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:

• 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.

A branch is called closed if its end node is carrying ${\bot }$, otherwise it is open.

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:

• 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

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

• In order to get tableau calculi for the other mentioned calculi, like $T$ or $K4$ one has to introduce additional rules:

$(T){\frac {X;\lnot \square P}{X;\square P;P}}$

which obviously reflects reflexivity of the reachability relation, or

$(K4){\frac {\square X;\lnot \square P}{X;\square X;\lnot P}}$

reflecting transitivity.