Mathematical Logics Modal Logic: K and more * Fausto Giunchiglia and - PowerPoint PPT Presentation

Mathematical Logics Modal Logic: K and more * Fausto Giunchiglia and Mattia Fumagallli University of Trento *Originally by Luciano Serafini and Chiara Ghidini Modified by Fausto Giunchiglia and Mattia Fumagalli 0

Properties of accessibility relation Formulas can be used to shape the “form” of the structure, as in the examples expressed before or to impose properties on the accessibility relation R . Temporal logic: if the accessibility relation is supposed to represent a temporal relation, and wRw l means that w l is a future world w.r.t. w , then R must be a transitive relation. That is if w l is a future world of w , then any future world of w l is also a future world of w . Logic of knowledge: if the accessibility relation is used to represent the knowledge of an agent A , and wRw l represents the fact that w l is a possible situation coherent with its actual situation w , then R must be reflexive, since w is always coherent with itself. 1

Typical Properties of R The following table summarizes the most relevant properties of the accessibility relation, which have been studied in modal logic, and for which it has been provided a sound and complete axiomatization Properties of R ∀ w.R ( w, w ) R is reflexive ∀ w v u. ( R ( w, v ) ∧ R ( v , u ) ⊃ R ( w, u )) R is transitive ∀ w v . ( R ( w, v ) ⊃ R ( v , w )) R is symmetric ∀ w v u. ( R ( w, v ) ∧ R ( w, u ) ⊃ R ( v , u )) R is Euclidean ∀ w. ∃ vR ( w, v ) R is serial ∀ w v .R ( w, v ) ⊃ ∃ u. ( R ( w, u ) ∧ R ( u, v )) R is weakly dense ∀ w v u. ( R ( w, v ) ∧ R ( v , u ) ⊃ v = u ) R is partly functional ∀ w ∃ ! v.R ( w, v ) R is functional ∀ u v w. ( R ( u, v ) ∧ R ( u, w ) ⊃ R is weakly connected R ( v , w ) ∨ v = w ∨ R ( w, v )) ∀ u v w. ( R ( u, v ) ∧ R ( u, w ) ⊃ R is weakly directed ∃ t ( R ( v , t ) ∧ R ( w, t ))) 2 We will investigate only the ones in red color.

R is reflexive The axiom T If a frame is reflexive (we say that a frame has a property, when the relation R has such a property) then the formulas T □ φ ⊃ φ holds. (Or alternatively φ ⊃ ◊ φ .) 3

R is reflexive - soundness Let M be a model on a reflexive frame F = ( W , R ) and w any world in W . We prove that M, w ⊨ □ φ ⊃ φ . Since R is reflexive then wRw 1 Suppose that M, w ⊨ □ φ (Hypothesis) 2 From the satisfiability condition of □ , M, w ⊨ □ φ , and wRw imply 3 that M, w ⊨ φ (Thesis) 4 Since from (Hypothesis) we have derived (Thesis), we can conclude that M, w ⊨ □ φ ⊃ φ . 4

R is reflexive - completeness Suppose that a frame F = ( W , R ) is not reflexive. If R is not reflexive then there is a w ∈ W which does not access to 1 itself. I.e., for some w ∈ W it does not hold that wRw . be any model on F , and let φ be the propositional formula Let M 2 p . Let V the set p true in all the worlds of W but w where p is set to be false. From the fact that w does not access to itself, we have that in all 3 the worlds w accessible from w , p is true, i.e, ∀ w', wRw', M, w' ⊨ p . Form the satisfiability condition of □ we have that M, 4 w ⊨ □ p . since M, w ⊨ p , we have that M, w ⊨ □ p ⊃ p . 5 5

R is symmetric The axiom B If a frame is symmetric then the formula B φ ⊃ □◊ φ holds. 6

R is symmetric - soundness Let M be a model on a symmetric frame F = ( W , R ) and w any world in W . We prove that M , w ⊨ φ ⊃ □◊ φ . Suppose that M , w ⊨ φ (Hypothesis) 1 we want to show that M , w ⊨ □◊ φ (Thesis) 2 Form the satisfiability conditions of □ , we need to prove that for every 3 world w l accessible from w , M , w l ⊨ ◊ φ . Let w l , be any world accessible from w , i.e., wRw l 4 from the fact that R is symmetric, we have that w l Rw 5 From the satisfiability condition of ◊ , from the fact that w l Rw and that 6 M , w ⊨ φ , we have that M , w l ⊨ ◊ φ . so for every world w l accessible from w , we have that M , w l ⊨ ◊ φ . 7 From the satisfiability condition of □ , M , w ⊨ □◊ φ (Thesis) 8 Since from (Hypothesis) we have derived (Thesis), we can conclude that 9 M , w ⊨ φ ⊃ □◊ φ . 7

R is symmetric - completeness Suppose that a frame F = ( W , R ) is not Symmetric. 1 If R is not symmetric then there are two worlds w, w l ∈ W such that wRw l and not w l Rw Let M be any model on F , and let φ be the propositional formula p . Let V 2 the set p false in all the worlds of W but w where p is set to be true. From the fact that w l does not access to w , it means that in all the 3 worlds accessible from w l , p is false, i.e. there is no world w ll accessible from w l wuch that M, w ll ⊨ p . 4 by the satisfiability conditions of ◊ , we have that M, w l ⊭ ◊ p . 5 Since there is a world w l accessible from w , with M, w ⊭ ◊ p , form 6 the satisfiability condition of □ we have that M, w ⊭ □◊ p . since M, w ⊨ p , and M, w ⊭ □◊ p . we have that M, w ⊭ p ⊃ □◊ p . 7 8

R is serial The axiom D If a frame is serial then the formula D □ φ ⊃ ◊ φ holds. 9

R is serial - soundness Let M be a model on a serial frame F = ( W , R ) and w any world in W . We prove that M , w ⊨ □ φ ⊃ ◊ φ . Since R is serial there is a world w l ∈ W with wRw l 1 Suppose that M, w ⊨ □ φ (Hypothesis) 2 From the satisfiability condition of □ , M, w ⊨ □ φ implies that M , w l ⊨ φ 3 Since there is a world w l accessible from w that satisfies φ , from the 4 satisfiability conditions of ◊ we have that M, w ⊨ ◊ φ (Thesis) . Since from (Hypothesis) we have derived (Thesis), we can conclude 5 that M, w ⊨ □ φ ⊃ ◊ φ . 10

R is serial - completeness Suppose that a frame F = ( W , R ) is not Serial. If R is not serial then there is a w ∈ W which does not have any 1 accessible world. I.e., for all w l it does not hold that wRw l . Let M be any model on F . 2 Form the satisfiability condition of □ and from the fact that w does 3 not have any accessible world, we have that M , w ⊨ □ φ . Form the satisfiability condition of ◊ and from the fact that w does 4 not have any accessible world, we have that M , w ⊨ ◊ φ . this implies that M , w ⊨ □ φ ⊃ ◊ φ 5 11

R is transitive The axiom 4 If a frame is transitive then the formula 4 □ φ ⊃ □□ φ holds. 12

R is transitive - soundness Let M be a model on a transitive frame F = ( W , R ) and w any world in W . We prove that M , w ⊨ □ φ ⊃ □□ φ . Suppose that M , w ⊨ □ φ (Hypothesis). 1 We have to prove that M , w ⊨ □□ φ (Thesis) 2 From the satisfiability condition of □ , this is equivalent to prove that for 3 all world w l accessible from w M , w l ⊨ □ φ . Let w l be any world accessible from w . To prove that M , w l ⊨ □ φ we have to 4 prove that for all the world w ll accessible from w l , M , w ll ⊨ φ . Let w ll be a world accessible from w l , i.e., w l Rw ll . 5 From the facts wRw l and w l Rw ll and the fact that R is transitive, we have 6 that wRw ll . Since M , w ⊨ □ φ , from the satisfiability conditions of □ we have that 7 M , w ll ⊨ φ . Since M , w ll ⊨ φ for every world w ll accessible from w l , then M , w l ⊨ □ φ . and 8 therefore M , w ⊨ □□ φ . (Thesis) 9 Since from (Hypothesis) we have derived (Thesis), we can conclude that M , w ⊨ □ φ ⊃ □□ φ . 10 13

R is transitive - completeness Suppose that a frame F = ( W , R ) is not transitive. If R is not transitive then there are three worlds w, w l , w ll ∈ W , such 1 that wRw l , w l Rw ll but not wRw ll . Let M be any model on F, and let φ be the propositional formula p . Let 2 V the set p true in all the worlds of W but w ll where p is set to be false. From the fact that w does not access to w ll , and that w ll is the only 3 world where p is false, we have that in all the worlds accessible from w , p is true. This implies that M , w ⊨ □ p . 4 On the other hand, we have that w l Rw ll , and w ll ⊨ p implies that 5 M , w l ⊨ □ φ . and since wRw l , we have that M , w ⊨ □□ p . 6 In summary: M , w ⊨ □□ p , and M , w ⊨ □ P ; from which we have 7 that M , w ⊨ □ p ⊃ □□ p . 14

R is euclidean The axiom 5 If a frame is euclidean then the formula 5 ◊ φ ⊃ □◊ φ holds. 15

R is euclidean - soundness Let M be a model on a euclidean frame F = ( W , R ) and w any world in W . We prove that M , w ⊨ ◊ φ ⊃ □◊ φ . Suppose that M , w ⊨ ◊ φ (Hypothesis). 1 The satisfiability condition of ◊ implies that there is a world w l accessible from 2 w such that M , w l ⊨ φ . 3 We have to prove that M , w ⊨ □◊ φ (Thesis) From the satisfiability condition of □ , this is equivalent to prove that for all world 4 w ll accessible from w M , w ll ⊨ ◊ φ , let w ll be any world accessible from w . The fact that R is euclidean, the fact that wRw l 5 implies that w ll Rw l . Since M , w l ⊨ φ , the satisfiability condition of ◊ implies that M , w ll ⊨ ◊ φ . 6 and therefore M , w ⊨ □◊ φ . (Thesis) 7 Since from (Hypothesis) we have derived (Thesis), we can conclude that 8 M , w ⊨ □ φ ⊃ □◊ φ . 16