PDL without converse operator -

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I found this exercise (5.14) in the book "Dynamic Logic" by Harel, Kozen and Tiuryn. I have no clue how to construct the mentioned model in the exercise body.



Construct the Kripke model, such that the operator $<alpha>$ is not continous.



In PDL with - (converse operator), the map $phi rightarrow <alpha>phi$ is continuous with respect to the order of logical implication. That is, if $K$ is a Kripke frame, $A$ a (finite or infinite) set of formulas, and $phi$ a formula such that $m_K(phi) = sup_psi in Am_K(psi)$ then $sup_psi in Am_K(<alpha>psi)$ exists and is equal to $m_K(<alpha>phi)$.



We define $m_K(psi)$ as a set of states in $K$ satisfying formula $psi$.



If program $alpha$ maps state $s_1$ to state $s_2$ then program $alpha-$ ($alpha$ with converse operator) maps $s_2$ to $s_1$.







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  • 1




    what is the converse operator?
    – Apostolos
    Aug 26 at 16:08










  • I have added the definition of converse operator.
    – Adam Mata
    Aug 26 at 17:24














up vote
1
down vote

favorite












I found this exercise (5.14) in the book "Dynamic Logic" by Harel, Kozen and Tiuryn. I have no clue how to construct the mentioned model in the exercise body.



Construct the Kripke model, such that the operator $<alpha>$ is not continous.



In PDL with - (converse operator), the map $phi rightarrow <alpha>phi$ is continuous with respect to the order of logical implication. That is, if $K$ is a Kripke frame, $A$ a (finite or infinite) set of formulas, and $phi$ a formula such that $m_K(phi) = sup_psi in Am_K(psi)$ then $sup_psi in Am_K(<alpha>psi)$ exists and is equal to $m_K(<alpha>phi)$.



We define $m_K(psi)$ as a set of states in $K$ satisfying formula $psi$.



If program $alpha$ maps state $s_1$ to state $s_2$ then program $alpha-$ ($alpha$ with converse operator) maps $s_2$ to $s_1$.







share|cite|improve this question


















  • 1




    what is the converse operator?
    – Apostolos
    Aug 26 at 16:08










  • I have added the definition of converse operator.
    – Adam Mata
    Aug 26 at 17:24












up vote
1
down vote

favorite









up vote
1
down vote

favorite











I found this exercise (5.14) in the book "Dynamic Logic" by Harel, Kozen and Tiuryn. I have no clue how to construct the mentioned model in the exercise body.



Construct the Kripke model, such that the operator $<alpha>$ is not continous.



In PDL with - (converse operator), the map $phi rightarrow <alpha>phi$ is continuous with respect to the order of logical implication. That is, if $K$ is a Kripke frame, $A$ a (finite or infinite) set of formulas, and $phi$ a formula such that $m_K(phi) = sup_psi in Am_K(psi)$ then $sup_psi in Am_K(<alpha>psi)$ exists and is equal to $m_K(<alpha>phi)$.



We define $m_K(psi)$ as a set of states in $K$ satisfying formula $psi$.



If program $alpha$ maps state $s_1$ to state $s_2$ then program $alpha-$ ($alpha$ with converse operator) maps $s_2$ to $s_1$.







share|cite|improve this question














I found this exercise (5.14) in the book "Dynamic Logic" by Harel, Kozen and Tiuryn. I have no clue how to construct the mentioned model in the exercise body.



Construct the Kripke model, such that the operator $<alpha>$ is not continous.



In PDL with - (converse operator), the map $phi rightarrow <alpha>phi$ is continuous with respect to the order of logical implication. That is, if $K$ is a Kripke frame, $A$ a (finite or infinite) set of formulas, and $phi$ a formula such that $m_K(phi) = sup_psi in Am_K(psi)$ then $sup_psi in Am_K(<alpha>psi)$ exists and is equal to $m_K(<alpha>phi)$.



We define $m_K(psi)$ as a set of states in $K$ satisfying formula $psi$.



If program $alpha$ maps state $s_1$ to state $s_2$ then program $alpha-$ ($alpha$ with converse operator) maps $s_2$ to $s_1$.









share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 26 at 17:24

























asked Aug 26 at 14:03









Adam Mata

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  • 1




    what is the converse operator?
    – Apostolos
    Aug 26 at 16:08










  • I have added the definition of converse operator.
    – Adam Mata
    Aug 26 at 17:24












  • 1




    what is the converse operator?
    – Apostolos
    Aug 26 at 16:08










  • I have added the definition of converse operator.
    – Adam Mata
    Aug 26 at 17:24







1




1




what is the converse operator?
– Apostolos
Aug 26 at 16:08




what is the converse operator?
– Apostolos
Aug 26 at 16:08












I have added the definition of converse operator.
– Adam Mata
Aug 26 at 17:24




I have added the definition of converse operator.
– Adam Mata
Aug 26 at 17:24















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