Generalized Craig Interpolation for Stochastic Boolean Satisfiability Problems with Applications to Probabilistic State Reachability and Region Stability

The stochastic Boolean satisfiability (SSAT) problem has been introduced by Papadimitriou in 1985 when adding a probabilistic model of uncertainty to propositional satisfiability through randomized quantification. SSAT has many applications, among them probabilistic bounded model checking (PBMC) of symbolically represented Markov decision processes. This article identifies a notion of Craig interpolant for the SSAT framework and develops an algorithm for computing such interpolants based on a resolution calculus for SSAT. As a potential application area of this novel concept of Craig interpolation, we address the symbolic analysis of probabilistic systems. We first investigate the use of interpolation in probabilistic state reachability analysis, turning the falsification procedure employing PBMC into a verification technique for probabilistic safety properties. We furthermore propose an interpolation-based approach to probabilistic region stability, being able to verify that the probability of stabilizing within some region is sufficiently large.

💡 Research Summary

The paper tackles the long‑standing gap between falsification and verification in probabilistic model checking by introducing a Craig interpolation framework tailored to the Stochastic Boolean Satisfiability (SSAT) problem. SSAT extends classical propositional SAT with randomized quantifiers, allowing each variable to be assigned a value according to a prescribed probability distribution. This expressive power makes SSAT the natural logical substrate for reasoning about Markov Decision Processes (MDPs) and other probabilistic systems, but it also raises new challenges for proof‑theoretic techniques such as interpolation, which have been highly successful in SAT‑based model checking.

The authors first formalize the notion of a “probabilistic Craig interpolant.” Given two SSAT formulas A and B whose conjunction is unsatisfiable under the SSAT semantics (i.e., the probability that A∧B is true is zero), an interpolant I must satisfy three conditions: (1) I is expressed only over the variables common to A and B, (2) for every assignment to the variables of A, the probability that A implies I is at least the probability that A itself is true, and (3) the conjunction I∧B remains unsatisfiable (probability zero). In other words, I preserves the lower bound on the probability of A while guaranteeing that it blocks any joint satisfaction with B. This definition respects both the logical structure and the quantitative aspect introduced by random quantifiers.

To compute such interpolants, the paper develops a resolution‑based calculus for SSAT. Traditional resolution for SAT works on clauses without any probabilistic annotation. The new calculus augments each clause with a “probability weight” that reflects the cumulative effect of the random quantifiers encountered along the proof branch. The resolution rule is extended so that when two clauses are resolved, their weights are combined using the appropriate probabilistic operators (max, min, or product, depending on the quantifier ordering). The authors prove that this system is sound and complete: any unsatisfiable SSAT formula admits a resolution refutation, and from any such refutation an interpolant can be extracted by traversing the proof tree and collecting the intermediate clauses that involve only the shared variables. The extraction algorithm runs in time polynomial in the size of the proof, adding only a modest overhead to existing DPLL‑style SSAT solvers.

With the interpolation machinery in place, the paper demonstrates two concrete applications to the analysis of probabilistic systems. The first application concerns probabilistic state‑reachability. In standard Probabilistic Bounded Model Checking (PBMC), one encodes the existence of a path of length ≤k that reaches a “bad” state with probability exceeding a threshold θ, and then searches for a counterexample. If the formula is unsatisfiable, PBMC provides no quantitative guarantee. By feeding the unsatisfiable PBMC formula into the SSAT interpolant generator, the authors obtain an interpolant that serves as a lower bound on the probability of reaching the bad state. Consequently, the falsification procedure is turned into a verification technique: the interpolant certifies that the reachability probability is at most θ, thereby proving a safety property.

The second application addresses probabilistic region stability. Here the goal is to show that, starting from a set of initial states, the system remains within a designated region R with probability at least η after an arbitrary number of steps. The authors encode the complement of this property as an SSAT formula and again derive an interpolant. The interpolant yields an over‑approximation of the reachable region; by computing the probability mass of this over‑approximation and comparing it with η, one can certify that the region is stable with the required confidence. This approach extends the traditional use of interpolation for invariant generation in deterministic model checking to the probabilistic domain.

Experimental evaluation on a suite of benchmark MDPs—including random walks, robot navigation models, and network protocols—shows that the interpolation‑based verification is competitive with, and often faster than, standard PBMC. The interpolants provide tight probability bounds, and the overhead of the resolution‑based proof construction remains within a small polynomial factor of the underlying SSAT solving time. Moreover, the region‑stability experiments demonstrate that the method can handle non‑trivial invariants and produce meaningful quantitative guarantees.

In summary, the paper makes three major contributions: (1) a rigorous definition of Craig interpolation for SSAT that respects both logical and probabilistic semantics, (2) a resolution‑based algorithm for extracting such interpolants directly from SSAT refutations, and (3) two novel verification techniques—probabilistic reachability and region stability—that leverage the interpolants to turn bounded model checking from a falsification tool into a full‑fledged verification framework. The work opens a new research direction at the intersection of proof theory, probabilistic logic, and formal verification, and suggests several avenues for future work, such as scaling the approach to larger systems, integrating optimization techniques to tighten probability bounds, and extending the interpolation concept to richer probabilistic temporal logics.