2.1 Descriptive frames for normal modal logics

The formulas, $\varphi $ , of propositional unimodal logics are built from propositional variables $p_i \in \mathcal {V}$ , for some countably-infinite set $\mathcal {V} = \{p_i \mid i < \omega \}$ , and constants $\top $ , $\bot $ using the Boolean connectives $\neg $ , $\land $ , and the unary possibility operator $\Diamond $ . The other Booleans and the necessity operator $\Box $ dual to $\Diamond $ are defined as standard abbreviations. We also use $\Diamond ^+\varphi = \varphi \lor \Diamond \varphi $ , $\Box ^+\varphi =\varphi \land \Box \varphi $ , and $\Diamond \Gamma =\{\Diamond \varphi \mid \varphi \in \Gamma \}$ , for a set $\Gamma $ of formulas. By a signature we mean any set $\sigma \subseteq \mathcal {V}$ , denoting by $\textit {sig}(\varphi )$ the (finite) set of variables in a formula $\varphi $ . If $\textit {sig}(\varphi ) \subseteq \sigma $ , we call $\varphi $ a $\sigma $ -formula. We denote by $\textit {sub}(\varphi )$ the set of subformulas of $\varphi $ together with their negations, and let $|\varphi |=|\textit {sub}(\varphi )|$ .

A (normal) modal logic, L, is any set of formulas that contains all Boolean tautologies, the modal axiom $ \Box (p_0 \to p_1) \to (\Box p_0 \to \Box p_1) , $ and is closed under the rules of modus ponens, uniform substitution of formulas in place of variables, and necessitation $\varphi /\Box \varphi $ . The smallest such logic goes by the moniker $\mathsf {K}$ . Given a set $\Gamma $ of formulas and a modal logic L, the smallest modal logic to contain L and $\Gamma $ is denoted by $L\oplus \Gamma $ . We write $L\oplus \varphi $ for $L\oplus \{\varphi \}$ . For example,

$$ \begin{align*} \mathsf{K4} & = \mathsf{K}\oplus \Box p_0 \to \Box\Box p_0, \\ \mathsf{K4.3} & = \mathsf{K4} \oplus \Box (\Box^+ p_0 \to p_1) \lor \Box (\Box^+ p_1 \to p_0),\\ \mathsf{GL.3} & = \mathsf{K4.3} \oplus \Box( \Box p_0 \to p_0) \to \Box p_0,\\ \mathsf{Log}\{(\mathbb N,<)\} & = \mathsf{K4.3} \oplus \Diamond\top \oplus \Box (\Box p_0 \to p_0) \to (\Diamond\Box p_0 \to \Box p_0). \end{align*} $$

All logics considered in this article are extensions of $\mathsf {K4.3}$ .

We interpret formulas in (general) frames $\mathfrak {F} = (W,R,\mathcal {P})$ , where R is a binary (accessibility) relation on a nonempty set W (of worlds or, more neutrally, points) and $\mathcal {P} \subseteq 2^W$ contains $\emptyset $ , W and is closed under $\cap $ , $\neg $ , and the operator

$$\begin{align*}\Diamond^{\mathfrak F} X = \{ x \in W \mid \exists y\in X \, x R y \}. \end{align*}$$

The structure $\mathfrak F^+ = (\mathcal {P}, \cap ,\neg ,\emptyset , W, \Diamond ^{\mathfrak F})$ is a Boolean algebra $(\mathcal {P}, \cap ,\neg ,\emptyset , W)$ with a normal and additive operator $\Diamond ^{\mathfrak F}$ (BAO, for short). If $\mathfrak F^+$ is generated by a set $\mathcal X\subseteq \mathcal {P}$ as a BAO, we say that the frame $\mathfrak F$ (or the set $\mathcal {P}$ ) is generated by $\mathcal X$ . If $|\mathcal X|=n$ , for some $n<\omega $ , we call $\mathfrak F n$ -generated or finitely generated. The elements of $\mathcal {P}$ are called internal sets in $\mathfrak F$ . If $\mathcal {P} = 2^W$ , $\mathfrak {F}$ is known as a Kripke frame; in this case, we drop $\mathcal {P}$ and write $\mathfrak {F}=(W,R)$ . A frame $\mathfrak {F}=(W,R,\mathcal {P})$ is descriptive if the following conditions hold: for any $x,y \in W$ and any $\mathcal {X} \subseteq \mathcal {P}$ ,

1. (dif) $x = y$ iff $\forall X \in \mathcal {P}\, (x \in X \leftrightarrow y \in X)$ ,

2. (tig) $xRy$ iff $\forall X\in \mathcal {P} \, (y \in X \to x \in \Diamond ^{\mathfrak F} X)$ ,

3. (com) if $\mathcal {X} \subseteq \mathcal {P}$ has the finite intersection property (fip, for short)—that is, $\bigcap \mathcal {X}' \ne \emptyset $ , for every finite $\mathcal {X}' \subseteq \mathcal {X}$ —then $\bigcap \mathcal {X} \ne \emptyset $ .

(Frames with (dif) are called differentiated, with (tig) tight, and with (com) compact.) Every BAO is isomorphic to $\mathfrak F^+$ , for some descriptive frame $\mathfrak F$ . A finite frame is descriptive iff it is a Kripke frame [Reference Chagrov and Zakharyaschev9, Section 8].

Given a signature $\sigma $ , a $\sigma $ -model based on a frame $\mathfrak {F}=(W,R,\mathcal {P})$ is a pair $\mathfrak {M}=(\mathfrak {F},\mathfrak {v})$ with a valuation $\mathfrak {v} \colon \sigma \to \mathcal {P}$ . The atomic $\sigma $ -type of $x\in W$ in $\mathfrak {M}$ is

$$\begin{align*}\textit{at}^\sigma_{\mathfrak{M}}(x) = \{ p_i\mid p_i \in\sigma,\ x\in\mathfrak v(p_i)\}\cup \{ \neg p_i \mid p_i \in\sigma,\ x\notin\mathfrak v(p_i)\}. \end{align*}$$

We omit $\sigma =\mathcal V$ , saying simply model and writing $\textit {at}_{\mathfrak {M}}(x)$ . The value of a formula $\varphi $ in $\mathfrak M$ is the set $\mathfrak {v}(\varphi ) \in \mathcal {P}$ computed inductively in the obvious way starting from $\mathfrak v(p_i)$ , $\mathfrak v(\top ) = W$ and $\mathfrak v(\bot ) = \emptyset $ . A set $X\subseteq W$ is definable in $\mathfrak {M}$ if $X=\mathfrak {v}(\varphi )$ , for some formula $\varphi $ , in which case $X \in \mathcal {P}$ . If every internal set $X\in \mathcal {P}$ is definable in $\mathfrak M$ , we say that $\mathfrak {F}$ is $\mathfrak {M}$ -generated. Every $\mathfrak F$ with countable $\mathcal {P}$ is clearly $\mathfrak M$ -generated, for some model $\mathfrak M$ .

A formula $\varphi $ is true at x in $\mathfrak M$ , written $\mathfrak M,x \models \varphi $ , if $x \in \mathfrak {v}(\varphi )$ . The $\sigma $ -type of x in $\mathfrak {M}$ is the set $t^\sigma _{\mathfrak {M}}(x)$ of all $\sigma $ -formulas that are true at x in $\mathfrak M$ . For a set X of points in $\mathfrak M$ , we let $t_{\mathfrak {M}}^{\sigma }(X)=\big \{t_{\mathfrak {M}}^{\sigma }(x)\mid x\in X\big \}$ . As before, we drop $\sigma =\mathcal V$ .

A set $\Gamma $ of formulas is finitely satisfiable in $\mathfrak M$ if, for every finite subset $\Gamma ' \subseteq \Gamma $ , there is $x' \in W$ such that $\Gamma ' \subseteq t_{\mathfrak {M}}(x')$ ; $\Gamma $ is satisfiable in $\mathfrak M$ if $\Gamma \subseteq t_{\mathfrak {M}}(x)$ , for some $x \in W$ . Using these definitions and notations, we can equivalently reformulate conditions (dif), (tig), and (com) for $\mathfrak M$ -generated frames as follows: for any $x,y \in W$ and any set $\Gamma $ of formulas,

1. (dif) $x = y$ iff $t_{\mathfrak {M}}(x) = t_{\mathfrak {M}}(y)$ ,

2. (tig) $xRy$ iff $\Diamond t_{\mathfrak {M}}(y) \subseteq t_{\mathfrak {M}}(x)$ iff $\{\varphi \mid \Box \varphi \in t_{\mathfrak {M}}(x)\}\subseteq t_{\mathfrak {M}}(y)$ ,

3. (com) if $\Gamma $ is finitely satisfiable in $\mathfrak M$ , then $\Gamma $ is satisfiable in $\mathfrak M$ .

A frame $\mathfrak F$ satisfies $\Gamma $ if there is a model $\mathfrak M$ based on $\mathfrak F$ satisfying $\Gamma $ . Further, $\varphi $ is valid in $\mathfrak F$ , written $\mathfrak F \models \varphi $ , if $\mathfrak M,x \models \varphi $ , for any model $\mathfrak M$ based on $\mathfrak F$ and any $x \in W$ . We call $\mathfrak F$ a frame for a logic L and write $\mathfrak F \models L$ if $\mathfrak F \models \varphi $ , for all $\varphi \in L$ . Conversely, any class $\mathcal {S}$ of general frames determines the modal logic $\mathsf {Log}\, \mathcal {S} = \{\varphi \mid \forall \mathfrak F \in \mathcal {S} \ \mathfrak F \models \varphi \}$ . We write $\mathsf {Log}(\mathfrak F)$ for $\mathsf {Log}(\{\mathfrak F\})$ .

A set $\Gamma $ of formulas is L-consistent if $(\bigwedge \Gamma '\to \bot )\notin L$ , for any finite $\Gamma '\subseteq \Gamma $ . We require the following well-known fact (see, e.g., [Reference Chagrov and Zakharyaschev9, Section 8.6]).

By Lemma 2.1, every modal logic L is determined by the class of all descriptive frames for L. A logic L is Kripke complete if L is determined by the class of all Kripke frames for L. L is d-persistent (aka canonical) if $(W,R,\mathcal {P}) \models L$ implies $(W,R) \models L$ , for any descriptive frame $(W,R,\mathcal {P})$ . L has the fmp if it is determined by its finite (Kripke) frames.

The smallest logic $\mathsf {K4.3}$ we are interested in is d-persistent; its descriptive and Kripke frames $\mathfrak F=(W,R,\mathcal {P})$ are transitive and weakly connected, that is,

$$ \begin{align*} & \forall x,y,z \in W\, (xRy \land yRz \to xRz),\\ & \forall x,y,z \in W\, (xRy\land xRz \to y = z \lor yRz \lor zRy). \end{align*} $$

$\mathsf {GL.3}$ , on the contrary, is not d-persistent yet has the fmp. In fact, all extensions of $\mathsf {K4.3}$ are Kripke complete [Reference Fine12].

A frame $\mathfrak F' = (W',R',\mathcal {P}')$ is a subframe of a frame $\mathfrak F = (W,R,\mathcal {P})$ if $W' \subseteq W$ , $R' = {R}\mathop {\restriction }_{W'}=R\cap (W'\times W')$ , and $\mathcal {P}' \subseteq \mathcal {P}$ . For every internal set $V\in \mathcal {P}$ , the frame ${\mathfrak F}\mathop {\restriction }_{V}=\big (V,{R}\mathop {\restriction }_{V},{\mathcal {P}}\mathop {\restriction }_{V}\big )$ with ${\mathcal {P}}\mathop {\restriction }_{V}=\{V\cap X\mid X\in \mathcal {P}\}$ is a subframe of $\mathfrak F$ . For a model $\mathfrak M=(\mathfrak F,\mathfrak v)$ , we let ${\mathfrak M}\mathop {\restriction }_{V}=({\mathfrak F}\mathop {\restriction }_{V},{\mathfrak v}\mathop {\restriction }_{V})$ , where ${\mathfrak v}\mathop {\restriction }_{V}(p)=V\cap \mathfrak v(p)$ . Given a frame $\mathfrak F = (W,R,\mathcal {P})$ with transitive R and a point $x\in W$ , we define the frame $\mathfrak F_x = (W_x,R_x,\mathcal {P}_x)$ by taking $W_x = \{y \in W \mid xR^+y\}$ , where $R^+$ is the reflexive closure of R (that is, $R^+=R\cup \{(y,y) \mid y\in W\}$ ), $R_x = {R}\mathop {\restriction }_{W_x}$ , and $\mathcal {P}_x = {\mathcal {P}}\mathop {\restriction }_{W_x}$ . We call $\mathfrak {F}$ rooted if $\mathfrak F = \mathfrak F_x$ , for some $x \in W$ , in which case x is called a root of $\mathfrak F$ . Note that $\mathfrak F_x$ is not necessarily a subframe of $\mathfrak F$ , but we have:

(1) $$ \begin{align} \text{if } \mathfrak F \text{ is descriptive and transitive, then } \mathfrak F_x \text{ is descriptive as well.} \end{align} $$

Indeed, suppose $\mathfrak F = (W,R,\mathcal {P})$ is descriptive and $x\in W$ . Conditions (dif) and (tig) for $\mathfrak F_x$ are straightforward and left to the reader. To establish (com), consider any $\mathcal {X}_x \subseteq \mathcal {P}_x$ with the fip. Then

$$ \begin{align*} \mathcal{X} = \{V \in \mathcal{P} \mid V \cap W_x \in \mathcal{X}_x\} \cup \{V \in \mathcal{P} \mid W_x \subseteq V\} \end{align*} $$

also has the fip, and so $\bigcap \mathcal {X} \ne \emptyset $ . To prove that $\bigcap \mathcal {X}_x \ne \emptyset $ , it suffices to show that $\bigcap \{V \in \mathcal {P} \mid W_x \subseteq V\} \subseteq W_x$ . To this end, suppose on the contrary that $y \in \bigcap \{V \in \mathcal {P} \mid W_x \subseteq V\}$ and $y \notin W_x$ . Then (dif) and (tig) give $Z,Y \in \mathcal {P}$ such that $x\in Z$ , $y\notin Z$ , $y \in Y$ , and $x \in \Box \neg Y$ . It follows that $Z\cup \neg Y \in \mathcal {P}$ , $W_x \subseteq Z\cup \neg Y$ , and so $y \in Z\cup \neg Y$ , which is a contradiction.

2.2 The structure of linear finitely-generated descriptive frames

From now on, all frames $\mathfrak F=(W,R,\mathcal {P})$ are assumed to be rooted frames for $\mathsf {K4.3}$ , so their relation R is always transitive and connected:

(2) $$ \begin{align} \forall x,y \in W \, \big(xRy \lor x=y \lor yRx \big). \end{align} $$

A cluster in $\mathfrak F$ is any set of the form $C(x) = \{x\} \cup \{ y \in W \mid xRy \land y R x\}$ with $x\in W$ . If x is irreflexive, i.e., $xRx$ does not hold, $C(x)$ is called a degenerate cluster and depicted as $\bullet $ ; a reflexive x (for which $xRx$ ) is depicted as $\circ $ . A non-degenerate cluster with $k \ge 1$ (reflexive) points is depicted as . The next example will be used many times in what follows.

Example 2.2. Consider the frame $\mathfrak F=(W_k,R_{k\bullet },\mathcal {P}_k)$ , where $0 < k < \omega $ ,

$$ \begin{align*} & W_k=A_k\cup\{b_n\mid n < \omega \},\qquad A_k= \{a_0,\dots, a_{k-1} \},\\ & xR_{k\bullet} y\ \text{ iff }\ \text{either } x = a_i \text{ or } x=b_n,\ y=b_m,\text{ and }m<n, \end{align*} $$

and $\mathcal {P}_k$ is generated by the sets $X_i = \{a_i\} \cup \{b_n\mid n < \omega ,\ n \equiv i \ (\text {mod}\ k) \}$ , for $i <k$ , and $\{b_n\}$ , for $n < \omega $ . (For instance, $\mathcal {P}_1$ consists of all finite subsets of $\{b_n\mid n<\omega \}$ and their complements in $W_1$ .) The underlying Kripke frame $(W_k,R_{k\bullet })$ is shown in the picture below, where all $\ast $ are $\bullet $ .

It is not hard to see that

(3) $$ \begin{align} \text{for any } X\in\mathcal{P}_k, X \text{ is infinite iff } A_k\cap X\ne\emptyset, \end{align} $$

and so $A_k\notin \mathcal {P}_k$ . For every nonempty $X\in \mathcal {P}_k$ , the set $\Diamond ^{\mathfrak F}X$ is cofinite in $W_k$ . Using these observations, it is readily checked that $\mathfrak F$ is a descriptive frame; we denote it by . Clearly, is $\mathfrak M$ -generated for $\mathfrak M$ with $\mathfrak {v}(p_i)=X_i$ if $i<k$ , and $\mathfrak {v}(p_i)=\emptyset $ otherwise. The descriptive frame $(W_k,R_{k\circ },\mathcal {P}_k)$ with $R_{k\circ }=R_{k\bullet }\cup \{ (b_n,b_n)\mid n<\omega \}$ is denoted by ; $(W_k,R_{k\circ })$ looks like in the picture above, with all $\ast =\circ $ . Note that but , cf. Example 2.10 $(a)$ .

The next lemma, originating in [Reference Fine12], will play a key role in our subsequent constructions. Let $\mathfrak {M}$ be a model based on a rooted frame $\mathfrak {F} = (W,R,\mathcal {P})$ for $\mathsf {K4.3}$ , and let $\Gamma $ be a set of formulas. A point $x\in W$ is called $\Gamma $ -maximal in $\mathfrak {M}$ if $\mathfrak {M},x \models \Gamma $ , and whenever $xRy$ and $\mathfrak {M},y\models \Gamma $ , then $yRx$ . We denote by $\max _{\mathfrak {M}} \Gamma $ the set of all $\Gamma $ -maximal points in $\mathfrak {M}$ .

Given a rooted frame $\mathfrak F = (W, R,\mathcal {P})$ for $\mathsf {K4.3}$ , let $R^s = \{(x,y) \in R \mid (y,x) \notin R\}$ be the strict R-accessibility in $\mathfrak F$ . Sometimes it will be convenient to view $(W,R)$ as a strict linear order $\mathfrak F_c = (W_c, <_R)$ of clusters, where $W_c = \{C(x) \mid x \in W\}$ and $C(x) <_R C(y)$ iff $xR^sy$ . A cluster C is final in $\mathfrak F$ if there is no cluster $C'$ with $C<_R C'$ . A cluster C is a root cluster if there is no cluster $C'$ with $C'<_R C$ , in which case $C<_R C'$ for every $C'\ne C$ in $\mathfrak F$ ; the root cluster in $\mathfrak F$ is unique. A cluster $C'$ is an immediate successor of a cluster C in $\mathfrak F$ if $C<_R C'$ and there is no $C"$ with $C<_R C"<_R C'$ , in which case C is an immediate predecessor of $C'$ . A sequence $C_n$ , $n<\omega $ , of clusters in $\mathfrak F_c$ is an infinite ascending chain if $C_n <_R C_{n+1}$ , for all $n < \omega $ . $\mathfrak F_c$ is converse well-founded if it has no infinite ascending chain of clusters.

The next lemma follows from, e.g., the more general [Reference Chagrov and Zakharyaschev9, Theorems 10.34 and 10.35].

Proof. Let $\mathfrak F=(W,R,\mathcal {P})$ , let $\le _R$ be the reflexive closure of $<_R$ , and let $\mathcal G$ be a finite set generating $\mathcal {P}$ with $|\mathcal G| = n$ . For $x,y\in W$ , we write $x\sim _{\mathcal G} y$ in case $x\in G$ iff $y\in G$ , for all $G\in \mathcal G$ , and denote by $[x]_{\mathcal G}$ the $\sim _{\mathcal G}$ -class of x. Clearly, $\big |\big \{[x]_{\mathcal G}\mid x\in W\big \}\big |\le 2^{|{\mathcal G}|}=2^n$ .

$(a)$ Suppose on the contrary that $C(x_i)$ , $i<\omega $ , is an infinite ascending chain in $\mathfrak F_c$ . Call $x \in W$ a middle-point if $C(x_0)\le _R C(x)\le _R C(x_i)$ , for some $i<\omega $ . Let $V_x=\{[y]_{\mathcal G}\mid y \text { a middle-point with } xR y\}$ . Since $V_x\supseteq V_y$ whenever $xR y$ and each $V_x$ is finite, there is $m<\omega $ such that $V_y=V_{x_m}$ , for every middle-point y with $C(x_m)\le _R C(y)$ . By induction on the construction of $X\in \mathcal {P}$ from the generators in $\mathcal G$ , it is readily seen that

(4) $$ \begin{align} \text{if } y,z \text{ are middle-points, } C(x_m)\le_RC(y), & C(x_m)\le_RC(z), \text{ and } y\sim_{\mathcal G}z,\\ & \text{then } y\in X \text{ iff } z\in X, \text{ for all } X\in\mathcal{P}.\nonumber \end{align} $$

(Indeed, the only non-trivial case is when $X=\Diamond ^{\mathfrak F}Y$ , $yRz$ and $y\in \Diamond ^{\mathfrak F}Y$ . Then there is $x\in Y$ with $yRx$ . If $zRx$ , we are done. Otherwise, x is a middle-point. As $V_y=V_z$ , there is a middle-point $x'$ with $zRx'$ and $x\sim _{\mathcal G} x'$ . By IH, $x'\in Y$ .) As there are finitely many $\sim _{\mathcal G}$ -classes, there exist $k\ne \ell \geq m$ such that $x_k\sim _{\mathcal G}x_\ell $ , and so $x_k\in X$ iff $x_\ell \in X$ , for all $X\in \mathcal {P}$ , by (4). But this contradicts (dif).

$(b)$ It is straightforward to show that if $C(x)=C(y)$ and $x\sim _{\mathcal G}y$ , then $x\in X$ iff $y\in X$ , for all $X\in \mathcal {P}$ . So by (dif), every cluster in $\mathfrak F$ has $\le 2^{|{\mathcal G}|}$ points.

Note that the existence of maximal points (Lemma 2.3) in models based on rooted finitely generated descriptive frames for $\mathsf {K4.3}$ also follows from Lemma 2.4. Another consequence is that such a frame $\mathfrak F$ contains a unique final cluster, and any non-root cluster in $\mathfrak F$ has an immediate predecessor. If $\mathfrak F_c^{-1}= (W_c,>_R)$ is isomorphic to an ordinal $\gamma $ and $\alpha \le \gamma $ , we denote by $C_\alpha ^{\mathfrak F}$ the cluster that is the image of $\alpha $ under this isomorphism. If $\alpha $ is a non-zero limit ordinal, we call $C_\alpha ^{\mathfrak F}$ a limit cluster. A non-final cluster is a limit cluster iff it does not have an immediate successor. By (dif) and Lemma 2.4 $(b)$ , we also have the following.

Now, suppose $\mathfrak M$ is a model based on a rooted finitely $\mathfrak M$ -generated descriptive frame $\mathfrak {F}= (W,R,\mathcal {P})$ for $L\supseteq \mathsf {K4.3}$ . Given a formula $\mu $ , a cluster C is called $\mu $ -maximal in $\mathfrak {M}$ if there is a point in C that is $\{\mu \}$ -maximal in $\mathfrak M$ . Further, C is maximal in $\mathfrak {M}$ if it is $\mu $ -maximal in $\mathfrak M$ , for some $\mu $ , and C is $\sigma $ -maximal in $\mathfrak {M}$ , for a signature $\sigma $ , if there is such a $\sigma $ -formula $\mu $ . Every definable in $\mathfrak M$ cluster is clearly maximal in $\mathfrak M$ . The next lemma says that the converse is also true.

Proof. $(a)$ If $C(x)$ is degenerate, then $\Diamond t_{\mathfrak M}(x) \not \subseteq t_{\mathfrak M}(x)$ by (tig). So there is a formula $\mu $ with $\mathfrak M,x\models \mu $ but $\mathfrak M,x\not \models \Diamond \mu $ .

$(b,\Rightarrow )$ Let $C(x)$ be maximal in $\mathfrak M$ with $\mathfrak M,x \models \mu $ and $\mathfrak M,y \not \models \mu $ whenever $xR^sy$ . Suppose $C(x)$ is a limit cluster. Let $S=\{C\in W_c\mid C(x)<_R C\}$ with $y_C\in C$ , for $C\in S$ . Consider

$$\begin{align*}\Gamma = \bigcup_{C\in S}\Diamond t_{\mathfrak M}(y_C) \cup \{\psi \mid \Box\psi \in t_{\mathfrak M}(x)\} \cup \{ \Box\neg\mu \}. \end{align*}$$

Clearly, $\Gamma $ is finitely satisfiable in $\mathfrak M$ , and so, by (com), $\Gamma \subseteq t_{\mathfrak M}(y)$ , for some y. Thus, by (tig), $x R y R y_C$ for all $C\in S$ , and so $yR^s y_C$ for all $C\in S$ and $yRx$ . But we also have $\mathfrak M,y\models \Box \neg \mu $ , contrary to $\mathfrak M,x\models \mu $ .

$(b,\Leftarrow )$ The (unique) final cluster is maximal in $\mathfrak M$ for $\top $ . Suppose $C(y)$ is an immediate successor of $C(x)$ . If $C(y)$ is degenerate, then $C(y)$ is maximal in $\mathfrak M$ by $(a)$ , and so there is $\mu $ with $\mathfrak M,y \models \mu \land \neg \Diamond \mu $ . It follows that $C(x)$ is $\Diamond (\mu \land \neg \Diamond \mu )$ -maximal in $\mathfrak M$ . If $C(y)$ is non-degenerate and $C(x)$ is not maximal in $\mathfrak M$ , then $\Diamond t_{\mathfrak M}(x) \subseteq t_{\mathfrak M}(y)$ , and so $yRx$ by (tig), contrary to $xR^s y$ .

$(c,\Leftarrow )$ Let $C(x)$ be $\mu $ -maximal in $\mathfrak M$ . If $C(x)$ is degenerate, it is defined by $\mu \land \neg \Diamond \mu $ . If $C(x)$ is the non-degenerate root cluster, then $\Diamond \mu $ defines $C(x)$ . Otherwise, take the immediate predecessor $C(y)$ of $C(x)$ . By $(b)$ , $C(y)$ is $\tau $ -maximal in $\mathfrak M$ , for some $\tau $ , so $\Box ^+ \neg \tau \land \Diamond \mu $ defines $C(x)$ . $(c,\Rightarrow )$ is obvious.

We require a few important consequences of Lemmas 2.4 and 2.6.

Given a rooted finitely $\mathfrak M$ -generated descriptive frame $\mathfrak F = (W, R,\mathcal {P})$ for $\mathsf {K4.3}$ , let $m_{\mathfrak F}$ be the largest ordinal $\le \omega $ with degenerate $C_n^{\mathfrak F}$ for all $n<m_{\mathfrak F}$ . We call the (possibly empty) interval $Z=\bigcup _{n<m_{\mathfrak F}}C_n^{\mathfrak F}$ the tail of $\mathfrak F$ . We may assume that $Z=\{z_n\mid n< m_{\mathfrak F}\}$ , where all $z_n$ are irreflexive and $z_{n}R z_{n-1}$ , $0<n<m_{\mathfrak F}$ . If Z is infinite, then $Z\ne W$ (as $\mathfrak F$ is rooted). If $Z\ne W$ , we call $C_{m_{\mathfrak F}}^{\mathfrak F}$ the head of Z. In particular, if $Z=\emptyset $ , its head is the final (non-degenerate) cluster $C_{0}^{\mathfrak F}$ ; if $Z\ne W$ and $Z\ne \emptyset $ is finite, its head is the immediate predecessor of $C_{m_{\mathfrak F}-1}^{\mathfrak F}=\{z_{m_{\mathfrak F}-1}\}$ ; and if Z is infinite, its head is the limit cluster $C_{\omega }^{\mathfrak F}$ . Thus, by Lemma 2.6,

(5) $$ \begin{align} \text{the head of a tail is always non-degenerate.} \end{align} $$

2.3 Building linear models from pieces

Definition 2.8. The ordered sum $\mathfrak F_{0} \lhd \dots \lhd \mathfrak F_{n-1} = (W, R,\mathcal {P})$ of rooted frames $\mathfrak F_{i}= ( W_i,R_i,\mathcal {P}_i)$ , $i<n$ , for $\mathsf {K4.3}$ with pairwise disjoint $W_i$ is defined by

$$ \begin{align*}W = \bigcup_{i<n}W_{i},\ \ R = \bigcup_{i<n}R_{i} \cup \bigcup_{i<j<n} (W_{i}\times W_{j}),\ \ \mathcal{P} = \{ X_{0}\cup \dots \cup X_{n-1} \mid X_{i}\in \mathcal{P}_{i}\}. \end{align*} $$

It is not hard to see that if the $\mathfrak F_i$ are descriptive, then $\mathfrak F_{0} \lhd \dots \lhd \mathfrak F_{n-1}$ is also descriptive. If $\mathfrak M_i = (\mathfrak F_i, \mathfrak v_i)$ , then $\mathfrak M=\mathfrak M_{0} \lhd \dots \lhd \mathfrak M_{n-1}$ is the model based on $\mathfrak F_{0} \lhd \dots \lhd \mathfrak F_{n-1}$ with the valuation $\mathfrak v(p) = \bigcup _{ i < n} \mathfrak v_i(p)$ , for any $p \in \mathcal {V}$ . We call the $\mathfrak M_i \lhd $ -components of $\mathfrak M$ .

Now, let $\mathfrak F = (W, R,\mathcal {P})$ be a rooted frame for $\mathsf {K4.3}$ . An interval in $\mathfrak {F}$ is any subset $I\subseteq W$ such that $xRyRz$ and $x,z \in I$ imply $y \in I$ , for all $x,y,z \in W$ . If $I\cap C\ne \emptyset $ , for a cluster C, then clearly $C\subseteq I$ . An interval I is closed if there are clusters $C,C'$ such that $I=C\cup C'\cup \bigcup \{ D\mid C<_R D <_R C'\}$ , in which case we write $I=[C,C']$ . Given two closed intervals $I,I'$ in $\mathfrak F$ , we write $I\prec _{\mathfrak F}I'$ if I and $I'$ are disjoint and $xRx'$ , for all $x\in I$ , $x'\in I'$ . Notice that if I is a closed internal interval in $\mathfrak F$ , then ${\mathfrak F}\mathop {\restriction }_{I}$ is also a rooted frame for $\mathsf {K4.3}$ . Also, if $\mathfrak F$ is descriptive, then ${\mathfrak F}\mathop {\restriction }_{I}$ is descriptive as well. And if $\mathfrak F$ is finitely $\mathfrak M$ -generated, for some model $\mathfrak M$ , then ${\mathfrak F}\mathop {\restriction }_{I}$ is finitely ${\mathfrak M}\mathop {\restriction }_{I}$ -generated. We clearly have the following.

2.4 Canonical formulas

To check whether a frame validates a given finitely axiomatisable logic, we use the canonical formulas of [Reference Bezhanishvili and Bezhanishvili4, Reference Chagrov and Zakharyaschev9, Reference Wolter38, Reference Zakharyaschev and Alekseev43] whose basic properties are summarised below in the context of $\mathsf {K4.3}$ ; for more details consult [Reference Chagrov and Zakharyaschev9, Section 16.3]. Every logic $L \supseteq \mathsf {K4.3}$ can be represented in the form

(6) $$ \begin{align} L = \mathsf{K4.3} \oplus \{\alpha(\mathfrak G_j,\mathfrak D_j,\bot) \mid j \in J_L\}, \quad \text{for some index set } J_L, \end{align} $$

where each $\alpha (\mathfrak G_j, \mathfrak D_j,\bot )$ is a canonical formula based on a finite rooted Kripke frame $\mathfrak G_j = (V_j, S_j)$ for $\mathsf {K4.3}$ and a (possibly empty) set $\mathfrak D_j\subseteq V_j$ of irreflexive non-root points in $\mathfrak G_j$ . If L is finitely axiomatisable, its canonical axiomatisation (6) with finite $J_L$ can be constructed effectively, given any finite set of axioms.

Let $\mathfrak F = (W,R,\mathcal {P})$ be any rooted finitely generated descriptive frame for $\mathsf {K4.3}$ . By Theorem 2.4, $\mathfrak F$ contains a unique final cluster, and any non-root cluster in $\mathfrak F$ has an immediate predecessor. The formulas $\alpha (\mathfrak G_j, \mathfrak D_j,\bot )$ are defined so that $\mathfrak F\not \models \alpha (\mathfrak G_j, \mathfrak D_j,\bot )$ iff there is an injection $f \colon V_j \to W$ such that the following conditions hold: for all $x,y \in V_j$ ,

1. (cf₁) $x S_j y$ iff $f(x) R f(y)$ (so x is irreflexive iff $f(x)$ is);

2. (cf₂) if $C(x)$ is the final cluster in $\mathfrak G_j$ , then $C(f(x))$ is the final cluster in $\mathfrak F$ ;

3. (cf₃) if $x \in \mathfrak D_j$ and $C(y)$ is the immediate predecessor of $C(x) = \{x\}$ in $\mathfrak G_j$ , then $C(f(y))$ is the immediate predecessor of $C(f(x))=\{f(x)\}$ in $\mathfrak F$ ;

4. (cf₄) $\{f(x)\} \in \mathcal {P}$ .

Intuitively, every frame $\mathfrak F$ with $\mathfrak F \not \models \alpha (\mathfrak G_j, \mathfrak D_j,\bot )$ can be obtained by inserting certain chains of clusters immediately before some clusters $C(x)$ in $\mathfrak G_j$ , provided that $x \notin \mathfrak D_j$ , and by enlarging some non-degenerate clusters in $\mathfrak G_j$ .

Canonical formulas of the form $\alpha (\mathfrak G, \emptyset ,\bot )$ axiomatise exactly the cofinal subframe logics whose frames are closed under taking cofinal subframes. We remind the reader [Reference Chagrov and Zakharyaschev9] that a subframe $\mathfrak F' = (W',R',\mathcal {P}')$ of a frame $\mathfrak F = (W,R,\mathcal {P})$ is called cofinal if $W'$ is cofinal in $\mathfrak F$ in the sense that, for any $x \in W'$ and $y \in W$ , whenever $x R y$ then either $y \in W'$ or there is $z \in W'$ with $y R z$ . Cofinal subframe logics enjoy the fmp, and so are decidable if finitely axiomatisable [Reference Zakharyaschev42]. Example 2.10 shows the canonical axioms of some extensions of $\mathsf {K4.3}$ .

Example 2.10. $(a)$ We prove that

$$\begin{align*}\mathsf{GL.3} = \mathsf{K4.3} \oplus \Box( \Box p_0 \to p_0) \to \Box p_0 = \mathsf{K4.3} \oplus \alpha (\circ, \emptyset, \bot) \oplus \alpha (\circ \lhd \bullet, \emptyset, \bot). \end{align*}$$

Let $\mathfrak F = (W,R,\mathcal {P})$ be a rooted finitely generated descriptive frame for $\mathsf {K4.3}$ . By Lemma 2.7, W is countable, and so $\mathfrak F$ is $\mathfrak M$ -generated, for some model $\mathfrak M=(\mathfrak {F},\mathfrak {v})$ . We claim that the following are equivalent:

1. 1. $\mathfrak M\not \models \mathsf {GL.3}$ ;

2. 2. there is a formula $\psi $ with a non-degenerate $\psi $ -maximal cluster in $\mathfrak M$ ;

3. 3. there is a non-degenerate non-limit cluster in $\mathfrak F$ ;

4. 4. $\mathfrak F\not \models \alpha (\circ , \emptyset , \bot ) \land \alpha (\circ \lhd \bullet , \emptyset , \bot )$ .

1. $\Rightarrow $ 2. Suppose $\mathfrak M,x\not \models \Box (\Box \varphi \to \varphi )\to \Box \varphi $ , for some formula $\varphi $ . Then the $\neg \big (\Box (\Box \varphi \to \varphi )\to \Box \varphi \big )$ -maximal cluster C in $\mathfrak M$ is non-degenerate.

2. $\Leftrightarrow $ 3. by Lemma 2.6.

2. $\Rightarrow $ 4. Suppose the $\psi $ -maximal cluster $C_\psi $ in $\mathfrak M$ is non-degenerate, for some $\psi $ . If $C_\psi $ is the final cluster of $\mathfrak F$ , then the injection f mapping $\circ $ to a point in $C_\psi $ satisfies $\mathbf {(cf_1)}$ – $\mathbf {(cf_4)}$ , and so $\mathfrak F\not \models \alpha (\circ , \emptyset , \bot )$ . If $C_\psi $ is not the final cluster, then the $\neg \psi $ -maximal cluster $C_{\neg \psi }$ is the final cluster in $\mathfrak F$ . If $C_{\neg \psi }$ is non-degenerate, then again $\mathfrak F\not \models \alpha (\circ , \emptyset , \bot )$ ; otherwise $\mathfrak F\not \models \alpha (\circ \lhd \bullet , \emptyset , \bot )$ as witnessed by f sending $\bullet $ to the point in the final cluster and $\circ $ to a point in $C_\psi $ .

4. $\Rightarrow $ 1. If $\mathfrak F\not \models \alpha (\circ \lhd \bullet , \emptyset , \bot )$ , then take an injection f from $\circ \lhd \bullet $ to $\mathfrak F$ satisfying $\mathbf {(cf_1)}$ – $\mathbf {(cf_4)}$ . By $\mathbf {(cf_4)}$ and Lemma 2.6, $\{f(\circ )\}=\mathfrak {v}(\varphi )$ , for some $\varphi $ . As $f(\circ )R f(\circ )$ by $\mathbf {(cf_1)}$ , it is easy to see that $\mathfrak M,f(\circ )\not \models \Box (\Box \neg \varphi \to \neg \varphi )\to \Box \neg \varphi $ . The case when $\mathfrak F\not \models \alpha (\circ , \emptyset , \bot )$ is similar.

$(b)$ Similarly, we can prove that

$$ \begin{align*} \mathsf{Log}\{(\mathbb N,<)\} & = \mathsf{K4.3} \oplus \Diamond\top \oplus \Box (\Box p_0 \to p_0) \to (\Diamond\Box p_0 \to \Box p_0)\\ & = \mathsf{K4.3} \oplus \alpha (\bullet, \emptyset, \bot) \oplus \alpha (\circ \lhd \circ, \emptyset, \bot) \end{align*} $$

by showing that, for every $\mathfrak M$ and $\mathfrak F$ as above, the following are equivalent:

1. – $\mathfrak M\not \models \mathsf {Log}\{(\mathbb N,<)\}$ ;

2. – either the final cluster in $\mathfrak F$ is degenerate or there is a non-degenerate non-limit cluster different from the final cluster in $\mathfrak F$ ;

3. – $\mathfrak F\not \models \alpha (\bullet , \emptyset , \bot ) \land \alpha (\circ \lhd \circ , \emptyset , \bot )$ .

$(c)$ A prominent example of a non-cofinal subframe logic is $\mathsf {K4.3} \oplus \Diamond p \to \Diamond \Diamond p$ with dense frames, whose canonical axioms

$$\begin{align*}\mathsf{K4.3} \oplus \alpha (\bullet \lhd \bullet^a, \{a\}, \bot) \oplus \alpha (\bullet \lhd \bullet^a \lhd \circ, \{a\}, \bot) \oplus \alpha (\bullet \lhd \bullet^a \lhd \bullet, \{a\}, \bot) \end{align*}$$

forbid any two consecutive degenerate clusters in finitely generated descriptive frames for the logic (see also Lemma 5.6).

4.1 The quasi-polysize bisimilar model property

Given a finite signature $\delta $ , a $\delta $ -model $\mathfrak M=(\mathfrak F,\mathfrak w)$ is called simple if either $\mathfrak F$ is finite or , for $0<k<\omega $ and $\ast \in \{\bullet ,\circ \}$ , and, for every $p \in \delta $ , there is $A_p\subseteq \{0,\dots ,k-1\}$ with $\mathfrak {w}(p)=\bigcup _{i\in A_p}X_i$ , where the $X_i$ are the infinite generators of the internal sets in defined in Example 2.2. Thus, even though is infinite, any simple $\delta $ -model based on it is fully determined by the finitary information provided by the sets $A_p$ , $p\in \delta $ , that is, by the atomic $\delta $ -types of the points in the -cluster. A $\delta $ -model is called quasi-finite if it is the ordered sum of finitely-many simple models.

Our first result is as follows.

We actually prove a stronger Theorem 4.5 that prescribes more structure for the pair of quasi-finite models witnessing the lack of an interpolant, which makes it easy to deduce the existence of a $\sigma $ -bisimulation between the models. The prescribed structure is easily checkable, which is used in the proof of the main Theorem 4.9. To formulate our ‘structural’ theorem, we require a few definitions.

For $0 < m < \omega $ , let $m^{<} = \underbrace {\bullet \lhd \dots \lhd \bullet }_m$ . An atomic frame takes one of the forms

(10)

The size $\|{\mathfrak {F}}\|$ of an atomic $\mathfrak {F}$ is defined by taking $\|{m^{<}}\|=m$ , , and . If $\mathfrak M=\mathfrak M_0\lhd \dots \lhd \mathfrak M_{n-1}$ , for some $0<n<\omega $ and simple $\delta $ -models $\mathfrak M_j$ based on atomic frames $\mathfrak F_j$ , $j<n$ , then we set $\|{\mathfrak {M}}\|=\|{\mathfrak F_0\lhd \dots \lhd \mathfrak F_{n-1}}\|= \|{\mathfrak F_0}\|+\dots +\|{\mathfrak F_{n-1}}\|$ .

The following lemma justifies this definition.

Proof. First, we show that, for every $\ell <N$ , there is a global $\sigma $ -bisimulation between $\mathfrak N_1^\ell $ and $\mathfrak N_2^\ell $ . This is clear for $(\mathfrak N_1^\ell ,\mathfrak N_2^\ell )$ of type $({a})$ , in which case the identity function on $\mathfrak H$ is a $\sigma $ -bisimulation.

If $(\mathfrak N_1^\ell ,\mathfrak N_2^\ell )$ is of type $({b})$ , then the final clusters $C_i$ of $\mathfrak N_i^\ell $ , $i=1,2$ , are non-degenerate. Thus, $\boldsymbol {\beta }_1\cup \boldsymbol {\beta }_2$ is a global $\sigma $ -bisimulation between $\mathfrak N_1^\ell $ and $\mathfrak N_2^\ell $ ,

$$ \begin{align*} & \boldsymbol{\beta}_1=\big\{(y_1,y_2) \mid {y_1 \text{ in } \mathfrak N_1^\ell, y_2 \text{ in } C_2, {at}^\sigma_{\mathfrak N_1^\ell}(y_1)={at}^\sigma_{\mathfrak N_2^\ell}(y_2)} \big\},\\ & \boldsymbol{\beta}_2=\big\{(y_1,y_2) \mid {y_2 \text{ in } \mathfrak N_2^\ell, y_1 \text{ in } C_1, {at}^\sigma_{\mathfrak N_1^\ell}(y_1)={at}^\sigma_{\mathfrak N_2^\ell}(y_2)} \big\}. \end{align*} $$

If $(\mathfrak N_1^\ell ,\mathfrak N_2^\ell )$ is of type $({c})$ , suppose $\mathfrak N_i^\ell =\mathfrak N_i^{0,\ell }\lhd \dots \lhd \mathfrak N_i^{\boldsymbol {n}_i-1,\ell }$ , for $0<\boldsymbol {n}_i<\omega $ and $i=1,2$ . By $({c})$ .1, $\mathfrak N_1^{\boldsymbol {n}_1-1,\ell }$ and $\mathfrak N_2^{\boldsymbol {n}_2-1,\ell }$ are simple models based on the same atomic frame of the form or . As in Example 2.2, let $A_k=\{a_s\mid s<k\}$ and $W_k=A_k\cup \{b_n\mid n<\omega \}$ (containing all the points of ). We claim that

(11) $$ \begin{align} {at}^\sigma_{\mathfrak N_1^\ell}(b_n)=\textit{at}^\sigma_{\mathfrak N_2^\ell}(b_n),\quad\text{for all } n<\omega. \end{align} $$

Indeed, suppose $n<\omega $ and let $s<k$ be such that $n\equiv s$ (mod k). As $\mathfrak N_i^{\boldsymbol {n}_i-1,\ell }$ is a simple model, we have

$$\begin{align*}{at}^\sigma_{\mathfrak N_i^\ell}(b_n)={at}^\sigma_{\mathfrak N_i^{\boldsymbol{n}_i-1,\ell}}(b_n)= {at}^\sigma_{\mathfrak N_i^{\boldsymbol{n}_i-1,\ell}}(a_s)=\textit{at}^\sigma_{\mathfrak N_i^\ell}(a_s),\quad\text{for } i=1,2, \end{align*}$$

and so (11) follows from $({c})$ .2. Now let

$$ \begin{align*} & \boldsymbol{\beta}_1=\big\{(y_1,y_2) \mid {y_1 \text{ in } \mathfrak N_1^{0,\ell}\lhd\dots\lhd\mathfrak N_1^{\boldsymbol{n}_1-2,\ell}, y_2\in A_k, {at}^\sigma_{\mathfrak N_1^\ell}(y_1)={at}^\sigma_{\mathfrak N_2^\ell}(y_2)} \big\},\\ & \boldsymbol{\beta}_2=\big\{(y_1,y_2) \mid {y_2 \text{ in } \mathfrak N_2^{0,\ell}\lhd\dots\lhd\mathfrak N_2^{\boldsymbol{n}_2-2,\ell}, y_1\in A_k, {at}^\sigma_{\mathfrak N_1^\ell}(y_1)={at}^\sigma_{\mathfrak N_2^\ell}(y_2)} \big\}. \end{align*} $$

By (11) and $({c})$ .2–3, $\boldsymbol {\beta }_1\cup \boldsymbol {\beta }_2\cup \big \{(b_n,b_n)\mid n<\omega \big \}$ is a global $\sigma $ -bisimulation between $\mathfrak N_1^\ell $ and $\mathfrak N_2^\ell $ . Finally, if $\boldsymbol {\beta }^0$ is a global $\sigma $ -bisimulation between $\mathfrak N_1^0$ and $\mathfrak N_2^0$ , then $\boldsymbol {\beta }^0\cup \{(x_1,x_2)\}$ is also a global $\sigma $ -bisimulation between $\mathfrak N_1^0$ and $\mathfrak N_2^0$ because $\textit {at}^\sigma _{\mathfrak N_1}(x_1) = \textit {at}^\sigma _{\mathfrak N_2}(x_2)$ . The union of the constructed global bisimulations is a (global) bisimulation $\boldsymbol {\beta }$ between $\mathfrak N_1$ and $\mathfrak N_2$ with $x_1\boldsymbol {\beta } x_2$ , as required.