Double-Negation Translation of Intuitionistic Modal Logics in Coq Miriam Polzer & Ulrich Rabenstein November 7, 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 1 / 16
Intuitionistic Modal Logic i □ Z ϕ ::= ⊤ | ⊥ | a | ϕ 1 → ϕ 2 | ϕ 1 ∨ ϕ 2 | ϕ 1 ∧ ϕ 2 | □ ϕ a ∈ Vars The intuitionistic modal logic i □ Z : All intuitionistic tautologies and axiom Z Closure under MP and substitution Closure under generalization: If A valid, then □ A valid. □ ( A → B ) → ( □ A → □ B ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 2 / 16
Intuitionistic Modal Logic i □ Z ϕ ::= ⊤ | ⊥ | a | ϕ 1 → ϕ 2 | ϕ 1 ∨ ϕ 2 | ϕ 1 ∧ ϕ 2 | □ ϕ a ∈ Vars Kripke-Semantics for i □ Z : Nonempty set of worlds Two relations: Intuitionistic relation R i , preorder Modal relation R m A frame condition , e.g. ∀ w 1 w 2 , ( ∃ w 3 , w 1 R i w 3 ∧ w 3 R m w 2 ) ⇒ ( ∃ w ′ 3 , w 1 R m w ′ 3 ∧ w ′ 3 R i w 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 2 / 16
Natural Deduction for i □ Z Some of the rules . . . i □ Z ⊢ G ⇒ A G ′ is permutation of G (Perm) i □ Z ⊢ G ′ ⇒ A (In) i □ Z ⊢ A , G ⇒ A (Ax) s is a substitution i □ Z ⊢ G ⇒ s ( Z ) i □ Z ⊢ G ⇒ A ∨ B i □ Z ⊢ A , G ⇒ C i □ Z ⊢ B , G ⇒ C ( ∨ E ) i □ Z ⊢ G ⇒ C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 3 / 16
Natural Deduction for i □ Z The only rule for box, as proposed by Bellin, De Paiva and Ritter: i □ Z ⊢ A 1 . . . A n ⇒ B i □ Z ⊢ G ⇒ □ A 1 i □ Z ⊢ G ⇒ □ A n . . . ( □ IE ) i □ Z ⊢ G ⇒ □ B Classical counterpart of an intuitionistic modal logic: cl □ Z := i □ ( Z ∧ ( ¬¬ a → a )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 4 / 16
Glivenko’s translation A glv := ¬¬ A Theorem Formula A is a classical tautology if and only if A glv is an intuitionistic tautology. i □ Z ⊢ A glv ⇔ cl □ Z ⊢ A ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 5 / 16
Glivenko’s translation i □ Z ⊢ A glv ⇔ cl □ Z ⊢ A Example □ ( ¬¬ p → p ) ∈ cl □ but ¬¬ □ ( ¬¬ p → p ) ̸∈ i □ a V ( p ) = { c } Intuitionistic Relation c b Modal Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 6 / 16
The translation triangle Translation Properties characterization cl □ ⊢ A ↔ A t adequateness ∀ A , cl □ Z ⊢ A ⇔ i □ Z ⊢ A t (..for a certain class of axioms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 7 / 16
The translation triangle Translation Properties characterization cl □ ⊢ A ↔ A t adequateness ∀ A , cl □ Z ⊢ A ⇔ i □ Z ⊢ A t (..for a certain class of axioms) Translations: Glivenko: A glv := ¬¬ A Kolmogorov: ¬¬ in front of every subformula Refined Gödel-Gentzen: simplfy Kolmogorov from the outside Kuroda: simplify Kolmogorov from the inside . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 7 / 16
The translation triangle The Triangle ggr glv kur kol For any translations t 1 , t 2 in The Triangle : i □ Z ⊢ ( A t 1 ↔ A t 2 ) ⇒ sufficient to show adequateness for one translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 8 / 16
Double Negation Tautologies For any translations t 1 , t 2 in The Triangle : i □ Z ⊢ ( A t 1 ↔ A t 2 ) ⇒ sufficient to show adequateness for one translation Technical work: i □ Z ⊢ ¬¬ ( ¬¬ A ∧ ¬¬ B ) ↔ ¬¬ ( A ∧ B ) i □ Z ⊢ ¬¬ ( ¬¬ A ∨ ¬¬ B ) ↔ ¬¬ ( A ∨ B ) i □ Z ⊢ ¬¬ ( ¬¬ A → ¬¬ B ) ↔ ( ¬¬ A → ¬¬ B ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 9 / 16
Reduction theorem Theorem (Reduction theorem) For t in the triangle ( ∀ A , cl □ Z ⊢ A ⇔ i □ Z ⊢ A t ) ⇔ i □ Z ⊢ Z t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 10 / 16
Reduction theorem Theorem (Reduction theorem) For t in the triangle ( ∀ A , cl □ Z ⊢ A ⇔ i □ Z ⊢ A t ) ⇔ i □ Z ⊢ Z t Proof. ⇒ Obviously cl □ Z ⊢ Z , by the premise i □ Z ⊢ Z t . ⇐ Let i □ Z ⊢ Z t . ← Let i □ Z ⊢ A t , then cl □ Z ⊢ A t and thus cl □ Z ⊢ A . → Induction on cl □ Z ⊢ A . On the blackboard... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 10 / 16
Envelopes Definition ∀ Z , i □ Z ⊢ ¬¬ sub ¬¬ ( A ) → A t A is a pre-envelope iff ∀ Z , i □ Z ⊢ A t → ¬¬ sub ¬¬ ( A ) A is a post-envelope iff ∀ Z , i □ Z ⊢ A t ↔ ¬¬ sub ¬¬ ( A ) A is a ¬¬ -envelope iff ∀ A , i □ Z ⊢ A t ↔ ¬¬ sub ¬¬ ( A ) Z is a Kuroda-envelope iff Example Box-free formulas are ¬¬ -envelopes. Shallow formulas with no disjunction under box are ¬¬ -envelopes. Implication-free formulas are pre-envelopes. Negations of pre-envelopes are post-envelopes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 11 / 16
Adequateness conditions Theorem Let B be a post-envelope and C be a pre-envelope then ∀ A , cl □ ( B → C ) ⊢ A ⇔ i □ ( B → C ) ⊢ A t . Let Z be a ¬¬ -envelope or a Kuroda-envelope, then ∀ A , cl □ Z ⊢ A ⇔ i □ Z ⊢ A t . Proof. follows from the reduction of adequateness and the definition of envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 12 / 16
Glivenko-Translation Definition Kuroda-axiom □ ¬¬ A → ¬¬ □ A Theorem 1 Assuming Kuroda-axiom, glv becomes equivalent to kur,kol and ggr. 2 i □ Z ⊢ □ ¬¬ A → ¬¬ □ A ⇒ ( ∀ A , cl □ Z ⊢ A ⇔ i □ Z ⊢ A t ) Proof. 1 by straightforward induction 2 since Kuroda-axiom is a Kuroda-envelope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 13 / 16
Our experience with Coq help for generating goals and premises during inductions proofs by auto with hint databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 14 / 16
Our experience with Coq help for generating goals and premises during inductions proofs by auto with hint databases Lemma weakening: forall A B G Z, KIbox Z G B -> KIbox Z (A :: G) B. intros. induction H; eauto 3 with KIboxDB. Qed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 14 / 16
Our experience with Coq help for generating goals and premises during inductions proofs by auto with hint databases Lemma weakening: forall A B G Z, KIbox Z G B -> KIbox Z (A :: G) B. intros. induction H; eauto 3 with KIboxDB. Qed. Lemma eq_kol_kur : forall Z f, KIbox Z [] ((kol f) <<->> (kur f)). unfold kur. intros; induction f; simpl. - apply eq_split; split; apply imp_id. - eauto with eq_impDB. - eauto with eq_andDB. - eauto with eq_orDB. - apply eq_dneg. eq_dest IHf. eq_split; apply box_imp; assumption. - apply tt_dneg. - apply ff_dneg. Qed. a framework for permutations was a big help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 14 / 16
Recommend
More recommend
