Predicates, ghost predicates and higher order predicates

Motivation

Predicates play an important role in specifying functions and
procedures. Since predicates are mainly used within assertions (i.e. within
contracts) it is often not necessary that predicates are computable. We often
use predicates to express not computable properties and give them a concise
name instead of repeating a long and clumsy boolean expression over and over
again.

Predicates can be identified with sets. A set associated with a predicate is
the set of all objects which satisfy the predicate.

Basic definitions

A predicate over a type T is a boolean valued function T->BOOLEAN. But the
fact that all functions are potentially partial is disturbing for
predicates. We want that predicates are total, i.e. they can be applied to all
arguments without any restriction.

In order to distinguish between predicates and functions we use the notation

  p: T?

to express that “p” is a predicate over the type T. E.g. we can define a
predicate which says that a specific natural number is at least “5”.

  is_five_or_more(n:NATURAL): BOOLEAN
    ensure
      Result = (5<=n)
    end

The function “is_five_or_more” has the type NATURAL?, i.e. it is a predicate
over natural numbers.

A predicate can have more than one argument. Predicates with more than one
argument can be used to define relations.

  [A,B,C]?     -- The total function [A,B,C]->BOOLEAN which describes
               -- a relation between the objects of type A, B and C

  [A,B]?       -- Binary relation between objects of type A and objects of
               -- type B

  [A,A]?       -- Binary relation between objects of the same type

The function “<=” is a predicate with two arguments of type NATURAL
(i.e. binary relation). It has the type [NATURAL,NATURAL]? and the following
definition.

  <= (a,b:NATURAL): ghost BOOLEAN
    ensure
      Result = some(x:NATURAL) a + x = b
    end

Note that if we have declared “<=” as a ghost predicate. In this case the
specification is sufficient. There is no need to give an actual
implementation.

If we specified “<=” as a computable predicate (i.e. with result type BOOLEAN
instead of “ghost BOOLEAN”), the compiler would expect an implementation to
verify the computability of the predicate.

Higher order predicates

A predicate can have other predicates as arguments. Predicates over predicates
are called higher order predicates. E.g. we can define a first oder predicate

  is_even(n:NATURAL): BOOLEAN
    ensure
      Result = (n mod 2 = 0)
    end

which tells whether a natural number is even. The predicate “is_even” has type
NATURAL?. Instead of writing an explicit function to implement the predicate
we can use an anonymous function/predicate object as well. The predicate object

  (x:NATURAL) -> (n mod 2 = 0)

is equivalent to “is_even”.

The predicate “is_even” has some properties which can be described by higher
order predicates. E.g. we can define the higher order predicate
“is_even_prop_1”.

  is_even_prop_1(p:NATURAL?): ghost BOOLEAN
    ensure
      Result = all(n:NATURAL) p[n] = (n mod 2 = 0)
    end

The predicate “is_even_prop_1” has the type NATURAL??, i.e. it is a predicate
over predicates.

The predicate “is_even” (which is of type NATURAL?) satisfies this predicate
(which is of type NATURAL??), i.e. the assertions

  is_even_prop_1(is_even)

  is_even_prop_1(x->(n mod 2 = 0))

are valid. All predicates which satisfy “is_even_prop_1” are equivalent to
“is_even”. Therefore “is_even_prop_1” can serve as a specification of
“is_even”.

It is easy to find another predicate which is satisfied by “is_even”.

  is_even_prop_2(p:NATURAL?): ghost BOOLEAN
    ensure
      Result = all(n:NATURAL) p[n] = some(m:NATURAL) 2*m = n
    end

This predicate says that “p[n]” is true if and only if there is a natural
number “m” which satisfies “2*m=n”. “is_even_prop_2” is still a full
specification of the predicate “is_even”. I.e. all predicates satisfying
“is_even_spec_2” are equivalent to “is_even”.

In order to get a better feeling about what can be expressed by predicates
which can try to find some inductive properties of “is_even”. We know that the
number “0” is even and that the evenness of a number “n” implies the evenness
of “n+2”. This is expressed by the following predicate

  is_even_prop_3(p:NATURAL?): ghost BOOLEAN
    ensure
      Result = (p[0] and all(n:NATURAL) p[n] => p[n+2])
    end

It is evident that “is_even” satisfies “is_even_prop_3” i.e. the assertion

  is_even_prop_3(is_even)

is valid. But not all predicates satisfying “is_even_prop_3” are equivalent to
“is_even”. A predicate which returns true for all natural number satisfies
“is_even_prop_3” as well i.e. we have the valid assertion

  is_even_prop_3(x->True)

Therefore “is_even_prop_3” cannot be considered as a full specification of
“is_even”.

However we feel that “is_even_prop_3” captures some essentials of
“is_even”. It is just not strong enough to be a full specification.

The module “predicate”

There is a module “predicate” which defines the base class and the basic
functions of predicates. The module “predicate” has an abstract class
PREDICATE[G]. All predicates of type T? inherit from the type PREDICATE[T].

  -- file: predicate.e
  immutable deferred class
    PREDICATE[G]
  end

The basic function is to ask whether an element satisfies a predicate

  feature
    [] (e:G): BOOLEAN
      deferred end
  end

I.e. for a predicate “p:NATURAL?” the call “p[5]” returns whether the number
“5” satisfies the predicate “p”.

Since we can identify a predicate with the set of all values which satisfy the
predicate, all set operations like union, intersection, complement etc. have
their equivalent within the class predicate.

  feature   -- set operations
    or (f,g:CURRENT): CURRENT
        -- Union of the sets described by `f' and `g'.
      ensure
        Result = x->(f[x] or g[x])
      end

    and (f,g:CURRENT): CURRENT
        -- Intersection of the sets described by `f' and `g'.
      ensure
        Result = x->(f[x] and g[x])
      end

    not (f:CURRENT): CURRENT
        -- Complement of the set described by `f'.
      ensure
        Result = x->(not f[x])
      end
  end

Note that the above function are computable functions and not ghost functions.

There are two predicate constants: One which cannot be satisfied by any object
and one which is satisfied by all objects.

  feature    -- Predicate constants
    False: CURRENT         -- overrides default type of "False"!
      ensure
        Result = x->(False:BOOLEAN)  -- explicit type necessary here!
      end

    True: CURRENT
      ensure
        Result = x->(True:BOOLEAN)
      end
  end

In Modern Eiffel all constants (numeric constants, character constants, string
constants and boolean constants) can be used as names for constant functions
(i.e. 0-ary functions). In case of ambiguity the corresponding type has to be
provided.

  False               -- Boolean constant `False'
  False: CHARACTER?   -- Predicate over CHARACTERs which is false for all
                      -- arguments.

  local
    a           := False          -- a has type BOOLEAN
    b: BOOLEAN  := True
    p: NATURAL? := False
    q           := True:STRING?   -- q has type STRING?

    d: BOOLEAN  := False:STRING?  -- ILLEGAL!! Type mismatch!!
  then
    ...
  end

The default type of the constants “True” and “False” is BOOLEAN. Within the
module “predicate” the default type is CURRENT (i.e. PREDICATE[G]). I.e. if we
want to use one of the constants within the module “predicate” with type
BOOLEAN, we have to specify the type explicitely.

Outside of the module “predicate” the constants “True” and “False” have their
default type. An explicit type has to be given in case that the default type
should be overridden.

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