1 / 8

Interval Hybrid Temporal Logic

Interval Hybrid Temporal Logic. Altaf Hussain. What is that about …. Definition of Interval hybrid temporal logic minimal interval structures van Benthem minimal interval structures Sound and complete Tableau calculus Technical results. Interval Hybrid temporal logic.

dustin-allen
Download Presentation

Interval Hybrid Temporal Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Interval Hybrid Temporal Logic Altaf Hussain

  2. What is that about … • Definition of Interval hybrid temporal logic • minimal interval structures • van Benthem minimal interval structures • Sound and complete Tableau calculus • Technical results

  3. Interval Hybrid temporal logic Hybrid languages are modal languages which use formulas to refer to specific points in a model. • Basic modal language built over propositional variables  = {p, q, r, …}. • Let  = {i, j, k, …} be a nonempty set – elements are called nominals and are used to name states • Let (  ) =  • (  ) – called the set of atoms and define the interval hybrid logic over (  ) as follows:  = i | p |  |  |  | @i where {<D>,<U>,<F>,<P>}, i  , p  

  4. Interval Hybrid temporal logic Interpretation • Let M = (W, >,<,<|,|>,V), where w  W; >,<, … are binary relations on W and V:    (W);i , V(i) is a singleton subset of W. • Then M, w |= p iffw  V(p) M, w |=  iffM, w |  M, w |=  iffM, w |=  and M, w |=  M, w |= <D> iff w’(w|>w’ and M, w’ |= ) M, w |= <U> iff w’(w<|w’ and M, w’ |= ) M, w |= <F> iff w’(w<w’ and M, w’ |= ) M, w |= <P> iff w’(w>w’ and M, w’ |= ) M, w |= iiffw  V(i) M, w |= @i  iffM, w’ |=  where w’ is the denotation of i under V.

  5. Interval Hybrid temporal logic Minimal interval structure • F = (W, >,<,<|,|>) is Minimal interval structure if it satisfies: • For >, < Irreflexivity, Transitivity • For |>, <| Reflexivity, Transitivity, Antisymetry and • xy(x < y  u(u<|x u<y)) Right monotonicity • xy(x < y  u(u<|y x<u)) Left monotonicity • Examples • W = {[n,m] | n  m and n,m  Z} • <| = {([n,m],[k,l], k  n  m  l} • |> = {([n,m],[k,l], n  k  l  m} • < = {([n,m],[k,l], n  m < k  l} • > = {([n,m],[k,l], k  l < n  m} • W = {(n,m) | n < m and n,m  Q} • <| = {((n,m),(k,l), k  n < m  l} • |> = {((n,m),(k,l), n  k < l  m} • < = {((n,m),(k,l), n < m  k < l} • > = {((n,m),(k,l), k < l  n < m}

  6. Tableau system • Tableau system is sound and complete • Tree where nodes are labeled by formulas and which is built via certain tableau rules • Some rules (all in the article): ,  denote formulas and s, t, u and a denote nominals @s  |-- @s  [] @s  |-- @s  [] @s () |-- @s , @s  []  @s () |--  @s  |  @s  [ ] @s @t  |-- @t  [@]  @s @t  |--  @t  [@] @s t, @t u |-- @s u [bridge] @s <D>t |-- @t <U>s [C(<|)]

  7. Tableau system • Show that @s (<D>i  )  @i (<U>  <U>s) is provable. • So we show that there is a closed tableau whose root node is: @k(@s (<D>i  )  @i (<U>  <U>s))  @k(@s (<D>i  ) ,  @k(@i (<U>  <U>s)) [@] @k(@s (<D>i  ) ,  (@i (<U>  <U>s)) [] @k@s <D>i, @k@s  [ ]  @i <U> @i <U>s [@] @s <D>i, @s  @s <D>i, @s  [C(<|)] @i <U>s [bridge] @i <U>s @i <U> 

  8. Technical results • Complexity of the satisfiability problem for minimal interval structures is EXPTIME • The logic minimal interval structures are decidable

More Related