4 We shall apply all the previous machinery to the study of regular languages
5 and the constructions of the associated finite automata. *)
7 include "tutorial/chapter6.ma".
9 (* The type re of regular expressions over an alphabet $S$ is the smallest
10 collection of objects generated by the following constructors: *)
12 inductive re (S: DeqSet) : Type[0] ≝
13 z: re S (* empty: ∅ *)
14 | e: re S (* epsilon: ϵ *)
15 | s: S → re S (* symbol: a *)
16 | c: re S → re S → re S (* concatenation: e1 · e2 *)
17 | o: re S → re S → re S (* plus: e1 + e2 *)
18 | k: re S → re S. (* kleene's star: e* *)
20 interpretation "re epsilon" 'epsilon = (e ?).
21 interpretation "re or" 'plus a b = (o ? a b).
22 interpretation "re cat" 'middot a b = (c ? a b).
23 interpretation "re star" 'star a = (k ? a).
25 notation < "a" non associative with precedence 90 for @{ 'ps $a}.
26 notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
27 interpretation "atom" 'ps a = (s ? a).
29 notation "`∅" non associative with precedence 90 for @{ 'empty }.
30 interpretation "empty" 'empty = (z ?).
32 (* The language sem{e} associated with the regular expression e is inductively
33 defined by the following function: *)
35 let rec in_l (S : DeqSet) (r : re S) on r : word S → Prop ≝
40 | c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2)
41 | o r1 r2 ⇒ (in_l ? r1) ∪ (in_l ? r2)
42 | k r1 ⇒ (in_l ? r1) ^*].
44 notation "\sem{term 19 E}" non associative with precedence 75 for @{'in_l $E}.
45 interpretation "in_l" 'in_l E = (in_l ? E).
46 interpretation "in_l mem" 'mem w l = (in_l ? l w).
48 lemma rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*.
53 Pointed Regular expressions
55 We now introduce pointed regular expressions, that are the main tool we shall
56 use for the construction of the automaton.
57 A pointed regular expression is just a regular expression internally labelled
58 with some additional points. Intuitively, points mark the positions inside the
59 regular expression which have been reached after reading some prefix of
60 the input string, or better the positions where the processing of the remaining
61 string has to be started. Each pointed expression for $e$ represents a state of
62 the {\em deterministic} automaton associated with $e$; since we obviously have
63 only a finite number of possible labellings, the number of states of the automaton
66 Pointed regular expressions provide the tool for an algebraic revisitation of
67 McNaughton and Yamada's algorithm for position automata, making the proof of its
68 correctness, that is far from trivial, particularly clear and simple. In particular,
69 pointed expressions offer an appealing alternative to Brzozowski's derivatives,
70 avoiding their weakest point, namely the fact of being forced to quotient derivatives
71 w.r.t. a suitable notion of equivalence in order to get a finite number of states
72 (that is not essential for recognizing strings, but is crucial for comparing regular
75 Our main data structure is the notion of pointed item, that is meant whose purpose
76 is to encode a set of positions inside a regular expression.
77 The idea of formalizing pointers inside a data type by means of a labelled version
78 of the data type itself is probably one of the first, major lessons learned in the
79 formalization of the metatheory of programming languages. For our purposes, it is
80 enough to mark positions preceding individual characters, so we shall have two kinds
81 of characters •a (pp a) and a (ps a) according to the case a is pointed or not. *)
83 inductive pitem (S: DeqSet) : Type[0] ≝
84 pz: pitem S (* empty *)
85 | pe: pitem S (* epsilon *)
86 | ps: S → pitem S (* symbol *)
87 | pp: S → pitem S (* pointed sysmbol *)
88 | pc: pitem S → pitem S → pitem S (* concatenation *)
89 | po: pitem S → pitem S → pitem S (* plus *)
90 | pk: pitem S → pitem S. (* kleene's star *)
92 (* A pointed regular expression (pre) is just a pointed item with an additional
93 boolean, that must be understood as the possibility to have a trailing point at
94 the end of the expression. As we shall see, pointed regular expressions can be
95 understood as states of a DFA, and the boolean indicates if
96 the state is final or not. *)
98 definition pre ≝ λS.pitem S × bool.
100 interpretation "pitem star" 'star a = (pk ? a).
101 interpretation "pitem or" 'plus a b = (po ? a b).
102 interpretation "pitem cat" 'middot a b = (pc ? a b).
103 notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
104 notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
105 interpretation "pitem pp" 'pp a = (pp ? a).
106 interpretation "pitem ps" 'ps a = (ps ? a).
107 interpretation "pitem epsilon" 'epsilon = (pe ?).
108 interpretation "pitem empty" 'empty = (pz ?).
110 (* The carrier $|i|$ of an item i is the regular expression obtained from i by
111 removing all the points. Similarly, the carrier of a pointed regular expression
112 is the carrier of its item. *)
114 let rec forget (S: DeqSet) (l : pitem S) on l: re S ≝
120 | pc E1 E2 ⇒ (forget ? E1) · (forget ? E2)
121 | po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
122 | pk E ⇒ (forget ? E)^* ].
124 (* Already in the library
125 notation "| term 19 C |" with precedence 70 for @{ 'card $C }.
127 interpretation "forget" 'card a = (forget ? a).
129 lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
132 lemma erase_plus : ∀S.∀i1,i2:pitem S.
133 |i1 + i2| = |i1| + |i2|.
136 lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
140 Comparing items and pres
142 Items and pres are very concrete datatypes: they can be effectively compared,
143 and enumerated. In particular, we can define a boolean equality beqitem and a proof
144 beqitem_true that it refects propositional equality, enriching the set (pitem S)
147 let rec beqitem S (i1,i2: pitem S) on i1 ≝
149 [ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
150 | pe ⇒ match i2 with [ pe ⇒ true | _ ⇒ false]
151 | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1==y2 | _ ⇒ false]
152 | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1==y2 | _ ⇒ false]
153 | po i11 i12 ⇒ match i2 with
154 [ po i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
156 | pc i11 i12 ⇒ match i2 with
157 [ pc i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
159 | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ false]
162 lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2).
164 [#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
165 |#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
166 |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
167 [>(\P H) // | @(\b (refl …))]
168 |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
169 [>(\P H) // | @(\b (refl …))]
170 |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
171 normalize #H destruct
172 [cases (true_or_false (beqitem S i11 i21)) #H1
173 [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
174 |>H1 in H; normalize #abs @False_ind /2/
176 |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
178 |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
179 normalize #H destruct
180 [cases (true_or_false (beqitem S i11 i21)) #H1
181 [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
182 |>H1 in H; normalize #abs @False_ind /2/
184 |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
186 |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
187 normalize #H destruct
188 [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
192 definition DeqItem ≝ λS.
193 mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
195 (* We also add a couple of unification hints to allow the type inference system
196 to look at (pitem S) as the carrier of a DeqSet, and at beqitem as if it was the
197 equality function of a DeqSet. *)
199 unification hint 0 ≔ S;
200 X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
201 (* ---------------------------------------- *) ⊢
204 unification hint 0 ≔ S,i1,i2;
205 X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
206 (* ---------------------------------------- *) ⊢
207 beqitem S i1 i2 ≡ eqb X i1 i2.
210 Semantics of pointed regular expressions
212 The intuitive semantic of a point is to mark the position where
213 we should start reading the regular expression. The language associated
214 to a pre is the union of the languages associated with its points. *)
216 let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝
222 | pc r1 r2 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2)
223 | po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2)
224 | pk r1 ⇒ (in_pl ? r1) · \sem{forget ? r1}^* ].
226 interpretation "in_pl" 'in_l E = (in_pl ? E).
227 interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
229 definition in_prl ≝ λS : DeqSet.λp:pre S.
230 if (\snd p) then \sem{\fst p} ∪ {ϵ} else \sem{\fst p}.
232 interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
233 interpretation "in_prl" 'in_l E = (in_prl ? E).
235 (* The following, trivial lemmas are only meant for rewriting purposes. *)
237 lemma sem_pre_true : ∀S.∀i:pitem S.
238 \sem{〈i,true〉} = \sem{i} ∪ {ϵ}.
241 lemma sem_pre_false : ∀S.∀i:pitem S.
242 \sem{〈i,false〉} = \sem{i}.
245 lemma sem_cat: ∀S.∀i1,i2:pitem S.
246 \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
249 lemma sem_cat_w: ∀S.∀i1,i2:pitem S.∀w.
250 \sem{i1 · i2} w = ((\sem{i1} · \sem{|i2|}) w ∨ \sem{i2} w).
253 lemma sem_plus: ∀S.∀i1,i2:pitem S.
254 \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
257 lemma sem_plus_w: ∀S.∀i1,i2:pitem S.∀w.
258 \sem{i1 + i2} w = (\sem{i1} w ∨ \sem{i2} w).
261 lemma sem_star : ∀S.∀i:pitem S.
262 \sem{i^*} = \sem{i} · \sem{|i|}^*.
265 lemma sem_star_w : ∀S.∀i:pitem S.∀w.
266 \sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
269 (* Below are a few, simple, semantic properties of items. In particular:
270 - not_epsilon_item : ∀S:DeqSet.∀i:pitem S. ¬ (\sem{i} ϵ).
271 - epsilon_pre : ∀S.∀e:pre S. (\sem{i} ϵ) ↔ (\snd e = true).
272 - minus_eps_item: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
273 - minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
274 The first property is proved by a simple induction on $i$; the other
275 results are easy corollaries. We need an auxiliary lemma first. *)
277 lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = ϵ → w1 = ϵ.
278 #S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
280 lemma not_epsilon_lp : ∀S:DeqSet.∀e:pitem S. ¬ (ϵ ∈ e).
281 #S #e elim e normalize /2/
282 [#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H
283 >(append_eq_nil …H…) /2/
284 |#r1 #r2 #n1 #n2 % * /2/
285 |#r #n % * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/
289 lemma epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → \snd e = true.
290 #S * #i #b cases b // normalize #H @False_ind /2/
293 lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → ϵ ∈ e.
294 #S * #i #b #btrue normalize in btrue; >btrue %2 //
297 lemma minus_eps_item: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
299 [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
304 lemma minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
306 [>sem_pre_true normalize in ⊢ (??%?); #w %
307 [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
308 |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]