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Non determinism through type isomophism

Alejandro Díaz-Caro, profile picture
Alejandro Díaz-Caro

The document discusses non-determinism through type isomorphism, presenting a series of equivalences and properties rooted in second-order propositional logic. It emphasizes the relationship between isomorphic propositions and their proofs, describing reduction rules and conjunction handling. Examples demonstrate the usage of these concepts in structured proofs and type equivalences.

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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
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 ])) 2 / 10
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 ])) 2 / 10
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 ] 3 / 10
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 3 / 10
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 3 / 10
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 3 / 10
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 3 / 10
What about ∧-elimination? Γ t+r:A∧B Γ π1 (t + r) : A 4 / 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) !! 4 / 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) !! Workaround: Church-style. Project w.r.t. a type If Γ t : A then πA (t + r) → t 4 / 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) !! 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 4 / 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) !! 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 4 / 10
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 ) 5 / 10
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) 5 / 10
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) 5 / 10
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 )) 5 / 10
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 5 / 10
Example (I) λx A∧B .x : (A ∧ B) ⇒ (A ∧ B) 6 / 10
Example (I) λx A∧B .x : (A ∧ B) ⇒ (A ∧ B) (A ∧ B) ⇒ (A ∧ B) ≡ ((A ∧ B) ⇒ A) ∧ ((A ∧ B) ⇒ B) 6 / 10
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 6 / 10
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 6 / 10
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 6 / 10
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) 6 / 10
Example (II) TF = λx B .λy B .(x + y ) TF : B ⇒ B ⇒ (B ∧ B) 7 / 10
Example (II) TF = λx B .λy B .(x + y ) TF : B ⇒ B ⇒ (B ∧ B) B ⇒ B ⇒ (B ∧ B) ≡ (B ⇒ B ⇒ B) ∧ (B ⇒ B ⇒ B) 7 / 10
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 7 / 10
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 7 / 10
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 7 / 10
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 7 / 10
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) 7 / 10
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 7 / 10
Confluence (some ideas) Of course, a non-deterministic calculus is not confluent! Counterexample πA (x A + y A ) x & xA yA | | ' w ? 8 / 10
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] 8 / 10
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 9 / 10
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 10 / 10
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) 10 / 10
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? 10 / 10
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] 10 / 10
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) 10 / 10
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 10 / 10
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 ) 10 / 10
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] 10 / 10

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Non determinism through type isomophism

  • 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 ])) 2 / 10
  • 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 ])) 2 / 10
  • 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 ] 3 / 10
  • 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 3 / 10
  • 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 3 / 10
  • 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 3 / 10
  • 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 3 / 10
  • 9. What about ∧-elimination? Γ t+r:A∧B Γ π1 (t + r) : A 4 / 10
  • 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) !! 4 / 10
  • 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 4 / 10
  • 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 4 / 10
  • 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 4 / 10
  • 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 ) 5 / 10
  • 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) 5 / 10
  • 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) 5 / 10
  • 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 )) 5 / 10
  • 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 5 / 10
  • 19. Example (I) λx A∧B .x : (A ∧ B) ⇒ (A ∧ B) 6 / 10
  • 20. Example (I) λx A∧B .x : (A ∧ B) ⇒ (A ∧ B) (A ∧ B) ⇒ (A ∧ B) ≡ ((A ∧ B) ⇒ A) ∧ ((A ∧ B) ⇒ B) 6 / 10
  • 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 6 / 10
  • 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 6 / 10
  • 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 6 / 10
  • 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) 6 / 10
  • 25. Example (II) TF = λx B .λy B .(x + y ) TF : B ⇒ B ⇒ (B ∧ B) 7 / 10
  • 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) 7 / 10
  • 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 7 / 10
  • 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 7 / 10
  • 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 7 / 10
  • 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 7 / 10
  • 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) 7 / 10
  • 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 7 / 10
  • 33. Confluence (some ideas) Of course, a non-deterministic calculus is not confluent! Counterexample πA (x A + y A ) x & xA yA | | ' w ? 8 / 10
  • 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] 8 / 10
  • 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 9 / 10
  • 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 10 / 10
  • 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) 10 / 10
  • 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? 10 / 10
  • 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] 10 / 10
  • 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) 10 / 10
  • 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 10 / 10
  • 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 ) 10 / 10
  • 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] 10 / 10