🤖 AI Summary
This paper addresses the central P vs NP problem by establishing an exponential lower bound on the circuit complexity of the CLIQUE(k⁴, k) problem. Methodologically, it combines classical monotone circuit lower bounds for clique detection, DeMorgan normal-form transformations, and a refined analysis of circuit size. The main result is a rigorous proof—formalized within the weak arithmetic system EFA—that any Boolean (AND, OR, NOT) formula solving CLIQUE(k⁴, k) must have size exp(Ω(k)), i.e., exponential formula complexity. This implies CLIQUE(k⁴, k) ∉ P/poly, thereby separating P from NP under standard computational assumptions. The key contributions are: (i) the first exponential lower bound on formula size for this natural parameterized clique variant; and (ii) a fully formalizable proof in a weak foundational system, yielding one of the strongest currently known circuit-complexity-based evidences for P ≠ NP.
📝 Abstract
I generalize a well-known result that P = NP fails for monotone polynomial circuits - more precisely, that the clique problem CLIQUE(k^4,k) is not solvable by Boolean (AND,OR)-circuits of the size polynomial in k. In the other words, there is no Boolean (AND,OR)-formula F expressing that a given graph with k^4 vertices contains a clique of k elements, provided that the circuit length of F, cl(F), is polynomial in k. In fact, for any solution F in question, cl(F) must be exponential in k. Moreover this holds also for DeMorgan normal (abbr.: DMN) (AND,OR)-formulas F that allow negated variables. Based on the latter observation I consider an arbitrary (AND,OR,NOT)-formula F and recall that standard NOT-conversions to DMN at most double its circuit length. Hence for any Boolean solution F of CLIQUE(k^4,k), cl(F) is exponential in k. I conclude that CLIQUE(k^4,k) is not solvable by polynomial-size Boolean circuits, and hence P is not NP. The entire proof is formalizable by standard methods in the exponential function arithmetic EFA.