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.