Double-Negation Translation of Intuitionistic Modal Logics in Coq - - PowerPoint PPT Presentation

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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

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Intuitionistic Modal Logic i□Z

ϕ ::= ⊤ | ⊥ | a | ϕ1 → ϕ2 | ϕ1 ∨ ϕ2 | ϕ1 ∧ ϕ2 | □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

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Intuitionistic Modal Logic i□Z

ϕ ::= ⊤ | ⊥ | a | ϕ1 → ϕ2 | ϕ1 ∨ ϕ2 | ϕ1 ∧ ϕ2 | □ϕ a ∈ Vars Kripke-Semantics for i□Z: Nonempty set of worlds Two relations: Intuitionistic relation Ri, preorder Modal relation Rm A frame condition, e.g. ∀w1w2, (∃w3, w1Riw3 ∧ w3Rmw2) ⇒ (∃w′

3, w1Rmw′ 3 ∧ w′ 3Riw2)

Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 2 / 16

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Natural Deduction for i□Z

Some of the rules . . .

i□Z ⊢ G ⇒ A (Perm) G′ is permutation of G 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

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Natural Deduction for i□Z

The only rule for box, as proposed by Bellin, De Paiva and Ritter:

i□Z ⊢ G ⇒ □A1 . . . i□Z ⊢ G ⇒ □An

i□Z ⊢ A1 . . . An ⇒ B

(□IE)

i□Z ⊢ G ⇒ □B Classical counterpart of an intuitionistic modal logic: cl□Z := i□(Z ∧ (¬¬a → a))

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Glivenko’s translation

Aglv := ¬¬A

Theorem

Formula A is a classical tautology if and only if Aglv is an intuitionistic tautology. i□Z ⊢ Aglv ⇔ cl□Z ⊢ A?

Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 5 / 16

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Glivenko’s translation

i□Z ⊢ Aglv ⇔ cl□Z ⊢ A

Example

□(¬¬p → p) ∈ cl□ but ¬¬□(¬¬p → p) ̸∈ i□

a b c

V (p) = {c} Intuitionistic Relation Modal Relation

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The translation triangle

Translation Properties

characterization cl□ ⊢ A ↔ At adequateness ∀A, cl□Z ⊢ A ⇔ i□Z ⊢ At (..for a certain class of axioms)

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The translation triangle

Translation Properties

characterization cl□ ⊢ A ↔ At adequateness ∀A, cl□Z ⊢ A ⇔ i□Z ⊢ At (..for a certain class of axioms) Translations: Glivenko: Aglv := ¬¬A Kolmogorov: ¬¬ in front of every subformula Refined Gödel-Gentzen: simplfy Kolmogorov from the outside Kuroda: simplify Kolmogorov from the inside

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The translation triangle

ggr kol kur The Triangle glv For any translations t1, t2 in The Triangle: i□Z ⊢ (At1 ↔ At2) ⇒ sufficient to show adequateness for one translation

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Double Negation Tautologies

For any translations t1, t2 in The Triangle: i□Z ⊢ (At1 ↔ At2) ⇒ 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) . . .

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Reduction theorem

Theorem (Reduction theorem)

For t in the triangle (∀A, cl□Z ⊢ A ⇔ i□Z ⊢ At) ⇔ i□Z ⊢ Z t

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Reduction theorem

Theorem (Reduction theorem)

For t in the triangle (∀A, cl□Z ⊢ A ⇔ i□Z ⊢ At) ⇔ i□Z ⊢ Z t

Proof.

⇒ Obviously cl□Z ⊢ Z, by the premise i□Z ⊢ Z t. ⇐ Let i□Z ⊢ Z t.

← Let i□Z ⊢ At, then cl□Z ⊢ At and thus cl□Z ⊢ A. → Induction on cl□Z ⊢ A. On the blackboard...

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Envelopes

Definition

A is a pre-envelope iff ∀Z, i□Z ⊢ ¬¬sub¬¬(A) → At A is a post-envelope iff ∀Z, i□Z ⊢ At → ¬¬sub¬¬(A) A is a ¬¬-envelope iff ∀Z, i□Z ⊢ At ↔ ¬¬sub¬¬(A) Z is a Kuroda-envelope iff ∀A, i□Z ⊢ At ↔ ¬¬sub¬¬(A)

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.

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Adequateness conditions

Theorem

Let B be a post-envelope and C be a pre-envelope then ∀A, cl□(B → C) ⊢ A ⇔ i□(B → C) ⊢ At. Let Z be a ¬¬-envelope or a Kuroda-envelope, then ∀A, cl□Z ⊢ A ⇔ i□Z ⊢ At.

Proof.

follows from the reduction of adequateness and the definition of envelopes

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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 ⊢ At)

Proof.

1 by straightforward induction 2 since Kuroda-axiom is a Kuroda-envelope. . .

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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

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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.

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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

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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.

we should have thought of good tactics earlier. . . a framework for permutations was a big help

Miriam Polzer & Ulrich Rabenstein DNegMod November 7, 2016 14 / 16

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Future Work

extend the signature of the logic, i.e. add diamond or non-unary

• perators

integrate this code with Tadeusz Ruitenburg development and possibly extend it (or reproof it with ND)

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References

Ferreira, Gilda, and Oliva, Paulo. “On various negative translations.” arXiv preprint arXiv:1101.5442 (2011). Hara, Masaki “Intuitionistic Propositional Calculus” https://github.com/qnighy/IPC-Coq Tadeusz Litak’s formalization of W. Ruitenburg JSL 1984 paper “On the Period of Sequences (An(p)) in Intuitionistic Propositional Calculus” Litak, Tadeusz. “Constructive modalities with provability smack.” Leo Esakia on Duality in Modal and Intuitionistic Logics. Springer Netherlands,

• 2014. 187-216.

Bellin, Gianluigi, Valeria De Paiva, and Eike Ritter. “Extended Curry-Howard correspondence for a basic constructive modal logic.” In Proceedings of Methods for Modalities. 2001. Litak, Tadeusz. “Double Negation and Modality” ALCOP 2016.

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