1 (*<*)theory PDL imports Base begin(*>*)
3 subsection{*Propositional Dynamic Logic --- PDL*}
6 The formulae of PDL are built up from atomic propositions via
7 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:%
10 \footnote{The customary definition of PDL
11 \cite{HarelKT-DL} looks quite different from ours, but the two are easily
12 shown to be equivalent.}
15 datatype formula = Atom atom
22 This resembles the boolean expression case study in
24 A validity relation between
25 states and formulae specifies the semantics:
28 consts valid :: "state \<Rightarrow> formula \<Rightarrow> bool" ("(_ \<Turnstile> _)" [80,80] 80)
31 The syntax annotation allows us to write @{term"s \<Turnstile> f"} instead of
32 \hbox{@{text"valid s f"}}.
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\<^sup>* \<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 \emph{A}ll ne\emph{X}t states whereas
46 @{term"EF f"} means that there \emph{E}xists some \emph{F}uture state in which @{term f} is
47 true. The future is expressed via @{text"\<^sup>*"}, the reflexive transitive
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. It too is defined by recursion over the syntax:
54 consts mc :: "formula \<Rightarrow> state set"
56 "mc(Atom a) = {s. a \<in> L s}"
58 "mc(And f g) = mc f \<inter> mc g"
59 "mc(AX f) = {s. \<forall>t. (s,t) \<in> M \<longrightarrow> t \<in> mc f}"
60 "mc(EF f) = lfp(\<lambda>T. mc f \<union> (M\<inverse> `` T))"
63 Only the equation for @{term EF} deserves some comments. Remember that the
64 postfix @{text"\<inverse>"} and the infix @{text"``"} are predefined and denote the
65 converse of a relation and the image of a set under a relation. Thus
66 @{term "M\<inverse> `` T"} is the set of all predecessors of @{term T} and the least
67 fixed point (@{term lfp}) of @{term"\<lambda>T. mc f \<union> M\<inverse> `` T"} is the least set
68 @{term T} containing @{term"mc f"} and all predecessors of @{term T}. If you
69 find it hard to see that @{term"mc(EF f)"} contains exactly those states from
70 which there is a path to a state where @{term f} is true, do not worry --- this
71 will be proved in a moment.
73 First we prove monotonicity of the function inside @{term lfp}
74 in order to make sure it really has a least fixed point.
77 lemma mono_ef: "mono(\<lambda>T. A \<union> (M\<inverse> `` T))"
83 Now we can relate model checking and semantics. For the @{text EF} case we need
88 "lfp(\<lambda>T. A \<union> (M\<inverse> `` T)) = {s. \<exists>t. (s,t) \<in> M\<^sup>* \<and> t \<in> A}"
91 The equality is proved in the canonical fashion by proving that each set
92 includes the other; the inclusion is shown pointwise:
97 apply(simp)(*<*)apply(rename_tac s)(*>*)
100 Simplification leaves us with the following first subgoal
101 @{subgoals[display,indent=0,goals_limit=1]}
102 which is proved by @{term lfp}-induction:
105 apply(erule lfp_induct)
108 (*pr(latex xsymbols symbols);*)
110 Having disposed of the monotonicity subgoal,
111 simplification leaves us with the following goal:
113 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymor}\isanewline
114 \ \ \ \ \ \ \ \ \ 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
115 \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
117 It is proved by @{text blast}, using the transitivity of
118 \isa{M\isactrlsup {\isacharasterisk}}.
121 apply(blast intro: rtrancl_trans)
124 We now return to the second set inclusion subgoal, which is again proved
132 After simplification and clarification we are left with
133 @{subgoals[display,indent=0,goals_limit=1]}
134 This goal is proved by induction on @{term"(s,t)\<in>M\<^sup>*"}. But since the model
135 checker works backwards (from @{term t} to @{term s}), we cannot use the
136 induction theorem @{thm[source]rtrancl_induct}: it works in the
137 forward direction. Fortunately the converse induction theorem
138 @{thm[source]converse_rtrancl_induct} already exists:
139 @{thm[display,margin=60]converse_rtrancl_induct[no_vars]}
140 It says that if @{prop"(a,b):r\<^sup>*"} and we know @{prop"P b"} then we can infer
141 @{prop"P a"} provided each step backwards from a predecessor @{term z} of
142 @{term b} preserves @{term P}.
145 apply(erule converse_rtrancl_induct)
149 @{subgoals[display,indent=0,goals_limit=1]}
150 is solved by unrolling @{term lfp} once
153 apply(subst lfp_unfold[OF mono_ef])
156 @{subgoals[display,indent=0,goals_limit=1]}
157 and disposing of the resulting trivial subgoal automatically:
163 The proof of the induction step is identical to the one for the base case:
166 apply(subst lfp_unfold[OF mono_ef])
171 The main theorem is proved in the familiar manner: induction followed by
172 @{text auto} augmented with the lemma as a simplification rule.
175 theorem "mc f = {s. s \<Turnstile> f}"
177 apply(auto simp add: EF_lemma)
182 @{term AX} has a dual operator @{term EN}
183 (``there exists a next state such that'')%
184 \footnote{We cannot use the customary @{text EX}: it is reserved
185 as the \textsc{ascii}-equivalent of @{text"\<exists>"}.}
186 with the intended semantics
187 @{prop[display]"(s \<Turnstile> EN f) = (EX t. (s,t) : M & t \<Turnstile> f)"}
188 Fortunately, @{term"EN f"} can already be expressed as a PDL formula. How?
190 Show that the semantics for @{term EF} satisfies the following recursion equation:
191 @{prop[display]"(s \<Turnstile> EF f) = (s \<Turnstile> f | s \<Turnstile> EN(EF f))"}
196 theorem main: "mc f = {s. s \<Turnstile> f}"
198 apply(auto simp add: EF_lemma)
201 lemma aux: "s \<Turnstile> f = (s : mc f)"
202 apply(simp add: main)
205 lemma "(s \<Turnstile> EF f) = (s \<Turnstile> f | s \<Turnstile> Neg(AX(Neg(EF f))))"
206 apply(simp only: aux)
208 apply(subst lfp_unfold[OF mono_ef], fast)