Tutorial Modeling Infinite Datatypes: Difference between revisions

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__NOTOC__
__NOTOC__
This tutorial describes how to model (and how not to model) infinite datatypes so that they can be animated with ProB. We illustrate this using a Stack datatype.
This tutorial describes how to model (and how not to model) infinite datatypes so that they can be animated with ProB.  
(You may also want to look at the manual page on [[Recursively_Defined_Functions |recursive functions]].)
We illustrate this by modeling a Stack datatype.


This does not work
Unfortunately, the following B machine does not work with ProB:
  MACHINE StackAxioms1
  MACHINE StackAxioms1
  SETS Stack
  SETS Stack
Line 13: Line 15:
  END
  END


To do: explain why: infinite Stack would be required; ProB does not support infinite deferred sets. We need to do a more constructive definition and use the infinite set ProB knows about: INTEGER.
The problem with the above model is that the <tt>PROPERTIES</tt> can only be satisfied if <tt>Stack</tt> is infinite. However, ProB does not presently support infinite deferred sets.  
We need to do a more constructive definition and use the infinite set ProB knows about: <tt>INTEGER</tt>. This is done in the following machine, which is included in the <tt>examples/Tutorial</tt> directory of the [[Download|ProB distribution]].


  MACHINE StackConstructive
  MACHINE StackConstructive
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   push = %(x,s).(x:RANGE & s:Stack | s \/ {card(s)+1|->x}) &
   push = %(x,s).(x:RANGE & s:Stack | s \/ {card(s)+1|->x}) &
   pop: Stack <-> Stack &
   pop: Stack <-> Stack &
   pop = %s.(s:Stack| {card(s)} <<| s) &
   pop = %s.(s:Stack\ {empty} | {card(s)} <<| s) &
   tops: Stack <-> RANGE &
   tops: Stack <-> RANGE &
   tops = %s.(s:Stack| s(card(s)))
   tops = %s.(s:Stack\ {empty} | s(card(s)))
  ASSERTIONS
  ASSERTIONS
  /* the assertions cannot be checked by ProB, they will trigger
  /* the assertions can now be checked by ProB, they used to trigger
    the expansion of the infinite functions above */
  the expansion of the infinite functions above */
   tops: Stack \ {empty} --> RANGE;
   tops: Stack \ {empty} --> RANGE;
   push: (RANGE*Stack) --> Stack \ {empty};
   push: (RANGE*Stack) --> Stack \ {empty};
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   t <-- Top = PRE cur /= empty THEN t := tops(cur) END
   t <-- Top = PRE cur /= empty THEN t := tops(cur) END
  END
  END
As the screenshot illustrates, this model can now be animated with ProB. Observe how the functions push, pop and tops are kept symbolic in the State View.
[[File:StackConstructiveProB.png|500px]]

Latest revision as of 13:08, 2 November 2023

This tutorial describes how to model (and how not to model) infinite datatypes so that they can be animated with ProB. (You may also want to look at the manual page on recursive functions.) We illustrate this by modeling a Stack datatype.

Unfortunately, the following B machine does not work with ProB:

MACHINE StackAxioms1
SETS Stack
CONSTANTS pop, push,empty
PROPERTIES
 empty:Stack &
 pop : Stack \ {empty} --> Stack &
!s,x . (s : Stack & x : NAT => pop(push(s |-> x)) = s) 
 & push: Stack*NAT --> Stack \ {empty}
END

The problem with the above model is that the PROPERTIES can only be satisfied if Stack is infinite. However, ProB does not presently support infinite deferred sets. We need to do a more constructive definition and use the infinite set ProB knows about: INTEGER. This is done in the following machine, which is included in the examples/Tutorial directory of the ProB distribution.

MACHINE StackConstructive
/* A machine that shows how to model a stack of type RANGE so that it can
   be animated and validated with ProB */
/* We could have used the sequence operations instead;
   our intention was also to show how you can model this in Event-B */
DEFINITIONS Stack == (INTEGER <-> RANGE)
SETS RANGE
CONSTANTS empty, push, pop, tops
PROPERTIES
  empty : Stack & empty = {} &
  push : (RANGE * Stack) <-> Stack &
  push = %(x,s).(x:RANGE & s:Stack | s \/ {card(s)+1|->x}) &
  pop: Stack <-> Stack &
  pop = %s.(s:Stack\ {empty} | {card(s)} <<| s) &
  tops: Stack <-> RANGE &
  tops = %s.(s:Stack\ {empty} | s(card(s)))
ASSERTIONS
/* the assertions can now be checked by ProB, they used to trigger
  the expansion of the infinite functions above */
  tops: Stack \ {empty} --> RANGE;
  push: (RANGE*Stack) --> Stack \ {empty};
  pop: Stack \ {empty} --> Stack
VARIABLES cur
INVARIANT
 cur: Stack 
 & cur : seq(RANGE) /* a slightly stronger invariant */
INITIALISATION cur := empty
OPERATIONS
  Push(yy) = PRE yy:RANGE THEN cur:= push(yy,cur) END;
  Pop = PRE cur /= empty THEN cur := pop(cur) END;
  t <-- Top = PRE cur /= empty THEN t := tops(cur) END
END

As the screenshot illustrates, this model can now be animated with ProB. Observe how the functions push, pop and tops are kept symbolic in the State View.

StackConstructiveProB.png