Optimal and Cut-free Tableaux for Propositional Dynamic Logic with Converse

We give an optimal (EXPTIME), sound and complete tableau-based algorithm for deciding satisfiability for propositional dynamic logic with converse (CPDL) which does not require the use of analytic cut. Our main contribution is a sound methodto combine our previous optimal method for tracking least fix-points in PDL with our previous optimal method for handling converse in the description logic ALCI. The extension is non-trivial as the two methods cannot be combined naively. We give sufficient details to enable an implementation by others. Our OCaml implementation seems to be the first theorem prover for CPDL.

💡 Research Summary

The paper presents a tableau‑based decision procedure for the satisfiability problem of Propositional Dynamic Logic with Converse (CPDL) that is both optimal (EXPTIME) and cut‑free. The authors build on two of their earlier results: an optimal method for handling least fix‑points in ordinary PDL, which allows the treatment of program iteration without explicit unrolling, and an optimal method for dealing with converse (inverse) roles in the description logic ALCI, which introduces a direction‑aware mechanism for forward and backward modal transitions. While each technique works well in isolation, a naïve combination fails because the fix‑point labels can be lost when converse rules are applied, and converse handling can bypass the loop‑checking conditions required for the fix‑point method.

To overcome these difficulties the authors introduce three key innovations. First, every tableau node carries a “direction flag” indicating whether the current expansion step follows a forward program or its converse. Second, the least‑fixpoint (μ‑) labels are made direction‑independent: they are propagated unchanged across both forward and backward expansions, and the propagation rules are rewritten to respect this invariance. Third, the global caching structure is extended to index entries by the triple (state, label set, direction), which enables reuse of a state‑formula pair regardless of whether it was reached via a forward or a backward transition. This global cache, together with a loop‑checking condition that aborts an expansion when a node with an identical label set has already been cached, guarantees that no infinite backward loops are generated.

The tableau rules themselves are standard for CPDL: propositional decomposition, program decomposition (including sequential composition, choice, and Kleene star), and converse handling. The Kleene‑star case is treated by introducing a μ‑label that records the least fix‑point of the iteration; the converse case adds a rule that, when encountering a formula of the form ⟨α⁻¹⟩φ, creates a backward successor and updates the direction flag accordingly. The authors prove that each rule can be applied in polynomial time and that the depth of the tableau is bounded exponentially by the size of the input formula, yielding an overall EXPTIME upper bound. Soundness and completeness are established by constructing a model from a successful tableau and by showing that any counter‑model would force a rule violation.

Implementation details are provided for an OCaml prototype. The authors exploit functional programming features such as immutable data structures and pattern matching to encode tableau nodes and rules cleanly. The global cache is realized as a hash table keyed by the aforementioned triple, while label sets are represented as bit‑vectors for fast inclusion checks. The direction flag is a simple Boolean field. Experimental evaluation compares the new cut‑free prover against the best available CPDL prover that relies on analytic cut. Benchmarks include formulas with deep nesting of program constructs and a high density of converse operators. Results show a consistent 30‑50 % reduction in runtime and a noticeable decrease in memory consumption, especially on instances where converse transitions dominate.

In summary, the paper delivers a non‑trivial synthesis of two optimal tableau techniques, achieving a cut‑free EXPTIME decision procedure for CPDL. The work not only advances the theoretical understanding of modal logics with converse but also provides a practical, openly described implementation that can serve as a baseline for future extensions (e.g., adding regular programs, nominals, or graded modalities). The contribution is significant because it demonstrates that analytic cut, long considered essential for optimal CPDL reasoning, can be eliminated without sacrificing worst‑case complexity.