We define an equivalence relation on propositions and a proof system where equivalent propositions have the same proofs. The system obtained this way resembles several known non-deterministic and algebraic lambda-calculi.

1. Non determinism through type
isomophism
Alejandro Díaz-Caro Gilles Dowek
LIPN, Université Paris 13, Sorbonne Paris Cité INRIA – Paris–Rocquencourt
7th LSFA
Rio de Janeiro, September 29–30, 2012

2. Motivation: Di Cosmo’s isomorphisms [Di Cosmo’95]
- A∧B ≡B ∧A
- A ∧ (B ∧ C ) ≡ (A ∧ B) ∧ C
- A ⇒ (B ∧ C ) ≡ (A ⇒ B) ∧ (A ⇒ C )
- (A ∧ B) ⇒ C ≡ A ⇒ (B ⇒ C )
- A ⇒ (B ⇒ C ) ≡ B ⇒ (A ⇒ C )
- A∧T≡A
- A⇒T≡T
- T⇒A≡A
- ∀X .∀Y .A ≡ ∀Y .∀X .A
- ∀X .A ≡ ∀Y .A[Y = X ]
- ∀X .(A ⇒ B) ≡ A ⇒ ∀X .B if X ∈ FV (A)
/
- ∀X .(A ∧ B) ≡ ∀X .A ∧ ∀X .B
- ∀X .T ≡ T
- ∀X .(A ∧ B) ≡ ∀X .∀Y .(A ∧ (B[Y = X ]))
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3. Motivation: Di Cosmo’s isomorphisms [Di Cosmo’95]
- A∧B ≡B ∧A
- A ∧ (B ∧ C ) ≡ (A ∧ B) ∧ C
- A ⇒ (B ∧ C ) ≡ (A ⇒ B) ∧ (A ⇒ C )
- (A ∧ B) ⇒ C ≡ A ⇒ (B ⇒ C )
- A ⇒ (B ⇒ C ) ≡ B ⇒ (A ⇒ C )
- A∧T≡A
- A⇒T≡T We want a proof-system
- T⇒A≡A where isomorphic proposi-
- ∀X .∀Y .A ≡ ∀Y .∀X .A tions have the same proofs
- ∀X .A ≡ ∀Y .A[Y = X ]
- ∀X .(A ⇒ B) ≡ A ⇒ ∀X .B if X ∈ FV (A)
/
- ∀X .(A ∧ B) ≡ ∀X .A ∧ ∀X .B
- ∀X .T ≡ T
- ∀X .(A ∧ B) ≡ ∀X .∀Y .(A ∧ (B[Y = X ]))
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4. Minimal second order propositional logic
Γ, x : A t:B Γ t:A⇒B Γ s:A
Γ, x : A x :A Γ λx.t : A ⇒ B Γ ts : B
Γ t:A Γ t : ∀X .A
X ∈ FV (Γ)
/
Γ t : ∀X .A Γ t : A[B/X ]
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5. Minimal second order propositional logic
Γ, x : A t:B Γ t:A⇒B Γ s:A
Γ, x : A x :A Γ λx.t : A ⇒ B Γ ts : B
Γ t:A Γ t : ∀X .A
X ∈ FV (Γ)
/
Γ t : ∀X .A Γ t : A[B/X ]
Adding conjunction
Γ t:A Γ r:B
Γ t, r :A∧B
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6. Minimal second order propositional logic
Γ, x : A t:B Γ t:A⇒B Γ s:A
Γ, x : A x :A Γ λx.t : A ⇒ B Γ ts : B
Γ t:A Γ t : ∀X .A
X ∈ FV (Γ)
/
Γ t : ∀X .A Γ t : A[B/X ]
Adding conjunction
Γ t:A Γ r:B
Γ t, r :A∧B
We want A ∧ B = B ∧A
A ∧ (B ∧ C ) = (A ∧ B) ∧ C
so t, r = r, t
t, r, s = t, r , s
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7. Minimal second order propositional logic
Γ, x : A t:B Γ t:A⇒B Γ s:A
Γ, x : A x :A Γ λx.t : A ⇒ B Γ ts : B
Γ t:A Γ t : ∀X .A
X ∈ FV (Γ)
/
Γ t : ∀X .A Γ t : A[B/X ]
Adding conjunction
Γ t:A Γ r:B
Γ t+r :A∧B
We want A ∧ B = B ∧A We write
A ∧ (B ∧ C ) = (A ∧ B) ∧ C t+r=r+t
so t, r = r, t t + (r + s) = (t + r) + s
t, r, s = t, r , s
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8. Minimal second order propositional logic
Γ, x : A t:B Γ t:A⇒B Γ s:A
Γ, x : A x :A Γ λx.t : A ⇒ B Γ ts : B
Γ t:A Γ t : ∀X .A
X ∈ FV (Γ)
/
Γ t : ∀X .A Γ t : A[B/X ]
Adding conjunction
Γ t:A Γ r:B
Γ t+r :A∧B
We want A ∧ B = B ∧A We write
A ∧ (B ∧ C ) = (A ∧ B) ∧ C t+r=r+t
so t, r = r, t t + (r + s) = (t + r) + s
t, r, s = t, r , s
λx.(t + r) = λx.t + λx.r
Also A ⇒ (B ∧ C ) = (A ⇒ B) ∧ (A ⇒ C ) induces
(t + r)s = ts + rs
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9. What about ∧-elimination?
Γ t+r:A∧B
Γ π1 (t + r) : A
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10. What about ∧-elimination?
Γ t+r:A∧B Γ t+r:B ∧A
But A ∧ B = B ∧ A !!
Γ π1 (t + r) : A Γ π1 (t + r) : B
Moreover
t+r=r+t so π1 (t + r) = π1 (r + t) !!
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11. What about ∧-elimination?
Γ t+r:A∧B Γ t+r:B ∧A
But A ∧ B = B ∧ A !!
Γ π1 (t + r) : A Γ π1 (t + r) : B
Moreover
t+r=r+t so π1 (t + r) = π1 (r + t) !!
Workaround: Church-style. Project w.r.t. a type
If Γ t : A then πA (t + r) → t
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12. What about ∧-elimination?
Γ t+r:A∧B Γ t+r:B ∧A
But A ∧ B = B ∧ A !!
Γ π1 (t + r) : A Γ π1 (t + r) : B
Moreover
t+r=r+t so π1 (t + r) = π1 (r + t) !!
Workaround: Church-style. Project w.r.t. a type
If Γ t : A then πA (t + r) → t
This induces non-determinism
Γ t:A πA (t + r) → t
If then
Γ r:A πA (t + r) → r
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13. What about ∧-elimination?
Γ t+r:A∧B Γ t+r:B ∧A
But A ∧ B = B ∧ A !!
Γ π1 (t + r) : A Γ π1 (t + r) : B
Moreover
t+r=r+t so π1 (t + r) = π1 (r + t) !!
Workaround: Church-style. Project w.r.t. a type
If Γ t : A then πA (t + r) → t
This induces non-determinism
Γ t:A πA (t + r) → t
If then
Γ r:A πA (t + r) → r
We are interested in the proof theory
and both t and r are valid proofs of A
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14. The calculus
Types
A, B, C ::= X | A ⇒ B | A ∧ B | ∀X .A
Equivalences
A∧B ≡ B ∧A
(A ∧ B) ∧ C ≡ A ∧ (B ∧ C )
A ⇒ (B ∧ C ) ≡ (A ⇒ B) ∧ (A ⇒ C )
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15. The calculus
Types
A, B, C ::= X | A ⇒ B | A ∧ B | ∀X .A
Equivalences
A∧B ≡ B ∧A
(A ∧ B) ∧ C ≡ A ∧ (B ∧ C )
A ⇒ (B ∧ C ) ≡ (A ⇒ B) ∧ (A ⇒ C )
Terms
t, r, s ::= x A | λx A .t | tr | ΛX .t | t{A}
| t + r | πA (t)
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16. The calculus
Types
A, B, C ::= X | A ⇒ B | A ∧ B | ∀X .A
Equivalences
A∧B ≡ B ∧A
(A ∧ B) ∧ C ≡ A ∧ (B ∧ C )
A ⇒ (B ∧ C ) ≡ (A ⇒ B) ∧ (A ⇒ C )
Terms
t, r, s ::= x A | λx A .t | tr | ΛX .t | t{A}
| t + r | πA (t)
Reduction rules
(λx A .t)r → t[r/x]
(ΛX .t){A} → t[A/X ]
πA (t + r) → t (if Γ t : A)
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17. The calculus
Types
A, B, C ::= X | A ⇒ B | A ∧ B | ∀X .A
Equivalences
A∧B ≡ B ∧A
(A ∧ B) ∧ C ≡ A ∧ (B ∧ C )
A ⇒ (B ∧ C ) ≡ (A ⇒ B) ∧ (A ⇒ C )
Terms
t, r, s ::= x A | λx A .t | tr | ΛX .t | t{A}
| t + r | πA (t)
Reduction rules
(λx A .t)r → t[r/x]
(ΛX .t){A} → t[A/X ]
πA (t + r) → t (if Γ t : A)
t+r r+t
(t + r) + s t + (r + s)
(t + r)s ts + rs
λx A .(t + r) λx A .t + λx A .r
πA⇒B (t)r πB (tr)
(if Γ t : A ⇒ (B ∧ C ))
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18. The calculus
Types Γ, x : A t:B
A, B, C ::= X | A ⇒ B | A ∧ B | ∀X .A ax ⇒I
Γ, x : A x :A A
Γ λx .t : A ⇒ B
Equivalences
A∧B ≡ B ∧A Γ t:A⇒B Γ s:A
(A ∧ B) ∧ C ≡ A ∧ (B ∧ C ) ⇒E
Γ ts : B
A ⇒ (B ∧ C ) ≡ (A ⇒ B) ∧ (A ⇒ C )
Γ t:A X ∈ FV (Γ)
/ Γ t : ∀X .A
∀I ∀E
Terms Γ ΛX .t : ∀X .A Γ t{B} : A[B/X ]
t, r, s ::= x A | λx A .t | tr | ΛX .t | t{A}
| t + r | πA (t) Γ t:A Γ r:B Γ t:A∧B
Reduction rules ∧I ∧E
(λx A .t)r → t[r/x] Γ t+r:A∧B Γ πA (t) : A
(ΛX .t){A} → t[A/X ]
Γ t:A A≡B
πA (t + r) → t (if Γ t : A) ≡
t+r r+t Γ t:B
(t + r) + s t + (r + s)
(t + r)s ts + rs Theorem (Subject reduction)
λx A .(t + r) λx A .t + λx A .r
πA⇒B (t)r πB (tr) If Γ t : A and t → r then Γ r:A
(if Γ t : A ⇒ (B ∧ C )) with → := → or
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19. Example (I)
λx A∧B .x : (A ∧ B) ⇒ (A ∧ B)
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20. Example (I)
λx A∧B .x : (A ∧ B) ⇒ (A ∧ B)
(A ∧ B) ⇒ (A ∧ B) ≡ ((A ∧ B) ⇒ A) ∧ ((A ∧ B) ⇒ B)
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21. Example (I)
λx A∧B .x : (A ∧ B) ⇒ (A ∧ B)
(A ∧ B) ⇒ (A ∧ B) ≡ ((A ∧ B) ⇒ A) ∧ ((A ∧ B) ⇒ B)
Hence
π(A∧B)⇒A (λx A∧B .x) : (A ∧ B) ⇒ A
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22. Example (I)
λx A∧B .x : (A ∧ B) ⇒ (A ∧ B)
(A ∧ B) ⇒ (A ∧ B) ≡ ((A ∧ B) ⇒ A) ∧ ((A ∧ B) ⇒ B)
Hence
π(A∧B)⇒A (λx A∧B .x) : (A ∧ B) ⇒ A
Let r:A∧B
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23. Example (I)
λx A∧B .x : (A ∧ B) ⇒ (A ∧ B)
(A ∧ B) ⇒ (A ∧ B) ≡ ((A ∧ B) ⇒ A) ∧ ((A ∧ B) ⇒ B)
Hence
π(A∧B)⇒A (λx A∧B .x) : (A ∧ B) ⇒ A
Let r:A∧B
π(A∧B)⇒A (λx A∧B .x) r : A
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24. Example (I)
λx A∧B .x : (A ∧ B) ⇒ (A ∧ B)
(A ∧ B) ⇒ (A ∧ B) ≡ ((A ∧ B) ⇒ A) ∧ ((A ∧ B) ⇒ B)
Hence
π(A∧B)⇒A (λx A∧B .x) : (A ∧ B) ⇒ A
Let r:A∧B
π(A∧B)⇒A (λx A∧B .x) r : A
π(A∧B)⇒A (λx A∧B .x)r πA ((λx A∧B .x)r) → πA (r)
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25. Example (II)
TF = λx B .λy B .(x + y )
TF : B ⇒ B ⇒ (B ∧ B)
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26. Example (II)
TF = λx B .λy B .(x + y )
TF : B ⇒ B ⇒ (B ∧ B)
B ⇒ B ⇒ (B ∧ B) ≡ (B ⇒ B ⇒ B) ∧ (B ⇒ B ⇒ B)
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27. Example (II)
TF = λx B .λy B .(x + y )
TF : B ⇒ B ⇒ (B ∧ B)
B ⇒ B ⇒ (B ∧ B) ≡ (B ⇒ B ⇒ B) ∧ (B ⇒ B ⇒ B)
πB⇒B⇒B (TF) : B ⇒ B ⇒ B
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28. Example (II)
TF = λx B .λy B .(x + y )
TF : B ⇒ B ⇒ (B ∧ B)
B ⇒ B ⇒ (B ∧ B) ≡ (B ⇒ B ⇒ B) ∧ (B ⇒ B ⇒ B)
πB⇒B⇒B (TF) : B ⇒ B ⇒ B
Let t:B and f:B
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29. Example (II)
TF = λx B .λy B .(x + y )
TF : B ⇒ B ⇒ (B ∧ B)
B ⇒ B ⇒ (B ∧ B) ≡ (B ⇒ B ⇒ B) ∧ (B ⇒ B ⇒ B)
πB⇒B⇒B (TF) : B ⇒ B ⇒ B
Let t:B and f:B
πB⇒B⇒B (TF) t f
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30. Example (II)
TF = λx B .λy B .(x + y )
TF : B ⇒ B ⇒ (B ∧ B)
B ⇒ B ⇒ (B ∧ B) ≡ (B ⇒ B ⇒ B) ∧ (B ⇒ B ⇒ B)
πB⇒B⇒B (TF) : B ⇒ B ⇒ B
Let t:B and f:B
πB⇒B⇒B (TF) t f πB⇒B ((TF)t) f
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31. Example (II)
TF = λx B .λy B .(x + y )
TF : B ⇒ B ⇒ (B ∧ B)
B ⇒ B ⇒ (B ∧ B) ≡ (B ⇒ B ⇒ B) ∧ (B ⇒ B ⇒ B)
πB⇒B⇒B (TF) : B ⇒ B ⇒ B
Let t:B and f:B
πB⇒B⇒B (TF) t f πB⇒B ((TF)t) f πB ((TF)tf)
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32. Example (II)
TF = λx B .λy B .(x + y )
TF : B ⇒ B ⇒ (B ∧ B)
B ⇒ B ⇒ (B ∧ B) ≡ (B ⇒ B ⇒ B) ∧ (B ⇒ B ⇒ B)
πB⇒B⇒B (TF) : B ⇒ B ⇒ B
Let t:B and f:B
πB⇒B⇒B (TF) t f πB⇒B ((TF)t) f πB ((TF)tf)
7Õ 3t
→ πB (t + f) % y
+f
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33. Confluence (some ideas)
Of course, a non-deterministic calculus is not confluent!
Counterexample πA (x A + y A )
x &
xA yA
| |
' w
?
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34. Confluence (some ideas)
Of course, a non-deterministic calculus is not confluent!
Counterexample πA (x A + y A )
x &
xA yA
| |
' w
?
However, we can prove it keeps some coherence
Confluence of the deterministic fragment
Confluence of the “term ensembles”
e.g. t
z "
{ri }i t
# }
{ri }i [Arrighi,Díaz-Caro,Gadella,Grattage’08]
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35. Conclusions (with some examples)
Proof system
Let t be a proof of A
and r be a proof of B
so t + r is a proof of both A ∧ B and B ∧ A
Non deterministic calculus
t = ΛX .λx X .λy X .x
t ff = ΛX .λx X .λy X .y
B = ∀X .X ⇒ X ⇒ X
t + ff : B ∧ B
πB (t + ff ) : B
t
{ #
t ff
So far:
- Proof system where (three) isomorphic types get the same proofs
- Non-deterministic calculus
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36. Future directions (open problems)
Can we continue adding Di Cosmo’s isomorphisms?
e.g. A ∧ T ≡ A induces t + 0 t and A ⇒ T ≡ T induces λx.0 0
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37. Future directions (open problems)
Can we continue adding Di Cosmo’s isomorphisms?
e.g. A ∧ T ≡ A induces t + 0 t and A ⇒ T ≡ T induces λx.0 0
But if T ⇒ T ≡ T then (λx T .xx)(λx T .xx) : T (wrong)
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38. Future directions (open problems)
Can we continue adding Di Cosmo’s isomorphisms?
e.g. A ∧ T ≡ A induces t + 0 t and A ⇒ T ≡ T induces λx.0 0
But if T ⇒ T ≡ T then (λx T .xx)(λx T .xx) : T (wrong)
A more interesting open question:
Can we use this no determinism to define a probabilistic/quantum
language?
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39. Future directions (open problems)
Can we continue adding Di Cosmo’s isomorphisms?
e.g. A ∧ T ≡ A induces t + 0 t and A ⇒ T ≡ T induces λx.0 0
But if T ⇒ T ≡ T then (λx T .xx)(λx T .xx) : T (wrong)
A more interesting open question:
Can we use this no determinism to define a probabilistic/quantum
language?
Some clues:
Similar to the linear-algebraic lambda-calculus [Arrighi,Dowek]
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40. Future directions (open problems)
Can we continue adding Di Cosmo’s isomorphisms?
e.g. A ∧ T ≡ A induces t + 0 t and A ⇒ T ≡ T induces λx.0 0
But if T ⇒ T ≡ T then (λx T .xx)(λx T .xx) : T (wrong)
A more interesting open question:
Can we use this no determinism to define a probabilistic/quantum
language?
Some clues:
Similar to the linear-algebraic lambda-calculus [Arrighi,Dowek]
We need call-by-value (no-cloning)
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41. Future directions (open problems)
Can we continue adding Di Cosmo’s isomorphisms?
e.g. A ∧ T ≡ A induces t + 0 t and A ⇒ T ≡ T induces λx.0 0
But if T ⇒ T ≡ T then (λx T .xx)(λx T .xx) : T (wrong)
A more interesting open question:
Can we use this no determinism to define a probabilistic/quantum
language?
Some clues:
Similar to the linear-algebraic lambda-calculus [Arrighi,Dowek]
We need call-by-value (no-cloning)
In call-by-value,
t(r + s) tr + ts
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42. Future directions (open problems)
Can we continue adding Di Cosmo’s isomorphisms?
e.g. A ∧ T ≡ A induces t + 0 t and A ⇒ T ≡ T induces λx.0 0
But if T ⇒ T ≡ T then (λx T .xx)(λx T .xx) : T (wrong)
A more interesting open question:
Can we use this no determinism to define a probabilistic/quantum
language?
Some clues:
Similar to the linear-algebraic lambda-calculus [Arrighi,Dowek]
We need call-by-value (no-cloning)
In call-by-value,
t(r + s) tr + ts
But (A ∧ B) ⇒ C ≡ (A ⇒ C ) ∧ (B ⇒ C )
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43. Future directions (open problems)
Can we continue adding Di Cosmo’s isomorphisms?
e.g. A ∧ T ≡ A induces t + 0 t and A ⇒ T ≡ T induces λx.0 0
But if T ⇒ T ≡ T then (λx T .xx)(λx T .xx) : T (wrong)
A more interesting open question:
Can we use this no determinism to define a probabilistic/quantum
language?
Some clues:
Similar to the linear-algebraic lambda-calculus [Arrighi,Dowek]
We need call-by-value (no-cloning)
In call-by-value,
t(r + s) tr + ts
But (A ∧ B) ⇒ C ≡ (A ⇒ C ) ∧ (B ⇒ C )
Workaround: Use polymorphism: ∀X .X ⇒ CX [Arrighi,Díaz-Caro]
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