The Complexity of Propositional Implication

The question whether a set of formulae G implies a formula f is fundamental. The present paper studies the complexity of the above implication problem for propositional formulae that are built from a systematically restricted set of Boolean connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post’s lattice and show that the implication problem is efficentily solvable only if the connectives are definable using the constants {false,true} and only one of {and,or,xor}. The problem remains coNP-complete in all other cases. We also consider the restriction of G to singletons.

💡 Research Summary

The paper investigates the computational complexity of the propositional implication problem under systematic restrictions on the set of Boolean connectives used in the formulas. Given a finite set B of Boolean functions, the problem IMP(B) asks whether a finite set Γ of B‑formulas entails a B‑formula φ (i.e., whether every assignment satisfying all formulas in Γ also satisfies φ). By employing Post’s lattice—the complete classification of Boolean clones—the authors reduce the infinite family of possible restrictions to a finite set of representative clones and provide a full dichotomy (actually a tetrarchy) of complexity classes for IMP(B).

The main classification theorem (Theorem 4.1) states:

1. coNP‑complete under AC⁰ many‑one reductions when the clone generated by B contains any of S₀₀, S₁₀, or D₂. These clones correspond to the presence of 0‑separating, 1‑separating, or monotone self‑dual functions, respectively. The hardness proof reduces the coNP‑complete TAUT‑DNF problem to IMP(B) by encoding a DNF formula and its “dual” using the available connectives, preserving logical equivalence via constant‑depth circuits.

2. ⊕L‑complete under AC⁰ many‑one reductions when B generates a linear clone L₂ (i.e., all functions are affine over GF(2) and both 0‑ and 1‑producing). In this case each formula can be rewritten as a linear equation modulo 2. The authors show that Γ⊢φ fails iff Γ∪{φ⊕t, t} is inconsistent, which translates to the solvability of a system of linear equations over Z₂. Since solving such systems is ⊕L‑complete (e.g., via the MOD‑GAP₂ problem), IMP(B) inherits ⊕L‑completeness.

3. **AC⁰