Efficient Satisfiability Checking for Coalgebraic Fixpoint Logics with Arithmetic Modalities

Core Concepts

The paper presents a generic algorithm for exponential-time satisfiability checking in coalgebraic µ-calculi, without requiring the existence of tractable sets of modal tableau rules.

Abstract

The paper discusses the coalgebraic µ-calculus, a generic framework for fixpoint logics over systems with non-standard branching types. Previous work on the coalgebraic µ-calculus established an exponential-time upper bound on satisfiability checking, but relied on the availability of well-behaved modal tableau rules. The authors present a new approach that avoids this requirement. The key idea is to analyze the complexity of the one-step satisfiability problem, which is the satisfiability problem for the nesting-free next-step fragment of the logic. If this one-step problem is in ExpTime, then satisfiability checking for the full coalgebraic µ-calculus is also in ExpTime. The authors show that this criterion is satisfied in several important cases, including: The graded µ-calculus and its extension with polynomial inequalities The probabilistic µ-calculus extended with polynomial inequalities The technical development uses automata-theoretic tools, including tracking automata and parity games, to construct a generic satisfiability checking algorithm that can decide satisfiability on-the-fly, without fully expanding the tableau.

Stats

The paper does not contain any explicit numerical data or statistics. It focuses on theoretical complexity results.

Quotes

"The coalgebraic µ-calculus provides a generic semantic framework for fixpoint logics over systems whose branching type goes beyond the standard relational setup, e.g. probabilistic, weighted, or game-based." "We prove the same upper complexity bound under more general assumptions, specifically regarding the complexity of the (much simpler) satisfiability problem for the underlying one-step logic, roughly described as the nesting-free next-step fragment of the logic." "Our leading example applications are on the one hand the graded µ-calculus and its extension with (monotone) polynomial inequalities (including Presburger modalities, i.e. monotone linear inequalities), and on the other hand the extension of the (two-valued) probabilistic µ-calculus with (monotone) polynomial inequalities."

Key Insights Distilled From

Coalgebraic Satisfiability Checking for Arithmetic $μ$-Calculi

by Dani... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2212.11055.pdf

Deeper Inquiries

What are the potential applications of the coalgebraic µ-calculus beyond the examples discussed in the paper

The coalgebraic µ-calculus has a wide range of potential applications beyond the examples discussed in the paper. One key application is in the field of formal verification and model checking of complex systems. By providing a generic semantic framework for fixpoint logics over systems with diverse branching types, the coalgebraic µ-calculus can be used to analyze and verify properties of systems in various domains such as software, hardware, and biological systems. It can be applied to verify liveness properties, safety properties, and other temporal specifications in systems with probabilistic, weighted, or game-based transitions. Additionally, the coalgebraic µ-calculus can be used in the analysis of concurrent systems, distributed systems, and communication protocols to ensure correctness and reliability. Its flexibility and expressiveness make it a valuable tool in formal methods and automated reasoning.

How could the satisfiability checking algorithm be further optimized or adapted to specific instances of the coalgebraic µ-calculus

The satisfiability checking algorithm for the coalgebraic µ-calculus presented in the paper could be further optimized or adapted to specific instances of the calculus by considering the following approaches: Efficiency Improvements: The algorithm could be optimized by incorporating heuristics or pruning techniques to reduce the search space and improve the overall efficiency of the satisfiability checking process. This could involve identifying redundant computations or exploring more efficient data structures for storing intermediate results. Specialized Handling: For specific instances of the coalgebraic µ-calculus where certain properties or structures are known in advance, the algorithm could be tailored to exploit these characteristics for faster decision-making. This could involve customizing the algorithm to take advantage of domain-specific knowledge or constraints. Parallelization: To speed up the satisfiability checking process, the algorithm could be parallelized to leverage the computational power of multiple processors or cores. By dividing the workload and processing multiple tasks simultaneously, the overall runtime of the algorithm could be significantly reduced. Adaptive Strategies: Implementing adaptive strategies within the algorithm could allow it to dynamically adjust its approach based on the characteristics of the input formula or the complexity of the problem instance. This adaptability could lead to more efficient and effective satisfiability checking in diverse scenarios.

Are there any other important cases of coalgebraic µ-calculi where tractable modal tableau rules are not known, and which could benefit from the approach presented in this paper

There are several important cases of coalgebraic µ-calculi where tractable modal tableau rules are not known, and which could benefit from the approach presented in this paper. Some of these cases include: Real-valued Weights with Non-linear Arithmetic: Instances of the coalgebraic µ-calculus involving systems with real-valued weights and non-linear arithmetic pose challenges in terms of satisifiability checking. By applying the generic algorithm proposed in the paper, these instances could potentially benefit from a more efficient and on-the-fly satisfiability checking process. Complex Weighted Transition Systems: Systems with complex weighted transitions, such as those found in optimization problems or resource allocation scenarios, may lack tractable modal tableau rules. The algorithmic approach introduced in the paper could offer a systematic and effective way to handle satisifiability checking in these cases. Hybrid Coalgebraic Logics: Hybrid coalgebraic logics that combine different modalities and branching types may present challenges in terms of satisifiability checking. By extending the algorithm to accommodate the complexities of hybrid logics, a broader range of coalgebraic µ-calculi instances could benefit from improved satisfiability checking capabilities.