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src/Doc/Tutorial/CTL/Base.thy

author | wenzelm |

Mon, 07 Oct 2013 21:24:44 +0200 | |

changeset 54313 | da2e6282a4f5 |

parent 48985 | 5386df44a037 |

child 58620 | 7435b6a3f72e |

permissions | -rw-r--r-- |

native executable even for Linux, to avoid surprises with file managers opening executable script as text file;

(*<*)theory Base imports Main begin(*>*) section{*Case Study: Verified Model Checking*} text{*\label{sec:VMC} This chapter ends with a case study concerning model checking for Computation Tree Logic (CTL), a temporal logic. Model checking is a popular technique for the verification of finite state systems (implementations) with respect to temporal logic formulae (specifications) \cite{ClarkeGP-book,Huth-Ryan-book}. Its foundations are set theoretic and this section will explore them in HOL\@. This is done in two steps. First we consider a simple modal logic called propositional dynamic logic (PDL)\@. We then proceed to the temporal logic CTL, which is used in many real model checkers. In each case we give both a traditional semantics (@{text \<Turnstile>}) and a recursive function @{term mc} that maps a formula into the set of all states of the system where the formula is valid. If the system has a finite number of states, @{term mc} is directly executable: it is a model checker, albeit an inefficient one. The main proof obligation is to show that the semantics and the model checker agree. \underscoreon Our models are \emph{transition systems}:\index{transition systems} sets of \emph{states} with transitions between them. Here is a simple example: \begin{center} \unitlength.5mm \thicklines \begin{picture}(100,60) \put(50,50){\circle{20}} \put(50,50){\makebox(0,0){$p,q$}} \put(61,55){\makebox(0,0)[l]{$s_0$}} \put(44,42){\vector(-1,-1){26}} \put(16,18){\vector(1,1){26}} \put(57,43){\vector(1,-1){26}} \put(10,10){\circle{20}} \put(10,10){\makebox(0,0){$q,r$}} \put(-1,15){\makebox(0,0)[r]{$s_1$}} \put(20,10){\vector(1,0){60}} \put(90,10){\circle{20}} \put(90,10){\makebox(0,0){$r$}} \put(98, 5){\line(1,0){10}} \put(108, 5){\line(0,1){10}} \put(108,15){\vector(-1,0){10}} \put(91,21){\makebox(0,0)[bl]{$s_2$}} \end{picture} \end{center} Each state has a unique name or number ($s_0,s_1,s_2$), and in each state certain \emph{atomic propositions} ($p,q,r$) hold. The aim of temporal logic is to formalize statements such as ``there is no path starting from $s_2$ leading to a state where $p$ or $q$ holds,'' which is true, and ``on all paths starting from $s_0$, $q$ always holds,'' which is false. Abstracting from this concrete example, we assume there is a type of states: *} typedecl state text{*\noindent Command \commdx{typedecl} merely declares a new type but without defining it (see \S\ref{sec:typedecl}). Thus we know nothing about the type other than its existence. That is exactly what we need because @{typ state} really is an implicit parameter of our model. Of course it would have been more generic to make @{typ state} a type parameter of everything but declaring @{typ state} globally as above reduces clutter. Similarly we declare an arbitrary but fixed transition system, i.e.\ a relation between states: *} consts M :: "(state \<times> state)set"; text{*\noindent This is Isabelle's way of declaring a constant without defining it. Finally we introduce a type of atomic propositions *} typedecl "atom" text{*\noindent and a \emph{labelling function} *} consts L :: "state \<Rightarrow> atom set" text{*\noindent telling us which atomic propositions are true in each state. *} (*<*)end(*>*)