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PRISM Manual | ThePRISMLanguage / PTAs
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<p><a class='wikilink' href='Main.html'>The PRISM Language</a> /
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</p><h1>PTAs</h1>
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<p class='vspace'>So far in this section, we have mainly focused on three types of models: DTMCs, MDPs and CTMCs.
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PRISM also supports a fourth: <em>probabilistic timed automata</em> (PTAs), which extend MDPs with the ability to model real-time behaviour. This is done in the style of <em>timed automata</em> [<a class='wikilink' href='../Main/References.html#AD94'>AD94</a>], by adding real-valued <em>clocks</em> which increase with time and can be reset. For background material on PTAs, see for example [<a class='wikilink' href='../Main/References.html#NPS13'>NPS13</a>].
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You can also find several example PTA models included in the PRISM distribution. Look in the <code>examples/pta</code> directory.
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</p>
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<p class='vspace'>Before describing how PTA features are incorporated into the PRISM modelling language, we give a simple example. Here is a small PTA:
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</p>
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<div class='vspace'></div><div class="img"><img src='../uploads/pta.png' alt='' title='' /></div>
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<p class='vspace'>and here is a corresponding PRISM model:
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<div class='vspace'></div>
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<div class='sourceblock ' id='sourceblock1'>
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<div class='sourceblocktext'><div class="prism"><span class="prismkeyword">pta</span><br/>
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<br/>
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<span class="prismkeyword">module</span> <span class="prismident">M</span><br/>
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<br/>
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<span class="prismident">s</span> : [<span class="prismnum">0</span>..<span class="prismnum">2</span>] <span class="prismkeyword">init</span> <span class="prismnum">0</span>;<br/>
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<span class="prismident">x</span> : <span class="prismkeyword">clock</span>;<br/>
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<br/>
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<span class="prismkeyword">invariant</span><br/>
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(<span class="prismident">s</span>=<span class="prismnum">0</span> => <span class="prismident">x</span><=<span class="prismnum">2</span>) &<br/>
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(<span class="prismident">s</span>=<span class="prismnum">2</span> => <span class="prismident">x</span><=<span class="prismnum">3</span>)<br/>
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<span class="prismkeyword">endinvariant</span><br/>
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<br/>
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[<span class="prismident">send</span>] <span class="prismident">s</span>=<span class="prismnum">0</span> & <span class="prismident">x</span>>=<span class="prismnum">1</span> -> <span class="prismnum">0.9</span>:(<span class="prismident">s</span>'=<span class="prismnum">1</span>)&(<span class="prismident">x</span>'=<span class="prismnum">0</span>) + <span class="prismnum">0.1</span>:(<span class="prismident">s</span>'=<span class="prismnum">2</span>)&(<span class="prismident">x</span>'=<span class="prismnum">0</span>);<br/>
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[<span class="prismident">retry</span>] <span class="prismident">s</span>=<span class="prismnum">2</span> & <span class="prismident">x</span>>=<span class="prismnum">2</span> -> <span class="prismnum">0.95</span>:(<span class="prismident">s</span>'=<span class="prismnum">1</span>) + <span class="prismnum">0.05</span>:(<span class="prismident">s</span>'=<span class="prismnum">2</span>)&(<span class="prismident">x</span>'=<span class="prismnum">0</span>);<br/>
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<br/>
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<span class="prismkeyword">endmodule</span><br/>
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</div></div>
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<div class='sourceblocklink'><a href='http://www.prismmodelchecker.org/manual/ThePRISMLanguage/PTAs?action=sourceblock&num=1' type='text/plain'>[$[Get Code]]</a></div>
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</div>
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<p class='vspace'>For modelling PTAs in PRISM, there is a new datatype, <strong><code>clock</code></strong>, used for variables that are clocks. These must be local to a particular module, not global. Other types of PRISM variables can be defined in the usual way. In the example above, we use just a single integer variable <code>s</code> to represent the locations of the PTAs.
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</p>
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<p class='vspace'>In a PTA, transitions can include a <em>guard</em>, which constrains when it can occur based on the current value of clocks, and <em>resets</em>, which specify that a clock's values should be set to a new (integer) value. These are both specified in PRISM commands in the usual way: see, for example, the inclusion of <code>x>=1</code> in the guard for the <code>send</code>-labelled command and the updates of the form <code>(x'=0)</code> which reset the clock <code>x</code> to 0.
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</p>
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<p class='vspace'>The other new addition is an <code>invariant</code> construct, which is used to specify an expression describing the clock <em>invariants</em> for each PRISM module. These impose restrictions on the allowable values of clock variables, depending on the values of the other non-clock variables. The <code>invariant</code> construct should appear between the variable declarations and the commands of the module. Often, clock invariants are described separately for each PTA location; hence, the invariant will often take the form of a conjunction of implications, as in the example model above, but more general expressions are also permitted. In the example, the clock <code>x</code> must satisfy <code>x<=2</code> or <code>x<=3</code> when local variables <code>s</code> is 0 or 2, respectively. If <code>s</code> is 1, there is no restriction (since the invariant is effectively <code>true</code> in this case).
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</p>
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<p class='vspace'>Expressions that include reference to clocks, whether in guards or invariants, must satisfy certain conditions to facilitate model checking. In particular, references to clocks must appear as conjunctions of <em>simple clock constraints</em>, i.e. conjunctions of expressions of the form <code>x~c</code> or <code>x~y</code> where <code>x</code> and <code>y</code> are clocks, <code>c</code> is an integer-valued expression and <code>~</code> is one of <code><</code>, <code><=</code>, <code>>=</code>, <code>></code>, <code>=</code>).
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</p>
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<p class='vspace'>There are also some additional restrictions imposed on PTA models that are dependent on which of the PTA model checking <a class='wikilink' href='../ConfiguringPRISM/ComputationEngines.html#pta'>engines</a> is in use.
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</p>
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<p class='vspace'>For the <strong>stochastic games</strong> and <strong>backwards reachability</strong> engines:
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</p>
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<div class='vspace'></div><ul><li>Modules cannot read the local variables of other modules and global variables are not permitted.
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<div class='vspace'></div></li><li>The model must also have a single initial state (i.e. the <code>init...endinit</code> construct is not permitted).
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</li></ul><p class='vspace'>For the <strong>digital clocks</strong> engine:
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</p>
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<div class='vspace'></div><ul><li>Clock constraints cannot use strict comparison operators, e.g. <code>x<=5</code> is allowed, but <code>x<5</code> is not.
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<div class='vspace'></div></li><li>Diagonal clock constraints are not allowed, i.e. those containing references to two clocks, such as <code>x<=y</code>.
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</li></ul><p class='vspace'>Finally, PRISM makes several assumptions about PTAs, regardless of the engine used.
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</p>
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<div class='vspace'></div><ul><li>Firstly PTAs should not exhibit <em>timelocks</em>, i.e. the possibility of reaching a state where no transitions are possible and time cannot elapse beyond a certain point (due to invariant conditions). PRISM checks for timelocks and reports an error if one is found.
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<div class='vspace'></div></li><li>Secondly, PTAs should be <em>well-formed</em> and <em>non-zeno</em> (see e.g. [<a class='wikilink' href='../Main/References.html#KNSW07'>KNSW07</a>] for details). Currently, PRISM does not check automatically that these assumptions are satisfied.
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<h3><a class='wikilink' href='../Main/Main.html'>PRISM Manual</a></h3>
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<p><strong><a class='wikilink' href='Main.html'>The PRISM Language</a></strong>
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</p><ul><li><a class='wikilink' href='Main.html'>Introduction</a>
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