Quelle  PDL.thy

(*<*)theory PDL imports Base begin(*>*)

subsection‹Propositional Dynamic Logic --- PDL›

text‹\index{PDL|(}
  formulae of PDL are built up from atomic propositions via
  and conjunction and the two temporal
  ‹AX› and ‹EF›\@. Since formulae are essentially
  trees, they are naturally modelled as a datatype:%
 footnote{The customary definition of PDL
 cite‹"HarelKT-DL"› looks quite different from ours, but the two are easily
  to be equivalent.}
 ›

datatype formula = Atom "atom"
                  | Neg formula
                  | And formula formula
                  | AX formula
                  | EF formula

text‹\noindent
  resembles the boolean expression case study in
 S\ref{sec:boolex}.
  validity relation between states and formulae specifies the semantics.
  syntax annotation allows us to write ‹s ⊨ f› instead of
 hbox{‹valid s f›}. The definition is by recursion over the syntax:
 ›

primrec valid :: "state ==> formula ==> bool"   ("(_ ⊨ _)" [80,80] 80)
where
"s ⊨ Atom a = (a ∈ L s)" |
"s ⊨ Neg f = (¬(s ⊨ f))" |
"s ⊨ And f g = (s ⊨ f ∧ s ⊨ g)" |
"s ⊨ AX f = (∀t. (s,t) ∈ M ⟶ t ⊨ f)" |
"s ⊨ EF f = (∃t. (s,t) ∈ M^* ∧ t ⊨ f)"

text‹\noindent
  first three equations should be self-explanatory. The temporal formula
 term‹AX f› means that term‹f› is true in \emph{A}ll ne\emph{X}t states whereas
 term‹EF f› means that there \emph{E}xists some \emph{F}uture state in which term‹f› is
 . The future is expressed via ‹^*›, the reflexive transitive
 . Because of reflexivity, the future includes the present.

  we come to the model checker itself. It maps a formula into the
  of states where the formula is true. It too is defined by
  over the syntax:›

primrec mc :: "formula ==> state set" where
"mc(Atom a) = {s. a ∈ L s}" |
"mc(Neg f) = -mc f" |
"mc(And f g) = mc f ∩ mc g" |
"mc(AX f) = {s. ∀t. (s,t) ∈ M ⟶ t ∈ mc f}" |
"mc(EF f) = lfp(λT. mc f ∪ (M^-1 `` T))"

text‹\noindent
  the equation for term‹EF› deserves some comments. Remember that the
  ‹^-1› and the infix ‹``› are predefined and denote the
  of a relation and the image of a set under a relation. Thus
 term‹M<sup>-1</sup> `` T› is the set of all predecessors of term‹T› and the least
  point (term‹lfp›) of term‹λT. mc f ∪ M^-1 `` T› is the least set
 term‹T› containing term‹mc f› and all predecessors of term‹T›. If you
  it hard to see that term‹mc(EF f)› contains exactly those states from
  there is a path to a state where term‹f› is true, do not worry --- this
  be proved in a moment.

  we prove monotonicity of the function inside term‹lfp›
  order to make sure it really has a least fixed point.
 ›

lemma mono_ef: "mono(λT. A ∪ (M^-1 `` T))"
apply(rule monoI)
apply blast
done

text‹\noindent
  we can relate model checking and semantics. For the ‹EF› case we need
  separate lemma:
 ›

lemma EF_lemma:
  "lfp(λT. A ∪ (M^-1 `` T)) = {s. ∃t. (s,t) ∈ M^* ∧ t ∈ A}"

txt‹\noindent
  equality is proved in the canonical fashion by proving that each set
  the other; the inclusion is shown pointwise:
 ›

apply(rule equalityI)
 apply(rule subsetI)
 apply(simp)(*<*)apply(rename_tac s)(*>*)

txt‹\noindent
  leaves us with the following first subgoal
 {subgoals[display,indent=0,goals_limit=1]}
  is proved by term‹lfp›-induction:
 ›

 apply(erule lfp_induct_set)
  apply(rule mono_ef)
 apply(simp)
txt‹\noindent
  disposed of the monotonicity subgoal,
  leaves us with the following goal:
 begin{isabelle}
  {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymor}\isanewline
  \ \ \ \ \ \ \ \ x\ {\isasymin}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ {\isacharparenleft}lfp\ {\isacharparenleft}\dots{\isacharparenright}\ {\isasyminter}\ {\isacharbraceleft}x{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isacharparenright}\isanewline
  \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
 end{isabelle}
  is proved by ‹blast›, using the transitivity of
 isa{M\isactrlsup {\isacharasterisk}}.
 ›

 apply(blast intro: rtrancl_trans)

txt‹
  now return to the second set inclusion subgoal, which is again proved
 :
 ›

apply(rule subsetI)
apply(simp, clarify)

txt‹\noindent
  simplification and clarification we are left with
 {subgoals[display,indent=0,goals_limit=1]}
  goal is proved by induction on term‹(s,t)∈M^*›. But since the model
  works backwards (from term‹t› to term‹s›), we cannot use the
  theorem @{thm[source]rtrancl_induct}: it works in the
  direction. Fortunately the converse induction theorem
 {thm[source]converse_rtrancl_induct} already exists:
 {thm[display,margin=60]converse_rtrancl_induct[no_vars]}
  says that if prop‹(a,b)∈r^*› and we know prop‹P b› then we can infer
 prop‹P a› provided each step backwards from a predecessor term‹z› of
 term‹b› preserves term‹P›.
 ›

apply(erule converse_rtrancl_induct)

txt‹\noindent
  base case
 {subgoals[display,indent=0,goals_limit=1]}
  solved by unrolling term‹lfp› once
 ›

 apply(subst lfp_unfold[OF mono_ef])

txt‹
 {subgoals[display,indent=0,goals_limit=1]}
  disposing of the resulting trivial subgoal automatically:
 ›

 apply(blast)

txt‹\noindent
  proof of the induction step is identical to the one for the base case:
 ›

apply(subst lfp_unfold[OF mono_ef])
apply(blast)
done

text‹
  main theorem is proved in the familiar manner: induction followed by
 ‹auto› augmented with the lemma as a simplification rule.
 ›

theorem "mc f = {s. s ⊨ f}"
apply(induct_tac f)
apply(auto simp add: EF_lemma)
done

text‹
 begin{exercise}
 term‹AX› has a dual operator term‹EN›
 ``there exists a next state such that'')%
 footnote{We cannot use the customary ‹EX›: it is reserved
  the \textsc{ascii}-equivalent of ‹∃›.}
  the intended semantics
 {prop[display]"(s ⊨ EN f) = (∃t. (s,t) ∈ M ∧ t ⊨ f)"}
 , term‹EN f› can already be expressed as a PDL formula. How?

  that the semantics for term‹EF› satisfies the following recursion equation:
 {prop[display]"(s ⊨ EF f) = (s ⊨ f | s ⊨ EN(EF f))"}
 end{exercise}
 index{PDL|)}
 ›
(*<*)
theorem main: "mc f = {s. s ⊨ f}"
apply(induct_tac f)
apply(auto simp add: EF_lemma)
done

lemma aux: "s ⊨ f = (s ∈ mc f)"
apply(simp add: main)
done

lemma "(s ⊨ EF f) = (s ⊨ f | s ⊨ Neg(AX(Neg(EF f))))"
apply(simp only: aux)
apply(simp)
apply(subst lfp_unfold[OF mono_ef], fast)
done

end
(*>*)