Canonical Logic Programs are Succinctly Incomparable with Propositional Formulas
This work addresses a foundational question in computational logic and complexity theory, providing a separation result that is incremental but clarifies the relative expressive power of these formalisms.
The paper tackled the problem of comparing the succinctness of canonical logic programs (CP) and propositional formulas (PF), showing that the PARITY problem has polynomial representations in PF but only exponential ones in CP, establishing their succinct incomparability. This result implies an exponential size blowup when translating PF to CP without new variables.
\emph{Canonical (logic) programs} (CP) refer to normal logic programs augmented with connective $not\ not$. In this paper we address the question of whether CP are \emph{succinctly incomparable} with \emph{propositional formulas} (PF). Our main result shows that the PARITY problem, which can be polynomially represented in PF but \emph{only} has exponential representations in CP. In other words, PARITY \emph{separates} PF from CP. Simply speaking, this means that exponential size blowup is generally inevitable when translating a set of formulas in PF into an equivalent program in CP (without introducing new variables). Furthermore, since it has been shown by Lifschitz and Razborov that there is also a problem that separates CP from PF (assuming $\mathsf{P}\nsubseteq \mathsf{NC^1/poly}$), it follows that CP and PF are indeed succinctly incomparable. From the view of the theory of computation, the above result may also be considered as the separation of two \emph{models of computation}, i.e., we identify a language in $\mathsf{NC^1/poly}$ which is not in the set of languages computable by polynomial size CP programs.