--- a/src/HOLCF/IOA/meta_theory/Automata.thy Wed Jul 16 11:34:42 1997 +0200
+++ b/src/HOLCF/IOA/meta_theory/Automata.thy Thu Jul 17 12:43:32 1997 +0200
@@ -13,17 +13,24 @@
types
('a,'s)transition = "'s * 'a * 's"
- ('a,'s)ioa = "'a signature * 's set * ('a,'s)transition set"
+ ('a,'s)ioa = "'a signature * 's set * ('a,'s)transition set *
+ (('a set) set) * (('a set) set)"
consts
(* IO automata *)
- state_trans ::"['a signature, ('a,'s)transition set] => bool"
- input_enabled::"['a signature, ('a,'s)transition set] => bool"
- asig_of ::"('a,'s)ioa => 'a signature"
- starts_of ::"('a,'s)ioa => 's set"
- trans_of ::"('a,'s)ioa => ('a,'s)transition set"
- IOA ::"('a,'s)ioa => bool"
+
+ asig_of ::"('a,'s)ioa => 'a signature"
+ starts_of ::"('a,'s)ioa => 's set"
+ trans_of ::"('a,'s)ioa => ('a,'s)transition set"
+ wfair_of ::"('a,'s)ioa => ('a set) set"
+ sfair_of ::"('a,'s)ioa => ('a set) set"
+
+ is_asig_of ::"('a,'s)ioa => bool"
+ is_starts_of ::"('a,'s)ioa => bool"
+ is_trans_of ::"('a,'s)ioa => bool"
+ input_enabled ::"('a,'s)ioa => bool"
+ IOA ::"('a,'s)ioa => bool"
(* reachability and invariants *)
reachable :: "('a,'s)ioa => 's set"
@@ -31,15 +38,18 @@
(* binary composition of action signatures and automata *)
asig_comp ::"['a signature, 'a signature] => 'a signature"
- compatible ::"[('a,'s)ioa, ('a,'t)ioa] => bool"
+ compatible ::"[('a,'s)ioa, ('a,'t)ioa] => bool"
"||" ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr 10)
- (* hiding *)
+ (* hiding and restricting *)
+ hide_asig :: "['a signature, 'a set] => 'a signature"
+ hide :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
restrict_asig :: "['a signature, 'a set] => 'a signature"
restrict :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
(* renaming *)
- rename:: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
+ rename_set :: "'a set => ('c => 'a option) => 'c set"
+ rename :: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
syntax
@@ -83,24 +93,34 @@
(* --------------------------------- IOA ---------------------------------*)
-state_trans_def
- "state_trans asig R ==
- (!triple. triple:R --> fst(snd(triple)):actions(asig))"
-
-input_enabled_def
- "input_enabled asig R ==
- (!a. (a:inputs(asig)) --> (!s1. ? s2. (s1,a,s2):R))"
asig_of_def "asig_of == fst"
starts_of_def "starts_of == (fst o snd)"
-trans_of_def "trans_of == (snd o snd)"
+trans_of_def "trans_of == (fst o snd o snd)"
+wfair_of_def "wfair_of == (fst o snd o snd o snd)"
+sfair_of_def "sfair_of == (snd o snd o snd o snd)"
+
+is_asig_of_def
+ "is_asig_of A == is_asig (asig_of A)"
+
+is_starts_of_def
+ "is_starts_of A == (~ starts_of A = {})"
+
+is_trans_of_def
+ "is_trans_of A ==
+ (!triple. triple:(trans_of A) --> fst(snd(triple)):actions(asig_of A))"
+
+input_enabled_def
+ "input_enabled A ==
+ (!a. (a:inputs(asig_of A)) --> (!s1. ? s2. (s1,a,s2):(trans_of A)))"
+
ioa_def
- "IOA(ioa) == (is_asig(asig_of(ioa)) &
- (~ starts_of(ioa) = {}) &
- state_trans (asig_of ioa) (trans_of ioa) &
- input_enabled (asig_of ioa) (trans_of ioa))"
+ "IOA A == (is_asig_of A &
+ is_starts_of A &
+ is_trans_of A &
+ input_enabled A)"
invariant_def "invariant A P == (!s. reachable A s --> P(s))"
@@ -121,44 +141,73 @@
(outputs(a1) Un outputs(a2)),
(internals(a1) Un internals(a2))))"
-
par_def
- "(ioa1 || ioa2) ==
- (asig_comp (asig_of ioa1) (asig_of ioa2),
- {pr. fst(pr):starts_of(ioa1) & snd(pr):starts_of(ioa2)},
+ "(A || B) ==
+ (asig_comp (asig_of A) (asig_of B),
+ {pr. fst(pr):starts_of(A) & snd(pr):starts_of(B)},
{tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))
- in (a:act ioa1 | a:act ioa2) &
- (if a:act ioa1 then
- (fst(s),a,fst(t)):trans_of(ioa1)
+ in (a:act A | a:act B) &
+ (if a:act A then
+ (fst(s),a,fst(t)):trans_of(A)
else fst(t) = fst(s))
&
- (if a:act ioa2 then
- (snd(s),a,snd(t)):trans_of(ioa2)
- else snd(t) = snd(s))})"
+ (if a:act B then
+ (snd(s),a,snd(t)):trans_of(B)
+ else snd(t) = snd(s))},
+ wfair_of A Un wfair_of B,
+ sfair_of A Un sfair_of B)"
+
(* ------------------------ hiding -------------------------------------------- *)
restrict_asig_def
"restrict_asig asig actns ==
- (inputs(asig) Int actns, outputs(asig) Int actns,
+ (inputs(asig) Int actns,
+ outputs(asig) Int actns,
internals(asig) Un (externals(asig) - actns))"
+(* Notice that for wfair_of and sfair_of nothing has to be changed, as
+ changes from the outputs to the internals does not touch the locals as
+ a whole, which is of importance for fairness only *)
restrict_def
- "restrict ioa actns ==
- (restrict_asig (asig_of ioa) actns, starts_of(ioa), trans_of(ioa))"
+ "restrict A actns ==
+ (restrict_asig (asig_of A) actns,
+ starts_of A,
+ trans_of A,
+ wfair_of A,
+ sfair_of A)"
+
+hide_asig_def
+ "hide_asig asig actns ==
+ (inputs(asig) - actns,
+ outputs(asig) - actns,
+ internals(asig) Un actns)"
+
+hide_def
+ "hide A actns ==
+ (hide_asig (asig_of A) actns,
+ starts_of A,
+ trans_of A,
+ wfair_of A,
+ sfair_of A)"
(* ------------------------- renaming ------------------------------------------- *)
+rename_set_def
+ "rename_set set ren == {b. ? x. Some x = ren b & x : set}"
+
rename_def
"rename ioa ren ==
- (({b. ? x. Some(x)= ren(b) & x : inp ioa},
- {b. ? x. Some(x)= ren(b) & x : out ioa},
- {b. ? x. Some(x)= ren(b) & x : int ioa}),
- starts_of(ioa) ,
+ ((rename_set (inp ioa) ren,
+ rename_set (out ioa) ren,
+ rename_set (int ioa) ren),
+ starts_of ioa,
{tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))
in
- ? x. Some(x) = ren(a) & (s,x,t):trans_of(ioa)})"
+ ? x. Some(x) = ren(a) & (s,x,t):trans_of ioa},
+ {rename_set s ren | s. s: wfair_of ioa},
+ {rename_set s ren | s. s: sfair_of ioa})"
end