The long-term partnership that President Obama signed with the Afghan government commits the U.S. to a role in the troubled nation for at least a dozen more years, leaving critics fuming over the uncertain costs of a conflict that already has stretched for a decade.

Details of the agreement are still vague, as is the eventual cost of the mission, which could be as much as $4 billion annually.

The agreement was reached even as Afghanistan remains tumultuous, with U.S.-led efforts to make peace with the Taliban spawning turmoil within the militant group’s own ranks.

Bombings Wednesday marked the start of the Taliban’s spring offensive — the annual beginning of renewed attacks. Some of the movement’s field commanders are angry at their leadership for negotiating.

Anthony Cordesman, a leading military analyst at the Center for Strategic and International Studies, said the $4 billion estimate is only a tentative figure and that the U.S. wants to provide only half of that. The Afghans would contribute $1 billion, and other countries would provide the remaining $1 billion.

“There are no firm numbers right now,” Mr. Cordesman said. “You’re talking about the ability to plan for the security situation in 2015.”

But the decision to commit the U.S. to at least another 12 years of involvement is sitting poorly with those on Mr. Obama’s political left, who question how it jibes with his pledge during the 2008 presidential campaign to end the war.

“America has been lulled to sleep by the mind-boggling elongation of a war 7,000 miles away,” said Rep. Dennis J. Kucinich, an Ohio Democrat who ran for president twice on a platform of opposing the wars in Afghanistan and Iraq. “The plain fact is we are not exiting Afghanistan, despite the appearances which the White House is trying to create. We are staying. Have we learned nothing from 10 years of quagmire?”

The agreement got a better reception from those on the right. Sen. Lindsey Graham, South Carolina Republican, said it signaled a transition but not an abandonment of Afghanistan.

Mr. Obama signed the deal on his surprise nighttime visit to Afghanistan on Tuesday and Wednesday, outlining a mission that calls for NATO troops to be withdrawn as planned in 2014, though the U.S. will be allowed to leave military forces in the country only to train Afghan troops — which could cost up to $4 billion a year — or to pursue al Qaeda terrorists.

Administration officials and national security sources said Mr. Obama and NATO allies will hammer out more details during a summit in Chicago this month, and want to have an international reconstruction and economic aid plan ready this fall.

Mr. Cordesman said continued U.S. involvement is critical because the Afghan economy needs outside support and would collapse if the U.S. pulled out abruptly. American spending on its commitment in Afghanistan is seven times the country’s own national budget.

“You can’t cut down that much money quickly without creating a nightmare of an economic impact — the country has a very limited ability to raise revenue,” he said. “The effect would be extremely serious. It would put more pressure on Pakistan and Iran, and they would react and it wouldn’t be by admitting more Afghan refugees.”

The president’s trip also could reassure Afghans that the U.S. would not repeat history by turning its attention elsewhere as it did after secretly helping the mujahedeen defeat Russia in the late 1980s.

The chaos that resulted from that led to the Taliban’s emergence and eventual rule, which ended only after a U.S.-led invasion following the Sept. 11 terrorist attacks.

Since then, Taliban fighters have waged a pitched battle punctuated by attacks like the suicide bombings in Kabul on Wednesday, just after Mr. Obama’s visit. The attack left seven people dead and wounded some schoolchildren.

The militants planned the attack “hurriedly after finding out about Obama’s ‘surprise visit’ to Afghanistan,” Taliban spokesman Zabiullah Mujahed said in a statement.

The Taliban broke off peace talks with the United States in March after the Obama administration failed to release five high-value Taliban operatives from the U.S. military detention facility at Guantanamo Bay, Cuba. Their release was a key demand of the Taliban when it signaled its willingness to start official peace talks with the United States in January.

The delay in releasing the prisoners has strengthened the hands of Taliban members opposed to peace.

However, a U.S. official who briefed reporters Tuesday said the prisoners’ release had been put on hold for “reasons that appear to have to do with internal political turbulence among the Taliban.”

“It’s quite clear to us that there’s a range of interest among Taliban in reconciliation, and there’s quite a bit of internal political turbulence within the Taliban on that score,” the senior Obama administration official said.

The Taliban said Wednesday’s bombing was the start of the spring offensive.

After more than a decade of war, the U.S. has tallied nearly $570 billion in military, reconstruction and aid spending on Afghanistan, according to www.costofwar.com, run by the National Priorities Project. For 2012, $111.1 billion has been allocated, and Mr. Obama has requested $86 billion for 2013.

The group estimates the cost of war in Iraq since 2003 at $807 billion, pushing the total for both wars to $1.38 trillion. Of those funds, some $30 billion to $60 billion was squandered in waste and abuse, the nonpartisan Commission on Wartime Contracting reported last year.

Jo Comerford, the project’s executive director, said the continued U.S. presence in Afghanistan is expected to be larger than in Iraq, where the U.S. is spending $10 billion this year, so the spending in Afghanistan could balloon far past the $4 billion annual estimate.

“With all eyes on the federal budget right now, that price tag is going to have to mean budget priorities are going to have to shift to accommodate it,” she said. “Right now, the Congress is not in a position where Congress and the American people can just borrow a bit more money, we are at a moment where spending is finite.”

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