representing permutations without permutations
play

Representing permutations without permutations The expressive power - PowerPoint PPT Presentation

Nov 24, 2022 •157 likes •1.4k views

Representing permutations without permutations The expressive power of sequential intersection Pierre VIAL Equipe Gallinette Inria (LS2N CNRS) June 10, 2018 Non-idempotent typing P. Vial 0 1 /33 Outline Non-idempotent Rigid vs.

  1. Intersection types (Coppo-Dezani 80) Type constructors: o ∈ O , → and ∧ (intersection). Strict types: no ∧ on the right h.s. of → ( e.g. , ( A ∧ B ) → A , not A → ( B ∧ C )) � no intro/elim. rules for ∧ ( A ∧ B ) ∧ C ∼ A ∧ ( B ∧ C ), A ∧ B ∼ B ∧ A ( assoc. and comm. ) � subtyping or permutation rules e.g., Γ , x : A 1 ∧ A 2 ∧ . . . ∧ A n ⊢ t : B ρ ∈ S n perm Γ , x : A ρ (1) ∧ . . . ∧ A ρ ( n ) ⊢ t : B Idempotency? A ∧ A ∼ A (Coppo-D) or not (Gardner 94-de Carvalho 07) idem: typing = qualitative info non-idem: qual. and quant. 1 Non-idempotent intersection types Non-idempotent typing P. Vial 8 /33

  2. Intersection types (Coppo-Dezani 80) Type constructors: o ∈ O , → and ∧ (intersection). Strict types: no ∧ on the right h.s. of → ( e.g. , ( A ∧ B ) → A , not A → ( B ∧ C )) � no intro/elim. rules for ∧ ( A ∧ B ) ∧ C ∼ A ∧ ( B ∧ C ), A ∧ B ∼ B ∧ A ( assoc. and comm. ) � subtyping or permutation rules e.g., Γ , x : A 1 ∧ A 2 ∧ . . . ∧ A n ⊢ t : B ρ ∈ S n perm Γ , x : A ρ (1) ∧ . . . ∧ A ρ ( n ) ⊢ t : B Idempotency? A ∧ A ∼ A (Coppo-D) or not (Gardner 94-de Carvalho 07) idem: typing = qualitative info non-idem: qual. and quant. Collapsing A ∧ B ∧ C into [ A, B, C ] ( multiset ) � no need for perm rules etc. [ A, B, A ] = [ A, B, A ] � = [ A, B ] [ A, B, A ] = [ A, B ] + [ A ] 1 Non-idempotent intersection types Non-idempotent typing P. Vial 8 /33

  3. System R 0 (Gardner-de Carvalho) (Strict Types) τ, σ := o ∈ O | I → τ (Intersection Types) I := [ σ i ] i ∈ I Strict types � syntax directed rules : Γ; x : [ σ i ] i ∈ I ⊢ t : τ x : [ τ ] ⊢ x : τ ax Γ ⊢ λx.t : [ σ i ] i ∈ I → τ abs Γ ⊢ t : [ σ i ] i ∈ I → τ (Γ i ⊢ u : σ i ) i ∈ I app Γ + i ∈ I Γ i ⊢ t u : τ Remark Relevant system (no weakening) In app -rule, pointwise multiset sum e.g. , ( x : [ σ ]; y : [ τ ]) + ( x : [ σ, τ ]) = x : [ σ, σ, τ ]; y : [ τ ] 1 Non-idempotent intersection types Non-idempotent typing P. Vial 9 /33

  4. Head Normalization ( λ ) r s λx t 1 t 1 x @ @ @ t q t q head redex head variable @ @ λxp λxp Head Normal Form Head Reducible Term 1 Non-idempotent intersection types Non-idempotent typing P. Vial 10 /33

  5. Head Normalization ( λ ) r s λx t 1 t 1 x @ @ @ t q t q head redex head variable @ @ λxp λxp Head Normal Form Head Reducible Term t is head normalizing (HN) if ∃ reduction path from t to a HNF. 1 Non-idempotent intersection types Non-idempotent typing P. Vial 10 /33

  6. Head Normalization ( λ ) r s λx t 1 t 1 x @ @ @ t q t q head redex head variable @ @ λxp λxp Head Normal Form Head Reducible Term t is head normalizing (HN) if ∃ reduction path from t to a HNF. The head reduction strategy : reducing head redexes while it is possible. 1 Non-idempotent intersection types Non-idempotent typing P. Vial 10 /33

  7. Head Normalization ( λ ) t is head normalizing (HN) if ∃ reduction path from t to a HNF. The head reduction strategy : reducing head redexes while it is possible. 1 Non-idempotent intersection types Non-idempotent typing P. Vial 10 /33

  8. Head Normalization ( λ ) obvious t is HN the head reduction strategy ( ∃ path from t to a HNF) terminates on t true but not obvious t is head normalizing (HN) if ∃ reduction path from t to a HNF. The head reduction strategy : reducing head redexes while it is possible. 1 Non-idempotent intersection types Non-idempotent typing P. Vial 10 /33

  9. Head Normalization ( λ ) obvious t is HN the head reduction strategy ( ∃ path from t to a HNF) terminates on t true but not obvious The head reduction strategy : reducing head redexes while it is possible. 1 Non-idempotent intersection types Non-idempotent typing P. Vial 10 /33

  10. Head Normalization ( λ ) obvious t is HN the head reduction strategy ( ∃ path from t to a HNF) terminates on t true but not obvious Intersection types come to help! The head reduction strategy : reducing head redexes while it is possible. 1 Non-idempotent intersection types Non-idempotent typing P. Vial 10 /33

  11. Subject Reduction and Subject Expansion A good intersection type system should enjoy: Subject Expansion (SE) : Subject Reduction (SR) : Typing is stable under anti- Typing is stable under reduction. reduction. SE is usually not verified by simple or polymorphic type systems 1 Non-idempotent intersection types Non-idempotent typing P. Vial 11 /33

  12. Subject Reduction and Subject Expansion A good intersection type system should enjoy: Subject Expansion (SE) : Subject Reduction (SR) : Typing is stable under anti- Typing is stable under reduction. reduction. SE is usually not verified by simple or polymorphic type systems t is typable typing the term. states + SE SR + extra arg. t can reach a Some reduction strategy terminal state normalizes t obvious 1 Non-idempotent intersection types Non-idempotent typing P. Vial 11 /33

  13. Properties ( R 0 ) Weighted Subject Reduction Reduction preserves types and environments, and. . . . . . head reduction strictly decreases the nodes of the deriv. tree ( size ). Subject Expansion Anti-reduction preserves types and environments. Theorem (de Carvalho) Let t be a λ -term. Then equivalence between: 1 t is typable (in R 0 ) 2 t is HN 3 the head reduction strategy terminates on t ( � certification!) Bonus (quantitative information) If Π types t , then size (Π) bounds the number of steps of the head red. strategy on t 1 Non-idempotent intersection types Non-idempotent typing P. Vial 12 /33

  14. Subject reduction and expansion in R 0 From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 2 ] ⊢ x : σ 2 Π a Π b Π 2 1 1 Γ; x :[ σ 1 , σ 2 , σ 1 ] ⊢ r : τ abs ∆ a ∆ b Γ ⊢ λx.r : [ σ 1 , σ 2 , σ 1 ] → τ 1 ⊢ s : σ 1 ∆ 2 ⊢ s : σ 2 1 ⊢ s : σ 1 app Γ + ∆ a 1 + ∆ b 1 + ∆ 2 ⊢ ( λx.r ) s : τ 1 Non-idempotent intersection types Non-idempotent typing P. Vial 13 /33

  15. Subject reduction and expansion in R 0 From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 2 ] ⊢ x : σ 2 Π a Π b Π 2 1 1 Γ; x :[ σ 1 , σ 2 , σ 1 ] ⊢ r : τ abs ∆ a ∆ b Γ ⊢ λx.r : [ σ 1 , σ 2 , σ 1 ] → τ 1 ⊢ s : σ 1 ∆ 2 ⊢ s : σ 2 1 ⊢ s : σ 1 app Γ + ∆ a 1 + ∆ b 1 + ∆ 2 ⊢ ( λx.r ) s : τ 1 Non-idempotent intersection types Non-idempotent typing P. Vial 13 /33

  16. Subject reduction and expansion in R 0 From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 2 ] ⊢ x : σ 2 Π a Π b Π 2 1 1 Γ; x :[ σ 1 , σ 2 , σ 1 ] ⊢ r : τ abs ∆ a ∆ b Γ ⊢ λx.r : [ σ 1 , σ 2 , σ 1 ] → τ 1 ⊢ s : σ 1 ∆ 2 ⊢ s : σ 2 1 ⊢ s : σ 1 app Γ + ∆ a 1 + ∆ b 1 + ∆ 2 ⊢ ( λx.r ) s : τ 1 Non-idempotent intersection types Non-idempotent typing P. Vial 13 /33

  17. Subject reduction and expansion in R 0 From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 2 ] ⊢ x : σ 2 Π a Π b Π 2 1 1 Γ; x :[ σ 1 , σ 2 , σ 1 ] ⊢ r : τ abs ∆ a ∆ b Γ ⊢ λx.r : [ σ 1 , σ 2 , σ 1 ] → τ 1 ⊢ s : σ 1 ∆ 2 ⊢ s : σ 2 1 ⊢ s : σ 1 app Γ + ∆ a 1 + ∆ b 1 + ∆ 2 ⊢ ( λx.r ) s : τ 1 Non-idempotent intersection types Non-idempotent typing P. Vial 13 /33

  18. Subject reduction and expansion in R 0 From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 1 ] ⊢ x : σ 1 By relevance and non-idempotence ! ax x :[ σ 2 ] ⊢ x : σ 2 Π a Π b Π 2 1 1 Γ; x :[ σ 1 , σ 2 , σ 1 ] ⊢ r : τ abs ∆ a ∆ b Γ ⊢ λx.r : [ σ 1 , σ 2 , σ 1 ] → τ 1 ⊢ s : σ 1 ∆ 2 ⊢ s : σ 2 1 ⊢ s : σ 1 app Γ + ∆ a 1 + ∆ b 1 + ∆ 2 ⊢ ( λx.r ) s : τ 1 Non-idempotent intersection types Non-idempotent typing P. Vial 13 /33

  19. Subject reduction and expansion in R 0 From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 2 ] ⊢ x : σ 2 Π a Π b Π 2 1 1 Γ; x :[ σ 1 , σ 2 , σ 1 ] ⊢ r : τ abs ∆ a ∆ b Γ ⊢ λx.r : [ σ 1 , σ 2 , σ 1 ] → τ 1 ⊢ s : σ 1 ∆ 2 ⊢ s : σ 2 1 ⊢ s : σ 1 app Γ + ∆ a 1 + ∆ b 1 + ∆ 2 ⊢ ( λx.r ) s : τ 1 Non-idempotent intersection types Non-idempotent typing P. Vial 13 /33

  20. Subject reduction and expansion in R 0 From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 1 ] ⊢ x : σ 1 ax x :[ σ 2 ] ⊢ x : σ 2 Π a Π b Π 2 1 1 Γ; x :[ σ 1 , σ 2 , σ 1 ] ⊢ r : τ abs ∆ a ∆ b Γ ⊢ λx.r : [ σ 1 , σ 2 , σ 1 ] → τ 1 ⊢ s : σ 1 ∆ 2 ⊢ s : σ 2 1 ⊢ s : σ 1 app Γ + ∆ a 1 + ∆ b 1 + ∆ 2 ⊢ ( λx.r ) s : τ 1 Non-idempotent intersection types Non-idempotent typing P. Vial 13 /33

  21. Subject reduction and expansion in R 0 From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] Π b 1 Π a 1 ∆ b 1 ⊢ s : σ 1 Π 2 ∆ a 1 ⊢ s : σ 1 ∆ 2 ⊢ s : σ 2 Γ + ∆ a 1 + ∆ b 1 + ∆ 2 ⊢ r [ s/x ] : τ 1 Non-idempotent intersection types Non-idempotent typing P. Vial 13 /33

  22. Subject reduction and expansion in R 0 From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] Π b 1 Π a 1 ∆ b 1 ⊢ s : σ 1 Non-determinism of SR Π 2 ∆ a 1 ⊢ s : σ 1 ∆ 2 ⊢ s : σ 2 Γ + ∆ a 1 + ∆ b 1 + ∆ 2 ⊢ r [ s/x ] : τ 1 Non-idempotent intersection types Non-idempotent typing P. Vial 13 /33

  23. Subject reduction and expansion in R 0 From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] Π a 1 Π b 1 ∆ a 1 ⊢ s : σ 1 Non-determinism of SR Π 2 ∆ b 1 ⊢ s : σ 1 ∆ 2 ⊢ s : σ 2 Γ + ∆ a 1 + ∆ b 1 + ∆ 2 ⊢ r [ s/x ] : τ 1 Non-idempotent intersection types Non-idempotent typing P. Vial 13 /33

  24. Outline Non-idempotent Rigid vs. non-rigid Context intersection types paradigms Using type A once or twice Proof red. not the same deterministic vs. non-deterministic Question 1 Rigid collapses on non-rigid Is this collapse surjective? ( A, A, B ) and ( A, B, A ) collapse on [ A, A, B ] In rigid fw., red. paths Is it possible to capture red. Question 2 captured by permutations paths without perm. ? ( A, B, A ) �→ ( A, A, B ) All this in a coinductive fw. (no productivity)! 1 Non-idempotent intersection types Non-idempotent typing P. Vial 14 /33

  25. Plan 1 Non-idempotent intersection types 2 System S (Sequential Interection) 3 Encoding Reduction Paths 4 Perspectives 2 System S (Sequential Interection) Non-idempotent typing P. Vial 15 /33

  26. Motivations Multiset intersection: ⊕ syntax-direction ⊖ non-determinism of proof red. ⊖ lack tracking: [ σ, τ, σ ] = [ σ ? , τ ] + [ σ ? ]. 2 System S (Sequential Interection) Non-idempotent typing P. Vial 16 /33

  27. Motivations Multiset intersection: ⊕ syntax-direction ⊖ non-determinism of proof red. ⊖ lack tracking: [ σ, τ, σ ] = [ σ ? , τ ] + [ σ ? ]. Klop’s Problem: can the set of ∞ -WN terms be characterized by an ITS ? Def: t is ∞ -WN iff its B¨ ohm tree does not contain ⊥ Tatsuta [07]: an inductive ITS cannot do it. Can a coinductive ITS characterize the set of ∞ -WN terms? 2 System S (Sequential Interection) Non-idempotent typing P. Vial 16 /33

  28. Motivations Multiset intersection: ⊕ syntax-direction ⊖ non-determinism of proof red. ⊖ lack tracking: [ σ, τ, σ ] = [ σ ? , τ ] + [ σ ? ]. Klop’s Problem: can the set of ∞ -WN terms be characterized by an ITS ? Def: t is ∞ -WN iff its B¨ ohm tree does not contain ⊥ Tatsuta [07]: an inductive ITS cannot do it. Can a coinductive ITS characterize the set of ∞ -WN terms? Answer: Impossible without tracking (need for a validity criterion). system R ( i.e. R 0 with a coinductive type grammar) does not work YES , with inter. = sequences + validity criterion . 2 System S (Sequential Interection) Non-idempotent typing P. Vial 16 /33

  29. Sequential intersection Strict Types: S k , T ::= o ∈ O | ( k · S k ) k ∈ K → T Sequence Types ( k · S k ) k ∈ K Example: (7 · o 1 , 3 · o 2 , 2 · o 1 ) → o o 1 o 2 o 1 o 7 3 2 1 7, 3, 2, 1 = “tracks” → Tracking: (3 · σ, 5 · τ, 9 · σ ) = (3 · σ, 5 · τ ) ⊎ (9 · σ ) vs. [ σ, τ, σ ] = [ σ , τ ] + [ σ ] 2 System S (Sequential Interection) Non-idempotent typing P. Vial 17 /33

  30. Sequential intersection Strict Types: S k , T ::= o ∈ O | ( k · S k ) k ∈ K → T Sequence Types ( k · S k ) k ∈ K Example: (7 · o 1 , 3 · o 2 , 2 · o 1 ) → o o 1 o 2 o 1 o 7 3 2 1 7, 3, 2, 1 = “tracks” → Tracking: (3 · σ, 5 · τ, 9 · σ ) = (3 · σ, 5 · τ ) ⊎ (9 · σ ) vs. [ σ, τ, σ ] = [ σ , τ ] + [ σ ] 2 System S (Sequential Interection) Non-idempotent typing P. Vial 17 /33

  31. Sequential intersection Strict Types: S k , T ::= o ∈ O | ( k · S k ) k ∈ K → T Sequence Types ( k · S k ) k ∈ K Example: (7 · o 1 , 3 · o 2 , 2 · o 1 ) → o o 1 o 2 o 1 o 7 3 2 1 7, 3, 2, 1 = “tracks” → Tracking: (3 · σ, 5 · τ, 9 · σ ) = (3 · σ, 5 · τ ) ⊎ (9 · σ ) vs. [ σ, τ, σ ] = [ σ , τ ] + [ σ ] 2 System S (Sequential Interection) Non-idempotent typing P. Vial 17 /33

  32. Sequential intersection Strict Types: S k , T ::= o ∈ O | ( k · S k ) k ∈ K → T Sequence Types ( k · S k ) k ∈ K Example: (7 · o 1 , 3 · o 2 , 2 · o 1 ) → o o 1 o 2 o 1 o 7 3 2 1 7, 3, 2, 1 = “tracks” → Tracking: (3 · σ, 5 · τ, 9 · σ ) = (3 · σ, 5 · τ ) ⊎ (9 · σ ) vs. [ σ, τ, σ ] = [ σ , τ ] + [ σ ] 2 System S (Sequential Interection) Non-idempotent typing P. Vial 17 /33

  33. Sequential intersection Strict Types: S k , T ::= o ∈ O | ( k · S k ) k ∈ K → T Sequence Types ( k · S k ) k ∈ K Example: (7 · o 1 , 3 · o 2 , 2 · o 1 ) → o o 1 o 2 o 1 o 7 3 2 1 7, 3, 2, 1 = “tracks” → Tracking: (3 · σ, 5 · τ, 9 · σ ) = (3 · σ, 5 · τ ) ⊎ (9 · σ ) vs. [ σ, τ, σ ] = [ σ ? , τ ] + [ σ ? ] 2 System S (Sequential Interection) Non-idempotent typing P. Vial 17 /33

  34. Derivations of S C ; x : ( S k ) k ∈ K ⊢ t : T x : ( k · T ) ⊢ x : T ax C ⊢ λx.t : ( S k ) k ∈ K → T abs C ⊢ t : ( S k ) k ∈ K → T ( D k ⊢ u : S k ) k ∈ K app C ⊎ ( ⊎ k ∈ K D k ) ⊢ t u : T 2 System S (Sequential Interection) Non-idempotent typing P. Vial 18 /33

  35. Derivations of S C ; x : ( S k ) k ∈ K ⊢ t : T x : ( k · T ) ⊢ x : T ax C ⊢ λx.t : ( S k ) k ∈ K → T abs C ⊢ t : ( S k ) k ∈ K → T ( D k ⊢ u : S k ) k ∈ K app C ⊎ ( ⊎ k ∈ K D k ) ⊢ t u : T System S features pointers (called bipositions ). 2 System S (Sequential Interection) Non-idempotent typing P. Vial 18 /33

  36. Derivations of S C ; x : ( S k ) k ∈ K ⊢ t : T x : ( k · T ) ⊢ x : T ax C ⊢ λx.t : ( S k ) k ∈ K → T abs C ⊢ t : ( S k ) k ∈ K → T ( D k ⊢ u : S k ) k ∈ K app C ⊎ ( ⊎ k ∈ K D k ) ⊢ t u : T System S features pointers (called bipositions ). Every S -derivation collapses on a R -derivation. 2 System S (Sequential Interection) Non-idempotent typing P. Vial 18 /33

  37. Derivations of S C ; x : ( S k ) k ∈ K ⊢ t : T x : ( k · T ) ⊢ x : T ax C ⊢ λx.t : ( S k ) k ∈ K → T abs C ⊢ t : ( S k ) k ∈ K → T ( D k ⊢ u : S k ) k ∈ K app C ⊎ ( ⊎ k ∈ K D k ) ⊢ t u : T System S features pointers (called bipositions ). Every S -derivation collapses on a R -derivation. Subject reduction is deterministic in S ( � = R ). 2 System S (Sequential Interection) Non-idempotent typing P. Vial 18 /33

  38. Infinitary Typing Theorem (V,LiCS17) A ∞ -term t is ∞ -WN iff t is S -typable in some way. � Klop’s Problem solved The hereditary head reduction strategy is complete for infinitary weak normalization. 2 System S (Sequential Interection) Non-idempotent typing P. Vial 19 /33

  39. Infinitary Typing Theorem (V,LiCS17) A ∞ -term t is ∞ -WN iff t is S -typable in some way. � Klop’s Problem solved The hereditary head reduction strategy is complete for infinitary weak normalization. Bonus (positive answer to TLCA Problem #20) System S also provides a type-theoretic characterization of the hereditary permutations (not possible in the inductive case, Tatsuta [LiCS07]). 2 System S (Sequential Interection) Non-idempotent typing P. Vial 19 /33

  40. Infinitary Typing Theorem (V,LiCS17) A ∞ -term t is ∞ -WN iff t is S -typable in some way. � Klop’s Problem solved The hereditary head reduction strategy is complete for infinitary weak normalization. Bonus (positive answer to TLCA Problem #20) System S also provides a type-theoretic characterization of the hereditary permutations (not possible in the inductive case, Tatsuta [LiCS07]). Theorem (V,LiCS18) Every term is typable in systems R and S (non-trivial). One can extract from the R -typing the order (arity) of any λ -term. In the infinitary relational model, no term has an empty denotation. 2 System S (Sequential Interection) Non-idempotent typing P. Vial 19 /33

  41. The Problem of the Collapse Coinductive typing (without validity criterion): allow to type all normalizing terms + some unproductive terms e.g. , Ω. 2 System S (Sequential Interection) Non-idempotent typing P. Vial 20 /33

  42. The Problem of the Collapse Coinductive typing (without validity criterion): allow to type all normalizing terms + some unproductive terms e.g. , Ω. Necessity to replace R (multiset inter.) with S (sequence inter) 2 System S (Sequential Interection) Non-idempotent typing P. Vial 20 /33

  43. The Problem of the Collapse Coinductive typing (without validity criterion): allow to type all normalizing terms + some unproductive terms e.g. , Ω. Necessity to replace R (multiset inter.) with S (sequence inter) But do we lose some derivations? Question: given Π a R -derivation, is there a S -deriv. P collapsing on Π? if true, the infinitary relational model is fully described by system S 2 System S (Sequential Interection) Non-idempotent typing P. Vial 20 /33

  44. The Problem of the Collapse Coinductive typing (without validity criterion): allow to type all normalizing terms + some unproductive terms e.g. , Ω. Necessity to replace R (multiset inter.) with S (sequence inter) But do we lose some derivations? Question: given Π a R -derivation, is there a S -deriv. P collapsing on Π? if true, the infinitary relational model is fully described by system S Easy in the case of normal forms ( i.e. when Π types a NF), not in other cases. 2 System S (Sequential Interection) Non-idempotent typing P. Vial 20 /33

  45. Difficulties [ ] → . . . → [ ] → o In the productive cases (HN,WN,SN, ∞ -WN), in i.t.s., one t 1 types the normal forms and uses x subject expansion. @ t q normalizing terms ⊆ typable terms @ Here, no form of productivity/stabilization. o λx p We develop a corpus of methods inspired by first order model λx 1 theory (last part of the talk). . . . → . . . → . . . → o 2 System S (Sequential Interection) Non-idempotent typing P. Vial 21 /33

  46. Outline Non-idempotent Rigid vs. non-rigid Context intersection types paradigms Using type A once or twice Proof red. not the same deterministic vs. non-deterministic Question 1 Rigid collapses on non-rigid Is this collapse surjective? ( A, A, B ) and ( A, B, A ) collapse on [ A, A, B ] In rigid fw., red. paths Is it possible to capture red. Question 2 captured by permutations paths without perm. ? ( A, B, A ) �→ ( A, A, B ) All this in a coinductive fw. (no productivity)! 2 System S (Sequential Interection) Non-idempotent typing P. Vial 22 /33

  47. Plan 1 Non-idempotent intersection types 2 System S (Sequential Interection) 3 Encoding Reduction Paths 4 Perspectives 3 Encoding Reduction Paths Non-idempotent typing P. Vial 23 /33

  48. Deterministic Subject Reduction From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r P 1 P 2 s s reduces into. . . λx @ T 3 Encoding Reduction Paths Non-idempotent typing P. Vial 24 /33

  49. Deterministic Subject Reduction From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r P 1 P 2 s s reduces into. . . λx (2 · S, 7 · S ) → T @ T 3 Encoding Reduction Paths Non-idempotent typing P. Vial 24 /33

  50. Deterministic Subject Reduction From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r P 1 P 2 s s S [7] reduces into. . . λx S [2] (2 · S, 7 · S ) → T @ T 3 Encoding Reduction Paths Non-idempotent typing P. Vial 24 /33

  51. Deterministic Subject Reduction From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r T P 1 P 2 s s S [7] reduces into. . . λx S [2] (2 · S, 7 · S ) → T @ T 3 Encoding Reduction Paths Non-idempotent typing P. Vial 24 /33

  52. Deterministic Subject Reduction From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r ax x : (2 · S ) ax x : (7 · S ) T P 1 P 2 s s S [7] reduces into. . . λx S [2] (2 · S, 7 · S ) → T @ T 3 Encoding Reduction Paths Non-idempotent typing P. Vial 24 /33

  53. Deterministic Subject Reduction From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r [ s/x ] P r P 1 s ax x : (2 · S ) s : S P 2 s ax x : (7 · S ) s : S T T P 1 P 2 s s S [7] reduces into. . . λx S [2] (2 · S, 7 · S ) → T @ T 3 Encoding Reduction Paths Non-idempotent typing P. Vial 24 /33

  54. How to encode reduction paths? System S : one red. path, poor dynamic behavior. System R : rich dynamic behavior, impossible to express red. paths (lack of tracking) Idea 1: use iso. of types (iso of lab. trees) o 3 o 1 o 2 o 1 o 3 o 2 8 3 → 1 7 2 → 1 o 2 o 1 o 2 o 1 8 4 → 1 5 3 → 1 T 1 = (8 · o 2 , 4 · (8 · o 3 , 3 · o 1 ) → o 2 ) → o 1 T 2 = (5 · (7 · o 1 , 2 · o 3 ) → o 2 , 3 · o 2 ) → o 1 Idea 2: replace app (syntactic eq.) with app h (eq. up to iso) ( D k ⊢ u : S ′ ( S k ) k ∈ K ≡ ( S ′ C ⊢ t : ( S k ) k ∈ K → T k ) k ∈ K ′ k ) k ∈ K ′ app h C ⊎ ( ⊎ k ∈ K D k ) ⊢ t u : T Hybrid system S h : every R -deriv. is a S h -collapse (easy). 3 Encoding Reduction Paths Non-idempotent typing P. Vial 25 /33

  55. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r P 5 P 8 s s reduces into. . . λx @ T 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  56. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r P 5 P 8 s s reduces into. . . λx (2 · S 2 , 7 · S 7 ) → T @ T 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  57. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r P 5 P 8 s s 8 [8] reduces into. . . λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  58. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r T P 5 P 8 s s 8 [8] reduces into. . . λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  59. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r ax x : (2 · S 2 ) ax x : (7 · S 7 ) T P 5 P 8 s s 8 [8] reduces into. . . λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  60. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r ax x : (2 · S 2 ) ax x : (7 · S 7 ) T P 5 P 8 s s 8 [8] reduces into. . . λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 �≡ S 7 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  61. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r ax x : (2 · S 2 ) ax x : (7 · S 7 ) T P 5 P 8 s s 8 [8] reduces into. . . λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 �≡ S 7 say S 2 ≡ S ′ 8 , S 7 ≡ S ′ 5 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  62. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r ax x : (2 · S 2 ) ax x : (7 · S 7 ) T P 5 P 8 s s 8 [8] reduces into. . . λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 �≡ S 7 say S 2 ≡ S ′ 8 , S 7 ≡ S ′ 5 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  63. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r [ s/x ] P r P 8 s ax x : (2 · S 2 ) s : S ′ P 5 8 s ax x : (7 · S 7 ) s : S ′ 5 T T ′ P 5 P 8 s s 8 [8] reduces into λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 �≡ S 7 say S 2 ≡ S ′ 8 , S 7 ≡ S ′ 5 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  64. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r [ s/x ] P r P 8 s ax x : (2 · S 2 ) s : S ′ P 5 8 s ax x : (7 · S 7 ) s : S ′ 5 T T ′ P 5 P 8 s s 8 [8] reduces into λx S ′ S ′ 5 [5] with T ′ ≡ T (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 �≡ S 7 say S 2 ≡ S ′ 8 , S 7 ≡ S ′ 5 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  65. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r ax x : (2 · S 2 ) ax x : (7 · S 7 ) T P 5 P 8 s s 8 [8] reduces into. . . λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  66. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r ax x : (2 · S 2 ) ax x : (7 · S 7 ) T P 5 P 8 s s 8 [8] reduces into. . . λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 ≡ S 7 s.t. S 2 ≡ S 7 ≡ S ′ 5 ≡ S ′ 8 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  67. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r ax x : (2 · S 2 ) ax x : (7 · S 7 ) T P 5 P 8 s s 8 [8] reduces into λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 ≡ S 7 s.t. S 2 ≡ S 7 ≡ S ′ 5 ≡ S ′ 8 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  68. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r [ s/x ] P r P ? s ax x : (2 · S 2 ) s : S ′ P ? ? s ax x : (7 · S 7 ) s : S ′ ? T T ′ P 5 P 8 s s 8 [8] reduces into λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T @ T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 ≡ S 7 s.t. S 2 ≡ S 7 ≡ S ′ 5 ≡ S ′ 8 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  69. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r [ s/x ] P r P ? s ax x : (2 · S 2 ) s : S ′ P ? ? s ax x : (7 · S 7 ) s : S ′ ? T T ′ P 5 P 8 s s 8 [8] reduces into λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T φ interface @ (2 · S 2 , 7 · S 7 ) ˜ → (5 · S ′ 5 , 8 · S ′ 8 ) T with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 ≡ S 7 s.t. S 2 ≡ S 7 ≡ S ′ 5 ≡ S ′ 8 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  70. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r [ s/x ] P r P 8 s ax x : (2 · S 2 ) s : S ′ P 5 8 s ax x : (7 · S 7 ) s : S ′ 5 T T ′ P 5 P 8 s s 8 [8] reduces into λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T φ interface @ (2 · S 2 , 7 · S 7 ) ˜ → (5 · S ′ 5 , 8 · S ′ 8 ) T case φ : 2 �→ 8 , 7 �→ 5 with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 ≡ S 7 s.t. S 2 ≡ S 7 ≡ S ′ 5 ≡ S ′ 8 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  71. Pseudo-Subject Reduction in S h From a typing of ( λx.r ) s . . . to a typing of r [ s/x ] P r [ s/x ] P r P 5 s ax x : (2 · S 2 ) s : S ′ P 8 5 s ax x : (7 · S 7 ) s : S ′ 8 T T ′ P 5 P 8 s s 8 [8] reduces into λx S ′ S ′ 5 [5] (2 · S 2 , 7 · S 7 ) → T φ interface @ (2 · S 2 , 7 · S 7 ) ˜ → (5 · S ′ 5 , 8 · S ′ 8 ) T case φ : 2 �→ 5 , 7 �→ 8 with (2 · S 2 , 7 · S 7 ) ≡ (5 · S ′ 5 , 8 · S ′ 8 ) Assume S 2 ≡ S 7 s.t. S 2 ≡ S 7 ≡ S ′ 5 ≡ S ′ 8 3 Encoding Reduction Paths Non-idempotent typing P. Vial 26 /33

  72. Operable Derivations hybrid deriv. + interfaces for each app h -rule = operable derivation 3 Encoding Reduction Paths Non-idempotent typing P. Vial 27 /33

  73. Operable Derivations hybrid deriv. + interfaces for each app h -rule = operable derivation system S op : deterministic with hard-coded red. paths. S -derivations: identity interfaces ( trivial op. deriv.) 3 Encoding Reduction Paths Non-idempotent typing P. Vial 27 /33

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Local convergence for random permutations The case of uniform pattern-avoiding permutations

Local convergence for random permutations The case of uniform pattern-avoiding permutations

Local convergence for random permutations The case of uniform pattern-avoiding permutations Jacopo Borga, Institut fr Mathematik, Universitt Zrich July 9, 2018 Dartmouth College, Hanover, New Hampshire Our goal 1. Scaling limits: Our

2.18k views • 175 slides
JUST THE MATHS SLIDES NUMBER 19.2 PROBABILITY 2 (Permutations and combinations) by

JUST THE MATHS SLIDES NUMBER 19.2 PROBABILITY 2 (Permutations and combinations) by

JUST THE MATHS SLIDES NUMBER 19.2 PROBABILITY 2 (Permutations and combinations) by A.J.Hobson 19.2.1 Introduction 19.2.2 Rules of permutations and combinations 19.2.3 Permutations of sets with some objects alike UNIT 19.2 -

400 views • 11 slides
Posets and Permutations in the Duplication-Loss Model: Minimal Permutations with d Descents.

Posets and Permutations in the Duplication-Loss Model: Minimal Permutations with d Descents.

Posets and Permutations in the Duplication-Loss Model: Minimal Permutations with d Descents. Mathilde Bouvel Elisa Pergola GASCom 2008 liafa Definitions Duplication-Loss Characterization Posets Enumeration Conclusion Outline of the talk 1

620 views • 31 slides
A Note on 5-bit Quadratic Permutations Classification Duan Boilov Begl Bilgin Hac

A Note on 5-bit Quadratic Permutations Classification Duan Boilov Begl Bilgin Hac

A Note on 5-bit Quadratic Permutations Classification Duan Boilov Begl Bilgin Hac Ali ahin March 6, 2017 Motivation 2/14 Permutations are main nonlinear part of symmetric primitives Quadratic permutations can be used to

381 views • 22 slides
Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1 Definitions A sequence a 1 , a 2 , . . . , a k of distinct integers is alternating if a 1 > a 2 < a 3 > a 4 < , and reverse alternating if

1.14k views • 98 slides
Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1 Basic

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1 Basic

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1 Basic definitions A sequence a 1 , a 2 , . . . , a k of distinct integers is alternating if a 1 > a 2 < a 3 > a 4 < , and reverse

1.59k views • 119 slides
Random permutations with logarithmic cycle Random permutations weights Classical measures The

Random permutations with logarithmic cycle Random permutations weights Classical measures The

Random permutations with logarithmic cycle weights Setting the scene Random permutations with logarithmic cycle Random permutations weights Classical measures The weighted measure The cycle counts C m Dirk Zeindler The cycle containing

1.6k views • 149 slides
Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. Basic

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. Basic

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. Basic definitions A sequence a 1 , a 2 , . . . , a k of distinct integers is alternating if a 1 > a 2 < a 3 > a 4 < , and reverse alternating

1.4k views • 119 slides
Algorithms and Pattern Avoiding Permutations Mikl os B ona Department of Mathematics

Algorithms and Pattern Avoiding Permutations Mikl os B ona Department of Mathematics

Algorithms and Pattern Avoiding Permutations Mikl os B ona Department of Mathematics University of Florida Gainesville FL 32611-8105 bona@ufl.edu May 27, 2013 Permutations A permutation is an arrangement of the integers { 1 , 2 ,

573 views • 40 slides
MATH 105: Finite Mathematics 6-4: Permutations Prof. Jonathan Duncan Walla Walla College Winter

MATH 105: Finite Mathematics 6-4: Permutations Prof. Jonathan Duncan Walla Walla College Winter

Types of Permutations Factorials Applications of Permutations Conclusion MATH 105: Finite Mathematics 6-4: Permutations Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Types of Permutations Factorials Applications of

959 views • 62 slides
s II, Acta Math. Acad. Sci. Hungar. 18 (1967), s S 151164; III, ibid. 18 (1967),

s II, Acta Math. Acad. Sci. Hungar. 18 (1967), s S 151164; III, ibid. 18 (1967),

Extremal problems Permutations How many permutations in a set (or group) with prescribed distances? Peter J. Cameron

524 views • 3 slides
Packing patterns in restricted permutations Lara Pudwell faculty.valpo.edu/lpudwell Special

Packing patterns in restricted permutations Lara Pudwell faculty.valpo.edu/lpudwell Special

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing patterns in restricted permutations Lara Pudwell faculty.valpo.edu/lpudwell Special Session on Patterns in Permutations AMS Fall

828 views • 50 slides
Enumeration of Pin-Permutations Fr ed erique Bassino Mathilde Bouvel Dominique Rossin ees

Enumeration of Pin-Permutations Fr ed erique Bassino Mathilde Bouvel Dominique Rossin ees

Enumeration of Pin-Permutations Fr ed erique Bassino Mathilde Bouvel Dominique Rossin ees AL Journ EA 2009 liafa Introduction Pin-permutations Decomposition tree Characterization Generating function Conclusion Main result of

665 views • 40 slides
Embeddings of Metrics on Strings and Permutations Graham Cormode joint work with S.

Embeddings of Metrics on Strings and Permutations Graham Cormode joint work with S.

Embeddings of Metrics on Strings and Permutations Graham Cormode joint work with S. Muthukrishnan, Cenk Sahinalp Miss Hepburn runs the gamut of emotions from A to B Dorothy Parker, 1933 Permutations and Strings Strings Web pages,

922 views • 42 slides
Discrete Mathematics in Computer Science Permutations Malte Helmert, Gabriele R oger

Discrete Mathematics in Computer Science Permutations Malte Helmert, Gabriele R oger

Discrete Mathematics in Computer Science Permutations Malte Helmert, Gabriele R oger University of Basel Permutations as Functions A permutation rearranges objects. Consider for example sequence o 2 , o 1 , o 3 , o 4 Lets rearrange the

634 views • 48 slides
Permutations and Combinations Rosen, Chapter 5.3 Motivating question In a family of 3, how

Permutations and Combinations Rosen, Chapter 5.3 Motivating question In a family of 3, how

10/17/2016 Permutations and Combinations Rosen, Chapter 5.3 Motivating question In a family of 3, how many ways can we arrange the members of the family in a line for a photograph? 1 10/17/2016 Permutations A permutation of a set of

591 views • 34 slides
Event Recording in Smart Room Ivan Galov, Rustam Kadirov, Andrew Vasilev, Dmitry Korzun

Event Recording in Smart Room Ivan Galov, Rustam Kadirov, Andrew Vasilev, Dmitry Korzun

Event Recording in Smart Room Ivan Galov, Rustam Kadirov, Andrew Vasilev, Dmitry Korzun Petrozavodsk State University Department of Computer Science Yaroslavl State University This project is supported by grant KA179 of Karelia ENPI joint

382 views • 14 slides
Password Authenticated Key Agreement for Contactless Smart Cards Dennis Kgler 1 , Heike Neumann

Password Authenticated Key Agreement for Contactless Smart Cards Dennis Kgler 1 , Heike Neumann

Password Authenticated Key Agreement for Contactless Smart Cards Dennis Kgler 1 , Heike Neumann 2 , Sebastian Stappert 2 , Markus Ullmann 1 , Matthias Vgeler 2 1 Bundesamt fr Sicherheit in der Informationstechnik 2 NXP Semiconductors Outline

174 views • 16 slides
Cloud Tutorial: AWS IoT CSE 520S Spring, Jan. 16, 2020 Ruixuan Dai XaaS: Basics in Cloud

Cloud Tutorial: AWS IoT CSE 520S Spring, Jan. 16, 2020 Ruixuan Dai XaaS: Basics in Cloud

Cloud Tutorial: AWS IoT CSE 520S Spring, Jan. 16, 2020 Ruixuan Dai XaaS: Basics in Cloud Computing Cloud Computing Cloud computing provides shared pool of configurable computing resource to end users on demand Three service models q IaaS

510 views • 47 slides
Smart programming languages, smart program analysis Varmo Vene Institute of Cybernetics at TUT

Smart programming languages, smart program analysis Varmo Vene Institute of Cybernetics at TUT

Smart programming languages, smart program analysis Varmo Vene Institute of Cybernetics at TUT & University of Tartu Introduction A quote from classics Everyone knows that debugging is twice as hard as writing a program in the first

371 views • 26 slides
Missing-data masks in all-combinations multi-band decoding It is shown that for MAP decoding

Missing-data masks in all-combinations multi-band decoding It is shown that for MAP decoding

morris@idiap.ch http://www.idiap.ch/ Missing-data masks in all-combinations multi-band decoding It is shown that for MAP decoding with all-comb experts multi-band expert weighting can make use of same soft missing-data mask as used with

477 views • 5 slides
Highly Granular Calorimeters: Technologies and Results Yong Liu Johannes Gutenberg-Universitt

Highly Granular Calorimeters: Technologies and Results Yong Liu Johannes Gutenberg-Universitt

Highly Granular Calorimeters: Technologies and Results Yong Liu Johannes Gutenberg-Universitt Mainz on behalf of the CALICE Collaboration Instrumentation for Colliding Beam Physics (INSTR17) Mar. 1, 2017, BINP Novosibirsk Highly granular

753 views • 35 slides
The bursty cosmic dawn Outline 1 Introduction Umberto Maio Motivations Leibniz Institute for

The bursty cosmic dawn Outline 1 Introduction Umberto Maio Motivations Leibniz Institute for

Introduction Method Simulations and observations The End The bursty cosmic dawn Outline 1 Introduction Umberto Maio Motivations Leibniz Institute for Astrophysics Potsdam (Germany) 2 Method INAF Osservatorio Astronomico di Trieste

344 views • 20 slides
Iterative Algorithms for Polynomial Eigenvalue Decomposition and Applications Stephan Weiss UDRC

Iterative Algorithms for Polynomial Eigenvalue Decomposition and Applications Stephan Weiss UDRC

Iterative Algorithms for Polynomial Eigenvalue Decomposition and Applications Stephan Weiss UDRC Loughborough, Surrey, Strathclyde & Cardiff Consortium Department of Electonic & Electrical Engineering University of Strathclyde,

871 views • 56 slides

More recommend