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Hybrid linear logic, revisited

Published online by Cambridge University Press:  22 April 2019

KAUSTUV CHAUDHURI Open the ORCID record for KAUSTUV CHAUDHURI [Opens in a new window] ,
JOËLLE DESPEYROUX ,
CARLOS OLARTE Open the ORCID record for CARLOS OLARTE [Opens in a new window]  and
ELAINE PIMENTEL Open the ORCID record for ELAINE PIMENTEL [Opens in a new window]
KAUSTUV CHAUDHURI
Affiliation:
Inria & LIX/École Polytechnique, Palaiseau, France Email: kaustuv.chaudhuri@inria.fr
JOËLLE DESPEYROUX
Affiliation:
INRIA and CNRS, I3S, Sophia-Antipolis, France Email: joelle.despeyroux@i3s.unice.fr
CARLOS OLARTE
Affiliation:
ECT – Universidade Federal do Rio Grande do Norte, Natal, Brazil Email: carlos.olarte@gmail.com
ELAINE PIMENTEL
Affiliation:
Departamento de Matemática – Universidade Federal do Rio Grande do Norte, Natal, Brazil Email: elaine.pimentel@gmail.com
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Abstract

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HyLL (Hybrid Linear Logic) is an extension of intuitionistic linear logic (ILL) that has been used as a framework for specifying systems that exhibit certain modalities. In HyLL, truth judgements are labelled by worlds (having a monoidal structure) and hybrid connectives (at and ↓) relate worlds with formulas. We start this work by showing that HyLL's axioms and rules can be adequately encoded in linear logic (LL), so that one focused step in LL will correspond to a step of derivation in HyLL. This shows that any proof in HyLL can be exactly mimicked by a LL focused derivation. Another extension of LL that has extensively been used for specifying systems with modalities is Subexponential Linear Logic (SELL). In SELL, the LL exponentials (!, ?) are decorated with labels representing locations, and a pre-order on such labels defines the provability relation. We propose an encoding of HyLL into SELL⋒ (SELL plus quantification over locations) that gives better insights about the meaning of worlds in HyLL. More precisely, we identify worlds as locations, and show that a flat subexponential structure is sufficient for representing any world structure in HyLL. This shows that HyLL's monoidal structure is not reflected in LL derivations, hence not increasing the expressiveness of LL, from a proof theoretical point of view. We conclude by proposing the notion of fixed points in multiplicative additive HyLL (μHyMALL), which can be encoded into multiplicative additive linear logic with fixed points (μMALL). As an application, we propose encodings of Computational Tree Logic (CTL) into both μMALL and μHyMALL. In the former, states are represented as atoms in the linear context, hence reflecting a more operational view of CTL connectives. In the latter, worlds represent states of the transition system, thus exhibiting a pleasant similarity with the semantics of CTL.

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