Sahlqvist formula

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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 first-order definable class of Kripke frames.

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[edit]

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 AB, 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[edit]

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[edit]

\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 but is not equivalent to any Sahlqvist formula.

Kracht's theorem[edit]

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 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].

References[edit]

  • L. A. Chagrova, 1991. An undecidable problem in correspondence theory. Journal of Symbolic Logic 56:1261-1272.
  • Marcus Kracht, 1993. How completeness and correspondence theory got married. In de Rijke, editor, Diamonds and Defaults, pages 175-214. Kluwer.
  • Henrik Sahlqvist, 1975. Correspondence and completeness in the first- and second-order semantics for modal logic. In Proceedings of the Third Scandinavian Logic Symposium. North-Holland, Amsterdam.