Exponential Complexity of Satisfiability Testing for Linear-Size Boolean Formulas [chapter]

Evgeny Dantsin, Alexander Wolpert
2013 Lecture Notes in Computer Science  
The exponential complexity of the satisfiability problem for a given class of Boolean circuits is defined to be the infimum of constants α such that the problem can be solved in time poly(m) 2 αn , where m is the circuit size and n is the number of input variables [IP01]. We consider satisfiability of linear Boolean formula over the full binary basis and we show that the corresponding exponential complexities are "interwoven" with those of k-CNF SAT in the following sense. For any constant c,
more » ... t fc be the exponential complexity of the satisfiability problem for Boolean formulas of size at most cn. Similarly, let s k be the exponential complexity of k-CNF SAT. We prove that for any c, there exists a k such that fc ≤ s k . Since the Sparsification Lemma [IPZ01] implies that for any k, there exists a c such that s k ≤ fc, we have sup c {fc} = sup k {s k }. (In fact, we prove this equality for a larger class of linear-size circuits that includes Boolean formulas.) Our work is partly motivated by two recent results. The first one is about a similar "interweaving" between linearsize circuits of constant depth and k-CNFs [SS12]. The second one is that satisfiability of linear-size Boolean formulas can be tested exponentially faster than in O(2 n ) time [San10, ST12] .
doi:10.1007/978-3-642-38233-8_10 fatcat:ritdjkikvbfvzgic4aiymzqere