Sahlqvist theorem for modal fixed point logic

• Published in: Theoretical computer science Vol. 424; pp. 1 - 19
• Main Authors: Bezhanishvili, Nick, Hodkinson, Ian
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 23-03-2012
• Subjects: Completeness; Correspondence; Descriptive frame; Modal mu-calculus; Correspondence; Descriptive frame; Modal mu-calculus; Completeness
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2011.11.026

We define Sahlqvist fixed point formulas. By extending the technique of Sambin and Vaccaro we show that (1) for each Sahlqvist fixed point formula φ there exists an LFP-formula χ(φ), with no free first-order variable or predicate symbol, such that a descriptive μ-frame (an order-topological structure that admits topological interpretations of least fixed point operators as intersections of clopen pre-fixed points) validates φ iff χ(φ) is true in this structure, and (2) every modal fixed point logic axiomatized by a set Φ of Sahlqvist fixed point formulas is sound and complete with respect to the class of descriptive μ-frames satisfying {χ(φ):φ∈Φ}. We also give some concrete examples of Sahlqvist fixed point logics and classes of descriptive μ-frames for which these logics are sound and complete.