No edit summary |
No edit summary |
||
Line 62: | Line 62: | ||
Currently there are four cases when ProB tries to keep a function such as <tt>f = %x.(PRED|E)</tt> symbolically rather than computing an explicit representation: | Currently there are four cases when ProB tries to keep a function such as <tt>f = %x.(PRED|E)</tt> symbolically rather than computing an explicit representation: | ||
* the domain of the function is obviously infinite; this is the case for predicates such as <tt>x:NATURAL</tt> | * the domain of the function is obviously infinite; this is the case for predicates such as <tt>x:NATURAL</tt>; in version 1.3.7-beta5 or later this has been considerably improved. Now ProB also keeps those lambda abstractions or set comprehensions symbolic where the constraint solver cannot reduce the domain of the parameters to a finite domain. As such, e.g., <tt>{x,y,z| x*x + y*y = z*z}</tt> or <tt>{x,y,z| z:seq(NATURAL) & x^y=z}</tt> are now automatically kept symbolic. | ||
* <tt>f</tt> is declared to be an <tt>ABSTRACT_CONSTANT</tt> and the equation is part of the <tt>PROPERTIES</tt> with <tt>f</tt> on the left. | * <tt>f</tt> is declared to be an <tt>ABSTRACT_CONSTANT</tt> and the equation is part of the <tt>PROPERTIES</tt> with <tt>f</tt> on the left. | ||
* the preference <tt>SYMBOLIC</tt> is set to true (e.g., using a <tt>DEFINITION</tt> <tt>SET_PREF_SYMBOLIC == TRUE</tt>) | * the preference <tt>SYMBOLIC</tt> is set to true (e.g., using a <tt>DEFINITION</tt> <tt>SET_PREF_SYMBOLIC == TRUE</tt>) | ||
* a pragma is used to mark the lambda abstraction as symbolic as follows <tt>f = /*@ symbolic */ %x.(PRED|E)</tt>; this requires ProB version 1.3.5-beta10 or higher. | * a pragma is used to mark the lambda abstraction as symbolic as follows <tt>f = /*@ symbolic */ %x.(PRED|E)</tt>; this requires ProB version 1.3.5-beta10 or higher. | ||
= Recursive Function Definitions in ProB = | = Recursive Function Definitions in ProB = |
Take the following function:
CONSTANTS parity PROPERTIES parity : (NATURAL --> {0,1}) & parity(0) = 0 & !x.(x:NATURAL => parity(x+1) = 1 - parity(x))
Here, ProB will complain that it cannot find a solution for parity. The reason is that parity is a function over an infinite domain, but ProB tries to represent the function as a finite set of maplets.
There are basically four solutions to this problem:
parity : (NAT --> {0,1}) & parity(0) = 0 & !x.(x:NAT & x<MAXINT => parity(x+1) = 1 - parity(x))
parity : (NATURAL --> {0,1}) & parity(0) = 0 & !x.(x:NATURAL1 => parity(x) = 1 - parity(x-1))
parity : INTEGER <-> INTEGER & parity = {0|->0} \/ %x.(x:NATURAL1|1-parity(x-1))
Note, you have to remove the check parity : (NATURAL --> {0,1}), as this will currently cause expansion of the recursive function. We describe this new scheme in more detail below.
parity : (NATURAL --> INTEGER) & parity = %x.(x:NATURAL|x mod 2)
You can experiment with those by using the Eval console of ProB, experimenting for example with the following complete machine. Note, you should use ProB 1.3.5-beta2 or higher.
(You can also type expressions and predicates such as parity = %x.(x:NATURAL|x mod 2) & parity[1..10] = res directly into the online version of the Eval console).
MACHINE InfiniteParityFunction CONSTANTS parity PROPERTIES parity : NATURAL --> INTEGER & parity = %x.(x:NATURAL|x mod 2) VARIABLES c INVARIANT c: NATURAL INITIALISATION c:=0 OPERATIONS Inc = BEGIN c:=c+1 END; r <-- Parity = BEGIN r:= parity(c) END; r <-- ParityImage = BEGIN r:= parity[0..c] END; r <-- ParityHistory = BEGIN r:= (%i.(i:1..c+1|i-1) ; parity) END END
You may also want to look at the tutorial page on modeling infinite datatypes.
Currently there are four cases when ProB tries to keep a function such as f = %x.(PRED|E) symbolically rather than computing an explicit representation:
As of version 1.3.5-beta7 ProB now accepts recursively defined functions. For this:
Here is a full example:
MACHINE Parity ABSTRACT_CONSTANTS parity PROPERTIES parity : INTEGER <-> INTEGER & parity = {0|->0} \/ %x.(x:NATURAL1|1-parity(x-1)) END
With such a recursive function you can:
However, composing it with another function (([1,2] ; parity)) does not work yet. This works for non-recursive infinite functions, as described above. Future versions of ProB will probably allow such constructs for recursive functions as well.
Also, you have to be careful to avoid accidentally expanding these functions. For example, trying to check parity : INTEGER <-> INTEGER or parity : INTEGER +-> INTEGER will cause ProB to try and expand the function. Future versions of ProB may overcome this limitation.
There are the following further restrictions: