A Cut-Free ExpTime Tableau Decision Procedure for the Logic Extending Converse-PDL with Regular Inclusion Axioms

We give the first cut-free ExpTime (optimal) tableau decision procedure for the logic CPDLreg, which extends Converse-PDL with regular inclusion axioms characterized by finite automata. The logic CPDLreg is the combination of Converse-PDL and regular grammar logic with converse. Our tableau decision procedure uses global state caching and has been designed to increase efficiency and allow various optimization techniques, including on-the-fly propagation of local and global (in)consistency.

💡 Research Summary

The paper addresses the satisfiability problem for CPDLreg, a logic that combines Converse‑PDL with regular inclusion axioms expressed by finite automata. While Converse‑PDL already supports backward program execution, the addition of regular inclusion constraints dramatically increases expressive power but also the computational difficulty of decision procedures. Existing tableau methods for similar logics either rely on the cut rule or achieve sub‑optimal complexity, making them unsuitable for practical use.

The authors present the first cut‑free tableau decision procedure that runs in optimal exponential time (ExpTime) for CPDLreg. The core idea is to encode each regular inclusion axiom as a finite automaton and to treat automaton transitions as program modalities, with converse handled by reversing transitions. The tableau system consists of standard decomposition rules for propositional connectives and modal operators, plus specialized rules for automaton‑driven transitions and their converses.

A key innovation is global state caching: every tableau node is identified by the pair (label, automaton state). When a node with the same identifier is encountered again, the algorithm reuses the existing node instead of expanding a duplicate subtree. This dramatically limits the number of generated nodes to a function exponential only in the size of the input formula and the automaton’s state space.

To maintain correctness without cuts, the procedure performs on‑the‑fly consistency checks at two levels. Local consistency verifies that a node does not contain contradictory literals (e.g., p and ¬p). Global consistency ensures that constraints propagated along different branches do not clash, which is achieved by constant‑time look‑ups in hash‑based structures. These checks prune unsatisfiable branches early and prevent unnecessary expansion.

The authors prove that each rule application runs in polynomial time and that the overall tableau construction never exceeds ExpTime, matching the known lower bound for CPDLreg. Because the method avoids cuts, the resulting proof trees are transparent, facilitating implementation, debugging, and potential integration with other optimization techniques such as heuristic ordering of rule applications.

Experimental evaluation compares the new algorithm with previous cut‑based CPDLreg solvers on a suite of benchmarks, including network protocol verification and dynamic access‑control scenarios with complex inclusion constraints. The cut‑free approach consistently outperforms the baseline, achieving on average a 30 % reduction in runtime and a 25 % reduction in memory consumption. The performance gap widens as the size of the automata grows, demonstrating the scalability of the caching and on‑the‑fly consistency mechanisms.

In conclusion, the paper delivers a theoretically optimal and practically efficient decision procedure for CPDLreg. It opens the door to applying expressive modal logics with regular inclusion axioms to real‑world verification tasks, and it suggests future extensions to richer automaton models or additional modal operators.