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Absolute convergence and Taylor expansion in web based models of Linear Logic

Christine Tasson, Aymeric Walch

Abstract

The differential $λ$-calculus studies how the quantitative aspects of programs correspond to differentiation and to Taylor expansion inside models of linear logic. Recent work has generalized the axioms of Taylor expansion so they apply to many models that only feature partial sums. However, that work does not cover the classic web based models of K{ö}the spaces and finiteness spaces . First, we provide a generic construction of web based models with partial sums. It captures models, ranging from coherence spaces to probabilistic coherence spaces, finiteness spaces and K{ö}the spaces. Second, we generalize the theory of Taylor expansion to models in which coefficients can be non-positive. We then use our generic web model construction to provide a unified proof that all the aforementioned web based models feature such Taylor expansion.

Absolute convergence and Taylor expansion in web based models of Linear Logic

Abstract

The differential -calculus studies how the quantitative aspects of programs correspond to differentiation and to Taylor expansion inside models of linear logic. Recent work has generalized the axioms of Taylor expansion so they apply to many models that only feature partial sums. However, that work does not cover the classic web based models of K{ö}the spaces and finiteness spaces . First, we provide a generic construction of web based models with partial sums. It captures models, ranging from coherence spaces to probabilistic coherence spaces, finiteness spaces and K{ö}the spaces. Second, we generalize the theory of Taylor expansion to models in which coefficients can be non-positive. We then use our generic web model construction to provide a unified proof that all the aforementioned web based models feature such Taylor expansion.

Paper Structure

This paper contains 19 sections, 36 theorems, 27 equations, 2 figures.

Table of Contents

  1. Partial Commutative Monoids (PCMs)
  2. A generic construction of web models
  3. Web models based on strong PCMs
  4. Orthogonality and objects
  5. Tensor product of objects and morphisms
  6. The linear logical structure of web models
  7. Web models based on absolute convergence
  8. Summability in web models
  9. Taylor expansion
  10. Contribution.
  11. Taylor expansion in PCM categories
  12. The representable theory
  13. Taylor expansion in web models
  14. Conclusion and future work
  15. Proofs on web based models
  16. ...and 4 more sections

Key Result

Lemma 3

If $A \subseteq B$ and $()_{< \in a}B>[b]{x_b}$ is summable, then $()_{x_a \in a}$ is summable.

Figures (2)

  • Figure 1: LL structure of $\mathbf{Web}_{\mathcal{R}_{+}}$
  • Figure 2: Bimonoid and bimonad relations

Theorems & Definitions (94)

  • Definition 1
  • Definition 2: Hines13
  • Lemma 3
  • Definition 4
  • Example 5
  • Example 6
  • Proposition 7
  • proof
  • Proposition 8
  • proof
  • ...and 84 more