\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.

Canonical (logic) programs (CP) refer to the class of normal programs (LP) augmented with connective not not , and  are equally expressive as propositional formulas (PF). In this paper we address the question of whether CP and PF are succinctly incomparable. Our main result shows that the PARITY problem only has exponential CP representations, while it can be polynomially represented in PF.  In other words, PARITY separates PF from CP. Simply speaking, this means that exponential size blowup is generally inevitable  when  translating a set of PF formulas into a (logically) equivalent CP program (without introducing new variables). Furthermore,  since it has been shown by Lifschitz and  Razborov  that there is also a problem which separates CP from PF (assuming P ⊈ NC 1 poly), it follows that the two formalisms are indeed succinctly incomparable.