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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. [ABSTRACT FROM AUTHOR]

Subexponential logic is a variant of linear logic with a family of exponential connectives – called subexponentials – that are indexed and arranged in a pre-order. Each subexponential has or lacks associated structural properties of weakening and contraction. We show that a classical propositional multiplicative subexponential logic (MSEL) with one unrestricted and two linear subexponentials can encode the halting problem for two register Minsky machines, and is hence undecidable. We then show how the additive connectives can be directly simulated by giving an encoding of propositional multiplicative additive linear logic (MALL) in an MSEL with one unrestricted and four linear subexponentials. [ABSTRACT FROM AUTHOR]

Logical frameworks and meta-languages form a common substrate for representing, implementing and reasoning about a wide variety of deductive systems of interest in logic and computer science. Their design and implementation and their use in reasoning tasks ranging from the correctness of software to the properties of formal computational systems have been the focus of considerable research over the last two decades. [ABSTRACT FROM AUTHOR]