[Paper Review] Algebraic totality, towards completeness

Christine Tasson|ArXiv.org|Dec 11, 2009

Logic, programming, and type systems1 references4 citations

TL;DR

This paper introduces an algebraic and topological model of classical linear logic using finiteness spaces equipped with totality candidates—closed affine subspaces not containing zero. It establishes completeness for the fragment $τ^{n} \to \mathcal{B}$ using algebraic geometry, proving that all total propositional functions over booleans are definable in a simply typed lambda calculus with conditionals.

ABSTRACT

Finiteness spaces constitute a categorical model of Linear Logic (LL) whose objects can be seen as linearly topologised spaces, (a class of topological vector spaces introduced by Lefschetz in 1942) and morphisms as continuous linear maps. First, we recall definitions of finiteness spaces and describe their basic properties deduced from the general theory of linearly topologised spaces. Then we give an interpretation of LL based on linear algebra. Second, thanks to separation properties, we can introduce an algebraic notion of totality candidate in the framework of linearly topologised spaces: a totality candidate is a closed affine subspace which does not contain 0. We show that finiteness spaces with totality candidates constitute a model of classical LL. Finally, we give a barycentric simply typed lambda-calculus, with booleans ${\mathcal{B}}$ and a conditional operator, which can be interpreted in this model. We prove completeness at type ${\mathcal{B}}^n o{\mathcal{B}}$ for every n by an algebraic method.

Motivation & Objective

• To develop an algebraic and topological framework for classical linear logic based on finiteness spaces and linearly topologised vector spaces.
• To introduce a new notion of totality candidate as a closed affine subspace not containing zero, providing a geometric-algebraic characterization of totality.
• To establish completeness for the type $\mathcal{B}^n \to \mathcal{B}$ in a simply typed lambda calculus with booleans and conditionals.
• To bridge linear logic semantics with algebraic geometry by interpreting proof terms as polynomial functions over a field.
• To show that stability and totality are independent concepts in this model, as exemplified by the non-stable but total $\mathtt{POr}$ function.

Proposed method

• Define finiteness spaces as linearly topologised vector spaces over a field $\mathbbm{k}$, with morphisms as continuous linear maps.
• Introduce totality candidates as closed affine subspaces not containing the origin, forming a model of classical linear logic.
• Construct a barycentric simply typed lambda calculus with booleans $\mathcal{B}$ and a conditional operator.
• Interpret terms as polynomial functions in $\mathbbm{k}[X_1, \dots, X_{2n}]$, with semantics defined via evaluation on supports.
• Use Hilbert's Nullstellensatz and polynomial division to prove that any total function vanishing on the variety $X_{2i-1} + X_{2i} = 1$ is a linear combination of the defining polynomials.
• Apply algebraic reconstruction to express any total proof term as a combination of basic boolean terms and conditional constructs.

Experimental results

Research questions

• RQ1 Can a categorical model of classical linear logic be constructed using algebraic and topological structures on linearly topologised vector spaces?
• RQ2 What is the algebraic and topological characterization of totality candidates in this setting?
• RQ3 Is completeness achievable for the fragment $\mathcal{B}^n \to \mathcal{B}$ in a simply typed lambda calculus with conditionals?
• RQ4 How do stability and totality relate in this model—can non-stable functions be total?
• RQ5 Can algebraic geometry techniques be used to prove completeness in linear logic semantics?

Key findings

• Finiteness spaces with totality candidates form a model of classical linear logic, where totality is characterized as a closed affine subspace not containing zero.
• The model supports a simply typed lambda calculus with booleans and a conditional operator, whose semantics is fully captured by polynomial functions.
• Completeness holds for all types $\mathcal{B}^n \to \mathcal{B}$: every total function is definable as a term in the calculus.
• The proof relies on the fact that any polynomial vanishing on the variety $\{X_{2i-1} + X_{2i} = 1 \mid 1 \leq i \leq n\}$ is in the ideal generated by $X_{2i-1} + X_{2i} - 1$, via Hilbert's Nullstellensatz.
• The non-stable $\mathtt{POr}$ function is total and definable, showing that totality does not imply stability in this model.
• A barycentric term construction allows expressing any total proof as a combination of basic boolean terms and conditionals, with explicit polynomial reconstruction.