doc-src/TutorialI/CTL/document/PDL.tex
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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}s{\isachardot}\ s\ {\isasymin}\ lfp\ {\isacharparenleft}{\isasymlambda}T{\isachardot}\ A\ {\isasymunion}\ M{\isasyminverse}\ {\isacharcircum}{\isacharcircum}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\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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