  In Descriptive Complexity, there is a vast amount of literature on decision
problems, and their classes such as \textbf{P, NP, L and NL}. ~ However,
research on the descriptive complexity of optimisation problems has been
limited. Optimisation problems corresponding to the \textbf{NP} class have been
characterised in terms of logic expressions by Papadimitriou and Yannakakis,
Panconesi and Ranjan, Kolaitis and Thakur, Khanna et al, and by Zimand.
Gr\"{a}del characterised the polynomial class \textbf{P} of decision problems.
In this paper, we attempt to characterise the optimisation versions of
\textbf{P} via expressions in second order logic, many of them using universal
Horn formulae with successor relations. The polynomially bound versions of
maximisation (maximization) and minimisation (minimization) problems are
treated first, and then the maximisation problems in the "not necessarily
polynomially bound" class.
