Towards a library of formalised undecidable problems in Coq: The - PowerPoint PPT Presentation

Towards a library of formalised undecidable problems in Coq: The undecidability of intuitionistic linear logic Yannick Forster and Dominique Larchey-Wendling LOLA 2018 July 12 saarland university computer science Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 1

Decidability A problem P : X → P is decidable if . . . Classically Fix a model of computation M : there is a decider in M ∃ u : T . ∀ x : X . ( ux ⊲ T ∧ Px ) ∨ ( ux ⊲ F ∧ ¬ Px ) For the cbv λ -calculus Type Theory ∃ f : X → B . ∀ x : X . Px ↔ fx = true dependent version ∀ x : X . { Px } + {¬ Px } (Coq, Agda, Lean, . . . ) Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 2

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 ) ✭ ¬ ( ∀ x : X . { Px } + {¬ Px } ) ✭✭✭✭✭✭✭✭✭✭✭ ¬ ( ∀ x : X . { Px } + {¬ Px } Type Theory In reality: most proofs are by reduction Definition P undecidable := Halting problem reduces to P Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 3

Reduction A problem is a type X and a unary predicate P : X → P A reduction of ( X , P ) to ( Y , Q ) is a function f : X → Y s.t. ∀ x . Px ↔ Q ( fx ) Write P � Q Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 4

An undecidability proof for intuitionistic linear logic 2 SM mTM FOL BBI 1 2 3 4 cbv λ TM PCP BPCP BSM MM eILL ILL FXP FAM Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 5

Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 6

PCP Symbols a , b , c : N Strings x , y , z : lists of symbols C 2 xfor nf FLo d 018 inO LoC 2018 d F ord inOxf Card c : pairs of strings Stacks A : lists of cards FLo C 2 018 inO xfor d F LoC 2018 inOxf ord FLoC 2018 inOxford [] 1 := ǫ [] 2 := ǫ FLoC 2018 inOxford ( x / y :: A ) 1 := x ( A 1 ) ( x / y :: A ) 2 := y ( A 2 ) PCP ( P ) := ∃ A ⊆ P . A � = [] ∧ A 1 = A 2 Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 7

PCP � BPCP Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 7

generalised BPCP BPCP: generalised PCP: Symbols a , b , c : B Symbols a , b , c : X Strings x , y , z : lists of symbols Strings x , y , z : lists of symbols Card c : pairs of strings Card c : pairs of strings Stacks A , R : lists of string Stacks A , R : lists of stacks [] 1 := ǫ [] 2 := ǫ ( x / y :: A ) 1 := x ( A 1 ) ( x / y :: A ) 2 := y ( A 2 ) BPCP ( P : Stack B ) := ∃ A ⊆ P . A � = [] ∧ A 1 = A 2 PCP X ( P : Stack X ) := ∃ A ⊆ P . A � = [] ∧ A 1 = A 2 Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 8

PCP � BPCP f : N ∗ → B ∗ f ( a 1 . . . a n : N ∗ ) := 1 a 1 0 . . . 1 a n 0 Lift f to cards and stack by pointwise application To prove: PCP P ↔ BPCP ( f P ) Define inverse function g , easy Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 9

Contribution PCP BPCP BSM MM eILL ILL Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 8

BPCP � BSM Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 8

Binary stack machines n stacks of 0s and 1s ( list bool ) for a fixed n instructions (with 0 � x < n and b ∈ bool and i ∈ N ) bsm instr ::= POP x i j | PUSH x b | HALT state: ( PC ∈ N , � S ∈ ( list bool ) n ) Small step semantics ( HALT is blocking): POP x i : if x is empty, then PC ← j else pop b from stack x ; if b is 0 then PC ← i else PC ← PC + 1; push b on stack x ; PC ← PC + 1; PUSH x b : BSM program B i , j : i : bsm instr i ; i + 1 : . . . ; j : bsm instr j → ∗ ( j + 1, � BSM ( B i , j , � S ) := ∃ � S ′ . B : ( i , � S ) − S ′ ) Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 9

BPCP � BSM Keep stacks for top and bottom row Hard code every card as PUSH instructions Iterate all possible stacks Check for stack equality Definition compare_stacks x y i p q := (* i *) [POP x (4+i) (7+i) ; (* 1+i *) POP y q q ; (* 2+i *) PUSH x Zero ; POP x i i ; (* 4+i *) POP y i q ; (* 5+i *) PUSH y Zero ; POP y q i ; (* 7+i *) POP y q p ; (* 8+i *) PUSH x’ Zero ; POP x’ q q ]. Lemma For all stack configurations v, → ∗ ( r , w ) compare stacks x y i p q : ( i , v ) − where r = p if the value of x is the value of y and r = q otherwise. The value of all stacks apart from x and y in w is equal to the value of all stacks in v. Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 10

Contribution PCP BPCP BSM MM eILL ILL Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 9

BSM � MM Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 9

Minsky Machines n registers { x 1 , . . . , x n } of value in N for a fixed n instructions (with x ∈ { x 1 , . . . , x n } and i ∈ N ) mm instr ::= INC x | DEC x i v ∈ N n ) Small step semantics, state: ( PC ∈ N , � INC x : x ← x + 1; PC ← PC + 1; if x is 0 then PC ← i else x ← x − 1; PC ← PC + 1; DEC x i : MM program M i , j : i : mm instr i ; i + 1 : . . . ; j : mm instr j → ∗ ( j + 1, � MM ( M i , j , � v ) := M : ( i , � v ) − 0 ) Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 10

BSM � MM Certified Compiler Stacks are registers, interpret bitstring as binary number Implement DIV2 , MOD2 , MUL2 . . . for push and pop operations Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 11

Contribution PCP BPCP BSM MM eILL ILL Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 10

MM � eILL Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 10

Intuitionistic Linear Logic We “restrict” to the ( !, ⊸ , & ) fragment, system G-ILL Γ ⊢ A A , ∆ ⊢ B [ id ] [ cut ] A ⊢ A Γ , ∆ ⊢ B Γ , A ⊢ B Γ , ! A , ! A ⊢ B ! Γ ⊢ B Γ ⊢ B [ ! L ] [ ! R ] [ w ] [ c ] Γ , ! A ⊢ B ! Γ ⊢ ! B Γ , ! A ⊢ B Γ , ! A ⊢ B Γ , A ⊢ C Γ , B ⊢ C Γ ⊢ A Γ ⊢ B [ & 1 [ & 2 [ & R ] L ] L ] Γ , A & B ⊢ C Γ , A & B ⊢ C Γ ⊢ A & B Γ ⊢ A ∆ , B ⊢ C Γ , A ⊢ B [ ⊸ L ] [ ⊸ R ] Γ , ∆ , A ⊸ B ⊢ C Γ ⊢ A ⊸ B Full linear logic faithfully embedded into that fragment ILL ( Γ , A ) := provable ( Γ ⊢ A ) Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 11

eILL Elementary sequents: ! Σ , g 1 , . . . , g k ⊢ d ( g i , a , b , c , d variables) Σ contains commands : ◮ ( a ⊸ b ) ⊸ c , correponding to INC ◮ a ⊸ ( b ⊸ c ) , correponding to DEC ◮ ( a & b ) ⊸ c , correponding to FORK goal directed rules for eILL (sound and complete w.r.t. G-ILL): TPS (even N k ) is (sound and) complete for eILL. Hence a fragment of both ILL and BBI Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 12

Encoding Minsky machines in eILL Given M as a list of MM instructions ◮ for every register x i in M , two logical variables x i and x i ◮ for every position/state ( PC = i ) in M , a variable q i the state ( i , � v ) is represented by ! Σ ; ∆ � v ⊢ q i ◮ where if � v = ( p 1 , . . . , p n ) then ∆ � v = p 1 . x 1 , . . . , p n . x n ◮ Variables: { x 1 , . . . , x n } ⊎ { x 1 , . . . , x n } ⊎ { q 0 , q 1 , . . . } ◮ the commands in Σ are determined by instructions in M . . . x ← x + 1 i : INC x ∈ M ! Σ , x , ∆ ⊢ q i + 1 ( ( x ⊸ q i + 1 ) ⊸ q i ∈ Σ ) PC ← i + 1 ! Σ , ∆ ⊢ q i Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 13

MM to eILL, (continued) Decrement if x = 0 then PC ← j i : DEC x j ∈ M else x ← x − 1; PC ← i + 1 corresponds to two proofs x > 0 and x = 0: . . . (Ax) ! Σ , x ⊢ x ! Σ , ∆ ⊢ q i + 1 ( x ⊸ ( q i + 1 ⊸ q i ) ∈ Σ ) ! Σ , x , ∆ ⊢ q i . . . . . . ( x �∈ ∆ ) ! Σ , ∆ ⊢ x ! Σ , ∆ ⊢ q j ( ( x & q j ) ⊸ q i ∈ Σ ) ! Σ , ∆ ⊢ q i Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 14

Zero test x �∈ ∆ in eILL ! Σ ; ∆ ⊢ x provable iff x �∈ ∆ Proof for y , ∆ with y � = x : . . . (Ax) ! Σ , y ⊢ y ! Σ , ∆ ⊢ x ( y ⊸ ( x ⊸ x ) ∈ Σ ) ! Σ , y , ∆ ⊢ x Proof for empty context ∆ = ∅ : (Ax) ! Σ , x ⊢ x ( ( x ⊸ x ) ⊸ x ∈ Σ ) ! Σ , ∅ ⊢ x Y. Forster and D. Larchey-Wendling A library of undecidable problems in Coq LOLA 2018 – July 12 15