Logic for Computer Scientists/Modal Logic/Modal Logic Tableaux

Modal Logic Tableaux
In classical propositional logics we introduced a tableau calculus (Definition) for a logic $$L$$  as  a finitely branching tree whose nodes are  formulas from $$L$$. Given a set $$\Phi$$ of formulae from $$L$$, a tableau for $$\Phi$$ was constructed by a (possibly infinite) sequence of applications of a tableau rule schema:

$$ \rho \frac{\psi}{D_1 \mid D_2 \mid \cdots \mid D_n} $$

where the premise $$\psi$$ as well as the denominators $$D_1, \cdots ,D_n$$ are sets of formulae; $$\rho$$ is the name of the rule. We introduce $$K$$-tableau with the help of the following rules:

$$(\land) \frac{X; P\land Q} {X; P; Q} $$          $$ (\lor) \frac{X; \lnot(P\land Q)} {X; \lnot P; \mid X; \lnot Q}$$

$$(\bot) \frac{X;P; \lnot P}{\bot}$$                 $$(\lnot)  \frac{X; \lnot\lnot P} {X; P}$$

$$(\theta) \frac{X; Y}{X}$$ $$(K) \frac{\square X; \lnot \square P} {X;\lnot P}$$

A tableau for a set $$X$$ of formulae is a finite tree with root $$X$$ whose nodes carry finite formulae sets. The rules for extending a tableau are given by: A branch is called closed if its end node is carrying $${\bot}$$, otherwise it is open.
 * choose a leaf node $$n$$ with label $$Y$$, where $$n$$ is not an end node, and choose a rule $$\rho$$, which is applicable to $$n$$;
 * if $$\rho $$ has $$k$$ denominators then create $$k$$ successor nodes for $$n$$, with successor $$i$$ carrying an appropriate instance of denominator  $$D_i$$;
 * if a successor $$s$$ carries a set $$Z$$ and $$Z$$ has already appeared on  the branch from the root to $$s$$ then $$s$$ is an end node.

As in the classical case, a formulae $$A$$ is a theorem in modal logic $$K$$, iff there is a closed $$K$$-tableau for the set $$\lnot A$$.

As an example take the formula $$\square (p \to q) \to (\square p \to \square q)$$: its negation is certainly unsatisfiable, because the formula is an instance of our previously given $$K$$-axiom.

Some remarks are in order: $$(K\theta) \frac{Y; \square X; \lnot  \square P} {X;\lnot  P} $$ $$(T)\frac{ X; \lnot  \square P} {X; \square P; P}$$ which obviously reflects reflexivity of the reachability relation, or $$(K4) \frac{ \square X; \lnot   \square P} {X; \square X; \lnot P}$$ reflecting  transitivity.
 * The $$\theta$$- rule, called the thinning rule, is necessary in order to construct the premisses for the application of the $$K$$-rule.
 * Both can be combined by a new rule
 * In order to get tableau calculi for the other mentioned calculi, like $$T$$ or $$K4$$ one has to introduce additional rules: