Multirole Logic as a Foundation for Global Coordination

Session types are protocols describing valid communications among parties. Two-party sessions enjoy duality, in which one party's action is always the dual of the other party (e.g. send/receive, choose/offer). Correspondence to logics is being actively studied in which cut reduction is communication and the duality in classical linear logic captures duality in two-party sessions. However in multiparty sessions, duality no longer holds, and the two-premiss cut rule is insufficient to express multiparty communication. Several prior studies proposed multi-cut/coherent-cut as a generalization, with coherence as a side condition guarding cuts. Instead of resorting to classical logic, we propose a new form of logic, Multirole Logic, where propositions are not limited to only two interpretations (itself and its negation), but multiple interpretations annotated by a set of roles. Such generalization naturally gives rise to a (complete) cut rule for multiple propositions and a (partial) cut rule that leaves a residual proposition. And we proved the admissibility of cut rule, thus generalizing the celebrated results of Gentzen. We report that our Multirole Logic is much more general and it provides a foundation for global coordination, including but not limited to multiparty session types.

Transcript

1. Multirole Logic as a Foundation for Global Coordination Hongwei Xi and Hanwen Wu Boston University NEPLS’16 at UMass visit multirolelogic.org for more
2. Outline • Gentzen’s 1935 “Main Theorem” is important. ◦ Sequent calculus is symmetric. ◦ But only has dual “role” ▪ Formulae on the left/right side of turnstyle ▪ Connectives have left/right rules • Multirole Logic generalized the logic. ◦ Formulae have many roles, calculus have many sides ◦ Cut among multiple sequents can be eliminated • Foundation for Global Coordination ◦ Cut reduction as global coordination ◦ Motivated by a logical interpretation of multiparty session types
3. Gentzen’s 1934 Sequent Calculus, Classical, Two-Sided
4. Gentzen’s 1934 Sequent Calculus, Classical, One-Sided
5. Gentzen’s 1935 Main Theorem Cut rule can be eliminated. • Consistency. • Sub-formula property. • Correspondence to normal proof in natural deduction.
6. Multirole Logic, Axiom
7. Multirole Logic, Cut, a guess
8. Multirole Logic, Cut, the answer
9. Multirole Logic, Classical, One-Sided
10. Multirole Logic, Cut w/ Spill
11. Multirole Logic • Generalized cut elimination for a cut among more than two sequents. • Admit Cut-with-spill • Degrade into classical logic when there are only two roles. • Negation can not be internalized, or cut-with-spill will not hold. • Also works for linear logic. • Dual multirole logic.
12. Foundation, Cut Reduction
13. Foundation for Global Coordination • In Curry-Howard Correspondence, proof normalization as beta reduction for lambda calculus. • Here, cut reduction as beta reduction for pi calculus. • More • Wadler, Carbone • Caries, Pfenning
14. Q&A Hongwei Xi and Hanwen Wu Boston University NEPLS’16 at UMass visit multirolelogic.org for more