(* Title: Turing machines+ −
Author: Xu Jian <xujian817@hotmail.com>+ −
Maintainer: Xu Jian+ −
*)+ −
+ −
theory turing_basic+ −
imports Main+ −
begin+ −
+ −
section {* Basic definitions of Turing machine *}+ −
+ −
datatype action = W0 | W1 | L | R | Nop+ −
+ −
datatype cell = Bk | Oc+ −
+ −
type_synonym tape = "cell list \<times> cell list"+ −
+ −
type_synonym state = nat+ −
+ −
type_synonym instr = "action \<times> state"+ −
+ −
type_synonym tprog = "instr list \<times> nat"+ −
+ −
type_synonym tprog0 = "instr list"+ −
+ −
type_synonym config = "state \<times> tape"+ −
+ −
fun nth_of where+ −
"nth_of xs i = (if i \<ge> length xs then None+ −
else Some (xs ! i))"+ −
+ −
lemma nth_of_map [simp]:+ −
shows "nth_of (map f p) n = (case (nth_of p n) of None \<Rightarrow> None | Some x \<Rightarrow> Some (f x))"+ −
apply(induct p arbitrary: n)+ −
apply(auto)+ −
apply(case_tac n)+ −
apply(auto)+ −
done+ −
+ −
fun + −
fetch :: "instr list \<Rightarrow> state \<Rightarrow> cell \<Rightarrow> instr"+ −
where+ −
"fetch p 0 b = (Nop, 0)"+ −
| "fetch p (Suc s) Bk = + −
(case nth_of p (2 * s) of+ −
Some i \<Rightarrow> i+ −
| None \<Rightarrow> (Nop, 0))"+ −
|"fetch p (Suc s) Oc = + −
(case nth_of p ((2 * s) + 1) of+ −
Some i \<Rightarrow> i+ −
| None \<Rightarrow> (Nop, 0))"+ −
+ −
lemma fetch_Nil [simp]:+ −
shows "fetch [] s b = (Nop, 0)"+ −
apply(case_tac s)+ −
apply(auto)+ −
apply(case_tac b)+ −
apply(auto)+ −
done+ −
+ −
fun + −
update :: "action \<Rightarrow> tape \<Rightarrow> tape"+ −
where + −
"update W0 (l, r) = (l, Bk # (tl r))" + −
| "update W1 (l, r) = (l, Oc # (tl r))"+ −
| "update L (l, r) = (if l = [] then ([], Bk # r) else (tl l, (hd l) # r))" + −
| "update R (l, r) = (if r = [] then (Bk # l, []) else ((hd r) # l, tl r))" + −
| "update Nop (l, r) = (l, r)"+ −
+ −
abbreviation + −
"read r == if (r = []) then Bk else hd r"+ −
+ −
fun step :: "config \<Rightarrow> tprog \<Rightarrow> config"+ −
where + −
"step (s, l, r) (p, off) = + −
(let (a, s') = fetch p (s - off) (read r) in (s', update a (l, r)))"+ −
+ −
fun steps :: "config \<Rightarrow> tprog \<Rightarrow> nat \<Rightarrow> config"+ −
where+ −
"steps c p 0 = c" |+ −
"steps c p (Suc n) = steps (step c p) p n"+ −
+ −
+ −
abbreviation+ −
"step0 c p \<equiv> step c (p, 0)"+ −
+ −
abbreviation+ −
"steps0 c p n \<equiv> steps c (p, 0) n"+ −
+ −
lemma step_red [simp]: + −
shows "steps c p (Suc n) = step (steps c p n) p"+ −
by (induct n arbitrary: c) (auto)+ −
+ −
lemma steps_add [simp]: + −
shows "steps c p (m + n) = steps (steps c p m) p n"+ −
by (induct m arbitrary: c) (auto)+ −
+ −
lemma step_0 [simp]: + −
shows "step (0, (l, r)) p = (0, (l, r))"+ −
by (case_tac p, simp)+ −
+ −
lemma steps_0 [simp]: + −
shows "steps (0, (l, r)) p n = (0, (l, r))"+ −
by (induct n) (simp_all)+ −
+ −
+ −
+ −
fun+ −
is_final :: "config \<Rightarrow> bool"+ −
where+ −
"is_final (s, l, r) = (s = 0)"+ −
+ −
lemma is_final_eq: + −
shows "is_final (s, tp) = (s = 0)"+ −
by (case_tac tp) (auto)+ −
+ −
lemma after_is_final:+ −
assumes "is_final c"+ −
shows "is_final (steps c p n)"+ −
using assms + −
apply(induct n) + −
apply(case_tac [!] c)+ −
apply(auto)+ −
done+ −
+ −
lemma not_is_final:+ −
assumes a: "\<not> is_final (steps c p n1)"+ −
and b: "n2 \<le> n1"+ −
shows "\<not> is_final (steps c p n2)"+ −
proof (rule notI)+ −
obtain n3 where eq: "n1 = n2 + n3" using b by (metis le_iff_add)+ −
assume "is_final (steps c p n2)"+ −
then have "is_final (steps c p n1)" unfolding eq+ −
by (simp add: after_is_final)+ −
with a show "False" by simp+ −
qed+ −
+ −
(* if the machine is in the halting state, there must have + −
been a state just before the halting state *)+ −
lemma before_final: + −
assumes "steps0 (1, tp) A n = (0, tp')"+ −
shows "\<exists> n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')"+ −
using assms+ −
proof(induct n arbitrary: tp')+ −
case (0 tp')+ −
have asm: "steps0 (1, tp) A 0 = (0, tp')" by fact+ −
then show "\<exists>n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')"+ −
by simp+ −
next+ −
case (Suc n tp')+ −
have ih: "\<And>tp'. steps0 (1, tp) A n = (0, tp') \<Longrightarrow>+ −
\<exists>n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')" by fact+ −
have asm: "steps0 (1, tp) A (Suc n) = (0, tp')" by fact+ −
obtain s l r where cases: "steps0 (1, tp) A n = (s, l, r)"+ −
by (auto intro: is_final.cases)+ −
then show "\<exists>n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')"+ −
proof (cases "s = 0")+ −
case True (* in halting state *)+ −
then have "steps0 (1, tp) A n = (0, tp')"+ −
using asm cases by (simp del: steps.simps)+ −
then show ?thesis using ih by simp+ −
next+ −
case False (* not in halting state *)+ −
then have "\<not> is_final (steps0 (1, tp) A n) \<and> steps0 (1, tp) A (Suc n) = (0, tp')"+ −
using asm cases by simp+ −
then show ?thesis by auto+ −
qed+ −
qed+ −
+ −
(* well-formedness of Turing machine programs *)+ −
abbreviation "is_even n \<equiv> (n::nat) mod 2 = 0"+ −
+ −
fun + −
tm_wf :: "tprog \<Rightarrow> bool"+ −
where+ −
"tm_wf (p, off) = (length p \<ge> 2 \<and> is_even (length p) \<and> + −
(\<forall>(a, s) \<in> set p. s \<le> length p div 2 + off \<and> s \<ge> off))"+ −
+ −
abbreviation+ −
"tm_wf0 p \<equiv> tm_wf (p, 0)"+ −
+ −
abbreviation exponent :: "'a \<Rightarrow> nat \<Rightarrow> 'a list" ("_ \<up> _" [100, 99] 100)+ −
where "x \<up> n == replicate n x"+ −
+ −
consts tape_of :: "'a \<Rightarrow> cell list" ("<_>" 100)+ −
+ −
defs (overloaded)+ −
tape_of_nat_abv: "<(n::nat)> \<equiv> Oc \<up> (Suc n)"+ −
+ −
fun tape_of_nat_list :: "'a list \<Rightarrow> cell list" + −
where + −
"tape_of_nat_list [] = []" |+ −
"tape_of_nat_list [n] = <n>" |+ −
"tape_of_nat_list (n#ns) = <n> @ Bk # (tape_of_nat_list ns)"+ −
+ −
fun tape_of_nat_pair :: "'a \<times> 'b \<Rightarrow> cell list" + −
where + −
"tape_of_nat_pair (n, m) = <n> @ [Bk] @ <m>" + −
+ −
+ −
defs (overloaded)+ −
tape_of_nl_abv: "<(ns::'a list)> \<equiv> tape_of_nat_list ns"+ −
tape_of_nat_pair: "<(np::'a\<times>'b)> \<equiv> tape_of_nat_pair np"+ −
+ −
fun + −
shift :: "instr list \<Rightarrow> nat \<Rightarrow> instr list"+ −
where+ −
"shift p n = (map (\<lambda> (a, s). (a, (if s = 0 then 0 else s + n))) p)"+ −
+ −
fun + −
adjust :: "instr list \<Rightarrow> instr list"+ −
where+ −
"adjust p = map (\<lambda> (a, s). (a, if s = 0 then (Suc (length p div 2)) else s)) p"+ −
+ −
lemma length_shift [simp]: + −
shows "length (shift p n) = length p"+ −
by simp+ −
+ −
lemma length_adjust [simp]: + −
shows "length (adjust p) = length p"+ −
by (induct p) (auto)+ −
+ −
+ −
(* composition of two Turing machines *)+ −
fun+ −
tm_comp :: "instr list \<Rightarrow> instr list \<Rightarrow> instr list" ("_ |+| _" [0, 0] 100)+ −
where+ −
"tm_comp p1 p2 = ((adjust p1) @ (shift p2 (length p1 div 2)))"+ −
+ −
lemma tm_comp_length:+ −
shows "length (A |+| B) = length A + length B"+ −
by auto+ −
+ −
lemma tm_comp_wf[intro]: + −
"\<lbrakk>tm_wf (A, 0); tm_wf (B, 0)\<rbrakk> \<Longrightarrow> tm_wf (A |+| B, 0)"+ −
by (auto simp: tm_wf.simps shift.simps adjust.simps tm_comp_length tm_comp.simps)+ −
+ −
+ −
lemma tm_comp_step: + −
assumes unfinal: "\<not> is_final (step0 c A)"+ −
shows "step0 c (A |+| B) = step0 c A"+ −
proof -+ −
obtain s l r where eq: "c = (s, l, r)" by (metis is_final.cases) + −
have "\<not> is_final (step0 (s, l, r) A)" using unfinal eq by simp+ −
then have "case (fetch A s (read r)) of (a, s) \<Rightarrow> s \<noteq> 0"+ −
by (auto simp add: is_final_eq)+ −
then have "fetch (A |+| B) s (read r) = fetch A s (read r)"+ −
apply(case_tac [!] "read r")+ −
apply(case_tac [!] s)+ −
apply(auto simp: tm_comp_length nth_append)+ −
done+ −
then show "step0 c (A |+| B) = step0 c A" by (simp add: eq) + −
qed+ −
+ −
lemma tm_comp_steps: + −
assumes "\<not> is_final (steps0 c A n)" + −
shows "steps0 c (A |+| B) n = steps0 c A n"+ −
using assms+ −
proof(induct n)+ −
case 0+ −
then show "steps0 c (A |+| B) 0 = steps0 c A 0" by auto+ −
next + −
case (Suc n)+ −
have ih: "\<not> is_final (steps0 c A n) \<Longrightarrow> steps0 c (A |+| B) n = steps0 c A n" by fact+ −
have fin: "\<not> is_final (steps0 c A (Suc n))" by fact+ −
then have fin1: "\<not> is_final (step0 (steps0 c A n) A)" + −
by (auto simp only: step_red)+ −
then have fin2: "\<not> is_final (steps0 c A n)"+ −
by (metis is_final_eq step_0 surj_pair) + −
+ −
have "steps0 c (A |+| B) (Suc n) = step0 (steps0 c (A |+| B) n) (A |+| B)" + −
by (simp only: step_red)+ −
also have "... = step0 (steps0 c A n) (A |+| B)" by (simp only: ih[OF fin2])+ −
also have "... = step0 (steps0 c A n) A" by (simp only: tm_comp_step[OF fin1])+ −
finally show "steps0 c (A |+| B) (Suc n) = steps0 c A (Suc n)"+ −
by (simp only: step_red)+ −
qed+ −
+ −
lemma tm_comp_fetch_in_A:+ −
assumes h1: "fetch A s x = (a, 0)"+ −
and h2: "s \<le> length A div 2" + −
and h3: "s \<noteq> 0"+ −
shows "fetch (A |+| B) s x = (a, Suc (length A div 2))"+ −
using h1 h2 h3+ −
apply(case_tac s)+ −
apply(case_tac [!] x)+ −
apply(auto simp: tm_comp_length nth_append)+ −
done+ −
+ −
lemma tm_comp_exec_after_first:+ −
assumes h1: "\<not> is_final c" + −
and h2: "step0 c A = (0, tp)"+ −
and h3: "fst c \<le> length A div 2"+ −
shows "step0 c (A |+| B) = (Suc (length A div 2), tp)"+ −
using h1 h2 h3+ −
apply(case_tac c)+ −
apply(auto simp del: tm_comp.simps)+ −
apply(case_tac "fetch A a Bk")+ −
apply(simp del: tm_comp.simps)+ −
apply(subst tm_comp_fetch_in_A)+ −
apply(auto)[4]+ −
apply(case_tac "fetch A a (hd c)")+ −
apply(simp del: tm_comp.simps)+ −
apply(subst tm_comp_fetch_in_A)+ −
apply(auto)[4]+ −
done+ −
+ −
lemma step_in_range: + −
assumes h1: "\<not> is_final (step0 c A)"+ −
and h2: "tm_wf (A, 0)"+ −
shows "fst (step0 c A) \<le> length A div 2"+ −
using h1 h2+ −
apply(case_tac c)+ −
apply(case_tac a)+ −
apply(auto simp add: prod_case_unfold Let_def)+ −
apply(case_tac "hd c")+ −
apply(auto simp add: prod_case_unfold)+ −
done+ −
+ −
lemma steps_in_range: + −
assumes h1: "\<not> is_final (steps0 (1, tp) A stp)"+ −
and h2: "tm_wf (A, 0)"+ −
shows "fst (steps0 (1, tp) A stp) \<le> length A div 2"+ −
using h1+ −
proof(induct stp)+ −
case 0+ −
then show "fst (steps0 (1, tp) A 0) \<le> length A div 2" using h2+ −
by (auto simp add: steps.simps tm_wf.simps)+ −
next+ −
case (Suc stp)+ −
have ih: "\<not> is_final (steps0 (1, tp) A stp) \<Longrightarrow> fst (steps0 (1, tp) A stp) \<le> length A div 2" by fact+ −
have h: "\<not> is_final (steps0 (1, tp) A (Suc stp))" by fact+ −
from ih h h2 show "fst (steps0 (1, tp) A (Suc stp)) \<le> length A div 2"+ −
by (metis step_in_range step_red)+ −
qed+ −
+ −
lemma tm_comp_pre_halt_same: + −
assumes a_ht: "steps0 (1, tp) A n = (0, tp')"+ −
and a_wf: "tm_wf (A, 0)"+ −
obtains n' where "steps0 (1, tp) (A |+| B) n' = (Suc (length A div 2), tp')"+ −
proof -+ −
assume a: "\<And>n. steps (1, tp) (A |+| B, 0) n = (Suc (length A div 2), tp') \<Longrightarrow> thesis"+ −
obtain stp' where fin: "\<not> is_final (steps0 (1, tp) A stp')" and h: "steps0 (1, tp) A (Suc stp') = (0, tp')"+ −
using before_final[OF a_ht] by blast+ −
from fin have h1:"steps0 (1, tp) (A |+| B) stp' = steps0 (1, tp) A stp'"+ −
by (rule tm_comp_steps)+ −
from h have h2: "step0 (steps0 (1, tp) A stp') A = (0, tp')"+ −
by (simp only: step_red)+ −
+ −
have "steps0 (1, tp) (A |+| B) (Suc stp') = step0 (steps0 (1, tp) (A |+| B) stp') (A |+| B)" + −
by (simp only: step_red)+ −
also have "... = step0 (steps0 (1, tp) A stp') (A |+| B)" using h1 by simp+ −
also have "... = (Suc (length A div 2), tp')" + −
by (rule tm_comp_exec_after_first[OF fin h2 steps_in_range[OF fin a_wf]])+ −
finally show thesis using a by blast+ −
qed+ −
+ −
lemma tm_comp_fetch_second_zero:+ −
assumes h1: "fetch B s x = (a, 0)"+ −
and hs: "tm_wf (A, 0)" "s \<noteq> 0"+ −
shows "fetch (A |+| B) (s + (length A div 2)) x = (a, 0)"+ −
using h1 hs+ −
apply(case_tac x)+ −
apply(case_tac [!] s)+ −
apply(auto simp: tm_comp_length nth_append)+ −
done + −
+ −
lemma tm_comp_fetch_second_inst:+ −
assumes h1: "fetch B sa x = (a, s)"+ −
and hs: "tm_wf (A, 0)" "sa \<noteq> 0" "s \<noteq> 0"+ −
shows "fetch (A |+| B) (sa + length A div 2) x = (a, s + length A div 2)"+ −
using h1 hs+ −
apply(case_tac x)+ −
apply(case_tac [!] sa)+ −
apply(auto simp: tm_comp_length nth_append)+ −
done + −
+ −
+ −
lemma tm_comp_second_same:+ −
assumes a_wf: "tm_wf (A, 0)"+ −
and steps: "steps0 (1, l, r) B stp = (s', l', r')"+ −
shows "steps0 (Suc (length A div 2), l, r) (A |+| B) stp + −
= (if s' = 0 then 0 else s' + length A div 2, l', r')"+ −
using steps+ −
proof(induct stp arbitrary: s' l' r')+ −
case 0+ −
then show ?case by (simp add: steps.simps)+ −
next+ −
case (Suc stp s' l' r')+ −
obtain s'' l'' r'' where a: "steps0 (1, l, r) B stp = (s'', l'', r'')"+ −
by (metis is_final.cases)+ −
then have ih1: "s'' = 0 \<Longrightarrow> steps0 (Suc (length A div 2), l, r) (A |+| B) stp = (0, l'', r'')"+ −
and ih2: "s'' \<noteq> 0 \<Longrightarrow> steps0 (Suc (length A div 2), l, r) (A |+| B) stp = (s'' + length A div 2, l'', r'')"+ −
using Suc by (auto)+ −
have h: "steps0 (1, l, r) B (Suc stp) = (s', l', r')" by fact+ −
+ −
{ assume "s'' = 0"+ −
then have ?case using a h ih1 by (simp del: steps.simps) + −
} moreover+ −
{ assume as: "s'' \<noteq> 0" "s' = 0"+ −
from as a h + −
have "step0 (s'', l'', r'') B = (0, l', r')" by (simp del: steps.simps)+ −
with as have ?case+ −
apply(simp add: ih2[OF as(1)] step.simps del: tm_comp.simps steps.simps)+ −
apply(case_tac "fetch B s'' (read r'')")+ −
apply(auto simp add: tm_comp_fetch_second_zero[OF _ a_wf] simp del: tm_comp.simps)+ −
done+ −
} moreover+ −
{ assume as: "s'' \<noteq> 0" "s' \<noteq> 0"+ −
from as a h+ −
have "step0 (s'', l'', r'') B = (s', l', r')" by (simp del: steps.simps)+ −
with as have ?case+ −
apply(simp add: ih2[OF as(1)] step.simps del: tm_comp.simps steps.simps)+ −
apply(case_tac "fetch B s'' (read r'')")+ −
apply(auto simp add: tm_comp_fetch_second_inst[OF _ a_wf as] simp del: tm_comp.simps)+ −
done+ −
}+ −
ultimately show ?case by blast+ −
qed+ −
+ −
lemma tm_comp_second_halt_same:+ −
assumes "tm_wf (A, 0)" + −
and "steps0 (1, l, r) B stp = (0, l', r')"+ −
shows "steps0 (Suc (length A div 2), l, r) (A |+| B) stp = (0, l', r')"+ −
using tm_comp_second_same[OF assms] by (simp)+ −
+ −
end+ −
+ −