Quelle Halting.thy Sprache: Isabelle

(*
Undecidability of the special Halting problem:
  Does a program applied to an encoding of itself terminate?
Author: Fabian Huch
*)

theory Halting
  imports "HOL-IMP.Big_Step"
begin

definition "halts c s \ (\s'. (c, s) \ s')"

text \<open>A simple program that does not halt:\<close>
definition "loop \ WHILE Bc True DO SKIP"

lemma loop_never_halts[simp]: "\ halts loop s"
  unfolding halts_def
proof clarify
  fix s' assume "(loop, s) \ s'"
  then show False
    by (induction loop s s' rule: big_step_induct) (simp_all add: loop_def)
qed

section \<open>Halting Problem\<close>

text \<open>
Given any encoding \<open>f\<close> (of programs to states), there is no Program \<open>H\<close> such that
for all programs \<open>c\<close>, \<open>H\<close> terminates in a state \<open>s'\<close> which has at variable \<open>x\<close> the
answer whether \<open>c\<close> halts.\<close>

theorem halting:
  "\H. \c. \s'. (H, f c) \ s' \ (halts c (f c) \ s' x > 0)"
  (is "\H. ?P H")
proof clarify
  fix H assume assm: "?P H"

  \<comment> \<open>inverted H: loops if input halts\<close>
  let ?inv_H = "H ;; IF Less (V x) (N 1) THEN SKIP ELSE loop"

  \<comment> \<open>compute in \<open>s'\<close> whether inverted \<open>H\<close> halts when applied to itself\<close>
  obtain s' where
    s'_def: "(H, f ?inv_H) \ s'" and
    s'_halts: "halts ?inv_H (f ?inv_H) \ (s' x > 0)"
    using assm by blast

  \<comment> \<open>contradiction: if it terminates, loop must have terminated; if not, SKIP must have looped!\<close>
  show False
  proof(cases "halts ?inv_H (f ?inv_H)")
    case True

    then have "halts (IF Less (V x) (N 1) THEN SKIP ELSE loop) s'"
      unfolding halts_def using big_step_determ s'_def by fast

    then have "halts loop s'"
      using s'_halts True halts_def by auto

    then show False by auto
  next
    case False

    then have "\ halts SKIP s'"
      using s'_def s'_halts halts_def by force

    then show False using halts_def by auto
  qed
qed

end

quality100%

¤ Dauer der Verarbeitung: 0.15 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.