3 research outputs found
A Probabilistic Higher-Order Fixpoint Logic
We introduce PHFL, a probabilistic extension of higher-order fixpoint logic,
which can also be regarded as a higher-order extension of probabilistic
temporal logics such as PCTL and the -calculus. We show that PHFL is
strictly more expressive than the -calculus, and that the PHFL
model-checking problem for finite Markov chains is undecidable even for the
-only, order-1 fragment of PHFL. Furthermore the full PHFL is far more
expressive: we give a translation from Lubarsky's -arithmetic to PHFL,
which implies that PHFL model checking is -hard and -hard.
As a positive result, we characterize a decidable fragment of the PHFL
model-checking problems using a novel type system
Fair Simulation for Nondeterministic and Probabilistic Buechi Automata: a Coalgebraic Perspective
Notions of simulation, among other uses, provide a computationally tractable
and sound (but not necessarily complete) proof method for language inclusion.
They have been comprehensively studied by Lynch and Vaandrager for
nondeterministic and timed systems; for B\"{u}chi automata the notion of fair
simulation has been introduced by Henzinger, Kupferman and Rajamani. We
contribute to a generalization of fair simulation in two different directions:
one for nondeterministic tree automata previously studied by Bomhard; and the
other for probabilistic word automata with finite state spaces, both under the
B\"{u}chi acceptance condition. The former nondeterministic definition is
formulated in terms of systems of fixed-point equations, hence is readily
translated to parity games and is then amenable to Jurdzi\'{n}ski's algorithm;
the latter probabilistic definition bears a strong ranking-function flavor.
These two different-looking definitions are derived from one source, namely our
coalgebraic modeling of B\"{u}chi automata. Based on these coalgebraic
observations, we also prove their soundness: a simulation indeed witnesses
language inclusion
A Probabilistic Higher-order Fixpoint Logic
We introduce PHFL, a probabilistic extension of higher-order fixpoint logic,
which can also be regarded as a higher-order extension of probabilistic
temporal logics such as PCTL and the -calculus. We show that PHFL is
strictly more expressive than the -calculus, and that the PHFL
model-checking problem for finite Markov chains is undecidable even for the
-only, order-1 fragment of PHFL. Furthermore the full PHFL is far more
expressive: we give a translation from Lubarsky's -arithmetic to PHFL,
which implies that PHFL model checking is -hard and -hard.
As a positive result, we characterize a decidable fragment of the PHFL
model-checking problems using a novel type system