HARRISBURG - In the halls of the Capitol these days, it has become the dreaded $4 billion question: Can the new governor stick to his campaign pledge to cut billions of dollars out of the state budget without raising a single tax?

Mathematically, most everyone agrees, it can be done.

But what Gov. Corbett's state will look like after his administration puts away its paring knife is quite a different question.

Other recession-riven states have confronted enormous budget gaps by increasing sales or income taxes, or "sin" taxes on sales of tobacco products and the like.

Corbett and his fellow Republicans in Harrisburg, however, have promised not to raise taxes. Absent a radical departure from that campaign pledge, the choices could come down to who in Pennsylvania gets hurt the most.

"There are slim pickings for real easy things to cut this time around," said Sharon Ward, executive director of the Pennsylvania Budget and Policy Center, a liberal-leaning Harrisburg think tank. "There is no low-hanging fruit."

Corbett has been mostly mum on details of how he plans to plug the deficit, projected at anywhere between $4 billion and $5 billion.

He has ticked off a list of cost-cutting measures, such as reducing the state government's car fleet, identifying waste and fraud, and forcing the legislature to give up money it holds in reserve to run its operations in the event of a budget impasse.

But all those changes add up to millions, not billions, in savings.

To reel in big numbers, elected officials and budget experts say, Corbett will have to look at the three areas that make up more than 85 percent of the state's $28 billion-plus budget: education, public welfare, and corrections.

As Ward and others point out, part of the calculation is the check that won't be in the mail from Washington anymore. In June, about $2.6 billion in federal stimulus funding for Pennsylvania runs out.

All this leaves Corbett's administration with few places to turn.

"The biggest expenditures," Ward said, "are education, health care, and public safety. I don't believe that is where we should be doing most of the cutting. But that is where the big money is."

Few expect Corbett, a career prosecutor, to reach deeply into the corrections budget - say, to halt pricey construction of prisons to relieve severe overcrowding.

That leaves education and welfare, and top lawmakers from Corbett's party have been anything but shy in pointing to these as targets.

"We are going to have to have substantial cuts in those areas," said Senate Majority Leader Dominic Pileggi (R., Delaware), who last year made headlines when he said he did not see how Corbett could possibly cut $4 billion from the budget.

Still, said Pileggi, any such discussion must begin with the premise that the state cannot spend more than it takes in.

"We'll start with that, and then we can discuss what our priorities as a state should be," he said. "That is really where the debate should and will be."

In education, Corbett could look to roll back the roughly $6 billion in state funding that goes to early-childhood education and to instruction in public schools after years of increases under his Democratic predecessor. Ed Rendell considered those areas a critical priority for the state.

Corbett could also try to ax the $690 million that subsidizes Pennsylvania State, Temple, and Lincoln Universities and the University of Pittsburgh, which are not part of the State System of Higher Education.

The public-welfare picture is murkier, however, because federal guidelines leave little wiggle room in many of the programs the department administers, including Medicaid and cash assistance programs for low-income families.

Corbett could try to follow states that reduced reimbursements to hospitals and other providers of health care to Medicaid enrollees, or that dropped optional Medicaid services, such as dental care.

Governors in other states have also sharply rolled back funding for mental-health services, drug- and alcohol-abuse prevention programs, and women's and family health programs, according to a 2010 study by the National Conference of State Legislatures.

The prospect of such reductions makes public-health advocates send up this warning flare: Cutting the social safety net for children and needy families may produce short-term savings but cost the state more in the future.

For instance, they argue, cutting drug programs could swell prison costs if those who could have been helped turn to crime.

"Hopefully, the greater conversation behind closed doors is the connection of the dots," said child advocate Cathleen Palm of Berks County. "Hopefully, they say, 'OK, if we cut this item here, what is the ripple effect? And what does it end up costing us in the long term?' "

Steve Crawford, who was Rendell's chief of staff, cautioned that any deep cuts in state aid for education, health, and welfare would shift the tax burden to beleaguered school districts, counties, and municipalities.

Crawford said: "So in the end, how much money are you really saving taxpayers?"

Jonathan Stein, general counsel at Community Legal Services of Philadelphia, said Corbett and the legislature should insist on a mix of cuts and revenue boosters to make ends meet.

Stein pointed to one item on the long list of products and industries exempt from the sales tax or other state taxes.

"Why must we be the only state not to tax smokeless tobacco?" he asked. "We should be looking at a balanced approach to budgeting."

Corbett has been adamant about not raising any taxes, especially during the recovery from a recession.

In his campaign, he opposed taxing the extraction of natural gas from the Marcellus Shale formation - even as Democrats and an increasing number of Republicans said it was time for such a tax. (Every other major gas-drilling state has such a levy.) More recently, however, Corbett said he would consider proposals to impose a "fee" on gas drillers, who were among the biggest donors to his campaign.

As for raising new revenue, Corbett has called for privatizing the Liquor Control Board and selling off state liquor stores. But there is plenty of opposition to that idea, and few expect it will be accomplished in time to raise money for the 2011-12 budget - if at all.

That leaves Corbett with his paring knife.

He has the advantage that his party now controls both chambers of the legislature. But an embarrassing spat last week among legislators reminded everyone that Corbett's agenda - much like Rendell's in recent years - could be bollixed up if partisan squabbles erupt.

The fracas began with accusations that debate had been stifled on a series of bills meant, paradoxically, to usher in more transparency in government. A committee hearing devolved into shouting ("You should be ashamed!" some Democrats cried), with some lawmakers tossing papers in the air and others storming out.

Still, Republicans in both chambers say holding the line on taxes is heeding the call of the people, even if the debate in the Capitol turns uncivil.

"It won't be pretty or pleasant, but it's what voters told us they wanted this past election," said State Sen. Jake Corman (R., Centre), chairman of the Appropriations Committee.

He added: "Tom Corbett didn't hide what he was going to do, and he won the election significantly. It's our job now to listen to the voters."

Contact staff writer Angela Couloumbis at 717-787-5934 or acouloumbis@phillynews.com.