Tutorial Modeling Infinite Datatypes

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.