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/chapter7.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) ^*].
45 notation "\sem{term 19 E}" non associative with precedence 75 for @{'in_l $E}.
46 interpretation "in_l" 'in_l E = (in_l ? E).
47 interpretation "in_l mem" 'mem w l = (in_l ? l w).
49 lemma rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*.
54 Pointed Regular expressions
56 We now introduce pointed regular expressions, that are the main tool we shall
57 use for the construction of the automaton.
58 A pointed regular expression is just a regular expression internally labelled
59 with some additional points. Intuitively, points mark the positions inside the
60 regular expression which have been reached after reading some prefix of
61 the input string, or better the positions where the processing of the remaining
62 string has to be started. Each pointed expression for $e$ represents a state of
63 the {\em deterministic} automaton associated with $e$; since we obviously have
64 only a finite number of possible labellings, the number of states of the automaton
67 Pointed regular expressions provide the tool for an algebraic revisitation of
68 McNaughton and Yamada's algorithm for position automata, making the proof of its
69 correctness, that is far from trivial, particularly clear and simple. In particular,
70 pointed expressions offer an appealing alternative to Brzozowski's derivatives,
71 avoiding their weakest point, namely the fact of being forced to quotient derivatives
72 w.r.t. a suitable notion of equivalence in order to get a finite number of states
73 (that is not essential for recognizing strings, but is crucial for comparing regular
76 Our main data structure is the notion of pointed item, that is meant whose purpose
77 is to encode a set of positions inside a regular expression.
78 The idea of formalizing pointers inside a data type by means of a labelled version
79 of the data type itself is probably one of the first, major lessons learned in the
80 formalization of the metatheory of programming languages. For our purposes, it is
81 enough to mark positions preceding individual characters, so we shall have two kinds
82 of characters •a (pp a) and a (ps a) according to the case a is pointed or not. *)
84 inductive pitem (S: DeqSet) : Type[0] ≝
85 pz: pitem S (* empty *)
86 | pe: pitem S (* epsilon *)
87 | ps: S → pitem S (* symbol *)
88 | pp: S → pitem S (* pointed sysmbol *)
89 | pc: pitem S → pitem S → pitem S (* concatenation *)
90 | po: pitem S → pitem S → pitem S (* plus *)
91 | pk: pitem S → pitem S. (* kleene's star *)
93 (* A pointed regular expression (pre) is just a pointed item with an additional
94 boolean, that must be understood as the possibility to have a trailing point at
95 the end of the expression. As we shall see, pointed regular expressions can be
96 understood as states of a DFA, and the boolean indicates if
97 the state is final or not. *)
99 definition pre ≝ λS.pitem S × bool.
101 interpretation "pitem star" 'star a = (pk ? a).
102 interpretation "pitem or" 'plus a b = (po ? a b).
103 interpretation "pitem cat" 'middot a b = (pc ? a b).
104 notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
105 notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
106 interpretation "pitem pp" 'pp a = (pp ? a).
107 interpretation "pitem ps" 'ps a = (ps ? a).
108 interpretation "pitem epsilon" 'epsilon = (pe ?).
109 interpretation "pitem empty" 'empty = (pz ?).
111 (* The carrier $|i|$ of an item i is the regular expression obtained from i by
112 removing all the points. Similarly, the carrier of a pointed regular expression
113 is the carrier of its item. *)
115 let rec forget (S: DeqSet) (l : pitem S) on l: re S ≝
121 | pc E1 E2 ⇒ (forget ? E1) · (forget ? E2)
122 | po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
123 | pk E ⇒ (forget ? E)^* ].
125 (* Already in the library
126 notation "| term 19 C |" with precedence 70 for @{ 'card $C }.
128 interpretation "forget" 'card a = (forget ? a).
130 lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
133 lemma erase_plus : ∀S.∀i1,i2:pitem S.
134 |i1 + i2| = |i1| + |i2|.
137 lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*.
141 Comparing items and pres
143 Items and pres are very concrete datatypes: they can be effectively compared,
144 and enumerated. In particular, we can define a boolean equality beqitem and a proof
145 beqitem_true that it refects propositional equality, enriching the set (pitem S)
148 let rec beqitem S (i1,i2: pitem S) on i1 ≝
150 [ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
151 | pe ⇒ match i2 with [ pe ⇒ true | _ ⇒ false]
152 | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1==y2 | _ ⇒ false]
153 | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1==y2 | _ ⇒ false]
154 | po i11 i12 ⇒ match i2 with
155 [ po i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
157 | pc i11 i12 ⇒ match i2 with
158 [ pc i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
160 | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ false]
163 lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2).
165 [#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
166 |#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
167 |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
168 [>(\P H) // | @(\b (refl …))]
169 |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
170 [>(\P H) // | @(\b (refl …))]
171 |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
172 normalize #H destruct
173 [cases (true_or_false (beqitem S i11 i21)) #H1
174 [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
175 |>H1 in H; normalize #abs @False_ind /2/
177 |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
179 |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
180 normalize #H destruct
181 [cases (true_or_false (beqitem S i11 i21)) #H1
182 [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
183 |>H1 in H; normalize #abs @False_ind /2/
185 |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
187 |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
188 normalize #H destruct
189 [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
193 definition DeqItem ≝ λS.
194 mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
196 (* We also add a couple of unification hints to allow the type inference system
197 to look at (pitem S) as the carrier of a DeqSet, and at beqitem as if it was the
198 equality function of a DeqSet. *)
200 unification hint 0 ≔ S;
201 X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
202 (* ---------------------------------------- *) ⊢
205 unification hint 0 ≔ S,i1,i2;
206 X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
207 (* ---------------------------------------- *) ⊢
208 beqitem S i1 i2 ≡ eqb X i1 i2.
211 Semantics of pointed regular expressions
213 The intuitive semantic of a point is to mark the position where
214 we should start reading the regular expression. The language associated
215 to a pre is the union of the languages associated with its points. *)
217 let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝
223 | pc r1 r2 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2)
224 | po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2)
225 | pk r1 ⇒ (in_pl ? r1) · \sem{forget ? r1}^* ].
227 interpretation "in_pl" 'in_l E = (in_pl ? E).
228 interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
230 definition in_prl ≝ λS : DeqSet.λp:pre S.
231 if (snd ?? p) then \sem{fst ?? p} ∪ {ϵ} else \sem{fst ?? p}.
233 interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
234 interpretation "in_prl" 'in_l E = (in_prl ? E).
236 (* The following, trivial lemmas are only meant for rewriting purposes. *)
238 lemma sem_pre_true : ∀S.∀i:pitem S.
239 \sem{〈i,true〉} = \sem{i} ∪ {ϵ}.
242 lemma sem_pre_false : ∀S.∀i:pitem S.
243 \sem{〈i,false〉} = \sem{i}.
246 lemma sem_cat: ∀S.∀i1,i2:pitem S.
247 \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
250 lemma sem_cat_w: ∀S.∀i1,i2:pitem S.∀w.
251 \sem{i1 · i2} w = ((\sem{i1} · \sem{|i2|}) w ∨ \sem{i2} w).
254 lemma sem_plus: ∀S.∀i1,i2:pitem S.
255 \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
258 lemma sem_plus_w: ∀S.∀i1,i2:pitem S.∀w.
259 \sem{i1 + i2} w = (\sem{i1} w ∨ \sem{i2} w).
262 lemma sem_star : ∀S.∀i:pitem S.
263 \sem{i^*} = \sem{i} · \sem{|i|}^*.
266 lemma sem_star_w : ∀S.∀i:pitem S.∀w.
267 \sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
270 (* Below are a few, simple, semantic properties of items. In particular:
271 - not_epsilon_item : ∀S:DeqSet.∀i:pitem S. ¬ (\sem{i} ϵ).
272 - epsilon_pre : ∀S.∀e:pre S. (\sem{i} ϵ) ↔ (\snd e = true).
273 - minus_eps_item: ∀S.∀i:pitem S. \sem{i} ≐ \sem{i}-{[ ]}.
274 - minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} ≐ \sem{e}-{[ ]}.
275 The first property is proved by a simple induction on $i$; the other
276 results are easy corollaries. We need an auxiliary lemma first. *)
278 lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = ϵ → w1 = ϵ.
279 #S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
281 lemma not_epsilon_lp : ∀S:DeqSet.∀e:pitem S. ¬ (ϵ ∈ e).
282 #S #e elim e normalize
285 |#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H
286 >(append_eq_nil …H…) /2/
287 |#r1 #r2 #n1 #n2 % * /2/
288 |#r #n % * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/
292 lemma epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → snd … e = true.
293 #S * #i #b cases b // normalize #H @False_ind /2/
296 lemma true_to_epsilon : ∀S.∀e:pre S. snd … e = true → ϵ ∈ e.
297 #S * #i #b #btrue normalize in btrue; >btrue %2 //
300 lemma minus_eps_item: ∀S.∀i:pitem S. \sem{i} ≐ \sem{i}-{ϵ}.
302 [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
307 lemma minus_eps_pre: ∀S.∀e:pre S. \sem{fst ?? e} ≐ \sem{e}-{ϵ}.
309 [>sem_pre_true normalize in ⊢ (??%?); #w %
310 [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
311 |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]