(* Title: thys/Turing.thy

   Author: Jian Xu, Xingyuan Zhang, and Christian Urban

*)

chapter {* Turing Machines *}

theory Turing2

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 \<times> 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))"

fun 

  fetch :: "tprog \<Rightarrow> state \<Rightarrow> cell \<Rightarrow> instr"

where

  "fetch tm 0 b = (Nop, 0)"

| "fetch tm (Suc s) b = 

     (case nth_of tm s of

        Some i \<Rightarrow> (case b of Bk \<Rightarrow> fst i | Oc \<Rightarrow> snd i)

      | None \<Rightarrow> (Nop, 0))"

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 = 

     (let (a, s') = fetch p s (read r) in (s', update a (l, r)))"

fun steps :: "config \<Rightarrow> tprog \<Rightarrow> nat \<Rightarrow> config"

  where

  "steps cf p 0 = cf" |

  "steps cf p (Suc n) = steps (step cf p) p n"

(* well-formedness of Turing machine programs *)

fun 

  tm_wf :: "tprog \<Rightarrow> bool"

where

  "tm_wf p = (1 \<le> length p \<and> (\<forall>((_, s1), (_, s2)) \<in> set p. s1 \<le> length p \<and> s2 \<le> length p))"

(* short-hand notation for tapes *)

abbreviation cell_replicate :: "'a \<Rightarrow> nat \<Rightarrow> 'a list" ("_ \<up> _" [100, 99] 100)

  where "x \<up> n \<equiv> replicate n x"

class tape_of =

  fixes tape_of :: "'a \<Rightarrow> cell list" ("<_>" [64] 67)

instantiation nat :: tape_of 

begin

fun tape_of_nat where 

  "<(n::nat)> = Oc \<up> (Suc n)"

instance ..

end

instantiation list :: (tape_of) tape_of 

begin

fun tape_of_list :: "'a list \<Rightarrow> cell list" 

  where 

  "<[]> = []" |

  "<[n]> = <n>" |

  "<n # ns> = <n> @ [Bk] @ <ns>"

instance ..

end

instantiation prod :: (tape_of, tape_of) tape_of 

begin

fun tape_of_prod :: "'a \<times> 'b \<Rightarrow> cell list" 

  where 

  "<(n, m)> = <n> @ [Bk] @ <m>" 

instance ..

end

definition 

  "std_tape tp \<equiv> \<exists>k (n::nat) l. snd tp = (Bk \<up> k, <n> @ Bk \<up> l)"

fun

  is_final :: "config \<Rightarrow> bool"

where

  "is_final cf = (fst cf = 0)"

end