3 research outputs found
Fixpoint Games on Continuous Lattices
Many analysis and verifications tasks, such as static program analyses and
model-checking for temporal logics reduce to the solution of systems of
equations over suitable lattices. Inspired by recent work on lattice-theoretic
progress measures, we develop a game-theoretical approach to the solution of
systems of monotone equations over lattices, where for each single equation
either the least or greatest solution is taken. A simple parity game, referred
to as fixpoint game, is defined that provides a correct and complete
characterisation of the solution of equation systems over continuous lattices,
a quite general class of lattices widely used in semantics. For powerset
lattices the fixpoint game is intimately connected with classical parity games
for -calculus model-checking, whose solution can exploit as a key tool
Jurdzi\'nski's small progress measures. We show how the notion of progress
measure can be naturally generalised to fixpoint games over continuous lattices
and we prove the existence of small progress measures. Our results lead to a
constructive formulation of progress measures as (least) fixpoints. We refine
this characterisation by introducing the notion of selection that allows one to
constrain the plays in the parity game, enabling an effective (and possibly
efficient) solution of the game, and thus of the associated verification
problem. We also propose a logic for specifying the moves of the existential
player that can be used to systematically derive simplified equations for
efficiently computing progress measures. We discuss potential applications to
the model-checking of latticed -calculi and to the solution of fixpoint
equations systems over the reals
A Universal Attractor Decomposition Algorithm for Parity Games
An attractor decomposition meta-algorithm for solving parity games is given
that generalizes the classic McNaughton-Zielonka algorithm and its recent
quasi-polynomial variants due to Parys (2019), and to Lehtinen, Schewe, and
Wojtczak (2019). The central concepts studied and exploited are attractor
decompositions of dominia in parity games and the ordered trees that describe
the inductive structure of attractor decompositions.
The main technical results include the embeddable decomposition theorem and
the dominion separation theorem that together help establish a precise
structural condition for the correctness of the universal algorithm: it
suffices that the two ordered trees given to the algorithm as inputs embed the
trees of some attractor decompositions of the largest dominia for each of the
two players, respectively.
The universal algorithm yields McNaughton-Zielonka, Parys's, and
Lehtinen-Schewe-Wojtczak algorithms as special cases when suitable universal
trees are given to it as inputs. The main technical results provide a unified
proof of correctness and deep structural insights into those algorithms.
A symbolic implementation of the universal algorithm is also given that
improves the symbolic space complexity of solving parity games in
quasi-polynomial time from ---achieved by Chatterjee,
Dvo\v{r}\'{a}k, Henzinger, and Svozil (2018)---down to , where is
the number of vertices and is the number of distinct priorities in a parity
game. This not only exponentially improves the dependence on , but it also
entirely removes the dependence on