(*<*)theory PDL = Base:(*>*)
subsection{*Propositional dynamic logic---PDL*}
text{*\index{PDL|(}
The formulae of PDL are built up from atomic propositions via the customary
propositional connectives of negation and conjunction and the two temporal
connectives @{text AX} and @{text EF}. Since formulae are essentially
(syntax) trees, they are naturally modelled as a datatype:
*}
datatype formula = Atom atom
| Neg formula
| And formula formula
| AX formula
| EF formula
text{*\noindent
This is almost the same as in the boolean expression case study in
\S\ref{sec:boolex}, except that what used to be called @{text Var} is now
called @{term Atom}.
The meaning of these formulae is given by saying which formula is true in
which state:
*}
consts valid :: "state \<Rightarrow> formula \<Rightarrow> bool" ("(_ \<Turnstile> _)" [80,80] 80)
text{*\noindent
The concrete syntax annotation allows us to write @{term"s \<Turnstile> f"} instead of
@{text"valid s f"}.
The definition of @{text"\<Turnstile>"} is by recursion over the syntax:
*}
primrec
"s \<Turnstile> Atom a = (a \<in> L s)"
"s \<Turnstile> Neg f = (\<not>(s \<Turnstile> f))"
"s \<Turnstile> And f g = (s \<Turnstile> f \<and> s \<Turnstile> g)"
"s \<Turnstile> AX f = (\<forall>t. (s,t) \<in> M \<longrightarrow> t \<Turnstile> f)"
"s \<Turnstile> EF f = (\<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<Turnstile> f)";
text{*\noindent
The first three equations should be self-explanatory. The temporal formula
@{term"AX f"} means that @{term f} is true in all next states whereas
@{term"EF f"} means that there exists some future state in which @{term f} is
true. The future is expressed via @{text"\<^sup>*"}, the transitive reflexive
closure. Because of reflexivity, the future includes the present.
Now we come to the model checker itself. It maps a formula into the set of
states where the formula is true and is defined by recursion over the syntax,
too:
*}
consts mc :: "formula \<Rightarrow> state set";
primrec
"mc(Atom a) = {s. a \<in> L s}"
"mc(Neg f) = -mc f"
"mc(And f g) = mc f \<inter> mc g"
"mc(AX f) = {s. \<forall>t. (s,t) \<in> M \<longrightarrow> t \<in> mc f}"
"mc(EF f) = lfp(\<lambda>T. mc f \<union> M\<inverse> `` T)"
text{*\noindent
Only the equation for @{term EF} deserves some comments. Remember that the
postfix @{text"\<inverse>"} and the infix @{text"``"} are predefined and denote the
converse of a relation and the application of a relation to a set. Thus
@{term "M\<inverse> `` T"} is the set of all predecessors of @{term T} and the least
fixed point (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M\<inverse> `` T"} is the least set
@{term T} containing @{term"mc f"} and all predecessors of @{term T}. If you
find it hard to see that @{term"mc(EF f)"} contains exactly those states from
which there is a path to a state where @{term f} is true, do not worry---that
will be proved in a moment.
First we prove monotonicity of the function inside @{term lfp}
*}
lemma mono_ef: "mono(\<lambda>T. A \<union> M\<inverse> `` T)"
apply(rule monoI)
apply blast
done
text{*\noindent
in order to make sure it really has a least fixed point.
Now we can relate model checking and semantics. For the @{text EF} case we need
a separate lemma:
*}
lemma EF_lemma:
"lfp(\<lambda>T. A \<union> M\<inverse> `` T) = {s. \<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<in> A}"
txt{*\noindent
The equality is proved in the canonical fashion by proving that each set
contains the other; the containment is shown pointwise:
*}
apply(rule equalityI);
apply(rule subsetI);
apply(simp)(*<*)apply(rename_tac s)(*>*)
txt{*\noindent
Simplification leaves us with the following first subgoal
@{subgoals[display,indent=0,goals_limit=1]}
which is proved by @{term lfp}-induction:
*}
apply(erule lfp_induct)
apply(rule mono_ef)
apply(simp)
(*pr(latex xsymbols symbols);*)
txt{*\noindent
Having disposed of the monotonicity subgoal,
simplification leaves us with the following main goal
\begin{isabelle}
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymor}\isanewline
\ \ \ \ \ \ \ \ \ x\ {\isasymin}\ M{\isasyminverse}\ {\isacharbackquote}{\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}
which is proved by @{text blast} with the help of transitivity of @{text"\<^sup>*"}:
*}
apply(blast intro: rtrancl_trans);
txt{*
We now return to the second set containment subgoal, which is again proved
pointwise:
*}
apply(rule subsetI)
apply(simp, clarify)
txt{*\noindent
After simplification and clarification we are left with
@{subgoals[display,indent=0,goals_limit=1]}
This goal is proved by induction on @{term"(s,t)\<in>M\<^sup>*"}. But since the model
checker works backwards (from @{term t} to @{term s}), we cannot use the
induction theorem @{thm[source]rtrancl_induct} because it works in the
forward direction. Fortunately the converse induction theorem
@{thm[source]converse_rtrancl_induct} already exists:
@{thm[display,margin=60]converse_rtrancl_induct[no_vars]}
It says that if @{prop"(a,b):r\<^sup>*"} 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
The base case
@{subgoals[display,indent=0,goals_limit=1]}
is solved by unrolling @{term lfp} once
*}
apply(rule ssubst[OF lfp_unfold[OF mono_ef]])
txt{*
@{subgoals[display,indent=0,goals_limit=1]}
and disposing of the resulting trivial subgoal automatically:
*}
apply(blast)
txt{*\noindent
The proof of the induction step is identical to the one for the base case:
*}
apply(rule ssubst[OF lfp_unfold[OF mono_ef]])
apply(blast)
done
text{*
The main theorem is proved in the familiar manner: induction followed by
@{text auto} augmented with the lemma as a simplification rule.
*}
theorem "mc f = {s. s \<Turnstile> f}";
apply(induct_tac f);
apply(auto simp add:EF_lemma);
done;
text{*
\begin{exercise}
@{term AX} has a dual operator @{term EN}\footnote{We cannot use the customary @{text EX}
as that is the ASCII equivalent of @{text"\<exists>"}}
(``there exists a next state such that'') with the intended semantics
@{prop[display]"(s \<Turnstile> EN f) = (EX t. (s,t) : M & t \<Turnstile> f)"}
Fortunately, @{term"EN f"} can already be expressed as a PDL formula. How?
Show that the semantics for @{term EF} satisfies the following recursion equation:
@{prop[display]"(s \<Turnstile> EF f) = (s \<Turnstile> f | s \<Turnstile> EN(EF f))"}
\end{exercise}
\index{PDL|)}
*}
(*<*)
theorem main: "mc f = {s. s \<Turnstile> f}";
apply(induct_tac f);
apply(auto simp add:EF_lemma);
done;
lemma aux: "s \<Turnstile> f = (s : mc f)";
apply(simp add:main);
done;
lemma "(s \<Turnstile> EF f) = (s \<Turnstile> f | s \<Turnstile> Neg(AX(Neg(EF f))))";
apply(simp only:aux);
apply(simp);
apply(rule ssubst[OF lfp_unfold[OF mono_ef]], fast);
done
end
(*>*)