(* Title: HOL/IOA/meta_theory/IOA.thy
ID: $Id$
Author: Tobias Nipkow & Konrad Slind
Copyright 1994 TU Muenchen
The I/O automata of Lynch and Tuttle.
*)
IOA = Asig +
types
'a seq = "nat => 'a"
'a oseq = "nat => 'a option"
('a,'b)execution = "'a oseq * 'b seq"
('a,'s)transition = "('s * 'a * 's)"
('a,'s)ioa = "'a signature * 's set * ('a,'s)transition set"
consts
(* IO automata *)
state_trans::"['action signature, ('action,'state)transition set] => bool"
asig_of ::"('action,'state)ioa => 'action signature"
starts_of ::"('action,'state)ioa => 'state set"
trans_of ::"('action,'state)ioa => ('action,'state)transition set"
IOA ::"('action,'state)ioa => bool"
(* Executions, schedules, and traces *)
is_execution_fragment,
has_execution ::"[('action,'state)ioa, ('action,'state)execution] => bool"
executions :: "('action,'state)ioa => ('action,'state)execution set"
mk_trace :: "[('action,'state)ioa, 'action oseq] => 'action oseq"
reachable :: "[('action,'state)ioa, 'state] => bool"
invariant :: "[('action,'state)ioa, 'state=>bool] => bool"
has_trace :: "[('action,'state)ioa, 'action oseq] => bool"
traces :: "('action,'state)ioa => 'action oseq set"
NF :: "'a oseq => 'a oseq"
(* Composition of action signatures and automata *)
compatible_asigs ::"('a => 'action signature) => bool"
asig_composition ::"('a => 'action signature) => 'action signature"
compatible_ioas ::"('a => ('action,'state)ioa) => bool"
ioa_composition ::"('a => ('action, 'state)ioa) =>('action,'a => 'state)ioa"
(* binary composition of action signatures and automata *)
compat_asigs ::"['action signature, 'action signature] => bool"
asig_comp ::"['action signature, 'action signature] => 'action signature"
compat_ioas ::"[('action,'s)ioa, ('action,'t)ioa] => bool"
"||" ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa" (infixr 10)
(* Filtering and hiding *)
filter_oseq :: "('a => bool) => 'a oseq => 'a oseq"
restrict_asig :: "['a signature, 'a set] => 'a signature"
restrict :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
(* Notions of correctness *)
ioa_implements :: "[('action,'state1)ioa, ('action,'state2)ioa] => bool"
(* Instantiation of abstract IOA by concrete actions *)
rename:: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
defs
state_trans_def
"state_trans asig R == \
\ (!triple. triple:R --> fst(snd(triple)):actions(asig)) & \
\ (!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)"
ioa_def
"IOA(ioa) == (is_asig(asig_of(ioa)) & \
\ (~ starts_of(ioa) = {}) & \
\ state_trans (asig_of ioa) (trans_of ioa))"
(* An execution fragment is modelled with a pair of sequences:
* the first is the action options, the second the state sequence.
* Finite executions have None actions from some point on.
*******)
is_execution_fragment_def
"is_execution_fragment A ex == \
\ let act = fst(ex); state = snd(ex) \
\ in !n a. (act(n)=None --> state(Suc(n)) = state(n)) & \
\ (act(n)=Some(a) --> (state(n),a,state(Suc(n))):trans_of(A))"
executions_def
"executions(ioa) == {e. snd e 0:starts_of(ioa) & \
\ is_execution_fragment ioa e}"
reachable_def
"reachable ioa s == (? ex:executions(ioa). ? n. (snd ex n) = s)"
invariant_def "invariant A P == (!s. reachable A s --> P(s))"
(* Restrict the trace to those members of the set s *)
filter_oseq_def
"filter_oseq p s == \
\ (%i.case s(i) \
\ of None => None \
\ | Some(x) => if p x then Some x else None)"
mk_trace_def
"mk_trace(ioa) == filter_oseq(%a.a:externals(asig_of(ioa)))"
(* Does an ioa have an execution with the given trace *)
has_trace_def
"has_trace ioa b == \
\ (? ex:executions(ioa). b = mk_trace ioa (fst ex))"
normal_form_def
"NF(tr) == @nf. ? f. mono(f) & (!i. nf(i)=tr(f(i))) & \
\ (!j. j ~: range(f) --> nf(j)= None) & \
\ (!i. nf(i)=None --> (nf (Suc i)) = None) "
(* All the traces of an ioa *)
traces_def
"traces(ioa) == {trace. ? tr. trace=NF(tr) & has_trace ioa tr}"
(*
traces_def
"traces(ioa) == {tr. has_trace ioa tr}"
*)
compat_asigs_def
"compat_asigs a1 a2 == \
\ (((outputs(a1) Int outputs(a2)) = {}) & \
\ ((internals(a1) Int actions(a2)) = {}) & \
\ ((internals(a2) Int actions(a1)) = {}))"
compat_ioas_def
"compat_ioas ioa1 ioa2 == compat_asigs (asig_of(ioa1)) (asig_of(ioa2))"
asig_comp_def
"asig_comp a1 a2 == \
\ (((inputs(a1) Un inputs(a2)) - (outputs(a1) Un outputs(a2)), \
\ (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)}, \
\ {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr)) \
\ in (a:actions(asig_of(ioa1)) | a:actions(asig_of(ioa2))) & \
\ (if a:actions(asig_of(ioa1)) then \
\ (fst(s),a,fst(t)):trans_of(ioa1) \
\ else fst(t) = fst(s)) \
\ & \
\ (if a:actions(asig_of(ioa2)) then \
\ (snd(s),a,snd(t)):trans_of(ioa2) \
\ else snd(t) = snd(s))})"
restrict_asig_def
"restrict_asig asig actns == \
\ (inputs(asig) Int actns, outputs(asig) Int actns, \
\ internals(asig) Un (externals(asig) - actns))"
restrict_def
"restrict ioa actns == \
\ (restrict_asig (asig_of ioa) actns, starts_of(ioa), trans_of(ioa))"
ioa_implements_def
"ioa_implements ioa1 ioa2 == \
\ ((inputs(asig_of(ioa1)) = inputs(asig_of(ioa2))) & \
\ (outputs(asig_of(ioa1)) = outputs(asig_of(ioa2))) & \
\ traces(ioa1) <= traces(ioa2))"
rename_def
"rename ioa ren == \
\ (({b. ? x. Some(x)= ren(b) & x : inputs(asig_of(ioa))}, \
\ {b. ? x. Some(x)= ren(b) & x : outputs(asig_of(ioa))}, \
\ {b. ? x. Some(x)= ren(b) & x : internals(asig_of(ioa))}), \
\ 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)})"
end