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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 [[Kripke semantics#Canonical models|canonical]], and corresponds to a [[first-order logic|first-order]] definable class of [[Kripke semantics|Kripke frames]].
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| 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.
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| == Definition ==
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| Sahlqvist formulas are built up from implications, where the consequent is ''positive'' and the antecedent is of a restricted form.
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| * A ''boxed atom'' is a propositional atom preceded by a number (possibly 0) of boxes, i.e. a formula of the form <math>\Box\cdots\Box p</math> (often abbreviated as <math>\Box^i p</math> for <math>0 \leq i < \omega</math>).
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| * A ''Sahlqvist antecedent'' is a formula constructed using ∧, ∨, and <math>\Diamond</math> from boxed atoms, and negative formulas (including the constants ⊥, ⊤).
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| * A ''Sahlqvist implication'' is a formula ''A'' → ''B'', where ''A'' is a Sahlqvist antecedent, and ''B'' is a positive formula.
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| * A ''Sahlqvist formula'' is constructed from Sahlqvist implications using ∧ and <math>\Box</math> (unrestricted), and using ∨ on formulas with no common variables.
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| == Examples of Sahlqvist formulas ==
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| ; <math>p \rightarrow \Diamond p</math>
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| : Its first-order corresponding formula is <math>\forall x \; Rxx</math>, and it defines all '''[[Reflexive relation|reflexive frames]]'''
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| ; <math>p \rightarrow \Box\Diamond p</math>
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| : Its first-order corresponding formula is <math>\forall x \forall y [Rxy \rightarrow Ryx]</math>, and it defines all '''[[Symmetric relation|symmetric frames]]'''
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| ; <math>\Diamond \Diamond p \rightarrow \Diamond p</math> or <math>\Box p \rightarrow \Box \Box p</math>
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| : Its first-order corresponding formula is <math>\forall x \forall y \forall z [(Rxy \land Ryz) \rightarrow Rxz]</math>, and it defines all '''[[Transitive relation|transitive frames]]''' | |
| ; <math>\Diamond p \rightarrow \Diamond \Diamond p</math> or <math>\Box \Box p \rightarrow \Box p</math>
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| : Its first-order corresponding formula is <math>\forall x \forall y [Rxy \rightarrow \exists z (Rxz \land Rzy)]</math>, and it defines all '''[[Dense order|dense frames]]'''
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| ; <math>\Box p \rightarrow \Diamond p</math>
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| : Its first-order corresponding formula is <math>\forall x \exists y \; Rxy</math>, and it defines all '''right-unbounded frames''' (also called serial)
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| ; <math>\Diamond\Box p \rightarrow \Box\Diamond p</math>
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| : Its first-order corresponding formula is <math>\forall x \forall x_1 \forall z_0 [Rxx_1 \land Rxz_0 \rightarrow \exists z_1 (Rx_1z_1 \land Rz_0z_1)]</math>, and it is the [[Church–Rosser theorem|Church-Rosser property]].
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| ==Examples of non-Sahlqvist formulas==
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| ; <math>\Box\Diamond p \rightarrow \Diamond \Box p</math>
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| : This is the ''McKinsey formula''; it does not have a first-order frame condition.
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| ; <math>\Box(\Box p \rightarrow p) \rightarrow \Box p</math>
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| : The ''Löb axiom'' is not Sahlqvist; again, it does not have a first-order frame condition.
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| ; <math>(\Box\Diamond p \rightarrow \Diamond \Box p) \land (\Diamond\Diamond q \rightarrow \Diamond q)</math>
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| : The conjunction of the McKinsey formula and the (4) axiom has a first-order frame condition but is not equivalent to any Sahlqvist formula.
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| == Kracht's theorem ==
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| 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].
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| ==References==
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| * L. A. Chagrova, 1991. An undecidable problem in correspondence theory. ''Journal of Symbolic Logic'' 56:1261-1272.
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| * Marcus Kracht, 1993. How completeness and correspondence theory got married. In de Rijke, editor, ''Diamonds and Defaults'', pages 175-214. Kluwer.
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| * 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.
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| [[Category:Modal logic]]
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33 year-old Electrician (Special School ) Dominic from Wingham, has hobbies and interests which includes interior design, property developers in singapore and history. Recently has traveled to Ruins of Loropéni.
My webpage - New launch Sg