Proving Decidability of Intuitionistic Propositional Calculus on Coq
Proving decidabilityof Intuitionistic Propositional Calculus on Coq Masaki Hara (qnighy) University of Tokyo, first grade Logic Zoo 2013 にて
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
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
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.
Known results• cf. classical counterpart of this problem is co-NP complete. – Proof: find counterexample in boolean-valued semantics (SAT).
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 <.
methodology1. Eliminate cut rule of LJ2. Eliminate contraction rule3. Split → 𝑳 rule into 4 pieces4. Prove that every rule is strictly decreasing
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
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
Proof of termination• ordering of Proposition List – Use Multiset ordering (Dershowitz and Manna ordering)
Multiset Ordering• Multiset Ordering: a binary relation between multisets (not necessarily be ordering)• 𝐴> 𝐵⇔ Not empty A B
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
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)
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)
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.
Decidability• Now, decidability can be proved by induction on the size of sequent.
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
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
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
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
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
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