Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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 ,w herem is the circuit size and n is the number of input variables (IP01).  consider satisfiability of linear Boolean formula over the full binary basis and  show that the corresponding exponential complexities are "interwoven" with those of k-CNF SAT in the following sense. For any constant c, let fc be the exponential complexity of the satisfiability problem for Boolean formulas of size at most cn. Similarly, let sk be the exponential complexity of k-CNF SAT .W e prove that for anyc ,t here exists ak such that fc ≤ sk. Since the Sparsification Lemma (IPZ01) implies that for any k ,t here exists ac such that sk ≤ fc ,w e have sup c{fc} =s up k{sk}. (In fact,  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 "interaving" beten linear- size 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).
['Evgeny Dantsin', 'Alexander Wolpert']
I am writing a research paper on a idea Exponential Complexity of Satisfiability Testing for Linear-Size Boolean Formulas
[{"from": "human", "value": "I am writing a research paper on a idea Exponential Complexity of Satisfiability Testing for Linear-Size Boolean Formulas"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The exponential complexity of the satisfiability problem for a given class of Boolean circuits is defined to be the infimum of constants \u03b1 such that the problem can be solved in time poly(m)2 \u03b1n ,w herem is the circuit size and n is the number of input variables (IP01).  consider satisfiability of linear Boolean formula over the full binary basis and  show that the corresponding exponential complexities are \"interwoven\" with those of k-CNF SAT in the following sense. For any constant c, let fc be the exponential complexity of the satisfiability problem for Boolean formulas of size at most cn. Similarly, let sk be the exponential complexity of k-CNF SAT .W e prove that for anyc ,t here exists ak such that fc \u2264 sk. Since the Sparsification Lemma (IPZ01) implies that for any k ,t here exists ac such that sk \u2264 fc ,w e have sup c{fc} =s up k{sk}. (In fact,  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 \"interaving\" beten linear- size 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)."}]