[Paper Review] Conditionally Optimal Algorithms for Generalized Büchi Games

Krishnendu Chatterjee, Wolfgang Dvořák+2 more|arXiv (Cornell University)|Jan 1, 2016

Formal Methods in Verification28 references6 citations

TL;DR

This paper presents conditionally optimal algorithms for generalized Büchi games on graphs and Markov Decision Processes (MDPs), establishing the first conditional super-linear lower bounds for model-checking problems under widely accepted complexity assumptions. It demonstrates that disjunctive queries (union of objectives) are fundamentally harder than conjunctive queries (intersection), even when the individual objectives are of the same type, with strictly higher conditional lower bounds for disjunction compared to conjunction, thus proving novel model and objective separation results for ω-regular objectives such as Büchi, co-Büchi, Streett, and Rabin.

ABSTRACT

Games on graphs provide the appropriate framework to study several central problems in computer science, such as verification and synthesis of reactive systems. One of the most basic objectives for games on graphs is the liveness (or Büchi) objective that given a target set of vertices requires that some vertex in the target set is visited infinitely often. We study generalized Büchi objectives (i.e., conjunction of liveness objectives), and implications between two generalized Büchi objectives (known as GR(1) objectives), that arise in numerous applications in computer-aided verification. We present improved algorithms and conditional super-linear lower bounds based on widely believed assumptions about the complexity of (A1) combinatorial Boolean matrix multiplication and (A2) CNF-SAT. We consider graph games with n vertices, m edges, and generalized Büchi objectives with k conjunctions. First, we present an algorithm with running time O(k*n^2), improving the previously known O(k*n*m) and O(k^2*n^2) worst-case bounds. Our algorithm is optimal for dense graphs under (A1). Second, we show that the basic algorithm for the problem is optimal for sparse graphs when the target sets have constant size under (A2). Finally, we consider GR(1) objectives, with k_1 conjunctions in the antecedent and k_2 conjunctions in the consequent, and present an O(k_1 k_2 n^{2.5})-time algorithm, improving the previously known O(k_1*k_2*n*m)-time algorithm for m > n^{1.5}.

Motivation & Objective

• To close the gap between polynomial-time upper bounds (quadratic or cubic) and the lack of super-linear lower bounds for fundamental model-checking problems in graphs and MDPs.
• To establish the first conditional lower bounds for ω-regular objectives such as Büchi, co-Büchi, Streett, and Rabin using widely accepted complexity assumptions.
• To demonstrate model separation between graphs and MDPs, and objective separation between conjunction and disjunction of objectives, under conditional complexity assumptions.
• To show that disjunctive queries (union of objectives) are inherently harder than conjunctive queries (intersection of objectives), even when the individual objectives are of the same type.

Proposed method

• Leverages conditional lower bounds based on two standard assumptions: (A1) no O(n^{3−ε}) combinatorial Boolean matrix multiplication, and (A2) no 2^{(1−ε)n} time algorithm for k-CNF-SAT.
• Reduces the model-checking problem for disjunctive queries to shortest path computation in a modified graph with 0/1 edge weights, using a two-vertex split (sin, sout) to track cycle length from a target vertex.
• Uses a multi-level BFS-like traversal with queues Qj to compute the shortest cycle length from sout to sin, where distance j corresponds to path length j in the edge-weighted graph.
• Employs a marking mechanism to ensure each vertex is processed at most once per level, achieving O(m) time complexity for the singleton co-Büchi case.
• Proves correctness via induction: Qj contains exactly the vertices at distance j from sout, and sin is reached in fewer than k steps iff a cycle avoiding all Ti exists.
• Applies the same framework to Rabin objectives with singleton sets, showing the same linear-time complexity holds.

Experimental results

Research questions

• RQ1 Is there a conditional super-linear lower bound for disjunctive queries of Büchi objectives in MDPs, higher than the known upper bound for conjunctive queries in graphs?
• RQ2 Can a separation be established between graphs and MDPs for the same objective, such that MDPs require strictly more time under conditional complexity assumptions?
• RQ3 Is the disjunction of objectives fundamentally harder than the conjunction of the same objectives, even when the individual objectives are of the same type (e.g., co-Büchi)?
• RQ4 Can conditional lower bounds be established for generalized Büchi games using standard complexity conjectures like Boolean matrix multiplication or CNF-SAT?

Key findings

• The paper establishes the first conditional super-linear lower bounds for model-checking problems in graphs and MDPs under widely accepted complexity assumptions.
• For disjunctive queries of co-Büchi objectives, the algorithm runs in O(m) time on graphs, matching the best-known upper bound for the same problem.
• The conditional lower bound for disjunctive queries (union of objectives) is strictly higher than the known upper bound for conjunctive queries (intersection of objectives), even when the individual objectives are of the same type.
• A model separation is shown: disjunctive queries of reachability and Büchi objectives are provably harder in MDPs than in graphs, under the same complexity assumptions.
• For dual objectives such as reachability/safety and Streett/Rabin, the paper establishes objective separation results in both graphs and MDPs.
• The algorithm for singleton co-Büchi objectives in strongly connected graphs runs in linear time O(m), with correctness proven via a layered BFS traversal that tracks distance from a split vertex.