(* Specification of the following loop back device
g
--------------------
| ------- |
x | | | | y
------|---->| |------| ----->
| z | f | z |
| -->| |--- |
| | | | | |
| | ------- | |
| | | |
| <-------------- |
| |
--------------------
First step: Notation in Agent Network Description Language (ANDL)
-----------------------------------------------------------------
agent f
input channel i1:'b i2: ('b,'c) tc
output channel o1:'c o2: ('b,'c) tc
is
Rf(i1,i2,o1,o2) (left open in the example)
end f
agent g
input channel x:'b
output channel y:'c
is network
<y,z> = f`<x,z>
end network
end g
Remark: the type of the feedback depends at most on the types of the input and
output of g. (No type miracles inside g)
Second step: Translation of ANDL specification to HOLCF Specification
---------------------------------------------------------------------
Specification of agent f ist translated to predicate is_f
is_f :: ('b stream * ('b,'c) tc stream ->
'c stream * ('b,'c) tc stream) => bool
is_f f = !i1 i2 o1 o2.
f`<i1,i2> = <o1,o2> --> Rf(i1,i2,o1,o2)
Specification of agent g is translated to predicate is_g which uses
predicate is_net_g
is_net_g :: ('b stream * ('b,'c) tc stream -> 'c stream * ('b,'c) tc stream) =>
'b stream => 'c stream => bool
is_net_g f x y =
? z. <y,z> = f`<x,z> &
!oy hz. <oy,hz> = f`<x,hz> --> z << hz
is_g :: ('b stream -> 'c stream) => bool
is_g g = ? f. is_f f & (!x y. g`x = y --> is_net_g f x y
Third step: (show conservativity)
-----------
Suppose we have a model for the theory TH1 which contains the axiom
? f. is_f f
In this case there is also a model for the theory TH2 that enriches TH1 by
axiom
? g. is_g g
The result is proved by showing that there is a definitional extension
that extends TH1 by a definition of g.
We define:
def_g g =
(? f. is_f f &
g = (LAM x. cfst`(f`<x,fix`(LAM k.csnd`(f`<x,k>))>)) )
Now we prove:
(? f. is_f f ) --> (? g. is_g g)
using the theorems
loopback_eq) def_g = is_g (real work)
L1) (? f. is_f f ) --> (? g. def_g g) (trivial)
*)
Focus_ex = Stream +
types tc 2
arities tc:: (pcpo,pcpo)pcpo
consts
is_f ::
"('b stream * ('b,'c) tc stream -> 'c stream * ('b,'c) tc stream) => bool"
is_net_g :: "('b stream *('b,'c) tc stream -> 'c stream * ('b,'c) tc stream) =>
'b stream => 'c stream => bool"
is_g :: "('b stream -> 'c stream) => bool"
def_g :: "('b stream -> 'c stream) => bool"
Rf ::
"('b stream * ('b,'c) tc stream * 'c stream * ('b,'c) tc stream) => bool"
defs
is_f "is_f f == (!i1 i2 o1 o2.
f`<i1,i2> = <o1,o2> --> Rf(i1,i2,o1,o2))"
is_net_g "is_net_g f x y == (? z.
<y,z> = f`<x,z> &
(!oy hz. <oy,hz> = f`<x,hz> --> z << hz))"
is_g "is_g g == (? f.
is_f f &
(!x y. g`x = y --> is_net_g f x y))"
def_g "def_g g == (? f.
is_f f &
g = (LAM x. cfst`(f`<x,fix`(LAM k. csnd`(f`<x,k>))>)))"
end