(* Title: thys/Turing.thy Author: Jian Xu, Xingyuan Zhang, and Christian Urban*)header {* Turing Machines *}theory Turingimports Mainbeginsection {* Basic definitions of Turing machine *}datatype action = W0 | W1 | L | R | Nopdatatype cell = Bk | Octype_synonym tape = "cell list \<times> cell list"type_synonym state = nattype_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)donefun 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)donefun 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)))"abbreviation "step0 c p \<equiv> step c (p, 0)"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 "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)donelemma 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 simpqed(* 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 assmsproof(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 simpnext 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 qedqed(* 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> nat \<Rightarrow> instr list"where "adjust p e = map (\<lambda> (a, s). (a, if s = 0 then e else s)) p"abbreviation "adjust0 p \<equiv> adjust p (Suc (length p div 2))"lemma length_shift [simp]: shows "length (shift p n) = length p"by simplemma length_adjust [simp]: shows "length (adjust p n) = 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 = ((adjust0 p1) @ (shift p2 (length p1 div 2)))"lemma tm_comp_length: shows "length (A |+| B) = length A + length B"by autolemma tm_comp_wf[intro]: "\<lbrakk>tm_wf (A, 0); tm_wf (B, 0)\<rbrakk> \<Longrightarrow> tm_wf (A |+| B, 0)"by (auto)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) qedlemma tm_comp_steps: assumes "\<not> is_final (steps0 c A n)" shows "steps0 c (A |+| B) n = steps0 c A n"using assmsproof(induct n) case 0 then show "steps0 c (A |+| B) 0 = steps0 c A 0" by autonext 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)qedlemma 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 h3apply(case_tac s)apply(case_tac [!] x)apply(auto simp: tm_comp_length nth_append)donelemma 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 h3apply(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]donelemma 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 h2apply(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)donelemma 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 h1proof(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(* if A goes into the final state, then A |+| B will go into the first state of B *)lemma tm_comp_next: 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 blastqedlemma 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 hsapply(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 hsapply(case_tac x)apply(case_tac [!] sa)apply(auto simp: tm_comp_length nth_append)done lemma tm_comp_second: 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 stepsproof(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 blastqedlemma tm_comp_final: 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[OF assms] by (simp)end