Dependency Stochastic Boolean Satisfiability: A Logical Formalism for NEXPTIME Decision Problems with Uncertainty

TL;DR

This paper introduces Dependency SSAT (DSSAT), an extension of SSAT capable of representing NEXPTIME decision problems with uncertainty, such as Dec-POMDPs, and demonstrates its theoretical properties and applications.

Contribution

It extends SSAT to DSSAT, capturing NEXPTIME problems like Dec-POMDPs, and establishes its theoretical complexity and practical relevance.

Findings

DSSAT is NEXPTIME-complete.

Dec-POMDP reduces to DSSAT in polynomial time.

DSSAT can be applied to probabilistic circuit synthesis.

Abstract

Stochastic Boolean Satisfiability (SSAT) is a logical formalism to model decision problems with uncertainty, such as Partially Observable Markov Decision Process (POMDP) for verification of probabilistic systems. SSAT, however, is limited by its descriptive power within the PSPACE complexity class. More complex problems, such as the NEXPTIME-complete Decentralized POMDP (Dec-POMDP), cannot be succinctly encoded with SSAT. To provide a logical formalism of such problems, we extend the Dependency Quantified Boolean Formula (DQBF), a representative problem in the NEXPTIME-complete class, to its stochastic variant, named Dependency SSAT (DSSAT), and show that DSSAT is also NEXPTIME-complete. We demonstrate the potential applications of DSSAT to circuit synthesis of probabilistic and approximate design. Furthermore, to study the descriptive power of DSSAT, we establish a polynomial-time reduction from Dec-POMDP to DSSAT. With the theoretical foundations paved in this work, we hope to encourage the development of DSSAT solvers for potential broad applications.