"What we intend to do over the course of the months that lie ahead is to expand the security bubbles in various areas. In some areas, we have already reversed the momentum of the Taliban. In others, we still need to do that and we are intent on doing that."
Petraeus acknowledged there was still much more work to be done on other elements of the war, not just with security.
Of the effort to reintegrate anti-government insurgents into Afghan society in exchange for pledging to renounce al-Qaeda, putting down arms against the government, and abiding by the Afghan constitution, Petraeus said so far, only "small numbers" have come in to do so.
"But we do see the beginnings of it. And, of course, the formal program, the Afghan-led reintegration of reconcilables, that program is really just beginning to roll out now."
Soon, he said, Karzai would announce the membership of the national peace council created during June's peace jirga to reconcile with senior members of the Taliban and other anti-government insurgents.
"That will be another important step in this process," he said.
Petraeus said Afghan President Hamid Karzai's government has taken a number of actions against corruption, including after news that the Central Bank had removed Kabul Bank's chairman and CEO after it was found that tens of millions of dollars had been funneled into risky Dubai property investments, and loans were given to shareholders with close links to the Karzai Administration.
"In recent years, indeed even in recent months, various elements of President Karzai's government have taken a number of actions against corruption," Petraeus said.
"The chief justice has hired hundreds of judicial workers, put a number of judges in jail. The Minister of Finance has fired literally in recent weeks, actually, just dozens of customs officials," he said. "The commander of the western border police, is a brigadier general, is now in jail and was recently convicted a few weeks ago. A very important provincial police chief just put in jail. Another one fired, a governor fired and so forth. So, there's actually been quite a bit of activity in the realm of anti-corruption."
"Having said that, President Karzai is the first-- to state publicly that more needs to be done," Petraeus said.
Petraeus said the next major assessment of the strategy would come not during President Obama's review, but at a NATO summit in mid-November, where a discussion for the "prospects for transition" will take place.
What metaphor would Petraeus use for to describe the war effort? He pointed to a Frederick Remington print on the wall of his office in Kabul, called "The Stampede".