Proving Decidability of Intuitionistic Propositional Calculus on Coq

Proving decidability
of Intuitionistic Propositional Calculus
on Coq
Masaki Hara (qnighy)
University of Tokyo, first grade
Logic Zoo 2013 にて

1. Task & Known results
2. Brief methodology of the proof
1. Cut elimination
2. Contraction elimination
3. → 𝐿 elimination
4. Proof of strictly-decreasingness
3. Implementation detail
4. 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 <.

methodology
1. Eliminate cut rule of LJ
2. Eliminate contraction rule
3. Split → 𝑳 rule into 4 pieces
4. Prove that every rule is strictly decreasing

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
•

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

• →𝐿 (→ 𝑅 )
• ∧𝐿 (∧ 𝑅 )
• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2
• We eliminate cut rule first.

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

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) ■

Cut-free LJ
Γ⊢𝐺 𝐴,𝐴,Γ⊢𝐺
• 𝑤𝑒𝑎𝑘 𝑐𝑜𝑛𝑡𝑟
𝐴,Γ⊢𝐺 𝐴,Γ⊢𝐺

• →𝐿 (→ 𝑅 )
• ∧𝐿 (∧ 𝑅 )
• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2
•

Cut-free LJ
Γ⊢𝐺 𝐴,𝐴,Γ⊢𝐺
• 𝑤𝑒𝑎𝑘 𝑐𝑜𝑛𝑡𝑟
𝐴,Γ⊢𝐺 𝐴,Γ⊢𝐺

• →𝐿 (→ 𝑅 )
• ∧𝐿 (∧ 𝑅 )
• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2
• Contraction rule is not strictly decreasing

Contraction-free LJ
𝐴,Γ⊢𝐴 ⊥,Γ⊢𝐺
𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵
• →𝐿 (→ 𝑅 )
• ∧𝐿 (∧ 𝑅 )
• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2

Contraction-free LJ
• Implicit weak
– 𝑎𝑥𝑖𝑜𝑚 (𝑒𝑥𝑓𝑎𝑙𝑠𝑜)
• 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

Proof of weak rule
• Easily done by induction ■

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
– (→ 𝑤𝑒𝑎𝑘 )
𝐵,Γ⊢𝐺
• 2. prove contr rule by induction on proof
structure.■

Contraction-free LJ
𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺 𝐴,Γ⊢𝐵
• →𝐿 (→ 𝑅 )
• ∧𝐿 (∧ 𝑅 )
• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2
• This time, → 𝐿 rule is not decreasing

Terminating LJ
𝐴→𝐵,Γ⊢𝐴 𝐵,Γ⊢𝐺
• Split →𝐿 into 4 pieces
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

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

Terminating LJ
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
• ∨𝐿 ∨ 𝑅1 ∨ 𝑅2

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)

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 Ordering
Proof
• 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.

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
…

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

Permutation Compatibility (Coq)
Instance LJ_provable_compat :
Proper
(@Permutation _==>eq==>iff)
LJ_provable.

• Allows rewriting over Permutation equality

Permutation solver (Coq)
• Permutation should be solved automatically
Ltac perm :=
match goal with
…

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

• 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

• 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

• 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

Source code
• Source codes are:
• https://github.com/qnighy/IPC-Coq

おわり
1. Task & Known results
2. Brief methodology of the proof
1. Cut elimination
2. Contraction elimination
3. → 𝐿 elimination
4. Proof of strictly-decreasingness
3. Implementation detail
4. 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

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