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# Proving Decidability of Intuitionistic Propositional Calculus on Coq

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### Proving Decidability of Intuitionistic Propositional Calculus on Coq

1. 1. Proving decidabilityof Intuitionistic Propositional Calculus on Coq Masaki Hara (qnighy) University of Tokyo, first grade Logic Zoo 2013 にて
2. 2. 1. Task & Known results2. Brief methodology of the proof 1. Cut elimination 2. Contraction elimination 3. → 𝐿 elimination 4. Proof of strictly-decreasingness3. Implementation detail4. Further implementation plan
3. 3. Task• Proposition: 𝐴𝑡𝑜𝑚 𝑛 , ∧, ∨, →, ⊥• Task: Is given propositional formula P provable in LJ? – It’s known to be decidable. [Dyckhoff]• This talk: how to prove this decidability on Coq
4. 4. Known results• Decision problem on IPC is PSPACE complete [Statman] – Especially, O(N log N) space decision procedure is known [Hudelmaier]• These approaches are backtracking on LJ syntax.
5. 5. Known results• cf. classical counterpart of this problem is co-NP complete. – Proof: find counterexample in boolean-valued semantics (SAT).
6. 6. methodology• To prove decidability, all rules should be strictly decreasing on some measuring. 𝑆1 ,𝑆2 ,…,𝑆 𝑁• More formally, for all rules 𝑟𝑢𝑙𝑒 𝑆0 and all number 𝑖 (1 ≤ 𝑖 ≤ 𝑁), 𝑆 𝑖 < 𝑆0 on certain well-founded relation <.
7. 7. methodology1. Eliminate cut rule of LJ2. Eliminate contraction rule3. Split → 𝑳 rule into 4 pieces4. Prove that every rule is strictly decreasing
8. 8. Sequent Calculus LJ Γ⊢𝐺 𝐴,𝐴,Γ⊢𝐺 Γ⊢𝐴 𝐴,Δ⊢𝐺• 𝑤𝑒𝑎𝑘 𝑐𝑜𝑛𝑡𝑟 (𝑐𝑢𝑡) 𝐴,Γ⊢𝐺 𝐴,Γ⊢𝐺 Γ,Δ⊢𝐺• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴⊢𝐴 ⊥⊢𝐺 Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵•
9. 9. Sequent Calculus LJ Γ⊢𝐺 𝐴,𝐴,Γ⊢𝐺 Γ⊢𝐴 𝐴,Δ⊢𝐺• 𝑤𝑒𝑎𝑘 𝑐𝑜𝑛𝑡𝑟 (𝑐𝑢𝑡) 𝐴,Γ⊢𝐺 𝐴,Γ⊢𝐺 Γ,Δ⊢𝐺• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴⊢𝐴 ⊥⊢𝐺 Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵• We eliminate cut rule first.
10. 10. Cut elimination• 1. Prove these rule by induction on proof structure. Γ⊢𝐺 Δ,Δ,Γ⊢𝐺• 𝑤𝑒𝑎𝑘𝐺 𝑐𝑜𝑛𝑡𝑟𝐺 Δ,Γ⊢𝐺 Δ,Γ⊢𝐺 Γ⊢⊥• ⊥ 𝑅𝐸 Γ⊢𝐺 Γ⊢𝐴∧𝐵 Γ⊢𝐴∧𝐵• ∧ 𝑅𝐸1 ∧ 𝑅𝐸2 Γ⊢𝐴 Γ⊢𝐵 Γ⊢𝐴→𝐵• → 𝑅𝐸 𝐴,Γ⊢𝐵 Γ1 ⊢𝐴 𝐴,Δ1 ⊢𝐺1 Γ2 ⊢𝐵 𝐵,Δ2 ⊢𝐺2• If (𝑐𝑢𝑡 𝐴 ) and (𝑐𝑢𝑡 𝐵 ) for all Γ1 ,Δ1 ⊢𝐺1 Γ2 ,Δ2 ⊢𝐺2 Γ⊢𝐴∨𝐵 A,Δ⊢𝐺 𝐵,Δ⊢𝐺 Γ1 , Γ2 , Δ1 , Δ2 , 𝐺1 , 𝐺2 , then (∨ 𝑅𝐸 ) Γ,Δ⊢𝐺
11. 11. Cut elimination• 2. Prove the general cut rule Γ ⊢ 𝐴　𝐴 𝑛 , Δ ⊢ 𝐺 𝑐𝑢𝑡𝐺 Γ, Δ ⊢ 𝐺 by induction on the size of 𝐴 and proof structure of the right hand.• 3. specialize 𝑐𝑢𝑡𝐺 (n = 1) ■
12. 12. Cut-free LJ Γ⊢𝐺 𝐴,𝐴,Γ⊢𝐺• 𝑤𝑒𝑎𝑘 𝑐𝑜𝑛𝑡𝑟 𝐴,Γ⊢𝐺 𝐴,Γ⊢𝐺• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴⊢𝐴 ⊥⊢𝐺 Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵•
13. 13. Cut-free LJ Γ⊢𝐺 𝐴,𝐴,Γ⊢𝐺• 𝑤𝑒𝑎𝑘 𝑐𝑜𝑛𝑡𝑟 𝐴,Γ⊢𝐺 𝐴,Γ⊢𝐺• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴⊢𝐴 ⊥⊢𝐺 Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵• Contraction rule is not strictly decreasing
14. 14. Contraction-free LJ• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵
15. 15. Contraction-free LJ• Implicit weak – 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺• Implicit contraction 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 – →𝐿 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵 – (∧ 𝑅 ) Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 – ∨𝐿 𝐴∨𝐵,Γ⊢𝐺
16. 16. Contraction-free LJ• Implicit weak – 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺• Implicit contraction 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 – →𝐿 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵 – (∧ 𝑅 ) Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 – ∨𝐿 𝐴∨𝐵,Γ⊢𝐺
17. 17. Proof of weak rule• Easily done by induction ■
18. 18. Proof of contr rule• 1. prove these rules by induction on proof structure. 𝐴∧𝐵,Γ⊢𝐺 𝐴∨𝐵,Γ⊢𝐺 𝐴∨𝐵,Γ⊢𝐺 – ∧ 𝐿𝐸 ∨ 𝐿𝐸1 (∨ 𝐿𝐸2 ) 𝐴,𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 𝐴→𝐵,Γ⊢𝐺 – (→ 𝑤𝑒𝑎𝑘 ) 𝐵,Γ⊢𝐺• 2. prove contr rule by induction on proof structure.■
19. 19. Contraction-free LJ• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵
20. 20. Contraction-free LJ• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵• →𝐿 (→ 𝑅 ) 𝐴→𝐵,Γ⊢𝐺 Γ⊢𝐴→𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∧𝐿 (∧ 𝑅 ) 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵• This time, → 𝐿 rule is not decreasing
21. 21. Terminating LJ 𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺• Split →𝐿 into 4 pieces 𝐴→𝐵,Γ⊢𝐺 𝐶,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐺 1. → 𝐿1 𝐴𝑡𝑜𝑚 𝑛 →𝐶,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐺 𝐵→𝐶,Γ⊢𝐴→𝐵 C,Γ⊢𝐺 2. (→ 𝐿2 ) 𝐴→𝐵 →𝐶,Γ⊢𝐺 𝐴→ 𝐵→𝐶 ,Γ⊢𝐺 3. (→ 𝐿3 ) 𝐴∧𝐵 →𝐶,Γ⊢𝐺 𝐴→𝐶,𝐵→𝐶,Γ⊢𝐺 4. (→ 𝐿4 ) 𝐴∨𝐵 →𝐶,Γ⊢𝐺
22. 22. Correctness of Terminating LJ• 1. If Γ ⊢ 𝐺 is provable in Contraction-free LJ, At least one of these is true: – Γ includes ⊥, 𝐴 ∧ 𝐵, or 𝐴 ∨ 𝐵 – Γ includes both 𝐴𝑡𝑜𝑚(𝑛) and 𝐴𝑡𝑜𝑚 𝑛 → 𝐵 – Γ ⊢ 𝐺 has a proof whose bottommost rule is not the form of 𝐴𝑡𝑜𝑚 𝑛 →𝐵,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐴𝑡𝑜𝑚 𝑛 𝐵,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐺 (→ 𝐿 ) 𝐴𝑡𝑜𝑚 𝑛 →𝐵,𝐴𝑡𝑜𝑚(𝑛),Γ⊢𝐺• Proof: induction on proof structure
23. 23. Correctness of Terminating LJ• 2. every sequent provable in Contraction-free LJ is also provable in Terminating LJ.• Proof: induction by size of the sequent. – Size: we will introduce later
24. 24. Terminating LJ• 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜) 𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺 𝐶,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐺 𝐵→𝐶,Γ⊢𝐴→𝐵 C,Γ⊢𝐺• → 𝐿1 → 𝐿2 𝐴𝑡𝑜𝑚 𝑛 →𝐶,𝐴𝑡𝑜𝑚 𝑛 ,Γ⊢𝐺 𝐴→𝐵 →𝐶,Γ⊢𝐺 𝐴→ 𝐵→𝐶 ,Γ⊢𝐺 𝐴→𝐶,𝐵→𝐶,Γ⊢𝐺• → 𝐿3 → 𝐿4 𝐴∧𝐵 →𝐶,Γ⊢𝐺 𝐴∨𝐵 →𝐶,Γ⊢𝐺 𝐴,Γ⊢𝐵 𝐴,𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• →𝑅 ∧𝐿 (∧ 𝑅 ) Γ⊢𝐴→𝐵 𝐴∧𝐵,Γ⊢𝐺 Γ⊢𝐴∧𝐵 𝐴,Γ⊢𝐺 𝐵,Γ⊢𝐺 Γ⊢𝐴 Γ⊢𝐵• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2 𝐴∨𝐵,Γ⊢𝐺 Γ⊢𝐴∨𝐵 Γ⊢𝐴∨𝐵
25. 25. Proof of termination• Weight of Proposition – 𝑤 𝐴𝑡𝑜𝑚 𝑛 = 1 – 𝑤 ⊥ =1 – 𝑤 𝐴 → 𝐵 = 𝑤 𝐴 + 𝑤 𝐵 +1 – 𝑤 𝐴∧ 𝐵 = 𝑤 𝐴 + 𝑤 𝐵 +2 – 𝑤 𝐴∨ 𝐵 = 𝑤 𝐴 + 𝑤 𝐵 +1• 𝐴 < 𝐵 ⇔ 𝑤 𝐴 < 𝑤(𝐵)
26. 26. Proof of termination• ordering of Proposition List – Use Multiset ordering (Dershowitz and Manna ordering)
27. 27. Multiset Ordering• Multiset Ordering: a binary relation between multisets (not necessarily be ordering)• 𝐴> 𝐵⇔ Not empty A B
28. 28. Multiset Ordering• If 𝑅 is a well-founded binary relation, the Multiset Ordering over 𝑅 is also well-founded.• Well-founded: every element is accessible• 𝐴 is accessible : every element 𝐵 such that 𝐵 < 𝐴 is accessible
29. 29. Multiset OrderingProof• 1. induction on list• Nil ⇒ there is no 𝐴 such that 𝐴 < 𝑀 Nil, therefore it’s accessible.• We will prove: 𝐴𝑐𝑐 𝑀 𝐿 ⇒ 𝐴𝑐𝑐 𝑀 (𝑥 ∷ 𝐿)
30. 30. Multiset Ordering• 2. duplicate assumption• Using 𝐴𝑐𝑐(𝑥) and 𝐴𝑐𝑐 𝑀 (𝐿), we will prove 𝐴𝑐𝑐 𝑀 𝐿 ⇒ 𝐴𝑐𝑐 𝑀 (𝑥 ∷ 𝐿)• 3. induction on 𝑥 and 𝐿 – We can use these two inductive hypotheses. 1. ∀𝐾 𝑦, 𝑦 < 𝑥 ⇒ 𝐴𝑐𝑐 𝑀 𝐾 ⇒ 𝐴𝑐𝑐 𝑀 (𝑦 ∷ 𝐾) 2. ∀𝐾, 𝐾 < 𝑀 𝐿 ⇒ 𝐴𝑐𝑐 𝑀 𝐾 ⇒ 𝐴𝑐𝑐 𝑀 (𝑥 ∷ 𝐾)
31. 31. Multiset Ordering• 4. Case Analysis• By definition, 𝐴𝑐𝑐 𝑀 (𝑥 ∷ 𝐿) is equivalent to ∀𝐾, 𝐾 < 𝑀 (𝑥 ∷ 𝐿) ⇒ 𝐴𝑐𝑐 𝑀 (𝐾)• And there are 3 patterns: 1. 𝐾 includes 𝑥 2. 𝐾 includes 𝑦s s.t. 𝑦 < 𝑥, and 𝐾 minus all such 𝑦 is equal to 𝐿 3. 𝐾 includes 𝑦s s.t. 𝑦 < 𝑥, and 𝐾 minus all such 𝑦 is less than 𝐿• Each pattern is proved using the Inductive Hypotheses.
32. 32. Decidability• Now, decidability can be proved by induction on the size of sequent.
33. 33. Implementation Detail•
34. 34. IPC Proposition (Coq) Inductive PProp:Set :=• | PPbot : PProp | PPatom : nat -> PProp | PPimpl : PProp -> PProp -> PProp | PPconj : PProp -> PProp -> PProp | PPdisj : PProp -> PProp -> PProp.
35. 35. Cut-free LJ (Coq) Inductive LJ_provable : list PProp -> PProp -> Prop :=• | LJ_perm P1 L1 L2 : Permutation L1 L2 -> LJ_provable L1 P1 -> LJ_provable L2 P1 | LJ_weak P1 P2 L1 : LJ_provable L1 P2 -> LJ_provable (P1::L1) P2 | LJ_contr P1 P2 L1 : LJ_provable (P1::P1::L1) P2 -> LJ_provable (P1::L1) P2 …
36. 36. Exchange rule• Exchange rule : Γ, 𝐴, 𝐵, Δ ⊢ 𝐺 𝑒𝑥𝑐ℎ Γ, 𝐵, 𝐴, Δ ⊢ 𝐺 is replaced by more useful Γ⊢ 𝐺 ′ ⊢ 𝐺 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 Γ where Γ, Γ′ are permutation
37. 37. Permutation Compatibility (Coq)Instance LJ_provable_compat : Proper (@Permutation _==>eq==>iff) LJ_provable.• Allows rewriting over Permutation equality
38. 38. Permutation solver (Coq)• Permutation should be solved automaticallyLtac perm := match goal with…
39. 39. Further implementation plan•
40. 40. Further implementation plan• Refactoring (1) : improve Permutation- associated tactics – A smarter auto-unifying tactics is needed – Write tactics using Objective Caml• Refactoring (2) : use Ssreflect tacticals – This makes the proof more manageable
41. 41. Further implementation plan• Refactoring (3) : change proof order – Contraction first, cut next – It will make the proof shorter• Refactoring (4) : discard Multiset Ordering – If we choose appropriate weight function of Propositional Formula, we don’t need Multiset Ordering. (See [Hudelmaier]) – It also enables us to analyze complexity of this procedure
42. 42. Further implementation plan• Refactoring (5) : Proof of completeness – Now completeness theorem depends on the decidability• New Theorem (1) : Other Syntaxes – NJ and HJ may be introduced• New Theorem (2) : Other Semantics – Heyting Algebra
43. 43. Further implementation plan• New Theorem (3) : Other decision procedure – Decision procedure using semantics (if any) – More efficient decision procedure (especially 𝑂(𝑁 log 𝑁)-space decision procedure)• New Theorem (4) : Complexity – Proof of PSPACE-completeness
44. 44. Source code• Source codes are:• https://github.com/qnighy/IPC-Coq
45. 45. おわり1. Task & Known results2. Brief methodology of the proof 1. Cut elimination 2. Contraction elimination 3. → 𝐿 elimination 4. Proof of strictly-decreasingness3. Implementation detail4. Further implementation plan
46. 46. References• [Dyckhoff] Roy Dyckhoff, Contraction-free Sequent Calculi for Intuitionistic Logic, The Journal of Symbolic Logic, Vol. 57, No.3, 1992, pp. 795 – 807• [Statman] Richard Statman, Intuitionistic Propositional Logic is Polynomial-Space Complete, Theoretical Computer Science 9, 1979, pp. 67 – 72• [Hudelmaier] Jörg Hudelmaier, An O(n log n)-Space Decision Procedure for Intuitionistic Propositional Logic, Journal of Logic and Computation, Vol. 3, Issue 1, pp. 63-75