Computing Nested Fixpoints in Quasipolynomial Time

07/16/2019
by   Daniel Hausmann, et al.
0

It is well known that the winning region of a parity game with n nodes and k priorities can be computed as a k-nested fixpoint of a suitable function; straightforward computation of this nested fixpoint requires n^k/2+1 iterations of the function. The recent parity game solving algorithm by Calude et al. runs in quasipolynomial time and essentially shows how to compute the same fixpoint using only a quasipolynomial number of iterations. We show that their central idea naturally generalizes to the computation of k-nested fixpoints of any set-valued function; hence k-nested fixpoints of set functions that can be computed in quasipolynomial time can be computed in quasipolynomial time as well. While this result is of clear interest in itself, we focus in particular on applications to modal fixpoint logics beyond relational semantics. For instance, the model checking problems for the graded and the (two-valued) probabilistic μ-calculus -- with numbers coded in binary -- can be solved by computing nested fixpoints of functions that differ from the function for parity game solving, but still can be computed in quasipolynomial time; our result hence implies that model checking for these μ-calculi is in QP. A second implication of our result lies in satisfiability checking for generalized μ-calculi, including the graded, probabilistic and alternating-time variants; in a general setting that covers all the mentioned cases, our result immediately improves the upper time bound for satisfiability checking for fixpoint formulas of size n with alternation-depth k from 2^O(n^2k^2 n) to 2^O(nk n).

READ FULL TEXT

page 1

page 2

page 3

page 4

research
07/01/2022

A Survey on Satisfiability Checking for the μ-Calculus through Tree Automata

Algorithms for model checking and satisfiability of the modal μ-calculus...
research
01/15/2019

Optimal Satisfiability Checking for Arithmetic μ-Calculi

The coalgebraic μ-calculus provides a generic semantic framework for fix...
research
02/20/2012

Strong Backdoors to Nested Satisfiability

Knuth (1990) introduced the class of nested formulas and showed that the...
research
02/14/2019

Variability Abstraction and Refinement for Game-based Lifted Model Checking of full CTL (Extended Version)

Variability models allow effective building of many custom model variant...
research
12/21/2022

Coalgebraic Satisfiability Checking for Arithmetic μ-Calculi

The coalgebraic μ-calculus provides a generic semantic framework for fix...
research
10/26/2018

Fixpoint Games on Continuous Lattices

Many analysis and verifications tasks, such as static program analyses a...
research
05/05/2021

Higher-Order Model Checking Step by Step

We show a new simple algorithm that solves the model-checking problem fo...

Please sign up or login with your details

Forgot password? Click here to reset