Modal Logic XIII. Yanjing Wang

Transcription

1 Modal Logic XIII Yanjing Wang Department of Philosophy, Peking University Apr 16th, 2015 Advanced Modal Logic (2015 Spring)

2 1 Frames

3 Frames and validity Why do we study frames? Frames as tools for analysing modal logics (as a set of valid formulas), driven by syntactic approaches to modal logic. Can we characterize a logic by a class of frames via validity (Is system S sound and complete w.r.t. a class of frames)? How can we show S φ? Frames as structures to be described by modal logics via validity.what classes of frames are definable by modal logic? What about its expressive power over frames compared with classic logics?

4 Models, frames, satisfiability and validity local global local class global class models M, w φ M φ K pm φ K m φ frames F, w φ F φ K pf φ K f φ In terms of the validity over classes of pointed models: M, w φ {M, w} φ M φ {M, w w W M } φ K m φ {M, w M K m } φ F, w φ {M, w M is based on F, w} φ F φ {M, w M is based on F } φ K pf φ {M, w M, w is based on (F, w) K pf } φ K f φ {M, w M is based on F K f } φ

5 Models, frames, satisfiability and validity local global local class global class models M, w φ M φ K pm φ K m φ frames F, w φ F φ K pf φ K f φ φ defines M, w if for all N, v: N, v φ M, w N, v φ defines M if for all N: N φ M total N φ defines K pm if for all M, w: M, w φ M, w K pm φ defines K m if for all M: M φ M K m φ defines F, w if for all F, w : F, w φ F, w? F, w φ defines F if for all F : F φ F? F φ defines K pf if for all F, w: F, w φ F, w K pf φ defines K f if for all F : F φ F K f Each φ must define a class of models/frames, but may not define any model or frame.

6 Models, Frames, satisfiability and validity φ can be replaced by Φ. Natural questions: what kind of... can be define by φ/φ. We can also define relative definability, for example: φ defines K f within C if for all F C: F φ F K f. (C itself may not be definable..) Q: intersection of C and K? Q: Can we cast the definability of classes of frames on the definability of classes of models? φ defines K f?iff? φ defines {M, w M is based on F K f } K f is modally definable?iff? {M, w M is based on F K f } is modally definable

7 Frame Definability Example (modal definability of classes of frames) p p defines the class of reflexive frames, i.e., for all F : F φ F is reflexive. ( p p can also define this class) p p defines the class of serial frames (every world has a successor). Ex. What about ( p p) p? p p defines the class of frames which consists of isolated reflexive points. defines the class of frames which consist of isolated irreflexive points. Q: Can two non-equivalent modal formulas define the same class of frames? Some tricky things in proving such results: to show F : F K = F φ we need to find a

8 Frame Definability A frame can be viewed as a first-order structure for the language with equality and R but NO unary predicates P (first-order frame language). Example (first-order definability of classes of frames) x xrx defines the class of reflexive frames, i.e., for all F : F x xrx F is reflexive. x y xry defines the class of serial frames. x y (Rxy x = y) defines the class of frames consisting of isolated reflexive points. x y Rxy defines the class of frames consisting of isolated irreflexive points.

9 Frame and validity The validity of modal formulas on frames is essentially (monadic) second-order since we need predicate variables over sets of possible worlds. The second-order frame language here is based on the first-order one with a P-indexed collection of monadic predicate variables which can be quantified over. local global models M ST x (φ)[w] M xst x (φ) frames F P 1 P n ST x (φ)[w] F P 1 P n xst x (φ)

10 Frame and validity Theorem F, w φ F P 1 P n ST x (φ)[w] F φ F P 1 P n xst x (φ) Proof. (F, V), w φ F ST x (φ)[w, V(p 1 ) V(p n )]

11 Digression: Monadic Second Order Logic (MSO) (Büchi): A language of finite words is recognisable by a finite state automaton if and only if it is MSO definable over finite words. (How do you define the words with even number of symbols?) (Büchi): A language of infinite words is recognisable by a Nondeterministic Büchi Automaton if and only if it is MSO definable over infinite words. (Thatcher,Wright) A set of finite trees is recognizable by finite tree automata iff it is MSO definable over finite trees. (Rabin s theorem): MSO over n-successor infinite trees (SnS) is decidable. (Janin and Walukiewicz): Modal mu-calculus is the