Certified Undecidability of Intuitionistic Linear Logic via Binary - PowerPoint PPT Presentation

Certified Undecidability of Intuitionistic Linear Logic via Binary Stack Machines and Minsky Machines Yannick Forster and Dominique Larchey-Wendling CPP 2019 January 14 saarland university computer science Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 1

Decidability A problem P : X → P is decidable if . . . Classically Fix a model of computation M : there is a decider in M For the cbv λ -calculus ∃ u : T . ∀ x : X . ( ux ⊲ T ∧ Px ) ∨ ( ux ⊲ F ∧ ¬ Px ) Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 2

Decidability A problem P : X → P is decidable if . . . Classically Fix a model of computation M : there is a decider in M For the cbv λ -calculus ∃ u : T . ∀ x : X . ( ux ⊲ T ∧ Px ) ∨ ( ux ⊲ F ∧ ¬ Px ) Type Theory ∃ f : X → B . ∀ x : X . Px ↔ fx = true Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 2

Decidability A problem P : X → P is decidable if . . . Classically Fix a model of computation M : there is a decider in M For the cbv λ -calculus ∃ u : T . ∀ x : X . ( ux ⊲ T ∧ Px ) ∨ ( ux ⊲ F ∧ ¬ Px ) Type Theory ∃ f : X → B . ∀ x : X . Px ↔ fx = true dependent version (Coq, Agda, Lean, . . . ) dec P := ∀ x : X . { P x } + {¬ P x } Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 2

Undecidability A problem P : X → P is undecidable if . . . Classically If there is no decider u in M Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 3

Undecidability A problem P : X → P is undecidable if . . . Classically If there is no decider u in M For the cbv λ -calculus ¬ ∃ u : T . ∀ x : X . ( ux ⊲ T ∧ Px ) ∨ ( ux ⊲ F ∧ ¬ Px ) Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 3

Undecidability A problem P : X → P is undecidable if . . . Classically If there is no decider u in M For the cbv λ -calculus ¬ ∃ u : T . ∀ x : X . ( ux ⊲ T ∧ Px ) ∨ ( ux ⊲ F ∧ ¬ Px ) Type Theory ¬ ( ∀ x : X . { Px } + {¬ Px } ) Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 3

Undecidability A problem P : X → P is undecidable if . . . Classically If there is no decider u in M For the cbv λ -calculus ¬ ∃ u : T . ∀ x : X . ( ux ⊲ T ∧ Px ) ∨ ( ux ⊲ F ∧ ¬ Px ) ✭ ✭✭✭✭✭✭✭✭✭✭✭ Type Theory ¬ ( ∀ x : X . { Px } + {¬ Px } ) Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 3

Undecidability A problem P : X → P is undecidable if . . . Classically If there is no decider u in M For the cbv λ -calculus ¬ ∃ u : T . ∀ x : X . ( ux ⊲ T ∧ Px ) ∨ ( ux ⊲ F ∧ ¬ Px ) ✭ ✭✭✭✭✭✭✭✭✭✭✭ Type Theory ¬ ( ∀ x : X . { Px } + {¬ Px } ) In reality: most proofs are by reduction Definition P undecidable := Halting problem reduces to P Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 3

Inductive Undecidability via reductions a problem ( X , P ) : Σ ( X : Type ) , X → Prop Inductive definition of undecidability over Σ X ( X → Prop ) dec Q → dec P undec P undec Halt undec Q Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 4

Inductive Undecidability via reductions a problem ( X , P ) : Σ ( X : Type ) , X → Prop Inductive definition of undecidability over Σ X ( X → Prop ) dec Q → dec P undec P undec Halt undec Q Lemma (Incompatibility between decidability and undecidability) If dec P and undec P then dec Halt Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 4

Inductive Undecidability via reductions a problem ( X , P ) : Σ ( X : Type ) , X → Prop Inductive definition of undecidability over Σ X ( X → Prop ) dec Q → dec P undec P undec Halt undec Q Lemma (Incompatibility between decidability and undecidability) If dec P and undec P then dec Halt Turing Reductions: dec Q → dec P Many-one reduction from ( X , P ) to ( Y , Q ) ◮ computable function f : X → Y s.t. ∀ x . P x ↔ Q ( f x ) ◮ “computable” requirement dropped in CTT ◮ We write P � Q when such reduction exists Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 4

An undecidability proof for Intuitionistic Linear Logic Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 5

An undecidability proof for Intuitionistic Linear Logic Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 5

An undecidability proof for Intuitionistic Linear Logic Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 5

An undecidability proof for Intuitionistic Linear Logic TM PCP BPCP BSM MM eILL ILL ITP 18 LICS 10 LICS 10 Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 5

An undecidability proof for Intuitionistic Linear Logic 1 2 3 4 5 TM PCP BPCP BSM MM eILL ILL ITP 18 LICS 10 LICS 10 Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 5

Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 6

PCP X n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

PCP X n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC CPP C Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

PCP X n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC CPP 19 i C PP Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

PCP X n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC CPP 19 i n 19 in C PP Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

PCP X n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC CPP 19 i n Ca 19 in C PP Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

PCP X n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC CPP 19 i n Ca s 19 in C PP CasC Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

PCP X n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC CPP 19 i n Ca s Ca 19 in C PP CasC Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

PCP X n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC CPP 19 i n Ca s Ca is 19 in C PP CasC ais Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

PCP X n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC CPP 19 i n Ca s Ca is 19 in C PP CasC ais CPP 19 inCasCais CPP 19 inCasCais Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

PCP X Symbols a , b , c : symbols of type X Strings x , y , z : lists of symbols n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC Card x / y : pairs of strings Card set R : finite set of cards CPP 19 i n Ca s Ca is 19 in Stacks A : lists of cards C PP CasC ais CPP 19 inCasCais CPP 19 inCasCais Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

PCP X Symbols a , b , c : symbols of type X Strings x , y , z : lists of symbols n CPP is xuz 19 i Ca s 19 in C ais ofze PP CasC Card x / y : pairs of strings Card set R : finite set of cards CPP 19 i n Ca s Ca is 19 in Stacks A : lists of cards C PP CasC ais CPP 19 inCasCais CPP 19 inCasCais [] 1 := ǫ [] 2 := ǫ ( x / y :: A ) 1 := x ( A 1 ) ( x / y :: A ) 2 := y ( A 2 ) PCP ( R ) := ∃ A ⊆ R . A � = [] ∧ A 1 = A 2 Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 7

Contribution 1 PCP BPCP BSM MM eILL ILL Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 8

PCP � BPCP Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 8

PCP � BPCP PCP is PCP N BPCP is PCP B Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 9

PCP � BPCP PCP is PCP N BPCP is PCP B f : N ∗ → B ∗ f ( a 1 . . . a n : N ∗ ) := 1 a 1 0 . . . 1 a n 0 Lift f to cards, card sets and stack by pointwise application Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 9

PCP � BPCP PCP is PCP N BPCP is PCP B f : N ∗ → B ∗ f ( a 1 . . . a n : N ∗ ) := 1 a 1 0 . . . 1 a n 0 Lift f to cards, card sets and stack by pointwise application To prove: PCP R ↔ BPCP ( f R ) Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 9

PCP � BPCP PCP is PCP N BPCP is PCP B f : N ∗ → B ∗ f ( a 1 . . . a n : N ∗ ) := 1 a 1 0 . . . 1 a n 0 Lift f to cards, card sets and stack by pointwise application To prove: PCP R ↔ BPCP ( f R ) Define inverse function g , easy Y. Forster and D. Larchey-Wendling Certified Undecidability of ILL CPP 2019 – Jan 14 9