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
94 \begin{isamarkuptext}%
96 Now we can relate model checking and semantics. For the \isa{EF} case we need
100 \isacommand{lemma}\ EF{\isacharunderscore}lemma{\isacharcolon}\isanewline
101 \ \ {\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%
127 \begin{isamarkuptext}%
128 The main theorem is proved in the familiar manner: induction followed by
129 \isa{auto} augmented with the lemma as a simplification rule.%
132 \isacommand{theorem}\ {\isachardoublequote}mc\ f\ {\isacharequal}\ {\isacharbraceleft}s{\isachardot}\ s\ {\isasymTurnstile}\ f{\isacharbraceright}{\isachardoublequote}\isanewline
138 \begin{isamarkuptext}%
140 \isa{AX} has a dual operator \isa{EN}
141 (``there exists a next state such that'')%
142 \footnote{We cannot use the customary \isa{EX}: it is reserved
143 as the \textsc{ascii}-equivalent of \isa{{\isasymexists}}.}
144 with the intended semantics
146 \ \ \ \ \ s\ {\isasymTurnstile}\ EN\ f\ {\isacharequal}\ {\isacharparenleft}{\isasymexists}t{\isachardot}\ {\isacharparenleft}s{\isacharcomma}\ t{\isacharparenright}\ {\isasymin}\ M\ {\isasymand}\ t\ {\isasymTurnstile}\ f{\isacharparenright}%
148 Fortunately, \isa{EN\ f} can already be expressed as a PDL formula. How?
150 Show that the semantics for \isa{EF} satisfies the following recursion equation:
152 \ \ \ \ \ s\ {\isasymTurnstile}\ EF\ f\ {\isacharequal}\ {\isacharparenleft}s\ {\isasymTurnstile}\ f\ {\isasymor}\ s\ {\isasymTurnstile}\ EN\ {\isacharparenleft}EF\ f{\isacharparenright}{\isacharparenright}%
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