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| In [[logic]], '''general frames''' (or simply '''frames''') are [[Kripke frame]]s with an additional structure, which are used to model [[modal logic|modal]] and [[intermediate logic|intermediate]] logics. The general frame semantics combines the main virtues of [[Kripke semantics]] and [[algebraic semantics (mathematical logic)|algebraic semantics]]: it shares the transparent geometrical insight of the former, and robust completeness of the latter.
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| ==Definition==
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| A '''modal general frame''' is a triple <math>\mathbf F=\langle F,R,V\rangle</math>, where <math>\langle F,R\rangle</math> is a Kripke frame (i.e., ''R'' is a [[binary relation]] on the set ''F''), and ''V'' is a set of subsets of ''F'' which is closed under
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| *the Boolean operations of (binary) [[intersection (set theory)|intersection]], [[union (set theory)|union]], and [[complement (set theory)|complement]],
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| *the operation <math>\Box</math>, defined by <math>\Box A=\{x\in F;\,\forall y\in F\,(x\,R\,y\to y\in A)\}</math>.
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| The purpose of ''V'' is to restrict the allowed valuations in the frame: a model <math>\langle F,R,\Vdash\rangle</math> based on the Kripke frame <math>\langle F,R\rangle</math> is '''admissible''' in the general frame '''F''', if | |
| :<math>\{x\in F;\,x\Vdash p\}\in V</math> for every [[propositional variable]] ''p''.
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| The closure conditions on ''V'' then ensure that <math>\{x\in F;\,x\Vdash A\}</math> belongs to ''V'' for ''every'' formula ''A'' (not only a variable).
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| A formula ''A'' is '''valid''' in '''F''', if <math>x\Vdash A</math> for all admissible valuations <math>\Vdash</math>, and all points <math>x\in F</math>. A [[normal modal logic]] ''L'' is valid in the frame '''F''', if all axioms (or equivalently, all theorems) of ''L'' are valid in '''F'''. In this case we call '''F''' an ''L''-'''frame'''.
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| A Kripke frame <math>\langle F,R\rangle</math> may be identified with a general frame in which all valuations are admissible: i.e., <math>\langle F,R,\mathcal{P}(F)\rangle</math>, where <math>\mathcal P(F)</math> denotes the [[power set]] of ''F''.
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| ==Types of frames==
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| In full generality, general frames are hardly more than a fancy name for Kripke ''models''; in particular, the correspondence of modal axioms to properties on the accessibility relation is lost. This can be remedied by imposing additional conditions on the set of admissible valuations.
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| A frame <math>\mathbf F=\langle F,R,V\rangle</math> is called
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| *'''differentiated''', if <math>\forall A\in V\,(x\in A\Leftrightarrow y\in A)</math> implies <math>x=y</math>,
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| *'''tight''', if <math>\forall A\in V\,(x\in\Box A\Rightarrow y\in A)</math> implies <math>x\,R\,y</math>,
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| *'''compact''', if every subset of ''V'' with the [[finite intersection property]] has a non-empty intersection,
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| *'''atomic''', if ''V'' contains all singletons,
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| *'''refined''', if it is differentiated and tight,
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| *'''descriptive''', if it is refined and compact.
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| Kripke frames are refined and atomic. However, infinite Kripke frames are never compact. Every finite differentiated or atomic frame is a Kripke frame.
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| Descriptive frames are the most important class of frames because of the duality theory (see below). Refined frames are useful as a common generalization of descriptive and Kripke frames.
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| ==Operations and morphisms on frames==
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| Every Kripke model <math>\langle F,R,{\Vdash}\rangle</math> '''induces''' the general frame <math>\langle F,R,V\rangle</math>, where ''V'' is defined as
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| :<math>V=\big\{\{x\in F;\,x\Vdash A\};\,A\hbox{ is a formula}\big\}.</math>
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| The fundamental truth-preserving operations of generated subframes, p-morphic images, and disjoint unions of Kripke frames have analogues on general frames. A frame <math>\mathbf G=\langle G,S,W\rangle</math> is a '''generated subframe''' of a frame <math>\mathbf F=\langle F,R,V\rangle</math>, if the Kripke frame <math>\langle G,S\rangle</math> is a generated subframe of the Kripke frame <math>\langle F,R\rangle</math> (i.e., ''G'' is a subset of ''F'' closed upwards under ''R'', and ''S'' is the restriction of ''R'' to ''G''), and
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| :<math>W=\{A\cap G;\,A\in V\}.</math>
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| A '''p-morphism''' (or '''bounded morphism''') <math>f\colon\mathbf F\to\mathbf G</math> is a function from ''F'' to ''G'' which is a p-morphism of the Kripke frames <math>\langle F,R\rangle</math> and <math>\langle G,S\rangle</math>, and satisfies the additional constraint
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| :<math>f^{-1}[A]\in V</math> for every <math>A\in W</math>.
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| The '''disjoint union''' of an indexed set of frames <math>\mathbf F_i=\langle F_i,R_i,V_i\rangle</math>, <math>i\in I</math>, is the frame <math>\mathbf F=\langle F,R,V\rangle</math>, where ''F'' is the disjoint union of <math>\{F_i;\,i\in I\}</math>, ''R'' is the union of <math>\{R_i;\,i\in I\}</math>, and
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| :<math>V=\{A\subseteq F;\,\forall i\in I\,(A\cap F_i\in V_i)\}.</math>
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| The '''refinement''' of a frame <math>\mathbf F=\langle F,R,V\rangle</math> is a refined frame <math>\mathbf G=\langle G,S,W\rangle</math> defined as follows. We consider the [[equivalence relation]]
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| :<math>x\sim y\iff\forall A\in V\,(x\in A\Leftrightarrow y\in A),</math>
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| and let <math>G=F/{\sim}</math> be the set of equivalence classes of <math>\sim</math>. Then we put
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| :<math>\langle x/{\sim},y/{\sim}\rangle\in S\iff\forall A\in V\,(x\in\Box A\Rightarrow y\in A),</math>
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| :<math>A/{\sim}\in W\iff A\in V.</math>
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| ==Completeness==
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| Unlike Kripke frames, every normal modal logic ''L'' is complete with respect to a class of general frames. This is a consequence of the fact that ''L'' is complete with respect to a class of Kripke models <math>\langle F,R,{\Vdash}\rangle</math>: as ''L'' is closed under substitution, the general frame induced by <math>\langle F,R,{\Vdash}\rangle</math> is an ''L''-frame. Moreover, every logic ''L'' is complete with respect to a single ''descriptive'' frame. Indeed, ''L'' is complete with respect to its canonical model, and the general frame induced by the canonical model (called the '''canonical frame''' of ''L'') is descriptive.
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| ==Jónsson–Tarski duality==
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| [[File:Rieger-Nishimura ladder.svg|thumb|right|100px|The Rieger–Nishimura ladder: a 1-universal intuitionistic Kripke frame.]]
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| [[File:Rieger-Nishimura.svg|thumb|right|300px|Its dual Heyting algebra, the Rieger–Nishimura lattice. It is the free Heyting algebra over 1 generator.]]
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| General frames bear close connection to [[modal algebra]]s. Let <math>\mathbf F=\langle F,R,V\rangle</math> be a general frame. The set ''V'' is closed under Boolean operations, therefore it is a [[subalgebra]] of the power set [[Boolean algebra (structure)|Boolean algebra]] <math>\langle\mathcal P(F),\cap,\cup,-\rangle</math>. It also carries an additional unary operation, <math>\Box</math>. The combined structure <math>\langle V,\cap,\cup,-,\Box\rangle</math> is a modal algebra, which is called the '''dual algebra''' of '''F''', and denoted by <math>\mathbf F^+</math>.
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| In the opposite direction, it is possible to construct the '''dual frame''' <math>\mathbf A_+=\langle F,R,V\rangle</math> to any modal algebra <math>\mathbf A=\langle A,\wedge,\vee,-,\Box\rangle</math>. The Boolean algebra <math>\langle A,\wedge,\vee,-\rangle</math> has a [[Stone space]], whose underlying set ''F'' is the set of all [[ultrafilter]]s of '''A'''. The set ''V'' of admissible valuations in <math>\mathbf A_+</math> consists of the [[clopen set|clopen]] subsets of ''F'', and the accessibility relation ''R'' is defined by
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| :<math>x\,R\,y\iff\forall a\in A\,(\Box a\in x\Rightarrow a\in y)</math>
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| for all ultrafilters ''x'' and ''y''.
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| A frame and its dual validate the same formulas, hence the general frame semantics and algebraic semantics are in a sense equivalent. The double dual <math>(\mathbf A_+)^+</math> of any modal algebra is isomorphic to <math>\mathbf A</math> itself. This is not true in general for double duals of frames, as the dual of every algebra is descriptive. In fact, a frame <math>\mathbf F</math> is descriptive if and only if it is isomorphic to its double dual <math>(\mathbf F^+)_+</math>.
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| It is also possible to define duals of p-morphisms on one hand, and modal algebra homomorphisms on the other hand. In this way the operators <math>(\cdot)^+</math> and <math>(\cdot)_+</math> become a pair of [[contravariant functor]]s between the [[category (mathematics)|category]] of general frames, and the category of modal algebras. These functors provide a [[equivalence of categories|duality]] (called '''Jónsson–Tarski duality''' after [[Bjarni Jónsson]] and [[Alfred Tarski]]) between the categories of descriptive frames, and modal algebras.
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| ==Intuitionistic frames==
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| The frame semantics for intuitionistic and intermediate logics can be developed in parallel to the semantics for modal logics. An '''intuitionistic general frame''' is a triple <math>\langle F,\le,V\rangle</math>, where <math>\le</math> is a [[partial order]] on ''F'', and ''V'' is a set of [[upper set|upper subset]]s (''cones'') of ''F'' which contains the empty set, and is closed under
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| *intersection and union,
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| *the operation <math>A\to B=\Box(-A\cup B)</math>.
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| Validity and other concepts are then introduced similarly to modal frames, with a few changes necessary to accommodate for the weaker closure properties of the set of admissible valuations. In particular, an intuitionistic frame <math>\mathbf F=\langle F,\le,V\rangle</math> is called
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| *'''tight''', if <math>\forall A\in V\,(x\in A\Leftrightarrow y\in A)</math> implies <math>x\le y</math>,
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| *'''compact''', if every subset of <math>V\cup\{F-A;\,A\in V\}</math> with the finite intersection property has a non-empty intersection.
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| Tight intuitionistic frames are automatically differentiated, hence refined.
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| The dual of an intuitionistic frame <math>\mathbf F=\langle F,\le,V\rangle</math> is the [[Heyting algebra]] <math>\mathbf F^+=\langle V,\cap,\cup,\to,\emptyset\rangle</math>. The dual of a Heyting algebra <math>\mathbf A=\langle A,\wedge,\vee,\to,0\rangle</math> is the intuitionistic frame <math>\mathbf A_+=\langle F,\le,V\rangle</math>, where ''F'' is the set of all [[prime filter]]s of '''A''', the ordering <math>\le</math> is [[inclusion (set theory)|inclusion]], and ''V'' consists of all subsets of ''F'' of the form
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| :<math>\{x\in F;\,a\in x\},</math> | |
| where <math>a\in A</math>. As in the modal case, <math>(\cdot)^+</math> and <math>(\cdot)_+</math> are a pair of contravariant functors, which make the category of Heyting algebras dually equivalent to the category of descriptive intuitionistic frames.
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| It is possible to construct intuitionistic general frames from transitive reflexive modal frames and vice versa, see [[modal companion]].
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| ==References==
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| *Alexander Chagrov and Michael Zakharyaschev, ''Modal Logic'', vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.
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| *Patrick Blackburn, [[Maarten de Rijke]], and Yde Venema, ''Modal Logic'', vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 2001.
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| [[Category:Modal logic]]
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| [[Category:Model theory]]
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| [[Category:Duality theories]]
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