4 research outputs found
Kleene Algebras, Regular Languages and Substructural Logics
We introduce the two substructural propositional logics KL, KL+, which use
disjunction, fusion and a unary, (quasi-)exponential connective. For both we
prove strong completeness with respect to the interpretation in Kleene algebras
and a variant thereof. We also prove strong completeness for language models,
where each logic comes with a different interpretation. We show that for both
logics the cut rule is admissible and both have a decidable consequence
relation.Comment: In Proceedings GandALF 2014, arXiv:1408.556
A proof theory of right-linear (omega-)grammars via cyclic proofs
Right-linear (or left-linear) grammars are a well-known class of context-free
grammars computing just the regular languages. They may naturally be written as
expressions with (least) fixed points but with products restricted to letters
as left arguments, giving an alternative to the syntax of regular expressions.
In this work, we investigate the resulting logical theory of this syntax.
Namely, we propose a theory of right-linear algebras (RLA) over of this syntax
and a cyclic proof system CRLA for reasoning about them.
We show that CRLA is sound and complete for the intended model of regular
languages. From here we recover the same completeness result for RLA by
extracting inductive invariants from cyclic proofs, rendering the model of
regular languages the free right-linear algebra.
Finally, we extend system CRLA by greatest fixed points, nuCRLA, naturally
modelled by languages of omega-words thanks to right-linearity. We show a
similar soundness and completeness result of (the guarded fragment of) nuCRLA
for the model of omega-regular languages, employing game theoretic techniques.Comment: 34 pages, 3 figure