2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
8 \ / This file is distributed under the terms of the
9 \ / GNU General Public License Version 2
10 V_____________________________________________________________*)
12 include "turing/multi_universal/alphabet.ma".
13 include "turing/mono.ma".
15 (************************* turning DeqMove into a DeqSet **********************)
16 definition move_eq ≝ λm1,m2:move.
18 [R ⇒ match m2 with [R ⇒ true | _ ⇒ false]
19 |L ⇒ match m2 with [L ⇒ true | _ ⇒ false]
20 |N ⇒ match m2 with [N ⇒ true | _ ⇒ false]].
22 lemma move_eq_true:∀m1,m2.
23 move_eq m1 m2 = true ↔ m1 = m2.
25 [* normalize [% #_ % |2,3: % #H destruct ]
26 |* normalize [1,3: % #H destruct |% #_ % ]
27 |* normalize [1,2: % #H destruct |% #_ % ]
30 definition DeqMove ≝ mk_DeqSet move move_eq move_eq_true.
32 unification hint 0 ≔ ;
34 (* ---------------------------------------- *) ⊢
37 unification hint 0 ≔ m1,m2;
39 (* ---------------------------------------- *) ⊢
40 move_eq m1 m2 ≡ eqb X m1 m2.
43 (************************ turning DeqMove into a FinSet ***********************)
44 definition move_enum ≝ [L;R;N].
46 lemma move_enum_unique: uniqueb ? [L;R;N] = true.
49 lemma move_enum_complete: ∀x:move. memb ? x [L;R;N] = true.
53 mk_FinSet DeqMove [L;R;N] move_enum_unique move_enum_complete.
55 unification hint 0 ≔ ;
57 (* ---------------------------------------- *) ⊢
60 (*************************** normal Turing Machines ***************************)
62 (* A normal turing machine is just an ordinary machine where:
63 1. the tape alphabet is bool
64 2. the finite state are supposed to be an initial segment of the natural
66 Formally, it is specified by a record with the number n of states, a proof
67 that n is positive, the transition function and the halting function.
70 definition trans_source ≝ λn.
71 FinProd (initN n) (FinOption FinBool).
73 definition trans_target ≝ λn.
74 FinProd (FinProd (initN n) (FinOption FinBool)) FinMove.
76 record normalTM : Type[0] ≝
78 pos_no_states : (0 < no_states);
79 ntrans : trans_source no_states → trans_target no_states;
80 nhalt : initN no_states → bool
83 lemma decomp_target : ∀n.∀tg: trans_target n.
84 ∃qout,cout,m. tg = 〈qout,cout,m〉.
85 #n * * #q #c #m %{q} %{c} %{m} //
88 (* A normal machine is just a special case of Turing Machine. *)
90 definition nstart_state ≝ λM.mk_Sig ?? 0 (pos_no_states M).
92 definition normalTM_to_TM ≝ λM:normalTM.
93 mk_TM FinBool (initN (no_states M))
94 (ntrans M) (nstart_state M) (nhalt M).
96 coercion normalTM_to_TM.
98 definition nconfig ≝ λn. config FinBool (initN n).
100 (* A normal machine has a non empty graph *)
102 definition sample_input: ∀M.trans_source (no_states M).
103 #M % [@(nstart_state M) | %]
106 lemma nTM_nog: ∀M. graph_enum ?? (ntrans M) ≠ [ ].
107 #M % #H lapply(graph_enum_complete ?? (ntrans M) (sample_input M) ? (refl ??))
108 >H normalize #Hd destruct
111 (******************************** tuples **************************************)
113 (* By general results on FinSets we know that every function f between two
114 finite sets A and B can be described by means of a finite graph of pairs
115 〈a,f a〉. Hence, the transition function of a normal turing machine can be
116 described by a finite set of tuples 〈i,c〉,〈j,action〉〉 of the following type:
117 (Nat_to n × (option FinBool)) × (Nat_to n × (option FinBool) × move).
118 Unfortunately this description is not suitable for a Universal Machine, since
119 such a machine must work with a fixed alphabet, while the size on n is unknown.
120 Hence, we must pass from natural numbers to a representation for them on a
121 finitary, e.g. binary, alphabet. In general, we shall associate
122 to a pair 〈〈i,x〉,〈j,y,m〉〉 a tuple with the following syntactical structure
125 1. "|" is a special character used to separate tuples
126 2. w_i and w_j are list of booleans representing the states $i$ and $j$;
127 3. x and y are encoding
131 definition mk_tuple ≝ λqin,cin,qout,cout,mv.
132 bar::qin@cin::qout@[cout;mv].
134 definition tuple_TM : nat → list unialpha → Prop ≝
135 λn,t.∃qin,cin,qout,cout,mv.
136 only_bits qin ∧ only_bits qout ∧ cin ≠ bar ∧ cout ≠ bar ∧ mv ≠ bar ∧
137 |qin| = n ∧ |qout| = n ∧
138 t = mk_tuple qin cin qout cout mv.
140 (***************************** state encoding *********************************)
141 (* p < n is represented with a list of bits of lenght n with the p-th bit from
142 left set to 1. An additional intial bit is set to 1 if the state is final and
145 let rec to_bitlist n p: list bool ≝
146 match n with [ O ⇒ [ ] | S q ⇒ (eqb p q)::to_bitlist q p].
148 let rec from_bitlist l ≝
150 [ nil ⇒ 0 (* assert false *)
151 | cons b tl ⇒ if b then |tl| else from_bitlist tl].
153 lemma bitlist_length: ∀n,p.|to_bitlist n p| = n.
154 #n elim n normalize //
157 lemma bitlist_inv1: ∀n,p.p<n → from_bitlist (to_bitlist n p) = p.
158 #n elim n normalize -n
159 [#p #abs @False_ind /2/
161 cases (le_to_or_lt_eq … (le_S_S_to_le … lepn))
162 [#ltpn lapply (lt_to_not_eq … ltpn) #Hpn
163 >(not_eq_to_eqb_false … Hpn) normalize @Hind @ltpn
164 |#Heq >(eq_to_eqb_true … Heq) normalize <Heq //
169 lemma bitlist_lt: ∀l. 0 < |l| → from_bitlist l < |l|.
170 #l elim l normalize // #b #tl #Hind cases b normalize //
171 #Htl cases (le_to_or_lt_eq … (le_S_S_to_le … Htl)) -Htl #Htl
172 [@le_S_S @lt_to_le @Hind //
173 |cut (tl=[ ]) [/2 by append_l2_injective/] #eqtl >eqtl @le_n
177 definition nat_of: ∀n. Nat_to n → nat.
178 #n normalize * #p #_ @p
181 definition bits_of_state ≝ λn.λh:Nat_to n → bool.λs:Nat_to n.
182 map ? unialpha (λx.bit x) (h s::(to_bitlist n (nat_of n s))).
184 lemma only_bits_bits_of_state : ∀n,h,p. only_bits (bits_of_state n h p).
185 #n #h #p #x whd in match (bits_of_state n h p);
188 |elim (to_bitlist n (nat_of n p))
189 [@False_ind |#b #l #Hind #H cases H -H #H [>H % |@Hind @H ]]
193 (******************************** action encoding *****************************)
195 definition low_char ≝ λc.
201 definition low_mv ≝ λm.
208 (******************************** tuple encoding ******************************)
209 definition tuple_type ≝ λn.
210 (Nat_to n × (option FinBool)) × (Nat_to n × (option FinBool) × move).
212 definition tuple_encoding ≝ λn.λh:Nat_to n→bool.
214 let 〈inp,outp〉 ≝ p in
216 let 〈qn,an,m〉 ≝ outp in
217 let qin ≝ bits_of_state n h q in
218 let qout ≝ bits_of_state n h qn in
219 let cin ≝ low_char a in
220 let cout ≝ low_char an in
222 mk_tuple qin cin qout cout mv.
224 lemma is_tuple: ∀n,h,p. tuple_TM (S n) (tuple_encoding n h p).
225 #n #h * * #q #a * * #qn #an #m
226 %{(bits_of_state n h q)} %{(low_char a)}
227 %{(bits_of_state n h qn)} %{(low_char an)} %{(low_mv m)}
229 [%[%[%[%[% /2/ |% cases a normalize [|#b] #H destruct]
230 |% cases an normalize [|#b] #H destruct]
231 |% cases m normalize #H destruct]
232 |>length_map normalize @eq_f //]
233 |>length_map normalize @eq_f //]
236 definition tuple_length ≝ λn.2*n+4.
238 lemma length_of_tuple: ∀n,t. tuple_TM n t →
239 |t| = tuple_length n.
240 #n #t * #qin * #cin * #qout * #cout * #mv *** #_ #Hqin #Hqout #eqt
241 >eqt normalize >length_append >Hqin -Hqin normalize >length_append
245 definition tuples_list ≝ λn.λh.map … (λp.tuple_encoding n h p).