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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)

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)

- Chris Calabro
- 2009

The dissertation of Chris Calabro is approved, and it is acceptable in quality and form for publication on microfilm and electronically:

We resolve an open question by [3]: the exponential complexity of deciding whether a k-CNF has a solution is the same as that of deciding whether it has exactly one solution, both when it is promised and when it is not promised that the input formula has a solution. We also show that this has the same exponential complexity as deciding whether a given… (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)

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)

- Chris Calabro
- 2008

It is well known that 2-SAT ∈ P, but the analogous result for 2-TQBF is a bit less well known. I couldn't it find anywhere, so I wrote this. Later I was told by Sam Buss that it was shown to be in linear time in [PAT79]. My proof here at least is easier to understand, but less insightful.

- Chris Calabro
- 2007