Verification of PCP-Related Computational Reductions in Coq Yannick - PowerPoint PPT Presentation

Verification of PCP-Related Computational Reductions in Coq Yannick Forster, Edith Heiter, Gert Smolka ITP 2018 July 12 saarland university computer science Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 1

saarland Decidability university computer science A problem P : X → P is decidable if . . . Classically Fix a model of computation M : there is a decider in M Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 2

saarland Decidability university computer science 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 ) Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 2

saarland Decidability university computer science 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 Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 2

saarland Decidability university computer science 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, . . . ) ∀ x : X . { Px } + {¬ Px } Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 2

saarland Undecidability university computer science A problem P : X → P is undecidable if . . . Classically If there is no decider u in M Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 3

saarland Undecidability university computer science 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 ) Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 3

saarland Undecidability university computer science 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 } ) Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 3

saarland Undecidability university computer science 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 } ) Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 3

saarland Undecidability university computer science 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 practice: most proofs are by reduction Definition P undecidable := Halting problem reduces to P Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 3

saarland Reduction university computer science 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 Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 4

saarland university computer science Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 5

saarland PCP university computer science C 2 xfor nf FLo d 018 inO LoC 2018 d F ord inOxf Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

saarland PCP university computer science C 2 xfor nf FLo d 018 inO LoC 2018 d F ord inOxf FLo F Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

saarland PCP university computer science C 2 xfor nf FLo d 018 inO LoC 2018 d F ord inOxf FLo C 2 F LoC 2018 Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

saarland PCP university computer science C 2 xfor nf FLo d 018 inO LoC 2018 d F ord inOxf FLo C 2 018 inO F LoC 2018 inOxf Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

saarland PCP university computer science C 2 xfor nf FLo d 018 inO LoC 2018 d F ord inOxf FLo C 2 018 inO xfor F LoC 2018 inOxf Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

saarland PCP university computer science C 2 xfor nf FLo d 018 inO LoC 2018 d F ord inOxf FLo C 2 018 inO xfor d F LoC 2018 inOxf ord Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

saarland PCP university computer science C 2 xfor nf FLo d 018 inO LoC 2018 d F ord inOxf FLo C 2 018 inO xfor d F LoC 2018 inOxf ord FLoC 2018 inOxford FLoC 2018 inOxford Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

saarland PCP university computer science Symbols a , b , c : N Strings x , y , z : lists of symbols C 2 xfor nf FLo d 018 inO Card c : pairs of strings LoC 2018 d F ord inOxf Stacks A , P : lists of cards FLo C 2 018 inO xfor d A ⊆ P : list inclusion F LoC 2018 inOxf ord FLoC 2018 inOxford FLoC 2018 inOxford Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

saarland PCP university computer science Symbols a , b , c : N Strings x , y , z : lists of symbols C 2 xfor nf FLo d 018 inO Card c : pairs of strings LoC 2018 d F ord inOxf Stacks A , P : lists of cards FLo C 2 018 inO xfor d A ⊆ P : list inclusion 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 Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

saarland History university computer science TM halting � MPCP Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 7

saarland History university computer science TM halting � MPCP SR � PCP Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 7

saarland History university computer science TM halting � MPCP SR � PCP PCP � CFI Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 7

saarland History university computer science TM halting � MPCP SR � PCP PCP � CFI At best proof sketches, no inductions are given Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 7

saarland Contribution university computer science CFI TM SRH SR MPCP PCP CFP Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 8

saarland Contribution university computer science CFI TM SRH SR MPCP PCP CFP Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 8

saarland university computer science MPCP � PCP Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 8

saarland MPCP university computer science C 2 xfor FLo d 018 inO LoC 2018 F ord inOxf Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 9

saarland MPCP university computer science C 2 018 inO xfor FLo d LoC 2018 F ord inOxf Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 9

saarland MPCP university computer science C 2 018 inO xfor FLo d LoC 2018 F ord inOxf FLo C 2 018 inO xfor d F LoC 2018 inOxf ord FLoC 2018 inOxford FLoC 2018 inOxford Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 9

saarland MPCP university computer science C 2 018 inO xfor FLo d LoC 2018 F ord inOxf FLo C 2 018 inO xfor d F LoC 2018 inOxf ord FLoC 2018 inOxford FLoC 2018 inOxford MPCP ( x / y , P ) := ∃ A ⊆ x / y :: P . xA 1 = yA 2 Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 9