Sahlqvist formula

In modal logic, Sahlqvist formulas are a certain kind of modal formula with remarkable properties. The Sahlqvist correspondence theorem states that every Sahlqvist formula is canonical, and corresponds to a class of Kripke frames definable by a first-order formula.

Sahlqvist's definition characterizes a decidable set of modal formulas with first-order correspondents. Since it is undecidable, by Chagrova's theorem, whether an arbitrary modal formula has a first-order correspondent, there are formulas with first-order frame conditions that are not Sahlqvist [Chagrova 1991] (see the examples below). Hence Sahlqvist formulas define only a (decidable) subset of modal formulas with first-order correspondents.

Definition
Sahlqvist formulas are built up from implications, where the consequent is positive and the antecedent is of a restricted form.
 * A boxed atom is a propositional atom preceded by a number (possibly 0) of boxes, i.e. a formula of the form $$\Box\cdots\Box p$$ (often abbreviated as $$\Box^i p$$ for $$0 \leq i < \omega$$).
 * A Sahlqvist antecedent is a formula constructed using ∧, ∨, and $$\Diamond$$ from boxed atoms, and negative formulas (including the constants ⊥, ⊤).
 * A Sahlqvist implication is a formula A → B, where A is a Sahlqvist antecedent, and B is a positive formula.
 * A Sahlqvist formula is constructed from Sahlqvist implications using ∧ and $$\Box$$ (unrestricted), and using ∨ on formulas with no common variables.

Examples of Sahlqvist formulas

 * $$p \rightarrow \Diamond p$$
 * Its first-order corresponding formula is $$\forall x \; Rxx$$, and it defines all reflexive frames


 * $$p \rightarrow \Box\Diamond p$$
 * Its first-order corresponding formula is $$\forall x \forall y [Rxy \rightarrow Ryx]$$, and it defines all symmetric frames


 * $$\Diamond \Diamond p \rightarrow \Diamond p$$ or $$\Box p \rightarrow \Box \Box p$$
 * Its first-order corresponding formula is $$\forall x \forall y \forall z [(Rxy \land Ryz) \rightarrow Rxz]$$, and it defines all transitive frames


 * $$\Diamond p \rightarrow \Diamond \Diamond p$$ or $$\Box \Box p \rightarrow \Box p$$
 * Its first-order corresponding formula is $$\forall x \forall y [Rxy \rightarrow \exists z (Rxz \land Rzy)]$$, and it defines all dense frames


 * $$\Box p \rightarrow \Diamond p$$
 * Its first-order corresponding formula is $$\forall x \exists y \; Rxy$$, and it defines all right-unbounded frames (also called serial)


 * $$\Diamond\Box p \rightarrow \Box\Diamond p$$
 * Its first-order corresponding formula is $$\forall x \forall x_1 \forall z_0 [Rxx_1 \land Rxz_0 \rightarrow \exists z_1 (Rx_1z_1 \land Rz_0z_1)]$$, and it is the Church–Rosser property.

Examples of non-Sahlqvist formulas

 * $$\Box\Diamond p \rightarrow \Diamond \Box p$$
 * This is the McKinsey formula; it does not have a first-order frame condition.


 * $$\Box(\Box p \rightarrow p) \rightarrow \Box p$$
 * The Löb axiom is not Sahlqvist; again, it does not have a first-order frame condition.


 * $$(\Box\Diamond p \rightarrow \Diamond \Box p) \land (\Diamond\Diamond q \rightarrow \Diamond q)$$
 * The conjunction of the McKinsey formula and the (4) axiom has a first-order frame condition (the conjunction of the transitivity property with the property $$ \forall x[\forall y(Rxy \rightarrow \exists z[Ryz]) \rightarrow \exists y(Rxy \wedge \forall z[Ryz \rightarrow z = y])] $$) but is not equivalent to any Sahlqvist formula.

Kracht's theorem
When a Sahlqvist formula is used as an axiom in a normal modal logic, the logic is guaranteed to be complete with respect to the basic elementary class of frames the axiom defines. This result comes from the Sahlqvist completeness theorem [Modal Logic, Blackburn et al., Theorem 4.42]. But there is also a converse theorem, namely a theorem that states which first-order conditions are the correspondents of Sahlqvist formulas. Kracht's theorem states that any Sahlqvist formula locally corresponds to a Kracht formula; and conversely, every Kracht formula is a local first-order correspondent of some Sahlqvist formula which can be effectively obtained from the Kracht formula [Modal Logic, Blackburn et al., Theorem 3.59].