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

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
On the average number of reversals needed to sort signed permutations 1 Thaynara Arielly de Lima

On the average number of reversals needed to sort signed permutations 1 Thaynara Arielly de Lima

Introduction Sorting signed permutations Conclusion On the average number of reversals needed to sort signed permutations 1 Thaynara Arielly de Lima & 1 , 2 Mauricio Ayala-Rinc on atica 1 & Ci ao 2 Departamentos de Matem encia

795 views • 46 slides
Survey of Computer Science Computability & The Halting Problem Kevin Walsh

Survey of Computer Science Computability & The Halting Problem Kevin Walsh

Survey of Computer Science Computability & The Halting Problem Kevin Walsh kwalsh@holycross.edu Example List all possible orderings (permutations) of n letters Example: n = 3: A, B, C ABC, ACB, etc. 6 permutations n n! permutations

551 views • 29 slides
Decomposition of Permutations in a Finite Field SVETLA NIKOVA 1 , VENTZISLAV NIKOV 2 , AND VINCENT

Decomposition of Permutations in a Finite Field SVETLA NIKOVA 1 , VENTZISLAV NIKOV 2 , AND VINCENT

Decomposition of Permutations in a Finite Field SVETLA NIKOVA 1 , VENTZISLAV NIKOV 2 , AND VINCENT RIJMEN 1 1 IMEC COSIC, KU LEUVEN, BELGIUM 2 NXP SEMICONDUCTORS, BELGIUM Decomposition of Permutations in relation to Side Channel

415 views • 15 slides
Packing patterns in restricted permutations Lara Pudwell faculty.valpo.edu/lpudwell Rutgers

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

Introduction Packing with Classical Restrictions Packing in Alternating Permutations Wrapping up Packing patterns in restricted permutations Lara Pudwell faculty.valpo.edu/lpudwell Rutgers Experimental Math Seminar March 5, 2020 Packing

986 views • 79 slides
MA/CSSE 473 Day 16 Combinatorial Object Generation Permutations MA/CSSE 473 Day 16 No new

MA/CSSE 473 Day 16 Combinatorial Object Generation Permutations MA/CSSE 473 Day 16 No new

MA/CSSE 473 Day 16 Combinatorial Object Generation Permutations MA/CSSE 473 Day 16 No new announcements Student Questions Combinatorial Object Generation Intro Permutation generation Q1 Permutations Subsets

450 views • 15 slides
Permutations, Card Shuffling, and Representation Theory Franco Saliola, Universit du Qubec

Permutations, Card Shuffling, and Representation Theory Franco Saliola, Universit du Qubec

Permutations Card Shuffling Representation theory Permutations, Card Shuffling, and Representation Theory Franco Saliola, Universit du Qubec Montral Based on joint work with: Victor Reiner, University of Minnesota Volkmar Welker,

1.71k views • 146 slides
Realizing Site Permutations Stephane Durocher, Saeed Mehrabi, Debajyoti Mondal, Matthew Skala

Realizing Site Permutations Stephane Durocher, Saeed Mehrabi, Debajyoti Mondal, Matthew Skala

Realizing Site Permutations Stephane Durocher, Saeed Mehrabi, Debajyoti Mondal, Matthew Skala Site permutations We are sitting somewhere on the Euclidean plane. We want to describe our location in relation to some fixed, known sites. One

301 views • 17 slides
Prolific Permutations and Expected Breadth Cheyne Homberger Permutation Patterns 2018 Joint work

Prolific Permutations and Expected Breadth Cheyne Homberger Permutation Patterns 2018 Joint work

Prolific Permutations and Expected Breadth Cheyne Homberger Permutation Patterns 2018 Joint work with Simon Blackburn and Pete Winkler 1/24 Plotting Permutations Definition If is a permutation of length n , then the plot of is the set of

453 views • 43 slides

More recommend