37 research outputs found
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
Phase Semantics for Linear Logic with Least and Greatest Fixed Points
The truth semantics of linear logic (i.e. phase semantics) is often overlooked despite having a wide range of applications and deep connections with several denotational semantics. In phase semantics, one is concerned about the provability of formulas rather than the contents of their proofs (or refutations). Linear logic equipped with the least and greatest fixpoint operators (?MALL) has been an active field of research for the past one and a half decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively.
In this paper, we extend the phase semantics of multiplicative additive linear logic (a.k.a. MALL) to ?MALL with explicit (co)induction (i.e. ?MALL^{ind}). We introduce a Tait-style system for ?MALL called ?MALL_? where proofs are wellfounded but potentially infinitely branching. We study its phase semantics and prove that it does not have the finite model property
A Non-wellfounded, Labelled Proof System for Propositional Dynamic Logic
We define a infinitary labelled sequent calculus for PDL, G3PDL^{\infty}. A
finitarily representable cyclic system, G3PDL^{\omega}, is then given. We show
that both are sound and complete with respect to standard models of PDL and,
further, that G3PDL^{\infty} is cut-free complete. We additionally investigate
proof-search strategies in the cyclic system for the fragment of PDL without
tests
Cyclic Proofs and Jumping Automata
We consider a fragment of a cyclic sequent proof system for Kleene algebra, and we see it as a computational device for recognising languages of words. The starting proof system is linear and we show that it captures precisely the regular languages. When adding the standard contraction rule, the expressivity raises significantly; we characterise the corresponding class of languages using a new notion of multi-head finite automata, where heads can jump