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We consider the relationship between the complexities of-and those of restricted to formulas of constant density. Let be the infimum of those such that-on variables can be decided in time and be the infimum of those such that on variables and clauses can be decided in time. We show that. So, for any ,-can be solved in time independent of if and only if the(More)
We consider the satisfiability problem for circuits of limited size and/or depth. Say that an algorithm solving a Boolean satisfiability problem on n variables is improved iff it takes time O(2 cn) for some constant c < 1, i.e. iff it is exponentially better than a brute force search. We show an improved algorithm for the satisfiability problem for circuits(More)
Preface This is a set of class notes that we have been developing jointly for some years. We use them for the graduate cryptography courses that we teach at our respective institutions. Each time one of us teaches the class, he takes the token and updates the notes a bit. You might think that, within a three or four years, one would have a rather complete(More)
We provide some evidence that Unique k-SAT is as hard to solve as general k-SAT, where k-SAT denotes the satisfiability problem for k-CNFs with at most k literals in each clause and Unique k-SAT is the promise version where the given formula has 0 or 1 solutions. Namely, defining for each k 1, s k = inf{δ 0 | ∃ a O(2 δn)-time randomized algorithm for k-SAT}(More)
One way to quantify how dense a multidag is in long paths is to find the largest n, m such that whichever ≤ n edges are removed, there is still a path from an original input to an original output with ≥ m edges-the larger we can make n, m, the denser is the graph. For a given n, m, we would like to lower bound the size such a graph, say in edges, at least(More)
We relate the exponential complexities 2 s(k)n of $\textsc {$k$-sat}$ and the exponential complexity $2^{s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))n}$ of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ (the problem of evaluating quantified formulas of the form $\forall\vec{x} \exists\vec{y} \textsc {F}(\vec {x},\vec{y})$ where F is a(More)
We provide some evidence that Unique-SAT is as hard to solve as general-SAT, where-SAT denotes the satisfiability problem for-CNFs and Unique-SAT is the promise version where the given formula has or solutions. Namely, defining for each , a-time randomized algorithm for-SAT and, similarly, a-time randomized algorithm for Unique-SAT , we show that. As a(More)
The n × n puzzle game is played on a matrix of numbered tiles with 1 tile missing to allow tiles to shift. The goal is to order the tiles by a sequence of shifts. We provide a O(n 2)-time algorithm to decide when an initial configuration of the n × n puzzle game is solvable. We also provide an algorithm solving the game in O(n 3) moves and show that this is(More)