On geometry of interaction for polarized linear logic
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Bibliographic record
Abstract
We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP , which is the multiplicative fragment of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multi-points to various categorical models of GoI. Multi-points are shown to play an essential role in semantically characterizing the dynamics of proof networks in polarized proof theory. For example, they permit us to characterize the key feature of polarization, focusing , as well as being fundamental to our construction of concrete polarized GoI models. Our approach to polarized GoI involves following two independent studies, based on different categorical perspectives of GoI: (i) Inspired by the work of Abramsky, Haghverdi and Scott, a polarized GoI situation is defined in which multi-points are added to a traced monoidal category equipped with a reflexive object U . Using this framework, categorical versions of Girard's execution formula are defined, as well as the GoI interpretation of MLLP proofs. Running the execution formula is shown to characterize the focusing property (and thus polarities) as well as the dynamics of cut elimination. (ii) The Int construction of Joyal–Street–Verity is another fundamental categorical structure for modelling GoI. Here, we investigate it in a multi-pointed setting. Our presentation yields a compact version of Hamano–Scott's polarized categories, and thus denotational models of MLLP . These arise from a contravariant duality between monoidal categories of positive and negative objects, along with an appropriate bimodule structure (representing ‘non-focused proofs’) between them. Finally, as a special case of (ii) above, a compact model of MLLP is also presented based on Rel (the category of sets and relations) equipped with multi-points.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it