Linear approximation and Taylor expansion of -terms F. Olimpieri - PowerPoint PPT Presentation

Linear approximation and Taylor expansion of λ -terms F. Olimpieri Aix-Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France F. Olimpieri Linear approximation and Taylor expansion of λ -terms 1 / 5

The pure λ -calculus λ -terms We inductively define Λ : if x ∈ V then x ∈ Λ; If M ∈ Λ , then λ x . M ∈ Λ; if M , N ∈ Λ, then MN ∈ Λ . λ x . M stands for x �→ M . We can model functional evaluation : ( λ x . M ) N → M [ N / x ] F. Olimpieri Linear approximation and Taylor expansion of λ -terms 2 / 5

The pure λ -calculus λ -terms We inductively define Λ : if x ∈ V then x ∈ Λ; If M ∈ Λ , then λ x . M ∈ Λ; if M , N ∈ Λ, then MN ∈ Λ . λ x . M stands for x �→ M . We can model functional evaluation : ( λ x . M ) N → M [ N / x ] F. Olimpieri Linear approximation and Taylor expansion of λ -terms 2 / 5

The pure λ -calculus λ -terms We inductively define Λ : if x ∈ V then x ∈ Λ; If M ∈ Λ , then λ x . M ∈ Λ; if M , N ∈ Λ, then MN ∈ Λ . λ x . M stands for x �→ M . We can model functional evaluation : ( λ x . M ) N → M [ N / x ] F. Olimpieri Linear approximation and Taylor expansion of λ -terms 2 / 5

Linearity Intuitive Definition A function f is linear when it uses only once its argument during the computation. Linearity for functional evaluation: The identity function is linear. Let M ∈ Λ, then ( λ x . x ) M → M . The copy function is non-linear. Let M ∈ Λ, then ( λ x . xx ) M → MM . F. Olimpieri Linear approximation and Taylor expansion of λ -terms 3 / 5

Linearity Intuitive Definition A function f is linear when it uses only once its argument during the computation. Linearity for functional evaluation: The identity function is linear. Let M ∈ Λ, then ( λ x . x ) M → M . The copy function is non-linear. Let M ∈ Λ, then ( λ x . xx ) M → MM . F. Olimpieri Linear approximation and Taylor expansion of λ -terms 3 / 5

Linearity Intuitive Definition A function f is linear when it uses only once its argument during the computation. Linearity for functional evaluation: The identity function is linear. Let M ∈ Λ, then ( λ x . x ) M → M . The copy function is non-linear. Let M ∈ Λ, then ( λ x . xx ) M → MM . F. Olimpieri Linear approximation and Taylor expansion of λ -terms 3 / 5

Linear approximation of λ -terms Linear logic leads to the introduction of a resource sensitive approximation of programs. Intuitively, a n -linear approximant of a term M is a version of it that uses exactly n times the argument under evaluation. We denote as T ( M ) the set of linear approximants of M . Lemma Let M ∈ Λ and s ∈ T ( M ) . If s → t then there exists N ∈ Λ such that t ∈ T ( N ) and M → N. F. Olimpieri Linear approximation and Taylor expansion of λ -terms 4 / 5

Linear approximation of λ -terms Linear logic leads to the introduction of a resource sensitive approximation of programs. Intuitively, a n -linear approximant of a term M is a version of it that uses exactly n times the argument under evaluation. We denote as T ( M ) the set of linear approximants of M . Lemma Let M ∈ Λ and s ∈ T ( M ) . If s → t then there exists N ∈ Λ such that t ∈ T ( N ) and M → N. F. Olimpieri Linear approximation and Taylor expansion of λ -terms 4 / 5

Linear approximation of λ -terms Linear logic leads to the introduction of a resource sensitive approximation of programs. Intuitively, a n -linear approximant of a term M is a version of it that uses exactly n times the argument under evaluation. We denote as T ( M ) the set of linear approximants of M . Lemma Let M ∈ Λ and s ∈ T ( M ) . If s → t then there exists N ∈ Λ such that t ∈ T ( N ) and M → N. F. Olimpieri Linear approximation and Taylor expansion of λ -terms 4 / 5

Some results Theorem Let M ∈ Λ . M is computationally meaningful iff the computation for some s ∈ T ( M ) ends. We can define a Taylor expansion for λ -terms: Taylor formula 1 � Θ( M ) = m ( s ) s s ∈ T ( M ) F. Olimpieri Linear approximation and Taylor expansion of λ -terms 5 / 5

Some results Theorem Let M ∈ Λ . M is computationally meaningful iff the computation for some s ∈ T ( M ) ends. We can define a Taylor expansion for λ -terms: Taylor formula 1 � Θ( M ) = m ( s ) s s ∈ T ( M ) F. Olimpieri Linear approximation and Taylor expansion of λ -terms 5 / 5