Weak Completeness of Coalgebraic Dynamic Logics

• Main Authors: Hansen, Helle Hvid, Kupke, Clemens
• Format: Journal Article
• Language: English
• Published: 10-09-2015
• Subjects: Computer Science - Logic in Computer Science
• DOI: 10.48550/arxiv.1509.03017

EPTCS 191, 2015, pp. 90-104 We present a coalgebraic generalisation of Fischer and Ladner's Propositional
Dynamic Logic (PDL) and Parikh's Game Logic (GL). In earlier work, we proved a
generic strong completeness result for coalgebraic dynamic logics without
iteration. The coalgebraic semantics of such programs is given by a monad T,
and modalities are interpreted via a predicate lifting \^I whose transpose is a
monad morphism from T to the neighbourhood monad. In this paper, we show that
if the monad T carries a complete semilattice structure, then we can define an
iteration construct, and suitable notions of diamond-likeness and box-likeness
of predicate-liftings which allows for the definition of an axiomatisation
parametric in T, \^I and a chosen set of pointwise program operations. As our
main result, we show that if the pointwise operations are "negation-free" and
Kleisli composition left-distributes over the induced join on Kleisli arrows,
then this axiomatisation is weakly complete with respect to the class of
standard models. As special instances, we recover the weak completeness of PDL
and of dual-free Game Logic. As a modest new result we obtain completeness for
dual-free GL extended with intersection (demonic choice) of games.