Proof nets for sumproduct logic Willem Heijltjes LFCS School of - PowerPoint PPT Presentation

Inductive saturation A A + × 0 1 × 0 + × 0 1

Inductive saturation A A + × 0 1 × 0 + × 0 1

Inductive saturation A A + × 0 1 × 0 + × 0 1

Inductive saturation A A + × 0 1 × 0 + × 0 1

Inductive saturation A A + × 0 1 × 0 + × 0 1

Inductive saturation A A + × 0 1 × 0 + × 0 1

Inductive saturation A A + × 0 1 × 0 + × 0 1

Inductive saturation A A + × 0 1 × 0 + × 0 1

The soundness proof R S To prove: X − → Y ⇔ X − → Y given σ R = σ S ◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct + × ◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σ R = σ S after, but not before, adding an injection/projection

Matching injections and projections Equivalent nets may factor through different injections or projections, but to allow induction nets must at least have the same domain and codomain. R T + + + × × × S Idea: ‘highest’ links, and in particular rooted links, are most significant (downward movement in saturation is unrestricted)

If σ R contains a rooted link, so does some S ⇔ R A A + × 0 1 × 0 + × 0 1

If σ R contains a rooted link, so does some S ⇔ R A A + × 0 1 × 0 + × 0 1

If σ R contains a rooted link, so does some S ⇔ R A A + × 0 1 × 0 + × 0 1

If σ R contains a rooted link, so does some S ⇔ R A A + × 0 1 × 0 + × 0 1

If σ R contains a rooted link, so does some S ⇔ R A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

A A + × 0 1 × 0 + × 0 1

The soundness proof R S To prove: X − → Y ⇔ X − → Y given σ R = σ S ◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct + × ◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σ R = σ S after, but not before, adding an injection/projection

Injections into pointed objects σ R = σ S R ′ S ′ + + + × × × q q ′ + + + full

Injections into pointed objects A A + × 0 × 1 0 + × 0 1

Injections into pointed objects A A + × 0 × 1 0 + × 0 1

Injections into pointed objects A A + × 0 × 1 0 + × 0 1