1 (*<*)theory PDL = Base:(*>*)
3 subsection{*Propositional dynmic logic---PDL*}
6 The formulae of PDL are built up from atomic propositions via the customary
7 propositional connectives of negation and conjunction and the two temporal
8 connectives @{text AX} and @{text EF}. Since formulae are essentially
9 (syntax) trees, they are naturally modelled as a datatype:
12 datatype formula = Atom atom
19 This is almost the same as in the boolean expression case study in
20 \S\ref{sec:boolex}, except that what used to be called @{text Var} is now
23 The meaning of these formulae is given by saying which formula is true in
27 consts valid :: "state \<Rightarrow> formula \<Rightarrow> bool" ("(_ \<Turnstile> _)" [80,80] 80)
30 The concrete syntax annotation allows us to write @{term"s \<Turnstile> f"} instead of
33 The definition of @{text"\<Turnstile>"} is by recursion over the syntax:
37 "s \<Turnstile> Atom a = (a \<in> L s)"
38 "s \<Turnstile> Neg f = (\<not>(s \<Turnstile> f))"
39 "s \<Turnstile> And f g = (s \<Turnstile> f \<and> s \<Turnstile> g)"
40 "s \<Turnstile> AX f = (\<forall>t. (s,t) \<in> M \<longrightarrow> t \<Turnstile> f)"
41 "s \<Turnstile> EF f = (\<exists>t. (s,t) \<in> M^* \<and> t \<Turnstile> f)";
44 The first three equations should be self-explanatory. The temporal formula
45 @{term"AX f"} means that @{term f} is true in all next states whereas
46 @{term"EF f"} means that there exists some future state in which @{term f} is
47 true. The future is expressed via @{text"^*"}, the transitive reflexive
48 closure. Because of reflexivity, the future includes the present.
50 Now we come to the model checker itself. It maps a formula into the set of
51 states where the formula is true and is defined by recursion over the syntax,
55 consts mc :: "formula \<Rightarrow> state set";
57 "mc(Atom a) = {s. a \<in> L s}"
59 "mc(And f g) = mc f \<inter> mc g"
60 "mc(AX f) = {s. \<forall>t. (s,t) \<in> M \<longrightarrow> t \<in> mc f}"
61 "mc(EF f) = lfp(\<lambda>T. mc f \<union> M^-1 ^^ T)"
65 Only the equation for @{term EF} deserves some comments. Remember that the
66 postfix @{text"^-1"} and the infix @{text"^^"} are predefined and denote the
67 converse of a relation and the application of a relation to a set. Thus
68 @{term "M^-1 ^^ T"} is the set of all predecessors of @{term T} and the least
69 fixpoint (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M^-1 ^^ T"} is the least set
70 @{term T} containing @{term"mc f"} and all predecessors of @{term T}. If you
71 find it hard to see that @{term"mc(EF f)"} contains exactly those states from
72 which there is a path to a state where @{term f} is true, do not worry---that
73 will be proved in a moment.
75 First we prove monotonicity of the function inside @{term lfp}
78 lemma mono_ef: "mono(\<lambda>T. A \<union> M^-1 ^^ T)"
84 in order to make sure it really has a least fixpoint.
86 Now we can relate model checking and semantics. For the @{text EF} case we need
91 "lfp(\<lambda>T. A \<union> M^-1 ^^ T) = {s. \<exists>t. (s,t) \<in> M^* \<and> t \<in> A}"
94 The equality is proved in the canonical fashion by proving that each set
95 contains the other; the containment is shown pointwise:
98 apply(rule equalityI);
101 (*pr(latex xsymbols symbols);*)
103 Simplification leaves us with the following first subgoal
105 \ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
107 which is proved by @{term lfp}-induction:
110 apply(erule Lfp.induct)
113 (*pr(latex xsymbols symbols);*)
115 Having disposed of the monotonicity subgoal,
116 simplification leaves us with the following main goal
118 \ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ A\ {\isasymor}\isanewline
119 \ \ \ \ \ \ \ \ \ s\ {\isasymin}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ {\isacharparenleft}lfp\ {\isacharparenleft}{\dots}{\isacharparenright}\ {\isasyminter}\ {\isacharbraceleft}x{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isacharparenright}\isanewline
120 \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
122 which is proved by @{text blast} with the help of a few lemmas about
126 apply(blast intro: r_into_rtrancl rtrancl_trans);
129 We now return to the second set containment subgoal, which is again proved
135 (*pr(latex xsymbols symbols);*)
137 After simplification and clarification we are left with
139 \ \isadigit{1}{\isachardot}\ {\isasymAnd}s\ t{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}{\isacharsemicolon}\ t\ {\isasymin}\ A{\isasymrbrakk}\ {\isasymLongrightarrow}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}
141 This goal is proved by induction on @{term"(s,t)\<in>M^*"}. But since the model
142 checker works backwards (from @{term t} to @{term s}), we cannot use the
143 induction theorem @{thm[source]rtrancl_induct} because it works in the
144 forward direction. Fortunately the converse induction theorem
145 @{thm[source]converse_rtrancl_induct} already exists:
146 @{thm[display,margin=60]converse_rtrancl_induct[no_vars]}
147 It says that if @{prop"(a,b):r^*"} and we know @{prop"P b"} then we can infer
148 @{prop"P a"} provided each step backwards from a predecessor @{term z} of
149 @{term b} preserves @{term P}.
152 apply(erule converse_rtrancl_induct)
153 (*pr(latex xsymbols symbols);*)
157 \ \isadigit{1}{\isachardot}\ {\isasymAnd}t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}
159 is solved by unrolling @{term lfp} once
162 apply(rule ssubst[OF lfp_Tarski[OF mono_ef]])
163 (*pr(latex xsymbols symbols);*)
166 \ \isadigit{1}{\isachardot}\ {\isasymAnd}t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}\isadigit{1}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}
168 and disposing of the resulting trivial subgoal automatically:
174 The proof of the induction step is identical to the one for the base case:
177 apply(rule ssubst[OF lfp_Tarski[OF mono_ef]])
182 The main theorem is proved in the familiar manner: induction followed by
183 @{text auto} augmented with the lemma as a simplification rule.
186 theorem "mc f = {s. s \<Turnstile> f}";
188 apply(auto simp add:EF_lemma);