3 \def\isabellecontext{PDL}%
6 \isamarkupsubsection{Propositional Dynamic Logic --- PDL%
10 \begin{isamarkuptext}%
12 The formulae of PDL are built up from atomic propositions via
13 negation and conjunction and the two temporal
14 connectives \isa{AX} and \isa{EF}\@. Since formulae are essentially
15 syntax trees, they are naturally modelled as a datatype:%
16 \footnote{The customary definition of PDL
17 \cite{HarelKT-DL} looks quite different from ours, but the two are easily
18 shown to be equivalent.}%
21 \isacommand{datatype}\ formula\ {\isacharequal}\ Atom\ atom\isanewline
22 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Neg\ formula\isanewline
23 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ formula\ formula\isanewline
24 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ AX\ formula\isanewline
25 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ EF\ formula\isamarkupfalse%
27 \begin{isamarkuptext}%
29 This resembles the boolean expression case study in
31 A validity relation between
32 states and formulae specifies the semantics:%
35 \isacommand{consts}\ valid\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}state\ {\isasymRightarrow}\ formula\ {\isasymRightarrow}\ bool{\isachardoublequote}\ \ \ {\isacharparenleft}{\isachardoublequote}{\isacharparenleft}{\isacharunderscore}\ {\isasymTurnstile}\ {\isacharunderscore}{\isacharparenright}{\isachardoublequote}\ {\isacharbrackleft}{\isadigit{8}}{\isadigit{0}}{\isacharcomma}{\isadigit{8}}{\isadigit{0}}{\isacharbrackright}\ {\isadigit{8}}{\isadigit{0}}{\isacharparenright}\isamarkupfalse%
37 \begin{isamarkuptext}%
39 The syntax annotation allows us to write \isa{s\ {\isasymTurnstile}\ f} instead of
40 \hbox{\isa{valid\ s\ f}}.
41 The definition of \isa{{\isasymTurnstile}} is by recursion over the syntax:%
44 \isacommand{primrec}\isanewline
45 {\isachardoublequote}s\ {\isasymTurnstile}\ Atom\ a\ \ {\isacharequal}\ {\isacharparenleft}a\ {\isasymin}\ L\ s{\isacharparenright}{\isachardoublequote}\isanewline
46 {\isachardoublequote}s\ {\isasymTurnstile}\ Neg\ f\ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymnot}{\isacharparenleft}s\ {\isasymTurnstile}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
47 {\isachardoublequote}s\ {\isasymTurnstile}\ And\ f\ g\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymand}\ s\ {\isasymTurnstile}\ g{\isacharparenright}{\isachardoublequote}\isanewline
48 {\isachardoublequote}s\ {\isasymTurnstile}\ AX\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymforall}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\ {\isasymlongrightarrow}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isanewline
49 {\isachardoublequote}s\ {\isasymTurnstile}\ EF\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
51 \begin{isamarkuptext}%
53 The first three equations should be self-explanatory. The temporal formula
54 \isa{AX\ f} means that \isa{f} is true in \emph{A}ll ne\emph{X}t states whereas
55 \isa{EF\ f} means that there \emph{E}xists some \emph{F}uture state in which \isa{f} is
56 true. The future is expressed via \isa{\isactrlsup {\isacharasterisk}}, the reflexive transitive
57 closure. Because of reflexivity, the future includes the present.
59 Now we come to the model checker itself. It maps a formula into the set of
60 states where the formula is true. It too is defined by recursion over the syntax:%
63 \isacommand{consts}\ mc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}formula\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline
65 \isacommand{primrec}\isanewline
66 {\isachardoublequote}mc{\isacharparenleft}Atom\ a{\isacharparenright}\ \ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ a\ {\isasymin}\ L\ s{\isacharbraceright}{\isachardoublequote}\isanewline
67 {\isachardoublequote}mc{\isacharparenleft}Neg\ f{\isacharparenright}\ \ \ {\isacharequal}\ {\isacharminus}mc\ f{\isachardoublequote}\isanewline
68 {\isachardoublequote}mc{\isacharparenleft}And\ f\ g{\isacharparenright}\ {\isacharequal}\ mc\ f\ {\isasyminter}\ mc\ g{\isachardoublequote}\isanewline
69 {\isachardoublequote}mc{\isacharparenleft}AX\ f{\isacharparenright}\ \ \ \ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymforall}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\ \ {\isasymlongrightarrow}\ t\ {\isasymin}\ mc\ f{\isacharbraceright}{\isachardoublequote}\isanewline
70 {\isachardoublequote}mc{\isacharparenleft}EF\ f{\isacharparenright}\ \ \ \ {\isacharequal}\ lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ mc\ f\ {\isasymunion}\ {\isacharparenleft}M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
72 \begin{isamarkuptext}%
74 Only the equation for \isa{EF} deserves some comments. Remember that the
75 postfix \isa{{\isasyminverse}} and the infix \isa{{\isacharbackquote}{\isacharbackquote}} are predefined and denote the
76 converse of a relation and the image of a set under a relation. Thus
77 \isa{M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T} is the set of all predecessors of \isa{T} and the least
78 fixed point (\isa{lfp}) of \isa{{\isasymlambda}T{\isachardot}\ mc\ f\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T} is the least set
79 \isa{T} containing \isa{mc\ f} and all predecessors of \isa{T}. If you
80 find it hard to see that \isa{mc\ {\isacharparenleft}EF\ f{\isacharparenright}} contains exactly those states from
81 which there is a path to a state where \isa{f} is true, do not worry --- this
82 will be proved in a moment.
84 First we prove monotonicity of the function inside \isa{lfp}
85 in order to make sure it really has a least fixed point.%
88 \isacommand{lemma}\ mono{\isacharunderscore}ef{\isacharcolon}\ {\isachardoublequote}mono{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ {\isacharparenleft}M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
90 \isacommand{apply}{\isacharparenleft}rule\ monoI{\isacharparenright}\isanewline
92 \isacommand{apply}\ blast\isanewline
94 \isacommand{done}\isamarkupfalse%
96 \begin{isamarkuptext}%
98 Now we can relate model checking and semantics. For the \isa{EF} case we need
102 \isacommand{lemma}\ EF{\isacharunderscore}lemma{\isacharcolon}\isanewline
103 \ \ {\isachardoublequote}lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ {\isacharparenleft}M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isachardoublequote}\isamarkupfalse%
105 \begin{isamarkuptxt}%
107 The equality is proved in the canonical fashion by proving that each set
108 includes the other; the inclusion is shown pointwise:%
111 \isacommand{apply}{\isacharparenleft}rule\ equalityI{\isacharparenright}\isanewline
113 \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
115 \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isamarkupfalse%
118 \begin{isamarkuptxt}%
120 Simplification leaves us with the following first subgoal
122 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A%
124 which is proved by \isa{lfp}-induction:%
127 \isacommand{apply}{\isacharparenleft}erule\ lfp{\isacharunderscore}induct{\isacharparenright}\isanewline
129 \isacommand{apply}{\isacharparenleft}rule\ mono{\isacharunderscore}ef{\isacharparenright}\isanewline
131 \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}\isamarkupfalse%
133 \begin{isamarkuptxt}%
135 Having disposed of the monotonicity subgoal,
136 simplification leaves us with the following goal:
138 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymor}\isanewline
139 \ \ \ \ \ \ \ \ \ 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
140 \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
142 It is proved by \isa{blast}, using the transitivity of
143 \isa{M\isactrlsup {\isacharasterisk}}.%
146 \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtrancl{\isacharunderscore}trans{\isacharparenright}\isamarkupfalse%
148 \begin{isamarkuptxt}%
149 We now return to the second set inclusion subgoal, which is again proved
153 \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
155 \isacommand{apply}{\isacharparenleft}simp{\isacharcomma}\ clarify{\isacharparenright}\isamarkupfalse%
157 \begin{isamarkuptxt}%
159 After simplification and clarification we are left with
161 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ t{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ t\ {\isasymin}\ A{\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}%
163 This goal is proved by induction on \isa{{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}}. But since the model
164 checker works backwards (from \isa{t} to \isa{s}), we cannot use the
165 induction theorem \isa{rtrancl{\isacharunderscore}induct}: it works in the
166 forward direction. Fortunately the converse induction theorem
167 \isa{converse{\isacharunderscore}rtrancl{\isacharunderscore}induct} already exists:
169 \ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ P\ b{\isacharsemicolon}\isanewline
170 \isaindent{\ \ \ \ \ \ }{\isasymAnd}y\ z{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}y{\isacharcomma}\ z{\isacharparenright}\ {\isasymin}\ r{\isacharsemicolon}\ {\isacharparenleft}z{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ P\ z{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ y{\isasymrbrakk}\isanewline
171 \isaindent{\ \ \ \ \ }{\isasymLongrightarrow}\ P\ a%
173 It says that if \isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}} and we know \isa{P\ b} then we can infer
174 \isa{P\ a} provided each step backwards from a predecessor \isa{z} of
175 \isa{b} preserves \isa{P}.%
178 \isacommand{apply}{\isacharparenleft}erule\ converse{\isacharunderscore}rtrancl{\isacharunderscore}induct{\isacharparenright}\isamarkupfalse%
180 \begin{isamarkuptxt}%
184 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}%
186 is solved by unrolling \isa{lfp} once%
189 \isacommand{apply}{\isacharparenleft}subst\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharparenright}\isamarkupfalse%
191 \begin{isamarkuptxt}%
193 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharbackquote}{\isacharbackquote}\ T{\isacharparenright}%
195 and disposing of the resulting trivial subgoal automatically:%
198 \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isamarkupfalse%
200 \begin{isamarkuptxt}%
202 The proof of the induction step is identical to the one for the base case:%
205 \isacommand{apply}{\isacharparenleft}subst\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharparenright}\isanewline
207 \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
209 \isacommand{done}\isamarkupfalse%
211 \begin{isamarkuptext}%
212 The main theorem is proved in the familiar manner: induction followed by
213 \isa{auto} augmented with the lemma as a simplification rule.%
216 \isacommand{theorem}\ {\isachardoublequote}mc\ f\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymTurnstile}\ f{\isacharbraceright}{\isachardoublequote}\isanewline
218 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ f{\isacharparenright}\isanewline
220 \isacommand{apply}{\isacharparenleft}auto\ simp\ add{\isacharcolon}\ EF{\isacharunderscore}lemma{\isacharparenright}\isanewline
222 \isacommand{done}\isamarkupfalse%
224 \begin{isamarkuptext}%
226 \isa{AX} has a dual operator \isa{EN}
227 (``there exists a next state such that'')%
228 \footnote{We cannot use the customary \isa{EX}: it is reserved
229 as the \textsc{ascii}-equivalent of \isa{{\isasymexists}}.}
230 with the intended semantics
232 \ \ \ \ \ s\ {\isasymTurnstile}\ EN\ f\ {\isacharequal}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ t\ {\isasymTurnstile}\ f{\isacharparenright}%
234 Fortunately, \isa{EN\ f} can already be expressed as a PDL formula. How?
236 Show that the semantics for \isa{EF} satisfies the following recursion equation:
238 \ \ \ \ \ s\ {\isasymTurnstile}\ EF\ f\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymor}\ s\ {\isasymTurnstile}\ EN\ {\isacharparenleft}EF\ f{\isacharparenright}{\isacharparenright}%
260 %%% TeX-master: "root"