1 Introduction
Many logical properties have been shown to be undecidable for normal modal logics. The pioneering work by Thomason [Reference Thomason19] showed the undecidability of Kripke completeness. This was followed by a series of works by Chagrov [Reference Chagrov8, Reference Chagrov9, Reference Chagrov11], which introduced a general method for showing undecidability. This method can be applied to show the undecidability of various logical properties, including the finite model property, first-order definability, decidability, tabularity, and the coincidence of a fixed tabular logic. For a comprehensive overview and additional references, we refer to [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20] and [Reference Chagrov and Zakharyaschev7, Chapter 17].
Given the generality of Chagrov’s method, it might seem that all meaningful logical properties would be undecidable. Indeed, [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20] pointed out that “we know only two interesting decidable properties of finitely axiomatizable logics in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
: consistency and coincidence with
$\mathsf {K}$
.”
However, in this article, we show that the property of being a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
(the lattice of normal modal logics) is decidable, answering the open question [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20, Problem 2] in the affirmative. This property contrasts with the consistency and the coincidence with
$\mathsf {K}$
as there are continuum many union-splittings and continuum many non-union-splittings in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. Our proof idea is to give a semantic characterization for a logic
$\mathsf {K} + \varphi $
to be a union-splitting in terms of finite modal algebras. This also yields the decidability of being a splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. Moreover, we observe that, for a formula
$\varphi $
,
$\mathsf {K} + \varphi $
has a decidable axiomatization problem iff
$\varphi $
is a decidable formula iff
$\mathsf {K} + \varphi $
is a union-splitting or the inconsistent logic. Thus, the decidability of being a union-splitting implies that having a decidable axiomatization problem and being a (un)decidable formula are also decidable. The latter answers [Reference Chagrov and Zakharyaschev7, Problem 17.3] for
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
in the affirmative.
This article is organized as follows. Section 2 introduces preliminaries, in particular, those on the decision problem of logical properties. Section 3 proves the main theorem, the decidability of being a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. Section 4 discusses applications of the decidability result to axiomatization problems and (un)decidable formulas.
2 Preliminaries
We assume familiarity with the basics of normal modal logics and algebraic semantics (see, e.g., [Reference Blackburn, de Rijke and Venema4, Reference Chagrov and Zakharyaschev7]). We will only deal with normal modal logics, so we also call them logics. For a logic
$L_0$
,
$\mathop {\mathsf {NExt}}{L_0}$
denotes the complete lattice of all logics containing
$L_0$
, ordered by inclusion. Recall that
$\mathsf {K}$
is the least normal modal logic, and
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
is the lattice of all normal modal logics. We will identify a logical property P with the set of logics having P, and write
$L \in P$
if the logic L has P.
We refer to [Reference Bergman1, Reference Burris and Sankappanavar6] for the basics of universal algebra. Recall that a modal algebra
$\mathfrak {A}$
is a tuple 
, where
$(A, \lor , \land , \lnot , 0, 1)$
is a Boolean algebra and 
is a unary operation on A such that 
and 
. Valuations, satisfaction, and validity are defined as usual. We will always use A to denote the underlying set of the algebra
$\mathfrak {A}$
. A modal algebra
$\mathfrak {A}$
is subdirectly irreducible (s.i., for short) if, whenever
$f: \mathfrak {A} \to \Pi _{i \in I} \mathfrak {A}_i$
is an embedding such that each induced projection
$\pi _i \circ f: \mathfrak {A} \to \mathfrak {A}_i$
is surjective, there is some
$i \in I$
such that
$\pi _i \circ f$
is an isomorphism. It is well-known in universal algebra that every variety is generated by its s.i. members.
We will use the following notations. For a logic L, let
$\mathcal {V}(L) = \{\mathfrak {A}: \mathfrak {A} \models L\}$
, and for a modal algebra
$\mathfrak {A}$
, let
$\mathsf {Log}\ \mathfrak {A} = \{\varphi : \mathfrak {A} \models \varphi \}$
. The letters
$\mathcal {H}$
and
$\mathcal {S}$
, respectively, denote the closure operator of taking homomorphic images and subalgebras. For a class
$\mathcal {K}$
of algebras, let
$\mathcal {K}_{\mathrm {si}}$
be the class of s.i. members of
$\mathcal {K}$
and
$\mathcal {K}_{\mathrm {fsi}}$
be the finite s.i. members of
$\mathcal {K}$
. We abbreviate 
(n times 
) as 
and 
as 
. Similar abbreviations apply to the modal operation on modal algebras.
2.1 Decision problem of logical properties
We refer to [Reference Chagrov and Zakharyaschev7, Chapter 17] and [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20] for a detailed introduction and survey of the decision problem of logical properties for modal logic. Intuitively, the question is:
Is there an algorithm that, given a modal logic, decides whether it has a specific property?
There are different ways to formulate this question, depending on how logics are encoded as input. Note that an input must be a finite object, so it is certainly not possible to take all of the continuum many logics into account. The most general possible formulation is to consider all recursively axiomatizable logics, encoded by recursive functions that enumerate their theorems. However, Kuznetsov showed that this only leads to triviality, similar to Rice’s Theorem for partial recursive functions. Kuznetsov left the result unpublished, but one can find a proof in [Reference Chagrov and Zakharyaschev7, Section 17.1]. The result also holds for
$\mathop {\mathsf {NExt}}{\mathsf {K4}}$
,
$\mathop {\mathsf {NExt}}{\mathsf {S4}}$
, and other lattices of normal modal logics.
Theorem 2.1 (Kuznetsov).
Let P be a non-trivial property of recursively axiomatizable logics, that is, there are a recursively axiomatizable logic that has P and a recursively axiomatizable logic that does not have P. Then it is undecidable whether a recursively axiomatizable logic has P.
So, it is a convention to restrict ourselves to finitely axiomatizable logics. Since most logics we encounter in practice are finitely axiomatizable, this is not a serious drawback. A finitely axiomatizable logic will be encoded by a finite set of formulas axiomatizing the logic, or equivalently, a single formula axiomatizing the logic.
Definition 2.2. Let
$L_0$
be a normal modal logic. A logical property P is decidable in
$\mathop {\mathsf {NExt}}{L_0}$
iff the set
$\{\varphi : L_0 + \varphi \in P\}$
is decidable.
Quite a lot of properties are known to be undecidable in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
, including Kripke completeness, the finite model property, first-order definability, decidability, and tabularity (see, e.g., [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20]). On the contrary, only very few interesting properties were known to be decidable in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. The most interesting ones would be the consistency and the coincidence with
$\mathsf {K}$
[Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20].
2.2 Union-splittings and Jankov formulas
The notion of splittings comes from lattice theory.
Definition 2.3. Let X be a complete lattice. A splitting pair of X is a pair
$(x, y)$
of elements of X such that
$x \not \leq y$
and for any
$z \in X$
, either
$x \leq z$
or
$z \leq y$
. If
$(x, y)$
is a splitting pair of X, we say that y splits X and x is a splitting in X.
This notion has been an important tool in the study of lattices of logics. See [Reference Chagrov and Zakharyaschev7, Section 10.7] for a historical overview.
Definition 2.4. Let
$L_0$
and L be logics.
-
1. L is a splitting in
$\mathop {\mathsf {NExt}}{L_0}$
iff it is a lattice-theoretic splitting in the lattice
$\mathop {\mathsf {NExt}}{L_0}$
. -
2. L is a union-splitting in
$\mathop {\mathsf {NExt}}{L_0}$
iff it is the join of a set of splittings in
$\mathop {\mathsf {NExt}}{L_0}$
.
Blok [Reference Blok5] identified splittings and union-splittings in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
using Jankov formulas, and proved the finite model property of them. An alternative proof can be found in [Reference Bezhanishvili, Bezhanishvili and Iemhoff2]. Recall that a modal algebra
$\mathfrak {A}$
is of height
$\leq n$
if 
, or equivalently, 
;
$\mathfrak {A}$
is of finite height if it is of height
$\leq n$
for some
$n \in \omega $
. Note that 
is valid only in the trivial algebra, so a non-trivial algebra cannot be of height
$\leq 0$
.
Definition 2.5. Let
$\mathfrak {A}$
be a finite s.i. modal algebra of height
$\leq n$
. The Jankov formula
$\epsilon (\mathfrak {A})$
associated with
$\mathfrak {A}$
is
where
and
$$ \begin{align*} \Delta = \{p_a : a \in A, a \ne 1\}. \end{align*} $$
These formulas are called Jankov formulas because they have a similar semantic characterization as the formulas introduced by Jankov [Reference Jankov15] (see also [Reference de Jongh16] for superintuitionistic logics).
Proposition 2.6. Let
$\mathfrak {A}$
be a finite s.i. modal algebra of finite height and
$\mathfrak {B}$
be a modal algebra. Then
$\mathfrak {B} \not \models \epsilon (\mathfrak {A})$
iff
$\mathfrak {A}$
is a subalgebra of an s.i. homomorphic image of
$\mathfrak {B}$
.
Theorem 2.7 [Reference Blok5].
Let L be a logic.
-
1. L is a splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
iff L is axiomatizable by a Jankov formula of a finite s.i. modal algebra of finite height. -
2. L is a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
iff L is axiomatizable by Jankov formulas of finite s.i. modal algebras of finite height.
Theorem 2.8 [Reference Blok5].
Every union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
has the finite model property.
A finitely axiomatizable logic with the finite model property is decidable, known as Harrop’s theorem (see, e.g., [Reference Chagrov and Zakharyaschev7, Theorem 16.13]).
Corollary 2.9. Every finitely axiomatizable union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
is decidable.
3 Decidability of being a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
In this section, we prove the decidability of being a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. We first give a semantic characterization of union-splittings in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. Let
${\mathcal {FH}}$
be the class of modal algebras of finite height, that is,
Lemma 3.1. Let
$\mathfrak {A}$
be a modal algebra and
$\mathfrak {B}$
be a subalgebra of
$\mathfrak {A}$
. If
$\mathfrak {B}$
is of finite height, then so is
$\mathfrak {A}$
.
Proof. Suppose that
$\mathfrak {B}$
is of finite height. Then there is some
$n \in \omega $
such that 
. Since by assumption
$\mathfrak {B}$
is a subalgebra of
$\mathfrak {A}$
, we have
$0_{\mathfrak {A}} = 0_{\mathfrak {B}}$
, and so 
. Thus,
$\mathfrak {A}$
is of finite height.
Remark 3.2. In the proof, it suffices to assume that 
for all
$b \in \mathfrak {B}$
, where
$i: \mathfrak {B} \to \mathfrak {A}$
is the inclusion map. So, we may weaken the assumption so that
$\mathfrak {B}$
is a stable subalgebra of
$\mathfrak {A}$
(cf. [Reference Bezhanishvili, Bezhanishvili and Ilin3, Definition 2.1]). But this is not needed for our purpose.
Theorem 3.3. For any modal logic L, the following are equivalent:
-
1. L is a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
; -
2. L is axiomatized over
$\mathsf {K}$
by Jankov formulas of finite s.i. modal algebras of finite height; -
3. for any modal algebra
$\mathfrak {A}$
,
$\mathcal {H}(\mathfrak {A})_{\mathrm {si}} \cap {\mathcal {FH}} \subseteq \mathcal {V}(L)$
implies
$\mathfrak {A} \in \mathcal {V}(L)$
; -
4. for any finite modal algebra
$\mathfrak {A}$
,
$\mathcal {H}(\mathfrak {A})_{\mathrm {fsi}} \cap {\mathcal {FH}} \subseteq \mathcal {V}(L)$
implies
$\mathfrak {A} \in \mathcal {V}(L)$
.
Proof.
$(1) \Leftrightarrow (2)$
: This is the second item of Theorem 2.7.
$(2) \Rightarrow (3)$
: Suppose that
$L = \mathsf {K} + \{\epsilon (\mathfrak {B}_i): i \in I\}$
where each
$\mathfrak {B}_i$
is a finite s.i. modal algebra of finite height. Let
$\mathfrak {A}$
be a modal algebra such that
$\mathfrak {A} \not \models L$
. We show that
$\mathcal {H}(\mathfrak {A})_{\mathrm {si}} \cap {\mathcal {FH}} \not \subseteq \mathcal {V}(L)$
. Since
$\mathfrak {A} \not \models L$
,
$\mathfrak {A} \not \models \epsilon (\mathfrak {B}_i)$
for some
$i \in I$
. So,
$\mathfrak {B}_i$
is a subalgebra of an s.i. homomorphic image of
$\mathfrak {A}$
. Since
$\mathfrak {B}_i$
is of finite height, it follows from Lemma 3.1 that
$\mathfrak {A}'$
is also of finite height. Thus,
$\mathfrak {A}' \in \mathcal {H}(\mathfrak {A})_{\mathrm {si}} \cap {\mathcal {FH}}$
. However, since
$\mathfrak {B}_i$
is a subalgebra of
$\mathfrak {A}'$
and
$\mathfrak {A}'$
is an s.i. homomorphic image of itself,
$\mathfrak {A}' \not \models \epsilon (\mathfrak {B}_i)$
. Hence,
$\mathfrak {A}' \not \models L$
, namely,
$\mathfrak {A}' \notin \mathcal {V}(L)$
. So,
$\mathcal {H}(\mathfrak {A})_{\mathrm {si}} \cap {\mathcal {FH}} \not \subseteq \mathcal {V}(L)$
.
$(3) \Rightarrow (4)$
: This is clear because for any finite modal algebra
$\mathfrak {A}$
,
$\mathcal {H}(\mathfrak {A})_{\mathrm {fsi}} = \mathcal {H}(\mathfrak {A})_{\mathrm {si}}$
.
$(4) \Rightarrow (2)$
: Suppose that (4) holds. Let
$$\begin{align*}L' = \mathsf{K} + \{\epsilon(\mathfrak{B}): \mathfrak{B} \in {\mathcal{FH}}_{\mathrm{fsi}}, \mathfrak{B} \not\models L\}.\end{align*}$$
It suffices to show that
$L = L'$
.
Since
$L'$
is a union-splitting by Theorem 2.7, it has the finite model property by Theorem 2.8. If
$L \not \subseteq L'$
, then by the finite model property of
$L'$
, there is a finite modal algebra
$\mathfrak {A}$
such that
$\mathfrak {A} \models L'$
and
$\mathfrak {A} \not \models L$
. So, by (4), there is some
$\mathfrak {B} \in \mathcal {H}(\mathfrak {A})_{\mathrm {fsi}} \cap {\mathcal {FH}}$
such that
$\mathfrak {B} \not \models L$
. Since
$\mathfrak {B} \not \models \epsilon (\mathfrak {B})$
and
$\epsilon (\mathfrak {B}) \in L'$
by the definition of
$L'$
,
$\mathfrak {B} \not \models L'$
. Since
$\mathcal {H}$
preserves validity, we have
$\mathfrak {A} \not \models L'$
, which is a contradiction. Thus,
$L \subseteq L'$
.
If
$L' \not \subseteq L$
, then there is a modal algebra
$\mathfrak {A}$
such that
$\mathfrak {A} \models L$
and
$\mathfrak {A} \not \models L'$
. By the definition of
$L'$
,
$\mathfrak {A} \not \models \epsilon (\mathfrak {B})$
for some
$\mathfrak {B} \in {\mathcal {FH}}_{\mathrm {fsi}}$
such that
$\mathfrak {B} \not \models L$
. So,
$\mathfrak {B}$
is a subalgebra of an s.i. homomorphic image of
$\mathfrak {A}$
. Since
$\mathcal {H}$
and
$\mathcal {S}$
preserve validity,
$\mathfrak {A} \not \models L$
, which is a contradiction. Thus,
$L' \subseteq L$
, and therefore,
$L = L'$
.
In practice, the characterization is particularly useful when combined with modal duality. Recall that a finite modal algebra is of finite height iff its dual space (i.e., finite Kripke frame) is cycle-free.
Example 3.4.
-
1.
is the largest union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
.Let
$\mathfrak {A}$
be a finite modal algebra such that
$\mathfrak {A} \not \models \mathsf {KD}$
and
$\mathfrak {X} = (X, R)$
be its dual space. Then, there is a point
$x \in \mathfrak {X}$
such that , that is, x is a dead end in
$\mathfrak {X}$
. So,
$\{x\}$
is a finite rooted cycle-free closed upset of
$\mathfrak {X}$
, and thus corresponds to an algebra
$\mathfrak {A}' \in \mathcal {H}(\mathfrak {A})_{\mathrm {fsi}} \cap {\mathcal {FH}}$
. Also, it is clear that
$\mathfrak {A}' \not \models \mathsf {KD}$
since x is a dead end. Thus, for any finite modal algebra
$\mathfrak {A}$
,
$\mathfrak {A} \notin \mathcal {V}(\mathsf {KD})$
implies
$\mathcal {H}(\mathfrak {A})_{\mathrm {fsi}} \cap {\mathcal {FH}} \not \subseteq \mathcal {V}(\mathsf {KD})$
. Hence,
$\mathsf {KD}$
is a union-splitting by Theorem 3.3.Moreover, let L be a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. For any finite
$\mathsf {KD}$
-algebra
$\mathfrak {A}$
, since the dual space of
$\mathfrak {A}$
is serial, there is no cycle-free closed upset of
$\mathfrak {A}$
, which implies that
$\mathcal {H}(\mathfrak {A})_{\mathrm {fsi}} \cap {\mathcal {FH}} = \emptyset $
, thus
$\mathfrak {A} \in \mathcal {V}(L)$
by Theorem 3.3. Since
$\mathsf {KD}$
has the finite model property, it follows that
$\mathcal {V}(\mathsf {KD}) \subseteq \mathcal {V}(L)$
, namely,
$L \subseteq \mathsf {KD}$
. So,
$\mathsf {KD}$
is the largest union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. -
2.
is not a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. Let
$\mathfrak {X}$
be the finite modal space
and
$\mathfrak {A}$
be its dual algebra. Clearly, . On the other hand, since the only cycle-free upset of
$\mathfrak {X}$
is the irreflexive singleton, which validates , it holds that . So, is not a union-splitting by Theorem 3.3.
is the largest union-splitting in 
, that is, x is a dead end in 
is not a union-splitting in 
. On the other hand, since the only cycle-free upset of 
, it holds that 
. So, 
is not a union-splitting by Theorem The condition (4) in Theorem 3.3 is special in its finitary nature. Subframe logics ([Reference Fine13], see also [Reference Chagrov and Zakharyaschev7, Section 11.3]) and stable logics ([Reference Bezhanishvili, Bezhanishvili and Ilin3]) also have similar semantic characterizations. However, the following example shows that there is a logic
${L \in \mathop {\mathsf {NExt}}{\mathsf {K4}}}$
such that the class of finite rooted L-frames is closed under subframes, while L is not a subframe logic. Also, several characterizations of stable logics are obtained in [Reference Bezhanishvili, Bezhanishvili and Ilin3] (see also [Reference Ilin14]), but none of them is completely finitary like the condition (4) in Theorem 3.3. An example for stable logics similar to the one below can be found in [Reference Takahashi18, Example 5.9].
Example 3.5. Recall Kripke semantics for modal logic (see, e.g., [Reference Blackburn, de Rijke and Venema4, Reference Chagrov and Zakharyaschev7]). Original Jankov formulas were adapted to transitive modal logic by Fine [Reference Fine12], called Jankov–Fine formulas in [Reference Blackburn, de Rijke and Venema4, Section 3.4]. We use Jankov–Fine formulas for Kripke frames in this example. The Jankov–Fine formula
$\gamma (\mathcal {G})$
associated with a finite rooted frame
$\mathcal {G}$
has the following property: for any Kripke frame
$\mathcal {F}$
,
$\mathcal {F} \not \models \gamma (\mathcal {G})$
iff
$\mathcal {G}$
is a p-morphic image of a generated subframe of
$\mathcal {F}$
.
Let
$\mathcal {F}$
be the Kripke frame of negative integers with the order
$<$
and a reflexive root
$\omega $
at the bottom. Let
$L = \mathsf {Log}\ \mathcal {F}$
.
Let
$\mathcal {G}$
be a finite rooted frame such that
$\mathcal {G} \models L$
. Then
$\mathcal {G} \not \models \gamma (\mathcal {G})$
since
$\mathcal {G}$
is a p-morphic image of a generated subframe of itself, thus
$\gamma (\mathcal {G}) \notin L$
. So,
$\mathcal {F} \not \models \gamma (\mathcal {G})$
, which means that
$\mathcal {G}$
is a p-morphic image of a generated subframe of
$\mathcal {F}$
. Note that a p-morphism cannot identify two irreflexive points in
$\mathcal {F}$
or an irreflexive point with the reflexive point
$\omega $
in
$\mathcal {F}$
. So, a generated subframe of
$\mathcal {F}$
has itself as its only p-morphic image. Since
$\mathcal {G}$
is finite,
$\mathcal {G}$
must be a finite irreflexive chain. Thus, the class of finite rooted L-frames is the class of finite irreflexive chains, which is closed under subframes. However,
$\mathcal {F}$
has the single reflexive point as its subframe, which refutes L. So, the class of L-frames is not closed under subframes, hence L is not a subframe logic.
Now we turn to our main theorem, the decidability of being a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. This answers the open question [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20, Problem 2] in the affirmative.
Lemma 3.6. Let
$L_0$
be a logic and
$\varphi $
be a formula. If
$L_0 + \varphi = L_0 + \{\psi _i: i \in I\}$
, then there is a finite subset
$I' \subseteq I$
such that
$L_0 + \varphi = L_0 + \{\psi _i: i \in I'\}$
.
Proof. Let
$\Phi $
be a set of formulas. It is clear from the syntax that, for any
$\psi \in L_0 + \Phi $
, there is a finite subset
$\Phi ' \subseteq \Phi $
such that
$\psi \in L_0 + \Phi '$
. It follows that the closure operator
$\Phi \mapsto L_0 + \Phi $
on the set of formulas is algebraic and
$\mathop {\mathsf {NExt}}{L_0}$
is an algebraic lattice, and finitely axiomatizable logics are exactly compact elements in
$\mathop {\mathsf {NExt}}{L_0}$
(see, e.g., [Reference Bergman1, Theorem 2.30]).
If
$L_0 + \varphi = L_0 + \{\psi _i: i \in I\}$
, then since it is a compact element, there is a finite subset
$I' \subseteq I$
such that
$L_0 + \varphi \subseteq L_0 + \{\psi _i: i \in I'\}$
. Since
$L_0 + \{\psi _i: i \in I'\} \subseteq L_0 + \{\psi _i: i \in I\} = L_0 + \varphi $
, we obtain
$L_0 + \varphi = L_0 + \{\psi _i: i \in I'\}$
.
Lemma 3.7. It is decidable whether a finite modal algebra is of finite height.
Proof. Let
$\mathfrak {A}$
be a finite modal algebra. For any
$a \in \mathfrak {A}$
and any
$n, m \in \omega $
such that
$n < m$
, if 
, then 
. So, 
for any
$a \in \mathfrak {A}$
. It follows that
$\mathfrak {A}$
is of finite height iff there is an
$n < |A|$
such that 
, which is decidable.
Lemma 3.8. It is decidable whether a finite modal algebra is s.i.
Proof. Let
$\mathfrak {A}$
be a finite modal algebra. We use the following characterization of s.i. modal algebras in [Reference Rautenberg17]:
$\mathfrak {A}$
is s.i. iff
$\mathfrak {A}$
has an opremum, that is, an element
$c \neq 1$
such that for any
$a \neq 1$
, there is some
$n \in \omega $
such that 
. By the same argument as in the proof of Lemma 3.7, if 
for some
$n \in \omega $
, then there must be such an
$n < |A|$
. So, the existence of an opremum is decidable, and hence so is being s.i.
Theorem 3.9. Being a union-splitting is decidable in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. That is, it is decidable, given a formula
$\varphi $
, whether the logic
$\mathsf {K} + \varphi $
is a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
.
Proof. First, we show that the union-splitting problem is
$\Sigma ^0_1$
. By Theorem 3.3 and Lemma 3.6,
$\mathsf {K} + \varphi $
is a union-splitting iff there is a finite set
$\{\mathfrak {A}_i: i < n\}$
of finite s.i. modal algebras of finite height such that
$\mathsf {K} + \varphi = \mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\}$
. By Lemmas 3.7 and 3.8, it is decidable whether a finite modal algebra is s.i. and of finite height. So, finite s.i. modal algebras of finite height and finite sets of them can be effectively coded. A finitely axiomatized logic
$\mathsf {K} + \{\alpha _j: j \leq m\}$
is recursively enumerable by enumerating all proofs from the axioms
$\{\alpha _j: j \leq m\}$
, so, given a formula
$\beta $
, the problem whether
$\beta \in \mathsf {K} + \{\alpha _j: j \leq m\}$
is
$\Sigma ^0_1$
. Since
$\Sigma ^0_1$
is closed under intersection, given a formula
$\varphi $
and a finite set
$\{\mathfrak {A}_i: i < n\}$
of finite s.i. modal algebras of finite height, the problem whether
$\mathsf {K} + \varphi = \mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\}$
is
$\Sigma ^0_1$
, for it is equivalent to
$\varphi \in \mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\}$
and
$\epsilon (\mathfrak {A}_0) \in \mathsf {K} + \varphi $
and
$\dots $
and
$\epsilon (\mathfrak {A}_{n-1}) \in \mathsf {K} + \varphi $
. As
$\Sigma ^0_1$
is closed under existential quantification, it follows that the union-splitting problem is
$\Sigma ^0_1$
.
Next, we show that the union-splitting problem is
$\Pi ^0_1$
. By Theorem 3.3,
$\mathsf {K} + \varphi $
is not a union-splitting iff there is a finite modal algebra
$\mathfrak {A}$
such that
$\mathcal {H}(\mathfrak {A})_{\mathrm {fsi}} \cap {\mathcal {FH}} \subseteq \mathcal {V}(\mathsf {K} + \varphi )$
and
$\mathfrak {A} \notin \mathcal {V}(K + \varphi )$
. A finite modal algebra
$\mathfrak {A}$
only has finitely many s.i. homomorphic images of finite height, which can be computed by Lemmas 3.7 and 3.8. So, whether
$\mathsf {K} + \varphi $
is not a union-splitting is
$\Sigma ^0_1$
, hence the union-splitting problem is
$\Pi ^0_1$
.
Thus, the union-splitting problem is both
$\Sigma ^0_1$
and
$\Pi ^0_1$
, hence decidable.
We can also provide an intuitive description of an algorithm that decides union-splittings in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
as follows. We start enumerating all finite sets
${\{\epsilon (\mathfrak {A}_i): i < n\}}$
where each
$\mathfrak {A}_i$
is a finite s.i. modal algebra of finite height. During the enumeration, for each enumerated finite set, we start verifying whether
${\mathsf {K} + \varphi = \mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\}}$
holds. If
$\mathsf {K} + \varphi $
is a union-splitting, then eventually the enumeration will find a finite set
$\{\epsilon (\mathfrak {A}_i): i < n\}$
that axiomatizes
$\mathsf {K} + \varphi $
and the identification verification halts. (One might be concerned about the “nested” computation here, but this is fine because we have a computable bijection from
$\omega $
to
$\omega \times \omega $
, which is the main reason that
$\Sigma ^0_1$
is closed under existential quantification.) Simultaneously, we start enumerating all finite modal algebras. For each of them, we compute all its s.i. homomorphic images of finite height and check if any of them witnesses that
$\mathsf {K} + \varphi $
breaks the condition (4) in Theorem 3.3. If
$\mathsf {K} + \varphi $
is not a union-splitting, we will eventually find such a witness. Combining these two, the algorithm decides whether
$\mathsf {K}+\varphi $
is a union-splitting.
This algorithm has an advantage that it is constructive, in the sense that if
$\mathsf {K} + \varphi $
is a union-splitting, then the algorithm outputs a finite set
$\{\epsilon (\mathfrak {A}_i): i < n\}$
that axiomatizes
$\mathsf {K} + \varphi $
. From this, the decidability of being a splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
follows.
Theorem 3.10. Being a splitting is decidable in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. That is, it is decidable, given a formula
$\varphi $
, whether the logic
$\mathsf {K} + \varphi $
is a splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
.
Proof. First, we run the algorithm from Theorem 3.9. If
$\mathsf {K} + \varphi $
is not a union-splitting, then the algorithm outputs false, and we are done because
$\mathsf {K} + \varphi $
is not a splitting. Suppose that
$\mathsf {K} + \varphi $
is a union-splitting. Then the algorithm outputs an axiomatization
$\mathsf {K} + \varphi = \mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\}$
, where each
$\mathfrak {A}_i$
is a finite s.i. modal algebra of finite height. If
$n=0$
, then
$\mathsf {K} + \varphi = \mathsf {K}$
, which is not a splitting. So, we may assume
$n \geq 1$
.
If
$\mathsf {K} + \varphi $
is a splitting, then
$\mathsf {K} + \varphi = \mathsf {K} + \epsilon (\mathfrak {B})$
for some finite s.i. modal algebra
$\mathfrak {B}$
of finite height. For each
$\mathfrak {A}_i$
, since
$\mathfrak {A}_i \not \models \epsilon (\mathfrak {A}_i)$
, we have
$\mathfrak {A}_i \not \models \epsilon (\mathfrak {B})$
, so
$\mathfrak {B}$
is a subalgebra of an s.i. homomorphic image of
$\mathfrak {A}_i$
thus
$|B| \leq |A_i|$
. Let
$m = \min \{|\mathfrak {A}_i|: i < n\}$
. Then
$|B| \leq m$
. Thus,
$\mathsf {K} + \varphi $
is a splitting iff
$\mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\} = \mathsf {K} + \epsilon (\mathfrak {B})$
for some finite s.i. modal algebra
$\mathfrak {B}$
of finite height such that
$|B| \leq m$
.
There are only finitely many such
$\mathfrak {B}$
, and we can effectively enumerate all of them. Also, given a
$\mathfrak {B}$
, whether
$\mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\} = \mathsf {K} + \epsilon (\mathfrak {B})$
holds is decidable because both logics are decidable by Corollary 2.9. Thus, it is decidable whether
$\mathsf {K} + \varphi $
is a splitting.
Note that in other lattices of normal modal logics, say,
$\mathop {\mathsf {NExt}}{\mathsf {K4}}$
or
$\mathop {\mathsf {NExt}}{\mathsf {S4}}$
, the situation is very different. All finite
$\mathsf {K4}$
-algebras (resp.
$\mathsf {S4}$
-algebras), not only those of finite height, split the lattice
$\mathop {\mathsf {NExt}}{\mathsf {K4}}$
(resp.
$\mathop {\mathsf {NExt}}{\mathsf {S4}}$
). It is unknown if Theorem 2.8 holds in
$\mathop {\mathsf {NExt}}{\mathsf {K4}}$
or
$\mathop {\mathsf {NExt}}{\mathsf {S4}}$
, let alone the decidability of being a union-splitting.
4 Axiomatization problems and (un)decidable formulas
In this section, we apply our decidability result to axiomatization problems and (un)decidable formulas.
4.1 Axiomatization problems
Axiomatization problems are one of the simplest types of decision problems and have been widely studied (see, e.g., [Reference Chagrov and Zakharyaschev7, Chapter 17] and [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20]). They are called the problem of coincidence in [Reference Chagrov and Zakharyaschev7, Chapter 17]
Definition 4.1. Given a modal logic
$L_0$
and a formula
$\varphi $
, the axiomatization problem for
$L_0 + \varphi $
is, given a formula
$\psi $
, to decide whether
$L_0 + \psi = L_0 + \varphi $
.
In other words, the axiomatization problem for
$L_0 + \varphi $
is decidable iff the property “
$ = L_0 + \varphi $
” is decidable in
$\mathop {\mathsf {NExt}}{L_0}$
.
The following theorem can be proved by combining Chagrov’s method [Reference Chagrov8, Reference Chagrov9] and the proof of Blok’s dichotomy theorem [Reference Blok5]. This is claimed in [Reference Chagrov and Zakharyaschev7, Section 17.6] and a proof can be found in [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20, Theorem 7].
Theorem 4.2. The axiomatization problem for a logic
$\mathsf {K} + \varphi $
is decidable iff
$\mathsf {K} + \varphi $
is a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
or the inconsistent logic.
We provide a proof of the right-to-left direction using our algorithm that decides union-splittings. Because of the constructive nature of the algorithm, unlike the proof in [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20, Theorem 7], our proof yields an algorithm that, given a formula
$\varphi $
such that
$\mathsf {K} + \varphi $
is a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
or the inconsistent logic, outputs an algorithm that decides the axiomatization problem for
$\mathsf {K} + \varphi $
.
Lemma 4.3. Let
$\mathsf {K} + \varphi $
be a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. Then the axiomatization problem for
$\mathsf {K} + \varphi $
is decidable.
Proof. Let
$\mathsf {K} + \varphi $
be a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. Applying the algorithm from Theorem 3.9, we obtain an axiomatization
$\mathsf {K} + \varphi = \mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\}$
, where each
$\mathfrak {A}_i$
is a finite s.i. modal algebra of finite height. If
$n = 0$
, then
$\mathsf {K} + \varphi = \mathsf {K}$
, and the decidability of the axiomatization problem for
$\mathsf {K} + \varphi $
follows from the decidability of
$\mathsf {K}$
. So, we may assume
$n \geq 1$
.
Given a formula
$\psi $
,
$\mathsf {K} + \psi = \mathsf {K} + \varphi $
iff
$\psi \in \mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\}$
and
${\epsilon (\mathfrak {A}_i) \in \mathsf {K} + \psi }$
for
$i < n$
. Whether
$\psi \in \mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\}$
holds is decidable by Corollary 2.9. To decide whether
$\epsilon (\mathfrak {A}_i) \in \mathsf {K} + \psi $
holds, note that if
$\epsilon (\mathfrak {A}_i) \in \mathsf {K} + \psi $
then
$\mathfrak {A}_i \not \models \psi $
since
$\mathfrak {A}_i \not \models \epsilon (\mathfrak {A}_i)$
, and if
$\mathfrak {A}_i \not \models \psi $
, then
$\mathsf {K} + \psi \not \subseteq \mathsf {Log}\ \mathfrak {A}_i$
, so
$\mathsf {K} + \epsilon (\mathfrak {A}_i) \subseteq \mathsf {K} + \psi $
since
$(\mathsf {K} + \epsilon (\mathfrak {A}_i), \mathsf {Log}\ \mathfrak {A}_i)$
is a splitting pair, and thus
$\epsilon (\mathfrak {A}_i) \in \mathsf {K} + \psi $
. So, it suffices to check whether
$\mathfrak {A}_i \not \models \psi $
holds, which is decidable. Hence, it is decidable whether
$\mathsf {K} + \psi = \mathsf {K} + \varphi $
.
Lemma 4.4. The axiomatization problem for the inconsistent logic is decidable.
Proof. It is well-known that the consistency is decidable in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
(see, e.g., [Reference Chagrov and Zakharyaschev7, Theorem 17.2]), as the lattice
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
has only two co-atoms and both are decidable. The statement follows as the axiomatization problem for the inconsistent logic coincides with the problem of deciding inconsistency.
Proof of Theorem 4.2.
The right-to-left direction follows from Lemmas 4.3 and 4.4; see [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20, Theorem 7] for the other direction.
Corollary 4.5. It is decidable whether the axiomatization problem for
$\mathsf {K} + \varphi $
is decidable.
Moreover, the proof yields an algorithm that not only decides whether an axiomatization problem is decidable for a logic
$\mathsf {K} + \varphi $
, but also outputs an algorithm deciding the axiomatization problem.
4.2 (Un)decidable formulas
Undecidable formulas are introduced in [Reference Chagrov and Jablonskij10] (see also [Reference Chagrov and Zakharyaschev7, Section 16.4]).
Definition 4.6. Let
$L_0$
be a logic. A formula
$\varphi $
is called a (un)decidable formula in
$\mathop {\mathsf {NExt}}{L_0}$
if it is (un)decidable, given a formula
$\psi $
, whether
$\varphi \in L + \psi $
.
From the proof of Lemma 4.3, we observe that if
$\mathsf {K} + \varphi $
is a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
, then
$\varphi $
is a decidable formula in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
: we first compute an axiomatization
$\mathsf {K} + \varphi = \mathsf {K} + \{\epsilon (\mathfrak {A}_i): i < n\}$
, then for any formula
$\psi $
,
$\varphi \in \mathsf {K} + \psi $
iff
$\mathfrak {A}_i \not \models \psi $
for all i, which is a decidable condition. Also,
$\bot $
is a decidable formula in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
as the consistency is decidable. Moreover, the proof of Theorem 4.2 in [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20, Theorem 7] in fact established the converse: if
$\mathsf {K} + \varphi $
is neither a union-splitting nor the inconsistent logic in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
, then
$\varphi $
is an undecidable formula in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. Thus, we obtain the following theorem.
Theorem 4.7. A formula
$\varphi $
is a decidable formula in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
iff
$\mathsf {K} + \varphi $
is a union-splitting in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
or the inconsistent logic.
Corollary 4.8. It is decidable whether
$\varphi $
is a (un)decidable formula in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
.
Proof. This follows from Theorems 3.9 and Theorem 4.7, and the fact that the consistency is decidable.
This answers [Reference Chagrov and Zakharyaschev7, Problem 17.3] for
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
in the affirmative.
Moreover, we also obtain a somewhat mysterious equivalence between decidable axiomatization problems, decidable formulas, and union-splittings in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
. Given that the proof of the undecidability of various logical properties [Reference Wolter, Zakharyaschev, Blackburn, Benthem and Wolter20, Theorem 9] makes an essential use of union-splittings, union-splittings may have a deeper connection to the decision problem of logical properties.
Corollary 4.9. For any formula
$\varphi $
, the following are equivalent:
-
1. the axiomatization problem for
$\mathsf {K} + \varphi $
is decidable; -
2.
$\varphi $
is a decidable formula; -
3.
$\mathsf {K} + \varphi $
is a union-splittings in
$\mathop {\mathsf {NExt}}{\mathsf {K}}$
or the inconsistent logic.
Remark 4.10. A Kripke complete logic L is called strictly Kripke complete (see, e.g., [Reference Chagrov and Zakharyaschev7, Section 10.5]) if no other logic has the same class of Kripke frames as L. Blok [Reference Blok5] showed that a logic L is strictly Kripke complete iff L is a union-splitting or the inconsistent logic. Thus, strict Kripke completeness is equivalent to each item in Corollary 4.9 and is also decidable.
Acknowledgment
The author is very grateful to Nick Bezhanishvili for his supervision on the Master’s thesis and his valuable comments on this article.
Funding
The author was supported by the Student Exchange Support Program (Graduate Scholarship for Degree Seeking Students) of the Japan Student Services Organization and the Student Award Scholarship of the Foundation for Dietary Scientific Research. Open access funding provided by University of Amsterdam.
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