K4.2 Modal Logic: Transitive & Directed Frames

• K4.2 is a normal modal logic defined by extending K4 with the directedness axiom, resulting in transitive and directed Kripke frames.
• The logic shows PSPACE-completeness in its one-variable fragment, underlining that a single atomic proposition can encode complex behaviors.
• K4.2 serves as a critical boundary in modal complexity, highlighting that adding convergence does not simplify satisfiability problems.

Searching arXiv for the primary source on K4.2 and closely related work. (Rybakov et al., 16 Jul 2025): Logics with the axiom of convergence: complexity with a small number of variables in the language (extended version)

Authors: M. Bystrov, R. Kuznets

Abstract: It is known that many modal and superintuitionistic logics are PSPACE-hard in languages with a small number of variables; however, questions about the complexity of similar fragments of many logics obtained by adding various axioms to "standard" ones remain unexplored. We investigate the complexity of fragments of modal logics obtained by adding an axiom requiring the convergence of the accessibility relation in Kripke frames: S4.2, K4.2, Grz.2, and GL.2. The main result is that S4.2 and Grz.2 are PSPACE-complete in a language with two variables, while K4.2 and GL.2 are PSPACE-complete in a language with one variable.

URL: http://arxiv.org/abs/([2507.12343](/papers/2507.12343)) Searching for related arXiv work on small-variable modal-logic complexity and convergence logics. No additional strongly relevant arXiv results were found with a direct title/keyword match beyond the primary source (Rybakov et al., 16 Jul 2025). K4.2 is a normal modal propositional logic obtained by extending $\mathbf{K4}$ with an axiom of convergence, more precisely an axiom of directedness. In the formulation studied in "Logics with the axiom of convergence: complexity with a small number of variables in the language (extended version)" (Rybakov et al., 16 Jul 2025), K4.2 belongs to a family of logics with the axiom of convergence that also includes $\mathbf{S4.2}$, $\mathbf{GL.2}$, and $\mathbf{Grz.2}$. Its defining significance is twofold: semantically, it is characterized by transitive, directed Kripke frames; computationally, its satisfiability problem remains PSPACE-complete even when formulas are restricted to a single propositional variable (Rybakov et al., 16 Jul 2025).

1. Axiomatic definition and placement in the modal hierarchy

The ambient setting is standard normal modal propositional logic. The base system is $\mathbf{K}$, defined Hilbert-style as

$$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$

where $\mathbf{Cl}$ is classical propositional logic, and the closure operations are Modus Ponens, substitution, and necessitation (Rybakov et al., 16 Jul 2025).

K4.2 is built on $\mathbf{K4}$, where

$$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$

The additional axiom is the directedness axiom

$$\bm{dir}=\Diamond(p\wedge \Box q)\to\Box(p\vee \Diamond q).$$

Accordingly,

$\mathbf{S4.2}$0

Within the hierarchy considered in the paper, $\mathbf{S4.2}$1 extends $\mathbf{S4.2}$2 by adding convergence or directedness. $\mathbf{S4.2}$3 is stronger because it also assumes reflexivity, while $\mathbf{S4.2}$4 and $\mathbf{S4.2}$5 are provability-style extensions based on $\mathbf{S4.2}$6 and $\mathbf{S4.2}$7 respectively, again with the same directedness axiom (Rybakov et al., 16 Jul 2025). The paper explicitly treats K4.2 as the weakest among these logics that is both transitive and directed but not reflexive or well-founded by default.

A related axiom is the Geach axiom

$\mathbf{S4.2}$8

In reflexive settings such as $\mathbf{S4.2}$9 and $\mathbf{GL.2}$0, $\mathbf{GL.2}$1 is equivalent to $\mathbf{GL.2}$2. Over non-reflexive bases such as $\mathbf{GL.2}$3 and $\mathbf{GL.2}$4, however, the paper stresses that one must use $\mathbf{GL.2}$5 rather than $\mathbf{GL.2}$6 to capture the frame condition of interest (Rybakov et al., 16 Jul 2025).

2. Kripke semantics and the directedness condition

Kripke semantics is assumed in the standard form. A frame is a pair $\mathbf{GL.2}$7 with non-empty $\mathbf{GL.2}$8 and accessibility relation $\mathbf{GL.2}$9, and a model is $\mathbf{Grz.2}$0 with valuation $\mathbf{Grz.2}$1. Truth for the modal operators is given by

$\mathbf{Grz.2}$2

The $\mathbf{Grz.2}$3 component contributes transitivity: $\mathbf{Grz.2}$4 The added directedness condition says that whenever a world has two distinct successors, those successors can be joined further along: $\mathbf{Grz.2}$5 The paper describes this informally as the situation in which, if $\mathbf{Grz.2}$6 has outgoing arrows to $\mathbf{Grz.2}$7 and $\mathbf{Grz.2}$8, one can push forward from $\mathbf{Grz.2}$9 and $\mathbf{K}$0 to a common successor $\mathbf{K}$1 (Rybakov et al., 16 Jul 2025).

For K4.2, the semantic characterization is therefore transitive, directed frames. The paper states that each logic under consideration is characterized by some class of Kripke frames where the accessibility relation is convergent, but that, more precisely, it is directed. For K4.2, soundness and completeness are thus with respect to frames whose accessibility relation is both transitive and directed (Rybakov et al., 16 Jul 2025).

This distinction matters because the same modal pattern behaves differently in reflexive and non-reflexive settings. Over reflexive transitive logics, directedness coincides with convergence in the usual sense and is equivalent to the Geach axiom. Over non-reflexive transitive logics such as K4.2, the axiom

$\mathbf{K}$2

is the modal counterpart used to capture directedness.

3. Finite-variable fragments and the associated decision problem

The language is the usual propositional modal language with propositional variables, Boolean connectives, and the operators $\mathbf{K}$3 and $\mathbf{K}$4. For a logic $\mathbf{K}$5, its $\mathbf{K}$6-variable fragment is the set of formulas using at most $\mathbf{K}$7 distinct propositional variables (Rybakov et al., 16 Jul 2025).

The central problem studied for K4.2 is satisfiability in bounded-variable languages: given a formula $\mathbf{K}$8 with at most $\mathbf{K}$9 variables, is $$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$0 satisfiable in a frame validating $$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$1? Equivalently, validity is the complement problem. The paper considers satisfiability over the class of all Kripke frames validating the logic, not only finite frames (Rybakov et al., 16 Jul 2025).

The background motivation is that many modal and superintuitionistic logics are already known to be PSPACE-hard in languages with a small number of variables, but analogous questions for logics obtained by adding axioms of convergence had remained largely unexplored. K4.2 is one of the principal cases in which this gap is addressed (Rybakov et al., 16 Jul 2025).

For K4.2, the notable fact is that bounding the number of propositional variables does not simplify the problem in any essential way. The one-variable fragment is already computationally intractable at the PSPACE level.

4. Complexity of K4.2 and neighboring convergence logics

The main complexity result for K4.2 is that its satisfiability problem remains PSPACE-complete when formulas are restricted to a single propositional variable (Rybakov et al., 16 Jul 2025). The paper places this within two more general theorems:

Theorem 1. Every modal logic between $$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$2 and $$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$3 is PSPACE-hard in a language with two propositional variables.

Theorem 2. Every modal logic between $$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$4 and $$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$5 is PSPACE-hard in a language with one propositional variable.

Since

$$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$6

Theorem 2 yields one-variable PSPACE-hardness for K4.2. Combined with the standard PSPACE upper bound for the unrestricted logic, this gives PSPACE-completeness for one-variable satisfiability (Rybakov et al., 16 Jul 2025).

The comparative picture stated in the paper is as follows.

Logic	Defining extension	PSPACE-complete fragment
$$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$7	$$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$8	one variable
$$\mathbf{K} = \mathbf{Cl} \oplus \bigl(\Box(p\to q)\to(\Box p\to \Box q)\bigr),$$9	$\mathbf{Cl}$0	one variable
$\mathbf{Cl}$1	$\mathbf{Cl}$2	two variables
$\mathbf{Cl}$3	$\mathbf{Cl}$4	two variables

The contrast with $\mathbf{Cl}$5 and $\mathbf{Cl}$6 is tied, in the paper’s account, to their connection with the modal counterparts of the intuitionistic logic $\mathbf{Cl}$7, whose known PSPACE-hardness arguments require two variables. By contrast, K4.2 and GL.2 inherit enough of the frame-encoding power of $\mathbf{Cl}$8 and $\mathbf{Cl}$9 to remain PSPACE-hard with one variable (Rybakov et al., 16 Jul 2025).

5. Proof methods and the role of one-variable coding

The extended text does not present full proofs for K4.2-specific lower bounds, but it states that the proofs use methods for studying the complexity of modal and superintuitionistic logics, citing work by Ladner, Statman, Shapirovsky, and Chagrov (Rybakov et al., 16 Jul 2025). The high-level picture given for K4.2 is a generic embedding strategy.

The paper’s presentation suggests the following structure. One starts from a logic already known to be PSPACE-hard in the one-variable fragment, such as $\mathbf{K4}$0, and embeds formulas into the one-variable fragment of a target logic $\mathbf{K4}$1 with

$\mathbf{K4}$2

Because K4.2 lies in this interval, the same general reduction applies (Rybakov et al., 16 Jul 2025).

A typical technique is to simulate multiple propositional variables by patterns of accessibility involving a single propositional variable $\mathbf{K4}$3. Each source variable is represented by a formula $\mathbf{K4}$4 using only $\mathbf{K4}$5 together with modal structure, often via nesting of $\mathbf{K4}$6 and $\mathbf{K4}$7. The model transformation then ensures that the truth of $\mathbf{K4}$8 along suitable paths encodes the valuation of all source variables (Rybakov et al., 16 Jul 2025).

The paper further notes that the directedness axiom is helpful because it ensures certain confluence properties in the frame. This gives extra freedom to construct coding gadgets in which different paths can later rejoin at a common successor. The central point is negative rather than constructive: adding directedness does not decrease the complexity. One can still build enough structure in transitive directed frames to simulate the computations used in earlier PSPACE-hardness reductions (Rybakov et al., 16 Jul 2025).

For membership in PSPACE, the upper bound is inherited from known PSPACE algorithms for K4-type and related logics. The paper indicates that restricting the number of variables never increases complexity, so the standard PSPACE upper bound for K4.2 continues to apply to the one-variable fragment. The algorithmic intuition is the familiar one of depth-first tableau search or filtration-based model construction with polynomial space reuse (Rybakov et al., 16 Jul 2025).

6. General interval results, implications, and open problems

K4.2 is not treated as an isolated logic. Its complexity behavior is part of two infinite interval results. The first covers every modal logic between $\mathbf{K4}$9 and $$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$0, each of which is PSPACE-hard in two variables. The second covers every modal logic between $$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$1 and $$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$2, each of which is PSPACE-hard in one variable. Since

$$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$3

the one-variable PSPACE-hardness of K4.2 is a special case of a broader theorem about an entire interval of modal logics (Rybakov et al., 16 Jul 2025).

Several implications are drawn explicitly for K4.2. First, even one atomic proposition suffices for PSPACE-complete behavior once transitivity and directedness are available. This supports the view that the complexity lies largely in the shapes of transitive directed frames rather than in propositional combinatorics. Second, adding the nontrivial directedness axiom does not simplify satisfiability in any essential way: restricting to directed frames still leaves the problem PSPACE-complete (Rybakov et al., 16 Jul 2025).

The paper also records several open problems. One concerns the variable-free fragments of K4.2 and $$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$4: for $$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$5, $$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$6, and $$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$7, variable-free fragments are already PSPACE-hard, but for K4.2 it is unknown whether the variable-free fragment is PSPACE-hard, NP-complete, or easier. A second concerns the one-variable fragments of $$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$8 and $$\mathbf{K4} = \mathbf{K} \oplus \Box p \to \Box\Box p.$$9, for which two-variable PSPACE-hardness is known but one-variable hardness remains open. A third concerns superintuitionistic fragments obtained as preimages under Gödel translation, including convergent analogues of logics such as BPL and FPL, whose finite-variable complexity is left unresolved (Rybakov et al., 16 Jul 2025).

A plausible implication is that K4.2 occupies a particularly informative boundary point in the landscape of modal complexity. It is weaker than the reflexive and provability-style convergence logics considered alongside it, yet already exhibits maximal PSPACE hardness in the one-variable setting. In that sense, K4.2 functions as a canonical example of how strong computational complexity can persist under substantial semantic restriction.