A Variational Method for Analysing Unit Clause Search

We expose a variational method for analysing algorithms, as applied to analyse the algorithm UC, which is the Davis-Putnam procedure for a set of clauses of three literals. The variable from a unit clause or, if there is none, the first remaining variable from a fixed list is chosen. The algorithm UC finds all the solutions as a set of cylinders. Following the nomenclature of P. W. Purdom [J. Inform. Process., 13 (1990), pp. 449--455], we call this algorithm "unit clause backtracking with cylinders of solutions." First we give an expression for the number of nodes of the calculation trees of all the inputs. Then we use a variational method to calculate the base $\beta$ of the principal exponential part of the average time of calculation $T$. This "exponential base" is the maximum of three elementary functions $f_i$ of four real variables. These functions are defined on the product of the half positive real line by the 3-dimensional unit real cube. We finally obtain the following short statement. Let $v$ be the number of the variables. Let $c$ be the number of the clauses. Let $\gamma=c/v>1$. Let $\gamma$ be constant when $v$ grows to infinity. The principal exponential part of the average time of UC is $\beta^v$ where $$\beta=\OO({\max} ; 0\leq \lambda\leq 1) 2^\lambda{\bigl(1-{3\lambda^2\over 8}+{\lambda^3\over4}\bigr)}^\gamma.$$ We mean that $\lim\limits_{v\rightarrow\infty}T^{1\over v}=\beta$. As a first consequence of our method we match UC, with the algorithm B without rearrangement (i. e. with a fixed order for introducing the variables). This gives a proof to a conjecture of P. W. Purdom [Artif. Intell., 21 (1983), pp. 117-133].