Hybrid and Subexponential Linear Logics Technical Report

TL;DR

This paper compares HyLL and SELL, two logical frameworks based on linear logic, demonstrating that HyLL is as expressive as LL and exploring its encoding into SELL and the limitations for temporal operators.

Contribution

The paper provides an encoding of HyLL into LL, establishing its expressiveness equivalence, and explores its encoding into SELL, offering new insights into the meaning of worlds in HyLL.

Findings

HyLL can be encoded into LL with high adequacy

HyLL can be encoded into SELL with quantification over locations

Encoding of CTL operators in HyLL requires fixed points in LL

Abstract

HyLL (Hybrid Linear Logic) and SELL (Subexponential Linear Logic) are logical frameworks that have been extensively used for specifying systems that exhibit modalities such as temporal or spatial ones. Both frameworks have linear logic (LL) as a common ground and they admit (cut-free) complete focused proof systems. The difference between the two logics relies on the way modalities are handled. In HyLL, truth judgments are labelled by worlds and hybrid connectives relate worlds with formulas. In SELL, the linear logic exponentials (!, ?) are decorated with labels representing locations, and an ordering on such labels defines the provability relation among resources in those locations. It is well known that SELL, as a logical framework, is strictly more expressive than LL. However, so far, it was not clear whether HyLL is more expressive than LL and/or SELL. In this paper, we show an encoding of the HyLL's logical rules into LL with the highest level of adequacy, hence showing that HyLL is as expressive as LL. We also propose an encoding of HyLL into SELL $\doublecup$ (SELL plus quantification over locations) that gives better insights about the meaning of worlds in HyLL. We conclude our expressiveness study by showing that previous attempts of encoding Computational Tree Logic (CTL) operators into HyLL cannot be extended to consider the whole set of temporal connectives. We show that a system of LL with fixed points is indeed needed to faithfully encode the behavior of such temporal operators.