoint union of Li . If we will prove that any
+
+
LR
i is a p-morphic image of Fn+1 , then the frame Bn is p-morphic image
+
+
+
of Fn+1 t Fn+1 t Fn+1 as well . This contradicts the condition 2) for Bn+ .
It is easy to see that:
+
1) if clusters A1 , A2 ∈ LR
i have co-covering cluster A0 ∈ Bn then A0
itself belongs to the same Li .
R
+
2) if clusters A1 ∈ LR
1 , A2 ∈ L3 and A0 ∈ Bn is their co-covering
R
cluster, then A0 ∈ Li : A0 RA1 , A1 RK2 =⇒ A0 RK2 . For A1 ∈ LR
2 , A2 ∈
is
similarly.
LR
3
R
6 ∃A0 : A0 RA1 , A0 RA2 , else K1 , K2 have
3) ∀A1 ∈ LR
1 , A2 ∈ L2
common predecessor.
Hence any antichain of clusters from LR
i has one-element co-covering
cluster provided that it generates a λ-frame. As in the proof of Lemma
+
1 we can show that any LR
i is a p-morphic image of Fn+1 . We can also
R
show that for each i, S1 (Li ) has one-element cluster. Further we define
+
p-morphism from Fn+1
onto LR
i as before.
Consequently we have the following chain of p-morphisms:
+
+
+
+
2 R
2 +
+
R
R
Fn+1
t Fn+1
t Fn+1
→ LR
1 t L2 t L3 → Bn (or t1 Fn+1 → t1 Li → Bn if
+
+
Q
++
3
∈ FwQ (λ), hence
LR
3 = ∅). By virtue of (t1 Fn+1 ) ∈ Fw (λ), we have Bn
Bn ∈ FwQ (λ), a contradiction.
Therefore supposing that fixed clusters K1 , K2 ∈ Bn+ do not have
common predecessor we obtain Bn ∈ FwQ (λ), a contradiction.
Assume K0 ∈ Bn+ : K0 RK1 , K0 RK2 and K0 is R-maximal common
predecessor. Since the algebra B∗ generated by frame B ∗ is not more
than 10-generated by (iii) – (iv) so according to the condition 3) for Bn ,
B∗ ∈ FwQ (λ). Hence the frame B ∗ is a p-morphic image of Fn+1 t . . . t Fn+k
for some ni ∈ N by Theorem 1.
132
Let ϕ : Fn+1 t . . . t Fn+k → B ∗ be a p-morphism ”onto”. Then for some
ni , i ≤ k, there exists a cluster Cn ∈ Fn+i such that ϕ(Cn ) = K0 ∈ B ∗
and Cn is R–maximal among all C such that ϕ(C) = K0 . By property of
p-morphism there are clusters Cj : ϕ(Cj ) = Kj , j ≤ 2.
We consider all possible p-morphisms f of subspace CnR into itself for
which Cj = f (Cj ), j = 1, 2, for these fixed clusters and study the set of
images of CnR and their locations under those p-morphisms.
Since p-morphisms preserve the truth of formulas, f (CnR ) under any
mentioned above p-morphism f is a λ-frame and by construction of Fn+ ,
f (CnR ) ⊆ Fn+ .
Examine the set A = {A : A = f (Cn )} for all p-morphisms f described
above. Since the cluster Cn is of finite depth and the logic λ is of width 2,
the set A is finite. Therefore there exists some R-maximal cluster A0 ∈ A ∈
CnR . Since A = f (Cn ) and Cn sees fixed Cj such that ϕ(Cj ) = Kj , j = 1, 2
for fixed earlier Kj ∈ Bn+ , j = 1, 2, so by property of ϕ as being a pmorphism we obtain ϕ(A0 )RKj , j = 1, 2, and A0 has a depth less than
Cn . This contradicts the choice of Cn as R-maximal inverse image of fixed
Kj ∈ Bn+ .
Consequently supposing that the fixed antichain {K1 , K2 } ∈ Bn+ does
not have co-covering clusters and examining all possible cases of location
we obtain a contradiction: by our hypothesis the algebra B∗ ∈ FwQ (λ), but
we straight showed the converse.
Therefore any antichain {K1 , K2 } ∈ Bn+ has co-covering cluster if it
generates a λ-frame. We obtain a contradiction 2) of Bn+ from Lemma 1.
Consequently there is no such a sequence of algebras from Statement 4.
Thus we proved
Theorem 2. Any tabular modal logic of width 2 has a finite basis of
admissible inference rules.
According to Rybakov [2] we know that some modal logics with depth
and width strictly greater than 2 do not have finite and even independent
basis for admissible inference rules. In [3] we showed that any modal logic
of depth 2 has a finite basis. In this paper we have shown that any tabular
modal logic of width 2 also has a finite basis.
Therefore now we know how the existence of finite basis of admissible
inference rules depends on the depth and width of logics. It leads us to set
the question: what is a necessary and sufficient condition for any modal
tabular logic to have a finite basis by admissibility?
133
References
[1] V. V. Rybakov, Admissibility of logical inference rules,
ch. 3, p. 295.
[2] V. V. Rybakov, Tabular logics without finite basis of admissible
inference rules, Logic Colloquium ’94, France, 1994.
[3] V. V. Rimatskiy, Basis of admissible inference rules of tabular
modal logics of depth 2, Algebra i Logica, 1996, V. 35, N 5, pp. 612–623.
Mathematics Department
Krasnoyarsk University
av. Svobodnyi 79
660 041 Krasnoyarsk, Russia
134