doc-src/TutorialI/CTL/document/PDL.tex
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Sun Nov 26 10:48:38 2000 +0100 (2000-11-26)
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     1 %
     2 \begin{isabellebody}%
     3 \def\isabellecontext{PDL}%
     4 %
     5 \isamarkupsubsection{Propositional dynamic logic---PDL%
     6 }
     7 %
     8 \begin{isamarkuptext}%
     9 \index{PDL|(}
    10 The formulae of PDL are built up from atomic propositions via the customary
    11 propositional connectives of negation and conjunction and the two temporal
    12 connectives \isa{AX} and \isa{EF}. Since formulae are essentially
    13 (syntax) trees, they are naturally modelled as a datatype:%
    14 \end{isamarkuptext}%
    15 \isacommand{datatype}\ formula\ {\isacharequal}\ Atom\ atom\isanewline
    16 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Neg\ formula\isanewline
    17 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ formula\ formula\isanewline
    18 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ AX\ formula\isanewline
    19 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ EF\ formula%
    20 \begin{isamarkuptext}%
    21 \noindent
    22 This is almost the same as in the boolean expression case study in
    23 \S\ref{sec:boolex}, except that what used to be called \isa{Var} is now
    24 called \isa{Atom}.
    25 
    26 The meaning of these formulae is given by saying which formula is true in
    27 which state:%
    28 \end{isamarkuptext}%
    29 \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}%
    30 \begin{isamarkuptext}%
    31 \noindent
    32 The concrete syntax annotation allows us to write \isa{s\ {\isasymTurnstile}\ f} instead of
    33 \isa{valid\ s\ f}.
    34 
    35 The definition of \isa{{\isasymTurnstile}} is by recursion over the syntax:%
    36 \end{isamarkuptext}%
    37 \isacommand{primrec}\isanewline
    38 {\isachardoublequote}s\ {\isasymTurnstile}\ Atom\ a\ \ {\isacharequal}\ {\isacharparenleft}a\ {\isasymin}\ L\ s{\isacharparenright}{\isachardoublequote}\isanewline
    39 {\isachardoublequote}s\ {\isasymTurnstile}\ Neg\ f\ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymnot}{\isacharparenleft}s\ {\isasymTurnstile}\ f{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isanewline
    40 {\isachardoublequote}s\ {\isasymTurnstile}\ And\ f\ g\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymand}\ s\ {\isasymTurnstile}\ g{\isacharparenright}{\isachardoublequote}\isanewline
    41 {\isachardoublequote}s\ {\isasymTurnstile}\ AX\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymforall}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M\ {\isasymlongrightarrow}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}\isanewline
    42 {\isachardoublequote}s\ {\isasymTurnstile}\ EF\ f\ \ \ \ {\isacharequal}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymTurnstile}\ f{\isacharparenright}{\isachardoublequote}%
    43 \begin{isamarkuptext}%
    44 \noindent
    45 The first three equations should be self-explanatory. The temporal formula
    46 \isa{AX\ f} means that \isa{f} is true in all next states whereas
    47 \isa{EF\ f} means that there exists some future state in which \isa{f} is
    48 true. The future is expressed via \isa{{\isacharcircum}{\isacharasterisk}}, the transitive reflexive
    49 closure. Because of reflexivity, the future includes the present.
    50 
    51 Now we come to the model checker itself. It maps a formula into the set of
    52 states where the formula is true and is defined by recursion over the syntax,
    53 too:%
    54 \end{isamarkuptext}%
    55 \isacommand{consts}\ mc\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}formula\ {\isasymRightarrow}\ state\ set{\isachardoublequote}\isanewline
    56 \isacommand{primrec}\isanewline
    57 {\isachardoublequote}mc{\isacharparenleft}Atom\ a{\isacharparenright}\ \ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ a\ {\isasymin}\ L\ s{\isacharbraceright}{\isachardoublequote}\isanewline
    58 {\isachardoublequote}mc{\isacharparenleft}Neg\ f{\isacharparenright}\ \ \ {\isacharequal}\ {\isacharminus}mc\ f{\isachardoublequote}\isanewline
    59 {\isachardoublequote}mc{\isacharparenleft}And\ f\ g{\isacharparenright}\ {\isacharequal}\ mc\ f\ {\isasyminter}\ mc\ g{\isachardoublequote}\isanewline
    60 {\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
    61 {\isachardoublequote}mc{\isacharparenleft}EF\ f{\isacharparenright}\ \ \ \ {\isacharequal}\ lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ mc\ f\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}{\isadigit{1}}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}{\isachardoublequote}%
    62 \begin{isamarkuptext}%
    63 \noindent
    64 Only the equation for \isa{EF} deserves some comments. Remember that the
    65 postfix \isa{{\isacharcircum}{\isacharminus}{\isadigit{1}}} and the infix \isa{{\isacharcircum}{\isacharcircum}} are predefined and denote the
    66 converse of a relation and the application of a relation to a set. Thus
    67 \isa{M{\isasyminverse}\ {\isacharcircum}{\isacharcircum}\ T} is the set of all predecessors of \isa{T} and the least
    68 fixed point (\isa{lfp}) of \isa{{\isasymlambda}T{\isachardot}\ mc\ f\ {\isasymunion}\ M{\isasyminverse}\ {\isacharcircum}{\isacharcircum}\ T} is the least set
    69 \isa{T} containing \isa{mc\ f} and all predecessors of \isa{T}. If you
    70 find it hard to see that \isa{mc\ {\isacharparenleft}EF\ f{\isacharparenright}} contains exactly those states from
    71 which there is a path to a state where \isa{f} is true, do not worry---that
    72 will be proved in a moment.
    73 
    74 First we prove monotonicity of the function inside \isa{lfp}%
    75 \end{isamarkuptext}%
    76 \isacommand{lemma}\ mono{\isacharunderscore}ef{\isacharcolon}\ {\isachardoublequote}mono{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}{\isadigit{1}}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}{\isachardoublequote}\isanewline
    77 \isacommand{apply}{\isacharparenleft}rule\ monoI{\isacharparenright}\isanewline
    78 \isacommand{apply}\ blast\isanewline
    79 \isacommand{done}%
    80 \begin{isamarkuptext}%
    81 \noindent
    82 in order to make sure it really has a least fixed point.
    83 
    84 Now we can relate model checking and semantics. For the \isa{EF} case we need
    85 a separate lemma:%
    86 \end{isamarkuptext}%
    87 \isacommand{lemma}\ EF{\isacharunderscore}lemma{\isacharcolon}\isanewline
    88 \ \ {\isachardoublequote}lfp{\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isacharcircum}{\isacharminus}{\isadigit{1}}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A{\isacharbraceright}{\isachardoublequote}%
    89 \begin{isamarkuptxt}%
    90 \noindent
    91 The equality is proved in the canonical fashion by proving that each set
    92 contains the other; the containment is shown pointwise:%
    93 \end{isamarkuptxt}%
    94 \isacommand{apply}{\isacharparenleft}rule\ equalityI{\isacharparenright}\isanewline
    95 \ \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
    96 \ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}%
    97 \begin{isamarkuptxt}%
    98 \noindent
    99 Simplification leaves us with the following first subgoal
   100 \begin{isabelle}%
   101 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}x{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A%
   102 \end{isabelle}
   103 which is proved by \isa{lfp}-induction:%
   104 \end{isamarkuptxt}%
   105 \ \isacommand{apply}{\isacharparenleft}erule\ lfp{\isacharunderscore}induct{\isacharparenright}\isanewline
   106 \ \ \isacommand{apply}{\isacharparenleft}rule\ mono{\isacharunderscore}ef{\isacharparenright}\isanewline
   107 \ \isacommand{apply}{\isacharparenleft}simp{\isacharparenright}%
   108 \begin{isamarkuptxt}%
   109 \noindent
   110 Having disposed of the monotonicity subgoal,
   111 simplification leaves us with the following main goal
   112 \begin{isabelle}
   113 \ \isadigit{1}{\isachardot}\ {\isasymAnd}s{\isachardot}\ s\ {\isasymin}\ A\ {\isasymor}\isanewline
   114 \ \ \ \ \ \ \ \ \ 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
   115 \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M{\isacharcircum}{\isacharasterisk}\ {\isasymand}\ t\ {\isasymin}\ A
   116 \end{isabelle}
   117 which is proved by \isa{blast} with the help of transitivity of \isa{{\isacharcircum}{\isacharasterisk}}:%
   118 \end{isamarkuptxt}%
   119 \ \isacommand{apply}{\isacharparenleft}blast\ intro{\isacharcolon}\ rtrancl{\isacharunderscore}trans{\isacharparenright}%
   120 \begin{isamarkuptxt}%
   121 We now return to the second set containment subgoal, which is again proved
   122 pointwise:%
   123 \end{isamarkuptxt}%
   124 \isacommand{apply}{\isacharparenleft}rule\ subsetI{\isacharparenright}\isanewline
   125 \isacommand{apply}{\isacharparenleft}simp{\isacharcomma}\ clarify{\isacharparenright}%
   126 \begin{isamarkuptxt}%
   127 \noindent
   128 After simplification and clarification we are left with
   129 \begin{isabelle}%
   130 \ {\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}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}%
   131 \end{isabelle}
   132 This goal is proved by induction on \isa{{\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\isactrlsup {\isacharasterisk}}. But since the model
   133 checker works backwards (from \isa{t} to \isa{s}), we cannot use the
   134 induction theorem \isa{rtrancl{\isacharunderscore}induct} because it works in the
   135 forward direction. Fortunately the converse induction theorem
   136 \isa{converse{\isacharunderscore}rtrancl{\isacharunderscore}induct} already exists:
   137 \begin{isabelle}%
   138 \ \ \ \ \ {\isasymlbrakk}{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}{\isacharsemicolon}\ P\ b{\isacharsemicolon}\isanewline
   139 \ \ \ \ \ \ \ \ {\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
   140 \ \ \ \ \ {\isasymLongrightarrow}\ P\ a%
   141 \end{isabelle}
   142 It says that if \isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\isactrlsup {\isacharasterisk}} and we know \isa{P\ b} then we can infer
   143 \isa{P\ a} provided each step backwards from a predecessor \isa{z} of
   144 \isa{b} preserves \isa{P}.%
   145 \end{isamarkuptxt}%
   146 \isacommand{apply}{\isacharparenleft}erule\ converse{\isacharunderscore}rtrancl{\isacharunderscore}induct{\isacharparenright}%
   147 \begin{isamarkuptxt}%
   148 \noindent
   149 The base case
   150 \begin{isabelle}%
   151 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}%
   152 \end{isabelle}
   153 is solved by unrolling \isa{lfp} once%
   154 \end{isamarkuptxt}%
   155 \ \isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharbrackright}{\isacharparenright}%
   156 \begin{isamarkuptxt}%
   157 \begin{isabelle}%
   158 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ t{\isachardot}\ t\ {\isasymin}\ A\ {\isasymLongrightarrow}\ t\ {\isasymin}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharcircum}{\isacharcircum}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}%
   159 \end{isabelle}
   160 and disposing of the resulting trivial subgoal automatically:%
   161 \end{isamarkuptxt}%
   162 \ \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}%
   163 \begin{isamarkuptxt}%
   164 \noindent
   165 The proof of the induction step is identical to the one for the base case:%
   166 \end{isamarkuptxt}%
   167 \isacommand{apply}{\isacharparenleft}rule\ ssubst{\isacharbrackleft}OF\ lfp{\isacharunderscore}unfold{\isacharbrackleft}OF\ mono{\isacharunderscore}ef{\isacharbrackright}{\isacharbrackright}{\isacharparenright}\isanewline
   168 \isacommand{apply}{\isacharparenleft}blast{\isacharparenright}\isanewline
   169 \isacommand{done}%
   170 \begin{isamarkuptext}%
   171 The main theorem is proved in the familiar manner: induction followed by
   172 \isa{auto} augmented with the lemma as a simplification rule.%
   173 \end{isamarkuptext}%
   174 \isacommand{theorem}\ {\isachardoublequote}mc\ f\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymTurnstile}\ f{\isacharbraceright}{\isachardoublequote}\isanewline
   175 \isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ f{\isacharparenright}\isanewline
   176 \isacommand{apply}{\isacharparenleft}auto\ simp\ add{\isacharcolon}EF{\isacharunderscore}lemma{\isacharparenright}\isanewline
   177 \isacommand{done}%
   178 \begin{isamarkuptext}%
   179 \begin{exercise}
   180 \isa{AX} has a dual operator \isa{EN}\footnote{We cannot use the customary \isa{EX}
   181 as that is the ASCII equivalent of \isa{{\isasymexists}}}
   182 (``there exists a next state such that'') with the intended semantics
   183 \begin{isabelle}%
   184 \ \ \ \ \ s\ {\isasymTurnstile}\ EN\ f\ {\isacharequal}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ t\ {\isasymTurnstile}\ f{\isacharparenright}%
   185 \end{isabelle}
   186 Fortunately, \isa{EN\ f} can already be expressed as a PDL formula. How?
   187 
   188 Show that the semantics for \isa{EF} satisfies the following recursion equation:
   189 \begin{isabelle}%
   190 \ \ \ \ \ s\ {\isasymTurnstile}\ EF\ f\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymor}\ s\ {\isasymTurnstile}\ EN\ {\isacharparenleft}EF\ f{\isacharparenright}{\isacharparenright}%
   191 \end{isabelle}
   192 \end{exercise}
   193 \index{PDL|)}%
   194 \end{isamarkuptext}%
   195 \end{isabellebody}%
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