Overview
This work establishes an equivalence between two semantic approaches in linear logic: phase semantics and base-extension semantics. The equivalence is demonstrated through the definition of bidirectional maps and the construction of an isomorphism between the respective models. Additionally, the research extends base-extension semantics by defining clauses for the exponentials of linear logic.
Research Context
Linear logic employs a resource-sensitive notion of consequence, leading to various semantic presentations. Phase semantics is an algebraic approach that interprets formulas within phase models. These models consist of a commutative monoid, a fixed subset, and an orthogonality relation defined with respect to that subset, alongside a valuation.
Base-extension semantics offers a distinct and more recent approach. It defines validity through the inductive extension of a provability relation. This provability relation operates on a base, which is characterized as a set of inference rules over atomic propositions.
Approach
The research first focused on defining bidirectional maps between bases and phase models. These maps facilitate translation between the two semantic frameworks. Subsequent to defining these maps, the approach involved constructing an isomorphism. This isomorphism establishes a structural equivalence between a phase model and its image resulting from the composition of these bidirectional maps, and similarly, between a base and its image under the same composition.
As a separate contribution, the research extended the scope of base-extension semantics. This extension involved defining specific clauses within the base-extension semantics framework to accommodate the exponentials of linear logic.
Findings
- An equivalence was established between phase semantics and base-extension semantics in linear logic.
- Bidirectional maps were defined that translate between bases and phase models.
- An isomorphism was constructed between a phase model and its image after applying the composition of the defined maps.
- An isomorphism was constructed between a base and its image after applying the composition of the defined maps.
- Base-extension semantics clauses were defined for the exponentials of linear logic.