Razborov proved that the monotone function matching is not in mP. But can we compute matching using a polynomial size circuit with a few negations? Is there a P/poly circuit with $O(n^\epsilon)$ negations that computes matching? What is the trade-off between the number of negations and the size for matching?
How many negations do we need to compute monotone functions?
cc.complexity theory;circuit complexity;lower bounds;matching;monotone
Markov proved that any function of $n$ inputs can be computed with only $\lceil \log (n+1)\rceil$ negations. An efficient constructive version was described by Fisher. See also an exposition of the result from the GLL blog.More precisely:Theorem: Suppose $f : \{0,1\}^n \to \{0,1\}^m$ is computed by a circuit $C$ with $g$ gates, then it is also computed by a circuit $C^*$ with $2g + O(n^2 \log^2 n)$ gates and $\lceil \log (n+1) \rceil$ negations.The main idea is to add for each wire $w$ in $C$ a parellel wire $w'$ in $C^*$ that always carries the complement of $w$. The base case is for the input wires: Fisher describes how to construct an inversion circuit $I(x) = \overline x$ with $O(n^2 \log^2 n)$ gates and only  $\lceil \log (n+1) \rceil$ negations. For the AND gates of circuit $C$, we can augment $a = b \land c$ with $a' = b' \lor c'$, and likewise for OR gates. NOT gates in $C$ cost nothing, we just swap the roles of $w$ and $w'$ downstream of the NOT gate. In this way, the entire circuit besides the inverter subcircuit is monotone.A. A. Markov. On the inversion complexity of a system of functions. J. ACM, 5(4):331334, 1958.M. J. Fischer. The complexity of negation-limited networks - A brief survey. In  Automata Theory and Formal Languages, 7182, 1975