include "re/re.ma".
include "basics/lists/listb.ma".
+(*
+Moves
+
+We now define the move operation, that corresponds to the advancement of the
+state in response to the processing of an input character a. The intuition is
+clear: we have to look at points inside $e$ preceding the given character a,
+let the point traverse the character, and broadcast it. All other points must
+be removed.
+
+We can give a particularly elegant definition in terms of the
+lifted operators of the previous section:
+*)
+
let rec move (S: DeqSet) (x:S) (E: pitem S) on E : pre S ≝
match E with
- [ pz ⇒ 〈 `∅, false 〉
- | pe ⇒ 〈 ϵ, false 〉
- | ps y ⇒ 〈 `y, false 〉
- | pp y ⇒ 〈 `y, x == y 〉
+ [ pz ⇒ 〈 pz ?, false 〉
+ | pe ⇒ 〈 pe ? , false 〉
+ | ps y ⇒ 〈 ps ? y, false 〉
+ | pp y ⇒ 〈 ps ? y, x == y 〉
| po e1 e2 ⇒ (move ? x e1) ⊕ (move ? x e2)
| pc e1 e2 ⇒ (move ? x e1) ⊙ (move ? x e2)
| pk e ⇒ (move ? x e)^⊛ ].
[>(\P H) % [* // #bot @False_ind //| #H1 destruct /2/]
|% [@False_ind |#H1 cases (\Pf H) #H2 @H2 destruct //]
]
- |#i1 #i2 #HI1 #HI2 #w >move_cat
- @iff_trans[|@sem_odot] >same_kernel >sem_cat_w
- @iff_trans[||@(iff_or_l … (HI2 w))] @iff_or_r
+ |#i1 #i2 #HI1 #HI2 #w
+ (* lhs = w∈\sem{move S a (i1·i2)} *)
+ >move_cat
+ (* lhs = w∈\sem{move S a i1}⊙\sem{move S a i2} *)
+ @iff_trans[|@sem_odot] >same_kernel
+ (* lhs = w∈\sem{move S a i1}·\sem{|i2|} ∨ a∈\sem{move S a i2} *)
+ (* now we work on the rhs, that is
+ rhs = a::w1∈\sem{i1·i2} *)
+ >sem_cat_w
+ (* rhs = a::w1∈\sem{i1}\sem{|i2|} ∨ a::w∈\sem{i2} *)
+ @iff_trans[||@(iff_or_l … (HI2 w))]
+ (* rhs = a::w1∈\sem{i1}\sem{|i2|} ∨ w∈\sem{move S a i2} *)
+ @iff_or_r
+ (* we are left to prove that
+ w∈\sem{move S a i1}·\sem{|i2|} ↔ a::w∈\sem{i1}\sem{|i2|}
+ we use deriv_middot on the rhs *)
@iff_trans[||@iff_sym @deriv_middot //]
+ (* w∈\sem{move S a i1}·\sem{|i2|} ↔ w∈(deriv S \sem{i1} a) · \sem{|i2|} *)
@cat_ext_l @HI1
|#i1 #i2 #HI1 #HI2 #w >(sem_plus S i1 i2) >move_plus >sem_plus_w
@iff_trans[|@sem_oplus]
]
qed.
-notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
+notation > "x ↦* E" non associative with precedence 65 for @{moves ? $x $E}.
let rec moves (S : DeqSet) w e on w : pre S ≝
match w with
[ nil ⇒ e
]
qed.
-(* lemma not_true_to_false: ∀b.b≠true → b =false.
-#b * cases b // #H @False_ind /2/
-qed. *)
-
(************************ pit state ***************************)
definition pit_pre ≝ λS.λi.〈blank S (|i|), false〉.
(* bisimulation *)
definition cofinal ≝ λS.λp:(pre S)×(pre S).
\snd (\fst p) = \snd (\snd p).
-
+
+(* As a corollary of decidable_sem, we have that two expressions
+e1 and e2 are equivalent iff for any word w the states reachable
+through w are cofinal. *)
+
theorem equiv_sem: ∀S:DeqSet.∀e1,e2:pre S.
- \sem{e1} =1 \sem{e2} ↔ ∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
+ \sem{e1} ≐ \sem{e2} ↔ ∀w.cofinal ? 〈moves ? w e1,moves ? w e2〉.
#S #e1 #e2 %
[#same_sem #w
cut (∀b1,b2. iff (b1 = true) (b2 = true) → (b1 = b2))
|#H #w1 @iff_trans [||@decidable_sem] <H @iff_sym @decidable_sem]
qed.
+(* This does not directly imply decidability: we have no bound over the
+length of w; moreover, so far, we made no assumption over the cardinality
+of S. Instead of requiring S to be finite, we may restrict the analysis
+to characters occurring in the given pres. *)
+
definition occ ≝ λS.λe1,e2:pre S.
unique_append ? (occur S (|\fst e1|)) (occur S (|\fst e2|)).
//
qed.
+(* The following is a stronger version of equiv_sem, relative to characters
+occurring the given regular expressions. *)
+
lemma equiv_sem_occ: ∀S.∀e1,e2:pre S.
(∀w.(sublist S w (occ S e1 e2))→ cofinal ? 〈moves ? w e1,moves ? w e2〉)
-→ \sem{e1}=1\sem{e2}.
+→ \sem{e1}≐\sem{e2}.
#S #e1 #e2 #H @(proj2 … (equiv_sem …)) @occ_enough #w @H
qed.
+(*
+We say that a list of pairs of pres is a bisimulation if it is closed
+w.r.t. moves, and all its members are cofinal.
+*)
+
+(* the sons of p w.r.t a list of symbols l are all states reachable from p
+with a move in l *)
+
definition sons ≝ λS:DeqSet.λl:list S.λp:(pre S)×(pre S).
map ?? (λa.〈move S a (\fst (\fst p)),move S a (\fst (\snd p))〉) l.
definition is_bisim ≝ λS:DeqSet.λl:list ?.λalpha:list S.
∀p:(pre S)×(pre S). memb ? p l = true → cofinal ? p ∧ (sublist ? (sons ? alpha p) l).
+(* Using lemma equiv_sem_occ it is easy to prove the following result: *)
+
lemma bisim_to_sem: ∀S:DeqSet.∀l:list ?.∀e1,e2: pre S.
- is_bisim S l (occ S e1 e2) → memb ? 〈e1,e2〉 l = true → \sem{e1}=1\sem{e2}.
+ is_bisim S l (occ S e1 e2) → memb ? 〈e1,e2〉 l = true → \sem{e1}≐\sem{e2}.
#S #l #e1 #e2 #Hbisim #Hmemb @equiv_sem_occ
#w #Hsub @(proj1 … (Hbisim 〈moves S w e1,moves S w e2〉 ?))
lapply Hsub @(list_elim_left … w) [//]
]
qed.
-(* the algorithm *)
+(* This is already an interesting result: checking if l is a bisimulation
+is decidable, hence we could generate l with some untrusted piece of code
+and then run a (boolean version of) is_bisim to check that it is actually
+a bisimulation.
+However, in order to prove that equivalence of regular expressions
+is decidable we must prove that we can always effectively build such a list
+(or find a counterexample).
+The idea is that the list we are interested in is just the set of
+all pair of pres reachable from the initial pair via some
+sequence of moves.
+
+The algorithm for computing reachable nodes in graph is a very
+traditional one. We split nodes in two disjoint lists: a list of
+visited nodes and a frontier, composed by all nodes connected
+to a node in visited but not visited already. At each step we select a node
+a from the frontier, compute its sons, add a to the set of
+visited nodes and the (not already visited) sons to the frontier.
+
+Instead of fist computing reachable nodes and then performing the
+bisimilarity test we can directly integrate it in the algorithm:
+the set of visited nodes is closed by construction w.r.t. reachability,
+so we have just to check cofinality for any node we add to visited.
+
+Here is the extremely simple algorithm: *)
+
let rec bisim S l n (frontier,visited: list ?) on n ≝
match n with
[ O ⇒ 〈false,visited〉 (* assert false *)
]
].
#S #l #n cases n // qed.
-
+
+(* The integer n is an upper bound to the number of recursive call,
+equal to the dimension of the graph. It returns a pair composed
+by a boolean and a the set of visited nodes; the boolean is true
+if and only if all visited nodes are cofinal.
+
+The following results explicitly state the behaviour of bisim is the general
+case and in some relevant instances *)
+
lemma bisim_never: ∀S,l.∀frontier,visited: list ?.
bisim S l O frontier visited = 〈false,visited〉.
#frontier #visited >unfold_bisim //
#b cases b normalize //
qed.
+(* In order to prove termination of bisim we must be able to effectively
+enumerate all possible pres: *)
+
let rec pitem_enum S (i:re S) on i ≝
match i with
[ z ⇒ [pz S]
#S #e1 #e2 @(memb_compose … (λi,b.〈i,b〉))
// qed.
+(* We are ready to prove that bisim is correct; we use the invariant
+that at each call of bisim the two lists visited and frontier only contain
+nodes reachable from \langle e_1,e_2\rangle, hence it is absurd to suppose
+to meet a pair which is not cofinal. *)
+
definition all_reachable ≝ λS.λe1,e2:pre S.λl: list ?.
uniqueb ? l = true ∧
∀p. memb ? p l = true →
definition disjoint ≝ λS:DeqSet.λl1,l2.
∀p:S. memb S p l1 = true → memb S p l2 = false.
-lemma bisim_correct: ∀S.∀e1,e2:pre S.\sem{e1}=1\sem{e2} →
+lemma bisim_correct: ∀S.∀e1,e2:pre S.\sem{e1}≐\sem{e2} →
∀l,n.∀frontier,visited:list ((pre S)×(pre S)).
|space_enum S (|\fst e1|) (|\fst e2|)| < n + |visited|→
all_reachable S e1 e2 visited →
definition sub_sons ≝ λS,l,l1,l2.∀x:(pre S) × (pre S).
memb ? x l1 = true → sublist ? (sons ? l x) l2.
+(* For completeness, we use the invariant that all the nodes in visited are cofinal,
+and the sons of visited are either in visited or in the frontier; since
+at the end frontier is empty, visited is hence a bisimulation. *)
+
lemma bisim_complete:
∀S,l,n.∀frontier,visited,visited_res:list ?.
all_true S visited →
(bisim ? sig n [〈e1,e2〉] []).
theorem euqiv_sem : ∀Sig.∀e1,e2:re Sig.
- \fst (equiv ? e1 e2) = true ↔ \sem{e1} =1 \sem{e2}.
+ \fst (equiv ? e1 e2) = true ↔ \sem{e1} ≐ \sem{e2}.
#Sig #re1 #re2 %
[#H @eqP_trans [|@eqP_sym @re_embedding] @eqP_trans [||@re_embedding]
cut (equiv ? re1 re2 = 〈true,\snd (equiv ? re1 re2)〉)
definition exp2 ≝ a·(b·a)^*.
definition exp4 ≝ (b·a)^*.
+definition exp5 ≝ (a·(a·(a·b)^*·b)^*·b)^*.
+
+example
+ moves1: \snd (moves DeqNat [0;1;0] (•(blank ? exp2))) = true.
+normalize //
+qed.
+
+example
+ moves2: \snd (moves DeqNat [0;1;0;0;0] (•(blank ? exp2))) = false.
+normalize // qed.
+
+example
+ moves3: \snd (moves DeqNat [0;0;0;1;0;1;1;0;1;1] (•(blank ? exp5))) = true.
+normalize // qed.
+
+example
+ moves4: \snd (moves DeqNat [0;0;0;1;0;1;1;0;1;1;1;0] (•(blank ? exp5))) = false.
+normalize // qed.
+
definition exp6 ≝ a·(a ·a ·b^* + b^* ).
definition exp7 ≝ a · a^* · b^*.
example ex1 : \fst (equiv ? (exp8+exp9) exp9) = true.
normalize // qed.
+definition exp10 ≝ a·a·a·a·a·a·a·a·a·a·a·a·(a^* ).
+definition exp11 ≝ (a·a·a·a·a + a·a·a·a·a·a·a)^*.
+
+example ex2 : \fst (equiv ? (exp10+exp11) exp11) = false.
+normalize // qed.
+
+definition exp12 ≝
+ (a·a·a·a·a·a·a·a)·(a·a·a·a·a·a·a·a)·(a·a·a·a·a·a·a·a)·(a^* ).
+
+example ex3 : \fst (equiv ? (exp12+exp11) exp11) = true.
+normalize // qed.
+
+let rec raw (n:nat) ≝
+ match n with
+ [ O ⇒ a
+ | S p ⇒ a · (raw p)
+ ].
+
+let rec alln (n:nat) ≝
+ match n with
+ [O ⇒ ϵ
+ |S m ⇒ raw m + alln m
+ ].
+
+definition testA ≝ λx,y,z,b.
+ let e1 ≝ raw x in
+ let e2 ≝ raw y in
+ let e3 ≝ (raw z) · a^* in
+ let e4 ≝ (e1 + e2)^* in
+ \fst (equiv ? (e3+e4) e4) = b.
+
+example ex4 : testA 2 4 7 true.
+normalize // qed.
+example ex5 : testA 3 4 10 false.
+normalize // qed.
+example ex6 : testA 3 4 11 true.
+normalize // qed.
+example ex7 : testA 4 5 18 false.
+normalize // qed.
+example ex8 : testA 4 5 19 true.
+normalize // qed.
+example ex9 : testA 4 6 22 false.
+normalize // qed.
+example ex10 : testA 4 6 23 true.
+normalize // qed.
+
+definition testB ≝ λn,b.
+ \fst (equiv ? ((alln n)·((raw n)^* )) a^* ) = b.
+
+example ex11 : testB 6 true.
+normalize // qed.
+
+example ex12 : testB 8 true.
+normalize // qed.
+
+example ex13 : testB 10 true.
+normalize // qed.
+
+example ex14 : testB 12 true.
+normalize // qed.
+
+example ex15 : testB 14 true.
+normalize // qed.
+
+example ex16 : testB 16 true.
+normalize // qed.
+
+example ex17 : testB 18 true.
+normalize // qed.
projects / helm.git / blobdiff