Variable Binding, Symmetric Monoidal Pardon Closed Theories, and - PowerPoint PPT Presentation

Variable Binding, Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Variable Binding, Symmetric Monoidal Pardon Closed Theories, and Bigraphs Motivation Signatures and free smc categories Application to Richard Garner 1 bigraphs Tom Hirschowitz 2 elien Pardon 3 Aur´ 1 Cambridge University 2 CNRS, Universit´ 3 ENS Lyon e de Savoie Concur ’09

Variable Binding, Motivation Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Motivation Signatures and free smc categories ◮ In the paper: elementary, algebraic approach to variable Application to bigraphs binding in the presence of linearity. ◮ Here: additional, longer-term motivation, hoping for your feedback.

Variable Binding, Long term goal Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Obtain Motivation Signatures and free ◮ algebraic, smc categories ◮ geometric, and Application to bigraphs ◮ modular models of programming languages, with ◮ a clear separation between program and execution, e.g., 2-dimensional. What do algebraic, geometric, and modular mean?

Variable Binding, Algebra Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Motivation Universal algebra < Lawvere theories < locally presentable Signatures and free smc categories categories / sketches. Application to bigraphs Paradox How algebraic is process algebra? I here mean definition of their dynamics, not behavioural theories.

Variable Binding, Geometric Symmetric Monoidal Closed Theories, and Bigraphs Geometric models of the chosen algebraic structure? Garner, Hirschowitz, See John Baez’ talk at LICS ’09. Pardon Motivation Signatures and free smc categories Application to bigraphs

Variable Binding, Modular models Symmetric Monoidal Closed Theories, and Bigraphs Montanari and colleagues (1996) and Melli` es (2002) call for Garner, Hirschowitz, modular models of programming languages. Pardon ◮ Understanding programming languages as free double Motivation categories. Signatures and free smc categories ◮ Tiles, or cells, composing vertically and horizontally: Application to bigraphs f a b u v α c d . g ◮ f and g are programs. ◮ α is an execution or a reduction. ◮ u and v are side effects, or interactions with the environment.

Variable Binding, Modularity Symmetric Monoidal Closed Theories, and Bigraphs Such a double category is modular when any execution of a Garner, composed program Hirschowitz, Pardon f 1 f 2 Motivation a b ′ b Signatures and free smc categories u v α Application to bigraphs c d g decomposes as f 1 f 2 a b ′ b u α 1 α 2 v c d ′ d g 1 g 2 with g 2 g 1 = g .

Variable Binding, Expected benefits Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Motivation Signatures and free smc categories ◮ Montanari et al., Sassone-Sobocinski: bisimulation is a Application to automatically a congruence. bigraphs ◮ Melli` es: term tracing, rewriting. ◮ Hopefully compilation.

Variable Binding, Starting point Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Motivation Signatures and free smc categories Question Application to bigraphs What should the horizontal category be? Or: what should program composition be?

Variable Binding, Standard answer Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Category of contexts: Pardon ◮ objects: typing contexts Γ = ( x : A , y : B ), Motivation Signatures and free ◮ morphisms ∆ → Γ: assignments smc categories Application to [ x = e , y = f ] , bigraphs ◮ composition by substitution: σ Θ ∆ [ x = e , y = f ] [ x = e [ σ ] , y = f [ σ ]] Γ .

Variable Binding, Other answers? Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Do not plan to use that, for hand-waving reasons: Motivation Signatures and free Duplication belongs to the dynamics smc categories Application to Does not model actual plugging of program fragments. bigraphs Besides: Claim, or thesis Duplication in composition hinders geometric intuition. Hopefully: results will provide more.

Variable Binding, Linear substitution as program composition Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Motivation Signatures and free smc categories Proposal Application to bigraphs Linear substitution as program composition. I.e., the horizontal category is monoidal ( � = finite products).

Variable Binding, Linear substitution as program composition Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Who tried already? Motivation ◮ Linear programming languages. Signatures and free ◮ No dynamic duplication either. smc categories ◮ But a nice modular model by Melli` es, a hint that Application to bigraphs linearity favours geometric intuition. ◮ Bigraphs (Jensen, Milner, . . . ). ◮ No categorical semantics, esp. for the dynamics. ◮ Premonoidal or precartesian categories (Power, Robinson, Schuermann, . . . ). ◮ No clear separation between program and execution. ◮ Rather models program equivalence.

Variable Binding, Linear substitution as program composition: Symmetric Monoidal Closed issues Theories, and Bigraphs Garner, Hirschowitz, Pardon Motivation Signatures and free ◮ First issue: can we still handle languages with smc categories duplication? Application to bigraphs Hopefully yes: bound variables may be used several times, e.g., λ x . xx . ◮ Second issue: linearity is not stable under reduction. ( λ x . xx ) y − → yy .

Variable Binding, Linear substitution as program composition: Symmetric Monoidal Closed issues Theories, and Bigraphs Garner, Hirschowitz, Pardon Problem: ( λ x . xx ) y − → yy . Motivation Signatures and free Proposal smc categories Application to Give up reductions, use tiles: bigraphs ( λ x . xx ) y 1 1 c τ β 2 1 . y 1 y 2 The map c says that y is duplicated as y 1 and y 2 .

Variable Binding, Linear substitution as program composition: Symmetric Monoidal Closed issues Theories, and Bigraphs Garner, Hirschowitz, Pardon Problem: ( λ x . xx ) y − → yy . Motivation Signatures and free Proposal smc categories Application to Give up reductions, use tiles: bigraphs ( λ x . xx ) y f n 1 1 c n c f c τ β 2 n 2 1 . y 1 y 2 ( f , f ) The map c says that y is duplicated as y 1 and y 2 .

Variable Binding, Contents Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Sounds reasonable? Motivation Now, this paper: Signatures and free smc categories ◮ Linear substitution, in the horizontal category only (no Application to dynamics). bigraphs ◮ Algebraic structure: symmetric monoidal closed ( smc ) categories. ◮ Follow-up on Gadducci’s GS · Λ-theories, with: ◮ Use of recent, geometric presentation of the free smc category (Hughes). ◮ Application to bigraphs.

Variable Binding, Signatures Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Formulae from intuitionistic multiplicative linear logic Motivation ( imll ): Signatures and free smc categories A , B , . . . ∈ F ( X ) ::= x | I | A ⊗ B | A ⊸ B x ∈ X . Application to bigraphs Definition A ( smc ) signature is given by: ◮ a set X of sorts, and s , t → ◮ a graph Σ → F ( X ) .

Variable Binding, Example 1: λ -calculus, first take Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Motivation Signatures and free ◮ One sort t . smc categories ◮ Two operations (edges): Application to bigraphs λ → t · → t . ( t ⊸ t ) and ( t ⊗ t ) ◮ Remember hoas . ◮ Here internal to some smc category.

Variable Binding, The generated smc category Symmetric Monoidal Closed Theories, and Bigraphs ◮ Any Σ freely generates an smc category S (Σ). Garner, Hirschowitz, ◮ Objects: formulae. Pardon ◮ Morphisms: imll proof nets, modulo Trimble rewiring. Motivation ◮ Example, two interpretations for λ x . ( � 1 � 2 ): Signatures and free smc categories ( t ⊸ t ) ⊗ t t ⊗ ( t ⊸ t ) Application to bigraphs · · λ λ t t . ◮ Combinatorial scoping condition generalising the Danos-Regnier criterion for imll .

Variable Binding, Back to Example 1 Symmetric Monoidal Closed Theories, and Bigraphs Garner, Hirschowitz, Pardon Motivation Signatures and free ◮ Linearity: cannot model λ x . xx . smc categories ◮ Still: Application to bigraphs Proposition Morphisms I → t are in bijection with closed linear λ -terms.

Variable Binding, Example 2: λ -calculus, second take Symmetric Monoidal Closed Theories, and Bigraphs Garner, Two sorts: Hirschowitz, Pardon ◮ t for terms, and Motivation ◮ v for variables. Signatures and free Operations: smc categories Application to bigraphs λ → t · → t ( v ⊸ t ) ( t ⊗ t ) d c w v → t v → v ⊗ v v → I . Reminiscent from weak hoas and Montanari et al. (independently). This signature uses the unit I !