Tableau Theorem Prover for Intuitionistic Propositional Logic

In CS 510 - Mathematical Logic and Programming Languages at PSU, we studied tableau provers for classical logic following Smullyan's "First Order Logic" book.
Here, I explain the intuitionistic logic version of tableau provers, as covered in Fitting's book "Intuitionistic Logic: Model Theory and Forcing".

Transcript

1. Tableau Theorem Prover for Intuitionistic Propositional Logic Larry Diehl Portland State University CS 510 - Mathematical Logic and Programming Languages Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
2. Motivation Tableau for Classical Logic If ¬A is contradictory in all paths, then A ∨ ¬A lets us conclude A is a tautology. For satisfiability, running tableau on A yield a (classical model) evaluation context σ. Tableau seems awfully tied to classical logic, is intuitionistic tableau doomed!? Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
3. Classical vs Intuitionistic Logic Classical Logic The meaning of a proposition is its truth value. Satisfiability: Does evaluating it yield true? A ∨ ¬A ¬¬A ⊃ A A ⊃ ¬¬A Intuitionistic Logic The meaning of a proposition is its constructive content. Satisfiability: Can you write it as a program? A ⊃ ¬¬A Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
4. Proof Theory for Intuitionistic Logic Γ, A B Γ A ⊃ B ⊃I Γ A ⊃ B Γ A Γ B ⊃E Γ A Γ B Γ A ∧ B ∧I Γ A ∧ B Γ A ∧E1 Γ A ∧ B Γ B ∧E2 Γ A Γ A ∨ B ∨I1 Γ B Γ A ∨ B ∨I2 Γ, A C Γ, B C Γ C ∨E Γ, A ⊥ Γ ¬A ¬I Γ A Γ ¬A Γ ⊥ ¬E Γ I Γ ⊥ Γ A ⊥E Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
5. Proof Theory for Classical Logic ... intuitionistic rules plus ... Γ A Γ ¬¬A ¬¬I Γ ¬¬A Γ A ¬¬E ...or... Γ A ∨ ¬A Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
6. Model Theory for Classical Logic Boolean Algebra B, false, true, &&, ||, ! Classical truth is a boolean value. Satisfiability σ A ⇔ σ A ≡ true σ A ⇔ σ A ≡ false Evaluation σ p ⇔ σ p σ A ∧ B ⇔ σ A && σ B σ A ∨ B ⇔ σ A || σ B σ A ⊃ B ⇔ !(σ A) || σ B σ ¬A ⇔ !(σ A) Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
7. Model Theory for Intuitionistic Logic Kripke Model C, ≤, ∅, Intuitionistic truth is constructive evidence, or a program. Forcing (intuitionistic satisfiability) Γ p ⇔ Γ p p Γ A ∧ B ⇔ Γ A × Γ B Γ A ∨ B ⇔ Γ A Γ B Γ A ⊃ B ⇔ Γ ≤ ∆ ⇒ ∆ A ⇒ ∆ B Γ ¬A ⇔ Γ ≤ ∆ ⇒ ∆ A ⇒ ⊥ Γ A ⇔ Γ ¬A Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
8. Classical vs Intuitionistic Model Theory Many more intuitionistic models than classical models because intuitionistic implication and negation allow arbitrary intrinsically distinct functions. Much bigger search space for an intuitionistic theorem prover! Evaluation σ A ⊃ B ⇔ !(σ A) || σ B σ ¬A ⇔ !(σ A) Forcing Γ A ⊃ B ⇔ Γ ≤ ∆ ⇒ ∆ A ⇒ ∆ B Γ ¬A ⇔ Γ ≤ ∆ ⇒ ∆ A ⇒ ⊥ Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
9. Classical Tableau Calculus S, T(A ∧ B) S, TA, TB T∧ S, F(A ∧ B) S, FA | S, FB F∧ S, T(A ∨ B) S, TA | S, TB T∨ S, F(A ∨ B) S, FA, FB F∨ S, T(A ⊃ B) S, FA | S, TB T⊃ S, F(A ⊃ B) S, TA, FB F⊃ S, T(¬A) S, FA T¬ S, F(¬A) S, TA F¬ Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
10. Intuitionistic Tableau Calculus ST ⇔ {TA | TA ∈ S} S, T(A ∧ B) S, TA, TB T∧ S, F(A ∧ B) S, FA | S, FB F∧ S, T(A ∨ B) S, TA | S, TB T∨ S, F(A ∨ B) S, FA, FB F∨ S, T(A ⊃ B) S, FA | S, TB T⊃ S, F(A ⊃ B) ST, TA, FB F⊃ S, T(¬A) S, FA T¬ S, F(¬A) ST, TA F¬ Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
11. Classical Tableau Interpretation Gradually build an evaluation context σ for A (such that σ A), until tableau is finished or the model is contradictory. Judgments TA means A is true in the model. FA means A is false in the model. Inference Rules If the premise is true, then the conclusion is true. Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
12. Intuitionistic Tableau Interpretation Gradually build a “proof” of A (an “element” of Γ A), until tableau is finished or the model is contradictory. Judgments TA means we have a proof of A. FA means A we do not (yet) have a proof of A. Inference Rules If the premise is true, then the conclusion may be true. The conclusion is logically consistent with the premise. Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
13. Intuitionistic Tableau Calculus ST ⇔ {TA | TA ∈ S} S, T(A ∧ B) S, TA, TB T∧ S, F(A ∧ B) S, FA | S, FB F∧ S, T(A ∨ B) S, TA | S, TB T∨ S, F(A ∨ B) S, FA, FB F∨ S, T(A ⊃ B) S, FA | S, TB T⊃ S, F(A ⊃ B) ST, TA, FB F⊃ S, T(¬A) S, FA T¬ S, F(¬A) ST, TA F¬ Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
14. Closed Example A ⊃ A [{F(A⊃A)}], [{TA, FA}]. S, T(A ∧ B) S, TA, TB T∧ S, F(A ∧ B) S, FA | S, FB F∧ S, T(A ∨ B) S, TA | S, TB T∨ S, F(A ∨ B) S, FA, FB F∨ S, T(A ⊃ B) S, FA | S, TB T⊃ S, F(A ⊃ B) ST, TA, FB F⊃ S, T(¬A) S, FA T¬ S, F(¬A) ST, TA F¬ Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
15. Closed Example A ⊃ (A ∧ A) [{F(A⊃(A ∧ A))}], [{TA, F(A∧A))}], [{TA, FA}, {TA, FA}]. S, T(A ∧ B) S, TA, TB T∧ S, F(A ∧ B) S, FA | S, FB F∧ S, T(A ∨ B) S, TA | S, TB T∨ S, F(A ∨ B) S, FA, FB F∨ S, T(A ⊃ B) S, FA | S, TB T⊃ S, F(A ⊃ B) ST, TA, FB F⊃ S, T(¬A) S, FA T¬ S, F(¬A) ST, TA F¬ Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
16. Open Example A ∨ ¬A [{F(A∨¬A)}], [{FA, F(¬A)}], [{TA}]. S, T(A ∧ B) S, TA, TB T∧ S, F(A ∧ B) S, FA | S, FB F∧ S, T(A ∨ B) S, TA | S, TB T∨ S, F(A ∨ B) S, FA, FB F∨ S, T(A ⊃ B) S, FA | S, TB T⊃ S, F(A ⊃ B) ST, TA, FB F⊃ S, T(¬A) S, FA T¬ S, F(¬A) ST, TA F¬ Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
17. Closed Example A ⊃ (A ⊃ B) ⊃ B [{F(A⊃(A ⊃ B) ⊃ B)}], [{TA, F((A ⊃ B)⊃B)}], [{TA, T(A⊃B), FB}], [{TA, FA, FB}, {TA, TB, FB}]. S, T(A ∧ B) S, TA, TB T∧ S, F(A ∧ B) S, FA | S, FB F∧ S, T(A ∨ B) S, TA | S, TB T∨ S, F(A ∨ B) S, FA, FB F∨ S, T(A ⊃ B) S, FA | S, TB T⊃ S, F(A ⊃ B) ST, TA, FB F⊃ S, T(¬A) S, FA T¬ S, F(¬A) ST, TA F¬ Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
18. Classical vs Intuitionistic Tableau Search When looking for a closed tableau: Classical You can prioritize any rule to apply to S to shrink the search space. Intuitionistic You must try applying all rules to S, but can still prioritize some and backtrack if they fail. Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
19. “Open” Example ¬A ⊃ ¬A [{F(¬A⊃¬A)}], [{T(¬A), F(¬A)}], [{FA, F(¬A)}], [{TA}]. S, T(A ∧ B) S, TA, TB T∧ S, F(A ∧ B) S, FA | S, FB F∧ S, T(A ∨ B) S, TA | S, TB T∨ S, F(A ∨ B) S, FA, FB F∨ S, T(A ⊃ B) S, FA | S, TB T⊃ S, F(A ⊃ B) ST, TA, FB F⊃ S, T(¬A) S, FA T¬ S, F(¬A) ST, TA F¬ Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
20. Closed Example ¬A ⊃ ¬A [{F(¬A⊃¬A)}], [{T(¬A), F(¬A)}], [{T(¬A), TA}], [{FA, TA}]. S, T(A ∧ B) S, TA, TB T∧ S, F(A ∧ B) S, FA | S, FB F∧ S, T(A ∨ B) S, TA | S, TB T∨ S, F(A ∨ B) S, FA, FB F∨ S, T(A ⊃ B) S, FA | S, TB T⊃ S, F(A ⊃ B) ST, TA, FB F⊃ S, T(¬A) S, FA T¬ S, F(¬A) ST, TA F¬ Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
21. Using Classical vs Intuitionistic Tableau Classical To show that A is true: 1 Assume that A is false. 2 Build a tableau for ¬A. 3 If some sub-proposition is true and false, A must be true. Intuitionistic To show that A is provable: 1 Assume that A has not been proven. 2 Build a tableau for ¬A. 3 If some sub-proposition is proven and not yet proven, it must be impossible that A has not been proven. Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
22. Classical vs Intuitionistic Tableau Soundness Classical Have a model σ from the tableau conclusion, so check that σ A. Intuitionistic Have a tableau derivation of A, so construct an element of Γ A. Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
23. Intuitionistic Tableau Soundness Theorem Have a tableau derivation of A, so construct an element of Γ A. Fitting’s Proof By showing the contrapositive. Sadly, (¬B ⊃ ¬A) (A ⊃ B) intuitionistically. Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic
24. References Classical is to Intuitionistic as Smullyan is to Fitting Classical Tableau Book First Order Logic - Smullyan’68 Intuitionistic Tableau Book Intuitionistic Logic: Model Theory and Forcing - Fitting’69 Intuitionistic Tableau Optimization Papers An O(n log n)-Space Decision Procedure for Intuitionistic Propositional Logic - Hudelmaier’93 A Tableau Decision Procedure for Propositional Intuitionistic Logic - Avellone et. al.’06 Larry Diehl Tableau Theorem Prover for Intuitionistic Propositional Logic