5 research outputs found
Definability by Horn formulas and linear time on cellular automata
International audienceWe establish an exact logical characterization of linear time complexity of cellular automata of dimension d, for any fixed d: a set of pictures of dimension d belongs to this complexity class iff it is definable in existential second-order logic restricted to monotonic Horn formulas with built-in successor function and d + 1 first-order variables. This logical characterization is optimal modulo an open problem in parallel complexity. Furthermore, its proof provides a systematic method for transforming an inductive formula defining some problem into a cellular automaton that computes it in linear time
Categorical models of Linear Logic with fixed points of formulas
We develop a denotational semantics of muLL, a version of propositional
Linear Logic with least and greatest fixed points extending David Baelde's
propositional muMALL with exponentials. Our general categorical setting is
based on the notion of Seely category and on strong functors acting on them. We
exhibit two simple instances of this setting. In the first one, which is based
on the category of sets and relations, least and greatest fixed points are
interpreted in the same way. In the second one, based on a category of sets
equipped with a notion of totality (non-uniform totality spaces) and relations
preserving them, least and greatest fixed points have distinct interpretations.
This latter model shows that muLL enjoys a denotational form of normalization
of proofs.Comment: arXiv admin note: text overlap with arXiv:1906.0559