The model checking problem for intuitionistic propositional logic with one variable is AC1-complete

We show that the model checking problem for intuitionistic propositional logic with one variable is complete for logspace-uniform AC1. As basic tool we use the connection between intuitionistic logic and Heyting algebra, and investigate its complexity theoretical aspects. For superintuitionistic logics with one variable, we obtain NC1-completeness for the model checking problem.

💡 Research Summary

The paper investigates the computational complexity of the model‑checking problem for intuitionistic propositional logic (IPC) when the language is restricted to a single propositional variable, denoted IPC₁. The authors establish that this problem is complete for log‑space uniform AC¹, making it the first natural problem known to be AC¹‑complete without an explicit logarithmic bound baked into its definition.

The technical core relies on the classic Rieger‑Nishimura classification: every IPC₁ formula is equivalent to exactly one member of the infinite sequence of Rieger‑Nishimura formulas {⊥, ⊤, φ₁, ψ₁, φ₂, ψ₂,…}. Each such formula is uniquely identified by a pair (i, x), called its Rieger‑Nishimura index, where i is the “rank” of the formula. By analysing the lattice of equivalence classes (a free Heyting algebra on one generator) the authors prove a tight relationship between formula length and rank: the length of any IPC₁ formula is at least the i‑th Fibonacci number, which yields rank(φ) ≤ c·log|φ| for a constant c. Consequently, the index of any sub‑formula can be stored using only logarithmic space.

Using this structural insight, the authors design a recursive algorithm (Algorithm 1) that computes the RN‑index of a given formula by traversing its syntax tree and applying three lattice operations (∧, ∨, →) that correspond to the Heyting algebra operations u, t, _. These operations are pre‑computed in a constant‑size lookup table, so each recursive step runs in constant time. Because the recursion depth is bounded by rank(φ) = O(log|φ|), the whole computation can be performed by a log‑space deterministic machine equipped with a stack, i.e., a LOGdetCFL machine. This places the model‑checking problem for IPC₁ in LOGdetCFL ⊆ AC¹.

For the lower bound, the authors reduce the canonical AC¹‑hard problem ALOGSPACE