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We show that simple syntactic expressions such as existential second order(ESO) universal Horn formulae can express NP-hard optimisation problems. Thereis a significant difference between the expressibilities of decision problemsand optimisation problems. This is similar to the difference in computationtimes for the two classes of problems; for example, a 2SAT Horn formula can besatisfied in polynomial time, whereas the optimisation version in NP-hard. Itis known that all polynomially solvable decision problems can be expressed asESO universal ($\Pi_1$) Horn sentences in the presence of a successor relation.We show here that, on the other hand, if $P \neq NP$, optimisation problemsdefy such a characterisation, by demonstrating that even a $\Pi_0$ (quantifierfree) Horn formula is unable to guarantee polynomial time solvability. Finally,by connecting concepts in optimisation duality with those in descriptivecomplexity, we will show a method by which optimisation problems can be solvedby a single call to a "decision" Turing machine, as opposed to multiple callsusing a classical binary search setting.

Lower Bounds on Syntactic Logic Expressions for Optimization Problems and Duality using Lagrangian Dual to characterize optimality conditions